This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
cp) is an example of an mstantmt10n of (J) (a) and Cr.- @) exemplifies an instantiation of (I) (b). In the latter case, j 2 and b-1 b-2 b..
If\;=
=
st
=
=
110
POSITIVE PROOF-THEORETIC SEMANTICS
CHAPTER 8
Constraint (ii) amounts to requiring (tra)-eliminability. Therefore the rule schemata for introductions into databases for every II'P on M become:
111
f- (U1 11 ~'Pill · · · (Uru ~'Prll Tn) ··.)A "11~ "11~ (Ulu ~'P111 · · · (Uru ~'Prll Tn) · · .)s f- (Ultst ~'Pltst · · · (Urtst ~'Prtst Ttst) ··.)A "11~
(H) (a)
"11~ (Ultst ~'Pltst · · · (Urtst ~'Prtst Ttst) · · .)s,
r[(U1 11 ~'Plll · · · (Ur 11 ~'Prll Tn) · · .) + · · · · · · + (Ullst ~'Pllsl · · · (Urlst ~'Prlsl Tlst) · · .)] ~'P U
where in each case the replacement of A by B is with respect to Ak. Suppose now the rules for CA are instantiations of (I) (a) and (H) (a). By (rei) and the schemata (J) (a) we obtain
r[(U1t1 ~'Pttt · · · (Urtt f-v'Prtt Ttl) · · .) + · · · · · · + (Ultst ~'Pltst · · · (Urtst ~'Prtst Ttst) · · .)] ~'P U ff- f[f(A1, ... , An)] f-vcp U;
(H) (b)
f- (Ulit ~'Plil · · · (Urit ~'Pril Tii) · · ·)A
+ ···
f- (Ulit ~'Plil · · · (Urit ~'Pril Til) · · .)B
+ · ··
· · · + (Ulis;
r[(Ull ~'Pll · · · (Uzu ~'Piu 11) · · .)] ~'P U ff- f[j(A1, ... , An)] ~'P U,
~'Plisl · · · (Urisi ~'Prisi TisJ · · .)AI~CB
and
where uliki is assigned to the element IPliki (TikJ, ... , Uriki is assigned to IPriki ( . . . l.fJl iki (TikJ ... ) ' ull is assigned to IPll (Tl)' ... ' and Uzu is assigned to 'Plu ( ... l.fJll (Tz) ... ) . If n = 0, then for every 11 'P on M, (H) (a) is ~[T] ~"' U f- ~[T + f] ~'P U, ~[T] ~"' U f- ~[! + T] ~"' U, and (H) (b) is not instantiated. The schemata (I) (a), (J) (b) thus completely determine the schemata (H) (a) and (JJ) (b), and the rules of G~ plus these schemata determine how formulas j(A1, ... , An) may be introduced into arbitrary st -formula contexts.
· · · + (Ulis; ~'Plisl
· · · (Uris; ~'Prisi TisJ · · .)si~CA·
The schema (H) (a) and (tra) give for each II'P on M:
and
Let CA denote an £-formula which contains a certain occurrence of A as a subformula, and let C8 denote the result of replacing this occurrence of A in C by B. The degree of A (d(A)) is the number of occurrences of propositional connectives in A.
(UI; 1 ~'Plit · · · (Urit ~'Prit Til) · · .)s + · · · · · · (Uris; ~'Prisi TisJ · · .)B~cpCA f- Cs ~'P CA·
... + (Ulisi ~'Plisl
Hence f- CA "1 I~ CB. If the rules for CA are instantiations of (I) (b) and (H) (b), then by the induction hypothesis and the previous theorem, the schemata (H) (b) give f- CAI~(Un r--'Pn · · · (Utu ~'Piu Tz) ... )s and f- Csi~(Ull ~cp 11 • • • (Utu ~'Piu 11) ··.)A· By (I) (b), we get f- CA "11~ Cs. Q.E.D.
Theorem 8.10. Iff- A "11~ Bin G~ +(J)+(H), then f- CA "11~ Cs
in G~ +(I)+ (H).
Let TA denote an sf-v-formula which contains a certain occurrence of A as a subformula 6f a formula component ofT. Combining the previous two theorems we obtain the desired replacement result.
Proof. By induction on l = d(CA)- d(A). If l = 0, the proof is trivial. Suppose that the claim holds for every l ::::; m, and l = m + 1. Then CA has the form j(A1, ... ,An), where one of the AkA contains the occurrence of A in question and d(AkA) ::::; l. Suppose that f- A "11~ B. By the induction hypothesis, f- AkA "11~ Aks, and by the previous theorem,
Theorem 8.11. If f-A "11~ Bin G~ +(I)+ (H), then f- TA "11~ Ts
in G~ +(I)+ (H).
1
112
FUNCTIONAL COMPLETENESS FOR
CHAPTER 8
113
f--- +
8.5. FUNCTIONAL COMPLETENESS FOR f-v +
We want to show that for every llep on M, S1 = {:::)ep, EB, @, @), @, @} is functionally complete for f-v +. We first show functional completeness with respect to the positive proof-theoretic semantics. In order to be able to do this, we introduce higher-level sequent rules for the constants and connectives in S1. These rules conform to the schemata (I) (a) and (II) (a) resp. (I) (b) and (II) (b):
II'P,(T1,81) r-'P (T r-'P' U)
(r1 , 8I(tp'h)/T))[T f---'~'' U] f---'~' U
f- ~ f-vep @
(f-v @)
f- 0 f-vep @
(@f-v)
~[T] f-vep
U f- ~[T +@ f-vep U
~[T] f-vep
U f- ~[@+ T] f-vep U
(f-v EB)
~ f-vep
r
(EB f-v)
~[T
(f-v @)
~ f-vep
(@f-v)
~[T]
T
+ U] f-vep V f-
r
(r1, 01 (
'PH (f-v"-'~) (~~H
f'vc·
the rules of f-v + ~ f-v~ A r f-v"" B 1- ~ + r f-v"" (A EBB) ~[~A+ ""B] f-v C 1- ~["" (A \fJ B)] f-v C ~ f-v"" A 1- ~ f-v"" (A@B) ~ f-v"" B 1- ~ f-v"" (A@B) ~["" A] f-v C ~["" B] f-v C 1- ~["" (A@B)] f-v C ~ f-v"" A ~ f-v"" B 1- ~ f-v"" (A(S?)B) ~[~A] f-v C 1- ~["" (A(S?)B)] f-v C ~["" B] f-v C 1- ~[~ (A@B)] f-v C f f'v A 11'1'(~ = (T2,1h)) f-v"" B 1- ~(f] f-v"" (A::::>'~' B) (i.e. f = {Tt, J1), ~[f)= (Sub~,(T2 )T2, J), where J(x) = J;(x), ifx E I Ti l(i = 1,2)) (Sub~Tl(x),J) c 1- (Tl,J rl Tl l(x/ ~(A ::::>'1' B))) C, if IT I= {y,z},J(y) = A,J(z) =~ B, and y = 1, then T is called a higher-level sequent.
sf'
The notion of an S-subformula of an s[' -formula is defined analogously as for S~ -formulas. Every S-subformula with S-degree 0 ofT is again called a formula component ofT, and iffor every llcp on M, (T, 8) ~cp T, this is again abbreviated by (T, 8) I~T. Definition 8.21. Given a structured consequence relation ~' the sequent calculus G[ is defined by the introduction rules in Table XII, for every llcp' llcp' on M.
120
CHAPTER 8 Table XII. The higher-level calculus Gf.
(f---cp
r-- cp')
(- r--,•) r f--', T 11,,(~) f--', -U 1- ~[f] f--', -(T =>,• U) resp. (- =>,·f--') (Sub~r1 (x),8) f--', V 1- (r1,6 il r1 l(x/- (T =>, U))) f--', V, if 1r 1= {y,z},J(y) = T,8(z) ="' U, and y = t.p(r), and that
8.10.
~f-...- -T 1- ~ f'--ep"' T
~ f-...-ep -U 1- ~ f-...-ep -(T@U)
0 · · · 0 - Vis;A, 0 · · · 0 -Vis;B·
Applying (Ill) (a) we obtain 1- -CB --11f-...- -CA. If the rules for -CA are instantiations of (Ill) (b) and (IV) (b), then by the induction hypothesis and the previous theorem, the schemata (I II) (b) give I-V} A f'--ep -CB and -VjB f'--ep -CA. By (IV) (b), we obtain -CA --1if'--CB· Q.E.D.
Theorem 8.24. If A +(I)- (IV). in
(f'--rv) (rvf'--) (f'---rv) (-rvf'--) (f-...- - ~ep' )' (- ::Jep' r--)' (f-...- -EB )' (-EB f-...-)' (f-...--®)
(f-...- -EB )'
resp. (-EB
(f-...- -EB) ~ f-...-ep -T r ( -EB f-...-) ~[-T + -U]
f-...-)'
is interchangeable with
f-...-ep -U 1- ~
+r
f-...-ep -(T EB U)
resp.
f-...-ep V 1- ~[-(T 0 U)] f-...-ep V.
Consider e.g. the direction from (f-...- - ::Jep') to (f-...- - ~ep' )' • Let I TI I = {x}, 1r 2 1= {x,y}, 1r 3 1 = {z}, and h B from ~ 1 U ~ 2 . This disproof is a direct arguiS a ispro . · bl (") I Tennant's ment to the conclusion that A =>h B iS dis?r~va . e. u n . system there are merely introduction and ehmmat10n rules; there i~ no systematic distinction between introducing compound formu~as mto proofs and disproofs and eliminating them from proofs and dzs~roofs. Therefore it is not clear how one could obtain an interpretatwn of the constructive connectives from Tennant 's ~ule~ in term~ of. ~ro~fs and disproofs, similar to the 'BHK interpretat10~' of the mt~itioms tic connectives in terms of proofs (or constructions); see for mstance
1
[166], [167]. 31
'BHK' stands for 'Brouwer-Heyting-Kolmogorov'.
130
DISPROOFS AND CONTRARIETY
CHAPTER 9
If we aim at a pr~of-theor~tic, constructive account of the meaning of strong,_ co?str~cti~e negatiOn "", conjunction A, disjunction v, and c~nstructive Imphcatwn -:J h in terms of direct (or canonical) proofs and disp~oofs, we must not only define what it means for a formula A to be directly provable fro~ a finite set of formulas ~ in this language (~--+ A), but also what It means for A to be directly disprovable (or refutable) from.~ (~ +-A). This definition must proceed by induction on the c~mplexity of A. The case where A is atomic does not concern the prov~nce of logic. What is regarded as a direct proof or disproof of an_ atomic sentence depends on the context of argumentation, and in thi~ respe_ct _th~ standards and criteria may differ considerably across vanous disciplines and communities. For compound A the following clauses seem to be very natural: (a)
(b)
0 1
~ --7
2 3
~--+AVB
1 2 3 4
~+-""A
rvA
~--+At\B
~--+A -:Jh B
~+-AI\B ~+-AVB
~+-A
-:Jh B
if if if if
~+-A
if if if if
~--+A
~--+A&~--+B ~--+A
or
~--+
B
~u{A}--+B
~+-A
or
~
+- B
~+-A&~+-B ~--+A
& ~+-B.
An e~rl~. suggestion of treating the notion of disproof (or refutation) as pnmitive can be found in von Kutschera's [97) motivation of 'direct' and 'extended direct' propositional logic. These systems are precisely D. Nelsons's (see [4], [119]) constructive propositional logics N3 and N 4 respectiv_ely. In f~ct, the above clauses (a) 1 - 3 are nothing but the rules for mtroducmg /\, V, and -:Jh on the right of --7 in a standard sequent calc~lus presentation of N4 (and positive logic). Moreover, if cl~~se (a) 0 IS strengthened into an equivalence and viewed as a defi~ztwn_ of+- by_ means of--+ and "", then the clauses (b) 1 - 4 are the r~ght mtro~ucti?n ~ules _for negated negations, conjunctions, disjunctions? an~ Imphcatwns m N4 respectively. Negation introduction on th~ ~Ight IS thus only defined for negations of compound formulas. But this JUSt reflects what has already been said about refuting atomic sentences. In other words, negative atomic information has to be treated on a par with positive atomic information. The left introduction rules are such that they guarantee the elimi-
131
nation of principal cuts: (a')
1 2 3
AU{AAB}--tC A U {A VB}--+ C A u f U {A Jh B}--+ C
if if if
A u {A,B}--+ C A u {A}--+ C & A U {B}--+ C A--+ A & f u {B}--+ C
(b')
1 2 3
AU{,..,~A}--tC
if if if if
AU{A}--tC ~ u {""'A} --+ C & A U { ~ B} --+ C ~ U {,..,A,~B}--+ C ~ U {A, "'B}--+ C.
4
A U {"'(A 1\ B)} --+ C A U {"'(A V B)}--+ C A u {"'(A "Jh B)}--+ C
As L6pez-Escobar [102] has observed, this treatment of negation in terms of disproofs avoids an unpleasant problem caused by the nonconstructive nature of intuitionistic negation. The problem arises in the context of the BHK interpretation of the intuitionistic connectives in terms of canonical proofs. Since -.hA is defined as A -:Jh f, according to this interpretation, a proof of -.hA is a construction that converts every proof of A into a proof of f. Since there is no (possible) proof off, a proof of -.hA would convert any proof of A into a non-existent entity. If we assume that the existence of a proof of -.hA precludes the existence of a proof of A, then a proof of -.hA would convert a non-existent object into a non-existent object. This is at the very least obscure. Some advocates of the BHK interpretation seem to be aware of the problem. In addition to the notion of proof, Troelstra, for instance, uses the notions of "hypothetical proof" [167] and reduction of an "alleged proof ... to an absurdity" [166). Moreover, he admits that "the notion of contradiction is to be regarded as a primitive (unexplained) notion" [167, p. 9). Nevertheless, this does not solve the problem of transformations into non-existent objects. As a remedy, L6pez-Escobar [102] has suggested supplementing the BHK interpretation by the notion of (canonical) refutation. He gives the following disproof-interpretation of the intuitionistic connectives /\, V, and -:J h and the constructive negation ,. ._. (notation adjusted):
i.) the construction c refutes A 1\ B iff c is of the form (i, d) with i either 0 or 1 and if i = 0, then d refutes A and if i = 1 then d refutes B, ii.) the construction c refutes A VB iff c is of the form (d, e) and d refutes A and e refutes B, iii.) the construction c refutes A -:Jh B iff c is of the form (d, e) and d proves A and e refutes B, viii.) [t)he construction c refutes ""A iff c proves A.
133
CHAPTER 9
DISPROOFS AND CONTRARIETY
A proof of""' A is thus not interpreted as a proof of A ~h f, but rather as a refutation of A. This seems to be the most natural and intimate way of linking proofs and disproofs by means of negation. Lopez-Escobar uses the following notion of provable sequent with respect to which N 4 emerges as sound: { A1, ... , An} --7 A is valid iff there is a construction 1r such that 1r( c1, ... , cn) proves A, whenever c1, ... , Cn are constructions proving A1, ... , An (if 1 :::=; n). A sequent 0 --7 A is valid iff a construction exists that proves A. Moreover, Lopez-Escobar assumes that no construction both proves and disproves the same A. Note that {A, ""'A} --7 B is valid under the stronger assumption that no formula A is both provable and disprovable. 32 The interaction between proofs, negation and disproofs developed above does not have direct proofs and disproofs as disjoint classes of deductions. Instead, the difference between proofs and disproofs is an intentional one: what may be regarded as a disproof of something may be viewed as a proof of something else. If this something is A, the something else is ,. ._, A. Taking ,. .,., as primitive and using reductio ad contradictionem as the natural deduction introduction rule for ""' would also avoid the problem of transformations into non-existent objects. 33 We have that II is a direct proof of ""'A from the set .6. of undischarged assumptions iff II is of the form
is disprovable. In a private communication, Tennant replied that if an atomic basis contains the rule
132
A B then "a single application of that very rule constitutes . . . a dir~ct disproof of {A, B}". Such a rule registers the joint contrariety of I~S atomic premises. But then the disprovability of singleton sets of a~omic sentences amounts to the selfcontrariety of these atoms. Irrespective of whether there are selfcontrary atoms or not, in the present context nothing is assumed about the refutability of atoms. In IR each rule resulting in a disproof is an indirect rule, exemplifying the idea of disproof as reductio. For compound A, every such elimination rule instantiates the following schema:
A where the deduction V specifies the refutation conditions of A, and where the absence of a formula below the horizontal line presents a "logical dead-end", as Tennant puts it. For example, then the rv-elimination rule of IR states that if II is a proof of A from .6., then
II'
""A and II' is a deduction showing that .6. U {A} is inconsistent. However, according to Tennant,
.6. 11
""A
is a disproof of {"'"'A} U .6.. A proof of '""A, however, e?ds in a~ application of ,..._,-Introduction, and, according to Tennant, ~t thus ~ails to be a disproof, even though the introduction of'"" A reqmres a disproof of A, namely if 11 is a disproof of .6. U {A} then
[d]isproofs have no 'conclusion'. Disproofs arise only through the terminal application of elimination rules. They cannot arise from the application of introduction rules. Terminal application of introduction rules produces only proofs, not disproofs.[165, p. 206]
.6.
'
AD-(i)
11
Therefore, Tennant just cannot provide clauses for a disproof-interpretation, for defining the notion of disproof by saying that a direct disproof II of A is a direct proof of "'A would mean that II terminates in an application of an introduction rule, quod non. Tennant's approach contains direct proofs, but no direct disproofs of compound formulas, i.e. no deductions not merely revealing the inconsistency of some data, but rather leading to the conclusion that a certain compound formula
(i)
rvA is a proof of ,..._,A from .6., where 0 prefixed to the discharg_e stroke indicates that A must have been used in 11. 34 In contrast to this meaning assignment to ""', the meaning of negation as falsity is essentially captured by clauses (a) 0 and (b) 1. Obviously, Tennant's introduction and elimination rules for ""' are not the natural deduction counterparts of the sequent rules for negation in N 4. For instance, in combination with the introduction and
32
A more comprehensive critical discussion of the BHK interpretation can be found in [180], which suggests generalizing the BHK interpretation into a semantical framework for various constructive substructurallogics. 33 Note that this a problem Tennant is not concerned with.
34
,,
The notation ~,A means~ U {A}, where A f/- ~-
135
CHAPTER 9
DISPROOFS AND CONTRARIETY
rules for intuitionistic relevant implication :Jr, Tennant's negatiOn rules allow the principle of contraposition to be proved:
Note, however, that due to the failure of transitivity of deduction in IR, this does not imply that 1- {,..._,A, A} -t B. 35 Therefore, whereas Tennant's notion of disproof as reductio gives rise to IR, the egalitarian approach, which treats the notions of proof and disproof in their own right, leads to Nelson's constructive four-valued system N4. Although in [165] Tennant does appeal to disproofs, it still seems appropriate to quote from Pearce [127, p. 5] (notation adjusted):
134 elimi~ation
(A :Jr B)0-(iv)
AD-(ii)
BD-(i) (i)
=""=B==o-=(=i~=·i)=::;===~B~ (ii) ""A (iii) ""B ::::lr"-' A (iv) (A ::::lr B) :J ("-'B :J"-'A) where
0 prefixed to the discharge stroke indicates that the assumption
t~us marke~ ~eed
not have been used in the deduction, but may be It has been used. Interpreting ,. ._, as falsity in the sense of refutab_Il~ty al~ne; however, does not justify the provability of the ~ontraposit_w~ prm_ciple, and, indeed, this principle fails to be provable m N4. T~Is IS as It should be, since there need not necessarily be a co~structwn 1r such th,at if_1r' is a construction converting any proof of A I~to a proof of B, 1r ( 1r) IS a proof converting any disproof of B into ~ disproof of ~- ~he weaker contraposition rule A -t B 1-,...., B -trv A Is no~ an ad~ISSible r~le of ~4 either and fails to be supported by the dispro~f-mterpretatwn, for If there is a construction converting any proof o~ A m to a ~roof of B, there need not necessarily be a construction con:ertmg any ~Isproof o~ B into a disproof of A. In this respect, the not10~ of negatiOn as falsity differs from, for example, the notions of negatiOn advo_c~ted by Lenzen [101] and Restall [142], who consider the contrapositiOn rule as absolutely indispensable for any negation. Furthermore, e_very negation as inconsistency in Gabbay's [66] sense (see below) satisfies contraposition as a rule. A more specific criticism conc~rns ~enn~nt's first introduction rule for implication, which states that If II IS a disproof of ~ U {A}, then discharge~ ~f
~ AD-(i)
'
II
A
(i)
::::lr
B
is a proof of A :Jr B from ~ . Th"IS ru1e, It · seems, confers a nonconstructive meaning to :Jr. Using the rule we can derive ,...., A :Jr (A ::::lr B) for any formula B: ,..._,A0-(ii)
(A
AD-(i) (i)
::::lr B) (ii)
"'A ::::lr (A
=>r
B).
[N]either Dummett nor subsequent adherents to his anti-realist theory of meaning (... Tennant ... ), have gone beyond the notions of verification or proof as the sole conveyors of meaning. In particular, none of them takes the step of interpreting "'A as a (constructive) disproof of A. Tennant also inquires into the origin of our understanding of the meaning of negation. According to him, this origin "is to be found in our sense of contrariety", and contrariety among at least some atomic sentences of a language is a necessary condition of the language's learnability. This prerequisite of our grasp of the meaning of negation does not depend on the use of an explicit unary negation connective: "it is enough to have a few pairs of antonyms ... , or contraries of a more general kind". In fact, the presence of antonyms and contrary notions appears to be indispensable for concept formation and information acquisition in general. 36 Our grasp of the meaning of negation is thus based on predications which we use to communicate distinctions. One and the same physical object cannot, according to the same scale, be both huge and tiny; most actions fail to be simultaneously both moral and immoral, etc. If information is indeed a difference that makes a difference, contrariety among atomic sentences of a language is central to the notion of linguistic information processing. Interestingly, these considerations of the atomicity of negation have a formal counterpart in N4. In this system every formula has a unique negation normal form ( nnf) with respect to the congruence relation of strong equivalence, 37 i.e. negations can be pushed towards the atoms. 'Positivization', the replacement in nnf's of (strongly) negated atoms by new atoms not already in the language, results in a faithful embedding of N4 into positive (intuitionistic) logic 35 However, in [37] it has been observed that in IR, f- AA"' A-+ (AV B) =>r (A A B), "which does not seem justifiable from the point of view of inferential
relevance" [37, p. 257]. 36 A more detailed analysis of oppositeness of meaning between lexical items may be found in (104, eh. 9]. 37 In N 4 two formulas A, B are said to be strongly equivalent if not only 1- A-+ Band f- B-+ A, but also f- "'A-+ "'Band 1- "'B-+ "'A.
136
NEGATION AS FALSITY
CHAPTER 9
(see [127]). Following Tennant's explanation of our understanding of negation, atomicity of strong negation in N4 accounts for the equal importance of positive and negative atomic information. To put it in a slogan: literals have equal rights. Suppose that for any formula A, +A denotes the removal of negation from A by positivization of A's nnf, and +~ = {+A I A E D.}. According to Pearce [128], a negation in a logical system -t is hard iff
1-
~ -t
A iff 1-
+~ -t
+A.
(a)
(!3) ('y)
9.2. NEGATION AS FALSITY
Consider a sententiallanguage containing a unary operation *. 38 A twoplace relation -t between finite sets of formulas and single formulas in this language is called a single-conclusion consequence relation iff for all formulas A, B and finite sets ~' r of formulas:
(i) (ii)
1- A-t A ~ -t A, r U {A} -t B 1- D. U r -t B
(reflexivity) (cut)
A binary relation +- between finite sets of formulas and single formulas is called a single-conclusion *-refutation relation iff for all formulas A, B and finite sets ~' r of formulas: (i) (ii)
1- *A +- A 1- A+- *A D.+- A, r U {*A}+- B 1- ~ ur +- B
(!3') (1')
the relation -t defined by ~ -t A iff ~ +- *A is a single-conclusion consequence relation; for every formula A, not both 1- 0 +- A and 1- 0 +- *A; there is a formula A such that not 1- A +- A.
The conditions (a) and (a') express the idea ofnegatio~ ~a conn~c~ing link between proofs and refuations, whereas the remammg conditiOns reflect the contrariety between the notions of proof and disproof linked in this way. In particular, it is reasonable to assume that not every formula A is a refutation of A from {A} and that not every formula A is interderivable with its negation *A. If * satisfies both (a) and (a') for a single-conclusion consequence relation -t and a single-conclusion *-refutation relation +-, then negation as falsity is a vehicle for either keeping -t and dispensing with +or keeping+- and dispensing with -t. Then not only_ the ~ouble _negation law A -t * * A but also its converse * * A -t A IS easily denvable and we have the rule ~+-*A
r U {A}+- B
1- ~ U r +-B.
(*-reflexivity)
Analogously, A +- * * *A and * * *A +- A. Clearly,. the_ relatio? +defined by (a) is a single-conclusion *-refutation relatiOn Iff * satisfies
(*-cut)
1- A-t** A. Let us refer to an ordered pair (-t, +-} as a system, if -t is a single conclusion consequence relation and +- is a single-concl~sion *refutation relation. If S = ( -t, +-} consists of a single-concl uswn consequence relation -t and just any binary relation +- between finite sets formulas and single formulas, then if* satisfies (a) and (a'), S is a ~ys tem. (To see this, note that since *A-t *A, by (a), *A+- A, and smce A-t A, by (a'), A+- *A. If~+- A and r U {*A} +-f!, then, by (a), ~ -t *A and r u {*A} -t *B. Applying (cut) we obtam ~ U r -t *B,
We assume that membership in -t and +- is determined by a set of inference rules. If -t is a single conclusion consequence relation, then * is a negation as falsity in -t iff 38
the relation +- defined by ~ +- A iff ~ -t *A is a single-conclusion *-refutation relation; for every formula A, not both 1- 0 -t A and 1- 0 -t *A; there is a formula A s.t. not both 1- A-t *A, 1- *A-t A.
If+- is a single conclusion *-refutation relation, then * is a negation as falsity in +- iff
(a') Pearce observes that hard negation cannot be contrapositive. Akama [2] suggests using hardness as a defining characteristic of the notion of strong negation. In the next section a definition is given of the notion of negation as falsity.
137
We take advantage of context sensitivity and reuse the symbol '*' from the structural language of display logic.
138
NEGATION AS INCONSISTENCY
CHAPTER 9
and (o:) gives ~ U r +---B.) IfS= (~, +---) is a pair consisting of any two-place relation~ between finite sets offormulas and single formulas and a single-conclusion *-refutation relation +---, then S is a system if* satisfies (o:'). What can be said in favour of (*-reflexivity) and (*-cut)? In particular, one might wonder why a refutation relation should not also be a single-conclusion consequence relation, preserving falsity instead of truth. In fact, inverse consequence due to Slupecki et al. [157], [158] and also W6jcicki's [198] notion of dual consequence provide examples of such falsity-preserving relations. Obviously, if one reads ~ +--- A as 'if the formulas in ~ are false, then A is false', then +--- should turn out to be reflexive and transitive. However, this approach is inappropriate if we want to introduce negation as falsity (in the sense of refutability) by means of+---: if~ +--- A is defined as ~ ~ *A, then *A ~ *A implies *A +--- A, whereas *A +--- *A would imply *A ~ * * A. Moreover, the rule ~ ~ *A r u {A} ~ *B f- r u ~ ~ *B hardly supports interpreting * as a negation operator. The reading of ~ ~ A appropriate for our purpose is 'there is a proof of A from ~'. And if this means that there is a refutation of *A from ~' then A ~ A just translates into A +--- *A. Conversely, if ~ +--- A is interpreted as 'there is a disproof of A from ~' and if this means that there is a proof of *A from~' then *A+--- A translates into an instance of (reflexivity), and (*-cut) translates into (cut). The conditions (*-reflexivity) and (*cut) are hence the obvious counterparts of (reflexivity) and (cut), if* is introduced as negation as falsity. Note that a more uniform proof-theoretic definition of negation as falsity is available in terms of four-place sequents
with the following reading: if every formula in r is true and every formula in I: is false, then some formula in ~ is true or some formula in e is false. Blarney and Humberstone's reflexivity and cut rules from Chapter 2 now reappear as notational variants: (odd reflexivity) (even reflexivity)
f- A I 0-+ A I 0
(odd cut) (even cut)
r I I: -+ A, ~ I e A, r I I: -+ ~ I e r I I:-+~ I A, e r I A, I:-+~ I e
f-
0 I A-+ 01 A 11-
r I I: -+ ~ I e r I I:-+~ I e
139
As a counterpart of conditions (a) and (a') one obtains
(o:")
f- *A I 0 ~ 0 I A f- 01 *A~ A 10
10 ~ 01 *A 01 A~ *A 10
f- A f-
There are equally obvious counterparts of conditions (/3), (/3'), (1), and ('y').
