viii.--the logic of causal propositions

VIII.—THE LOGIC OF CAUSALPROPOSITIONS

BY ARTHUR W. BURKS

1. IntroductionIT is characteristic of a science that growth in its content isaccompanied by an increase in the technicality of its language.This is no accident, for the achievements of a well-developedscience cannot be completely expressed in the vague, ambiguous,and weak terminology of its early stages. For example, thelanguage of pre-Galilean physics is inadequate to express themodern conceptions of force and energy.

Unfortunately the remark just made about scientific languagesdoes not generally apply to philosophical ones. The notableexceptions are to be found in the fields of logic, both deductiveand inductive, where the techniques of symbolic logic have beenemployed to give precise and technical definitions of such conceptsas ' valid inference ', ' constructibility ', ' intensional', and' degree of confirmation '. This paper is intended as a contribu-tion along the same lines : it seeks to develop a language forexpressing causal propositions (contrary-to-fact conditionals,statements of causal laws, assertions of causal necessity andpossibility, etc.) which is more precise, explicit, and formal thanthe language of everyday discourse.

The new language is called the ' logic of causal propositions '.It will be developed by the following procedure. In section 2various types of causal propositions will be analysed as they occurin ordinary usage. To resolve ambiguities and vaguenesses aswell as to simplify the treatment, a new symbolism will beintroduced and the sentences being analysed translated into it.The logical principles governing inferences among causal andother propositions will then be formulated. Finally, in section3, the resultant symbolic language will be organized into a formalsystem.

It should be made clear at the outset that we are concernedwith the deductive interrelations of causal and non-causalpropositions, and that hence this is not a study in the logic ofconfirmation or the theory of probability. As the traditionaldivision of logic into inductive logic and deductive logic

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recognizes, there are two different kinds of relations into whichcausal propositions enter. In other words, to define ' cause ',' causal law', etc., completely, we must formulate both thedeductive and the inductive properties of these concepts. Inthis paper we will deal only with the former.

It might seem that we could best achieve our objective of atechnical language for discussing causality by ignoring presentusage and starting ab initio. This is not the case, however, sincethe primary motive for constructing a new language is to helpanswer questions (see the example in the next paragraph) whichhitherto have been of necessity phrased in the old language.Unless there is continuity between the new and the old we shallnot be answering the original questions but different ones. Onthe other hand, despite this continuity, there must be importantdifferences between the two languages, else one could not bemore technical than the other. Basically, in developing atechnical language out of a vague and ambiguous one we mustmake certain Bomewhat arbitrary decisions. Also, the symbolsof the new language should be chosen so as to be convenient froma formal point of view, whether or not they correspond directlyto the basic symbols of ordinary language; though of coursethe new symbolism should be sufficiently rich to make possibledefinitions of the old symbols (insofar as they are clear and precise)in terms of them.

The relations which obtain between a technical language andthe language of which it is an outgrowth may be illustrated bythe following example. Consider the question: Can causalpropositions be adequately translated into an extensional lan-guage (e.g., that of Prindpia MathenuUica) ? The first point tonote is that this question is a technical reformulation of a veryold metaphysical one: Can the concept of causal connexion bedenned in terms of ideas of matter-of-fact and constant conjunc-tion, i.e., can causal potentialities be reduced to actualities ?This issue was argued by the medieval nominalists and realists,but it is important from our point of view that stating it as theydid in a non-technical language they fused and confused it withmany other issues: questions concerning the ontological statusand value character of properties and concepts, of mathematicalprinciples, and of ethical laws. The second point to note is thatjust as symbolic logic aids in separating out this one issue ofcausal laws from all the others, it can profitably be employed tohelp solve it. For the logic of causal propositions tells us someof the properties of causal propositions, and hence providescriteria for judging any proposed extensional translation.

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THE LOGIC OF CAUSAL PROPOSITIONS 365

There is a matter related to this second point which warrantsattention. If the question stated in the previous paragraph hasan affirmative answer, i.e., if causal propositions are extensional,a knowledge of the extensional analysis of causal propositionswould make the development of the proposed logic unnecessary.So it might seem that the answer to the given question shouldbe regarded as a means to the proposed logic, rather than thereverse. Such is not the case, however, as the question underconsideration admits of no easy answer. For example, the simpletranslation of a law in terms of universal quantification andmaterial implication is inadequate (cf. 2.4). That an extensionaltranslation, if it exists, is complicated, is shown by the followingconsiderations.

