Chapter 11 REASONING AND ASPECTUAL-TEMPORAL CALCULUS Jean-Pierre Descl´es University of Paris-Sorbonne
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Chapter 11 REASONING AND ASPECTUAL-TEMPORAL CALCULUS Jean-Pierre Descl´es University of Paris-Sorbonne
In the present article, we propose a formal representation of the reasoning expressed in and by natural language sentences like: (1) The hunter has killed the deer. therefore: 1/ The deer has been killed. 2/ The deer is dead. 3/ The deer is no longer alive. 4/ The deer had been alive. (2) Peter has come out of the garage. therefore: 1/ Peter was in the garage. 2/ Peter is no longer in the garage. 3/ Peter has come outside the garage from the inside. (3) Peter is already back home. therefore: 1/ Peter was not at home sometime earlier. 2/ One could expect that Peter was not at home. (4) If Peter had been there, Mary would not have left. 1/ Since Peter was not there, Mary has left. D. Vanderveken (ed.), Logic, Thought & Action, 217–244. 2005 Springer. Printed in The Netherlands.
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(5) One more step and I will shoot. therefore: 1/ You have the intention of making one more step. 2/ I don’t shoot but I have the capacity of shooting. How can we infer the sentences (1.1), (1.2), (1.3) and (1.4) from (1)? What are the operations we must execute from the understanding of aspectual and temporal grammatical markers and from the understanding of lexical units? The same questions can be asked about (2), (3), (4) and (5).
1.
Theoretical Framework
In order to explain this kind of problem, one has to be able to build metalinguistic representations of the above sentences in a way that such inferences are automatic. The formal model of the metalinguistic representations that we choose is applicative (or functional), which means that it applies operators to different types of operands (Descl´ ´es, 1990). These applicative representations take Church’s lambda-calculus applicative formalisms with types, as well as Curry’s (1958) (see Appendix) Combinatory Logic with types. Since the above reasoning requires aspectual and temporal notions, we will use actualization intervals associated with predicates and sentence relations (that means the intervals of instants between which a predicative relation is considered as actualized or true). Indeed, the analyses of aspects and tenses that we have presented in different publications is based on topological representations (Descl´´es, 1980, 1990b, 1991, 1993; Guentcheva, ´ 1990; Maire-Reppert, Oh, Berri, 1993; Descles ´ & Guentch´´eva, 1990, 1995. . . ). Therefore, we attach topological operators interpreted on topological intervals of instants. We associate with an interval of instants two boundaries : a left boundary γ(I) and a right boundary δ(I). A boundary of a topological interval can be “open” (in this case, the boundary does not belong to the interval) or can be “closed” (in that case, the boundary belongs to the interval). An interval is closed when its left and right boundaries are closed; it is open when its left and right boundaries are open; it is semi-open when its left boundary is closed and its right boundary is open. The knowledge of lexical meanings (verbs in particular) requires knowledge of representation formalisms such as Sowa’s conceptual graphs. As far as we are concerned, we use the representations such as the semanticcognitive schemes which we have presented in several previous publications (Descl´´es, 1990a, 1994, Abraham, 1995). Each of these schemes represents the meaning of a predicate by a typed λ-expression.
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We have indicated that the applicative metalinguistic representions constitute a formalism on the basis of combinatory logic and λ-calculus. Combinatory logic with types was used for analysing grammatical problems such as passivization, reflexivization, typology of voices (Shaumyan, 1987; Descl´´es, Guentch´eva, Shaumyan, 1985,1986; Descl´es, 1990). We will argue here that this formalism is adequate to analyse the reasoning in natural languages by means of reductions (technically β-reductions). The method of this formalization is divided into several phases: 1/ Observation and analysis of linguistic data; 2/ Conceptualization by means of a concept network (for example, concerning all aspects: process, event, state, perfect, perfective, imperfective. . . ); 3/ Schematization and design of the schemes (for example, the use of semantic-cognitive schemes for the representation of predicate meanings); 4/ Mathematization of concepts, operations and intuitive schemes (for example, the use of topology and basic operations such as application); 5/ Construction of a formal language that must be adequate to formalize the intuitive conceptualizations; 6/ Interpretation of this formal language in a model (Tarski’s sense). Instead of starting from a pre-established formal language (such as, for example, Prior’s tense logic), we prefer defining and interpreting a formal metalanguage, this starting from a more or less mathematized model. This approach implies a conceptualization of intuitive notions (for example: progressivity, perfectivity, inchoativity. . . ), which we will later try to formulate in a mathematical way. Many of the logicians (such as, for example, in Montague’s approach) start from formal languages (a logic of tenses, a logic of modalities or a logic of indexical terms), and then build corresponding semantic models in order to provide to a certain extent approximations of natural languages. We take the opposite approach: first, we define the model which has already been mathematized (for example, a quasi-topological model of states, events and process or a model of speech-act operations); second, we express the concepts of the model in terms of operators of combinatory logic. This formal language is a metalanguage in the way it describes the semantics of grammatical categories of natural languages.
