Nomological Statements and Admissible Operations

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NOMOLOGICAL STATEMENTS STATEMENTS NOMOLOGICAL AND AND

ADMISSIBLE OPERATIONS OPERATIONS ADMISSIBLE

HANS REICHENBACH HANS REICHENBACH Pro/es8or Professor 0/ of Philo8ophy Philosophy in the the University University o/ of CaU/ornia, California, Los Los Angeles Angelm

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I INTRODUCTION INTRODUCTION

The problem problem of of aa ‘reasonable’ 'reasonable' implication The implication has has frequently frequently occupied occupied

logicians. Whereas in in conversational language this this kind kind of logicians. Whereas conversational language of propro-

positional operationisis regarded regardedas as having having a clear positional operation clear and and wellwelldefined meaning, logicians logicianshave have been been compelled to define define as as defined meaning, compelled to

implication aa term term of much wider wider meaning meaning;;and and it it appears appears extremely extremely implication of much difficult to go go back back from from this this implication in a wider to the difficult to implication in wider sense to the narrower and assumed for for implication implication in in aa narrower and very very specific specific meaning meaning assumed non-formalizedlanguage. language.We We face face here here aa discrepancy non-formalized discrepancy between between usage and and rules: whereas in in actual usage is quite able to usage everyone is to say whether an implication is reasonable, he would be at a loss to say whether an implication is reasonable, he would be at a loss to give rules which give which distinguish distinguish reasonable reasonable implications implications from from ununreasonable ones. ones. The The term term 'reasonable', to reasonable ‘reasonable’, therefore, therefore, is is aa challenge challenge to the logician logician for finding finding rules rules delineating delineating a usage usage that thatfollows follows unconscious rules. unconscious rules. The problem uncovering such such rules rules appears appears even The problem of of uncovering even more more difficult difficult when it is when it is realized realized that that 'reasonable' ‘reasonable’implications implications of of conversational conversational language are are not restricted language restricted to toimplications implications expressing expressing aa logical logical entailment. entailment, but include include what what may be called entailment, but called a physical entailment. For For the the first first kind, kind, we we may may use use as as an anillustration illustration the theimplication, implication, 'if all ‘if all men men are are mortal mortal and and Socrates Socrates is aa man, man, then then Socrates Socrates is is mortal'. may be be ifiustrated illustrated by by the theimplication, implication, mortal’. The The second second kind may 'if metal is is heated, heated, it it expands'. the latter latter kind ‘if aa metal expands’. Since Since the kind of of implication implication expresses what what is is called called aa law law of of nature, nature, whereas the the former may expresses be be said to to express express aa law law of of logic, logic, II have have proposed proposed to to include include both both kinds under the name of nomological implications. under the name of nornological implications. It It isiseasily easily seen seen that thatthe theproblem problemunder under consideration consideration is not specific for for implication, implication, but but concerns specific concerns all all propositional propositional operations operations alike. The The ‘or’, 'or', for instance, can have an an 'unreasonable' as aa alike. for instance, can have ‘unreasonable’as as well well as 'reasonable' meaning.To To say, say, ‘snow 'snow isis white white or or sugar is sour', ‘reasonable’ meaning. sour’, appears as unreasonable as saying saying ‘if 'if snow is not not white, unreasonable as snow is white, sugar sugar is is

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NTRODUCTfON

sour'; sow’; but but both both statements statements are are true truein inthe thesense senseof of the theoperations operations

in the the truth truth tables reasonable ‘or’ 'or' would defined in tcblesof of symbolic symboliclogic. A reasonable given in in the the Statement statement:: 'there rain in in the winter or be given ‘thereis is sufficient sufficient rain there is aa drought drought in in the thesummer', summer’,an anexclusive exclusivedisjunction disjunction which which for many many aacountry countryexpresses expressesaaconsequential consequential alternative. alternative. Since Since the operation operation is is made made reasonable reasonable by by the the compound compound statement statement whose major major operation operation itit is and which whose which expresses expresses aa law of nature or or of of logic, logic, we we face facehere herethe thegeneral generalproblem problemofofnomological nornological8tatestuteinto the two ments, class of of statements statements which which subdivides subdivides into two subsubments, a class classes of analytic classes of analytic and and synthetic synthetic nomological nomological statements. statements. The The statement confers a certain prerogative statement prerogative upon its its major major operation, operation, which may may be operation. It It will which be called called aa nomological nornological operation. will be be seen, seen, howhowever, that that the to supply ever, the operations operations so so defined defhed are are still still too too general general to supply 'reasonable' propositional propositionaloperations, operations,and and that that such ‘reasonable’ such operations operations must be must be defined defined as as aasubclass subclassofofnomological nornological operations. operations. Based in an earlier Based on these considerations, considerations, II have developed developed in earlier presentation 11 aa theory of statements. Since we are presentation of nomological nomological statements. are here concerned with an an explication of here concerned with of aa term, term, i.e., i.e., with with constructing constructing term proposed proposed to to take take over the functions vague term, term, aa precise precise term over the functions of aa vague we cannot cannot expect expect to arrive we arrive at at results resultswhich which cover cover the the usage usage of of the the vague term term without without exceptions; exceptions;ifif only onlyfor forthe the reason reasonthat that the vague vague term term is differently by different different persons. persons. All Allthat that can can be be vague is used used differently achieved, therefore, constructing aa formal is constructing achieved, therefore, is formal definition definition which which correspondstotothe the usage usageofofthe the vague vagueterm term at at least least in in aa high corresponds high percentage of of cases. cases.For For this this reason, reason, II thank thank those of my percentage my critics critics who have drawn drawn my my attention attention to who have to cases cases where where there there appears appears to to be be aa discrepancy explicans and explicandurn, ifif these are discrepancy between between explicans and explicandum, these terms terms are used to denote used to denote the the precise precise term term and and the the vague vague term, term, respectively. respectively. To their criticisms, criticisms, II added added my my own own and and found found more more such such disdisTo their crepancies. present monograph, monograph, II wish wish to develop an crepancies. In In the the present to develop an improved improved definition nature and andreasonable reasonable operations, operations, hoping hoping definition of of laws laws of of nature that the the percentage percentage of of cases cases of of adequate adequate interpretation interpretation is thus thus increased. the remaining remaining cases, cases, my my definition definition may may be be regarded regarded increased. For For the as should be be glad as aa proposal proposal for for future future usage usage of of the the term, term, and and II should glad if if it it In my book, book,Element3 Elements of Symbolic Logic, In my Log'ic,New NewYork, York, 1947, 1947,ohap. chap. VIII. VIII. This book book will will be be quoted as ESL. This ESL.

INTBODTYCTION INTRODUCTION

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appears possible to adjust adjust one's to the appears possible to one’sown own usage usage to the proposed proposed definition definition without sacrificing without sacrificing essential essential connotations. connotations. In its its fundamental fundamental idea, idea, the thenew new theory theory corresponds corresponds to the the old oId one; and II will here a short summary one; will therefore therefore give give here summary of of the older older theory insofar insofar as as it is taken theory taken over over into into the thepresent present one. one. The The truth truth tables tablesofofsymbolic symboliclogic logicrepresent representmetalinguistic metalinguistic statements expressing relations between between compound compound statements statements of statements expressing relations these the object language language and their elementary elementary statements. statements. Now Now these tables can be read tables read in in two two directions. directions. Going Going from the compound compound statement to the elementary statements, we we read read the the tables tables as a statement elementary statements, disjunction of disjunction of T-cases, T-cases, for for instance, instance, as asfollows: follows: b' is is true, true, then then ‘a’ 'a' is true and 'b' First F i r s t direction. d i r e c t i o n . If 'a ‘aD 3 b’ ‘b’ is is true, or 'a' ‘a’isisfalse false and and 'b' ‘b’is is true, true, or or 'a' ‘a’isisfalse false and and 'b' ‘b’isisfalse. false. Going from from the the elementary elementary statements statements to the the compound compound stateGoing ment, we ment, we read the the tables tables as asfollows: follows: If 'a' true, then Second ‘a’ is is true and 'b' ‘b’ is true, then 'a ‘a I)b’ Second direction. If is true. b' is is true. true. If 'a' true. If 'a' ‘a’isis false false and 'b' ‘b’ is is true, true, then then 'a ‘aD 3 by ‘u’isis false and 'b' false and ‘b’ is is false, false, then then 'a‘aD 3 b' b’ is is true. true. IIn n the the interpretation interpretation assumed assumed for for mathematical mathematical logic, logic, both both directions of reading are used. I speak here of an adjunctive interdirections of reading are used. I of an adjunctive pretation of correspondingly, of the the truth truthtables tablesand, and, correspondingly,ofofadjunctive adjunctive operations. to omit the possible, however, however, to the second second direction direction operations.1 It is possible, of the truth tables of reading reading the tables and and to to use use only only the thefirst firstdirection. direction. II then interpretation of of the the truth truth tables then speak speak of of aa connective connective interpretation tables and, and, correspondingly, of connective operations. correspondingly, of connective operations. It It is is important important to to realize realize that, that, for for all all'reasonable' ‘reasonabIe’operations operations of conversationallanguage, language,the thetruth truth tables tables are are adequate adequate if we read conversational them only in in the first them first direction, direction, i.e., interpret these these operations operations as as connective. Deviations Deviations from from a reasonable usage occur only when, connective. in addition, addition, the tables tables are are read read in in the the second second direction. direction. In other other words, reasonable reasonable operations operationsare are not not adjunctive, but connective. words, connective. For instance, implication: ‘If 'If a large instance, consider consider the reasonable reasonable implication: large sun spot spot turns turns up on the sun the day day of of the the concert, concert, the the short-wave short-wave radio to such The term 'adjunctive' ‘adjunctive’ corresponds corresponds to such terms terms as as'extensional', ‘extensional’, The 'truth-functional', 'material', which which have have been been used of ‘truth-functional’, ‘material’, used in presentations presentations of 11

logic. But But since since these these terms terms are are often often used used in in various various meanings, meanings,II prefer prefer to to logic. use the precisely precisely defined defined term 'adjunctive'. ‘adjunctive’.

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transmission of the concert will be seriously disturbed'. When we transmission of seriously disturbed’.

regard this statement regard this statement as true, before before the concert concert is given, given, we we

shall be quite willing to admit that willing to that any anyof ofthe thethree threepossible possiblecases cases stated stated for for the thefirst firstdirection directionmay mayoccur. occur.However, However,we wewould would refuse refuse

to regard if, say, no no sun spot turns up regard the the statement statement as as verified verified if, and the of the concert is not disthe short-wave short-wave radio-transmission radio-transmission of concert is dis-

turbed; to regard regard the implication turbed; and andwe we would would not even even be be willing to as verified even ifif aa sun sun spot turns verified even turns up up and andthe theradio radiotransmission transmission is disturbed, unless unless further further evidence evidence for for aa causal causal relation relation between between the the two two phenomena phenomena isisadduced. adduced.11 This This means means that we we use use here a connective implication, implication,but but not not an adjunctive connective adjunctive implication. implication. Similar examples are easily examples are easily given given for for the theother otherpropositional propositionaloperations. operations. It follows cannot be It follows that aadefinition definition of of reasonable reasonable operations operations cannot achieved by by changing the truth truth tables. changing the tables. These These tables tables are are adequate; adequate; however, we we have have to to renounce the the use of the the second second direction direction for for reading the tables. reading the tables. This This program program can can be be carried carried out out as as follows. follows. We We define connective operations operations as as a subclass define connective subclass of of the thecorresponding corresponding adjunctive operations. operations. Then, whenever whenever a connective connective operation operation is is adjunctive operation operation isis also alsotrue, true, and and the true, the the corresponding corresponding adjunctive use of the the first first direction direction of of reading reading the the truth truth tables use of tables is is thus thus assured. assured. However, However, the the second second direction direction is is excluded, excluded, because because aa verification verification

of compound connective connective statements statements requires than a verification of compound requires more more than verification of the the corresponding adjunctive statement. statement. IInn other of corresponding adjunctive other words, words, satissatisfying for an adjunctive fying the requirement requirement for adjunctive operation operation is merely merely aa necessary, not a sufficient for the verification necessary, not sufficient condition condition for verification of of the the corresponding connective connective operation. corresponding operation. Connective operations operations will be defined operations, Connective w i l l be defined as as nomological nomological operations, i.e., as major operations statements. In i.e., operations of of nomological nomological statements. I n a;Inomolognomological statement, all operations are are used, used, first, first, in in the ical all propositional propositional operations adjunctive sense; sense; i.e., i.e., the the statement statement must must be true in an an adjunctive adjunctive interpretation.But But in in addition, the statement has to satisfy interpretation. addition, the statement has satisfy certain another kind. certain requirements requirements of another kind. The The introduction introduction of of suitable suitable 1