(!3")
for every formula A, neither both f- 0 10 ~ 01 A and f- 0 10 ~ 0 I *A nor f-
('y")
0I 0~
A
I 0and
f-
0 I 0 ~ *A I 0
there is a formula A, such that neither both ff-
I 0 ~ *A I 0and 0 I A ~ 0 I *A and A
ff-
I 0 ~ A I 0 nor 0 I *A ~ 0 I A
*A
Definition 9.1. A unary operation * is called a nega_ti?n as f~lsity ,in a four-place consequence relation~ iff *satisfies conditions (a ), (/3 ),
and ('y"). Moreover, in this setting there are separate, symmetrical, and explicit introduction rules for strong negation, namely: ( ~'"" odd) ( ~'"" even) ('""~odd) ('""~even)
r 11: ~ ~ 1A, e r- r 11: ~ "'A, D.. I e r 11: ~ D.., A 1e r- r 11: ~ D.. 1"'A, e r 1A, 1: ~ D.. 1e r- "'A, r 11: ~ D.. I e A, r 11: ~ D.. 1e f- r 1""'A, 1: ~ D.. I e
However, with four-place (single-conclusion) consequenc: relat~ons we lose comparability with Gabbay's notion of ne~ation ~ mco_nsistency. Instead of developing a generalization of negatiOn as mconsi~tency to four-place Gentzen sequents, we shall introduce f~ur-place display sequent arrows in Section 13.3 and use this generalizatiOn of DL to redisplay Nelson's useful paraconsistent logic N4.
9.3. NEGATION AS INCONSISTENCY
Consider again a propositional language containing ~ unary conn~c tive *· Think of a logical system as being given by a ,smgle-~onclusiOn consequence relation ~ over this language. Gabb~y s (66]_ Idea for a syntactic definition of negation (as inconsistency) m a logica~ system is that f- A ~ *B iff A together with B leads to some undesirable C
140
from a set ()* of unwanted formulas. In this context, the object language counterpart of bunching premises together is conjunction /\, governed by its sequent rules in positive logic (i.e. (a) 1 and (a') 1). Hence, let us suppose that 1\ is already in the language or that it can conservatively be added. Gab bay defines * as a negation (as inconsistency) in -+ iff there is a non-empty set ()* of formulas which is not the same as the set of all formulas such that for every finite set ~of formulas and every formula A we have:
Moreover, ()* must not contain any theorems. If such a collection of unwanted formulas exists, it can always be taken as {C 11- 0 -+ *C}, since by (reflexivity) the latter set is non-empty, if * is a negation. There is an equivalent definition, which does not refer to (}*. Namely, * is a negation in -+ iff for every finite set ~ of formulas and every formula A the following holds:
h IT' is a disproof of ~' C, B. By the claim for disproofs, IT' is a wd. ere f f ~ B and hence by rv-lntroduction, IT is a proof of"" B Isproo o ' ' El' . t' then from~. If IT ends in an application of rv- Imma lOll, ~,c
IT'
""A
Lemma 9.2. Suppose C is a theorem of IR. Then in IR, (i) if II is a disproof of~ U {C}, then IT is a disproof of~' and (ii) if II is a proof of A from~ U {C}, then IT is a proof of A from~. Proof. By simultaneous induction on the construction of proofs and
disproofs in IR. For example, if II is a proof of A from {A}, then ~ = 0 and C = A, thus A is a theorem of IR. If II terminates in an application of rv-Introduction, then A = ""B and II has the form: ~
C BD-(i)
' 'II'
rvB 39
See (101 J for a critical discussion of Gab bay's definition in terms of intuitive criteria of adequacy.
A
. . f f ~ C U {"-'A} while IT' is a proof of A from ~'C. IS a disproo o ' ' f f A f ~ and However, by the claim for proofs, IT' is also a pr~o o rom "" A hence the above disproof figure also represents a disproof of ~ U { }. The remaining cases are equally simple. Q.E.D.
Observation 9.3. Negation in IRis a negation as inconsistency. Proof Put(}*= {rvAI\A I A is a formula}.lf ~-+"-'A, then ~U{A}-+ "" AI\.A follows by /\-Introduction. For the converse, note that for every
B,
rvBI\B ""B B rv(rvBI\B)
1- ~-+*A iff ::JC (1- 0-+ *C & 1- ~ u {A}-+ C). 39 In [69] the notion of negation as inconsistency (alias inferential negation) is extended to a novel kind of non-monotonic inference relations between structured databases, called structured consequence relations. In IR, the deducibility relation is not unrestrictedly transitive and hence fails to be a consequence relation. The basic idea of negation as inconsistency, however, does not depend on this assumption. We want to show that negation "" in IR is a negation as inconsistency, and for this purpose we only assume that (reflexivity) holds.
141
NEGATION AS INCONSISTENCY
CHAPTER 9
is a proof in IR of "" ("" B 1\ B) from 0. If now IT is a proof of ""B 1\ B from~
U {A}, then ~u{A}
IT
""(rvB 1\ B)
rvB 1\ B
. d' f of ~ U {A} U {"-' ("-' B 1\ B)} and, by the disproof part IS a Isproo bt · oof of "" A of the previous lemma and ""-Introduction, we o am a pr from ~. Q.E.D. Negation in minimal (intuitionistic, classical) senten~ial lo?ic, MPbL (IPL CPL) can also be shown to be a negation as m consistency . y identifying (); with the set of all explicit contradictions in the respectiVe language. . · t' fies contrapoA tated above every negation as mconsistency sa IS . . s· B B by defimtlon there ss ' sition as a rule. Suppose A -+ B · mce * -+ * ' . · ) is a C E (}* such that {*B' B} -+ C. Performing an applicatiOn of (cut . { B A} -+ C and hence *B -+ *A· Therefore, strong negawe obt am * ' · M reover every tion in Nelson's N4 is not a negation as inconsistency. o ' .. negation as inconsistency validates the law of exclu~ed contr~di~I~~' *(*A 1\ A). Since *A-+ *A, we have ~*A,~}-+ B, son;e(*A 1\ A). -+ B for some B E (} ' which means VJ -+ . (\ A H ence, * A '
o:
NEGATION AS FALSITY VS. NEGATION AS INCONSISTENCY 143 142
CHAPTER 9
bHence ' negation . . in Bel nap ' s [15] "use ful four-valued logic" also fails t o e a negatiOn m Gabbay's sense.
9.4. THE RELATION BETWEEN NEGATION AS FALSITY AND NEGATION AS INCONSISTENCY
Our aim · as mconsistency . . as fals. t is. to show that e.ver~ nega t IOn is a negation I y, I.e. we want to JUStify the following picture: negation as falsity negation as inconsistency
N4
IR MPL IPL CPL
If * is a negation as in;en~sfiency, · t relation-+ sequence
:e
A f- 0.\0.\A.
are give~ a single-conclusion con-
of formulas and sin~le for~~:: bym:~~ ~l~ti~n ~l betwe(en finite sets tio £ 1 ·t I e nmg cause a) for nega } . . n as a SI y. t remains to be shown that (-+ the defined ~ is in fact a single-conclusion * r ' ~ . IS a sy~tem, I.e. moreover, that conditions ({3) and (r) are relatwn, and,
sat~s~~~-atwn
J." l()_btservation 9.4. Every negation as inconsistency is 1a si y. a negation as
;roof. It must be shown that the defined
relation~
satisfies ( *-reflexiviy), (*-cut), ({3), and (r). (*-reflexivity): *A~ A is cl f A and the definition Since 0-+ *(*A 1\ A and ear rom * -+*A there is a C such that 0 -+ *C and {A} U {*j} *A} -+ *A 1\ A, and thus by the defi "t" f A -+ ' hence A -+ * * A ' m Ion o ~ ~ *A ( ) follows from the definition of~ 'd ( t) £ · *-cut : The (*-cut )-rule r U {A} ~ B Th" an cu or -+. Assume b. ~ *A and . IS means b. -+ A and r U {A} B A of (cut) gives b. u r -+ *B , w h"1ch means b. u r ~ -+ B* · n application · d ( Suppose that both 0 -+ A and 0 -+ *A £ A ' as reqmre . {3): E ()* such that A -+ B or some · Then there is a B theorem uod n . However, by (cut), -+ B, that is,()* contains a A A' q on. (1): Suppose that for every formula A A-+ *A d * -+ . Then there is a B E ()* such that A -+ B . 0 ' an ing_ (cut) to the latter and *A-+ A, we obtain 0 B-+t *Ah. Ap?ly()* IS non-empt ·t . · u t en, smce y, I must contam a theorem; a contradiction. Q.E.D.
of~-
Obviously, the construction in the proof does not work for any unary connective. Consider, for instance, 0 or 0• in the smallest normal modal logic K. In K, 0 and 0• do not satisfy (*-reflexivity). We have thus obtained a non-trivial generalization of Gabbay's definition. This definition of negation as falsity covers Nelson's strong negation, a recognized negation living beyond the realm of negation as inconsistency. In view of its atomicity, in [69) we have referred to strong negation in N4 as a rewrite connective. 40 This choice of terminology emphasizes the problem of ensuring that a generalization of Gabbay's definition of negation in order to capture strong negation in N 4 does not result in too general a notion of negation. One may thus wonder whether the notion of negation as falsity also encompasses unary connectives which fail to be 'negations' on intuitive grounds. The conditions (a), ({3), ('"y) and the basic properties of -+ and ~ may appear to be rather weak and perhaps insufficient. But in fact, positive normal modalities do not present counterexamples. Consider prefixes of the shape 0.\. If 0.\ in normal modal logic is to figure as a negation as falsity, then, m an axiomatic setting, the following rule must preserve validity:
~A,
~I.~
Thus, if B is any tautology, then f- 0.\0.\B. For the possible worlds models this implies that every world has at least one successor. If now ({3) were satisfied, then for every formula A, we would have ff A 1\ 0.\A. In particular, ff 0.\B 1\ 0.\0.\B. Hence ff 0.\B, quod non, since 0.\B cannot be falsified in serial Kripke models. For prefixes of the form 0.\, 0.\0.\(p V •P) enforces a property of the accessibility relation also validating 0.\(p V •p). Thus, f- (p V •p) 1\ 0.\(p V •p), in contradiction of ({3). In general, negative modalities do not fail to be negations as falsity. In [69) we have observed that if'"'"' is interpreted as 0•, all equivalences which axiomatize N4 in Hilbert-style hold in Kripke models in which the accessibility relation forms pairs {t, s I t'Rs and s'Rt}. In such models every negative modality is a negation as falsity. According to Lenzen's [101) catalogue of principles indispensable for any genuine negation, the notion of negation as falsity is not only too general, since it does not require the contraposition rule, it is also too restrictive, because it calls for double negation introduction: A f- * * A. As already remarked, in some systems of normal modal logic the latter fails to hold for*:= 0•. Lenzen, however, takes 0• to be a weak form of negation and hence rejects A f- * * A as a necessary property of negations. While
n
40
Note that in (69) N4 is called N.
144
CHAPTER 9
SEMANTICS-BASED NONMONOTONIC REASONING
on ~he one hand this attitude may be welcomed because negation as falsity turns out to impose a significant constraint, on the other hand the same type of objection can be raised against the weaker rule f-A
I
f- **A,
which together with contraposition as a rule and
(6) there is a formula A such that not A f- *A constitutes Lenzen's list, namely, for every tautology A, a theorem of K.
O...,O...,A is not
9.5. SEMANTICS-BASED NONMONOTONIC REASONING
In this section, we discuss Gabbay's idea of basing nonmonotonic inference on semantic consequence in IPL extended by a consistency operator and Turner's suggestion of replacing the intuitionistic base system by Kleene's three-valued logic 3. It is shown that a certain counterintuitive feature of these approaches can be avoided by using Nelson's constructive logic N3 instead of intuitionistic logic or Kleene's system. The _addition of a c~nsistency operator to the more fundamental paraconsistent constructive logic N4 is considered in [195]. In order to avoid fixpoint definitions of nonmonotonic inference Gabbay [65] suggested basing nonmonotonic deduction on semanti~ consequence in a logical system extended by a consistency operator M (see also [34] and [35]). MA is to be read as 'it is consistent to assume at this stage that A'.
Definition 9. 5. A formula A is said to nonmonotonically follow from a set of assumptions Ll = {A 1, ... , An} ( Llr--A) if there are formulas B1, ... , Bm such that A1, ... ,An ~ B1 Al, ... ,An,Bl ~B2
A1, ... ,An,Bl, ... ,Em~ A,
~nd the a~xiliary relation ~ is defined as follows: cl' ... ' ck ~ c If there exist extra assumptions D 1 • • . D · such that (i) {C C ' ' J 1' ... ' kl MDl, ... ' MDj} is consistent, and (ii) {Cl, ... , ck, MDl, ... 'MDj} F C.
145
This notion of nonmonotonic inference requires, of course, a clear semantics for consistency statements MA. Gabbay's idea is to interpret M as possibility with respect to the 'information ordering' !:; in Kripke models for intuitionistic logic, that is, MA is true at an information state t iff there is a state u such that t !:; u and A is true at u. Let us refer to the result of extending IPL by M as CG. Assuming that nonmonotonic inferences are appropriate only in the presence of incomplete information, Turner [169) suggested using G~b bay's definition of nonmonotonic inference based on a system of partzal, in effect Kleene's system 3. Turner considers model structures (I,!:;), where I is the set of all partial interpretations of the atomic sentences and !:; is a 'plausibility' relation on I, that is, a reflexive transitive relation such that t ~ u implies t ~ u, where ~ is the natural 'information ordering' on partial interpretations. Consistency statements MA are evaluated as true at an information state t E I like in CG, MA is defined to be false at t, if A is false at every information state u which is at least as plausible as t, and MA is evaluated as undefined at t otherwise. Let us call Turner's system CT. Gabbay's andTurner's approaches both sucessfully deal with various counterintuitive features of McDermott's and Doyle's [109] nonmonotonic formalism. In McDermott's and Doyle's logic, for instance, {-,Mp} is nonmonotonically inconsistent, since the non-derivability of -,p forces Mp to be assumed. Moreover, {(Mp :J q), ...,q} {Mp, ...,p}
is inconsistent; is satisfiable;
{M(p A q), -,p} M(p A q) fv Mp.
is satisfiable;
However Gabbay's and Turner's nonmonotonic systems suffer from an' . other weakness (see [103]), namely the fact that Mp :J p ...,p, since m CG as well as in CT, {(Mp :J p), M-,p} I= ...,p and {(Mp :J p), M-,p} is consistent. In the literature it has been suggested interpreting Mp :JP as 'p is true by default'. Intuitively Mp :J p ...,p clearly fails to be sound, no matter that also Mp :J p p, because ...,p should simply not be nonmonotonically derivable from the assumption that p is true by default. According to Lukaszewicz [103], this weakness renders it problematic to apply Gabbay's and Turner's definitions of nonmonotonic inference to formalizing common-sense reasoning. We shall see that if Gabbay's definition of nonmonotonic inference is based on semantic consequence in Kripke models for Nelson's system N3, then (Mp :J p) A M "'p F"' p fails to hold. N3 combines t_he ~dvant~ges_ of (i) having an intuitionistic and hence a genuine implicatiOn sahsfy~ng the Deduction Theorem and (ii) semantically being based on partial, three-valued interpretations. As we shall see, this is exactly what is
r.-
r.-
r.-
SEMANTICS-BASED NONMONOTONIC REASONING
CHAPTER 9
146
needed to overcome the problem with the approaches of Gabbay and Turner. Moreover, we shall discuss justifying the choice of a suitable base logic and in the course of this consideration discuss the suggestion of evaluating MA as true at an information state t iff the negation of A fails to be true at any possible extension of t. It will turn out that this rather strong notion of consistency is definable in N3 itself and, moreover, its definition in N3 directly expresses a natural and basic constraint on formalizing M. Before that, however, we shall look at the intuitionistic and the Kleene base logics. lntuitionistic base. Assume we have a non-empty set Atom of propositional variables. CG is the theory of the class of all intuitionistic Kripke models in the language {'h, M, ~h, 1\, V} over Atom. An intuitionistic Kripke frame is a structure (I,~), where I is a non-empty set and ~ is a reflexive transitive relation on I. An intuitionistic Kripke model is a structure (I,~' v), where (I,~) is an intuitionistic Kripke frame, v is a valuation function assigning to each propositional variable a subset of I, and for every p E Atom and every t, u E I: (persistence 0 ) ift ~ u, then t E v(p) implies u E v(p). Kripke [94] suggested the following 'informational' reading of frames (I,~): I is a set of information states and ~ is the relation of possible expansion of information states over time. Let M = (I,~' v) be an intuitionistic Kripke model, t E I and A a formula. M, t f= A (A is verified at t in M) is inductively defined as follows: M,tf=p M,tf=BI\C M,tf=BVC M,t F B -:Jh M,t F 'hB M,t F MB
c
iff iff iff iff iff iff
t E v(p), p E Atom M, t f= Band M, t f= C M, t f= B or M, t f= C (VuE J) if t ~ u, then M,u (Vu E J) if t ~ u, then M, u 3u E I, t ~ u and M, u f= B
~Mpp
9) gested basing Gabbay's defTurner (16 sug 1 d 1 gt·c 3 In K leene 3-valued base. · · r Kl ene's three-va ue o · inition of nonmonotomc m~ere~ce ?n e A B = "' A V B' 3 a non-constructive implication lS defined by ~ def r~sulting in the following truth table for ~:
u
0
1 u 1 u 1 1
0 u 1
1 1 u 0
. is to be read as 'undecided'. Since there where the thtrd truth value u d t h ld in Kleene's are no tautologies, the Deduction Theorem oes no o logic. ' tics for CT makes use of partial interp~e~ations. f>:Turner s seman. . . from the set of proposttional vanpartial interpretatwn lS a maplpmg { } The natural 'definedness 0 M, t lf=- A
(3t E I) s C t & M, t
Obviously, c implies b, and in the context of Nelson models for N3, also a implies b. However, consider again Mp ~hp. If p is falsified at every point in a model, then clause b renders Mp ~h p true at every point, in other words, p is true by default everywhere, which is absurd. But clause c, that has been suggested in (184), may also be viewed as problematic. This clause restores (persistence+) and (persistence-) for the entire language, and persistence of consistency statements may be an unwanted property. Nevertheless this persistent notion of consistency has some nice properties. Consider (*1) again. The intuition behind
{(Mp ~hp), M'"" p} ~ts'"" p.
CHOICE OF PARAMETERS
~part from (i) (implicitly) claiming that the semantic clause for M m C_G captu_res the_ notion of consistency in a plausible way and (ii) c?nsid~rably Impro~m~ on ~cDermott's and Doyle's approach, Gabbay give~ ~Ittle fur_ther JUStificatiOn for using an intuitionistic base system. AdditiOnal evidence for the suitableness of working with intuitionistic
(*1) seems to be this: I
I
L
152
CHAPTER9
CHOICE OF PARAMETERS
153
(*4) A fl•hE iff Af-vB. If on,e w~nts to confine :he meaning of M in the base system by a general . mam for~al equation', then the following is a natural equivalence (agam for consistent formulas A):
(*5) A, 'hE
f= f
w~er~ f may ~e defined
iff A
f=
ME,
asP 1\ 'hP, for some p E Atom. What one ob-
~am~ ~sa_ c~nsistency oper~tor slightly stronger than Gabbay's, namely mtmtiO~Istlc double negation 'h 'h· 41 The verification conditions for
'h'hA Imply those for Gabbay's MA:
t
f= 'h 'hA
iff iff only if
(Vu E I) t ~ u :::::} u ~ 'hA (VuE I) t ~ u :::::} ((3w E I) u ~wand w (3u E I) t ~ u and u f= A.
f= A)
Since in intuitionistic logic 'h 'h 'hA is equivalent to 'hA, one still has {(M~ ::J~ ~), _M:'hP} f= 'hP and therefore {(Mp ::Jh p)}f-v•hP· Now, in N3 mtmt10mst1c negation can be defined by -, h A -def A "' A , an d . -'h"' the consistency operator M given by clause c and falsification on the spot turns out to be definable by MA =def 'h ,..._, A:
t
f=+,..._, A
::J "'""'A
iff iff iff
(Vu E I) if t [;;;; u, then u f=+"' A implies u not (:3u E J) t [;;;; u and u f=+,...., A (VuE I) t [;;;; u implies u ~+,..._,A;
iff iff
t F=-"' A ::J ,....,,..._, A t f=+"' A and t f=- ,. . , . . , A tf=+rvA.
f=+ A
Note that, due to (persistence-) and the reflexivity of _, c for the defi ne d M we have
M, t
f=-
MA iff
M, t
f=+"' A
iff VuE I, t [;;;; u implies M, u
f=-
A,
that is, ~mer's falsification conditions amount to falsification on the spot. Obviously, the defined M satisfies (*5): I A semantic ~reatment of intuitionistic double negation as a modal opera- regar ds 'h'h as a necesstty · operator tor can be found m [40] · Note that Dosen D. However, he. notes that one "can prove DA0 A h. h = ..., ..., , w tc goes some way towards explamng why intuitively 0 ... has some features of possibility" ([40 p. 16), notation adjusted). ' 4
iff iff iff
A,"' E f=+ f A f=+,...., B ::J f A f=+ --, ""' E A f=+ ME.
The counterintuitive results of McDermott and Doyle and the problem with Gabbay's and Turner's systems do not arise in N3 either: the assumption sets {Mp =>h q,"' q}, {,...., Mp} are satisfiable, {Mp,"' p} and {M(p/\q),,....., p} are not satisfiable, M(pl\q)f-v Mp, and {(Mp ::Jh p),
Mrv p} ~!t 3
"-' p.
According to Clarke and Gabbay [35, p. 177 f.] (notation adjusted), "[i)t could be viewed as a justifiable criticism of the intuitionistic system that MC ::Jh C is equivalent to CV 'hC, i.e. neutral with respect to C or 'hC. This", they continue, "is not the usual intention behind defaults." Note that neither in C3d nor in C3, nor in N3 we have that MA ::Jh A is equivalent to AV ""'A. It seems as if the fact that on the basis of CG and CT we have Mp ::Jh p f-v 'hP and Mp ::Jh p f-v ""'p respectively has been considered the main obstacle to working with Gab bay's definition of nonmonotonic inference instead of fixpoint definitions. We have seen that this obstacle can easily be removed by using Nelsons constructive logic N3 as the underlying base system, that is, by working with C3d or C3. The beauty of Gabbay's definition results from the flexibility provided by the choice of the underlying base logic. In principle any logic given by a class of 'information models' will do. What is needed is some kind of information order [;;;; to interpret the consistency operator M. The various persistence conditions and the presence or absence of properties of [;;;; (like reflexivity, seriality, transitivity , etc.) give rise to a semanticsdriven landscape of subsystems of C3d, C3, but also of CG and CT. As the inspection of the anomalies of CG and CT has shown, in the absence of (persistence), the derivation of Mp ::Jh p f-v 'hP is blocked for CG, and in the absence of either (persistence+) or reflexivity of[;;;;, the derivation of Mp ::Jh p f-v ""'Pis blocked for CT. There thus exists a large variety of different notions of consistency and hence notions of nonmonotonic consequence, which may be compared, tested against benchmark problems, and applied in knowledge representation. Subintuitionistic subsystems of IPL obtained by dispensing with properties like transitivity of[;;;; will be considered in detail in Chapter 10. It should also be pointed out that in semantics-based nonmonotonic reasoning there is no need for the underlying base logic to be monotonic. It is well-known that in IPL the persistence property corresponds to the
154
CHAPTER 9
validity of the monotonicity axiom schema A ~h (B ~h A). If thus Mp ~h p f-v 'hP is avoided by abandoning the persistence requirement in CG, one obtains a relevance logical base system, see Chapter 10. There is nothing wrong with basing nonmonotonic inference on a nonmonotonic logic, since nomonotonicity as such is only a symptom, comparable to the absence of contraction or permutation of premise occurrences in substructurallogics. What is important, however, is the naturalness of the nonmonotonic inference mechanism or rather its formal representation. Gabbay's definition describes such a simple and natural mechanism. There is an obvious open problem, namely axiomatizing C3d and C3.
9.7. MODAL LOGIC OF CONSISTENCY OVER N4
In the present chapter, we have discussed the modal logic of consistency over IPL as suggested by Gabbay [65] and the modal logic of consistency over Kleene's three-valued logic 3 as defined by Turner [169]. It was observed that conterintuitive features of the resulting nonmonotonic consequence relations can be avoided in a very natural way by replacing IPL and 3 with N3, or by replacing IPL with a suitable subintuitionistic logic. While N3 is a system of partial logic, Nelson's N 4 is both partial and paraconsistent. Since in applications one has to cope not only with partial but also with inconsistent information, in general the monotonic logic of information on which nonmonotonic consequence is semantically based should be partial and paraconsistent. With strong negation primitive, the latter means that {A, '"'" A} does not entail arbitrary formulas. In [195] it is shown that C4, the modal logic of consistency over Nelson's paraconsitent logic N4, can be faithfully embedded into the modal logic S4. Moreover, it is shown that this embedding can be used to obtain cut-free display sequent calculi for both C4 and N4. From this presentation one can also straightforwardly obtain a complete display sequent calculus for C3.
CHAPTER 10
DISPLAYING AS TEMPORALIZING
This chapter is about display sequent calculi for subintuit~onisti~ l?gics, that is, logics obtained from intuitionistic sen~e.ntiall~gic by_givmg up or relaxing all or part of the following conditlO~s:_ ~I) persi~tence of atomic information, (ii) reflexivity of the accessibihty relat10n ~ (iii) transitivity of ~- The sequent calculi are obt_ain~d. fr~m. sequ~nt systems for certain 'temporalizations' of t~e ~ubmtmtwmstlc logics. However we will not consider full temporahzat10ns, but only one particular c~nstruction which is available because the temporalizing and the temporalized system are complete with respect to the same class of Kripke models. The subintuitionistic logics are m_otiv~~ed _b~ ext~nd ing the well-known informational interpretation of mtmtwmstic Knpke models.
10.1. SUBINTUITIONISTIC LOGICS
One particularly appealing feature of intuitionistic propositionallogic, IPL, is that it may be regarded as the logic of cumulative research, see [94]. It is sound and complete W:ith respect ~o the class of all no~-empty sets I of information states which are quasi-ordered by a relat10n ~ of 'possible expansion' of these states, and in which atomi~ formulas _established at a certain state are also verified at every possible expans10n of that state. There are thus three constraints which are imposed on the basic picture of information states related by ~: ~i)_ the pers~~:ence (alias heredity) of atomic information, (ii_) th~ _re~ex_Ivity, and (m) the transitivity of~- The persistence of every mtmtwmstlc formula emerges as the combined effect of (i) and (iii). Although a Kripke frame, that is a binary relation over a non-empty set, admittedly provides an extr~mely simple model of information dynamics, and, moreover, each of the conditions (i) - (iii), as well as their com~ina~ion~, m~y. be of value for reasoning about certain varieties of scientific mqmry, It IS nevertheless interesting and reasonable to consider giving up all or some of these conditions. Evidently, conceiving of information pro~ress _as a steady expansion of previously acquired insights is extremely Ide~hzed, and the basic model of such a progress should leave room for mcorporating revisions, contractions, and merges of information as well. If
155
T• 156
I
CHAPTER 10
I
persist~nce
is given up, !:;;; can no longer be understood as a relation of possible e~pa_nsion. This reading, however, may be replaced by more gen~rally thmkmg of !:;;; as describing a possible development of infor~atiOn s_tates. Development thus need not imply the persistence of mformat10n. . When talking about the development of information states one mig~t want to di~pense with the assumption that such states always possibly develop mto themselves. There might be information states which ~n practice simply 'must' be changed, say, in the light of overwhelmmg a_nd unde~i~ble evidence. In other words, it may make sense not to reqmre reflexivity of!:;;;. Even more obviously, ~:;;;read as possible development need not be transitive in general. If t C u and u c the intermediate state u may just be indispensable order to at w from t. _In~or~ation obtained at w may, for instance, depend on conceptual distmctiOns or certain findings which are available at w only b:cause w develops from u rather than from t directly. Moreover' one might doubt that information development can ever lead to 'deacl ends'. Such an optimistic attitude would amount to requiring seriality of 1:;;;.42 I~ seell_ls only fair to say that the evaluation of intuitionistic formu~as m Knpk~ models (J, !:;;;, v}, where (J, !:;;;} is a Kripke frame and v IS a ~otal assig~m.ent function to interpret the propositional variables, provides deep msight into the nature of the intuitionistic connectives. It sets apart intuitionistic conjunction and disjunction, which are evaluated 'on the spot', from intuitionistic negation 'h and intuitionistic
i~
a-;ri~~
~ Indee~, IPL is also characterized by a class of Kripke models where c IS J~st senal, but not necessarily reflexive and transitive. In these so-called rudimentary Kripke models (46], however, not only persistence is postulated ~or e:ery i~tuitionistic formula A, but also converse heredity, which says that If A IS venfie~ at e:ery state into which a given state t may develop (in one step), then A Is_venfied_at t already. Upon reflection this is a very strong, yet not completely Implausible constraint. Suppose, for instance, you are playing chess, a~d ev~ry _possibl~. move will put you into a winning position. Then you are m a ;vmnmg positlOn already. Dosen (44] has also shown that in order to ~haractenze IPL, converse heredity can be replaced by ancestrality: if t venfies A, ~he~ there is _a state u such that u ~ t and u verifies A. In Kripke models sati~fymg heredity and ancestrality both ~ and its inverse relation must be ser_Ial. Thus, according to this conception, development not only is always possible, but also never starts from scratch. In the absence of persistence, the requirement of symmetry would also make sense. However, with this condition we would leave the 'subintuitionistic sector'. .