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The theoretical framework in which we develop the following linguistic analyses is that of the Cognitive Applicative Grammar which can be regarded on the one hand as an extension of Shaumyan’s Universal Applicative Grammar (1987) with integrations of cognitive representations, and on the other hand as formalizations of speech-act operations from the works of Benveniste (1964), Searle and Vanderveken (1985) and Culioli (1994). In the Cognitive Applicative Grammar (Descl´ ´es, 1990a), there are three different representation levels with explicit processes of change of representations from one level into another. These three levels are: (i) the level of phenotype representations — or morpho-syntactic configurations -, its task is the analysis of the morpho-syntactic data of different languages; (ii) the level of genotype representations — or logico-grammatical operations -, its task is to exhibit the invariants and the grammatical functions of language; (iii) the level of semantic-cognitive representations, its purpose is, first, the analysis and the formal representations of the meanings of lexical units, and second, the interaction of language activities with other cognitive activities of human perception and action. The change of representations from one level into another is similar to the generalized compiling process of high-level programming languages. This compiling process changes units from one representation level to another by means of synthetical “reunitarization” (definition of new units from given units) or by means of analytical “decompositons” of a unit. This device is oriented by an “intelligent” mechanism called “contextual exploration”. The purpose of this mechanism is to resolve ambiguities by locating the relevant contextual information during different stages of the process, thus orienting the process towards a decision for solving ambiguities in some grammatical units. The goal of this complex device (compiling directed by contextual exploration) is to establish an explicit relationship between abstract representations and directly observable linguistic configurations.
2.
Conceptualizations of Aspect and Tense
We recall some of the theoretical elements of the aspect-tense model that we have developed and presented in several previous publications. A predicative relation λ or “propositional content” (which is called a “lexis” by Culioli (1994)) is organized by means of predicate operations.
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These predicate operations are very well analyzed and expressed in an applicative formalism. We can consider the following applicative expression with prefixation of the predicate operator: (*)
“to see” “a-deer” “the-hunter”
This applicative expression is obtained by means of two successive applictive operations. First, the predicate “to see” is an operator that applies to a first operand “deer”; the result is a new operator that applies to a second operand “hunter”. The result is the predicative relation (*). The building of this predicative relation is represented as follows: “to see” “to see” “a-deer”
“a-deer” > “the-hunter” >
“to see” “a-deer” “the-hunter”
Such a predicative relation is tenseless. Before inserting it in a reference space, the speaking subject perceives it in different ways, depending on whether he views it as a progressive process, an event or a resultative state. Thus the subject constructs an aspectualized predicative relation (Descl´´es, 1991) which is considered as true on a topological interval of instants I. We note this aspectual predicative relation as follows: (**)
ASP PI (“to see” “a-deer” “the-hunter”)
where the aspectual operator ASP applies to the predicative relation (*). According to whether the aspect is a state, an event or a process, the interval I will be respectively open, closed or semi-open. We say that the state, the event or the process is true (or actualized) on this topological interval I. Having aspectualized the predicative relation, the subject now has to insert it into his own time reference that is distinct from the external time reference (the clock time, the cosmic time, the calendar time. . . ). The subject will or will not consider the aspectualized predicative relation as concomitant with his own process of speaking, that cannot be reduced to a punctual instant because each process of speaking takes time. In this reference, T0 denotes “this first non-actualized instant”, which means the right boundary (unfinished) of the speaking process in progress (Descl´´es, 1980, 1990b, 1994; Descl´ ´es & Guentch´eva, 1990, 1995). We have therefore several sentences with the same propositional content (*), for example: (a) The hunter is looking at the deer at this moment (Unfinished) process concomitant with the speaking process in progress
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(b) Finally; the hunter has seen the deer, he is happy Resultative state of a finished anterior process (c) While the hunter was looking at the deer (Unfinished) process non concomitant with the speaking process in progress (d) The hunter saw the deer, and then he Event inserted into a series of events With topological representations (which means the diagrams of the actualization of the aspectual predicative relation (**) in accordance with a choice of different aspectual and temporal scope), we will have respectively the differents temporal diagrams given in figure 1. (a’) (b’)
(c’) (d’)
-------------[ [T0 [T0 --[----------] [T - - - [T0 --[ ] - - - - - - - - - - [T0 --[ Figure 1.