The case case that that 'a' The ‘a’is is true true and and'b' ‘b’isistrue trueisissometimes sometimesregarded regarded as aa verifying verifying

a reasonable however, ae as insufficient for a verireasonable implication, implication, sometimes, sometimes, however, insufficient for fication. If If this case is regarded as verifying the implication, implication, II speak speak of a regarded aa verifying the fication. semi-adjunctive ESL, §5 64. 64. serni-adjunctive implication. implication. See See ESL,

INTRODUCTION

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requirements requirements of this kind constitutes constitutes the the problem problem of of the the present present investigation. investigation. As As far as asanalytic analyticnomological nomological statements statements are are concerned, concerned, the the method outlined outlined here here has has found found an an application application in in Carnap’s Carnap's theory theory method of analytic implication. implication. Carnap Carnap has has pointed pointed out out that ifif an of an implicimplication stands in the place ation place of the major major operation operation of of aa tautology tautology or or analytic statement, itit can analytic statement, can be be regarded regarded as as an an explicans explicans for for the the relation of entailment. This will be be taken relation of logical logical entailment. This conception conception will taken over over into the present present theory. theory. However, However, what is to to be be added added is is aa correscorresponding definition definitionfor for physical physical entailment, entailment,and and with with it, ponding it, quite quite generally, for for synthetic nomological statements. Furthermore, Furthermore, it generally, nomological statements. it will be shown, as mentioned above, above, that that the will theclass classof ofnomological nomological operations is still too too wide wide to to supply supply what what may may be be called called ‘reasonable’ 'reasonable' operations is still operations. This This applies appliesboth bothto to the the synthetic synthetic and and to the analytic operations. analytic case;; in fact, case fact, not not all alltautological tautologicalimplications implications appear appearreasonable. reasonable. For instance, the tautological a3 3 b', For tautological implication, implication, 'a. ‘ a ,ti b’, can can scarcely scarcely be accepted be accepted as as reasonable. reasonable. The general form of of the theory to The general form to be be developed, developed, which which is the the same as as the the form of my previous theory, theory, can now be outlined as same as follows. First, aitclass follows. First, classofoforiginal originalnomological nomolog.ical statements statements is is defined; defined; then statements is is constructed constructed as then the theclass classof ofnomological nomoEogical statements as comprising comprising all those statements that are aredeductively deductively derivable derivable from sets of of statements statements of of the first first class. class. Among Among these, these, aa narrower narrower group group is is defined as nomological inthe thenarrower narrowersense. sense.ItIt isis this this group, group, also defined as nornological in also called the the group whichisis regarded regarded as as called group of of admissible admissible statements, statements, which supplying reasonable propositional propositional operations, operations, while while the the class of supplying reasonable of nomologicalstatements statementssupplies suppliesthe thelaws lawsofofnature nature and and the the laws nomological laws of logic. logic. As As in in the the previous stateof previous theory, theory, the the original original nomological nomological statements ments are included included in in the theadmissible admissiblestatements. statements. Furthermore, Furthermore, analytic, or tautological, tautological, statements statements are are included included in innomological nomological statements, a subclass of them being as in in the older statements, subclass of being admissible, admissible, as older 61 and theory. The 63 in ESL; theory. The new new definitions definitions replace replace §9 61 ESL; the the and §9 63 other sections of chapter chapter VIII VIII in other sections of in ESL ESL remain remain unchanged. unchanged. As As in in ESL, the theory ESL, theory isis developed developed only only for for the the simple simple calculus calcuhs of of functions. An An extension extension to to the the higher be functions. higher calculus calculus can can presumably presumably be constructed, but but would constructed, would require require further further investigation. investigation. Although Although the class class of of 'reasonable' ‘reasonable’ operations operations must must be be defined defined

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INTRODUCTION

as aa narrower operations, one one must must not narrower subclass subclass of of nomological nornological operations, conclude that the conclude that the latter latteroperations operationsappear appearcompletely completely 'unreason‘unreason-

able'. able’. It seems seems that that there there isisno nounique unique explicans explicans for for the term term 'reasonable';; the the requirements which we we tacitly tacitly include ‘reasonable’ requirements which include in this this

term differ with the context differ with context in in which which the the operation operation is is used. used. The The theory presented accounts for these variations by defining various by defining categories categories and indicating indicating their their specific specific characteristics characteristics and and appliappli-

cations. cations. As an an instrument instrument for As for carrying carrying out out this this construction, construction, aa distinction distinction between three orders of truth truth is between three orders of is introduced. introduced. Analytic Analytic truth truth supplies supplies the highest, truth the highest, or third third order, order, synthetic synthetic nomological nomological truth the second second order, and merely factual factual truth truth the thelowest, lowest, or or first firstorder. order. The The two two higher orders nornological truth truth and and higher orders of truth, truth, which whichconstitute constitutenomological embrace all nomological statements,are are thus thus set above embrace all nomological statements, above merely merely factual truth. This This distinction distinction is is used, used, in in turn, turn, for the factual truth. the definition definition of nomological statementsinin the narrower nomological statements narrower sense, sense, which which are are conconstructed in such a way way that if if their their essential essential parts are true taken separately, they are order than than the the statement separately, they are true true of of aa lower lower order statement itself. itself. By means means of this this method, method, certain certain rather rather strong strong requirements requirements of of reasonableness can be reasonableness can be satisfied. satisfied. An important statements in in the An important application application of nomological nomological statements wider is given given by wider sense sense is by the the definition definitionof of modalities. modalities. These These categories categories are not statements, but are not presupposed presupposed for for nomological nomological statements, are defined defined by their help help and constitute constitute a sort sort of of byproduct byproduct of of the theory theory of of nomologicalstatements. statements. The The modalities modalities are are usually referred, not nomological to a statement, statement, but but to tothe thesituation, situation,or orstate stateof ofaffairs, affairs,denoted denoted by by it; i.e., it; i.e., they they are areused used ininthe theobject objectlanguage. language. We We thus thus define: define: a isis necessary necessary ifif 'a' ‘a’isisnomological. nornological. a is is impossible impossible if 'a' ‘G7isisnomological. nomological. a is (contingent)ififneither neither‘a’ 'a' nor nor 'a' is merely merely possible possible (contingent) ‘6’is is nomononiological. logical. For 'merely ‘merely possible', possible’, the term 'possible' ‘possible’ is is often often used, used, but but somesometimes 'possible' refers to to the the disjunction of 'necessary' and 'merely times ‘possible’ refers disjunction of ‘necessary’ and ‘merely possible'. It It can be seen seen that that the these possible’. can easily easily be the term term 'nomological' ‘nomological’ of of these definitions must must be be interpreted interpreted as in the the wider definitions as nomological nornological in wider sense; sense; if we we attempted attempted to to interpret interpret ititasasnomological nornological in in the thenarrower narrower sense, we we would would be be led led into into serious serious difficulties. difficulties. For For instance, instance, certain sense, certain

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analytic analytic statements statements would would then then not notdescribe describenecessary necessarysituations. situations. Whereas the use Whereas the use of of analytic analytic statements statementsfor forthe thedefinition definitionofoflogical logical modalities is obvious, obvious, it it is the modalities is thesignificance significance of of the the given givendefinitions definitions that that they theyalso alsoallow allowfor forthe thedefinition definitionofofphysical physicalmodalities. modalities. These These two result according two kinds kinds of of modalities modalities result according as, in in the the above abovedefinitions, definitions, the term as analytic or term 'nomological' ‘nomological’ is is specified specified as or synthetic synthetic nomonomological, respectively. respectively. Furthermore, Furthermore, a distinction distinction between between absolute absolute logical,

must be be made; made; for for these these points points and and the and relative relative modalities modalities must further theory of modalities to ESL, further modalities II refer to ESL, §Q 65. 65. In In the thefollowing following presentation, we shall refer to to modalities for the the presentation, we shall occasionally occasionally refer modalities for purpose of illustrating illustrating nomological nomological statements. constructed for for the is constructed The The class class of of admissible admissibleimplications implications is purpose of satisfying very strong requirements requirements and thus thus of of expliexplicating reasonable implications implications in in the the narrowest narrowest sense of of the the term. Conversational language has has two two kinds of Conversational language of usage usage for forimplications implications subject to very exacting requirements: they are used subject very exacting requirements: they are used for for prepredictions, or they are contrary to fact. dictions, or are employed employed as conditionals conditionals contrary fact. It unctive implications cannot convey convey important important It isis obvious obvious that that adj adjunctive implications cannot information in in a predictive usage.InInorder orderto to know know that that the predictive usage. the imiminformation plication is true, true, we would would have have to to know know that that aa particular plication is particular T-case, T-case, which verifies verifiesit, it, is is true; true; but which but once once we we know know this this T-case, T-case, we we would wodd lose in in information information ifif we we merely merelystate state the the implication implication and and not not the lose T-case itself. itself. This This applies applieswhether whether we weknow knowthe the truth truth of the T-case T-case of the from past past observations or because we can can predict predict it. it. For because we For instance, instance, from we can can predict predict that that itit will tomorrowand and that that the we will be be Wednesday Wednesday tomorrow this conjunction by the adjunctive sun will will rise; replacing replacing this conjunction by adjunctive imimplication, ‘if 'if it is the sun will rise', we we say plication, is Wednesday Wednesday tomorrow tomorrow the will rise’, less than than we less we know, know, and and therefore therefore such such an an implication implication has no no practical use. practioal use. It It has hasoften oftenbeen beenemphasized emphasizedthat thatfor fora counter/actual a counterfactual usage, usage, likewise, adjunctive implications are completely inadequate. Nolikewise, adjunctive implications NObody would would say, say, 'if not white, sugar would be sour', body ‘if snow were were not would be sour’, although this this implication implication isis true true in the adjunctive although adjunctive sense. sense. But we we heated, it would 'if this metal would say, would say, ‘if metal -were heated, would expand'. expand’. Since Since in the conversational conversational language language is “rather clear clear and unambiguous unambiguous in usage of conditionals contrary to fact, we possess in this usage usage of conditionals contrary to fact, we possess in this usage aa sensitive test test for the sensitive the adequacy adequacy of of the theexplication explication of of reasonable reasonable