4
SUBINTUITIONISTIC LOGICS
157
implication ~h, which are evaluated 'dynamically' with their evaluation clauses referring to !:;;;. These verification conditions (together with persistence as postulated for propositional variables and derived for arbitrary formulas) illuminate why the various modal translations faithfully embed IPL into 84. The completeness theorem then shows that the class of all Kripke models satisfying (i) - (iii) in fact characterizes IPL. Whereas, however, the interpretation of the intuitionistic connectives in Kripke models is in the first place laid down by the verification clauses, the conditions (i) - (iii) appear as degrees of freedom which may or may not be postulated. Hence, if Kripke frames are ju~t the right kind of structures for the language Lh of IPL, then the basic 'intuitionistic' system seems to be the logic of the class K of all models based on any Kripke frame whatsoever in that language. We shall follow Dosen (47] and refer to this subintuitionistic system as K(a). Although the interpretation of Lh in Kripke models conveys a neat understanding of the intuitionistic connectives by treating conjunction and disjunction as Boolean, while treating negation and implication as intensional, it should be clear that this is not the only possible partition into Boolean and intensional operations of IPL. Sylvan (161], for instance, presents IPL as an extension of classical propositional logic formulated in Boolean negation --, and Boolean conjunction I\. See also the nonhereditary Kripke models for IPL in (44]. In K(a) an implication p ~h q is interpreted as 'strict implication' O(p ~ q) and 'hP is interpreted as 'strict negation' D(p ~ f) in the minimal normal modal propositional logic K. It is thus natural to regard K(a) as the logic embedded into K by the modal translation a which for each A E Lh replaces every implication and negation in A by its corresponding strict implication and strict negation, respectively. This is the perspective under which K(a) is studied by Dosen (47]. More generally, if A is a modal extension of ~lassi~al ~ropositional logic, A(a) is defined as {A E Lh I f-A a (A)}. Ax10matizatiO~S of K (a) and various extensions of it have been presented by Corsi (36]. The axiomatization of K(a) in Table XIII is taken from (47]. Both Corsi and Dosen show that K(a) is also complete with respect to the class K of Kripke models based on frames having a strongly generating sg d 1 · h' state, that is, a 'base state' from which every state can eve op wit m one step. This is the kind of models for subintuitionistic logics that independently has been investigated by G. Restall (138], who defines validity of an intuitionistic formula A in K 89 as verification of A at the base state of every model M E Ksg· Restall presents an axiomatization SJ of the intuitionistic formulas valid in Ksg in this sense. Since K(a) is
158
CHAPTER 10
SUBINTUITIONISTIC LOGICS
Table XIII. An ax:iomatization of K(a). Al A2 A3 A4 A5 A6 A7 A8 A9 AlO mp:Jh weakening adjunction
axiom schemata A :>hA ((A =>h B) 1\ (B =>h C)) ((C :>hA) 1\ (C =>h B)) (A 1\ B) :>hA (A 1\ B) =>h B A =>h (AV B) B :>h (A V B) ((A =>h C) 1\ (B =>h C)) (A 1\ (B V C)) :>h ((A A f =>hA rules A, (A =>h B) f- B A f- (B =>hA) A, B f-A 1\ B
=>h (A =>h C) =>h (C =>h (A
159
r,t,p
1\
B))
p,t =>h ((A V B) :>h C) B) V (A 1\ C))
p
Figure I. A lattice of subintuitionistic logics.
the set of all intuitionistic formulas verified at any state of any M E K clearly K (a) ~ SJ. Conversely, if A is not verified at some state t of ~o~e m~del from K, consider the submodel generated by t and extend Its p~ssible dev~lo_pment' relation to make t strongly generating. The res_ultmg mo_del1s m Ks 9 , and by induction on A it can be shown that A IS not venfied at t. Thus, SJ is just another axiomatization of the system K(a).
O~r aim_ is to fin~ decent sequent calculi for K(a) and some interesting axiOmatic d" 1 d' extensiOns of this system. Recently' R . G ore' [75] h as ' reISp a~e IPL, that is, has presented IPL as a system of display logic that differs from the diplay calculus for IPL I·n [16] . Th"IS proof sys. te~ DI~L, ~Irectly suggests Gentzen systems for the subintuitionistic ~o?Ics, smce It cap~~r~s persistence of atomic information (p), reflexI.VIty (r) and transitivity (t) of ~' respectively, by separate structural znference rules. In the present chapter, it is shown that the sequent sys~ems resulting from giving up all or part of these structural rules are m fa~t sound and complete with respect to the corresponding classes of Knpke models. Moreover, seriality (s) of~ can also be expressed by a P_urely structural rule. We arrive at the lattice of subintuititionistic logics exhibited in Figure I.
It can easily be verified that the systems in this lattice are pairwise distinct. Interestingly, our characterization results reveal a connection with Finger's and Gabbay's [61] notion of temporalizing (or adding a temporal dimension to) a given logic. Temporalization is a method for combining an arbitrary logical system with a propositional tense logic. It turns out that the display calulus DK(a) for K(a)is obtained from a sequent calculus for an extension of the combined system Kt(K(a)), which is the result of temporalizing K(a) by the minimal tense logic
Kt. The above lattice of subintuitionistic logics is shaped from a semantical point of view. This taxonomy is different from the substructural hierarchy of systems weaker than IPL, cf. [48], [180]. The possible worlds models developed by Dosen [42] for these systems may, however, also be viewed as information models, see [179], [180]. Whereas the route from K(a) to IPL is paved by conditions on Kripke models, the substructural subsystems of intuitionistic logic arise from the standard sequent calculus presentation of IPL by a systematic variation of the hitherto standard structural inference rules. It would be interesting, of course, to clarify the relations between both ways of 'going subintuitionistic'. Persistence in combination with transitivity, for example, corresponds to the monotonicity axiom A :Jh (B :Jh A). Indeed, display logic has been designed for the very purpose of Gentzenizing substructural logics (and combinations of such systems). It thus turns out to be general enough a schema for dealing with both routes to weakening IPL.
160
SOUNDNESS AND COMPLETENESS OF DK(a)
CHAPTER 10
161
10.2. SEQUENT SYSTEMS FOR SUBINTUITIONISTIC LOGICS
In [75], Rajeev Gore has defined a display calculus for IPL using the Godel-McKinsey-Tarski translation of IPL into S4. This sequent system, DIPL, contains separate structural rules for the persistence of propositional variables and the reflexivity and transitivity of the 'possible expansion' relation (;;;. It therefore proves suitable for defining sequent calculi for the subintuitionistic logics we are interested in. To prove the adequacy of these calculi, it proves useful to translate sequents into formulas of the language Ct(£), which is the language of tense logic defined over the set of formulas of the logical object language £as the set of atoms. In our case£= £h. To be precise:
7i(A) 71(I) 71(*X) 71(X 0 Y) 71 (•X)
= = =
A t •7z(X) 71 (X) A 71 (Y) (P)71(X)
7z(I) 7z(*X) 7z(X o Y) 7z(•X)
f
•71 (X) 72 (X) V 7z(Y) [F]7z(X).
Consider the partition of the introduction rules for the logical vocabulary in Table XIV. We can conveniently define various sequent ~ystems by combing these modules of operational rules on top of the logical and structural rules of Chapter 3, see Table XV.
Definition 10.1. Let Atom be a denumerable set of propositional variables. Lh is the smallest set b. such that -Atom~~'
Theorem 10.2.
- f, t E b., - if A, B E b., then (A A B), (A V B), (A
~h
B) E b..
Lt(Lh) is the smallest set b. such that - £h
~b.,
- if A, B E ~' then ·A, (A A B), (A V B), (A ~ B), [F]A, (P)A E b.. In Lh the unary operation 'h (intuitionistic negation) may be defined by 'hA :=A ~h f. In Lt(Ch) as in Lt the unary operations [P] and (F) can be defined by [P]A := •(P)•A, and (F)A := •[F]•A, respectively. When it comes to defining functions over the set of formulas of Lt(£), like the depth of parsing trees, one has to be careful, if£ and Lt share some logical vocabulary. The resulting 'double parsing problem' can be circumvented either by renaming in £, or by restricting the base clause in the inductive definition of Lt(£) to monolithic formulas, that is, formulas not built up from Boolean connectives, see [61]. The declarative meaning of the structural connectives I, *, •, and o in DL is given by the translation r from Chapter 3, which now sends sequents into formulas of Lt(Lh):
where
Ti
(i = 1, 2) is defined as follows:
f-Kt
Proof. See Chapter 3.
A iff f-nKt I~ A. Q.E.D.
Theorem 10.3. The systems DKt, DKt(K(a))' and DK(a) enjoy strong cut-elimination. Proof. These systems are proper display calculi, that is, their rules satisfy the conditions which in (16] have been shown to guarantee cutelimination and which in Chapter 4 have been proved to guarantee strong cut-elimination (for a certain set of primitive reductions). Q.E.D. Corollary 10.4. DKt, DKt(K(a))' and DK(a) enjoy the subformula property.
10.3. SOUNDNESS AND COMPLETENESS OF DK(a)
We want to prove completeness with respect to models (I,(;;;, v) b~ed on ordinary possible worlds (or Kripke) frames (I,~), wher~ I .Is a non-empty set, ~ ~ I xI and v: Atomxi --7 {0, 1}. ?ur aim IS ~o prove that for every intuitionistic formula A, I ~ A IS provable. m DK(a) iff A is valid in every Kripke model. We shall de~ne.a n~ti~n of valid sequent such that K f= X ~ Y ('X ~ Y is vahd m K ) Iff in DKt(K(a))' f- X ~ Y, in order to derive that for every formula
SOUNDNESS AND COMPLETENESS OF DK(o-)
162
163
CHAPTER 10
A E eh, K F I---+ A iff in DK(a) this notion of valid sequent is
Table XIV. Modules of introduction rules.
K
Boolean rules I
F X---+ y
I (-+ f)
We thus have to define K
I (t-+) ) (:::>-+)
DKt DKt(K(a))' DK(o-)
I---+ A. The obvious candidate for
f= A, for every A E et(eh)· define M, t f= A ('A is verified
(f -+)
I system
f-
Theorem 10.6. K
f= T(X---+ Y)
iff f-nKt(K(a))' X---+ Y.
Proof.
(3-.Al o ... o •An) 1- •A1 1\ ... 1\ •An -+ ·B (3-.A 1 o ... o •An) 1- -.(P)••A1 1\ ... 1\ •(P)·•An -+ -,(P)...,...,B (3-,A 1 o ... o •An) 1- -,(P)Al o ... o •(P)An o (P)B -+ f t 1- f contradiction
zo can be extended to a maximal £-consistent p E I. By definition and Lemma 10.17 (iii), p ~ t. .. (ii) :::::}: Let w ~ t and •(P)B :::5 t. Then (P)B ~ t and, by defimt10n ~f ~' [F](P)B ~ w. Hence •[F](P)B :::5 w. Since 1- •[F](P)B --+ •B, 1t follows by Lemma 10.17 (iii) that ·B :::S w. ~: Suppose not w ~ t. Then there is aB such that [F]B :::S wand ·B :::5 t. Since 1- ·B--+ •(P)[F]B, -,(P)[F]B :::5 t. Thus, it is not the case that •(P)B :::5 t :::::} -,B :::5 w. Q.E.D.
Lemma 10.21. For every t E I and every A E .Ch:
MKt(K(u))''
t FA
iff A :::St. Proof. If A E Atom, the claim holds by definition; if the main connective of A is a Boolean operation, use Lemma 10.17. A = [F]B. ~: If [F]B :::5 t, then, by the definitio? of ~' (Vu E I) t ~ u implies B :::5 u. By the induction hypothesiS, t f= [F]B. :::::}:
170
CHAPTER 10
CHAPTER 11
Suppose [F]B ~ t. ~hen •[F]B ::5 t and, by Lemma 10.18 X= [F]C ::5 t} U {•B}) IS £-consistent. Thus X has a · '1 f ~( { C I extension, say w. By the definition of C 't C Mmaxima -consistent th . d t. h _, - w. oreover --,B -< w B e m uc Ion ypothesis and the cas fi B I '. - . y hence t ~ [F]B. e or oo ean negatwn, w ~ B, A = (B -.:J h C). (Vu E J) t c
If (B -.:J C) ::5 t, then, by the definition of c . (B h -.:J C) -< u By th . d t" h _, and the clause for Boolean implication. (Vu E e/)nt ~ IO~ ytothesis (B ::> C), and hence t f= (B -.:J C) ' - u Imp Ies u f= [F](B -.:J C) ~ t d h h . :=;.: Suppose (B :>h C) ~ t. Then t ~ (B -.:Jh C). an ence t ~ [F](B ::> C), which is equivalent to -
A = (P)B.
(ii),
. {::} {::} {::}
TRANSLATION OF HYPERSEQUENTS INTO DISPLAY SEQUENTS
{=;
·
u Imp 1Ies
If (P)B -< t then b L · t and Y emma. 10.20. (1), there is a (P) - u. Hence, by the mductlon hypothesis . uppose B ~ t. Then --,(P)B ::5 t. By Lemma 10.20
{=:
u E I such that u c t (P)B ::::;.· S -
F
!:"
L .
B-.::
'
(VuE/) (u ~ t::::;. (--,(P)B ::5 t::::;. --,B-< u)) (VuE I) (u ~ t::::;. --,B-< u) {VuE/) (u ~ t::::;. u ~ t ~ (P)B
JJ)
A = 'hB. Treat 'hB as B :>h f. Q.E.D. Lemma 10. 22. Every max · 1 y . . £-consistent. Ima -consistent structure IS also maximal Proo{ Suppose xo is maximal Y-consistent. Since f- To (Y) 10 17 (ii) '72(Y) -< xo ---:l- Y, T2(Y) Z ~ xo. Then there are X . X -< 'xo . ow assume that L 1' 2 such that ~ z o X 1 y and 'X2 ---:l- 'T2(Y). Since X---+ y f- X---+ . ---:lT2(Y) 1\ 'T2(Y). Thus f- z oX o T2(Y), we obtam f- .zox1 oX2--;)1 X 2 ---+ f, and thus X 0 Is maximal fconsistent. Q.E.D.
~ X . Hence, by Lemma
J
Theorem 10 23 DKt(K( ))' · all Kripke model~· if K L Xa Yis complete with respect to the class of · r- ---+ , then ~ X ---+ Y. Proof. Suppose If X ---:l- y Th X . y . to a maximal Y-consist . en IS -consistent and can be extended xo S ent and hence also £-consistent xo that is X -< · uppose now for reductio that xo F X---+ Y. Then ' -
xo F 71(X) ::> 72(Y)
{::} 71(X) ::> 72(Y) :S
xo
by Lemma 10.21. Obviously,~ T1(X)o(T1(X) -.:J T2(Y))---+ To (Y) s· moreover ~X---+ 7 (X) d ~ 2 . mce T2(Y)) ---:l- Y. Thus1 (3Z ~ o 72(Y) ---+ Y, we obtain~ X o (71 (X) ::> - X ) ~ Z ---+ Y. In other words xo ~ y q uo d non. Q.E.D. ' ,
As we have seen in Chapter 2, the study of nonclassical and modal logics has led to the development of many generalizations of the ordinary notion of sequent. In view of the diversity of these types of proof systems it becomes increasingly important to investigate their interrelations and their advantages and disadvantages. In this chapter, we shall consider the relation between display logic and the method of hypersequents independently introduced by G. Pottinger (134] and A. Avron [7]. It will be shown that hypersequents can be simulated by display sequents. Whereas hypersequents are finite sequences of ordinary sequents, display sequents are obtained by enriching the structural language of sequents, thus turning the antecedent and the succedent of a sequent into 'Gentzen terms', see also [41]. As Avron [12, p. 3] points out, "although a hypersequent is certainly a more complex data structure than an ordinary sequent, it is not much more complicated, and goes in fact just one step further". The translation of hypersequents into display sequents illuminates how display sequents generalize ordinary sequents still further than hypersequents. In contradistinction to the application ofhypersequents to modal logics, the modal display calculus comprises introduction rules for the modal and tense logical operators which can arguably be interpreted as meaning assignments; see Chapter 1. Therefore the translation of hypersequents into display sequents has something to recommend preferring display logic over the hypersequential formalism. We shall consider three case-studies of translating hypersequents into display sequents, corresponding with three different interpretations of hypersequents in [12]. 11.1. HYPERSEQUENTIAL CALCULI
Hypersequents have systematically been studied in a series of papers by A. Avron [7], [9], [10], [12}. As explained in Chapter 2 already, a hypersequent is a sequence r1 ---+ ~1 I r2 ---+ ~2
I ... I r n
-+
L1n
of ordinary sequents (or sequents in which ~i and ri are sequences of formula occurrences) as their components. The symbol 'I' in the state-
171
172
ment of a hypersequent is intuitively to be understood as disjunction. Hypersequents allow one to draw a distinction between internal and external versions of structural rules. The internal rules deal with formulas within a certain component, whereas the external rules deal with components within a hypersequent. If G, H, HI, H 2 etc. are used as schematic letters for possibly empty hypersequents, one obtains, for example, the following external and internal expansion rules:
HI I H2 I H3 f- HI I H2 I H2 I H3 versus
HI 1r,A,6.-+ e 1H2
r- HI 1r,A,A,6.-+ e 1H2.
Cut only has an internal version:
GI 1 ri-+ 6-I,A 1HI G2l A,r2-+ 6.2l H2 GI 1G2l ri,r2-+ 6.I,6.2I HI 1H2 The method of hypersequents allows a cut-free presentation GSS of SS satisfying the subformula property. In this system, which is presented in Section 11.1.2, the introduction rules for 0 take the form
HI I 06. -+ A I H2 f- HI I 06. -+ DA I H2 HI 16., A -+ r 1H2 r- HI 1 6., DA -+ r 1H2, where 06. ={DB I BE 6.}. With these rules, internal monotonicity cannot be replaced by internal monotonicity for atoms only, as has been emphasized by Hacking [85]. In the terminology of Chapter 1, the right introduction rule for 0 fails to be even weakly explicit: it exhibits D in one of the premise sequents and also in the antecedent of the conclusion sequent into which DA is introduced. Therefore this rule cannot be viewed as assigning a meaning to 0 by specifying in a purely schematic way how a formula with 0 as the main connective is introduced into conclusions. In [12], Avron demonstrates the versatility and usefulness ofhypersequential calculi by presenting cut-free hypersequent systems for several logics for which an appropriate cut-free Gentzen calculus presentation turned out to be a problem for a long time. Among these systems are Lukasiewicz 3-valued logic L3, the modal logic SS, and Dummett's LC. (In this chapter, we shall use the names 'L3', 'SS', and 'LC' to denote certain familiar semantic specifications of these systems.) In the completeness proofs for the sequential presentations, A vron considers various interpretations of hypersequents. These interpretations give rise to different translations of hypersequents into display sequents. In Section 11.3, these translations will be dealt with separately.
173
HYPERSEQUENTIAL CALCULI
CHAPTER 11
11.1.1. GL3 The problem with L3 is to find a cut-free sequent calculus presentat~on such that the sequent arrow reflects the 'official' implication connective of this logic, i.e., f- AI, ... , An-+ B iff
(\/)
F AI
:J (A2 :J .. · :J (An :J B) ... ).
The n-contraction sequent systems of Prijatelj [135] for Lukasiewicz's many-valued logics, for example, fail to satisfy cut-eliminati?n·. A ~y persequential calculus GL3 for.L3 satisfying.(\/) an~ cut-eln:n_mati~n has been presented in [10]. This system consist~. of (I) the axwJ?atic rule f- A -+ A and the above internal cut rule, (n) the standard mternal structural rules, apart from internal contraction, (iii) the standard external structural rules, (iv) the rule (MX):
a 1ri r2 r3 -r ~11~2,~31 H ' G' 1r1,ri
G I ri,r~,r~ I H-r ~i,~~,~~ I H -r ~~,~i I r2,r~ -r ~2,~~ I fg,r~ -r ~3,~3'IH
and (v) the following introduction rules for the primitive operations of L3:
c 1r
-+ 6., A 1 H G 1r', B -+ 6.' 1H G 1r', r, (A :J B) -+ 6.', 6. 1 H
Glf,A-tB,6.IH G 1 r -+ (A :J B), 6. 1 H
Glf,A-+6-IH G 1 r -+ 6., ·A 1 H
c 1r
-+ 6., A 1 H G 1 r, ·A -+ 6. 1 H
c
c 1 A, r
-+ 6. 1 H 1 (AAB),r-+ 6.1 H
GIB,r-+6-IH -+ 6. 1 H
c 1 (A A B), r
G 1r -+ 6., A 1H G 1r -+ 6., B I H c 1 r -+ 6., (A A B) 1 H
Glf,A-+6-IH Glf,B-+6-IH c 1r, (A v B)-+ 6.1 H G
c 1r
r
-+ 6., A 1 H -+ 6., (A v B) 1 H
1
c 1r G
1
r
-+ 6., B 1 H -+ 6., (A v B) 1 H
11.1.2. GSS
It seems that no cut-free ordinary sequent system for SS is known. The cut-free sequent calculus presentations of SS due to Kanger [91], Sato [150], Belnap [16], Indrzejczak [89], and the present author [181 ], [182],
175
DISPLAY CALCULI
174
CHAPTER 11
for instance, all make use of a generalized notion of sequent or sequent calculus. Avron [12] presents a cut-free hypersequent calculus GS5 for SS. This system consists of the axiomatic rule f- A -+ A, the internal cut rule, the standard internal and external structural rules, the above hypersequential introduction rules including the introduction rules for D, and the 'modalized splitting rule': G 1 or1, r2-+ 0.:11, .6.2 1 H G 1 or1 -+ 0.:11 1 r2-+ .6.2 1 H 11.1.3. GLC
The superintuitionistic propositional logic LC axiomatized by Dummett [50] is the logic of the class of all linearly ordered intuitionistic Kripke models. As emphasized by Avron, the cut-free sequent system for LC presented in [160] has the disadvantage of comprising infinitely many schematic rules. This drawback is avoided in the hypersequent system GLC for LC in [9]. In GLC, components of hypersequents are constrained to be single-conclusion ordinary sequents. GLC consists of the axiomatic rule f- A -+ A, the internal single-conclusion cut rule, the standard external structural rules, the standard single-conclusion internal structural rules, hypersequential versions of the single-conclusion introduction rules of intuitionistic logic, and two further structural rules, the 'intuitionistic splitting rule' and the 'communication rule': Glf,L1-+AIH
G1 1 r1 -+ A1 1 H1 G2 1 r2-+ A2 1 H2 G1 1 G2 1 r1 -+ A2 1 r2-+ A1 1 H1 1 H2
11.2. DISPLAY CALCULI
As we have discussed at length, in DL complex structures are built up using certain structure connectives. In order to present as sequent systems logics combining logical operations with an essentially distinct inferential behaviour, there may be more than one family of structure operations. In the one-family case, the structural language of sequents comprises the nullary I, the unary operations * and •, and the binary operation o. Lukasiewicz 3-valued logic combines connectives from two different families of operations, namely on the one hand combining (extensional, additive) conjunction and disjunction and on the other
hand internal (intensional, multiplicative) negation and imp~ication. To distinguish the corresponding families of structure connectives, the · structure operatiOns may b e subscri"pted·· I.l ' o.~' *l·' and •i versus le, * and •c· In the present cont ext , h owever, 1.l -- I c, *l· = *c, and 0 ~'- ~ so that we are left with distinguishing between °i and 0 c· T~e ·~- Cl clear and simple inferential behaviour of the structure connecf Ives IS laid down by the following basic structural rules of DL: Basic structural rules (1.1) (1.2)
(2.1) (2.2) (3) (4)
X oi Y -+ Z -H- X -+ Z oi *Y -1f- Y -+ *X 0 i Z X oc Y -+ Z -11- X -+ Z Oc *Y -11- Y -+ *X Oc Z X -+ y Oi z -11- X oi *z-+ y -11- *y Oi X -+ z X -+ Y oc Z -11- X oc *z -+ Y -11- *y 0 c X -+ Z X -+ Y -1f- *y -+ *X -11- X -+
**Y
X-+ •Y -11- •X-+ Y
As before X 1 -+ Y1 -11- X 2 -+ Y2 abbreviates X1 -+ Y1 I- X2-+ Y2 and X -+ y 'I- X 1 -+ y 1_ If two sequents are interderivable by me~ns of 2 2 rules (1) __ (4), then these sequents are said to be structurally or display equivalent. . . . The intended declarative meaning of the structure connectives I.s m part context-sensitive, and the earlier translation 7 of sequents mto tense logical formlas is now defined as: 7(X-+ Y) := 71(X) ::::> 72(Y), where the translations 7i (i = 1, 2) are defined as follows: 7i(A) 71 (I) 71(*X) 71(X oi Y) 71(XocY) 71(•X)
A =
f
72(*X) (X 72 °i Y) 72 (X °c Y)
= =
' 71(X)
=
T2(X) + 72(Y) 72(X) V 72(Y)
72(•X)
=
(F]72(X)
t
=
•72(X)
=
71(X) 0 71 (y) 71(X)/\71 (y) (P)71(X)
=
A+ B is defined as -,A ::::> B, which in classical logic and thus H ere . A v B and A 0 B in normal tense logic is, of course, eqmvalent to . ' 1 . ll . is defined as the dual of A+ B (i.e. •(A ::::> ·B)), which c ass~ca. ~IS equivalent to A 1\ B. Moreover, if t and fare not assumed as pnmitive, t is defined asp::::> p, for some atom p, and f is define~ as •t. The rul~: (1) _ (4) obviously are correct under the 7-translat10n, and the pa
( (P), [F]) exemplifies the idea of a residuated pair of unary operations, see Chapter 3. Recall that DL derives its name from the fact that any substructure of a given display sequent s may be displayed as the entire antecedent or succedent, respectively, of a structurally equivalent sequent s'. In order to state this fact precisely, some definitions are needed. An occurrence of a substructure in a given structure is said to be positive (negative) if it is in the scope of an even (uneven) number of *'s. An antecedent (succedent) part of a sequent X -+ Y is a positive occurrence of a substructure of X or a negative occurrence of a substructure of Y (a positive occurrence of a substructure of Y or a negative occurrence of a substructure of X).