The ultimate data on which causal laws and theories rest—statements reporting the results of observations and experiments—can certainly be expressed in an extensional language. Todetermine whether or not causal laws and theories are them-selves extensional we must examine the relation between themand their evidence. Since the relation between empiricalevidence and a law of nature is inductive rather than deductivein character this kind of consideration brings in all the com-plexities of inductive logic and probability theory. Hence itseems reasonable to begin with a study of the deductive proper-ties of causal propositions.

2. Analysis of Causal Propositions1

In this section we shall analyse causal propositions and theirrelations to extensional propositions (material implications, ordi-nary disjunctions, conjunctions, quantified statements, etc.) andlogical propositions (strict implications, assertions of logicalnecessity, possibility, impossibility, etc.). The results will begiven in the form of logical principles which in section 3 will beorganized into a formal system. We shall introduce in thissection only those principles governing causal propositions whichare not included in the ordinary machinery of existing logic.We intend these new principles merely to supplement those ofexisting logic, and shall construct the deductive system of section3 so that it contains both.

1 In these analyses I am indebted to private discussions with C. H.Longford, Paul Henle, and Irving M. Copi.

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2.1. Contrary-to-Fact Implication

Consider a hypothetical, contrary-to-fact statement about thepast, expressed in the subjunctive mood ; e.g., someone says 01a person coming out of a gas-filled basement: ' If he had lit amatch the whole house would have exploded '. This is clearly acompound sentence of the 'if . . . then . . .' form. It shouldnot, however, be analysed as ' If (he had lit a match) then (thewhole house would have exploded)', for the antecedent andconsequent of this sentence are subjunctive rather than indicativeand hence are not amenable to existing logics. Rather, thesubjunctive element should be placed in the connective, giving' If it had been the case that (he lit a match) then it would havebeen the case that (the whole house exploded)'.

The logical manipulation of sentences is simplified if thetemporal reference is made by means of a fixed co-ordinatesystem rather than by the use of verb tenses. Let' L ' and ' H 'designate the repeatable properties of ' lighting a match ' and' exploding a house ' and ' a ' the approximate, extended spatio-temporal region referred to. Since the kind of implicationinvolved in this sentence is neither material nor strict (we calli t ' contrary-to-fact implication ' or ' counterfactual implication ')a new symbol ' s ' is introduced for it. The sentence thenbecomes ' La 8 Ha '.

There are two important points to note about our symbol' 8'. In the first place, we stipulate that ' s ' always representsa certain state as a fact, and is never used to express an attitudeof wishing or the like. (Contrast in this connexion the subjunc-tive sentences ' 0 were he only here \ ' May he return soon ',and ' Part we in friendship from your land'.) In the secondplace, ' 8' is always counterfactual. In contrast, ordinary sub-junctive sentences are sometimes counterfactual and sometimesnot. The sentence ' if the criminal had come by automobile hewould have left tracks ' uttered by a detective as he approachesthe scene of a crime is an example of the non-counterfactualsubjunctive, as is the sentence ' If they should miss the trainthey would have to wait an hour at the station '. Since we donot wish our symbolism to be subject to such ambiguity westipulate that ' 8 ' is always counterfactual and introduce anothersymbol in 2.2 for the non-counterfactual uses of the subjunctive.We can express the counterfactual character of ' s' by statingthat the following is a valid logical principle :

p s q . D ~ p .Further valid rules for ' s ' can best be stated in connexion with

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the concept of ' causal implication ', and hence will be given ina later subsection (2.3).

It is the function of logic to classify and formulate fallaciousforms of argumentation as well as valid ones. Ordinarily thisis done by some systematic method; e.g., the use of truth-tables for the propositional calculus. At this stage in the de-velopment of the logic of causal propositions, however, such amethod is by the nature of the case excluded. Hence we shallemploy the older method of listing invalid argument forms(formal fallacies). For example, ' La s Ha ' implies neither that' Ha ' is true nor that it is false, for while ' La s Ha ' implies thatunder the circumstances the truth of ' La' would have beensufficient to insure the truth of' Ha ' it does not imply that it wasnecessary (cf. 2.2). E.g., the house could have been explodedby the spark from a starting motor. Hence the following state-ment forms are fallacious (not universally valid):

(F,) p s q . 3 ~ q ,(F2) p s q . S . q s p .