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INTRODUCTION INTRODUOTION

implications, and we we shall shall often often make make use use of of it. it. For instance, implications, and instance, it is required for a conditional contrarytoto fact fact that that it required for conditional contrary it be be unique. unique. By By this property II mean mean that, that,ififthe theimplication implication 'a‘aD 3b' 6’ is is used used for aa conditional contrary contrary to fact, fact, the contrary 'a D conditional contrary implication implication ‘a 3 8’ cannot be so cannot so used. used. Obviously, Obviously, adjunctive adjunctive implication implication does not

satisfy the condition when it is used satisfy condition of uniqueness uniqueness when used countercounterfactually, ‘a’is false, both contrary contrary implications implications are factually, because, because, ifif 'a' true in It has out that in the the adjunctive adjunctive sense. sense. It has often often been been pointed pointed out this absence absence of of uniqueness uniqueness makes makes adjunctive adjunctive implications implications ininappropriate for for counterfactual use. In In the theory appropriate count’erfactual use. theory of of admissible admissible implications will therefore therefore be be an an important implications itit will important requirement requirement that two two contrary contrary implications implications cannot cannot be be both both admissible. admissible. The The present present theory theory satisfies satisfies this requirement, requirement , whereas whereas my previous previous theory theory could satisfy itit only to some could satisfy some extent. extent. Since the theory Since the theory to to be be developed developed is is rather rather tecimical technical and andinvolves involves much detail, of which whichisis at at first first not not easily it much detail, the the significance significance of easily seen, seen, it may be to outline the definition may be advisable advisable to outline the the major major ideas ideas on on which which the definition of original nomological statements statements is is based. of original nomological based. These These ideas ideas have have been been developed essentially essentially for for synthetic synthetic nomological nomological statements, statements, because because developed statements of statements of this this kind kind are are in in the the foreground foreground of of this this investigation; investigation; the application to tautologies is then rather easily given. application to tautologies is then rather easily given. The The leading leading idea idea in in the thedefinition definitionof of original originalnomological nomological statestatements ments of of the the synthetic synthetickind kindwifi, will,of of course, course, be be given given by by the the principle principle that that such such statements statementsmust must be be general general statements, statements, or or all-statements, all-statements, and must not be and be restricted restricted to to aasingle single case. case. We We know know from from the writings of David Hume that physical necessity, the necessity writings of David that physical necessity, necessity of the the laws laws of of nature, nature, springs springs from from generality, generality, that thatcausal causalconnection connection differs from mere mere coincidence coincidenceininthat that it it expresses differs from expresses a permanence permanence of of coincidence. Hume is the coincidence. Hume believed believedthat that this this generality generality isis all all that that is required He was was right right when he insisted insisted that that required for for causal causal connection. connection. He when he unverifiable additionsto to this this requirement requirement should should be be ruled ruled out; out; in unverifiable additions in fact, fact, any any belief belief in in hidden hidden ties ties between between cause cause and and effect effect represents represents surplus meaning meaning which which Occam's Occam’s razor razor would shave shave away. away. HowHowa surplus ever, it it turns turns out outthat thatgenerality generalityalone, alone,though thoughnecessary, necessary, is is not not all unreasonable unreasonable forms ruled out. out. sufficient to sufficient to guarantee guarantee that that all forms are are ruled We therefore introduce, introduce,inin addition addition to to generality, generality, aa set set of We shall shall therefore requirements restricting restricting the statement forms forms to be admitted. It goes goes requirements the statement to be admitted. It

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without saying saying that that these without these additions additions are are formulated formulated as as verifiable verifiable properties of of statements, statements, and and that, that, for aa given properties given statement, statement, we we can can always find find out out whether always whether it satisfies satisfies the the requirements. requirements. In his his early early writings writings on on mathematical mathematical logic, logic, Bertrand Bertrand Russell Russell has pointed ff(x)f (x)3 D g(xfl' has pointed out outthat thataageneral generalimplication implicationofofthe theform, form,'(x) ‘(x)[ g (z )] ’ eliminates the unreasonable eliminates unreasonable properties properties of adjunctive adjunctive implication implication to some extent, extent, but but that to that these these properties properties reappear reappear if if the the implicans implicans '/(x)' is always or the implicate ‘f(x)’ always false false or implicate 'g(x)' ‘g(x)’ is always always true. The The exclusion of of these these two two cases oases will willtherefore thereforebe bean an important important requirerequireexclusion ment within the ment the definition definition of of aareasonable reasonableimplication. implication. However, However, for a general statements, this this requirement for general theory theory of of nomological nomological statements, requirement must be generalized so as as to to be to other other opermust generalized so be applicable applicable likewise Iikewise to ations and totostatements among ations statementspossessing possessing several several operators, operators, among which there there may may be be existential existential operators. operators. It It can that for which can be be shown shown that for the latter latter case case an animplicans implicans which which is is not not always always false false does does not not exclude an an unreasonable unreasonable implication. implication. The construction construction of such aa more requirement is by means more comprehensive comprehensive requirement is achieved achieved by means of of aa formal formal property of of statements, statements,which whichisiscalled calledexhaustiveness exhaustiveness and which which will be defined defined in in group group EE of chapter 2. will be 2. (See (See also the discussion discussion of (4Oae-b) chapter 3.) (4Oa-b) inin chapter 3.) Even ifif an so far mentanimplication implication satisfies satisfies the requirements requirements so mentioned, it can ioned, it can have have forms forms that that are arenot notaccepted acceptedasasreasonable. reasonable. Assume that that during Assume during a certain time it it so so happens happens that that all allpersons persons in a certain in certain room room are are over over 30 30 years years old; old;then thenthe thegeneral generalimplicimplication, 'for all x, x , if x is a person person in in this room at at this this time, time, xx is is over over ation, ‘for all 30 years old’, old', is true in 30 years in the theadjunctive adjunctive sense, sense, and and its itsimplicans implicans is is not not always always false. false. Yet Yet this thisimplication implication does does not not appear appearreasonable, reasonable, as is is seen seen when when it is used used counterfactually: counterfactually: the the statement, statement, 'if ‘if another person had had been been in in this this room at at this time, time, he he would would have been been over over 30 years years old', old‘, would would not be be acceptable acceptable as true. true. This This example shows that that aa reasonable implication has has to to satisfy satisfy further further example shows reasonable implication requirements, which exclude exclude aa restriction requirements, which restriction of of the the implication implioation to to certain times times and and places places and andguarantee guaranteeits itsuniversal universalapplication. application. These requirements w will be explained explained in in group group P, F, chapter These requirements ill be chapter 2. 2. It of this this kind It should should be be noted noted that that requirements requirements of kind are rather strong strong and and are are adhered adhered to, to,ininconversational conversationallanguage, language, only only when when the implicational character of of the the statement is implicational character is explicitly explicitly stated,

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INTRODUCTION INTRODUCTION

for instance instance by 'if-then', 'implies', for by using using terms terms like like ‘if-then’, ‘implies’,etc. etc. No No objection, objection, however, isis raised raised when when the the statement statement is given however, given the wording: wording: 'all ‘all persons in in this room 30 years years old', old’,which which persons room at this this time time were were over over 30 form appears appears quite reasonable. In In the form the disguise disguise of of aa conversational conversational all-statement, therefore, therefore, we we accept accept adjunctive adjunctive implications, fact implications, aa fact which shows showsthat that these these implications implicationsare are not not merely merely aa creation creation of of which the the logician logician but but are arewidely widely used used in inconversational conversational language. language. The The present present investigation investigation into into the the nature natureof ofreasonable reasonable implications implications is is therefore restricted to to an therefore restricted an explicit explicit use use of of this thisoperation. operation. Similar Similar considerations apply apply to other considerations other propositional propositional operations. operations. It turns It turns out out that thatininorder ordertotocarry carrythrough throughthe therequirements requirements mentioned mentioned it is necessary necessary to introduce introduce rules rules which which eliminate eliminate redundant parts procedureof ofreduction, reduction, redundant partsof of statements statementsand anddefine defineaaprocedure by means by means of of which which aa statement statement is is transformed transformed into into simpler simpler forms. forms. This isis necessary, first, because This necessary, first, because a reasonable reasonable statement statement could could easily be be made made unreasonable unreasonableby byadding addingtotoitit redundant redundant parts; parts; for easily instance, ifif aa statement statement contains to a particular instance, contains no no terms terms referring referring to particular space-time region, we we could could add add to to it some space-time region, some tautology tautology containing containing such terms such terms without without changing changing the meaning meaning of of the thestatement. statement.1 Secondly,however, however,ititmay maybe be possible possibletotoinsert insert parts parts that are Secondly, are merely factually factually true true into a nomological statement in in such such a way merely nomological statement that the the requirements the statement statement still still satisfies satisfies the requirements mentioned mentioned previously. with certain previously. IIn n combination combination with certain other requirements, requirements, the reduction rules out out such forms;; and and II have reduction procedure procedure rules such forms have been been able able to to construct a proof that unreasonable parts of construct proof that unreasonable parts of aa certain certain kind kind cannot cannot be contained statements as defined in this contained in in original original nomological nomological statements defined in this presentation presentation (see (see theorem theorem 5). 5). All the the criteria criteria so so far far mentioned mentioned are are of of aa formal formalnature; nature; and and they they All are based on the assumption assumption that we we are able to find find out whether whether these formal relations hold. hold. For For instance, instance, itit is presupposed that formal relations presupposed that we are able able to to find find out out whether whether aa statement statement is to aa given we are is equipollent equipollent to given other statement, statement, whether whether it can be be written written in in syntactical syntactical forms forms of other it can of a certain certain kind, kind, such such as:an all-statement, as much much as as such such all-statement, etc. etc. In In as 1 This This objection objection was correctly aeainst previous theory by was raised raised correctly against my my previous J. C. vol. 55, J. C. C. C .McKinsey, McKinsey,American American Mathematical Mathematical Monthly, Monthly, vol. 5 5 , 1948, 1948, pp. 261-263; and y N. Goodman, Philos. Philos. Review Review 1948, 1948,vol. vol.57, 57,pp. pp.100—102. 100-102. 261—263; andbby N. Goodman,

XN'rRODUCTION INTRODUCTION

11 11

an assumption assumption is made, made, the the present present theory theory presupposes presupposes the the comcom-

pleteness pleteness of of the the lower lower functional functional calculus. calculus. However, However, since since aa general general decision procedure cannot cannot be decision procedure be constructed constructed for for this thiscalculus, calculus,we we cancannot not give give rules rules indicating indicating how how the the test test for forequipollence equipollence is is to to be be made. made. In I n principle, principle, therefore, therefore, there there may may exist exist statements statementsof of complicated complicated

forms for which we are actually forms for which we actually unable unable to decide decide whether whether they satisfy laid down; down; we we then then have satisfy the requirements requirements laid have to put put these these

statements into statements into aa category category under under the the heading, heading, 'at ‘atpresent present unknown unknown whether and hope hope that that some day they whether nomological', nornological’, and some day they will will be taken out out of of this this category, category, because because in in principle principle the thedecision decision can can be be made. made.

In practice, practice, however, however, we we shall shall encounter encounter no nosuch suchdifficulties, difficulties, because scientific laws laws have have rather rather simple syntactical forms and because scientific cannot as to structural cannot compete, compete, as structural form, form, with the the involved involved statestatements which the mathematical likes to to make the the subject which the mathematical logician logician likes subject of his his investigation. investigation. From formal properties properties II will will now now turn to to the thediscussion discussion of of aa

property which property which is independent independent of of form. form. Being Being laws laws of of nature, nature, nomologicalstatements, statements,ofofcourse, course,must mustbe betrue; true; they they must even nomological even be true,which whichisisaastronger strongerrequirement requirement than than truth truth alone. be verifiably verifiably true, alone. Some remarks about about this requirement Some remarks requirement must must now now be beadded. added.1 The requirement of truth isisnot requirement of notsufficient sufficient because because we we wish wish to to exclude from from nomological statements those all-statements exclude nomological statements all-statements which which are merely factually true, true, or or 'true merely factually ‘true by by chance'. chance’. This This kind kind of of statestatement may ment may obtain obtain even even ifif no noreference reference to toindividual individual space-time space-time regions isis made; made; for for instance, the statement, 'all regions instance, the ‘allgold gold cubes cubes are are smaller than than one cubic smaller cubic mile', mile’, may may possibly possibly be true. true. When When we we reject a statement of reject of this this kind kind as as not not expressing expressing a law law of of nature, we mean mean to say we say that that observable observable facts facts do not require require any any such such statement for for their and thus thus do statement their interpretation interpretation and do not not confer confer any any truth, or on it. it. If they or any any degree degree of probability, probability, on they did, did, ifif we we had had good inductive inductive evidence for for the the statement, good statement, we we would would be be willing willing to accept it, it. For For instance, instance, the the statement, statement, 'all ‘allsignals signals are are slower slower than In I n ESL, ESL, p. p. 369, 369, II used used the theterm term'demonstrably ‘demonstrablytrue'. true’.Since Since'demon‘demonstrable' proof, I will will now ngiv use use the the above above strable’ usually usually refers refers only only to deductive deductive proof,

it is now now generally generally term 'verifiable' ‘verifiable’alone alonewould would not not suffice suffice because it term. The term

used as true or used in the the neutral neutral meaning meaning 'verifiable ‘verifiable as or false'. false’.