Theorem 11.1. (Display Theorem, Belnap [16]) For every sequent s and every antecedent (succedent) part X of s there exists a sequent s' structurally equivalent to s such that X is the entire antecedent (succedent) of s'.
sequential calculus for L3, contains combining conj.m~c~ion a~d disjun~ tion and (internal) negation and implication as pnm1t1ve. Smce w~ ~1d not make these distinctions in previous chapters, we shall exphc1tly state the introduction rules to be considered, reusing the symbols 'A' and 'V': absurdity (-+ f)
X -+ I f- X -+ f
(f-+)
f-f-+1
Internal negation (-+ •)
X-+ *A f- X-+ ·A
(• -+)
*A-+ X f- ·A-+ X
Internal implication
X As has been emphasized in Chapter 3, the Display Theorem has both technical and conceptual significance. It allows an "'essentials-only' proof of cut elimination relying on easily established and maximally general properties of structural and connective rules" [18]; see Chapter 4. Moreover, the Display Theorem allows the removal of a certain amount of holism in the proof-theoretic semantics of the logical operations. If the meaning of an n-place connective f is specified by it's introduction rules, then these introduction rules must not exhibit any logical operations other than f. Otherwise f's meaning is - at least in part - holistic. This idea may be strengthened by postulating that the succedent (antecedent) of the conclusion sequent in a right (left) introduction rule must not exhibit any structural operation. The display property guarantees this segregation principle. In addition to the basic structural rules, every display sequent system to be considered will contain the logical rules (id) and (cut). Logical rules (id) (cut)
A -+ B f- X -+ A :J B X -+ A B -+ y f- A :J B -+ *X Oi
Oi
y
Combining conjunction (-+ /\) (/\ -+)
X-+A X-+Bf-X-+At\B A-+ X f-A 1\ B-+ X; B-+ X f-A 1\ B-+ X
Combining disjunction (-+V)
X -+ A f- X -+ A V B;
(V-+)
A-+X
X -+ B f- X -+ A V B
B-+Xf-AVB-+X
Forward-looking necessity (-+[F])
•X -+ A f- X -+ [F]A
([F] -+)
A -+ X f- [F]A -+ •X
Intuitionistic implication
f-A-+ A X-+ A
177
DISPLAY CALCULI
CHAPTER 11
176
A-+ Y f- X-+ Y
In the absence of unrestricted contraction and monotonicity rules, the distinction between internal (intensional, multiplicative) and combining (extensional, additive) conjunction and disjunction can be drawn. The display calculus to be defined in Subsection 11.2.1, like the hyper-
•X
Oi
A -+ B f- X -+ A :Jh B B -+ y f- A :Jh B -+ •( *X
X -+ A
Oi
Y)
As we have seen, introduction rules for the other normal tense logical modalities are also easily available, for example:
178
CHAPTER 11
Note that (Pi) is required in Lemma 11.2, because inst~ad of (--+ :J) and (::=>--+) we did not formulate order-sensitive introductiOn_ rules for the left- und right searching implications known from Categonal Grammar.
Backward-looking possibility
(--+ (P)) ( (P) --+)
X--+ A f- •X--+ (P)A A --+ •X f- (P)A --+ X
In addition to the display sequent rules introduced so far, we shall also consider further purely structural rules, where o is any of oi and oc:
{I)
X --+ Z -lf-- I
{Ai)
X1 oi {X2 oi X3) --+ Z -lf-- {X1 oi X2) X1 oi X2--+ Z f- X2 oi X1 --+ Z X Oi X--+ z f- X--+ z X1 --+ Z f- X1 oi X2--+ Z
{Pi) (Ci)
{Mi)
o
X --+ Z
X --+ Z -lf- X --+ I oi
o Z
X3 --+ Z
xl Oc {X2 Oc X3) --+ z -lf- (Xl Oc X2) Oc x3 --+ z {Pc) X1 Oc X2 -t Z f- X2 Oc X1 -t Z X Oc X --+ Z f- X --+ Z {Cc) {Me) X1 -t Z f- X1 Oc X2 -t Z {Mix) I--+ Z oc {*Xl oi *X2 oi *X3 oi Y1 oi Y2 oi Y3) oc Z', I --+ Z Oc {*X~ oi *X~ oi *X~ oi Y{ oi Y~ oi Y3) oc Z' f-f-- I--+ Z oc (*Xl oi *X~ oi Y1 oi Y{)oc oc( *x2 Oi *X~ oi y2 oi Oc ( *x3 oi *X~ oi y3 oi Y3) Oc Z' {Ac)
yn
{MN) (rP) (rT) (r4) {rB)
179
DISPLAY CALCULI
I --+ X f- •I --+ X p --+ X f- •p --+ X * • *X --+ Y f- X --+ Y * • *X --+ Y f- * • • * X --+ Y * • *(X o * • *Y) --+ Z f- Y o * • *X --+ Z
The rule {Mix) is a special rule to be used in the display presentation ofL3.
Lemma 11.2. In every display calculus extending the logical rules, the basic structural rules, the (I) rules, (Me), (Cc), (Pi) and the above introduction rules by purely structural rules we have: (i) f- X --+ T1(X) and f- T2(X) --+ X; (ii) f- X--+ Y implies f- T1{X)--+ Y, and f- Y--+ X implies f- y --+ T2(X). Proof. By induction on the complexity of X. Q.E.D.
11.2.1. DL3 We shall define a display sequent calculus for L3 satisfying condition ( (A::=> B) (A::=> B) : => ((B ::=>C) : => (A=> B)) (A::=> (B ::=>C)) : => (B : => (A=> C)) ((A::=> B) :J B) :J ((B :J A) :J A) (({(A::=> B)::=> A) ::=>A) : => (B =>C)) => (B =>C) (A 1\ B) ::=>A (A 1\ B) : => B (A :J B) : => ((A::=> C) : => (A=> (B 1\ C))) A :J (A V B) B :J (A V B) (A::=> C) :J ((B :J C) :J ((A V B) :J C)) {N1) (-,B ::=>-,A)::=> (A :J B)
{Il) {12) {I3) (I4) (15) {Cl) (C2) (C3) (Dl) {D2) (D3)
Inference rule: A, (A :J B)/ B
Theorem 11.4. (i) If f- A in HL3, then f- I --+ A in DL3. . (ii) If f- X --+ Y in DL3, then p T(X --+ Y) m L3.
180
Proof. (i) It suffices to show that the axiom schemata of HL3 can be proved in DL3, and that modus ponens preserves provability in DL3. The only non-trivial cases are given by the implicational schemata (I4) and (I5). Proofs in DL3 can be obtained by following Avron's [10] derivations of (I4) and (I5) in GL3 using the translation of hypersequents into display sequents presented in Section 11.3. This will give rise to display proofs with two final applications of (Cc). The sequents *A -+ (A ::::) B) and B -+ *I oc A oc (*A oi B) are easily derivable; for the latter use monotonicity and (Mix). In the case of (I4), leaving out some simple steps, we obtain: *A -+ (A ::::) B) B -+ *I Oc A oc (*A oi B) (A::::) B)::::) B-+ **A oi (*I Oc A Oc (*A oi B)) ((A::::) B)::::) B) oi (B::::) A)-+** A oi (*I Oc A Oc (*A oi B)) *(*I Oc A Oc (*A oi B)) oi ((A::::) B)::::) B) oi (B::::) A)-+ A *(*I Oc A oc (*A oi B))-+ (I4) *(I4) -+ *I oc A Oc (*A oi B) I Oc *(I4) Oc *(*A Oi B)-+ A I Oc *(I4) oc *(*A oi B) oi ((A::::) B)::::) B) oi (B::::) A)-+ A I Oc *(I4) Oc *(I4) -+ *A oi B I Oc *(I4) Oc *(I4) -+ A ::::) B B -+ B (I Oc (*(I4) oc *(I4))) oi ((A::::) B)::::) B)-+ B A-+ A (I Oc (*(I4) Oc *(I4))) oi ((A::::) B)::::) B) oi (B::::) A)-+ A
11.2.2. DS5 In normal modal logic enough structural assumptions are made to make the distinction between combining and internal disjunction and conjunction disappear. We are thus dealing with one-familiy logics and may just use o instead of oi or oc· Definition 11. 'l. The sequent system DS5 consists of the logical rules,
the basic structural rules, the rules (I), (A), (P), (C), (M), the introduction rules for the logical operations of S5, and the rules (MN), (rT), (r4), and (rB). Theorem 11.8. 1-nss X-+ Y iff F=ss r(X-+ Y).
In modal logics without the symmetry schema B, the completeness proof with respect to a display formulation uses the fact that for every frame complete normal modal logic, the extension by (P) is conservative. As we have seen in Chapter 4, the modal display calculus offers a modular sequent-style treatment of many normal modal logics. The rules (rT), (r4), and (rB), for instance, correspond with the familar axiom schemata T, 4, and B, respectively. Instead of (r4) and (rB) one could also use a structural rule corresponding with the modal axiom schema 5, namely
I Oc *(I4) Oc *(I4) -+ (I4) I -+ (I4) Oc {I4) Oc (I4) I-+ (I4) Also a proof of (I5) may be obtained by starting from B -+ *I oc A oc (*A oi B). (ii) By induction on proofs in DL3. Q.E.D. Corollary 11. 5. I-HL3 A iff 1-nL3 I -+ A.
Using Lemma 11.2, this result can be strengthened to strong completeness, cf. [77]. Theorem 11.6. 1-nL3 X-+ Y iff FL3 r(X-+ Y).
F TI(X)::::) in L3. By the previous corollary and completeness of HL3, we have 1-nL3 I-+ r1(X)::::) r2(Y) and thus 1-nL3 r 1(X)-+ r2(Y). Since by Lemma 11.2, 1- X-+ r1(X) and 1- r2(Y)-+ Y in DL3, two applications of the cut rule give 1- X-+ Y. Q.E.D. Proof.(=>): This is Theorem 11.4, (ii). c~=): Assume that
T2 (Y)
181
DISPLAY CALCULI
CHAPTER 11
* • *X -+ Y 1- • * • * X -+ Y.
11.2.3. DLC There is more than one way to display logics containing a constructive implication but lacking an involutive negation. One may, for example, retain the involutive structural operation * and use a modal translation of the logic under consideration, if such a translation is available and the modal logic in question is displayable, or one may work with a version of display logic due toR. Gore [75], [76] in which the structure operations * and o are replaced by two binary operations o1\ and ov with the following Ti-translations: T1(Xo"Y) TI(XovY)
= =
Tl (X) 1\ Tl (Y) Tl(X)...;.. Tl(Y)
r2(X o" Y) T2(X Oy Y)
=
T2(X) -:Jh T2(Y) T2(X) V T2(Y),
183
CHAPTER 11
DISPLAY CALCULI
where -;- is the residual of disjunction, namely the subtraction operation from dual intuitionistic logic. The basic structural rules for these operations are as follows:
Definition 11. 9. The sequent system D54.3 results from D55 by giving up (rB) and adding the structural rule
182
X -+ Y X
Oy
o/\
Z
--lf-
X
y-+
z
-If-
X -+
o/\
Y-+ Z
-If-
Y
o/\
X-+ Z
-If-
Y-tXo/\Z
y
-if-
X -+
z Oy y
+
X
Oy
z
and we would have the following introduction rules for
~h
Oy
(r.3)
z-+ y
and V.
Proof. See Chapter 4.
m(A 1\ B) m(A V B) m(A ~h B)
I--+ Z ov (X1 of\ At) ov Z' I--+ Y ov (X2 o/\ A2) ov Y' ff- I--+ Z ov Y ov (X1 o/\ A2) ov (X2 of\ At) ov Z' ov Y'
Dp, for every atom p
f m( A) 1\ m( B) m( A) V m( B) D(m(A)
~
m(B)).
Theorem 11.11. For every formula A in the language of LC, I- A in LC iff f- m(A) in 54.3.
B) V (B ~ A) can then be
I--+Aoi\A I--+Boi\B I --+ (A of\ B) ov (B of\ A) I ov (A of\ B)--+ (B of\ A) I ov (A of\ B) --+ (B ~A)
Definition 11.12. The sequent system DLC consists of the logical rules, the basic structural rules, the rules (I), (A), (P), (C), (M), the introduction rules for f, ~h, 1\, and V, and the structural rules (MN), (rT), (r4), (rP), and (.3). Let D54.3t be the result of adding the introduction rules for (P) and the following structural rule to D54.3:
I --+ (A ~ B) ov (B ~ A) I --+ (A ~ B) V (B ~ A)
I --+ X f- I --+
In the present context, however, working with the modal translation approach has the advantage of translating hypersequents into the same version of display logic in each of the examples considered. Axiomatically, the modal logic 54.3 results from the familiar axiomatic presentation of 54 (= KT4) by adding the .3 schema D(DA ~ DB) V D(DB :J DA). This schema corresponds with the weak connectedness of the accessibility relation R in modal Kripke models: ~
* • *X --+ Y.
Q.E.D.
m(p) m(f)
In order to prove the characteristic axiom schema of LC, the display version of Avron's communication rule may be used:
VrVsVt((Rrs 1\ Rrt)
--+ Y I- •
The Godel-McKinsey-Tarski translation of intuitionistic propositionallogic into 54 amounts to a faithful embedding of LC into 54.3. This translation m is defined as follows:
X --+ A B--+ Y f- (A ~h B) --+ X of\ Y X --+A of\ B f- X --+ (A ~h B)
~
* • *X
Theorem 11.10. f-ns4.3 X--+ Y iff FS4.3 T(X--+ Y).
A --+ X B --+ X f- (A V B) --+ X X --+ A ov B f- X --+ (A V B)
The characteristic axiom schema (A proved as follows:
X --+ Y, •X --+ Y,
* • *X.
In the next section, we shall consider the system DLC U D54.3t. Like the other display calculi defined in this paper, DLC U D54.3t enjoys cut-elimination and the subformula property with respect to the logical operations. This follows from the very general strong cut-elimination theorem for display calculi satisfying certain conditions on the shape of sequent rules in addition to eliminability of principal cuts; see Chapter 4. In particular, it follows that
((s = t) V Rst V Rts)).
Lemma 11.13. DLC U D54.3t is a conservative extension of both DLC and D54.3.
Since any generated subframe of a transitive weakly connected frame is connected, (i.e., Vs\lt((s = t) V Rst V Rts)), and generated subframes preserve validity, 54.3 is characterized by the class of all linearly ordered modal Kripke models, see for instance [73, p. 29f.].
Theorem 11.14. In DLC, f- X--+ Y iff
1
f= T(X--+ Y)
in LC.
184
Proof. Weak soundness and completeness is proved in [75], using Lemma 11.13. Strong completeness follows using Lemmata 11.2 and 11.13. Q.E.D.
Proof.
11.3. MAPPING HYPERSEQUENTS INTO DISPLAY SEQUENTS
11.3.1.
185
MAPPINGS INTO DISPLAY SEQUENTS
CHAPTER 11
{::}
f-oL3
p(H)
f-oLa f-oLa
I -+ Tz (po (H)) I-+ (A2 => ... => (An => B) ... ). The interpretation c/JH of a hypersequent H = s 1 I ... I sn with singleconclusion components Si is the formula cp81 V ... c/Jsn.
The structures used in the framework should be built from the formulae of the logic and should not be too complicated (for human understanding and for computer implementation). Most important- the subformula property they allow should be a real one.
Observation 11. 22. Let the translation r* of display sequents into formulas be defined like the earlier translatioin r, except that rt(A) = m( A), for formulas A in the language of LC. For every non-empty hypersequent H with single-conclusion components, r2((o(H)) = m(cfJH). Lemma 11.23. In DLC U DS4.3t: (i) f-- X -t ri(X) and f-- r2(X) -t X; (ii) f-- X -t Y implies f-- ri(X) -t Y, and f-- Y -t X implies f-- Y -t r2(X). Proof. By induction on the complexity of X. Q.E.D. Theorem 11.24. In DLC, f-- ((H) iff f--GLC H. Proof. ~ ~
~
~ ~
f--oLC ((H) f--oLCuDS4.3t I-t r2((o(H)) f--os4.3 I-t r2((o(H)) f--os4.3 I-t m(cfJH) f--Lc c/JH f--GLC H
Lemmata 11.23 (i), 11.13 Lemma 11.13 Observation 11.22 Theorems 11.11 and 11.14 (12, Section III.l.4] Q.E.D.
187
In a footnote to this paragraph, he remarks that "[a] use of "structural connectives" that can arbitrarily be nested usually violates this principle. It seems to me that this is the weak point of Belnap's framework of Display Logic". The point is that the subformula property for the logical operations implied by (strong) cut-elimination in display logic (see Chapter 4) fails to be of immediate use in proof-search and other applications of cut-elimination. As Avron [12] observes, also in hypersequent systems cut-elimination normally does not imply Craig interpolation. The subformula property of display calculi may, however, still be used to prove conservative extension results. As often, there is a price to be paid for greater generality. A major advantage of DL is the simple and natural treatment of modal operators it allows. The display introduction rules for these operators can be viewed as meaning assignments and in combination with purely structural necessitation rules give rise to cut-free sequent systems for the minimal normal modal and tense logics K and Kt. Modularity in the sequent-style presentation is achieved by a correspondence between many important modal axiom schemata and purely structural display sequent rules, see Chapter 4. Irrespective of the aspect of greater generality, the modal display calculus provides a reason for preferring the method of display sequents over the method of hypersequents at least in one important application area, namely modal and tense logic.
11.4. DISCUSSION 11.5. OTHER TRANSLATIONS INTO DL
We have seen that there are straightforward and perspicious translations of hypersequents into display sequents. The components of a hypersequent can be translated using the structure connectives of display logic which are employed in the introduction rules for the internal logical operations and the modalities, while the structure connective 'I' in a hypersequent is translated by 'oc', the structural analogue of combining disjunction (conjunction) in succedent (antecedent) position. Whereas hypersequents as defined by A vron may be expected to work neatly for one-family systems and logics comprising connectives from two different families of logical operations, DL has been developed as a framework for systematically combining connectives from n > 1 families of logical operations. In [12, p. 2], Avron requires of a "good" proof system to satisfy among other things the following condition:
Recently, translations of other kinds of proof systems into DL have also been considered in the literature. In [115], G. Mints has presented systems of indexed sequents for the normal modal propositional logics obtained by combining the axiom schemata B, T, and 4 over K. Mints gives a detailed proof of cut-elimination for these systems of indexed sequents and shows how they can be translated into theoremwise equivalent display sequent calculi. R. Gore [80] raises the questions whether higher-level sequents can be translated into display sequents and whether 'Basic Logic' [148] can be embedded into DL and vice versa. Display logic might thus serve as a background theory used to compare with each other various kinds of generalized sequent systems.
CHAPTER 12
PREDICATE LOGICS ON DISPLAY
... predicate logic can be regarded as a special kind of propositional modal logic
S. Kuhn [95, p. 145] This chapter provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Benthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic, DL, by introduction rules for the existential and the universal quantifier. These rules for Vx and :Jx are analogous to the display introduction rules for the modal operators 0 and 0 and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal 'modal' predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules. The chapter has two main aims, namely 1. presenting a uniform proof-theoretic schema for both substructural subsystems of classical first-order logic, FOL, and various subsystems of FOL obtained by relaxing Tarski's truth definition for the existential and universal quantifiers, and 2. introducing these quantifiers into the framework of DL.
Recently, DL has received a great deal of attention. The original approach has been refined, extended, and re-applied to well-known logics and applied to new ones, see also [80], [118], [140]. However, so far all these developments have remained at the level of propositional logic. The addition of quantifiers to DL is briefly discussed in [16]: Quantifiers may be added with the obvious rules: (UQ)
Aa f- X
X f- Aa
(x)Ax f- X
X f- (x)Ax
provided, for the right rule, that a does not occur free in the conclusion . . . . The rule for the existential quantifier would be dual. ... [A]s yet this addition provides no extra illumination. I think that is because these rules for quantifiers are "structure free" (no structure connectives are involved; ... ) . One upshot is that adding these quantifiers to modal logic brings along Barcan and its converse ... willy-nilly, which is an indication of an unrefined account; alternatives therefore need investigating. [16, p. 408 f.] 43 189
190
The crucial observation here is that the standard quantifier rules arc structure-free. In what follows a structural account is developed of existential and universal quantification in DL, which is based on ideas put forward by, for instance, R. Montague [117] and, more recently, J. van Benthem [20] (see also the references listed in [95]). The key to this structural and more refined account of quantification is to look at the quantifier prefixes :Jx and Vx as modal operators. It has often been observed that the modal operators D ('it is necessary that') and 0 ('it is possible that') can be regarded as universal and existential quantifiers respectively. This is actually already Leibniz' idea that a proposition is necessarily true, if it is true in every possible world, whereas a proposition is possibly true, if it is true in some possible world. The Kripke-style possible worlds semantics generalizes this idea by introducing a binary relation of accessibility between worlds. There is thus a widely shared conviction that the truth conditions for D and 0 depend on the meaning of the universal and the existential quantifier in the meta-language of classical predicate logic. But then, conversely, the meaning of Vx and :lx may be explicated in terms of necessitiy and possibility, i.e. in terms of accessibility between 'worlds'. Moreover, from this modal perspective one may expect an illumination of universal and existential quantification. In particular, one may expect that the lattice of normal modal logics leads to an interesting landscape of predicate logics, in which classical first-order logic is just one among other 'first-class' citizens. This is the perspective on predicate logic advocated by van Benthem [20], [21]. According to van Benthem, the modal perspective reveals the abstract semantical core of quantification. Let M be any first-order model and let a, (3, ... range over variable assignments in that model. The standard Tarskian truth definition for the existential quantifier is:
M
f= 3xA[a]
iff for some assignment f3 on IM
I: a
=x
f3 and M
f= A[f3]
In the more general semantics the concrete relations =x between variable assignments are replaced by abstract binary relations Rx of 'variable update' between 'states' a, (3, /, ... from a set of states S. Assuming an interpretation of atoms containing free variables, the truth 43
191
A SEQUENT CALCULUS FOR KFOL
CHAPTER12
Belnap concludes this paragraph with the remark: "Introducing a family for each constant helps". While I shall take up the suggestion for further investigation, I shall neither try to elaborate this remark nor attempt to relate it to the treatment of quantifiers presented in the present paper.
definition for the existential quantifier becomes: M
f= 3xA[a]
iff for some (3 E S : aRxf3 and M
F A[J)]
or in the standard notation of modal logic,
'
M, a
f= 3xA
iff for some (3 E S : aRxf3 and M, f3
FA
Thus to every individual variable x there is associated a transition relati~n Rx on states. The resulting minimal predicate logic, KFO~, is nothing but thew-modal version of the minimal normal mo_dallogic K. In order to obtain an axiomatization of KFOL, one may JUSt take any axiomatic presentation of K and replace every occurrence of 0 and D by one of 3x and Vx, respectively.
12.1. A SEQUENT CALCULUS FOR KFOL
The idea now is not only to explicate the truth conditions of the quantifiers from the modal perspective but also to obtain from it structuredependent sequent rules for :Jx and Vx. DL offers a structural accou~t of 0 and 0 and therefore suggests a structural account of the quantifiers. Recall from Chapter 3 M. Dunn's definition of a residuated pair.
Definition 3.1 Consider two partially ordered sets A = (A,~) and B = (B, ~') with functions f: A---+ Band g: B---+ A. The pair (!,g) is called
residuated a Galois connection a dual Galois connection a dual residuated pair
iff iff
(fa~' b iff a~ gb);
iff iff
(fa ~' b iff gb ~a); (b ~' fa iff gb ~ a).
(b ~'fa iff a~ gb);
In category-theoretic terms, a residuated pair provides an ex~mple _of adjointness between functors, and also the universal and the existential quantifiers can be understood as a pair of adjoint functors (see [72, Chapter 15]). For every binary relation Rx on a non-empty set S of states, we may define the following functions on the powerset of S:
VxA 3xvA VxvA :JxA
....-
{a I Vb (aRxb implies bE A)} {a I 3b (bRxa and bE A)} {a I Vb(bRxa impliesb EA)} {a I 3b (aRxb and bE A)}
192
DISPLAY OF PREDICATE LOGIC
CHAPTER 12
193
Table XVII. Introduction rules for Vx and :Jx. (--+Vx) (Vx --+)
•xX --+ A f- X --+ VxA A --+ X f- VxA --+ •xX
(--+ :Jx) (:Jx --+)
X --+ A f- * •x *X --+ :JxA * •x *A --+ X f- :JxA --+ X
That is, we have:
To exploit this Gentzen duality between Vx and 3x, for every individual variable x, we introduce a structure connective •x· The basic structural rules for •x are:
12.2.
DISPLAY OF PREDICATE LOGICS
Since our main interest is in the behaviour of the quantifiers Vx and :lx as modal operators, we shall concentrate on a first-order language £ without individual constants and equality. We assume that £ contains ::J, --,, and V as primitive, whereas the remaining Boolean connectives and 3 are defined as usual. Note that the above introduction rules for 3x can be derived under this definition.
X --+ •x Y -H- •xX --+ Y.
One driving force behind van Benthem's investigation into generalizations of Tarski's truth definition is the interest in decidable predicate logics. When it comes to extensions of KFOL, van Benthem
We obtain the structure-dependent introduction rules for Vx and 3x in DL stated in Table XVII. In addition to these introduction rules we need further structural assumption in order to take care of the necessitiation rules in axiomatic presentations of normal tense logics:
would like to find logics (1) that are reasonably expressive, (2) that share the important meta-properties of predicate logic (such as interpolation: effective axiomatizability, perhaps even 'Gentzenizability') and (3) that m1ght even improve on this, by being decidable. (20, p. 11]
Let us refer to the result of adding these sequent rules to the display calculus DCPL for classical propositionallogic defined in Chapter 3 as DKFOL. Then f-nKFOL I --+ A iff f-KFOL A. The proof is completely analogous to the display of K in Chapter 3. Derivations in DKFOL are also just re-written derivations of DK; consider, for instance, the following cut-free proof of the Vx-version of the K axiom schema: Px--+ Px VxPx --+ •xPx Vx(Px ::::> Qx) o VxPx--+ •xPx •x(Vx(Px ::::> Qx) o VxPx) --+ Px Qx--+ Qx Px ::::> Qx--+ * •x (Vx(Px ::::> Qx) o VxPx) o Qx Vx(Px ::::> Qx)--+ •x(* •x (Vx(Px ::::> Qx) o VxPx) o Qx) Vx(Px ::::> Qx) o VxPx--+ •x(* •x (Vx(Px ::::> Qx) o VxPx) o Qx) •x(Vx(Px ::::> Qx) o VxPx) --+ * •x (Vx(Px ::::> Qx) o VxPx) o Qx •x(Vx(Px ::J Qx) o VxPx) o •x(Vx(Px ::J Qx) o VxPx) --+ Qx
(C)
•x(Vx(Px ::J Qx) o VxPx) --+ Qx Vx(Px ::J Qx) o VxPx--+ VxQx Vx(Px ::J Qx) --+ VxPx ::J VxQx I o Vx(Px ::J Qx) --+ VxPx ::::> VxQx I--+ Vx(Px ::J Qx) ::::> VxPx ::::> VxQx
KFOL satisfies these requirements. In comparison to deciable fragments of FOL obtained by restricting the language of first-order logic, say, to admit only special quantifier prefixes, KFOL .is a deci~ab~e system in the full first-order language. Decidable predicate logics m the full language of first-order logic have also been known from the field of substructurallogics for a long time. The system which is nowadays referred to as BCK predicate logic, for example, also enjoys decidability; cf. [83], [177]. The question arises whether it is possible to pr~sent both the family of substructural subsystems of FOL and the family ?f extensions of KFOL within a single proof-theoretic schema. As we will see, the predicate logical extension of DL suggested in the present paper provides such a uniform proof-theoretic framework. Moreover, we not only obtain sequent systems (i) for the familiar substruct~ral subsystems of FOL - extending the non-associative Lambek logic, N~L, as the minimal substructural predicate logic - and (ii) for extensiOns of KFOL, but also (iii) for substructur3.1 subsystems of extensions of KFOL. The above cut-free proof of the K schema, for instance, makes use of the contraction rule (C), which is lacking in BCK predicate logic. The minimal displayable classical predicate logic is then the resu~t of depriving DKFOL of all its structural rules (apart f~om the ba:'Ic structural rules for the structure connectives and the logical rules (Id) and (cut), see Chapter 3). Let us refer to :his syst~m as D~FOLmin· We thus obtain three 'landscapes' of classical predicate logics:
194
CHAPTER12 FOL
A ROUTE FROM KFOL TO FOL
£
FOL
NLL
KFOL
KFOL
DKFOLmin
195
are equivalent to the axiom schemata 3y3xB :::::> 3x3yB and 3y\fxB :::::> \fx3yB respectively. The latter are Sahlqvist formulas, i.e. their corresponding first-order frame conditions can be straightforwardly computed. Instead of searching for correspondences between axiom schemata and frame conditions, in the present context we are interested in correspondences between axiom schemata and structural sequent rules. We say that a first-order axiom schema corresponds to a set of structural rules iff (i) the rules allow proving the schema, and (ii) in the presence of the schema, the structural rules are axiomatically derivable under a suitable translation T from sequents into formulas. Since in antecedent position a structure connective is interpreted as backward-looking possibility and, in general, we do not assume symmetry of the binary relations Rx, this translation sends sequents to formulas of a 'tense logical' counterpart of the first-order logic £ under consideration. If the tense logical counterpart Lt is a conservative extension of £, then for every first-order formula A, we obtain that A is provable in the display system iff A is provable axiomatically. The persistence of atomic information, for instance, can be expressed by the purely structural sequent rule: X -t Py f- •xX -t Py.
•x
Figure II. Three 'kites' of predicate logics.
Van Benthem [20] investigates capturing predicate logics intermediate
~etween ~FOL and FOL by frame conditions corresponding to additi~na~ axwm schemata. There are, however, familiar predicate logical prmc1ples extending KFOL, which are not reflected by conditions on frames alone. As van Benthem observes, simple instances of the axiom schema:
(*) A
:::::>
VxA,
if x does not occur free in A
taken fr?m the axiomatization of FOL in [59] fail to be naturally translatab~e mto frame conditions. The schema Py :::::> \fxPy, where Py is ~tom1c, ~or example, corresponds to persistence (or heredity) of atomic mformatwn not depending on x: M, a
f= Py and a.Rx/3,
then M, {3
f= Py.