2.2. Causal ImplicationConsider the following statement of a causal law : ' A beam

of electrons moving in a vacuum perpendicular to a magneticfield is deflected'. While we commonly symbolize such asentence by means of material implication this is not correct(cf. 2.4). The implication involved in a causal law is differentfrom both material implication and strict implication. We shallcall it ' causal implication ' and introduce the symbol ' c ' for it.The given sentence may then be symbolized by ' (x)(ExcDx)\We shall call any statement of the form ' (x)(Ex c Dx)' a causaluniversal. Causal implications also occur in statements thatare not universal, as ' Ea c Da '. A non-counterfactual sub-junctive (' If he should release that eraser it would fall', ' If hewere to strike me I would strike him') is also translated as acausal implication. In such cases there is no strictly logicaldifference between a sentence expressed in the indicative mood(' If he releases that eraser it will fall') and the correspondingsubjunctive sentence, so (in contrast to ordinary languages) wewill use the same symbolic form for them.

It should be noted that those symbolically simple predicateswhich designate ' dispositioas ', ' potentialities ', or ' powers 'contain causal implications implicitly. Thus ' That diamondwas hard ' implies ' If an attempt had been made to scratch thatdiamond with, e.g., steel, it would have failed ' and ' That table

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is green' implies ' If white light were shone on the table greenlight would be reflected'. All predicates except those desig-nating immediately experienced qualities (e.g., as in ' This bookappears red now') are of this kind.

We intend ' Ea c Da ' to assert that the conditions expressedby ' Ea ' are causally sufficient to make' Da' true. By' sufficientconditions' we mean a set of conditions, complete with respectto negative properties as well as positive ones [i.e., counteractingcauses must be explicitly mentioned) sufficient to cause the stateof affairs expressed by the consequent. Thus ' (x)(Ex c Dx)'should be interpreted as ' A beam of electrons moving in a vacuumperpendicular to a magnetic field and subject to no other forces isdeflected '. There are a number of consequences of the fact that' c' means ' causally sufficient' that need to be noted.

The first is that the argument from ' Ea c Da ' to ' Da c Ea ' isfallacious, and hence also

(F,) p c q . S . q c p .The second consequence is that the antecedent may containsuperfluous or even irrelevant conditions ; e.g., ' A beam ofelectrons of one milliampere intensity moving in a vacuum per-pendicular to a magnetic field and parallel to a nearby woodenrod is deflected' [' (x)(ExOxPx . c Dx)'] is true. In fact, onthis interpretation of the symbolism these are valid inferences :

EacDa. .-. EaOa.cDa.(x)(Ex c Dx). .-. (x)(ExPx. c Dx).

The leading principles of these inferences are(P,). p c q. 3 : pr . c q,(P3). (x)(fxcgx)3(x)(fxhx.cgx).

Because of these two principles ' c ' may not correspond tothe ordinary use of ' cause' in certain respects. Thus thephrase ' causal law' is probably not customarily applied to atrue causal universal such as ' (x)(ExOxPx. c Dx)' since theantecedent contains superfluous and irrelevant conditions. Also,by (P,) we can validly infer ' Ea ~ Ea . c Da ' from the truestatement ' Ea c Da', and it is doubtful whether ordinaryusage would sanction this inference (cf. 2.5). Fortunately it isnot necessary for us to decide exactly what ordinary usage is onthese points, for whatever the utility of the ordinary concept of' causal law ' in everyday affairs it has neither the precision northe logical simplicity requisite for a basic concept in our system.

The third consequence of the fact that ' c ' means ' causallysufficient' is that it expresses a transitive relation :

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(P4). p c q . q c r : D . p c r ,(P5). (x)(fx c gx). (x)(gx c hx): D (x)(fx c hx).