In a general way, an aspectualized predicative relation ASP I (Λ) will be actualized (or realized) on an interval I. If this relation is aspectualized as a state, or respectively as an event or as an unfinished process, it will be true respectively on an open interval O, on a closed interval F or on an interval J closed at the left boundary and open at the right boundary (Descl´´es 1990b). When I designates an interval (open or closed or semiopen), γ(I) and δ(I), the respective bounds are the left or the right of the interval I. If an interval is open, then the two bounds (at left and at right) do not belong to the interval. If the interval is closed, then the two bounds (at left and at right) belong to the interval. We have thus the three diagramatic representations of realization intervals according to the aspects of the predicative relations: a state, an event and an (unfinished) process. We have three more specific aspectual operators STATE, EVENT and PROC. C The intervals, with topological boundaries, where the aspectualised predicative relations are realized are shown by temporal diagrams (see figure 2): Remark: We consider (Descl´´es, 1980, 1995; Decl´´es & Guentcheve, 1995), the trichotomy state / event / process essential in the analy-
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Reasoning and Aspectual-Temporal Calculus ]
[ STATEO (Λ)
[ ] EVENT F (Λ)
[
[ PROC J (Λ)
Figure 2.
ses of aspects in natural languages (see also: Lyons (1977), Comrie (1979) or Mourelatos (1981)). Note that this trichotomy is different from Vendler’s classification of “state”, “activity”, “accomplisment” and “achievement”. Some authors (for instance: Smith (1991), KoceskaToscewa and Mazurkiewicz (1994), Karolak (1997), Kamp (1981, 1983), Vet (1995)) claim that the dichotomy state / event (punctual) is sufficient. However, several arguments against this theoretical viewpoint have been given (see Descl´´es and Guentch´eva, 1995)). We formulate different rules about states, events and processes. 1/ RULE on STATE: IF a state STATE O (Λ) relating to a predicative relation Λ is true on an open interval O, THEN for each sub-interval O’ of O, the state relating to the same predicative relation Λ remains true over O’: ST AT EO (Λ) & [O ⊃ O ] ⇒ ST AT EO (Λ) Remark: One finds a similar rule in Dowty (1979). 2/ RULE on EVENT: IF an event EVEN F (Λ) relating to a predicative relation Λ is true, THEN the predicative relation Λ is true at the right boundary δ(F): EV EN NF (Λ) ⇒ (Λ)δ(F) In general, an event is actualized only at the right boundary δ(F), which is the final boundary of the event actualized on a closed interval F. The same event relating to the same predicative relation Λ is not actualized at the right boundary of each of the sub-interval F’ of F, but only at δ(F). An event will only be true if the final instant δ(F) is reached. For example, the sentence John wrote a letter in one hour is a predicative relation which is true at its right boundary (final boundary) of a closed interval F, while this predicative relation is not true in a closed subinterval F’ of F. However, in some cases, an aspectualized predicative relation Λ, viewed as an event, can be true both for a closed interval F and for each sub-interval F’ of F. An example of this case is: John ran in the park yesterday afternoon.