In o~der to detect frame correspondences, van Bent hem treats ( *) inductively._ In one case, however, frame correspondence is only achieved by assummg ~x an? Vx to be S4 modalities. The case where A = \fyB or A= 3yB gives nse to two subcases: (*1) where y = x and (*2) where Y i= x. In the first case one obtains analogues of the well-known axiom schemata 4 and 5, namely [x]B :::::> [x][x]B and (x)B ::::> [x](x)B. These s~h_e~ata co~respond to the transitivity and Euclidicity of the accessibility relatiOns Rx- Case (*2) is less obvious. Van Benthem observes that in S4 the proof rules B ::::> \fxB f- \fyB\fx\fyB B ::::> VxB f- 3yB\fx3yB
Our aim now is displaying various predicate logics between KFOL and FOL by adding purely structural rules to certain fixed collections of logical, structural and operational sequent rules of display logic. 44 The introduction rules for Vx and 3x thus remain unaltered, and all variation is achieved at the level of structural rules. In the following we shall consider two particular routes from from KFOL to FOL.
12.3. A ROUTE FROM KFOL TO FOL There is a simple obstacle to presenting FOL as a purely axiomatic extension of KFOL. The problem arises with atomic £-formulas like Rxy and Ryx, since \fx\fyRxy :::::> \fx\fyRyx, though a theorem of FOL, fails to be a theorem of KFOL. Identifying Rxy and Ryx does not help, because then \fx3yRxy 1\ --.\fx3yRyx, for example, would not be satisfiable. The familiar operation [yjx]A of substituting y for every 44
As we saw in Chapter 1, there also exist sequent calculi for Kin the ordinary Gentzen-style. However, there is nothing that deserves to be called a correspondence theory with respect to standard modal Gentzen calculi, neither in terms of frame conditions nor in terms of axiom schemata.
197
CHAPTER 12
A ROUTE FROM KFOL TO FOL
free occurrence of x in A can be regarded as a syntactic device exactly for overcoming this obstacle. Consider the axiomatization of FOL in [59]:
for every x, y. An axiomatic presentation of KFOL* results from the axiomatization of KFO L by adding
196
1.1 all universal closures of tautology schemata and of the following quantifier schemata 1.2 - 1.4, 1.2 \t'x(A :::> B) :::> (\fxA :::> \t'xB), 1.3 (*), 1.4 \t'xA :::> [yjx]A, if y is free for x in A, 1.5 A, A :::> B 1- B. Schema 1.4, but not its instance \t'xA :::> A allows \fx\fyRxy :::> \t'x\t'yRyx to be axiomatically derived: 1 2 3 4 5 6 7 8 9 10 11
\t'x\fyRxy :::> Rz1z2 Vz1 \t'zz(\t'x\t'yRxy :::> Rz1zz) \t'z1 \t'zz\t'x\t'yRxy :::> Vz1 \t'zzRz1z2 \t'x\t'yRxy :::> Vz1 \t'zz\t'x\t'yRxy \t'x\t'yRxy :::> Vz1 \t'zzRz1zz \t'z1 \t'zzRz1zz :::> Ryx \t'x\fy(\fz1 \t'zzRz1z2 :::> Ryx) \t'x\t'y\fzl\t'zzRzlzz :::> \t'x\t'yRyx \t'zl\t'zzRzlzz :::> \t'x\t'y\t'zl\t'zzRz1z2 \t'z1 \t'zzRz1z2 :::> \t'x\t'yRyx \t'x\t'yRxy :::> \t'x\t'yRyx
1.4, propositionallogic 1.1, 1 1.2, 2 1.3, propositional logic propositionallogic, 4, 3 1.4, propositionallogic 1.1, 6 1.2, 7 1.3, propositionallogic propositionallogic, 8, 9 propositional logic, 5, 10
Note that clause 1.1 amounts to postulating a necessitation rule for prefixes \fx. Van Benthem (20] suggests treating the substitution operator as a further necessity-type normal modal operator. Thus, substitution is not regarded as a meta-language device of the syntactic presentation, but is treated as part of the object language of first-order (or poly-modal) logic. The new modality is denoted by [Sxy], and formulas [Sxy]A are interpreted by means of a doubly indexed 'accessibility relation' Ax,y:
M, a F [Sxy]A iff for all {3
E S:
if aAx,y/3 then M, f3
FA
If [Sxy] is indeed to be conceived of as substitution of variables, then in addition to the K schema and a necessitation rule for [Sxy], further principles have to be postulated. In particular, substitutions commute with negation, which means that the relations Ax,y are functions and [Sxy]A is equivalent to (Sxy)A (:= ---,[Sxy]---,A). The system KFOL* is formulated in the language £*, which extends C with operators [Sxy],
[Sxy](A :::>B) :::> ([Sxy]A :::> [Sxy]B); A 1- [Sxy]A; and
Py;
sub1
[Sxy]Px
sub2
[Sxy]Pz
sub3
[Sxy]---,A
sub4
[Sxy](A :::> B)
sub5
[Sxy]VxA
sub6
[Sxy]VzA
\t'z[Sxy]A, if z =1- x,y;
sub7
[Sxy]VyA
\fyA, if x does not occur free in A;
sub8
[Sxy]A
:::>
---,\fx---,A;
sub9
[Sxy][Sxy]A
=
[Sxy]A;
sub10
[Sxy][Syx]A
-
[Sxy]A.
-
Pz, if z =1- x; ---,[Sxy]A; ([Sxy]A :::> [Sxy]B);
-
\fxA;
The extension of the logical object language is mirrored by a corresponding extension of the structural language of sequents: every operator [Sxy] comes with a structural connective •x,y· The inferential meaning of •x,y is given by its basic structural rules analogous to the basic structural rules for •x:
X-+ •x,yY 1- •x,yX-+ Y; •x,yX -t Y 1- X -+ •x,yY In succedent position •x,y is to be read as the forward-looking necessity operator with respect to Ax,y, in antecedent position as the backwardlooking possibility operator with respect to Ax,y· Since [Sxy] is a normal modality, we also postulate:
(MN•x,y)
I-+ X 1- I-+ •x,yX X -t I 1- X -+ •x,yi·
Proposition 12.1. The schemata sub1- sub10 correspond to the following structural rules:
198
CHAPTER 12 rsub1.1 rsub1.2
X ----+ •x,yPx f- X ---+ Py; X ---+ Py f- X ---+ •x,yPx;
rsub2.1 rsub2.2
X ----+ •x,yPz f- X ---+ Pz, if z =/= x; X ----+ Pz f- X ---+ •x,yPz, if z =/= x;
rsub3.1 rsub3.2
X----+ •x,y * •x,yY f- X---+ *Y; X ----+ Y f- •x,y * •x,y * X ---+ Y;
rsub4
= rsub3.2;
rsub5.1 rsub5.2
X----+ •x,y •x Y f- X---+ •xY; X---+ •xY f- X---+ •x,y •x Y;
rsub6.1 rsub6.2
X ----+ •x,y •z Y f- X ---+ •z •x,y Y, if Z =/= x, y; X ---+ •z •x,y Y f- X ----+ •x,y •z Y, if Z =/= X, y;
rsub7.1
X----+ •x,y •y Y f- X----+ •yY,
rsub7.2
if x does not occur free in any formula in Y; X ----+ •y Y f- X ---+ •x,y •y Y, if x does not occur free in any formula in Y;
rsub8
X----+ •x,y
rsub9.1 rsub9.2
X-+ •x,y •x,y Y f- X----+ •x,yY; X ---+ •x,yY f- X ---+ •x,y •x,y Y;
rsub10.1 rsubl0.2
X ---+ •x,y •y,x Y f- X ----+ •x,yY; X---+ •x,yY f- X---+ •x,y •y,x Y.
* •xY f-
X----+ *Y;
Proof. Relegated to Section 12.7. Q.E.D.
We shall refer to the result of augmenting DKFOL by rsub1 - rsub10 the basic structural rules for the structure connectives • x,y, (M N • x,y ) ,' an d (---+ [Sxy]) •x,yX ---+ A f- X ---+ [Sxy]A ([Sxy] ---+) A---+ X f- [Sxy]A----+ •x,yX
A ROUTE FROM KFOL TO FOL
from DKFOL and DKFOL* by adjoining the following sequent rules:
f-KFOL*
A iff f-DKFOL* I-+ A.
X ---+ * •x *A f- X ----+ VxvA A ---+ X f- VxvA ----+ * •x *X I ---+ X f- I ---+ * •x *X X ---+ I f- X ---+ * •x *I
(---+ Vx) (Vxv----+)
(MN•x)t
An obvious question is whether DKUKtFOL* can be faithfully embedded into DKtFOL under a suitable translation
p(X---+ Y)
(yfxfVyRyy:::) Rxx (yfxfVyRyy:::) Rxy (yfxfVyRyy:::) Ryx
PI(X)---+ p2(Y)
(yfxfVzRzz:::) Rxx (yfxfVzRzz:::) Rxy (yfxfVzRzz:::) Ryx
(y/xfRyy:::) Rxx (yfxfRyy:::) Rxy (yfxfRyy:::) Ryx
It is unclear to me how this could be achieved by a systematic definition of (yfxt as a function on£, and I therefore prefer to deal with [yjx], which in contrast to [Sxy] is also defined for terms, as a meta-linguistic device. In the above axiomatization, it is 1.3 and 1.4 which extend KFOL. To obtain a characterization of 1.4, we extend [yjx] to a function on structures by defining:
[yjx]I [yjx](X o Y) [yjx] *X [yjx]•z X
I
[yjx]X o [yjx]Y *[yjx]X •z[yjx]X
Proposition 12.3. Schemata 1.3 and 1.4 correspond to
Proof. See Section 12.7. Q.E.D.
Consider now the languages Lt and£* U£t, which are obtained from £and£* by the addition of backward-looking universal quantifiers Vxv for every variable x. The systems DKtFOL and DKUKtFOL* resul~
:=
of sequents into sequents such that p2([Sxy]A) = [yjx]p2(A). We would then need a Gentzen dual of [y/x] to supply Pl([Sxy]A). As it seems, however, there is no natural such Gentzen dual (yfxt Consider, for example, VyRyy ::J Ryy, VzRzz ::J Ryy, and Ryy ::J Ryy, which are provable in FOL. Since Ryy = [yjx]Rxx, Ryy = [yjx]Rxy, and Ryy = [yjx]Ryx, the following formulas would have to be provable:
as DKFOL*. Corollary 12.2.
199
r 1.3
X ---+ Y f- X ---+
r1.4
X---+ •xY
f-
•x Y,
X---+ [yjx]Y,
if x does not occur free in any formula in Y; ify is free for x in every formula in Y.
201
CHAPTER 12
ANOTHER ROUTE TO FOL
Proof. Straightforward, using a translation 7(X -+ Y) := 71 (X) ::)
This work is closely related to the algebraic study of restricted firstorder logic, cf. [120). The addition of substitution operators to cylindric modal logic is dealt with in [107] and [174). Alechina and van Lambalgen [3], [98] investigate the fine structure of quantification resulting ~rom explicitly considering dependency on parameters, while Meyer-Vwl [110) considers substructural quantification in terms of choice processes. Kuhn [95) prefers to overcome the problem of representing atomic £-formulas by using the idea of a variable-free formulation of predicate logic. This idea, too, has a distinctive poly-modal flavour, and we turn to it in the next section.
200
72(Y) of sequents into .Ct-formulas analogous to the 7-translation defined in Chapter 3. The tense logical extension of KFOL under consideration is the system KtFOL, which results from KFOL by adding the following axiom schemata and rule: 45
A ::) \fx--,\f£•A; A ::) \fxv•\fx--,A; \fxv(A::) B) ::) (\fxvA::) \fxvB}; A / \fxvA. Q.E.D. Let b. ~ { 1. 3, 1.4}, and let b.' be the set of rules corresponding to the schemata in Ll.
Corollary 12.4. (i) f-KtFOLUll. A only if f-oKtFOLull.' I -+ A. (ii) f-oKtFOLull.' X -+ Y only if f-KtFOLull. 7(X -+ Y).
Corollary 12.5. For every £-formula A, cisely the case that f-KFOLull. A.
f-oKFOLull.'
I -+ A in pre-
Proof. By the previous corollary, for every .Crformula A, f-oKtFOLull.' I-+ A iff f-KtFOLull. A. The claim follows from the fact that KtFOLU b. is a conservative extension of KFO L U b.. (One can verify that both systems are characterized by the same class of models.) Q.E.D. Note that KtFOL U b. and KFOL U b. are propositional theories not closed under uniform substituion. Andreka, van Benthem, and Nemeti emphasize that the generalized semantics invites the introduction of new vocabulary, reflecting distinctions not usually found in first-order logic. Examples are irreducibly polyadic quantifiers :Jy binding tuples of variables y, or modal calculi of substitutions (5, p. 712). Indeed, extended modal formalisms open up many routes from KFOL to FOL, and there is an extensive literature on modal approaches toward first-order logic. The central reference is Y. Venema's work on cylindric modal logic [171 ), [173), in which restricted first-order logic in the language with equality is extended by a set of 'irreflexivity rules'. 45 A new axiomatization of the minimal normal tense logic Kt with slightly stronger interaction (between future and past tense) axiom schemata can be found in Chapter 13. This axiomatization was inspired by the display presentation of Kt.
12.4.
ANOTHER ROUTE TO
FOL
The generalized truth definition does not allow atomic £-formulas to be treated as atoms of the full system FOL. The idea behind the variablefree presentation of FOL is to consider only n-place atomic £-formulas pn (n > 0), in which n distinct variables occur exactly once and in a fixed order. From these atoms, which can be taken as n-sorted sentence letters, representations of the remaining atoms of .C are generated by applying certain necessity-type normal modal operators. These operators are: 46 [r] [s]
[i)
rotation switch identification
The vocabulary of the language £** comprises these modalities, the atoms pn, ::), ., and the quantifier prefix \fx. Every atom pn is a sortn formula. If A is a sort-n formula, then also \fxA, [r]A, [s]A, [i]A, and -,A are sort-n formulas. If m ::; n, A is a sort-n formula, and B a sort-m formula, then (A ::) B) and (B ::) A) are sort-n formulas. Note that the usual definitions give rise to sort-n truth and falsity constants tn and rn. In the structural language of sequents this is reflected by constants In. We use An, Bn, en etc. as schematic letters for sort-n formulas and sometimes omit superscripts if no confusion can arise. [zp A := [l]A, and [l]n+l A := [l][l]n A, for l E {[r), [s); [i]} and ~ 2:: :· Formulas [r]An, [s]An, and [i]An receive a natural mterpretatwn .m models M = ((D), V), where (D) is the set of all sequences of fimte or denumerably infinite length of elements from a non-empty domain 46
For the sake of greater uniformity, our notation diverges from that in (95).
,..,., "'
202
ANOTHER ROUTE TO FOL
CHAPTER 12
D, and V maps atoms pn to n-tuples of elements from D, cf. [95]. If d = (d1, ... , dn, .. . ) E (D), then a sort-n formula An is true at d in
(M, d
f= An)
M
according to the following inductive definition:
M,d F pn M,df=·B M,d f= (B :J C) M,d f=VxB M,d f= [r]B M,d f= [s]B 1 M,d f= [s]Bm, (2::; m) M,d F [i]B
iff iff iff iff iff iff iff iff
(d1, ... ,dn) E V(Pn);
not M,d f= B; M,d f= B implies M,d f= C; for every dE D, M, (d, d2, ... , dn)
f=
B;
M, (dn, d1, · · ·, dn-1) F B; M,d F B; M, (d2, d1, d3, ... , dn) F B; M,(dl,dl,d3, ... ,dn) F B.
Kuhn [95] defines further modal operators and uses them to give an axiomatization of FOL in the language £**. These definitions are rather tedious, although the defined connectives again receive a natural interpretation in models ((D), V). Our aim is verifying that in Kuhn's axiomatization, every axiom schema which is not minimally derivable corresponds to a purely structural sequent rule. As far as schemata exhibiting defined modalities are concerned, this is a completely straightforward matter, once the definitions have been restated for the structural companions •r, •8 , and •i of [r], [s], and [i] respectively. We shall therefore consider only one simple paradigmatic case to explain the general method of rewriting these axiom schemata as structural sequent rules. Suppose in the following that m = max( n, k).
and also In is a sort-n structure. If X is a sort-n structure, then so are *X, •rX, •sX, and •iX. If m ::; n, X is a sort-n structure, and y a sort-m structure, then (X o Y) and (Y o X) are sort-n structures. We use xn, yn, zn etc. as schematic letters for sort-n structures and sometimes omit superscripts if no confusion can arise. The structure constants 1n are governed by rules analogous to those for I, and the system DKFOL** results from DKFOL by adding (i) introduction rules for [r], [s], and [i] analogous to those for Vx, and (ii) necessitation and basic structural rules for •n •s, and •i analogous to those for •x· We now restate the earlier definitions for the structure connectives •r,
•s, and
•i·
Definition 12. 7.
The axiom schema we consider is [d]ij [db A
Proposition 12.8. The schema [dJij[d]jiA
Definition 12.6.
[rJ(m+l)-k([r][s])k-l(An 1\ tk) 2. [rJ;;lAn := [rJZ-lAn [r]k[r]j[i][r]j 1 [r];; 1 A if j > k 1 3. [dJjkA := { A[rh[r]j+l[i][r]j~drJ;; A if j < k
X X
1. [r]kAn :=
if
j = k
Formulas [r]nAn and [r];;:- 1 An are sort-n formulas, and [d]jkAn is a sortmax(j, k, n) formula. Let dt be the result of deleting the first k components of (d1, ... , dk, .. . ) =d. If the length of d is at least m, then
f= [r]kAn M, d f= [r];; 1 An M,d F [d]jkA M,d
iff iff iff
M,(dk,dl, ... ,dk-l,dt) f=An M, (dz, ... , dk, d1, dt) f= An M, (d1, ... , dj-1 1 di, dj} FAn.
In order to display poly-modal logics in the given vocabulary, we inductively define sort-n structures. For each n 2: 1, every sort-n formula
203
--t --t
= [d]ij A.
= [d]ijA corresponds to
•ij •ji Y I- •ijY; •ij Y I- •ji •ij Y.
Proof. Straightforward, if in succedent position •n •s, and ~i. are trans; lated as [r], [s], and [i] respectively, and in antecedent position as (r), (sr, and (ir respectively. Q.E.D.
In the primitive vocabulary, Kuhn's axiomatization contains the following minimally non-derivable schemata:
=
2.1 -.[l]A [l]-.A, for lE {[r], [s], [i]}; 2.2 [r]A 1 =A\ 2.3 [s]A 1 =: A1 . Proposition 12.9. The schemata
structural sequent rules:
2.1-2.3 correspond to the following
THE BARCAN FORMULA
CHAPTER 12
204
r2.1.1 r2.1.2 r2.2.1 r2.2.2 r2.3.1 r2.3.2
X--+ •t * •tY I- X--+ *Y; X --+ Y I- •t * •t *X --+ Y;
205
for an application of cut like
xi --+ • T yi 1- xi --+ y1.l
X --+ •x,y •y A
xi --+ yi I- x--+. T yl.l
rsub7.1
x1 --+. yi I- xi --+ yi. s
'
XI --+ yi I- X --+ •sYl.
•yX--+ y would require applying cut to the uppermost 'parametric ancestors' of A in the derivation of •yX --+ A and replacing every such ancestor by an occurrence of Y. If x occurs free in Y, the preservation of rule applications is, however, not given:
Proof. The cases of 2.2 and 2.3 are obvious. The case of the functionality schema 2.1 is analogous to that of sub3. Q.E.D.
Also on this route to FOL, the inferential meaning of the modal operators involved- in particular, the inferential meaning of the universal quantifier - is laid down 'once and for all' by their introduction rules. These rules exhibit only one occurrence of the operations in question either in antecedent or in succedent position in the conclusion sequent (and exhibit no other logical operation). Therefore these introduction rules may qualify as inferential meaning assignments; see Chapter 2.
X--+ •x,y •y Y(x) X--+ •yY(x) •yX --+ Y(x)
not rsub7.I
12.6. THE BARCAN FORMULA 12.5. STRONG CUT-ELIMINATION It is well-known that the Barcan formula
BF 'v'xDA ::J DVxA
In Chapter 4 it was proved that every displayable logic enjoys strong cut-elimination. This result shows that no matter in which order the elimination steps in Belnap's [16] general cut-elimination proof for DL are applied, every sufficiently long sequence of one-step reductions terminates in a cut-free proof. 47 Recall that a displayable logic is a system admitting a display presentation such that its rules satisfy certain conditions regarding their shape, and the introduction rules for the logical operations are such that principal cuts can be eliminated. In particular, the rules are supposed to be closed under certain substitutions of structures for formulas, a constraint rather sensitive to side-conditions on rules. The side-conditions on rsub7.1, rsub7.2, r1.3, and rl.4 violate this closure condition and therefore the 'wholesale' elimination theorem does not apply. Consider for instance rsub7.1. The reduction strategy
and its converse
BFc D'v'xA ::J VxDA
correspond to the assumptions of constant domains and persistence of individuals along the accessibility relation respectively; cf. for example [64]. As has been observed by Belnap [16], adding the obvious str~cture free rules, i.e. (UQ), for the universal quantifier to DL results m the provability of BF and BFc. Consider:
A--+A DA-+ •A (UQ) 'v'xDA-+ •A •'v'xDA --+ A (UQ) •'v'xDA-+ 'v'xA VxDA --+ D'v'xA I o 'v'xDA --+ D'v'xA I --+ 'v'xDA ::J D'v'xA
47 Provided stacks of configurations in which applications of structural rules are followed by a cut are reduced top-down.
I
l
A--+ A (UQ) 'v'xA--+ A D'v'xA--+ •A eD'v'xA--+ A D'v'xA--+ DA (UQ) D'v'xA --+ 'v'xDA I o D'v'xA --+ 'v'xDA I --+ D'v'xA ::J VxDA
207
CHAPTER 12
REMAINING PROOFS
The structural account of the quantifiers as modal operators blocks these proofs of BF and BFc. 48 In the generalized semantics neither BF nor BFc are valid. The latter principles beome provable in the presence of further structural sequent rules, however.
Note also that the display introduction rules for 3x and Vx may remain unchanged in display sequent calculi for intuitionistic predicate logic. As in the Kripke semantics for intuitionistic predicate logic, the non-classical behaviour of the existential and universal quantifer results from the non-classical interpretation of implication and negation. In Chapter 10 intuitionistic logic was displayed using a modal translation into S4. Also if one displays intuitionistic logic more directly by dispensing with the structure operation *, interpreting the structure operation o in succedent position as intuitionistic implication, =>h, and defining intuitionistic negation 'h by 'hA := A ~h f (see [76], [140]), the classical interdefinability of 3x and Vx fails. In the direct approach one obtains the following introduction rules for intuitionistic negation and implication:
206
Proposition 12.1 0. BF and BFc correspond to rBF rBFc
X--+ •x • Y I- X--+ • •x Y; X --+ .. x I- X --+ •x • Y.
Proof. Straightforward, if in succedent position • is translated as D, and in antecedent position as D's Gentzen dual Ov. Q.E.D. What have we achieved? In the preceding we have formulated structure-dependent introduction rules for the universal and the existential quantifer by treating them as modal operators. In this extension of Belnap's display formalism to quantifier logic, the rules for Vx and 3x proved to be general enough to avoid the derivability of the Barcan formula BF and its converse BFc. Moreover, we have considered two routes from the resulting minimal predicate logic DKFOL to full classical first-order logic. In the languages with V primitive, every additional axiom schema and also BF and BFc turned out to be expressible by purely structural sequent rules. It is also worth emphasizing that this extension of DL to predicate logic naturally 'Gentzenizes' various families of interesting substructural systems. Let DKFOL* . and ** mzn DK F OL min refer to the result of removing all structural rules apart from the basic structural rules of their structure connectives, (id) and (cut) from DKFOL* and DKFOL** respectively. In addition to the 'kites' of systems in Figure II, we obtain two further logical 'landscapes' depicted in Figure Ill. 48 Fitting [64] points out that what he takes to be the "most obvious" axiomatization of a minimal normal modal extension of FOL, namely one using the rule of universal generalization
(--+ 'h) (·h --+)
X o A --+ I I- X --+ 'hA X --+ A I- 'hA --+ X o I
(--+~h) (~h--+)
X
o A --+ B I- X --+ A ~h B X --+ A B --+ Y I- A ~h B --+ X
£*
£**
DKFOL*
DKFOL**
o
Y
DKFOL;',in
Figure Ill. More 'kites' of predicate logics.
A => B I- A => VxB, if x does not occur free in B,
allows proving BFc but not BF. The provability of both BFc and BF can be avoided by using an axiomatization like the one in [59], which refers to universal closures instead of postulating universal generalization, cf. [87, p. 179ff.]. It should be clear, however, that such axiomatizations are rather remote from reflecting in an axiomatic setting the idea of capturing the meaning of Vx and 3x by means of introduction and elimination rules as meaning assignments. In DL, however, the introduction rules for Vx and 3x may be argued to allow such an interpretation; cf. [181].
12.7. REMAINING PROOFS
In order to verify the correspondences to be established, we extend £* by the backward-looking quantifiers Vxu and [SxyJ and refer to the resuiting language as £!. The system K tFO L *, the tense logical version
1
208
of the minimal predicate logic with a modal substitution operator, results from KtFOL U KFOL* by adding the following axiom schemata and rule:
A-+A *A-+ *A -,A-+ *A \lx•A -+ •x * A •x\lx•A -+ *A A -+ * •x \lx•A [Sxy]A-+ •x,y * •x\fx--,A rsub8 [Sxy]A -+ *\lx•A [Sxy]A-+ --,\fx·A
A :J [Sxy]•[SxyJ•A; A :J [SxyJ•[Sxy]•A; [SxyJ(A :J B) :J ([SxyJA :J [Sxy]"B); A/ [SxyJ"A. We extend the tran~l~tion T from sequents into Lt-formulas (used in the proof of PropositiOn 12.3) to a translation T' from sequents into .c;- formulas by defining:
T!(•x,yX) T2(•x,yX)
•[SxyJ'Tl(X) [Sxy]T2(X).
Proof of Proposition 12.1. Consider first the direction from structural rules to axiom schemata. The left-to-right direction of sub4 is just the K schema for [Sxy], which is minimally derivable. The derivations of sub1, sub2, sub5- sub7, sub9, and sub10 are simple. In the case of the left-to-right direction of sublO, for example, we have:
For the right-to-left direction of sub4 it is enough to observe that 49 this schema is equivalent to the right-to-left direction of sub3. Like the cut-free proof of the K schema in Section 12.1, the derivation of ([Sxy]A :J [Sxy]B) :J [Sxy](A :J B) involves applications of the mantonicity and contraction rules. We now turn to the direction from axiom schemata to structural 1 rules. In most cases, the axiomatic derivation under the T -translation is accomplished merely by the transitivity of :J. The derivation of rsub3.1, rsub3.2, and rsub8, however, involves tense logical principles: rsub3.1
A-+A [Sxy]A -+ •x,yA rsublO [Sxy]A -+ •x,y •x,y A •x,y[Sxy]A-+ •x,yA •x,y •x,y [Sxy] -+A •x,y[Sxy]A-+ [Sxy]A [Sxy]A-+ [Sxy][Sxy]A
1 2 3 4
rsub3.2
Less obvious are the derivations of both directions of sub3:
A-+A [Sxy]A-+ •x,yA •x,y[Sxy]A-+ A *A -+ * •x,y [Sxy]A ·A-+* •x,y [Sxy]A [Sxy]•A -+ •x,y * •x,y[Sxy]A rsub3.1 [Sxy]•A-+ *[Sxy]A [Sxy]•A-+ •[Sxy]A
209
REMAINING PROOFS
CHAPTER 12
A -+ A rsub3.2 •x,y * •x,y * A -+ A * •x,y *A-+ [Sxy]A *[Sxy]x]A-+ •x,y *A •[Sxy]A-+ •x,y *A •x,y•[Sxy]A-+ *A •x,y•[Sxy]A-+ ·A •[Sxy]A-+ [Sxy]•A
The derivation of sub8 follows the pattern of the derivation of the rightto-left direction of sub3:
rsub8
r!(X) ::::> [Sxy]••[SxyJ•Tt (Y) [Sxy]••[SxyJ•Tt (Y) ::::> ::::> •[Sxy]•[SxyJ•Tt (Y) Tt (X) ::::> •[Sxy]•[SxyJ•TI(Y) T1(x) ::::> •T1(Y)
T1 (X) ::::> r2(Y) •T2 (Y) ::::> •T1 (X) •T2(Y) ::::> [SxyJ•[Sxy]••TI(X) -.[Sxy]•-.T2 (Y) ::::> •[Sxy]Tt (X) •[Sxy]••T2 (Y) ::::> [Sxy]•Tt (X) 6 -.r2(Y) ::::> [SxyJ1Sxy]•ri(X) 7 -.[SxyJ1Sxy]-.rl(X) ::::> T2(Y) 8 -.[SxyJ••[Sxy]•Tl (X) ::::> T2(Y)
1 2 3 4 5
1 2 3 4
T1 (X) ::::> [Sxy]••'v'Xv•Tl (Y) [Sxy]-.-.\I£•TI (Y) ::::> ::::> -,\fx-.\lxv•TI (Y) Tt (X) ::::> -,\fx-.\lxv•Tt (Y) T1 (x) ::::> •T1 (Y)
sub3 modus ponens, 1,2 KtFOL*, 3
CPL, 1 KtFOL *, CPL, 2 KtFOL*, 3
sub3, 4 KtFOL*, 5 CPL, 6 CPL, 7
sub8 modus ponens, 1,2 KtFOL*, 3 Q.E.D.