12 12

INTRODUCT%ON

or equally equally fast as as light light signals', signals’, is accepted accepted as a law law of of nature nature because observable observablefacts factsconfer conferaahigh highprobability probabilityupon uponit. it. It It is because the inductive not mere mere truth, which inductive verification, verification, not which makes makes an allallstatement statement a law of of nature. nature. In I n fact, fact, ifif we we could could prove prove that gold gold cubes cubes of giant size size would would condense condense under under gravitational gravitational pressure pressure into aasun-like sun-likegas gasball ballwhose whose atoms atoms were were all all disintegrated, disintegrated, we we would be be willing willing also alsoto to accept accept the the statement about would about gold gold cubes cubes among among the laws laws of of nature. nature. The reason is is easily easily explained. The inductive inductive inference inference extends

truth from from 'some' ‘some’ to 'all'; ‘all’;itittherefore thereforeallows allows for for aa predictive predictive as as

well as as counterfactual counterfactual use use of of implications. implications. We We saw saw that that these well these two two kinds kinds of of usage usage are are essential essential for for reasonable reasonable implications; implications ; therefore, therefore, if it qualifies for the the category if an an implication implication is is inductively inductively verified, verified, it qualifies for category of reasonable We discussed discussed the the example of an imreasonable implications. implications. We example of implication which is restricted to persons in a certain room during a plication which is restricted to in

certain certain time; when when we we reject reject such such an an implication implication for for countercounterfactual use, it isis because factual because this this implication implication is is not not verified verified through through inductive extension. extension. The The requirement requirementthat that the all-statement inductive all-statement be be verifiably true, true, therefore, therefore, guarantees guarantees the the kind kind of of truth truth with verifiably with which which we wish we wish to toestablish establishlaws lawsofofnature; nature;it guarantees it guarantees'inductive inductive generality. The word word 'verifiable' ‘verifiable’ includes includes a reference reference to to possibility. possibility. Since Since physical possibility is is aa category to be physical possibility be defined defmed in in terms terms of of nomonomological statements, statements, itit would would be be circular circular to to use, in the definition logical definition of such statements, statements, this category. the term category. For For this reason, reason, II defined defined the 'verifiably true' as verified at at some sometime, time, in in the the past past or or in in ‘verifiably true’ as meaning meaning verified the future. future. It Ithas hasbeen been argued argued against against this this definition definition that there there may be laws laws of of nature nature which which will will never never be be discovered discovered by human human beings. the present present investigation investigationII shall shall show show that that the the latter beings. 1IIn n the statement, indeed, indeed, can can be be given given aa meaning, meaning, and and that thatwe wecan candefine define statement, term veriflably verifiably true in in the the wider wider sense 8eme which covers this this meaning. meaning. aa term which covers But in shallbegin begin with with the the narrower narrower in order order to to define define this term, term, IIshall This objection was by Mr. Mr. Albert Albert This objection was raised raised against against my my theory theory in in a letter by Hofstadter, interesting objections objections answered in Hofstadter, which included some further further interesting the present paper. paper. The The same same objection objection was the present was made made by by G. G. D. D. W. W. Berry, Berry, Journ. of Symbolic Logic,vol. vol. 14, 14, 1949, p. 52. Symbolic Logic, 1949, p. 52. 1

IXTBODUOTION

13 13

term, and later to the and proceed proceed later the introduction introduction of of the the wider wider term term

(chapter (chapter 6). 6). Although Although inductive inductive verifiability verifiability is is presupposed presupposed for fornomological nomological statements, statements, the the definition definition of of such such statements statements can can be be given given without without entering into an an analysis analysis of of the the methods methods of of verification. verification. What What we we statements is is not aa are looking looking for in aa definition definition of of nomological nomological statements

method such statements, statements, but but a set method of of verifying verifying such set of of rules ruleswhich which guarantee is actually guarantee that inductive inductive verification verification is actually used used for for these these statements, in as much as they are synthetic. The requirements statements, as much as they are synthetic. The requirements laid laid down down in in the thedefinition definition of of nomological nomological statements, Statements, in fact, fact, represent represent a set set of of restrictions restrictions which which exclude exclude from from such such statements statements all synthetic forms that can without inductive all synthetic forms that can be be verified verified without inductive extension. extension.

More than that, More than that, the therestrictions restrictions single single out, out,among amonginductively inductively verified statements, a special group of all-statements verified statements, special group of all-statements associated associated with a very of probability; probabifity; and they very high high degree degree of they are are so so constructed constructed

that they they allow allow us us to to assume assume that that these these all-statements all-statements are true without Merelyfactual factualtruth, truth, though without exceptions. exceptions. Merely though in itself itself found found by inductive inductive inference, inference, is is thus thusdistinguished distinguishedfrom fromnomological nomological truth in and the in that that ititdoes doesnot notassert assert an aninductive inductive generality; generality; and

requirements introduced statements are all governed governed requirements introduced for for nomological nomological statements are all by the very very principle principle that factual factual truth truthmust mustnever neverbe besufficient sufficient to t o verify verify deductively deductively aa statement statement of this kind. kind. The is thus The predictive predictive usage usage of of admissible admissible implications implications is thus reduced reduced to the the predictive predictive use use of of inductive inductive inferences inferences equipped equipped with high degrees of ofprobability. probability. Their Their counterfactual counterfactual usage, degrees usage, likewise, likewise, appears appears justified by by this justified this interpretation, interpretation, although although this this usage usage imposes imposes even stronger even stronger requirements requirements upon upon implications implications than aa predictive predictive usage, as as will willbe beshown shownininchapter chapter7.7.ItIt isis its its origin usage, origin in inductive inductive extension, its .inductive generality, that makes an implication extension, its inductive generality, that makes an implication reasonable. reasonable. Since the the function function of of the requirements to be introduced Since requirements to introduced is thus negative negative rather than than positive, positive, inasmuch inasmuch as as these these requirements requirements are are merely restrictive, restrictive, itit is is not not necessary necessary to to give give in in this this presentation presentation a merely detailed discussion discussion of ofinductive inductive verification. verification.That That inductive inductive methods methods detailed exist and and are applied, is a familiar exist applied, is familiar fact; their study study belongs belongs in a theory of of induction induction and and probability, probabifity,and and as as far far as my own theory own conconception of of this this subject subject matter matter is concerned, ception concerned, II refer refer to another another

14 14

INTRODUCTION

publication.'1 However, However,II should shouldlike liketoto add add to to the present publication. present investigation a brief vestigation brief account of of the the methods methods of of inductive inductive verification verification in their relation relation to to general general implication; implication; this account account is given given in in the the appendix. appendix. Those who who have have studied the Those the construction construction of of artificial artificiallanguages languages are often sceptical as as to to the rules that that govern are the possibility possibility of finding rules govern conversational language. language. They They are are disappointed by the conversational the vagueness vagueness of the terms life, and and point point to terms used used in the the language language of everyday everyday life, the apparent inconsistencies in actual usage of language. Yet on apparent inconsistencies in actual usage of language. Yet closer inspection,ititturns turns out out that that aa natural closer inspection, natural language language is by by no no means as inconsistent as is believed. If If it is sometimes sometimes believed. it is is difficult difficult to to find rules, rules, one one must must not conclude find conclude that no no rules rules exist. exist. Physical Physical phenomena, too, do not phenomena, too, not always always openly openly display display the therules rulesfollowed followed by them; able to show them; but but physicists physicists have have been been able show that all all such such phenomena are controlled by very phenomena are controlled by very precise precise rules, rules, though though the the formulation these rules natural formulation of of these rules may may be be extremely extremely complicated. complicated. A natural language is aa complex complex system systemofofpsychological psychologicaland andsociological sociological language is phenomena, and one one cannot cannot expect expect its its laws laws tto phenomena, and o be visible visible tto o the the untrained eye. Those who are not afraid to search for its laws, untrained eye. Those who are not afraid search for its laws, however, have have been been surprised surprised to to discover discover that that rather however, rather precise precise laws laws can be constructed language, and and that, that, once can constructed into actual usage of language, once laws laws have have been been abstracted abstractedfrom fromsingle single examples, examples, they they cover cover large large parts of parts of usage usage practically practically without without exceptions. exceptions. Perhaps it Perhaps it is is possible possible to to explain explain the the hidden hidden precision precision of of language language by the by the fact fact that thatlanguage languagebehavior behavior isiscontinuously continuously tested tested and and corrected by by its its practical practical applications applications;; that, that, in particular, corrected particular, predicpredictions and contrary to tions and conditionals conditionals contrary to fact fact are areof ofgreatest greatestsignificance significance inexact in in the in everyday everyday life, life, and that that aalanguage language which which were were inexact use use of of such such concepts concepts would would soon soon be be led led into intoserious seriousconificts conflicts with with observational experiences. experiences.IfIf itit is required observational required for for aa reasonable reasonable imimplication to be plication to be applicable applicable to to predictions, predictions, the the usage usage of of reasonable reasonable implications isis not not aa matter of implications of taste, taste, or orof of social social convention, convention, but something eminently eminently practical practical;;and and if if we we have have developed developedaanatural natural something To Probability, second second edition, edition, Berkeley To my my book, book, The The Theory Theory of of Probability, Berkeley 1949; 1949; quoted ThP.This Thisbook book includes includes aadiscussion discussion of of induction induction for for predictive predictive quoted as ThP. usage and aa justification justification of of induction, induction, problems problems which which cannot be dealt dealt with with in the the present present monograph. monograph. 1

XN'TRODUCTION

16

feeling for the reasonableness of an implication, we have been feeling for reasonableness of been so so

conditioned by by the exigencies conditioned exigencies of of everyday everyday life. life. Thus Thus practical practical needs zceds have made made language language aa forceful forceful instrument instrument which which owes owes its its efficiency to its precision. The study of natural languages, thereeiEciency to its precision. The study natural languages, fore, offers to the fore, offers to bhe logician logician the the possibility possibility of of making making laws laws explicit explicit which, though unknown which, though unknown to the the language language user, user, implicitly implicitly control control his language language behavior behavior and and make make it it consistent. his consistent. The present study

is intended intended to be be aa contribution contribution to t o this this task. task.

II I1 FUNDAMENTAL TERMS TERMS FUNDAMENTAL

In statements we we shall shall refer refer to I n the thedefinition definition of of nomological nomological statements two kinds of of these these statements. two of properties properties of statements. First, First, we we shall shall speak speak of of properties properties which which remain remain invariant invariant for fortautological, tautological, or orequipolequipollent, lent, transformations, transformations, such such as as truth, truth,or orbeing beingsynthetic. synthetic.These Thesewill will be Terms used used for for the the formulation be called called invariant invariant properties. properties. Terms formulation of of these these properties properties wifi will be be called called I-terms. I-terms. Second, Second, we we shall shall speak speak of of properties whichaa statement statement has has only only in in aa particular properties which particular form form of of writing, do not not remain writing, and and which which do remain invariant invariant for for all all tautological, tautological, or or equipollent, transformations,such such as as being equipollent, transformations, being an an implication, implication, or or containing containing an an all-operator. all-operator. These These will .Rillbe be called calledvariant variantproperties. proprtka. Terms Terms used used for for the the formulation formulation of of such such properties properties will will be be called called V-terms.The The definition definition of of nomological nomologicalstatements statements will be be laid laid down V-terns. down in in certain certain requirements, requirements, which which we we distinguish distinguish correspondingly correspondingly as as I-requirements and and V-requirements. V-requirements. In I n the thebeginning, beginning, we we shall shalldeal dealonly onlywith withoriginal originalnomological nomological statements. statements. For For their their definition definitionboth bothkinds kindsof ofrequirements requirements will will be be used. is thus thus made In used. The The term term 'original ‘original nomological' nomological’ is made a. a V-term. In order to construct the requirements, it is advisable first to define order to construct the requirements, it is advisable first to define certain certain terms terms which which are are to to be beused. used. These These definitions definitions are are ordered ordered by by groups. groups. Notational Sentence name name variables, variables, belonging belonging to to N o t a t i o n a l remark. r e m a r k . Sentence the will beexpressed expressedbybythe theletters letters‘p’, 'p', ’q’, 'q', ‘r’, 'r', etc; the metalanguage, metalanguage, w i l l be etc; combinations such letters letters will will be be interpreted interpreted in in the sense combinations ofof such sense of autonymous such that that 'p. q' is a u t o n p o u s use of of operations operations (Carnap), (Carnap), such ‘p.q’ is the the name name of of the the conjunction conjunction of of pp and andq.q.Sentential Sententialvariables, variables,belonging belonging in in the will be be expressed expressedby bythe theletters letters ‘a’, 'a', 'b', the object object language, Ianguage, will ‘b’, 'c', ‘c’, etc.; etc. ; functional in the the object functional and and argument argumentvariables, variables,likewise likewise belonging belonging in object language, wifi be be expressed expressedby bythe the letters letters ‘f’, 'f', 'g', language, will ‘g’, 'x', ‘d, ‘y’, etc. etc. These These 'y', variables require the the use variables require uae of ofquotation quotation marks maxh within within aacontinuous continuous