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3/ RULE on PROCESS: IF an unfinished process PROC J (Λ) relating to a predicative relation Λ is true on a semi-open interval J, THEN for each semi-open interval J’ with the same beginning of J (γ(J) = γ(J’)), the predicative process PROC J (Λ) relating to the same predicative relation remains true for J’: CJ (Λ) P ROC CJ (Λ) & [J ⊃ J ] ⇒ P ROC Remark: for the justification of this rule, see Descl´ ´es and Guentch´eva (1995). 4/ RULE on PROCESS: IF an unfinished process PROC J (Λ) relating to a predicative relation Λ is true on an interval J, THEN when the process is finished (in French: “achev´ ´e”), it generates: (i) an event EVEN F (Λ) (relating to the same predicative relation Λ); this event is realized on a closed interval F which includes the smaller closed interval cl(J) including J, called the closure of J; (ii) a resultative state RES-STATE O (Λ) (relating to the same predicative relation Λ) which is true on an open O that is posterior and adjacent to the interval F: PROC J (Λ) ⇒ there is an EVEN F (Λ) and a RES-STATE O (Λ) such that [F ⊃ cl(J)] & [O is adjacent to F and posterior to F]
We represent this rule with the diagram of the figure 3. [ - - - - - - - - J - - - - - - - - - - >[ [ - - - - - - - - - cl(J) - - - - - - - - ] [ - - - - - - - - - - - F - - - - - - - - - - - - ]< - - - - - - - - O - - - - - - - - >[ Figure 3.
Remark: This rule is designed to capture the meaning of “perfect” in Indo-European languages (see Descl´´es, 1980; Guentch´eva, 1990).
3.
Analysis of Tenses and Aspects Let us start with the following sentence: (6)
The hunter is looking at the deer (at this moment).
The underlying logical form of the sentence is analyzed in the following steps: (i) Formation of an underlying predicative relation expressed by an applicative notation: ((“to see” “a-deer”) “the-hunter”) ; (ii) Aspectualization of this predicative relation as an unfinished process in
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progress, with the aspectual process operator PROC ; (iii) Inclusion of this process, called predicative process, into the speaking process in progress; (iv) Establishment of a concomitance relation between the unfinished speaking process and the unfinished predicative process. Generally, a predicative relation Λ that is viewed as an unfinished process is true on an interval J1 of instants, hence the process operator PROC J 1 and the predicative process PROC J 1 (Λ). This predicative process becomes an argument of the metalinguistic speaking predicate “SAY(. . . )S 0 ”, where the symbol S 0 denotes the abstract subject “EGO” of any speaking act: EGO is the origin of the system of persons (I, YOU, HE / SHE) as well as deictic spatial markers as HERE, THERE. The speaking process is true on the interval J0 of instants, hence the aspectual operator PROC J 0 and the following speech-act scheme (see Descl´´es, 1980): (***)
P ROCJ0 (SAY(. . .)S 0 )
Now we can express explicitly the underlying logical form of sentence (6). To simplify the notations, we use the symbol P2 to designate the transitive lexical predicate associated to the verb “to see” and respectively the symbols T1 and T2 for the terms “the hunter” and “the deer”. Formula (7) is an aspectual-tense representation of (6), expressed by a prefixed applicative notation: (7)
& (P ROCJ0 (SAY(P ROCJ1 (P2 T2 T1 ))S 0 ) ([δ(J1 ) = δ(J0 )])
This formula is a conjunction of the two constituents (7a) and (7b): (7a)
(P ROCJ0 (SAY(P ROCJ1 (P2 T2 T1 ))S 0 )([δ(J1 ) = δ(J0 )])
(7b)
[δ(J1 ) = δ(J0 )]
The predicative process PROC J 1 (P2 T2 T1 ) is in this way considered as being true on the interval J1 . Formula (7a) represents the embedding of the predicative process PROC J 1 (P2 T2 T1 ) into the speech-act process. Formula (7b) expresses a temporal constraint on the intervals of actualization of the two processes: In an unfinished process concomitant with the speech-act, the right boundaries δ(J1 ) of J1 and δ(J0 ) of J0 must be identical. Formula (7) can be read as follows: (7’)
“The speaking process “S 0 SAY. . . ”, which is true on an interval J0 , has as its argument a predicative process “P2 T2 T1 ” which is true on an interval J1 , with the temporal constraint that the two right boundaries of the two intervals J0 and J1 must be identical”
or, in others words:
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LOGIC, THOUGHT AND ACTION (7”)
“the predicative relation “P2 T2 T1 ” is conceived from an aspectual point of view as an unfinished process which is an argument of the speaking process “S 0 SAY that. . . ”, the final sections of the two processes are identical.”