In other words, schema sub4 corresponds to quasi-functionality ('each point is related to at most one point'). 49
210
CHAPTER 12
CHAPTER13
Proof of Corollary 12.2. Refer to DKUKtFOL*
[Sxy]} ([SxyJ -7) (MN•x,y)t (-7
X
-7
* •x,y *A f-
X
+
[SxyJA * •x,y *X
-7
A -7 X f- [SxyJA -7 I -7 X f- I -7 * •x,y *X X
-7
I f- X
-7
* •x,y *I
as DKtFOL*. Since f-KtFOL* A just in case f-oKtFOL* I -7 A the claim follows by KtFOL* (DKtFOL*) being a conservative exte~sion of KFOL* (DKFOL*). Q.E.D.
APPENDIX
Many important logical systems have proof-theoretic presentations of more than one type, say, an axiomatization, a natural deduction proof system, and a sequent calculus presentation. Usually, this is a rather fortunate situation. It may happen that certain axiom schemata are characterizable by algebraic or relational properties expressible in an interesting fragment of first-order logic, and that Gentzen-style proof systems lend themselves to automated deduction. As we have seen, display logic is an elegant and powerful refinement of Gentzen 's sequent calculus and meets quite a few methodological requirements of a more philosophical and a more technical nature. It would be nice to relate the modal display calculus to natural deduction proof systems for intensional logics, and in the present chapter we shall relate DL to generalized Fitch-style natural deduction systems for modal logics. Moreover, we shall show that DL may have repercussions on the axiomatic presentation of logical systems and define an apparently new axiomatization of Kt. Eventually, we shall consider a generalization of DL, namely four-place display sequents. We shall redisplay Nelson's system N4 and obtain a display sequent calculus for N3 by the addition of a purely structural sequent rule.
13.1. DL AND FITCH-STYLE NATURAL DEDUCTION
A full-circle theorem 50 for a given logic A says that certain proof systems S1 , ... , S4 for A of the four maybe most important types of inference systems (axiomatic, natural deduction, tableaux, sequent calculi) are all equivalent in the following sense: - Every proof of a formula A from formulas A 1 , ... , Ak in 8 1 can be transformed into a proof of A from A 1 , ... , Ak in S2; - every proof of A from A 1 , ... , An in S4 can be transformed into a proof of A from A1, ... , Ak in S1. 50 This term was pointed out to me by Tijn Borghuis, who found it in (13, Chapter XI].
211
212
CHAPTER 13
0
FITCH-STYLE NATURAL DEDUCTION Table XVIII. Additional axiom schemata and frame conditions.
Hilbert-style
<equent cakulu'
natuml d'ductioa
tableaux
Figure IV. A full circle.
To establish such a full circle, one has to make sure that in each case the data A 1 , ... , Ak are structured in the same way. If one, for instance, considers proofs from sets of assumptions, then these proofs have to be transformed into proofs from (representations of) the same data structures. Moreover, in the case of normal tense logic there is a whole lattice of logics rather than one designated logical system. The standard syntactic presentation of these systems is in Hilbert-style and is thus modular. If one wants to establish a full circle for the most important systems of normal tense logic, the problem is that in the literature there seem to be no modular natural deduction presentations of these systems, which are not obtained by simply adding axiom schemata to presentations of the minimal normal tense logical system Kt. We shall therefore first of all extend T. Borghuis' generalization of Fitch-style natural deduction [27], [28] to a modular proof theoretic framework for normal tense logic. We shall then relate these natural deduction systems to display calculi. As is clear from Chapter 6, a proper application of the tableau method within display logic, including a reduction to Kowalski clausal form, is available for logics of functional accessibility relations. More generally, display tableaux will therefore be obtained simply by inverting the rules of the display sequent systems for the logics under consideration.
0
(AI} (A2} (A3} (A4} (A5}
axiom schema [F]A :::> [F][F]A [F][F]A :::> [F]A A:::> (F}(P}A A:::> (P}(F}A (F) A :::> [F]( (F}A V A V (P}A)
(A6}
(P}A :::> [F]( (P}A V A V (F) A)
frame condition
VxVyVz((x < y 1\ y < z) :::> x < z) VxVy(x < y :::> 3z(z < y 1\ x < z)) Vx3y(x < y) Vx3y(y < x) VxVyVz((x < y 1\ x < z) :::> :::> (z < y V y < z V z = y)) VxVyVz((y < x 1\ z < x) :::> :::> (z < y V y < z V z = y))
contribution to the emergence of a general proof theory of intensional logic.
Hilbert-style. Consider once more the language of normal_ tense _logic in the vocabulary of classical propositional logic CPL (mcludmg t 'truth' and f 'falsity') together with the unary connectives [F] 'always in the future', (F) 'sometimes in the future', [P] 'always in the.~ast', and (P) 'sometimes in the past'. The minimal normal propos1t10nal tense logic Kt can be axiomatized as follows: AO Al A3 A5 Rl
all tautologies; (F}A := -.[F]-.A; A:::> [F](P}A; [F](A :::> B) :::> ([F]A :::> [F]B); A 1- [F]A;
RO A2 A4 A6 R2
A, A:::> B 1- B; (P}A := -.[P]-.A; A:::> [P](F}A;
[P](A :::> B) :::> ([P]A :::> [P]B); A 1- [P]A.
We also consider a number of further axiom schemata, which are presented in Table XVIII along with their defining first-order properties on Kripke frames (I, PN (
PN(I n1, A, .6.1), PN(I IT2, --.A, .6.2) =>
PN (
52
PN(\
F13
'B, .6.)
,G,A,u(A,I{A))u(A,I{B)))
,A :J B,.6. \{A})
PN(I IT,B,.6.) => PN(I
'A, .6.)
PN(I IT1,C,.6.1), PN(I IT2,C,.6.2), PN(I IT3,A v B,.6.3) =>
~~ B
F9
A2, .6.1 U .6.2)
A2
PN(I IT, A 1\ B, .6.) => PN F6
,B,A,uA,)
~
A1 F5
~:
PN (
,A,{f}) PN(I IT1,A1,.6.I),PN(I IT2,A2,.6.2) => PN (
PN(I n1,A,.6.1), PN(I n2,A :J B,.6.2) => PN(
F12 Fl
215
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13
'--.[F]-.A, (.6.1 u .6.2) \ {--.(F) A})
216
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13
F19
PN(\ Ill, B, ~I), PN(\ II2, ·B, ~2) PN (
~~
'·[P]•A,
•[P]•A
F20
F21
PN(\ IT,
PN(\ IT,
A,~) A,~)
=> =>
PN (
PN (
we shall associate the following tableau rules with the axiom schemata (A1) - (A6):
=>
(~1 u ~2) \ {·(P)A })
II [P)\ (F) A , [P)(F)A, [P](F)A
(T1) (T2) (T3) (T4) (T5) (T6)
~)
II [F)\ (P)A , (F)(P)A, [F)(P)A
=>
PN
(F2)
PN(\ IT,
A,~)
=>
PN
(F3)
PN(\ II,
A,~)
=>
PN
(F4)
PN(\ IT,
A,~)
=>
PN
(F5)
PN(\ IT, (F) A,~) PN
(F6)
(I ~)((F) A
PN(\ IT, (P)A, ~) PN
(I ~)((P)A
(I [F) (I [;rr~\~ (I ~F)(P)A
, [F)[F]A,
, (F)(P)A,
~)
(I ~P)(F)A
, (P)(F)A,
~)
[F)\ II , [F]A, A
~)
[F)[F)~) [F)~)
, [F)((F)A V A V (P)A),
~)
=> V A V (F)A)
, [F)((P)A V
A
V (F)
• *• *X *• *• X
the tree are sequents, and every branching is an instantiation of one of the reduction rules. A closed tableau is a finite tableau such that every leaf is of the form A ---+ A, I ---+ t, or f ---+ I. We write C/(11, X ---+ Y) if 11 is a closed tableau for X ---+ Y.
Display sequent calculi. Display sequent calculi are the proof systems of their corresponding tableau refutation systems. The display sequent calculi are obtained by reading the tableau rules 'bottom up' and adding axiom schemata (i.e., axiomatic sequent rules), namely 1- A ---+ A, 1- I ---+ t, and 1- f ---+ I. Thus, every proof in a display sequent calculus amounts to a closed display tableau. Recall from Chapter 4 that the sequent systems under consideration enjoy strong cut-elimination. We shall now define the transformations needed to form the circle.
=> V A V (P)A)
X---+ ••Y 1- X---+ •Y •X ---+ Y 1- • • X ---+ Y X---tYI-*•*•X---+Y X---tYI-•*•*X---tY
---+ Y 1- * • *X ---+ Y X---tY •X---+Y ---+ Y 1- * • *X ---+ Y X---tY •X---+Y A tableau for a sequent X ---+ Y is a tree with root X ---+ Y. The nodes of
The additional axiom schemata are taken into account by the following rules:
(Fl)
217
From axioms to natural deduction. We define the mapping ( · )H of axiomatic proofs into natural deduction proofs. Suppose PH(II, A,~), where~= {A1, ... , An}· Case 1: A = Aj E ~- Take any enumeration B1 ... Bk of~\ {Aj} and
A),~)
Display tableaux. Tableau calculi are refutation methods consisting of certain rules that manipulate sequents. The intuition behind tableaux is semantic in nature: if the premise sequent of a tableau rule has a countermodel, then so has at least one of its conclusion sequents. Display tableaux are built up from display sequents; see Chapter 3. The inverted versions of the basic structural rules, the introduction rules in Table II (Chapter 3) and the additional structural rules in Table V (Chapter 4) preserve countermodels in this sense. In particular,
B1 set IIN =
Bk
Aj· Case 2: A ~ ~' and A is an instantiation of one of the axiom schemata.
A1 Then IIN =
:
An 11''
where 11' is a certain natural deduction proof of A from 0. We shall present 11' here only for (a characteristic sample of) the tense logical
The effect of (·) H can be summarized as
axiom schemata:
(F) A •(F)A ·[F]·A (F)A :J •[F]·A
·[F]·A [F]·A (F) A •[F]·A :J (F)A
[F]
A
[F] I (P)A [F](P)A A :J [F](P)A
219
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13
218
A:JB A B
[F]B [F]A :J [F]B [F](A :J B) :J ([F]A :J [F]B)
[F]IA [F][F]A I [F]A :J [F][F]A
[F]IA [F]A [F][F]A :J [F]A
I
I1F)(P)A A :J (F)(P)A
I [F]( (F) A V A V (P)A) (F) A :J [F]( (F) A V A V (P)A)
From natural deduction to closed tableaux. We inductively define the mapping (· f of natural deduction proofs into closed tableaux. If ~={AI, ... , An}, then o.6. =(AI o ... o An)i if n = 1, then o~ =AI; and if~ = 0, then o~ =I. A proof of A from~= {A1, ... , An} will be mapped to a closed tableau foro~ -t A. 53 Fl:
ITT =A-+ A
F2:
rrr
=I-+ t
F3:
(F)A
Case 3: A (/. ~' A is not an instantiation of one of the axiom schemata, and A is obtained from earlier items in IT by means of (i) RO or (ii) R1 or R2. (i): In this case there is a formula C and there are proofs III, II2, such that PN(III, C, .6.I ~ ~) and PN(II 2 , C :J A, ~ 2 ~ ~). Take any enumeration BI ... Bk of~\ (~I U ~ 2 ) and set BI
(ii): In this case A = [F]B or A= [P]B, and there is a proof III such that PN(III, B, 0). Then rrN =
or
An [P] 1 rr1
A
o~ 1
Ill
o o~ 2 -+ A1 A A2
lh
F5:
We deal with only one rule. Suppose CT(II1, o~-+ A o~-+ A ITT = Il1 A 1\ B -+ A AoB-+A A-+A
F6:
We deal with only one rule. Suppose CT(II 1 , o~ -+ A).
1\
B).
o~-+
rrr =
A VB o~ -+ A o B o~ o *B-+ A Ill
53 This is justified since, due to the structural tableau (and sequent) rules, A 1 o ... o An can legitimately be viewed as the set {A 1 , ... , An}, and I can be viewed as 0.
220 F7:
221
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13 Suppose CT(II1, o~1 ---+C), CT(II 2, o~2 ---+C), and CT(II3, o~3 ---+A V B). IJT =
o o~ 2 ---+ B ---+ B o * o ~ 2 A -+ B o * o ~ 2 A o ~ 2 ---+ B
o~ 1
o~ 1
o~3 U (~1
\{A})
U (~2
II 1
\ {B})---+ C
o~ 2 ---+
*A oB II 2 ·A ---+ *A o B *A-+ *A oB A o *A---+ B *A---+ *A oB *A o *B---+ *A *A-+ *A A-t A
o~3oo(~1 \{A})oo(~2\{B})oi-tC
o~3 o o(~l \{A}) o o(~2 \ {B})---+ C o *I *C o o~ 3 o o(~ 1 \{A}) o o(~ 2 \ {B})---+ *I
o~3---+
(C o *(o~l \{A})) o (C o *(o~2 \ {B} )) Il3 AVB-t(Co*(o~l \{A}))o(Co*(o~2\{B})) A---+ C o *(o~l \{A}) B---+ C o *(o~2 \ {B})
Fll: Suppose CT(II 1, o~ 1 -+ A) and CT(II 2, o~2 ---+ ·A). Then, by the previous construction, there is a tableau II 3 such that CT(II3, o~1 U ~2 -+f). rrr = o(~1
F8:
---+ B o * o ~ 2 II 1 A ---+ B o * o ~ 2 o~ 2 -+
B *A oB II2 A : : > B -+ *A o B A-tA B-tB o~ 2 -+
o~
F9:
Suppose CT(II 1 ,o~---+ B). IJT =
\ {B})-+ ·B
o(~l
B
o~ 1
Ao
U (~2
o(~1 \ {B}) o o(~2 \ {B})---+ ·B \ {B}) o o(~2 \ {B})-+ (B ::::>f) (B ::::>f)-+ ·B o(~l \ {B}) U (~2 \ {B}) oB-+ f (B ::::>f) -+ *B : (B : : > f) o I ---+ *B II 3 (B : : > f) -+ *B o *I B---+ B f---+ *I f-+I
Suppose CT(II 1, o~ 1 ---+A) and CT(II 2 , o~ 2 ---+ A::::> B). IJT =
o~ 1 o o~ 2 -+
\ {B})
\{A}---+ A::::> B
F12:
Suppose CT(II 1 , o~ 1 -+A) and CT(II 2, o~ 2 ---+ •A). Then there is a tableau II 3 such that CT(II 3 , o~ 1 u ~2---+ f). IJT = o(~l
\ {•B})
U (~2
\ {•B})---+ B
o(~1
\ {·B}) o o(~2 \ {·B}) -+ B o(~1 \ {•B}) o o(~2 \ {•B})---+ ·B : : > f ·B : : > f-+ B o(~ 1 \ {·B}) o o(~2 \ {·B}) o ·B ---+ f (--,B : : > f) o I -+ B : (•B::::>f)-tBHI II 3 *B o (·B : : > f) ---+ *I *B o (·B : : > f) -+ I (--,B : : > f) ---+ * * B o I *B-+ ·B f---+ I *B---+ *B B-tB
222
223
FITCH-STYLE NATURAL DEDUCTION
CHAPTER 13
f -t -.[F]-.A ::::> f -t *[F]-.A ([F]-.A ::::> f) o I-t *[F]-.A [F]-.A ::::> f -t *[F]-.A o *I [F]-.A -t [F]-.A f -t *I
[F]-.A [F]-.A Fl3: Suppose CT(II 1 , o~ -t A). IIT =
o~ o
A 1 -t A rr1
::::>
Fl4: Suppose CT(II 1, o~ -t A),~= {A1, ... , An} =j:. 0. rrr = -t [F]A o(F]~ -t [F] o ~ • o [F]~ -to~
f-ti
o(F]~
[F] o ~ -t [F]A •[F] o ~ -t A [F] o ~ -t •A rr1
• o [F]~ o (*An o ... o *A2) -t A1 • o [F]~ -t A1 o(F]~ -t •A1
and let II5 =
-.[F]-.A -t (F)A *[F]-.A -t (F)A *(F)A -t [F]-.A • * (F)A -t -.A • * (F)A -t *A *(F)A -t • *A * • *A-t (F)A A-t A.
o(~ U ~2)
\ {[F]-.A} -t (F)A {[F]-.A} -t -.[F]-.A U ~2) \ {[F]-.A} -t [F]-.A ::::> f
o(~1 U ~2) \
Suppose CT(II 1,I -t A). Then rrr
=
I-t [F]A ei -t A rr1.
Fl5: Suppose CT(II 1, o~ -t A), ~ = { A1, ... , An} =j:. 0. rrr = o[P]~ -t
[P]A o(P]~ -t [P] o ~ • * o~ -t * o [P]~ o[P]~ -t
[P] o ~ -t [P]A • *A-t *(P] o ~
*• *o ~
A1 o (*An o ... o *A2) -t A1 A1 -t A1 I-t [P]A
= • *A -t *I rr1.
Fl6: Suppose CT(II 1 , o~ 1 -t B) and CT(II 2 , o~ 2 -t -.B). Then there is a tableau II3 such that CT(II 3, o~ 1 U ~ 2 -t f).
II4
F17: Analogous to the previous case. F18: Suppose CT(II 1 , o~ 1 -t B) and CT(II 2 , o~2 -t -.B). Then there is a tableau II 3 such that CT(II3, o~1 U ~2 -t f). Let II4 =
-.(F)A -.(F)A
::::> ::::>
f -t -.[F]-.A f -t *[F]-.A
-.(F)A -t [F]-.A e(-.(F)A) -t -.A •( -.(F)A) -t *A -.(F) A -t • *A *(F)A -t • *A) * • *A-t (F)A A-t A.
[P]A 1 -t * • * o ~ A 1 -to~
Suppose CT(II 1, I-t A). Then rrr
o(~1
II5
f -t *I f -t I
224 Fl9:
CHAPTER 13
FITCH-STYLE NATURAL DEDUCTION
225
Analogous to the previous case.
A-+ ((F)A o A) o (P)A A o *(P)A-+ (F) A o A and let II 3 = A-+ (F) A o A *(F)A o A-+ A A-+ A. •A -+ ((F) A o A) o (F) A *((F) A o A) o •A -+ (P)A Let Il4 = •A-+ (P)A
o~
F20: Suppose CT(II 1, o~ -+A). ITT =
-+ [P](F)A • * (F)A-+ * o 6 *(F)A-+ • * o6 * • * o ~ -+ (F)A Ill
F21: Analogous to the previous case. (Fl): Suppose CT(II1, o6-+ A),~= {A 1, ... ,An} f 0. liT=
A-+A
o[F]~
-+ [F][F]A • o [F]~ -+ [F]A • • o[F]~-+ A • o [F]~-+ •A o[F]~-+ • • A o[F]6-+ •A o[F]~ -+ [F] o ~ [F] o 6 -+ •A
(F)A-+ [F]((F)A V A V (P)A)
* • *A-+ [F]( (F) A V A V (P)A) and let II 5 =
• * • *A-+ (F)A V A V (P)A • * • *A -+ ((F) A V A) o (P)A • * • *A o *(P)A-+ (F) A V A • * • *A o *(P)A-+ (F)A o A • * • *A-+ ((F) A o A) o (P)A II2 Il3 Il4. o~-+
If 6 =
0, then we get rrr
=
[F]I-+ [F][F]A •[F]I -+ [F]A • • [F]I-+ A •[F]I-+ •A [F]I-+ • •A [F]I-+ •A I-+ A.
Then
rrr =
o~-+
[F]((F)A V A V (P)A) (F)A II 5
Ill
(F6): Similar to the previous case.
From this definition it is clear that Lemma 13.2. PN(IT, A, { A1, ... , An})
(F2): Similar to the previous case. o~-+ (F)(P)A * • * • o6-+ (F)(P)A • o ~-+ (P)A
=?
CT(ITT, A1 o ... o An -+ A).
Ill
From closed tableaux to sequent calculus. The mapping ( · )8 from closed tableaux into sequent calculus proofs is as simple as could be: just turn the closed tableaux and their sequents upside down. Obviously,
(F5): Assume CT(II1, o6-+ (F) A). Let II 2 =
Lemma 13.3. CT(IT,A1o ... oAn-+ A)=? Ps(IT 8 ,Alo ... oAn-+ A).
(F3): Suppose CT(II 1 ,o~-+ A). ITT = (F4): Similar to the previous case.
* • *A-+ ((F)A o A) o (P)A * • *A o *(P)A-+ (F) A o A * • *A -+ (F)A o A * • *A o *A-+ (F) A * • *A-+ (F)A A-+A
From sequent calculus to axioms. In order to inductively define the mapping (-)H from sequent calculus proofs into axiomatic proofs, we use the earlier translation T of sequents into tense logical formulas; see Chapter 3. The axiomatic sequents are thus translated into instantiations of the axiom schema A ::J A. Suppose Ps(IT, X -+ Y) and Ps(x,~Y', X1 o ... o Xk -+ Y'). By the induction hypothesis there is
1
226
CHAPTER 13
an axiomatic proof rrH of T(X-+ Y) from
A NEW AXIOMATIZATION OF KT
0.
(x,~Y' )H =
Tl(Xk)
rrH II',
where II' is a certain axiomatic proof of T(X' -+ Y') from T(X -+ Y). Sequent rules with two or three premise sequents are dealt with similarly. We shall not specify II' here; the existence of II' is clear from the possible worlds semantics.
227
While AO and RO are inherited from the underlying classical propositionallogic (CPL), and Al and A2 are just definitions, the remaining axiom schemata and rules exhibit a certain division of labour as far as the tense logical operations are concerned: A5, A6, Rl and R2 indicate that [F] and [P] are normal necessity-type operators, whereas A3 and A4 specify some interaction between the future and the past. The aim of this section is to present an alternate axiomatization of Kt, one which underlines the interaction between the forward and backward oriented modalities, but puts less emphasis on normality as a salient feature of the presentation. To achieve this, the K axiom schemata A5 and A6 are replaced by the following regularity rules: R3
(A=::> B) f- ([F]A
=::>
[F]B)
R4
(A=::> B) f-- ([P]A
=::>
[P]B).
Theorem 13. 5. The Full-Circle Theorem: 1. 2. 3. 4. 5.
R3 is obviously derivable from Rl, A5 and RO; R4 is derivable from R2, A6, and RO. The interaction between the future and the past is tightened by dispensing with the axiom schemata A3 and A4 in favour of the schemata:
PH(II,A,{Al, ... ,An}) implies PN(IIN, A, {A1, ... , An}) implies CT((IIN)T, A1 o ... o An -+A) implies Ps(((IIN)T) 8 ,Al o ... o An-+ A) implies PH((((IIN)T)S)H,A,{Al,···,An})
Proof. By the previous lemmata. Q.E.D.
Corollary 13.6. The tableau calculi presented are strongly complete with respect to their corresponding possible worlds semantics: A1, ... , An p A iff there exists a closed display tableau for A1 o ... o An -+A.
13.2. A NEW AXIOMATIZATION OF Kt The presentation of the minimal normal propositional tense logic Kt as a display calculus suggests an interesting alternate axiomatization of Kt. Consider again the standard axiomatization of Kt from the previous section: AO A1 A3 A5 R1
all tautologies; (F)A -,[F]-.A;
=
A:::::> [F](P)A; [F](A :::::> B) :::::> ([F]A A I- [F]A;
:::::>
[F]B);
RO A2 A4 A6 R2
A, A::::>B I- B; (P)A •[P]-.A; A:::::> [P](F)A; [P](A :::::> B) :::::> ([P]A A I- [P]A.
A7
[F]( (P)A =::>B)
=::>
(A=::> [F]B);
A8
[P]( (F) A
=::>
(A
A9
[P](A
=::>
[F]B)
=::> ( (P)A =::>
B);
AlO
[F](A
=::>
[P]B)
=::>
((F) A
B).
=::>
B)
=::>
[P]B); =::>
The latter schemata are obviously valid on Kripke frames and hence derivable in Kt. We have achieved our aim, if from AO, A7- AlO and RO - R5 we can derive A3- A6. The derivation of A3 and A4 is trivial. We may, for example, use Rl to obtain [F]( (P)A =::> (P)A) and then apply RO to the latter and the following instance of A7: [F]( (P)A =::> (P)A) =::> (A =::> [F](P)A). The derivation of A5 and A6 is less trivial. 54 Table XIX exhibits a detailed derivation of A5 which parallels an analogous derivation of A6.
=
:::::>
[P]B);
54
Note that it would be enough to derive the distribution of [F] and [P] over conjunction: [F](A 1\ B) :::::> ([F]A 1\ [F]B), [P](A 1\ B) :::::> ([P]A 1\ [P]B), cf. [33, Exercise 4.5 (b), p. 122].
228
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24
CHAPTER 13
N4 REDISPLAYED
Table XIX. An axiomatic derivation of A5.
The new axiomatization of Kt was extracted from a certain cutfree proof of the K axiom schema in the modal display calculus. The proof in question is instructive from the point of view of proof search in display logic, but there are shorter cut-free display proofs of K, from which a derivation of A5 and A6 can be obtained. The first fifteen lines of the previous derivation of A5, for example, may be replaced by the following deduction:
A :J (B V ·(A :J B)) [F]A :J [F](B V -.(A :J B)) [P]([F]A :J [F](B V •(A :J B))) [P]([F]A :J [F](B V •(A :J B))) :J :J ( (P)[F]A :J (B V ·(A :J B))) (P)[F]A :J (B V •(A :J B)) (A :J B) :J ( -.(P)[F]A V B) [F](A :J B) :J [F](-.(P)[F]A V B) ([F](A :J B) 1\ [F]A) :J [F](•(P)[F]A V B) [P](([F](A :J B) 1\ [F]A) :J [F](-.(P)[F]A V B)) [P](([F](A :J B) 1\ [F]A) :J [F](-.(P)[F]A V B)) :J :J ((P)([F](A :J B) 1\ [F]A) :J (•(P)[F]A V B)) (P)([F](A :J B) 1\ [F]A) :J (•(P)[F]A V B) (P)[F]A :J (B V -.(P)([F](A :J B) 1\ [F]A)) [F]((P)[F]A :J (B V •(P)([F](A :J B) 1\ [F]A))) [F]((P)[F]A :J (B V •(P)([F](A :J B) 1\ [F]A))) :J :J ([F]A :J [F](B V •(P)([F](A :J B) 1\ [F]A))) [F]A :J [F](B V •(P)([F](A :J B) 1\ [F]A)) ([F](A :J B) 1\ [F]A) :J :J (F](B V •(P)([F](A :J B) 1\ [F]A)) [P](([F](A :J B) 1\ [F]A) :J :J [F](B V •(P)([F](A :J B) 1\ [F]A))) [P](([F](A :J B) 1\ [F]A) :J :J [F](B V •(P)([F](A :J B) 1\ [F]A))) :J :J ( (P) ([F) (A :J B) 1\ [F) A) :J :J (B V •(P)([F](A :J B) 1\ [F]A))) (P)([F](A :J B) 1\ [F]A) :J :J (B V •(P)([F](A :J B) 1\ [F]A)) (P)([F](A :J B) 1\ [F]A) :J B [F]((P)([F](A :J B) 1\ [F]A) :J B) [F]((P)([F](A :J B) 1\ [F]A) :J B) :J (([F](A :J B) 1\ [F]A) :J [F]B) ([F](A :J B) 1\ [F]A) :J [F]B [F](A :J B) :J ([F]A :J [F]B)
CPL
R3, 1 R2, 2 A9 RO, 3, 4 CPL, 5 R3, 6 CPL, 7 R2, 8 A9
1 2 3 4
5 6
([F](A :J B) 1\ [F]A) :J [F]A [P](([F](A :J B) 1\ [F]A) :J [F]A) [P](([F](A :J B) 1\ [F]A) :J [F]A) :J :J ((P)(([F](A :J B) 1\ [F]A) :J A) (P)(([F](A :J B) 1\ [F]A) :J A (A :J B) :J (•(P)([F](A :J B) 1\ [F]A) V B) [F](A :J B) :J (F](-.(P)([F](A :J B) 1\ [F]A) V B)
229
CPL
R2, 1 A9 RO, 2, 3 CPL, 4
R3, 5
RO, 9, 10 CPL,ll
R1, 12 A7
13.3. N 4 REDISPLAYED
RO, 13, 14
In this section we shall restrict the structural language of display logic to containing three structure operations: the constant I, the unary *, and the binary o. Moreover, we shall increase the arity of the sequent arrow. A display sequent now has the form
CPL, 15
R2, 16
A9
where every Xi is a structure built up from the formulas, I, *, and o. Since we intend to assume commutativity of o in both antecedent and succedent position, we shall lay down only 'one-sided' basic structural rules, namely:
RO, 17, 18 CPL, 19 R1, 20
Basic structural rules
A7 RO, 21, 22 CPL, 23
(a) (b) (c) (d)
x1 o Y 1 x2-+ X3l x4 X1 1 x2 o Y-+ x3 1 X4 X1 1 X2-+ x3 o Y 1 X4 x1 1 x2-+ X3 1 x4 o Y
-11- x1 1 x2 o *Y-+ X3 1 x4 -H- X1 o *Y 1 X2-+ X3l x4 -H- x1 1 x2 -+ x3 1 X4 o *Y -11- x1 1 x2-+ X3 o *Y 1 X4
Two four-place sequents are said to be structurally equivalent if they are interderivable by means of rules (a)- (d). These rules make sense under the following translation T of sequents into formulas of the language of
1
230
CHAPTER 13
N4 REDISPLAYED
Table XX. Introduction rules for four-place sequents.