FUNDAMENTAL FUNDAMENTAL TERMS TERMS

17 17

on separate lines text; for formulae formulae on lines the quotation quotation marks marks wifi will be be omitted. quotation marks omitted. Likewise, Likewise, quotation marks will will be be omitted omitted after after aacolon colon in in the text. The metalinguistic is made made The metalinguistic variables variables will be be used used when when reference reference is to the inner inner structure structure of of the the sentences sentences denoted denoted by the the individual individual letters. The object language variables variables will will be be used, used, first, letters. fist, when no referenceisis made madeto to the the inner reference inner structure of the the sentence, sentence, or the the function, abbreviated abbreviated by by one one letter, letter, and all the structure function, structure referred referred to is is expressed expressed by by combination combination of of letters. letters.Secondly, Secondly,however, however, object language variables will will be be used used in in a mixed object language variables mixed conteftt, where where the structure structure of of the thesentences sentences isispartially partiallyexpressed, expressed, partially partially described in in words. described words. The between these these cases cases may may be ifiustrated The distinction distinction between illustrated by by examples. II shall shall write write:: 'a' b';; the examples. ‘a’ is is derivable derivable from 'a. ‘a.b’ the variable variable 'x' ‘z’ in '(x)t(x)' ‘ ( x ) f ( ~ )is ’ bound; etc. In I n these these cases, cases, no no reference reference is made made to the inner structure structure of of the the expressions expressions abbreviated abbreviated by one letter, letter, and the truth truthofofthe themetalinguistic metalinguistic sentence sentence is is visible visible from the structure expressed by by the the symbols. IInn contrast, to inner inner structure contrast, reference reference to structure of the referred to to is made the individual individual sentences sentences referred made in in such such statements as, statements as, 'p ‘ pisisderivable derivablefrom fromq', q’,for for which which II use use metalinguistic metalinguistic variables. The The truth truth of of such not visible visibIe from the variables. such a statement statement is not structure indicated by the statements can the symbols; symbols; therefore such statements only occur form, such such as: as: if p isis derivable only occur in conditional conditional form, derivable from from q, q, then ...; assume Slssume that pp isisnomological; nornological; etc. A A mixed mixed context context is is then ...; D g(x)' and ‘f’ '/' given by by aa statement of the form: given form: if if 'f(x) ‘ f ( x )3 g(z)’ is is analytic and with ‘g’, 'g', then then 'f' is identioal with ‘f’ or or 'g' ‘g’ is is composed composed of elementary elementary is not identical functions. IfIf such were formulated formulatedby by the the help functions. such statements statements were help of of metalinguistic have to to be be extended extended metalinguistic variables, variables, autonymous autonymous use use would would have to the to the parentheses; parentheses; although although this thiscould couldof of course course be beconsistently consistently done, done, II prefer prefer to to use useobject objectlanguage language variables variables and and quotes. quotes. The The decision forone oneoror the the other other method methodisis aa matter decision for matter of of style style and and personal taste) not not of who do do not not like like the the rather rather personal taste, of correctness. correctness. Those Those who wide use of of quotes quotes may may regard regard the the expression, wide use expression, “"/' f ’ is is composed composed of of elementary elementary functions', functions’, as as an an abbreviation abbreviationfor forthe thelonger longerexpression, expression, “f’ is is interpreted by by aafunction functionwhich whichisiscomposed composed of of elementary elementary "/' functions'. Likewise, the expression, "a' is an implication', functions’. Likewise, the expression, ‘‘a’ an implication’, can can be regarded "a' isis regarded as an abbreviation abbreviation for the longer longer expression, expression, “a’

18 18

rUNDAMENTAL 1E1%MS FUNDAMENTAL TERMS

interpreted by an implication'. interpreted implication’. In In this this way, way, the thewider wider use use‘of quotes can be regarded can regarded as an an abbreviated abbreviated mode mode of of speech speech translatable translatable into a narrower of quotes. Note that the into narrower use of quotes. Note the wider wider use use of of quotes quotes can occur only can only in in conditional conditional sentences. sentences. In aasynthetic synthetic statement, statement,sentential sentential and andfunctional functional variables variables express uninterpreted constants, constants, i.e., i.e., such such statements statements are are true express uninterpreted true only for specific specificvalues valuesof ofthese thesevariables. variables.IIn an analytic analytic statement, statement, only for n an sentential and functional in the functional variables represent represent free variables variables in sense that any value sense that value may be be given given to them them while while the statement statement remains true. true. A remains A notational notational distinction distinction between between these these two two cases casm will not not be made, the same letter may represent will made, because because the same letter represent a free free variable for the whole formula and and an uninterpreted variable for whole formula uninterpreted constant constant for for a part part of of it. it.Bound Boundfunctional functional variables variables will will not be be used used since since the the presentation presentation remains remains entirely entirely within within the thelower lowerfunctional functionalcalculus. calculus. A sentence is called propositionalterm term ifif it it has called an elementary elementary propositional has no no inner inner structure structure expressible expressible by by the the use use of of propositional propositional symbols; symbols ; otherwise A function together with with its otherwise it it is called called compound. compound. A function together its variables, such such as 'f(x, variables, ‘f(x, y)’, is called called aafunctional. functional.11 A function is is A function y)', is called elementary ifif itit does not stand for called elementary does not for aa combination combination of of other other functions; notational distinction functions ;otherwise otherwise it it is is called called compound. compound. A notational distinction between elementary and and compound compoundterms terms will will not not be be made; in between elementary in fact, owing such a owing to the thevagueness vagueness of of conversational conversational language, language, such distinction can scarcely be carried it is distinction can scarcely be carried out uniquely. uniquely. However, However, it usually to assume assume that, that, in usually sufficient sufficient to in aa certain certain context, context,some somerule rule has has been laying down down this this distinction; the rule itself been introduced introduced laying distinction; the itself is is irrelevant. irrelevant. Furthermore, Furthermore,ifif by by regarding regarding certain certain terms terms as as elementary, elementary, a statement statement can can be be shown shown to to be be tautological, tautological, or or to to be bederivable derivable from some some other other statement, statement, these relations will not be changed if if from will not the assumed assumed elementary elementary terms are are further further subdivided. subdivided. GROUP A. TRUTH AND GROUP A. TRUTH ANDTRANSFORMATIONS TRANSFORMATIONS Definition D e f i n i t i o n 1. 1 . A statement as statement isisverifiably verifiably true if if itit isisverified verifiedas practically true at some practically true some time time during during the past, past, present, present, or or future future history history of mankind. mankind. (I-term). (I-term). If time, but but regarded Ifaa statement statementisisregarded regarded as as verified verified at at some some time, regarded as at aa later later time, as falsified falsified at time, then then the thelater laterdecision decisiontakes takesprecedence, precedence, 1

1

ESL, p. p. 81. 81. ESL,

FUNDAMENTAL TEnMS TERMS

19 19

being based based on on aa more more comprehensive comprehensivebody body of of evidence. evidence. The The earlier earlier being

decision is regarded decision is regarded as as erroneous. erroneous. say that Definition When we we say that aastatement statementppcan canbc bewritten written D e f i n i t i o n 2. 2 . When asp' top', meant that that pp and as p’or orthat thatppisisequisignificant equisignificant to p’, it it is is meant and p' p’ contain contain certain elementary terms terms and and that, that, in certain elementary in these these elementary elementary terms, terms, p' p’ is is tautologicaily equivalenttotopp,, or or is equipollent to p (see tautologically equivalent equipollent to (see ESL, ESL, p. 108). (I-term). (I-term). p. GROUPB. REDUCTION GROUP B. REDUCTION

The procedure procedure of of reduction reduction serves serves to to eliminate eliminate redundant redundant parts parts The from formavoiding avoidingunnecessary unnecessarycompcompfrom aa statement statement and andto togive giveititaaform lications. It is lications. It is obvious obvious that the thedefinition definition of of such such aa procedure procedure is to some extent a matter that the some extent matter of of taste. taste. However, However, it it will will be seen that the definition given leads leads to to statement statement forms definition given forms which which appear appear appropriate appropriate both from standards of of taste and from general general standards and from from the the viewpoint viewpoint of

constructing propositional propositional operations operationsthat that appear reasonable, constructing reasonable, in particular, that can particular, implications implications that can be be interpreted interpreted asasconditionals conditionals contrary to fact. contrary fact. However, However, the thelatter latterconsequence consequencewill willbecome become visible only only in in later chapters visible chapters of of this this presentation. presentation. In order order to tocarry carrythrough through the thereduction reduction procedure procedure we we first first

define redundant redundant parts, parts, and define and then then define define aa procedure procedure of of contraction contraction which serves serves to to diminish diminish the the number number of of binary in a which binary operations operations in statement. The statement. The contraction contraction procedure procedure is is subdivided subdivided into into two two forms, forms, according as as the expressions referred referred to to are synthetic or analytic. according analytic. The term 'analytic' ‘analytic’will will always always be be used used synonymously synonymously with the the term ‘tautological’. term 'tautological'. Definition contained inina statement D e f i n i t i o n3.3A . unit A unit contained a statementp pisisany anycomcombination of of signs signs in in pp such that, that, ifif this in thiscombination combination is is enclosed enclosed in bination parentheses within the statement, the resulting total expression parentheses within the statement, resulting total expression is equisignificant to p . (V-term). (V-term). is equisignificant to p. Definition is closed includes, A unit is closedif ifit it includes,for forevery every arguarguD e f i n i t i o n4.4A . unit ment variable in it, it, aa corresponding operator. An operator operator ment variable occurring occurring in corresponding operator. is redundant ifif its its variable variable does does not not occur occur in in any any functional functional within within is its scope. its scope. For instance, in '(x)/(x)' For instance, in ‘(x)f(x)’the unit unit 'f(x)' ‘f(x)’isis not notclosed, closed, whereas whereas the total total expression is aa closed closed unit. unit. In In '(x)/(y)' the operator '(x)' the expression is ‘(x)f(y)’the ‘(x)’ is redundant. is

20 20

FWNDAMEN!PAL TERMS

Double negation negation lines lines are are redundant, redundant, except Definition D e f i n i t i o n 5. 5 . Double except if if such their scope is aa unit which is is binary-connected binary-connected to to aa unit unit u, such u,which their scope is unit u1 that u1 that u, isis equisignificant equisignificant to to u2. u,. The term 'binary-connected' ‘binary-connected’ refers to connection connection by means means of of of the exception a binary binary operation. operation. The The significance significance of exception made made in definition will be be explained explained presently. presently. Note Note that that the definition 55 will the term term 'scope ‘scope of line.'1 of a negation' negation’ is meant meant to to include include the negation negation line. Definition 6 . If uu is is aasynthetic synthetic unit, unit,then thenan anelementary elementary D e f i n i t i o n 6. propositional term, or or an elementary in u is propositional term, elementary function, function, occurring occurring in is redundant can be be written written without without this this term, or function, redundant ifif uu can function, and without replacing replacingitit by by some term or function without some term function not already already used used in in u. u. (V-term). (V-term). If than once If the the elementary elementary term, term, or orfunction, function, occurs occurs more more than onoe in in u, u, the phrase 'without the ‘without this term' term’ is is to tomean mean that thatall alloccurrences occurrences of of the term addition about about replacing the term term by by the term are a m eliminated. eliminated. The The addition replacing the another one another one is is necessary necessary because because variables variables can can of of course course be be given given c)' the the term term 'c' redundant, different names. names. For For instance, instance, in different in 'a. ‘ a .(a (avv c)’ ‘c’ is is redundant, whereas ‘a’ 'a' is not, although although 'a' ‘a’could could be be eliminated eliminated by by replacing replacing whereas it by 'b', which latter term, however, does not occur in the it by ‘b’, which latter term, however, does not occur in theoriginal original statement. statement. Definition 7 . A synthetic synthetic unit unit u,u,isiscontractible contractible ifif canceling canceling D e f i n i t i o n 7. binary-connectedunits unitswithin withinu1 u, together together with with the the sign sign of of their their binary-connected connecting operations operations leads leads to to a unit u2which which isisequisignificant equisignificant connecting unit u2 with If and and only with u1. %. If only if if adding adding negation negation lines lines on on units units inside inside %makes it possible to cancel other units, units, it it is possible to cancel other is admitted admitted and and required required for for the process areredundant. redundant. process of of contraction. contraction. The The canceled canceled units units are ((V-terms). V-terms). This definition of contraction, contraction, which which applies appliesonly only to to binary This dehition of binary operations, may be illustrated in application to the statement operations, may be illustrated application the statement (1)

Here u, u1 isis the the whole wholeformula. formula.IfIfthe the term term ‘a.F, 'a. which which is Here is binarybinaryconnected by the preceding implication sign, is canceled together connected by the preceding implication sign, is canceled together with sign and and aa negation negation line line is is added added on on the the term term with the the implication implication sign 1

ESL, ESL, p. 25. 25.