Now, we are going to define an aspectual operator by integrating the two elementary process operators PROC J 0 and PROC J 1. into a complex operator. The predicative relation “P2 T2 T1 ” becomes an argument of this complex operator. With this aim, we use the combinators B, B2 and C of the combinatory logic (see Appendix). We present the integration as in Gentzen’s “natural deduction” style but with “reunitarization” relations (or definitions of new units from an applicative combination of differents more primitives units). Thus, in the following integration process, step 9. expresses a reunitarization relation: the complex operator SA is defined in terms of the operator SAY and the operand S 0 .
Integration of the operator “Speech-act” (SA) The symbol SA designates, in the following integrative process, an operator which means that “S 0 performs a speech-act”. We call it an “enunciative operator”. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
P ROCJ0 (SAY(P ROCJ1 (P2 T2 T1 ))S 0 ) B2 P ROCJ SAY(P ROCJ1 (P2 T2 T1 ))S 0 C(B2 P ROCJ0 SAY)S 0 (P ROCJ1 (P2 T2 T1 )) C(CB2 SAYP ROCJ0 )S 0 (P ROCJ1 (P2 T2 T1 )) BC(CB2 SAY)P ROCJ0 S 0 (P ROCJ1 (P2 T2 T1 )) B(BC)(CB2 )SAYP ROCJ0 S 0 (P ROCJ1 (P2 T2 T1 )) C(B(BC)(CB2 )SAY)S 0 P ROCJ0 (P ROCJ1 (P2 T2 T1 )) B(C(B(BC)(CB2 ))SAYS 0 P ROCJ0 (P ROCJ1 (P2 T2 T1 )) [SA =def B(C(B(BC)(CB2 )SAYS 0 ] SAP ROCJ0 (P ROCJ1 (P2 T2 T1 )) B(SAP ROCJ0 )P ROCJ1 (P2 T2 T1 ) BBSAP ROCJ0 P ROCJ1 (P2 T2 T1 ) (SA0P ROCJ0 0P ROCJ1 )(P2 T2 T1 )
hyp intB2 intC intC intB intB intC intB def of SA repl, 8, 9 intB intB def of 0, 12
Comments: In step 1., it is shown that the predicative process is being embedded into the speech-act process; the first process is true on one interval J1 and the second is true on another J0 . Steps 2. to 8. introduce the combinators B, B2 and C which allow a combination of the speaking operators, hence the definition introduced in step 9. of the operator SA which means “the speaker performs a speaking-act” or “the speaker S 0 says that. . . ”. We deduce from the definition given at step 9., the expression given at step 10. by remplacement of the definiens. By combining the operators SA, PROC J 0 and PROC J 1 , we obtain a new complex operator (at step 12. or, equivalently, at step 13.) whose operand is P2 T2 T1 . The operators SA, PROC J 0 and PROC J 1
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are combined as compositions of functions. From this integrative process it follows that step 13 is an applicative integration of step 1. with the same meaning. The expression at step 13. is deduced from the expression given at step 1. At step 13., it is shown that the complex operator, “(SA 0 PROC J 0 0 PROC J 1 )” is a grammatical operator whose meaning is only aspectual. This applies to the predicative relation “(P2 T2 T1 )”. The two formulas in step 1. and step 13. are considered as being equivalent to the same aspectual meaning.
In the above deduction, we introduce a derived operator SA. Its meaning is: “the speaking subject S 0 says that”. We call it an “enunciative operator”. This enunciative operator is built by means of an application of the abstract operator (a derived abstract combinator) to the elementary operator (saying) and to the operand S 0 . Its formal definition is given by the following relation between a definiendum (at the left) and a definiens (at the right): (8)
[SA =def B(C(B(BC)(CB2 )SAYS 0 ]
At the end of the above reduction, the applicative expression obtained at step 13. represents the result of the application of a complex aspectual operator on the predicative relation (P2 T2 T1 ), that is to say: (9)
(SA 0 P ROCJ0 0 P ROCJ1 )(P2 T2 T1 )
Now, we return to the expression (6) again. We substitute in (7) the expression (7a) by (9), since the two expressions (7a) and (9) are equivalent to the same aspectual meaning, hence the expression (10): (10)
& ((SA 0 P ROCJ0 0 P ROCJ1 )(P2 T2 T1 ))([δ(1J ) = δ(0J )])
Thus, we can continue the reduction process in such a way as we define an integrated aspectual operator considered as a reunitarized operator. The operand of the first aspectual operator is a predicative relation but the operand of the second aspectual operator is a lexical predicate.