N4 and N3. 55
where 7i (i = 1,2,3,4) is defined as follows: 7l(A) 72(A)
73(A) 74(A)
71(1) 73(1)
72(1) 74(1)
7l(*X) 72( *X) 73(*X) 74(*X) 7l(X 0 Y) 72(X 0 Y) 73(X 0 Y) 74(X 0 Y)
A
"'A t f
72(X) 7l(X) 74(X) 73(X) 71 (X) !\ 71 (Y) '"'"' 71 (X)/\ '"'"' 7l{Y) 73(X) V 73(Y) '"'"'73(X)V '"'"'73(Y).
Here t is defined asp ~hp, for some p E Atom, and f is defined as ""'t. Again, an occurrence of a substructure in a given structure is said to be positive (negative) if it is in the scope of an even (uneven) number of *'s. An o-antecedent (e-antecedent) part of a sequent xl I x2 --7 x3 I x4 is a positive occurrence of a substructure of X 1 or a negative occurrence of a substructure of X 2 (a positive occurrence of a substructure of X2 or a negative occurrence of a substructure of XI). An o-succedent (esuccedent) part of x1 1 x2 --+ x3 1 x4 is a positive occurrence of a substructure of x3 or a negative occurrence of a substructure of x4 (a positive occurrence of a substructure of X4 or a negative occurrence of a substructure of X3).
Theorem 13. 7. (Display Theorem) For every sequent s and every aantecedent (e-antecedent) part X of s there exists a sequent s' structurally equivalent to s such that X is the entire o-antecedent (e-antecedent) of s'; and for every sequent sand every o-succedent (e-succedent) part X of s there exists a sequent s' structurally equivalent to s such that X is the entire o-succedent (e-succedent) of s'. Proof. Analogous to the proof of the Display Theorem in Chapter 3. Q.E.D. 55
231
We reuse the symbol 'o' and the letter 'T' from previous chapters with new meanings, because the context precludes any ambiguity.
x1 1 X2 -+ *A 1 x4 r- X1 1 X2 -+ ~A 1 x4 x1 1 x2-+ x3 1 **A r- X1 1 x2-+ x3 1 "'A *A 1 X2 -+ x3 1 x4 r- "'A 1 Xz -+ X3 1 x4 x1 1 **A-+ x3 1 x4 r- x1 1 "'A-+ x3 1 x4 x 1 X2-+ A 1 x4 Y 1 x2 -+ B 1 X4 rr- x o Y 1 x2 -+ A AB 1 x4 x1 1 X2 -+ x3 1 A oB r- X1 1 X2 -+ x3 1 A AB (-+ 1\ even) A oB 1 Xz -+ X3 1 X4 r- A AB I X2 -+ X3 I X4 (/\-+ odd) (/\-+ even) x1 1 A-+ X3l x x1 1 B -+ x3 1 Y rr- x1 1 A AB -+ x3 1 x o Y x 1 X2-+ A oB 1 x4 r- x1 1 X2-+ A v B 1 x4 (-+V odd) x1 1 x-+ x3 1 A x1 1 Y-+ x3 1 B r(-+V even) r- x1 1 x o Y -+ x3 1 A v B A 1 x2-+ x 1 x4 B 1 X2 -+ Y 1 x4 r(V-+ odd) r- A v B 1 x2 -+ x o Y 1 x4 x1 1 A oB -+ x3 1 x4 r- X1 1 A v B -+ x3 1 x4 (V-+ even) X1 o A 1 Xz-+ B 1 X4 r- X1 I X2-+ A ~h B I X4 (-+~h odd) (-+ ~h even) x 1 X2 -+ A 1 *x3 Y 1 X2 -+ x3 1 B rr- x o Y 1 x2 -+ x3 1 A ~h B xl I Xz-+ A I x4 B 1 x2 -+ x3 1 x4 r(~h-+ odd) r- A ~h B 1 x2 o *x1 -+ x3 1 X4 A o *B 1 *x1 -+ x3 1 x4 r- x1 1 A ~h B -+ x3 I x4 (~h-+ even)
( -+~ odd) ( -+""' even) (odd ~-+) (even "'-+ ) (-+ 1\ odd)
As logical rules we assume appropriate versions of identity and cut, namely: logical rules (odd id) (even id)
r- AII-+AII r- IIA-+IIA
(odd cut) (even cut)
X1 1 X2 -+ A 1 x4 A 1 Xz -+ X3 1 x4 r- X1 I X2 -+ X3 I X4 X1 1 X2 -+ X3 1 A X1 1 A-+ x3 1 X4 r- X1 1 Xz -+ X3 I X4
Moreover, we have the separate, symmetrical, explicit, and segregated introduction rules for the logical operation of N 4 and N3 in Table XX. In addition to the basic structural rules, the logical rules, and the introduction rules, we postulate a tight collection of further structural rules, stated in Table XXI. As a package, these rules allow one to derive plenty of other structural rules which one would like to consider sep-
232
Table XXI. Further structural rules for four-place sequents.
(It)
(r) (A) (P) (C)
(M)
Corollary 13.12. In 4DN4, {i) 1- x1 o *x2 1 1-+ TI(X1)/\ ""'T2(X2) 1 1. {ii) 1- T3(X3)v "-'T4(X4) 11-+ x3 o *x4 11.
Xz --+ X3 I X4 -H-I o X1 1 Xz --+ X3 1 X4 I Xz --+ X3 I X4 -H- X1 1 I o Xz --+ X 3 1 X4 I Xz --+ X3 I X4 -11- X1 1 Xz --+ I o X3 1 X4 I Xz --+ X3 I X4 -11- XI 1 Xz --+ X3 1 I o X4 X o (Y o Z) 1 Xz --+ X3 1 X4 -11- (X o Y) o z 1 X2 --t X 3 1 X 4 xi 1 x o (Y o Z) --+ x3 1 x4 -11- xi 1 (X o Y) o z--+ x3 1 X 4 X 0 y I Xz --+ x3 I x4 1- y 0 X I Xz --t x3 I x4 X1 1 x o Y --+ x3 1 x4 1- xi 1 Y ox --+ x3 1 x4 x ox I Xz --+ X3 I X4 1- X I Xz --+ X3 1 X4 xi 1 x ox --+ x3 1 x4 1- xi 1 x --+ x3 1 x4 X I Xz --+ X3 I X4 1- X o Y 1 Xz --+ X3 1 X4 xi 1 x --+ x3 1 x4 1- xi 1 x o Y --+ x3 1 x4
X1 XI xi XI
233
N 4 REDISPLAYED
CHAPTER 13
I
Proof. (i): We shall omit some obvious steps in the derivation: I I X 2 -+ I I 72 (X 2) 1 1 x2 -+ *T2(X2) 1 1 1 1 x2 -+""'T2(X2) 1 1 x1 11-+ T1(XI) 11 * x2 11 -+""'72{X2) 1I x1 o *x2 1 1-+ TI(XI)/\ ""'72(X2) 1 1 x1 1X2 11-+ TI(X1)/\ "'72(X2) 11 (ii): Similar. Q.E.D.
Theorem 13.13. In 4DN4, 1- xl N4, 1- T(Xl 1 x2-+ x3 1 X4).
arately in a synoptical treatment of substructural constructive logics (see also Chapter 6).
I x2-+ x3 I X4)
I x4
if and only if in
The previous corollary and two applications of (odd cut) give the desired result: from X 1 o *X2 I I-+ T1(XI)/\ "'T2(X2) I I and TI(XI)/\ "-' T2(X2) I I -+ T3(X3)V "-' T4(X4) I I we obtain xl 0 *x2 I I -+ T3(X3)V "'T4(X4) I I. This sequent together with T3(X3)V "'T4(X4) I I -+ x3 0 *x4 I I gives xl 0 *x2 I I -+ x3 0 *x4 I I which is display equivalent to xl I x2-+ x3 I x4. Q.E.D.
in
Proof. (i): By induction on axiomatic proofs in N4. (ii): By induction on proofs in 4DN4. Q.E.D.
Definition 13.14. The sequent system 4DN3 results from 4DN4 by the addition of the purely structural sequent rule
Corollary 13.10. 1- A in N4 iff 1- I I I-+ A I I in 4DN4. In combination with this weak completeness property, the next lemma is crucial for the proof of strong completeness.
1-AIA-+XIY. In 4DN3 we may define f as pi\ ""'P for some p E Atom, and t as ,.,_,f. The additional rule corresponds to the ex falso schema (AI\ ""'A) ::J h B, which in an axiomatic context marks the difference between N4 and N3. Hence,
Lemma 13.11. In 4DN4, (i) 1- x1 1 1-+ T1(XI) 1 I; (ii) 1- 1 1 x2-+ 1 1 T2(X2); (iii) 1- T3(X3) I I-+ x3 I I; and (iv) 1- I I T4(X4) -+I I x4. Proof. By induction on Xi.
-+ x3
Proof. (::::}:)This is Theorem 13.9 (ii). (-{::=:) Suppose in N4, 1- T(Xl I x2 -+ x3 I X4)· By Corollary 13.10, in 4DN4, 1- I I I-+ T(Xl I x2-+ x3 I X4) I I. Therefore in 4DN4,
Definition 13.8. The system 4DN4 is defined as the collection of the said display sequent rules. Theorem 13. 9. (i) If 1- A in N4, then 1- I I I-+ A I I in 4DN4. (ii) If 1- xl I x2-+ x3 I x4 in 4DN4, then 1- T(Xl N4.
I x2
Theorem 13.15. 4DN3 is strongly sound and complete with respect to N3.
Q.E.D.
l
234
CHAPTER 13
We shall not consider generalizing the strong cut-elimination theorem for DL from Chapter 4 to four-place sequents. Note, however, that in this higher-arity setting, the subformula property of the sequent systems for N 4 and N3 follows from cut-elimination by merely inspecting the introduction rules.
BIBLIOGRAPHY
1. 13.4. FUTURE WORK 2. The recent rapid development of modal proof theory is still ongoing. In order to approach a more stable situation, future work will have to focus on clarifying the relative advantages and disadvantages of and interrelations between major formalisms such as higher-arity sequent systems, higher-level sequent systems, relational proof systems, and DL. A translation of systems of indexed sequents into DL is presented in [115]; the relation between the method of hypersequents and DL is dealt with in Chapter 11. Research on display logic still has to address part of the agenda determined in Belnap's seminal paper [16]. In particular, the relation between the cut-elimination property of displayable logics and the interpolation property remains to be investigated. Moreover, in spite of the syntactic and semantic characterization of the properly displayable normal modal and tense logics, further case-studies of displayable nonclassical logics may still be of interest, because displayablity can, for instance, also be achieved via suitable modal or tense logical translations; see Chapters 10, 11 and [195]. Other issues that have not been addressed so far (or at least not in this book) are, for example, a detailed presentation of the relations between DL and type-theoretical grammars (see [118]), a systematic investigation of the practical use of DL for decidability problems, the computational complexity of display calculi, and the possibility of a formulas-as-types analysis of proofs in display sequent systems. Finally, there is plenty of work to be done on implementations of display calculi. An implementation of Gore's display calculus for relation algebra as an Isabelle theory can be found in [38].
3. 4. 5.
6. 7. 8.
9.
10. 11. 12.
13. 14.
15.
16.
M. D'Agostino and M. Mondadori. The Taming of the Cut. Classical Refutations with Analytic Cut. Journal of Logic and Computation, 4, 285-319, 1994. S. Akama. What is Strong Negation in a System? Draft, Teikyo University of Technology, 1995. N. Alechina and M. van Lambalgen. Generalized quantification as substructural logic. Journal of Symbolic Logic, 61, 1006-1044, 1996. A. Almukdad and D. Nelson. Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49, 231-233, 1984. H. Andreka, J. van Benthem and I. Nemeti. Back and Forth Between Modal Logic and Classical Logic. Journal of the Interest Group in Pure and Applied Logics, 3, 685-720, 1995. A. Avron. On modal systems having arithmetical interpretations. Journal of Symbolic Logic, 49, 935-942, 1984. A. Avron. A Constructive Analysis of RM. Journal of Symbolic Logic, 52, 939-951, 1987. A. Avron. Gentzenizing Schroeder-Heister's natural extension of natural deduction. Notre Dame Journal of Formal Logic, 31, 127-135, 1990. A. Avron. Using Hypersequents in Proof Systems for Non-classical Logics. Annals of Mathematics and Artificial Intelligence, 4, 225-248, 1991. A. Avron. Natural3-valued Logics- Characterization and Proof Theory. Journal of Symbolic Logic, 56, 276-294, 1991. A. Avron. Gentzen-Type Systems, Resolution and Tableaux. Journal of Automated Reasoning, 10, 265-281, 1993. A. Avron. The Method of Hypersequents in the Proof Theory of Propositional Non-Classical Logics. In: W. Hodges et al. (eds.), Logic: From Foundations to Applications, Oxford Science Publications, Oxford, 1996, 1-32. E. Earth and E. Krabbe. From Axiom to Dialogue. De Gruyter, Berlin, 1982. J. Barwise. An Introduction to First-Order Logic. In: J. Barwise (ed.), Handbook of Mathematical Logic, North Holland, Amsterdam, 5-46, 1977. N.D. Belnap. A Useful Four-Valued Logic. In: J.M. Dunn and G. Epstein (eds.), Modern Uses of Multiple- Valued Logic, Reidel, Dordrecht, 8-37, 1977. N.D. Belnap. Display Logic. Journal of Philosophical Logic, 11, 375-
235
236
BIBLIOGRAPHY
417, 1982. Reprinted with minor changes as §62 of A.R. Anderson, N.D. Belnap, and J.M. Dunn, Entailment: the logic of relevance and necessity. Vol. 2, Princeton University Press, Princeton, 1992. 17. N.D. Belnap. Linear Logic Displayed. Notre Dame Journal of Formal Logic, 31, 14-25, 1990. 18. N.D. Belnap. The Display Problem. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 79-92, 1996. 19. M. Benevides and T. Maibaum. A Constructive Presentation for the Modal Connective of Necessity (D). Journal of Logic and Computation, 2, 31-50, 1992. 20. J. van Benthem. Modal Foundations of Predicate Logic. Logic Journal of the Interest Group in Pure and Applied Logics, 5, 259-286, 1997. 21. J. van Benthem. Exploring Logical Dynamics. CSLI Publications, Stanford, 1996. 22. K. Bimbo and J.M. Dunn. Two Extensions of the Structurally Free Logic LC. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. 23. G. Birkhoff. Lattice Theory. American Mathematical Society, Providence, 19673 . 24. S. Blamey and L. Humberstone. A Perspective on Modal Sequent Logic. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 27, 763-782, 1991. 25. P. Bonatti. Sequent Calculi for Default and Autoepistemic Logics. In: P. Miglioli et al. (eds.), Theorem Proving with Analytic Tableaux and Related Methods, Proceedings TABLEAUX '96, Lecture Notes in AI 1071, Springer Verlag, Berlin, 127-142, 1996. 26. G. Boolos. Don't eliminate cut. Journal of Philosophical Logic, 13, 373-378, 1984. 27. T. Borghuis. Interpreting Modal Natural Deduction in Type Theory. In: M. de Rijke (ed.), Diamonds and Defaults, Kluwer Academic Publishers, Dordrecht, 67-102, 1993. 28. T. Borghuis. Coming to Terms with Modal Logic. PhD thesis, Department of Computer Science, University of Eindhoven, 1994. 29. R. Bull and K. Segerberg. Basic Modal Logic. In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. II, Extensions of Classical Logic, Reidel, Dordrecht, 1-88, 1984. 30. J. Burgess. Basic Tense Logic. In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. II, Extensions of Classical Logic, Reidel, Dordrecht, 89-133, 1984. 31. C. Cerrato. Modal sequents for normal modal logics. Mathematical Logic Quarterly, 39, 231-240, 1993. 32. C. Cerrato. Modal sequents. In: H. Wansing (ed.), Proof Theory of Modal Logic, 141-166, Kluwer Academic Publishers, Dordrecht, 1996. 33. B. Chellas. Modal Logic: An Introduction. Cambridge University Press, Cambridge, 1980.
BIBLIOGRAPHY 34.
35.
36. 37.
38.
39. 40. 41. 42. 43. 44.
45. 46. 47.
48. 49. 50. 51. 52.
237
M. Clarke. Intuitionistic Non-Monotonic Reasoning - further results. In: Y. Kodratoff (ed.), ECAI 88. Proc. 8th European Conference on Artificial Intelligence, Pitman Publishing, London, 525-527, 1988. M. Clarke and D. Gabbay. An Intuitionistic Basis for Non-Monotonic Reasoning. In: P. Smets et al. (eds.), Non-Standard Logics for Automated Reasoning, Academic Press, London, 163-178, 1988. G. Corsi. Weak Logics with Strict Implication. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 33, 389-406, 1987. G. Crocco and L. Farinas del Cerro. Structure, Consequence Relation and Logic. In: D. Gabbay (ed.), What is a Logical System?, Oxford University Press, Oxford, 239-259, 1994. J .E. Dawson and R. Gore. A Mechanised Proof System for Relation Algebra using Dispaly Logic. Australian National University, Canberra, 1998. K. Dosen. Sequent-systems for modal logic. Journal of Symbolic Logic, 50, 149-159, 1985. K. Dosen. Intuitionistic Double Negation as a Necessity Operator. Publications de L 'Institute Mathematique (Belgrade), 49, 15-20, 1984. K. Dosen. Sequent systems and groupoid models I. Studia Logica, 47, 353-389, 1988. K. Dosen. Sequent systems and groupoid models II. Studia Logica, 48, 41-65, 1989. K. Dosen. Logical Constants as Punctuation Marks. Notre Dame Journal of Formal Logic, 30, 362-381, 1989. K. Dosen. Ancestral Kripke Models and Nonhereditary Kripke Models for the Heyting Calculus. Notre Dame Journal of Formal Logic, 32, 580-597, 1991. K. Dosen. Modal Logic as Metalogic. Journal of Logic, Language and Information, 1, 173-201, 1992. K. Dosen. Rudimentary Kripke models for the intuitionistic propositional calculus. Annals of Pure and Applied Logic, 62, 21-49, 1993. K. Dosen. Modal Translations in K and D. In: M. de Rijke (ed.), Diamonds and Defaults, Kluwer Academic Publishers, Dordrecht, 103127, 1993. K. Dosen and P. Schroeder-Heister (eds.), Substructural Logics, Clarendon Press, Oxford, 1993. A. Dragalin. Mathematical Intuitionism. Introduction to Proof Theory. American Mathematical Society, Providence,1988. M. Dummett. A Prepositional Calculus with a Denumerable Matrix. Journal of Symbolic Logic, 24, 96-107, 1959. J. M. Dunn. A 'Gentzen' System for Positive Relevant Implication (Abstract). Journal of Symbolic Logic, 38, 356-357, 1973. J. M. Dunn. Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation and Various Logical Operations. In: J. van Eijk (ed.), Logics in AI, Proc. European Workshop
238
53.
54.
55.
56. 57.
58.
59. 60. 61. 62. 63. 64.
65.
66. 67.
68. 69.
BIBLIOGRAPHY JELIA 1990, Lecture Notes in Computer Science 478, 31-51, SpringerVerlag, Berlin, 1990. J.M. Dunn. Partial-Gaggles Applied to Logics with Restricted Structural Rules. In: K. Dosen and and P. Schroeder-Heister (eds.), Substructural Logics, Clarendon Press, Oxford, 63-108, 1993. J .M. Dunn. Perp and star: two treatments of negation. In: J. Tomberlin (ed.), Philosophical Perspectives (Philosophy of Language and Logic) 7, Ridgeview, Atascadero, 331-357, 1994. J.M. Dunn. Gaggle Theory Applied to Modal, Intuitionistic, and Relevance Logics. In: I. Max and W. Stelzner (eds.), Logik und Mathematik: Frege-Kolloquium 1993, de Gruyter, Berlin, 335-368, 1995. J.M. Dunn. Generalized Ortho Negation. In: H. Wansing (ed.), Negation. A Notion in Focus, de Gruyter, Berlin, 3-26, 1996. J.M. Dunn. A Comparison of Various Model-theoretic Treatments of Negation: A History of Formal Negation. In: D. Gabbay and H. Wansing (eds.), What is Negation, Kluwer Academic Publishers, Dordrecht, 2351, 1998. J.M. Dunn and R. Meyer. Combinators and Structurally Free Logic. Logic Journal of the Interest Group in Pure and Applied Logics, 5, 505-537, 1997. H. Enderton. A Mathematical Introduction to Logic. Academic Press, New York, 1972. M. Finger. Towards Structurally-free Theorem Proving. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. M. Finger and D. Gabbay. Adding a Temporal Dimension to a Logic System. Journal of Logic, Language and Information, 1, 203-233, 1992. M. Fitting. Tableau Methods of Proof for Modal Logics. Notre Dame Journal of Formal Logic, 13, 237-247, 1972. M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht, 1993. M. Fitting. Basic Modal Logic. In: D. Gabbay et al. (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 1, Logical Foundations. Oxford University Press, Oxford, 365-448, 1993. D. Gabbay. Intuitionistic Basis for Non-Monotonic Logic. In: Proc. 6th Conference on Automated Deduction., Lecture Notes in Computer Science 138, Springer-Verlag, Berlin, 260-273, 1982. D. Gabbay. What is Negation in a System?. In: F. Drake and J. Truss (eds.), Logic Colloquium '86, Elsevier, Amsterdam, 95-112, 1988. D. Gabbay. A general theory of structured consequence relations. In: K. Dosen and P. Schroeder-Heister (eds.), Substructural Logics, Clarendon Press, Oxford, 109-151, 1993. D. Gabbay. Labelled Deductive Systems: Volume 1. Foundations. Oxford University Press, Oxford, 1996. D. Gabbay and H. Wansing. What is Negation in a System? Negation in
BIBLIOGRAPHY
70.
71. 72. 73. 74.
75. 76.
77.
78.
79.
80. 81.
82. 83.
84. 85.
239
Structured Consequence Relations. In: A. FUhrmann and H. Rott (eds.), Logic, Action and Information, de Gruyter, Berlin, 328-350, 1996. G. Gentzen. Investigations into Logical Deduction. In: M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, North Holland, ~m sterdam, 1969, 68-131. English translation of: Untersuchungen uber das logische SchlieBen. Mathematische Zeitschrift, 39, I 176-210, II 405-431, 1934. L. Goble. Gentzen systems for modal logics. Notre Dame Journal of Formal Logic, 15, 455-461, 1974. R. Goldblatt. Topoi. The Categorical Analysis of Logic. North-Holland, Amsterdam, 1979. R. Goldblatt. Logics of Time and Computation. CSLI Publications, Stanford, 2nd edition, 1992. R. Gore. Cut-Free Sequent and Tableau Systems for Propositional Normal Modal Logics. PhD thesis, University of Cambridge Computer Laboratory, Technical Report No. 257, 1992. R. Gore. Intuitionistic Logic Redisplayed. Technical Report TR-ARP1-1995, Australian National University, 1995. . R. Gore. A uniform display system for intuitionistic and dual mtuitionistic logic. Technical Report TR-SRS-2-95, Australian National University, 1995. R. Gore. On the Completeness of Classical Modal Display Logic. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 137-140, 1996. R. Gore. Cut-free Dispaly Calculi for Relation Algebras. In: D. van Dalen and M. Bezem (eds.), CSL96: Selected Papers of the Annual Conference of the Association for Computer Science Logic, Lecture Notes in Computer Science 1258, Springer-Verlag, Berlin, 198-210, 1997. R. Gore. Tableau Methods for Modal and Temporal Logics. To appear in: M. D'Agostino, D. Gabbay, R. Hiihnle, and J. Posegga (eds.), Handbook of Tableau Methods, Kluwer Academic Publishers, Dordrecht, 1998. R. Gore.Substructural Logics on Display. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. R. Gore. Gaggles, Gentzen and Galois: Cut-free Display Calculi and RelationM Semantics for Algebraizable Logics. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. S. Gottwald. Mehrwertige Logik. Akademie-Verlag, Berlin, 1989. V.N. Grishin. A nonstandard logic and its application to set theory (in Russian). In: Studies in Formalized Languages and Nonclassical Logics (in Russian), Moscow, Nauka, 135-171, 1974. . . . Y. Gurevich. Intuitionistic Logic with Strong NegatiOn. Studw Logtca, 36, 49-59, 1977. I. Hacking, What is Logic?, The Journal of Philosophy 76 (1979), 285-
240
86. 87. 88. 89. 90.
91. 92.
93.
94.
9S. 96.
97.
98.
99. 100. 101. 102. 103.
BIBLIOGRAPHY 319. Reprinted in: D. Gabbay (ed.), What is a Logical System?, Oxford University Press, Oxford, 1994, 1-33. J.R. Hindley and J.P. Seldin. Introduction to Combinators and .ACalculus. Cambridge University Press, Cambridge, 1986. G. Hughes and M. Cresswell. An Introduction to Modal Logic. Methuen, London, 1968. A. Indrzejczak. Generalised Sequent Calculus for Propositional Modal Logics. Logica Trianguli, 1, 1S-32, 1997. A. Indrzejczak. Cut-free Double Sequent Calculus for SS. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. P. Jackson and H. Reichgelt. A general proof method for first-order modal logic. In: Proc. 9th International Joint Conference on Artificial Intelligence, 942-944, 1987. S. Kanger. Provability in Logic. Almqvist & Wiksell, Stockholm, 19S7. M. Kracht. Power and Weakness of the Modal Display Calculus. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 93-121, 1996. S. Kripke. Semantical analysis of modal logic I: Normal modal propositional calculi. Zeitschrift fiir mathematische Logik und Grundlagen der Mathematik, 9, 67-96, 1963. S. Kripke. Semantical analysis of intuitionistic logic I. In: J. Crossley and M. Dummett (eds.), Formal Systems and Recursive Functions, North-Holland, Amsterdam, 92-129, 196S. S. Kuhn. Quantifiers as Modal Operators. Studia Logica, 39, 14S-1S8, 1980. F. von Kutschera. Die Vollstandigkeit des Operatorensystems { ....,, 1\, V, :::>} fur die intuitionistische Aussagenlogik im Rahmen der Gentzensemantik. Archiv fiir Mathematische Logik und Grundlagenforschung, 11, 3-16, 1968. F. von Kutschera. Ein verallgemeinerter Widerlegungsbegriff fur Gentzenkalkule. Archiv fiir Mathematische Logik und Grundlagenforschung, 12, 104-118, 1969. M. van Lambalgen. Natural deduction for generalized quantifiers. In: J. van der Does and J. van Eijck (eds.), quantifiers, Logic, and Language, CSLI Publications, Stanford, 22S-236, 1996. J. Lambek. The mathematics of sentence structure. American Mathematical Monthly, 65, 1S4-170, 19S8. D. Leivant. On the proof theory of the modal logic for arithmetic provability. Journal of Symbolic Logic, 46, S31-S38, 1981. W. Lenzen. Necessary Conditions for Negation Operators. In: H. Wansing (ed.), Negation. A Notion in Focus, de Gruyter, Berlin, 37-S8, 1996. E.G.K. L6pez-Escobar. Refutability and Elementary Number Theory, Indagationes Mathematicae, 34, 362-374, 1972. W. Lukaszewicz. Non-Monotonic Reasoning. Formalization of Commonsense Reasoning. Ellis Horwood, Chichester, 1990.