TEEMS FUNDAMENTBL TERMS

21 21

the resulting 'c.d', ‘c.d’, the resulting form form (2) (2)

(a 3 b ) 3 c . d

represents the represents the the unit unitu2 u,which which is is equivalent equivalent to to u1. %. The The form form (2) (2) is is the reduced form of of (1). Another Another example example is is given given by by the the contraction reduced form contraction into 'aD of of '(aD ‘(a3b)b—_ ) = b' b’ into ‘63 b'; b ’ ; or or in in the thecontraction contraction of of 'a.(b ‘a.(bv 6)’ 'a'. Note that is not not a unit before into ‘a’. that in inthe thefirst firsttwo twoexamples examples u2 u,is before into the canceling, whereasitit is is so in in the the last lastexample. example. canceling, whereas The term inapplicable to to The term 'binary-connected' ‘binary-connected’makes makes definition definition 77 inapplicable expressions like‘a’, 'a', within within which whichthe the unit unit ‘a’ 'a' isis not not binary binary conconexpressions like nected. The by canceling nected. The reduction reduction of of such such expressions expressions is is achieved achieved by canceling

the double negation negation lines, lines, which whichare arenot not units, units, but but are are redundant redundant the according to to definition according definition 5. 5. It It isispossible possible to to set setup upeven evenstronger strongerrequirements requirements for for concontraction; traction ;for for instance, instance, the theintroduction introductionof ofparentheses parentheses may may enable enable us to to cancel cancel aa unit, unit, m as in in the the transition transition from from ‘a. 'a.bb v a.Ô' us a. C’ ttoo 'a.(b ‘a.( b vye)'. c)’. But definition But definition 77 appears appears sufficient sufficient for for our our purposes. purposes. Whereas definitions6-7 6—7refer referonly onlytoto synthetic synthetic units, units, the Whereas definitions following definition gives rules of contraction for analytic units. following definition gives rules of contraction for A separate separate treatment treatmentofofthese thesetwo twocases casesisisunavoidable, unavoidable,because because all all analytic units units are are equivalent to one another analytic equivalent to another and therefore therefore the condition of of equisignificance equisignificance does doesnot not supply supply aa sufficient sufficient restriction restriction condition for the reduction fact, if the reduction process. process. IIn n fact, the word word 'reduction' ‘reduction’ is is not not carefully modified,every everyanalytic analyticstatement statementcan canbebe ‘reduced’ 'reduced' to carefully modified, some such such simple form as as 'a some simple form ‘av a'. 6’.Although the operation operation of of concontraction, introduced 7, can can be be taken taken over for analytic analytic in definition definition 7, over for traction, introduced in units, itit will be modified soas as to to apply apply merely merely to to the will therefore therefore be modified so the major unit,asasfollows: follows: major operation operation of aa unit, Definition An analytic 8. An analytic unit unit u1 u, whose whose major major operation operation is is D e f i n i t i o n 8. binary is cancelingone one major major term, term, possibly after after contractible, ifif canceling is contractible, canceling or adding adding aa negation negation line line on on the the other other major major term, term, leads leads canceling or The canceled to an analytic canceled term term is is redundant. redundant. (V-terms). (V-terms). to analytic unit ug.The The following definitionapplies appliesboth bothto to synthetic synthetic and and analytic The following definition analytic statements. statements. Definition D e f i n i t i o n 9. 9. AAstatement, statement, or or aa unit unit in in aastatement, statement, is is reducedififititcontains containsno nocontractible contractible units units and and no no redundant redundant elemenreduced elementary terms, functions, or negation negation lines. lines. ((V-term). V-term). tary functions, operators, operators, or

22 22

FUNDAMENTAL TERMS

Examples for for synthetic statements: Examples statements : non-reduced form non-reduced form

reduced reduced form form

(3)

(aDb).(aDb)

a

(4)

{aD(cJb)].[aD(öDb)]

aDc

(5)

(6)

(7)

.

g(x) v

.

D a].

g(y) D

a] (x)[f(x) D g(x)]

D g(x). [h(y) D h(z)]}

The will here here be be understandable. understandable. On On the the The application application of of definition definition 8 will left-hand side is the the whole statement, and left-hand side of of (3), (3)) u1 is whole statement, and u2 u2results results by by adding everything adding aa negation negation line line on on 'a'. ‘a’.On On the theleft-hand left-handside sideof of (4), (4), everything is canceled followingthe thefirst first occurrence occurrenceofofthe theletter letter ‘c); 'c'; then then the canceled following the negation line on on top top of negation line of this this letter letter isiscanceled. canceled. This This statement statement can can also be be reduced by the help also reduced by help of of definition definition 6, 6, because because it contains contains term ‘b’. 'b'. In the redundant redundant elementary elementary term I n (5) (5)the the redundance redundance of an elementary function functionisis visible visibleonly onlyafter after the the statement statement is transelementary formed into aa one-scope one-scope form (also (also called prenex form). This and and the examples (6)—(7) show that the proof of the equivalence of the the examples (6)-(7) show that the proof of the equivalence of the reduced form form to to the original reduced original form form may involve tautological transtransformations concerning concerning operators. operators. In In (6), for instance, instance, the the operators formations (6)) for operators are moved to their functionals, and the statement moved close close to functionals, and statement then then assumes the form assumes the form (3). (3). Double negation lines lines are in Double negation in general general redundant, redundant, according according to to definition 5, and and thus aa reduced definition 5, reduced expression expression carries carries in in general general no no double negation lines. lines. An An exception exception isis given given by by the tautology double negation tautology >00 requires therefore aa proof proofthat that d'd' = 0 (161) requires P(B) P(B)= 1; therefore =0 can here be be given given by by showing showingthat that P(B) P(B) < < 1.1. For can here For this thiscase, case, relation relation (162) is no no longer longer derivable, derivable,and and the the value value PP(A) (162) is ( A ) is subject ttoo no no specific restrictions.But But this this case specific restrictions. case has has no nopractical practicalsignificance significance because we can can never never prove proveinductively inductivelythat that dd is is strictly strictly = = 0. because we 0.

APPxNOtX APPENDIX

131 131

It It should should be be noted noted that, that,although although (162) (162) allows allows us to go go from from (158) to to (159), relation (162) (162)isis not not sufficient to provide for a (158) (159), relation sufficient to provide for transition from (159) to (158). (158).For For the the latter transition, transition from (159) to transition, it must must that his leads be required required that dd 5 d', d’, and and (161) (161) shows shows that leads to the the condition condition 11— - P(B) P ( B ) 5 P(A), P ( A ) ,or or P(A) P ( A )+ +P(B) P ( B ) 2 11

(163) (163)

Conditions (162)and and (163) (163) are are compatible only for the Conditions (162) compatible only the special special case of the equality case of equality sign. sign. In I ngeneral, general, therefore, therefore, we we can can proceed proceed only in one direction. For instance, if we we know that that (162) (162) holds, holds, a proof of of (158) (158) isis aa proof proof of of (159), (159),but butnot not vice vice versa. versa. In In the usual proof application of inductive inductiveverification, verification,the theterms terms‘A’ 'A' and and ‘B’ 'B' are so application of so This explains defined that (162) is satisfied. defmed (162) is satisfied. This explains why why confirming confirming evidence for (158) (158) is is also alsoconfirming confirming evidence evidence for for (159), (159), whereas whereas

confirming evidence for for (159) is not confirming evidence not confirming confirming evidence evidence for for (158). (158). These considerations considerationssupply supplythe the answer answer to to a so-called paradox of of These so-called paradox confirmation pointed pointed out by confirmation byC. C.Hempe!. Hempel.1

For instance, although is not high that although the probability probability is that aahouse house probable that that something not not red is is highly probable is not a house. house. is red, it is d' is very Here d’ very small, small, whereas dd is not small, and making d' d’ even even smaller by further further confirming confirming evidence evidence has scarcely scarcely any any influence influence upon d. The ratio ratio between and d’ d' is given by by (16!) and is rather rather between dd and (161) and large, because the number number of large, because the of things things that thatare arenot notred, red,i.e., i.e.,11— - P(B), P(B), is much much larger larger than the thenumber number ofofhouses, houses, i.e., i.e., P(A). P ( A )Condition . Condition (162) isis here here satisfied. satisfied. In In this example, (162) example, of of course, course, nobody nobody would would assert to the assert that that all all houses houses are are red, red, because because too many instances instances to contrary consider the the statement, statement, 'all contrary are are known. known. However, However, consider ‘allbuildings buildings made by by man are lower than than 1300 feet'. Although Although this this statement made 1300 feet’. statement is is true up to to the thepresent present time, time, we we would would not be be willing willing to assert it for all times times;; the general probability that aa building building made made by man man be lower than than 1300 feet can can scarcely scarcely be be estimated estimated as as high high as as 1. Its Its be lower 1300 feet contrapositive, in contrast, it is contrapositive, in contrast, expresses expresses aa high high probabifity, probability, since since it is highly probable probable for for all all times times that that something than 1300 feet highly something not not lower lower than 1300 feet will not be will not be a building building made made by by man. man. A A confirming confirming instance instance for for this this contrapositive form,for for instance, instance,aa mountain mountain that that is higher than contrapositive form, 1

See See

ThP, ThP,p. p.

435. 435.

132 132

APmNDIx

1300 feet, 1300 feet,

will w ill not change change our estimate estimate of of the theoriginal original form. form. The third condition that no The third condition of of schematization schematization isis that no exceptions exceptions to to (155) be known knownand and that that we that aa class (155) be we have have no no evidence evidence that class C C can can be defined by us us such defined by such that P(A.C, B B)) < >00because becauseofof(163). (163).Therefore, Therefore, 1, requires P(A, = 0, = 00 requires d= P ( A ,C) C) = 0, or in in other other words: words: if if P(A, P ( A ,B) B)= = 1, exceptions C C can can occur occur only only in in a zero-frequency. This means means that that exceptions zero-frequency. This even in this case are not not impossible, but the the limit limit of of even case exceptions exceptions are impossible, but d> their relative must be their relative frequency frequency must be = 0. 0 .2 2 If d >0,0, exceptions exceptions can can occur in in aa higher however, is is subject subject to to occur higher frequency frequency P(A, P ( A ,C), C ) , which, which, however, (166).In In order order to to study study this this relation let us put the restriction restriction (166). relation let (167) (167)

'

P(A,C)=/, P(A, C ) = f,

P(A,B)—P(A.C,B)=e P(A, B ) - P ( A.C, B ) = e

equation (15). For the above ThP, p. 79, left left part of equation (15). For above form we have p. '79, interchanged 'B' and 'C'. interchanged ‘B’ ‘C’. a The class GC can can then still be infinite; infinite; see ThP, p. 72. 72.