Integrative process of the verbal aspectual operator. 1. 2. 3. 4. 5. 6. 7. 8. 9.
&((SA 0 P ROCJ0 0 P ROCJ1 )(P2 T2 T1 )) ([δ(J1 ) = δ(J0 )]) C&([δ(J1 ) = δ(J0 )]) ((SA 0 P ROCJ0 0 P ROCJ1 )(P2 T2 T1 )) B(C&([δ(J1 ) = δ(J0 )]) (SA 0 P ROCJ0 0 P ROCJ1 ) (P2 T2 T1 ) (C&([δ(J1 ) = δ(J0 )])) 0 (SA 0 P ROCJ0 0 P ROCJ1 )(P2 T2 T1 ) [UNF-PRST =def (C&([δ(J1 ) = δ(J0 )])) 0 (SA0P ROCJ0 0 P ROCJ1 )] UNF-PRST (P2 T2 T1 ) B2 UNF-PRST P2 T 2 T 1 [prog-prest =def B2 UNF-PRST] T (prog-prest P2 ) T2 T1
hyp intC intB def 0 def from 5 int B2 def from 8
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Comments: step 1. has exactly the same meaning as in expression (2). The temporal constraints have been added to the proper aspectual conditions. We rearrange the operators so as we can isolate the predicative relation (steps 2 to 4). At step 4. the two operators are combined as two composed functions. Then we introduce the definition of a grammaticalized aspectual-temporal operator “Unfinished-Present”, designated by UNF-PRST at step 5. At step 6., we isolate the two arguments of the predicative relation so that we can define an operator which applies only to the predicate, hence the definition of a verbal progressive present operator designated as prog-prest at step 8. and in the final expression at step 9. The operator prog-prest is the morphological trace of the deep grammatical operator UNF-PRST which (i) encodes the temporal constraints (that two intervals of actualization have the same right boundary) and also (ii) combines the two process operators with the enunciative predicate SA. Now, the predicate P2 becomes an argument of the morphological operator prog-prest, hence the “aspectualized and tensed” new predicate “prog-prest P2 ” derived from the lexical predicate P2 .
Finally, we obtain the expressions (11) and (12) which are equivalent to the expression (7); these two expressions have the same meaning as the expression (7): (11)
UNF-PRST(P2 T2 T1 )
(12)
(prog-prestP2 )T2 T1
We remark that the grammatical operator UNF-PRST takes the entire predicative relation (P2 T2 T1 ) as its operand whereas the verbal progressive present operator prog-prest takes only the lexical predicate P2 as its operand. Now we look at the sentence (6) The hunter is looking at a deer (at this moment). This sentence can be analyzed by means of the prefixed applicative expression (6’): (6’)
is-looking-at a-deer the-hunter
The verbal operator is-looking-at can be analyzed as a complex binary predicate, derived from a lexical predicate “look-at” by means of the morphological operator “progressive present” prog-prest. Thus we have the following definition: (6”)
[is-looking-at = prog-prest look-at]
From this definition we can deduce relations between a linguistic configuration The hunter is looking at a deer expressed at the phenotype level and the corresponding applicative expressions:
Reasoning and Aspectual-Temporal Calculus (6”’)
The hunter is looking at a deer
(6”’)
= is-looking-at a-deer the-hunter
(6”’)
= prog-prest (look-at) a-deer the-hunter
229
By a similar calculus but in a bottom-up way, we get the following reduction of an applicative expression with a verbal aspectual operator applied to a lexical predicate into an underlying applicative expression which describes the grammatical meaning of the verbal aspectual operator: 1. 2. 3. 4. 5.
(prog-prestP2 )T2 T1 [ [prog -prest = B2 UNF-PRST] T [UNF-PRST =def (C&([δ(J1 ) = δ(J0 )])) 0 (SA 0 P ROCJ0 0 P ROCJ1 )] &((SA 0 P ROCJ0 0 P ROCJ1 )(P2 B2 A1 )) ([δ(J1 ) = δ(J0 )]) &(P ROCJ0 (SAY(P ROCJ1 (P2 T2 T1 ))S 0 ) ([δ(J1 ) = δ(J0 )])
The expression obtained at step 5. is considered as the normal form of the expression given in step 1. These applicative expressions are obtained by successive reductions (in technical terms β-reductions), that is to say, successive eliminations of combinators and replacement by means of definition relations of complex operators. We use the symbol ‘β →’ to represent the relation of reduction between applicative expressions; we obtain: (13) (pro-prest P2 )T2 T1 β → &(P ROCJ0 (SAY(P ROCJ1 (P2 T2 T1 ))S S0 )([δ(J1 ) = δ(J0 )])
In replacing P2, T2 , T1 by their corresponding lexical units, we obtain: (14)(pro-prest look-at) a-deer the-hunter 0 1 0 1 0 β → &(UNF-PROCJ (SAY(UNF-PROCJ (look-at a-deer the-hunter))S )([δ(J ) = δ(J )])
4.