BIBLIOGRAPHY 104. 10S.
106.
107. 108. 109. 110. 111. 112.
113.
114. 11S. 116.
117.
118.
119. 120. 121.
241
J. Lyons. Semantics, Vol. I, Cambridge University Press, Cambridge, 1977. W. MacCaull. Relational Proof System for Linear and Other Substructural Logics. Journal of the Interest Group in Pure and Applied Logics, 5, 673-697, 1997. S. Martini and A. Masini. A computational interpretation of modal proofs. In: H. Wansing (ed.), Proof Theory of Modal Logic, 213-241, Kluwer Academic Publishers, Dordrecht, 1996. M. Marx and Y. Venema. A Modal Logic of Relations. Technical Report IR-396, Vrije Universiteit Amsterdam, 199S. A. Masini. 2-Sequent calculus: a proof theory of modalities. Annals of Pure and Applied Logic, 58, 229-246, 1992. D. McDermott and J. Doyle. Non-Monotonic Logic I. Journal of Artificial Intelligence, 13, 41-72, 1980. W. Meyer-Viol. Instantial Logic. PhD thesis, University of Amsterdam, Institute for Logic, Language and Computation, 199S. G. Mints (Mine). Cut-elimimation theorem in relevant logics. The Journal of Soviet Mathematics, 6, 422-428, 1976. G. Mints. Cut-free calculi of the SS type. Studies in constructive mathematics and mathematical logic. Part II. Seminars in Mathematics, 8, 79-82, 1970. G. Mints. Gentzen-type systems and resolution rules. Part I. Propositional Logic. In: P. Martin-Lofand G. Mints (eds.), COLOG-88, Lecture Notes in Computer Science 417, Springer-Verlag, Berlin, 198-231, 1990. G. Mints. A Short Introduction to Modal Logic. CSLI Lecture Notes 30, CSLI Publications, Stanford, 1992. G. Mints. Indexed systems of sequents and cut-elimination. Journal of Philosophical Logic, 26, 671-696, 1997. G. Mints, V. Orevkov, and T. Tammet. Transfer of Sequent Calculus Strategies to Resolution for S4. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 17-31, 1996. R. Montague. Logical Necessity, Physical Necessity, Ethics and Quantifiers. Inquiry, 2S9-269, 1960. Reprinted in: R. Thomason (ed.), Formal Philosophy. Selected Papers of Richard Montague, Yale University Press, New Haven, 71-83, 1974. M. Moortgat. Categorial Type Logics. In: A. ter Meulen and J. van Benthem (eds.), Handbook of Logic and Language, North-Holland, Amsterdam, 93-177, 1997. D. Nelson. Constructible falsity. Journal of Symbolic Logic, 14, 16-26, 1949. I. Nemeti. Algebraizations of Quantifier Logics, an introductory Overview. Studia Logica, 50, 48S-S69, 1991. H. Nishimura. A Study of Some Tense Logics by Gentzen's Sequential Method. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 16, 343-3S3, 1980.
242
122. 123. 124.
125. 126.
127.
128.
129.
130.
131.
132.
133.
134. 135. 136.
BIBLIOGRAPHY
BIBLIOGRAPHY M. Ohnishi and K. Matsumoto. Gentzen Method in Modal Calculi. Osaka Mathematical Journal, 9, 113-130, 1957. M. Ohnishi and K. Matsumoto. Gentzen Method in Modal Calculi, II. Osaka Mathematical Journal, 11, 115-120, 1959. E. Orlowska. Relational interpretation of modal logics. In: H. Andreka, D. Monk and I. Nemeti (eds.), Algebraic Logic. Colloquia Mathematica Societatis Janos Bolyai 54, North Holland, Amsterdam, 443-471, 1988. E. Orlowska. Relational proof systems for relevant logics. Journal of Symbolic Logic, 57, 1425-1440, 1992. E. Orlowska. Relational Proof Systems for Modal Logics. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers Dordrecht, 55-77, 1996. ' ~· Pear.ce. n Reasons for Choosing N. Technical report 14/91, Gruppe fur Log1k, Wissenstheorie und Information, Free University of Berlin 1991. ' D. Pearce. Reasoning with Negative Information 11: Hard Negation, Strong Negation and Logic Programs. In: D. Pearce and H. Wansing (eds.), Nonclassical Logics and Information Processing Lecture Notes in AI 619, Springer-Verlag, Berlin, 63-79, 1992. ' D. Pearce. Answer Sets and Constructive Logic, 11: Extended Logic Programs and Related Non-monotonic Formalisms. In: L. Pereira and A. Nerode (eds.), Logic Programming and Non-Monotonic Reasoning, MIT Press, Cambridge (Massachusetts), 457-475, 1993. D. Pearce and G. Wagner. Reasoning with Negative Information I: Strong Negation in Logic Programs. In L. Haaparanta et al. (eds.), Language, Knowledge, and Intentionality, (Acta Philosophica Fennica 49), Helsinki, 430-453, 1990. D. Pearce and G. Wagner. Logic Programming with Strong Negation. In: P. Schroeder-Heister (ed.), Proc. Workshop on Extensions of Logic Programming, Lecture Notes in AI 475, Springer-Verlag, Berlin, 311326, 1990. A. Pliuskeviciene. Cut-free Calculus for Modal Logics Containing the Barcan Ax~om. To appear in: M. Kracht, M. de Rijke, H. Wansing, and M. ZakharJaschev (eds.), Advances in Modal Logic '96, CSLI Publications, Stanford, 1998. A. PliuskeviCiene. Extended Disjunction and Existence Properties for Some Predicate Modal Logics. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. G. Pottinger. Uniform, cut-free formulations ofT, S4 and S5 (Abstract). Journal of Symbolic Logic, 48, 900-901, 1983. A. Prijatelj. Bounded Contraction and Gentzen-style Formulations of Lukasiewicz Logics. Studia Logica, 57, 437-456, 1996. W. Rautenberg. Klassische und nichtklassische Aussagenlogik. Vieweg, Braunschweig, 1979.
137. 138. 139. 140.
141. 142. 143. 144. 145.
146.
147. 148. 149.
150. 151. 152. 153.
154.
243
W. Rautenberg. Modal tableau calculi and interpolation. Journal of Philosophical Logic, 12, 403-423, 1983, 14, 229, 1985. G. Restall. Subintuitionistic Logics. Notre Dame Journal of Formal Logic, 35, 116-129, 1994. G. Rest all. A Useful Substructural Logic. Bulletin of the Interest Group in Pure and Applied Logic, 2, 137-148, 1994. G. Restall. Displaying and Deciding Substructural Logics 1: Logics with Contraposition. Technical Report TR-ARP-11-94, Australian National University, Canberra, 1994. To appear in: Journal of Philosophical Logic, 27, 1998. G. Restall. Display Logic and Gaggle Theory. Reports on Mathematical Logic, 29, 133-146, 1995 (published 1996). G. Restall. Combining Possibilities and Negations. Studia Logica, 59, 121-141, 1997. M. de Rijke. Extending Modal Logic. PhD thesis, University of Amsterdam, Institute for Logic, Language and Computation, 1993. D. Roorda. Resource Logics. PhD thesis, Institute for Logic, Language and Computation, University of Amsterdam, 1991. D. Roorda. Dyadic Modalities and Lambek Calculus. In: M. de Rijke (ed.), Diamonds and Defaults, Kluwer Academic Publishers, Dordrecht, 215-253, 1993. M. Ryan, P.-Y. Schobbens, and 0. Rodrigues. Counterfactuals and Updates as Inverse Modalities. In: Y. Shoam (ed.), TARK 'g6. Proceedings Theoretical Aspects of Rationality and Knowldege, Morgan Kaufmann, San Francisco, 163-173, 1996. G. Sambin and S. Valentini. The modal logic of provability. The sequential approach. Journal of Philosophical Logic, 11, 311-342, 1982. G. Sambin, G. Battilotti, and C. Faggian. Basic logic: reflection, symmetry, visibility. To appear in: Journal of Symbolic Logic. M. Sato. A Study of Kripke-type Models for Some Modal Logics by Gentzen's Sequential Method. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 13, 381-468, 1977. M. Sato. A cut-free Gentzen-type system for the modal logic S5. Journal of Symbolic Logic, 45, 67-84, 1980. P. Schroeder-Heister. A Natural Extension of Natural Deduction. Journal of Symbolic Logic 49, 1284-1300, 1984. P. Schroeder-Heister. Uniform proof-theoretic semantics for logical constants (Abstract). Journal of Symbolic Logic 56, 1142, 1991. K. Schroter. Methoden zur Axiomatisierung beliebiger Aussagen- und Priidikatenkalkiile. Zeitschrijt fur mathematische Logik und Grundlagen der Mathematik 1, 214-251, 1955. 0. Serebriannikov. Gentzen's Hauptsatz for Modal Logic with Quantifiers. In: I. Niiniluoto and E. Saarinen (eds.), Intensional Logic: Theory and Applications (= Acta Philosophica Fennica, 35), 79-88, 1982.
244 155. 156.
157. 158. 159. 160. 161. 162.
163. 164. 165.
166. 167. 168. 169. 170.
171. 172. 173. 174.
BIBLIOGRAPHY
BIBLIOGRAPHY T. Shimura. Cut-Free Systems for the Modal Logic S4.3 and S4.3Grz. Reports on Mathematical Logic, 25, 57-73, 1991. G. Shvarts. Gentzen style systems for K45 and K45D. In: A. Meyer and M. Taitslin (eds.), Logic at Batik '89, Lecture Notes in Computer Science 363, Springer-Verlag, Berlin, 245-256, 1989. J. Slupecki, G. Bryll and V. Wybraniec-Skardowska. The theory of rejected propositions I. Studia Logica 29, 75-119, 1971. J. Slupecki, G. Bryll and V. Wybraniec-Skardowska. The theory of rejected propositions 11. Studia Logica 30, 97-142, 1972. R. Smullyan. Analytic cut. Journal of Symolic Logic 33, 560-564, 1968. 0. Sonobo. A Gentzen-type Formulation of Some Intermediate Propositional Logics. Journal of Tsuda College, 7, 7-14, 1975. R. Sylvan. Intuitionistic Logic - Subsystem of, Extension of, or Rival to, Classical Logic. Philosophical Studies, 53, 147-151, 1988. M. Takano. Subformula property as a substitute for cut-elimination in modal propositional logics. Mathematica Japonica, 37, 1129-1145, 1992. N. Tennant. Perfect Validity, Entailment and Paraconsistency. Studia Logica, 43, 179-198, 1984. N. Tennant. Natural Deduction and Sequent Calculus for Intuitionistic Relevant Logic. Journal of Symbolic Logic, 52, 665-680, 1987. N. Tennant. Negation, Absurdity and Contrariety. In: D. Gabbay and H. Wansing (eds.), What is Negation?, Kluwer Academic Publishers, Dordrecht, 199-222, 1998. A. Troelstra. Arend Heyting and his Contribution to Intuitionism. Nieuw Archief voor Wiskunde, 24, 1-23, 1981. A. Troelstra and D. van Dalen. Constructivism in Mathematics, Vol. I, North-Holland, Amsterdam, 1988. A. Troelstra. Lectures on Linear Logic. CSLI Lecture Notes 29, CSLI Publications, Stanford, 1992. R. Thrner. Logics for Artificial Intelligence. Ellis Horwood , Chichester , 1984. A. Urquhart. Many-valued Logic. In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. Ill, Reidel, Dordrecht, 71116, 1986. Y. Venema. Many-Dimensional Modal Logic. PhD thesis, University of Amsterdam, Mathematical Institutue, 1991. Y. Venema. Meeting strength in substructurallogics. Studia Logica, 55, 3-32, 1995. Y. Venema. Cylindric Modal Logic. Journal of Symbolic Logic, 60, 591-623, 1995. Y. Venema. A modal logic of quantification and substitution. In: L. Czirmaz, D. Gabbay and M. de Rijke (eds.), Logic Colloquium '92, 293-309, CSLI Publications, Stanford, 1995.
175. 176. 177. 178. 179.
180. 181. 182.
183. 184. 185. 186. 187.
188. 189.
190.
191. 192.
193.
194.
245
G. Wagner. Logic Programming with Strong Negation and Inexact Predicates. Journal of Logic and Computation, 1, 835-859, 1991. G. Wagner. Vivid Logic. Know ledge-Based Reasoning with Two Kinds of Negation. Lecture Notes in AI 764, Springer-Verlag, Berlin, 1994. H. Wang. A Survey of Mathematical Logic. Science Press, Peking, 1962. H. Wansing. Functional Completeness for Subsystems of lntuitionistic Propositional Logic. Journal of Philosophical Logic, 22, 303-321, 1993. H. Wansing. Informational Interpretation of Substructural Propositional Logics. Journal of Logic, Language and Information, 2, 285-308, 1993. H. Wansing. The Logic of Information Structures. Lecture Notes in AI 681, Springer-Verlag, Berlin, 1993. H. Wansing. Sequent Calculi for Normal Modal Propositional Logics. Journal of Logic and Computation, 4, 125-142, 1994. H. Wansing. Strong Cut-Elimination for Constant Domain First-Order S5. Journal of the Interest Group in Pure and Applied Logics, 3, 797810, 1995. H. Wansing. Tarskian Structured Consequence Relations and Functional Completeness. Mathematical Logic Quarterly, 41, 73-92, 1995. H. Wansing. Semantics-based Nonmonotonic Inference. Notre Dame Journal of Formal Logic, 36, 44-54, 1995. H. Wansing. Strong cut-elimination in Display Logic. Reports on Mathematical Logic, 29, 117-131, 1995 (published 1996). H. Wansing. A new axiomatization of Kt. Bulletin of the Section of Logic, 25, 60-62, 1996. H. Wansing. A Proof-theoretic Proof of Functional Completeness for Many Modal and Tense Logics. In: H. Wansing (ed.), Proof Theory of Modal Logic, Kluwer Academic Publishers, Dordrecht, 123-136, 1996. H. Wansing (ed.). Proof Theory of Modal Logic, Dordrecht, Kluwer Academic Publishers, 1996. H. Wansing. A Full-Circle Theorem for Simple Tense Logic. In: M. de Rijke (ed.), Advances in Intensional Logic, Kluwer Academic Publishers, Dordrecht, 173-193, 1997. H. Wansing. Displaying as Temporalizing. Sequent Systems for Subintuitionistic Logics. In: S. Akama (ed.), Logic, Language and Computation, Kluwer Academic Publishers, Dordrecht, 159-178, 1997. H. Wansing. Modal tableaux based on residuation. Journal of Logic and Computation, 7, 719-731, 1997. H. Wansing. Negation as Falsity: a Reply to Tennant. In: D. Gabbay and H. Wansing (eds.), What is Negation?, Kluwer Academic Publishers, Dordrecht, 223-238, 1998. H. Wansing. Translation of Hypersequents into Display Sequents. To appear in: Journal of the Interest Group in Pure and Applied Logics, 6, 1998. H. Wansing. Predicate Logics on Display. To appear in: Studia Logica.
246 195. 196. 197. 198. 199. 200. 201.
BIBLIOGRAPHY H. Wansing. Displaying the Modal Logic of Consistency. To appear in: Journal of Symbolic Logic. H. Wansing {ed.). Two special issues on generalized sequent systems. Logic Journal of the Interest Group in Pure and Applied Logics, 1998. L. Wittgenstein. Philosophical Investigations. Blackwell, Oxford, 1953. R. W6jcicki. Dual Counterparts of Consequence Operations. Bulletin of the Section of Logic, 2, 54-57, 1973. R. Zach. Proof Theory of Finite-valued Logics. Diplomarbeit, Institut fiir Computersprachen, Technische Universitiit Wien, 1993. J.J. Zeman. Modal Logic. The Lewis-Modal Systems. Clarenden Press, Oxford, 1973. J. Zucker and R. Tragesser. The adequacy problem for inferential logic. Journal of Philosophical Logic, 7, 501-516, 1978.
INDEX
2-sequent 17 3 127, 144, 147 4DN3 233 4DN4 232f. (*-cut) 136 (*-reflexivity) 136, 143 A
adjoint functor 31, 191 Akama, S. 136 Alechina, N. 201 ancestrality 156 Andreka, H. 200 antiaxiom 115 anti-realism 7 ant-regular 56 A-scheme Avron, A. 6, 22 ff., 171 f., 174, 182, 184, 186 f.
B B 67 BCK 193 Barwise, J. 1 Basic Logic 187 Belnap, N.D. 1, 27, 34, 36 f., 40, 42, 47 f., 67, 82, 142, 173, 176, 189 f., 204 ff., 234 Benevides, M. 24 van Benthem, J.F.A.K. 14, 189 f., 193 f., 196, 200 BHK interpretation 129 ff. Birkhoff, G. 32 Blarney, S. 19 ff., 138 Bochvar, D.A. 150 Borghuis, T. 25, 211 f., 214
c C3 150, 153 f. C3d 150, 153 f. C4154
CG 145 f., 148, 153 f. CT 145, 147 f., 153 CLL 51, 55 f. CPL 5, 9, 11, 16, 18, 24, 40, 43, 46, 141 f., 213 f., 227 canonical model 166, 169 Cerrato, C. 15 f. Clarke, M. 153 classical logic 2 clause 79 Kowalski form of 79, 212 component 22, 171 compositionality 8 connective combining vs. internal 37f., 174 f. consequence relation dual138 inverse 115, 138 single-conclusion 136 *-refutation 136 structured 103 f. conservativity 13, 23, 40, 45, 55, 61, 67, 109, 118, 164, 183, 195 cons-regular 55 constituent 47, 88 constructible falsity 119 context 34 context-sensitivity 29, 33, 43, 175 contraction-elimination 83 contraction mapping 104 f. contraposition 134, 146, 149 contrapositive 33, 119 contrariety 135 correspondence 59, 78, 83, 165, 194 f., 203 f., 206, 213 corresponding structure 68 Corsi, G. 157 Curry, H.B. 4 cut4, 11,40, 136,138,172,178,204, 231
248
INDEX
INDEX
analytic 12 -elimination 12 f., 16 f., 21, 40, 82, 88, 173, 234 strong 48, 51 ff., 57, 87, 161, 187, 204 -formula 12, 48, 89 principal13, 67, 89, 131 Cut for M 104 -degree 117 elimination 105, 116 f. D DCPL 40, 46, 50, 191 DIPL 158, 160 DK 40, 43 ff., 55, 60 f., 191 DKf 76, 78 ff., 83 ff. DKt 40, 43 ff., 55, 57, 60 f., 67, 71, 73, 161 DKt(K(a)) 161 ff. DKt(K(a)) 161 ff. DK(a) 159, 161 ff. DKFOL 191 ff., 202, 206 DKFOL* 198 f., 210 DKFOL** 202 DKFOLmin 193 f. DKFOL;',.in 206 f. DKFOL;;in 206 f. DKtFOL 198 ff. DKtFOL* 210 DK U KtFOL* 198 f., 210 DL3 179 f., 184 f. DLC 183, 186 DLC U DS4.3t 183, 186 DPDL- 78 ff., 83 ff. DS4.3 183 DS4.3t 183 DS5 181, 183, 185 decidability 12, 64, 83, 193 Deduction Theorem 19, 105, 145 f. definedness ordering 147 direct propositionallogic 115 extended 115 disjunction property 119 displayable 56, 73 properly 48
display equivalence 29, 175, 229 display logic 23, 27 ff., 64 display property 36 f. Display Theorem 34 ff., 79, 176, 230 disproof interpretation 131 f. Dosen, K. 10, 16 f., 152, 156 f., 159 Dosen Principle 10 f., 14 f., 19, 21, 23, 25, 62 f. Doyle, J. 145, 150, 153 Dummett, M. 135, 172, 174, 185 Dunn, J.M. 27, 30, 32 f., 191 E e-antecedent / succedent 229 eigenconstant 90 eigenlabel 90 (even cut) 138 (even reflexivity) 138 explicitness (of a rule) 8, 11, 24, 65, 172 extension (of a label) 98
F FOL 189, 193 ff. Feys, R. 4 Finger, M. 14, 159 Fitting, M. 87, 98, 100, 206 formula Barcan 189, 205 converse 205 congruent 47, 55, 88 monolithic 160 primitive 58 f. dually 62 principal 47, 82, 88 Sahlqvist 195 sort-n 201 unwanted 128, 140 F-tableau 100 full-circle 211 f. Full Circle Theorem 226 functional completeness 65, 71 ff., 112 ff., 124 ff.
G Gt 107 f., 110 f., 113 Gt" 119 ff., 124 ff. GL3 173, 184 f. GLC 174, 186 GS5 172 ff., 185 Gabbay, D.M. 1, 13 ff., 87, 103, 105, 127 f., 134, 139 f., 142 ff., 150 ff., 159 gaggle 63 Galois connection 30, 191 dual 30, 191 GA-scheme 36 generality 13, 23 Gentzen, G. 7, 10 Gentzen sequent 4 Gentzen term 4, 27, 171 Gore, R. 14, 63, 158, 160, 181, 187, 234 Grefe, C. 64 G-tableau 64 H HL3 179 f., 185 Hacking, I. 172 height of a proof 53, 116 f. of a tableau 97 higher-arity proof system 3, 18 ff. higher-dimensional proof system 3, 17 f. higher-level proof system 3, 16 f., 66 Hilbert system 1, 11, 64 holism 8, 36, 176 Humberstone, L.l. 19 ff., 138 hypersequent 22, 171 hypersequent system 3, 22 ff., 171 ff.
IPL 127, 141 f., 144 f., 153 ff., 155 ff. IR 127 f., 133, 135, 140 ff. identification 201 identity (reflexivity) 4, 40, 136, 138, 176, 231
249
Identity 104 implication intuitionistic 128, 206 intuitionistic relevant 134 material 32 strict 157 lndrzejczak, A. 173 inductive proof 52 inductive tableau 95 information ordering 147 interpolation 187, 234 K K 5, 10 f., 16, 18, 24, 40, 43 ff., 59 f., 73, 98, 143, 187, 191, 195, 227 Kf64, 75 Kt 6, 12, 21, 33, 40, 43 ff., 57 f., 60, 62, 71 ff., 187, 200, 213 f., 226 ff. K4t 6, 12, 21 K4 5 f. K4B 6, 12 K45 6 KB 6, 12,98 KT 5, 98 KTB 6, 12, 98 KGrz 6 KD 11, 18 KDB 6, 12,98 KD4 5, 98 KD45 5 K(a) 157 ff., 165 ff. Kt(K(a)) 159 Kt(K(a))' 165 ff. KFOL 191, 193, 199 KFOL * 196 ff., 209 KtFOL 200, 207 KtFOL* 207, 209 KtFOL U KFOL* 207 Kanger, S. 88, 173 Kanger-style calculi 3 Kleene, S.C. 150 Kracht, M. 27, 35 f., 41, 43, 47, 57 ff., 61 ff.
250
INDEX
Kripke, S.A. 5, 87, 146 Kripke model 73 intuitionistic 146, 155 ff. rudimentary 156 Kuhn, S. 189, 201 ff. von Kutschera, F. 7, 109, 115, 130 L L3 172 f., 177 f., 184 LC 172, 174, 182 f., 185 f. labelled deductive system 3, 13, 87 van Lambalgen, M. 201 Lambek Calculus 105 language game 8 Leibniz, G.W. 190 Lenzen, W. 134, 143 Lopez- Escobar, E.G.K. 131 f. Lukasiewicz, J. 43, 150 Lukaszewicz, W. 145 M MPL 141 f. Maibaum, T. 24 many-valued logic 19, 173 Masini, A. 17 f. Matsumoto, K. 5 f., 12, 65 maximal ¥-consistent 167 McDermott, D. 145, 150, 153 Meyer-Viol, W. 201 mimicing structure 63 Mints, G. 27, 87, 187 modal literal 79 model structure 147 modularity 3, 7, 10 f., 14, 181, 187, 212 monotonicity 4 Montague, R. 14, 190 multiple sequent system 3 N N3 115, 127, 130, 144 ff., 149 f., 152 f. 211, 230, 234 N4 115 f., 127£., 130 ff., 134 ff., 139, 142 ff., 154, 211, 229 ff., 234 NLL 193 f.
INDEX
natural deduction 24 f., 211 f., 214 ff. negation as inconsistency 128, 139 ff. atomicity of 135, 143 Boolean 28, 157 hard 136 intuitionistic 128, 206 double 152 normal form 135 split 33 strict 157 strong, as falsity (refutation) 115, 127, 136 f., 142 f. Nelson, D. 13, 65, 115, 127 f., 130, 135, 139, 143 f., 149, 154, 211 Nelson model 149 Nemeti, I. 200 Nishimura, H. 6, 21 f. non-monotonic reasoning 103, 146 ff. normal form 35 reduced 35 Nor mal Form Theorem 36
Pottinger, G. 22, 171 Prijatelj, A. 173 principal move 49, 53, 89, 96 f. proof-theoretic semantics 7, 65, 108 f., 114, 120 f. pure 85, 92 sufficiently 92 P-scheme 36
Q QS5 87 quasi-functionality 209 R Rautenberg, W. 87 rank of a formula 82 of a proof 53 of a tableau 96 of a structure 35 (ref) 108 refutation (disproof) 115, 127 ff. relational proof system 3 replacement 71, 108, 110 f., 121 f. residual 31 f. residuated pair 30, 43, 191 dual 30, 191 residuation abstract law of 32, 63 Restall, G. 32 f., 62 ff., 134, 157 de Rijke, M. 31 Roorda, D. 57 rotation 201 rule branch extension 100 communication 174 double line 16 duality 15 export vs. import 214 invertible 42, 79 splitting 174
0 o-antecedentfsuccedent 229 occurrence negative vs. positive 34 (odd cut) 138 (odd reflexivity) 138 Ohnishi, M. 4 ff., 12, 65 p PDL- 77 f. parameter 47, 88 parametric ancestor 49, 92, 205 parametric move 49, 53 f., 56, 92, 97 partial interpretation 14 7 Pearce, D. 135 f. persistence (heridity) 13, 146, 148 ff., 155 f., 159, 166 converse 156 plausibility relation 147 positivization 135
S4.3 6, 182 f. S4Grz 6 S5 4 f., 10, 12, 16, 19, 21 f., 65, 98, 100, 172 ff., 185 SJ 157 f. Sambin, G. 11 Sato, M. 19, 173 Schroeder-Heister, P. 7, 108 Schroter, K. 18 S-degree 107, 119 Segerberg, K. 3, 10 segregation 36 f., 176 semantics independence 23 separation (of a rule) 8, 11 Serebriannikov, 0. 2 Shimura, T. 6 Shvarts, G. 5 simplicity 23 size of a proof 52 of a tableau 95 Slupecki, J. 115, 138 S-subformula 107, 119 strong completeness 45, 167 ff., 226, 233 strong normalization 51, 94 ff. strongly equivalent 135, 150 structural connective 4, 27 f. families of 27, 42, 174 f. structural rule 11, 19, 21, 61 additional 39, 76, 166, 178, 232 basic 28 f., 175, 229 external vs. internal 22, 172 structured database 104 subformula property 11 f., 22 f., 48, 161, 187 subintuitionistic logic 13, 155 ff. subordinate proof 214 substitution 48, 104 f., 195 ff. switch 201 Sylvan, R. 157 symmetry (of a rule) 8, 11
5 S4 4 f., 10, 16, 98, 154, 157, 160, 182 f., 194, 207
l
251
T TQS5 88, 97, 100
252
INDEX
Takano, M. 6, 12 Tarski, A. 189 temporal completeness 66 temporalization 159, 164 Tennant, N. 127 ff., 132 ff. three-valued logic 144 f., 147, 150, 154, 172 f. (tra) 108 -elimination 110 trace 32 Troelstra, A. 131 Thrner, R. 144 ff., 150, 152 typed .\-calculus 1, 51
u undecidability 63 f. uniqueness 10 V
Valentini, S. 11 Venema, Y. 200 Void 34
w Wittgenstein, L. 7 f. W6jcicki, R. 138