APPENDIX

133 133

C,, and We call call ff the thefrequency frequency of of the the exception exception C and ee the thedegree degree of of the the We exception Thedenominator denominatorinin (166) (166)can canthen then be be written written in in the . The the emeption CC. form e + d; for dd,, we we thus find form d ; solving solving (166) (166) for find

+

(168)

- f- e 5 d 1-f

This relation relation may be called /or exceptions. states This called the the restriction restriction for exceptions. ItIt states that, for that, for aasmall smallvalue value d, d,aahigh highdegree degreeee of of exception exception is is restricted restricted /, and to aa low low frequency frequency f, and aahigh highfrequency frequency /f of ofexceptions exceptions is is restricted to aa low e. The The highest highest degree degree of of exception exception is restricted low degree degree e. which case case (166) (166)furnishes furnishesf f5d;d ; assumed for for PP(A.C, B)) = = 00,, for which assumed ( A . C ,B lower of exception exception allow allow for for aa somewhat lower degrees degrees of somewhat higher higher f,f, which which however (168). however is is controlled controlled by by (168). These considerations considerationsshow showthat that aa high high probability probability P P(A, ( A , B) B ) is no guarantee for the absence If we guarantee for absence of of exceptions. exceptions, If we can prove prove that that P(A, B) is close to 1 within the interval d, we know that exceptions P ( A , B ) is close to 1 within the interval d, we know that exceptions are subject to restriction but they may restriction (168); but may exist. exist. Even Even ifif we we exceptions could could still still occur, occur, though though they they are are could show = 0, could show that that d = 0, exceptions limited to to a zero-frequency. It follows that a proof for for the the absence limited zero-frequency. It follows that of exceptions of exceptions must be be based based on onconsiderations considerations involving involving other other evidence than merely evidence for a high value of P(A, B). evidence than merely evidence for a high P ( A , B). It is that the laid It is in this this connection connection that the requirement requirement of of universality, universality, laid down for original original nomological nomologicalstatements, statements, assumes assumes aa most most important important down for function. If If an all-statement function. all-statement is restricted restricted to aa certain certain space-time space-time region, region, there there may may exist exist special special conditions conditions in in this this region region which which make make the all-statement all-statement true, true, whereas whereas itit may maybe befalse falsefor forother otherregions. regions. If we If we can can maintain, maintain, without without any any restriction restriction to to individuals individuals or or it appears individual regions, that that condition individual space-time space-time regions, condition (158) is true, it improbable that we improbable that we could could ever ever define define aa class class C C for for which which (164) (164) holds. Thus universality represents some someguarantee guaranteethat, that, as as is holds. Thus universality represents is required for for all-statements, not even required all-statements, not even aa zero-frequency zero-frequency class class of exceptions exists. exceptions exists. The rules laid down down in in the thedefinition definitionof of admissible admissible implications implications serve therefore therefore as an instrument instrument to to equip equipsuch suchimplications implications with with inductive validityyand the prospect inductive prospect of of truth truthwithout without exceptions. exceptions. It It isis for for this thisreason reason that thatsuch suchimplications implications can can be be used used for for prepredictions and for conditionals contrary to to fact. dictions and conditionals contrary fact.

BIBLIOGRAPHY C.,‘The 'The Given Given and and Perceptual BAYLIS,C., Perceptual Knowledge', Knowledge’,Philosophic PhilosophicThought Thought Buffalo: University University of of Buffalo Buffalo in France France and and the the United United States, States, Buffalo: Publications 181—201. 1950,pp. pp. 181-201. Publications in inPhilosophy, Philosophy,1950, BEARDSLEY, E. E. L., L., "Non-accidental" BEARDSLEY, ‘‘Won-accidental”and andcounter-factual counter-factual sentences', sentences’, Journal of of Philosophy, Philosophy, 46 46 (1949), (1949), pp. pp.573—591. 673-691. BURRS, A.,‘The 'The Logic Logic of of Causal Causal Propositions', (1951), pp. 363—382. BURKS,A,, Propositions’,Mind, Mind,6060 (1951), pp. 363-382. CARNAP, R.,‘Testability 'Testability and and Meaning', CARNAP, R., Meaning’, Philosophy Philosophy of of Science, Science, 33 (1936), (1936), pp. 419-471; 419-471; 44(1937), (1937),pp. pp.2—40. 2-40. Syntax of of Language, New York: York: Harcourt, Brace , The Logical Logical Syntux Language, New Brace and Co., Co., 1937, 1937, §69. 3 69. Meaning and Nece&sity, University of of Chicago , Meaning Necessity, University Chicago Press, Press, 1947. 1947. CHISROLM, R., R., ‘The 'The Contrary-to.fact Conditional', Mind, Mind, 55 55 (1946), CHISHOLM, Contrary-to-fact Conditional’, pp. 289-307. pp. 289—307. Diaas, Conditionals', pp.pp.513—527. DIGGS,B. B.J., J.,'Counterfactual ‘Counterfactual Conditionals’,Mind, Mind,61(1952), 61 (1952), 613-527, GOODMAN, N., 'A ‘A Query Query on on Confirmation', Confirmation’, Journal of of Philosophy, Philosophy, 43 43(1946), (1946), N., pp.383—385. 383-386. pp. , 'The ‘The Problem Problem of of Counterfactual Counterfactual Conditionals', Conditionals’, Journal Journal of of PhiloPhilosophy, SOPhy, 44 44 (1947), (1947),pp. pp.113—128. 113-128. HAMPSHIRE, 'Subjunctive Conditionals’, Conditionals', Analy8is, HAMPSHIRE,S.,S.,‘Subjunctive A d y 8 i s s ,99(1948), (1948),pp. pp.9—14. 9-14. HEMPEL,C.C.and P.,P., ‘Studies Explanation’, HEMPEL, and OPPENHEIY, OPPENIrEIM, 'Studiesininthe the Logic Logic of Explanation', Philosophy of of Science, Science, 15 15(1948), (1948),pp. pp.135—175. 135-175. LEWIS, I., Analysk Analysis of of Knowledge Knowledgeand and Valuation, Valuation, La La Salle, LEWIS, CC. . I., Salle, Illinois, Illinois, Open Open Court Court Publishing PublishingCo., Co.,1947, 1947,pp. pp.219—253. 219-263. and LANGFORD, H.,Symbolic SymbolicLogic, Logic,New NewYork York and and London: and LmaFoRD, C.C.H., London: The Century Century Co., Co., 1932, 1932, Ch. VII. VII. PErRCE,C.C.S., S.,Collected CollectedPapers, Papers,v.v.I1 II (Elements (Elements of PEIRCE, of Logic), Logic),Cambridge, Cambridge,Mass.: Mass. : Harvard Harvmd University University Press, Press, 1932, 1932, p. p. 199. 199. POPPER, K. K. R., R., 'A ‘ANote Noteon onNatural NaturalLaws Lawsand andso-called so-called"contrary-to-fact “contrary-to-fact Conditionals”’, Mind, Mind,5858(1949), (1949),pp. pp.62—66. 62-66. Conditionals", RAMSEY,F.F.P., P., Poundations Foundations of of Mathematics, Mathematic8,New NewYork: York: Harcourt, Harcourt, Brace Brace and and Co., CO.,1931, 1931,pp. pp.237—257. 237-257. REICHENBACH, H.,H.,Elements MacMiIlm Co., Co., REICHENBACH, ElementsofofSymbolic SymbolicLogic, Logic,New New York: York: MacMillan 1947, Ch. Ch. VIII, VIII, quoted as 1947, as ESL. ESL. of Probability, Probability, Berkeley Berkeley and and Los , Theory Theory of Los Angeles: Angeles: University Wniversity of of 1949, quoted ThP. California Press, 1949, quoted as ThP. California Press,

BIBLIOGRAPHY BIBLIOGIRAPHY

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N.,Principia Principia Mathematica, Mathematica, I, I, Cambridge RUSSEL,U. WHITEHEAD, A.A.N., B. and WHITEHEAD, RUSSEL, University Press, University Press,1925, 1926,pp. pp.20—22. 20-22. New York: York: W. W. W. Norton Norton and ,The The Principles Principles of of Mathematics, New and Co., Co., 1938, pp. pp.492—493. 492-493. SIMoN, H.A., A.,‘On 'On the the Definition Definition of of the the Causal Relation’, Relation', Journal of SIMON,H. of PhiloPhilosophy, 49, 49, 1952, 1952,pp. pp.517—528. 517-528. WEINBERG, Jvirus,‘Contrary-to-fact 'Contrary-to-fact Conditionals’, Conditionals', Journal Journal of WEINBERG,JULIUS, of Philosophy, Philosophy, 48 (1951), (1951), pp. pp.17—22. 17-22. WILL, F. L., L., ‘The 'The Contrary-to-fact Conditional', Mind, 56(1947), pp.pp. 236—249. WILL,F. Contrary-to-fact Conditional’, Mind, 56 (1947), 236-249.

TABLE OF OF THEOREMS THEOREMS

Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem

1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 10. 10

. . . .

1. 2. 3.

•

.

•

.

. . •

.

•

.

4. . . 5. . . 6. . . 7. . . 8. . . 9.. . . •

•

page Page .. 24 24 .. 31 31 31 . 31 .. 32 32 43 . 43 55 . 55 56 .. 56 64 . 64 65 65 66 66 .

. ....

Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem

page Page

....

11 11•

. . . . .... .... ... .... .... .... ....

12 . 12 13 . 13 14. 14 15 .. 15 16 . 16 17 . 17 18. 18 19 . 19 20. 20

....

69 69 69 69 74 74 76 76 76 76 77 77 77 77 78 78 78 78 81 81

Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem

21 . 21 22 . 22 23 23 . 24.. 24 25.. 25 26 26 . 27 27 . 28.. 28 29 . 29 30 30 .

. . •

.

•

•

•

.

•

.

. . . .

. . . . . . . .

..

. .

. .

•

•

page Page .. 82 82 . 88 88 . 103 103 . 105 105 . 107 107 .. 107 107 . 108 108 . 108 108 . 108 108 .. 112 112 .

.

TABLE OF OF DEFINITIONS DEFINITIONS

Definition Definition Definition Definition Definition Deftnition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition

1 .. 1 22 .. 3 3 .. 4 4..

... . . . . . .

.

.

.

.

.

.

.

.

.

.

.

•

.

.

.

.

•

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

•

.

.

.

.

.

.

.

.

.

.

•

.

.

.

.

.

.

. . .

5 . . . . 5.

6 .. 6 77 .. 8.. 8 9 .. 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17

. . .

. . .

...

. . . . . .

. . .

. . .

. . . . . . ... . . .

.. .

18 18 19 19 19 19 19 19

20 20 20 20 20 20 21 21 21 21 27 27 28 28 29 29 29 29 29 29 29 29 30 30 30 30

Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition Definition

18 .. 18 19 .. 19 20 20 .. 21 .. 21 22 22 .. 23 23 .. 24 24 .. 25 25 .. 25a 25a .. 26 .. 26 27 . 27 28 28 .. 29 29 30 30 .. 31 31 .. 32 32 .. 33 .. 33

.. . .

. . . . . . .

.

.

.

.

.

.

.

.

.

. .

. . . . .

.

.

.

.

•

. .

. .

. .

... . . . . .

•

.

.

•

.

.

•

.

.

.

.

.

.

.

.

.

.

. .

. . . .

. .

30 30 30 30 31 31 31 31 31 31 32 32 32 32 33 33 33 33 37 37 48 48 60 60 66 66 66 66 67 67 67 67 67 67

Definition 34 Definition Definition 35 Definition 35 Definition 36 Definition 36 Definition 37 37 Definition Definition Definition 38 38 Definition Definition 39 39 Definition 40 Definition 40 Definition Definition 41 41 Definition Definition 42 42 Definition Definition 43 43 Definition Definition 44 44 Definition Definition 45 45 Definition Definition 46 46 Definition Definition 47 47 Definition Definition 48 48

. . . . . . . . . •

.

•

•

.

.

... . . .

•

•

.

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . .

•

•

•

•

.

. •

•

.

. .

.

...

70 70 71 71 75 75 89 89 90 90 94 94 95 95 101 101 101 101 112 112 112 112 119 119 119 119 119 119 120 120

TABLE OF DEFINITIONS IN THE APPENDIX TABLE DEFINITIONS IN APPENDIX Definition I Definition II Definition I1

. .. . .

. . .

.

.

.

126 126 127 127

Deffi-ijtjon Ill Definition I11

. . . .

.

.

.

.

127 127

Definition IV I V .. . . 127 Definition 127

Definition V .. Definition Definition Definition V VII ..

..

.

.

.

.

. .