Formal Calculus on Aspectual-temporal Conditions Let us examine sentence (15): (15)
The hunter has seen a deer
(the aspectual value of the present perfect tense is a resultative state) By an analogous process, we have the following reduction: (16) prest-perffresult see) a-deer the-hunter 0 T1O (to see a-deer the-hunter))S 0 )([δ(O1 ) = δ(J0 )]) β → &(P ROCJ (SAY(RESU-PRST
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In the underlying normal form of the sentence, the tenseless predicative relation “to see a-deer the-hunter” (or in the infixed notation: “the-hunter to see a-deer”) is viewed by the speaking subject as a resultative state which is actualized on an interval O1 concomitant with the speech-act process. We have the temporal constraint: [δ(O1 ) = δ(J0 )]. The present resultative state RESU-PRST TO 1 (to see a-deer the-hunter) is adjacent to the occurrence of the event EVEN F 2 (see the-hunter a-deer) and is concomitant with the speaking act. In other words, the event EVEN F 2 (to see the-hunter a-deer) has an occurrence which is located before the speech-act process actualized on the interval J0 . The reulting state is true on the interval O1 ; this open interval O1 is adjacent to the closed interval F2 and is located after F2 ; the two right bounds of O1 and J0 are identical. In order to formalize this resultativity and to relate it to the interval J0 of actualization of the speech-act process, it is necessary to add further conditions. Let Λ be an arbitrary predicative relation which is considered as resultative state on the interval O; the “present resultative state of Λ is then defined as follows (see Descl´´es, 1980): (17)
RESU-PRST O (Λ) ⇔def there exists EV EN NF (Λ) such as:
(i) δ(F) is a “continuous cut” (in Dedekind’s sense) in the union of the closed interval F and the open interval O, hence: δ(F) = γ(O); (ii) [δ(O)F < J0 ] (the interval F is before the instant J0 ) (iii) δ(O) = δ(J0 ). We represent this continuous cut δ(F) by means of the diagram given in figure 4: < - - - - - - - - - - F - - - - - - - - - - >< - - - - - - - - - - - O - - - - - - - - - - - > ] [ [ δ(F) =γ(O) Figure 4.
The continuous cut δ(F) (in Dedekind’s sense) means that the two intervals F and O are disjoint: the right boundary δ(F) of the closed interval F is identical with the left boundary γ(O) of the interval O; since the two intervals F and O are topological intervals it follows that the interval F is necessarily closed and O is necessarily open. In example (15), we have the diagram (figure 5) with intervals corresponding to the realizations of the event (a) “the hunter saw a deer” and the resultative state (b) “the hunter has seen a deer”.
Reasoning and Aspectual-Temporal Calculus <Event [
F
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(to see a-deer the-hunter)> ] [ δ (F) =γ(O) Figure 5.
The definition of the operator RESU-PRST is more complex than the definition of UNF-PRST. T The reduction of the applicative representation to its normal form is as follows: (18)
(prest-perf-result P2 )T 2 T 1 →β &{&(P ROCJ0 (SAY (RESU-PRST1O (P P2 T 2 T 1 ))S 0 )([δ(O1 ) = δ(J 0 )]))} P2 T 2 T 1 ))([F 2 < J 0 ])([δ(F 2 ) = γ(O1 )])([δ(O1 ) = δ(J0 )]))} {&(EVEN2F (P
Now, look at the sentence (19): (19) (Yesterday), the hunter was looking at the deer. (with the value of the progressive past tense: past unfinished process)
With an analogous calculus, we get the following reduction: (20)
(pro-pastt-proc P2 )T 2 T 1 →β &(P ROCJ0 (SAY(P ROCJ1 (P P2 T 2 T 1 ))S 0 )([δ(1J )