127 127 128 128

INDEX admissible: admissible : 75; 75 ; fully, 71, 71, 75, 75, 79; 79; fully, t o p, p, 95; 96; relative to semi-, serni-, 71, 71, 73, 73, 74, 74, 75, 75, 78, 78, 79 79 all-statement all-statement: proper, proper, 40 40 analytic, 19 19 antecedent: antecedent : major, major, 96, 96, 98; 98; minor, minor, 96, 96, 98 98

contradiction, contradiction, 29 29 contraposition,77, 77,78, 102,106, 106,123 123 contraposition, 78, 102,

binary-connected, 20, 21 2I binary-connected, 20,

47, 47, 48, 48, 68; 68; elongated, 29 29

calculus: calculus : of functions, 5; of functions, 5; lower functional, 11, 18; 18; lower functional, 11, completeness of, of, 11; 11; completeness propositions, 25, 25, 60 00 of propositions, Carnap, R., Carnap, E., 55

causal causal relation, relation, 4 C-form: 27, 28, 28, 41, 61; 61; C-form: 27, elongated, 29 elongated, 29 class:: class attribute, 126; attribute, 126; reference, reference, 126; 126; open, 127 open, 127 closed, closed, 19 19 conditional contrary 7, 8, 8, conditional contrary to to fact: fact: 7, 14, 19, 19, 68, 68, 83, 83, 86, 86, 112, 112, 120, 120, 124; 124; 14, primary, 96; 96; primary, reasonable, reasonable, 90; 90; regular, regular, 88; 88; secondary, 96 secondary, 96 confirmation: confknation: paradox of, paradox of, 131 131 conjunctive: conjunctive: non-, 71 non-, 71 consequent, 96 consequent, 96 contractible, 20, contractible, 20, 21 21 contraction, 20, contraction, 20, 23 23

counterfactuals: counterfactuals :92 92

88, 90, 99, of non-interference, non-interference, 88. 99, 101 101

datum: datum : observational, 85 observational, 85 description, definite, definite, 35, 35, 36 36 description, D-form: D-form: 27, 27, 28, 28, 41, 41, 43, 43, 44, 44, 45, 45, 46, 46,

entailment: entailment : logical, 1; 1; physical, 1, 1, 56 equipollence: equipollence: 11; 11; of meanings, meanings, 119 119 equipollent, equipollent, 10, 10, 16, 16, 19 19

equisignificance, 21 equisignificance, 21 equisignificant, equisignificant, 19, 19,20 20 equivalence: equivalence : relative, 112; 112; relative admissible, admissible, 112 112 exhaustive: exhaustive :

in elementary terms, 30, 50; 30, 50; in major terms, terms, 30, 30, 49; 49; non-, 45; non-, quasi-, 67, 67, 69, 69, 70, 70, 72, 72, 73; 73; unrestrictedly, 37, unrestrictedly, 37, 38; 38;

exhaustiveness: 9,43, 48, 49, 50, 52, exhaustiveness: 9,43,48,49,50,52, 55, 55, 56, 56, 71; 71; non-, 43, 44 expansion:: expansion q-expansion, 37; r-expansion, 44 44 r-expansion, explicandum, 2 explicandum, explicans, 2, 5, 6, explicans, 6, 124 124 explication, 2, 124 explication, 2, 124

i38 i38

extension:: 31; extension 31; conjunctive, 31; conjunctive, 31 ; disjunctive, disjunctive, 31 31 false, 120 120

frequency interpretation, interpretation, 91 91 functions:: functions calculus of, of, 5; calculus 5; lower, 11, lower, 11, 18 18 Goodman, Goodman, N., N., 34, 34,106 106

Hempel, C, C., 36, 36, 131 131 implicans, 48, 49, 49, 50, 50, 52, 52, 57, 57, 58, 58, 75 75 implicans, 48, implicate, 9, 50, 50, 52, 52, 57, 57, 58, 58, 74 74 implicate, 9, implication: 1, 4; absolute, 109; 109; accent-, accent-,120, 120, 122; 122; 4, 7, 7, 8, 8,10, 10,115; 115; adjunctive, 4, semi-, semi-, 122; 122; admissible, 90, 106; 106; admissible, 7, 7, 13, 13, 90, relative, 121; relative, 121; semi-, 106; semi-, 106; analytic, analytic, 5, 70; arrow-, 120, arrow-, 120, 123; 123; connective, 4; 4; contrary, contrary, 8, 8, 81, 81, 82, 82, 83; 83; contrary relative, 108; contrary 108; converse, 51, 52, 53, 55, 73, converse, 51, 73. 75; double, 51, 56, 66, 58, 58, 111; 111; double, 51, general, 14, 14, 49; nomological, 1; nomologicd, 1; proper, 120, 120, 121; 121; reasonable, 1, 9, 10, 10, 14, 14, 42, 42, 73, 73, reasonable, 82, 82, 101; 101; on-, un-, 23, 23, 49, 49, 68; 68; ,relative, 97, 97, 98, 98, 99, 99, 109, 109, 112; 112; separable, 101; separable, 101; serial, 96, 103, serial, 103, 105; 105; synthetic, 69, 82; tautological, 5, 82, 82, 113, 113, 116; 116; tautological, 5, with antecedent, 88 88 with contrary contrary antecedent, impossible, 66 impossible, indeterminate, 121 121 35, 38 38 individual-term, 32, 32, 34, 34, 35, individual-term,

INDEX INDEX

inductive: inductive : extension, 13; 13; generality, 12, 12, 13; 13; inference, inference, 12; 12 ; verification, 12, 12,13, 13, 14 14 interpretation: interpretation : adjunctive, 3; 3; connective, 33 invariance principle, 80 invariance principle, 80 I-requirement, 16,40, 40, 48, 48, 67 67 I-requirement, 16,

Kalish, D., P., 36 Kalish, 36

knowledge: knowledge : advanced, advanced, 127 127 known:: known statistically, 127 127

language: language : artificial, 14; 14; conversational, 7, 10, conversational, 10, 14, 14, 18, 18, 34,

123; 123; natural, 14, natural, 14, 15, 15, 34 34 law: of of logic, logic, 1; 1; law: of nature, 1,1,12, of nature, 12,124, 124,125 125 logic:: logic three-valued, three-valued, 122 122

matrix: 24; 24; complete, 26, 27, 27, 39, 39, 51, 51, 63 63

meaning: meaning : alternating, 117; 117 ; logical, 122; logical, 122 ; physical, 120; physical, 120; restricted, 120 120 meaningless: meaningless: 120, 120, 121; 121; physically, 121, 121, 122 122 modalities:: 6; modalities 6; absolute, 7; 7; logical, 7; logical, 7; physical, 7; 7; relative, 77 relative, necessary, 6 necessary, nomological: nornological: 6; 6; absolute, 57; 57 ; analytic, 2, 2, 5; analytic,

rNDEx INDEX

derivative, 60; 60; in the narrower narrower sense, sense, 5, 6, 6, 60; 60; in the wider in wider sense, sense, 6, 6, 60, 60, 66; 66; original-, 5, original-, 5, 10, 10,35, 35, 38, 38, 41, 41, 48, 48, 53, 53, 55, 56, 61, 64, 82; 55, 82; relative-, 95; 95; synthetic, synthetic,2, 2, 5, 5,64, 64, 71 71 observational: observational : datum, 85; 85; procedure, 85 procedure, 85 operand, operand, 43, 43, 45 45 operation: operation : adjunctive, 3, 3, 4; admissible, admissible, 124; 124; binary, 20; 20; connective, connective, 3, 3, 4, 4, 60; 60; nomological, 2,4, nomological, 2, 4, 5, 5, 6, 6, 57; 57; relative relative nomological, nornological, 57; 57 ; propositional, 56; propositional, reasonable, 2,3,4,5,56,57,66,124 2,3,4,5,56,57,66,124 reasonable, operator: operator: 43, 43, 45; 45; all-, 29, 29, 45, 45, 60, 60, 61; 61; commutative, commutative, 27; 27; 9, 37, 37, 41, 41, 45, 45, 61; 61; existential, 9, iota-, iota-, 35 35 operator-derivable, 25 operator-derivable, 25 paradox paradox of of confirmation, confirmation, 131 131 permissible, permissible, 90, 90, 101, 101,112 112 posit, 91 91 possible: possible : merely, 6; 6; physically, physically, 85 85 probability::13; probability 13; direct, 126, 126, 127; 127; genuine, genuine, 89, 89, 90, 90, 128; 128; indirect, 127 127 procedure: procedure : observational, observational, 85 85 properties: properties : invariant, 16; 16; variant, variant, 16 16 propositions: propositions : calculus calculus of, of, 25, 25, 60 60

139 139

propositionally derivable, derivable, 25 25 propositionally Quine, Quine, W., W., 106 106 reasonable within reasonable within a certain certain context, context,

74 74 reconstruction rational, 34 34 reconstruction:: rational, reduced, 21, 22, 22, 24, 24, 45 45 reduced, 21, reduction, 10, 19, 19, 45 45 reduction, 10, redundant, 19, 19,20, 20, 63 63 redundant, reference class: class :126; 126 ; reference open, open, 127 127 residual: 29, 29, 37, 37, 45, 45, 48, 48, 53, 53, 61; 61; residual: conjunctive, conjunctive, 31, 31, 63; 63; disjunctive, 31, 31, 63; 63; disjunctive, self-contained, 41 self-contained, 41 retroaction, 99 retroaction, 99 Russell, B., 9, 9, 49, 49, 50 50 Russell, B.,

schematization: 129; 129; schematization: condition condition of, of, 129, 129,130 130 self-contained, 31, 31, 32 32 self-contained, separability:: separability conditions conditions of, of, 100, 100,106 106 separable, 101,105, 105, 112 112 separable, 101, statement: statement : admissible, 5, admissible, 5 , 67, 67, 75, 75, 101; 101; fully-, 67; fully-, 67; 68; fully- by by derivation, derivation, 68; fullysemi-, semi-, 67; 67; all-, all-, 8, 8, 10, 10, 12; 12; analytic, 7; 7; analytic, compound, 3, 4; elementary, 3; 3; nomological, nomological, 2, 2, 4, 4, 5, 5, 38, 38, 57, 57, 61; 61; absolute, absolute, 57; 57 ; analytic, 2, 5; derivative, 60; derivative, 60; in narrower sense, sense, 5; in the narrower in the in the wider wider sense, sense, 60; 60; original, 5, 5 , 10, 10, 35, 35, 38, 38, 41, 41, 48, 48, 53, 56, 56, 61, 64, 82; 82; synthetic, 2, 2, 5, 5, 64, 64, 71; 71; purely existential, existential, 41; 41 ; tautological, 5; 5; universal, universal, 33 33

140 140 supplementable, supplementable, 70,

synthetic, synthetic, 19 19

INDEX INDEX

73, 73, 78 78

tautology: tautology: 5, 5, 8, 8, 10, 10, 23; 23; admissible, admissible, 69; 69 ; inadmissible, inadmissible,69; 69 ; terms: terms :elementary, elementary,47 47 true: true: 120; 120; of first first order, order, 66; 66; of second order, 66, 66, 79; 79; of third third order, order, 66, 66, 79; 79; verifiably, verifbbly, 11, 11, 18, 18, 30, 67; 67; verifiably in the the wider wider sense, sense, 85, 85, 87 87

truth: 11, truth: 11, 12; 12; analytic, analytic, 6; 6; factual, 6, 13; factual, 6, 13; orders orders of, of, 6; 6; nomological, 6, 13 nomological, 6, 13

universal, universal, 9, 9, 33 33 variables: 45;; variables : 45

argument, argument,16, 16, 26, 26, 45; 45;

bound, bound, 38; 38;

free, free, 18, 18, 38, 38, 60; 60;

functional, functional, 16, 16, 18; 18;

metalinguistic, metalinguistic, 17; 17;

object language, 17; object language, 17;

propositional, 71;; propositional, 24, 24, 71 sentential, sentential,

16, 18 18

verifiably verifiably true, true, 11, 11, 18, 18, 30, 30, 67 67 verification: inductive, inductive, 12, 12, 13, 13, 14 14 V-requirements, V-requirements,16, 16, 48, 48, 67, 67, 70 70 V-terms, V-terms,28 28 wholeness property, property, 74 74 weight, weight, 89 89