Note that the acceptance of these principles allows us to predicatethe relation of causal implication of events which are not con-tiguous in time and space. (In such cases, of course, there is acertain oversimplification in our use of the same individualvariable or constant in both the antecedent and the consequentof an implication.) In fact, we do not intend ' c ' to connotetemporal sequence at all; i.e., it may be used to express causalrelations of a non-temporal sort {e.g., Ohm's Law, the ExclusionPrinciple). As a result transposition is valid:

(P6) p c q . 3 . ~ q c ~ p ,for if a cause is sufficient to produce an effect then the efEect isa causally necessary condition of the cause and the absence ofthe effect causally implies the absence of the cause. Similarlywe have:

(PT) p q . c r : a : p ~ r . c ~ q .

Because of these deviations some of the arguments which arevalid by the rules of the logic of causal propositions do not soundvalid when expressed in ordinary languages. Thus ' If it rainshe'll wear his raincoat; therefore if he doesn't wear his raincoatit won't rain' sounds invalid ; rain could cause him to wear araincoat but bis not wearing a raincoat has no causal influenceon the weather. Nevertheless if he does not wear a raincoatwe can infer on causal grounds from the given premise that itwon't rain. The matter is further complicated by the fact thatsuch assertions of causal implication are elliptical, many of therelevent conditions being omitted from the antecedent; thispoint will be taken up later in this section.

The fourth consequence of the fact that ' c ' means ' causallysufficient' has to do with the validity of the principle of exporta-tion.1 Suppose two conditions are causally sufficient to producean effect. For example, if an object satisfies the two conditionsof being heavier than air and being released, it falls

[(x)(HxRx.cFx)J.

It does not follow either causally or materially from the fact thatone condition is satisfied that the second is causally sufficient toproduce the effect. Thus

HaRa.cFa, Ha. .-. RacFa1 A. W. Burks, and I. M. Copi, " Lewis Carroll's Barber Shop Paradox ",

MIND, vol. 59 (1950), pp. 219-222.

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370 ARTHUR W. BURKS :

is an invalid argument, and so this statement form is fallacious :(F4) p q . c r : 3 : p 3 . q c r .

On the other hand,(Pg) p q . c r : 3 : p c . q 3 r

is a valid statement form, for it does follow, and on causalgrounds, that where the first condition is satisfied the second willin fact produce the effect.

An important implication is that statements of ordinary usageinvolving causal implication must be construed as elliptical toavoid this fallacy. Suppose someone asserts ' If this object isreleased here and now (a) then (causally) it will fall', when infact it is true that ' Ha '. It might seem that we should trans-late this statement into ' Ra c Fa ' and regard it as having beenderived from the premises ' HaRa . c Fa ' and ' Ha '. But thesuggested argument commits fallacy (F4). To avoid thisfallacy we must regard the English sentence as elliptical for alonger statement whose antecedent mentions or refers to condi-tions sufficient to bring about the consequent. An ellipticalsentence is necessarily vague, so that it is impossible to decidewith certainty what its meaning is. A plausible translation of' If this object is released here and now (a) then (causally) it willfall' is ' HaRa . c Fa ' ; i.e., the given sentence is elliptical fora statement which explicitly states conditions sufficient to causethe consequent. It seems clear that most ordinary assertions ofcausal implication are elliptical in this or a closely similar way.

2.3. Causal Implication and Contrary-to-Fact Implication

We can further explicate the meanings of ' c ' and ' 5 ' by con-sidering their interrelations. A typical use of a causal law is toinfer a contrary-to-fact implication from it:

(x)(ExcDx), ~ E a .-. EasDa.Thus there is a valid form of inference involving the denial ofthe antecedent of a causal implication:

(P9) p c q . ~ p : 3 . p s q .Similarly,

~ E a , ~ (EasDa) .-. ~(x)(ExcDx)and

(PI0) ~ fy . ~ (fy s gy): 3 ~ (x)(fx c gx)are valid.

It is natural to ask : Does the converse of (P9) hold ? Equiva-lently, is it a valid principle that

(P,,) p s q . = : ~ p . p c q ?

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It does seem that when we use the subjunctive in the counter-factual sense (' s ') we mean to be asserting both a causal relation(' c') and to be denying that the antecedent is true. A moregeneral sort of confirmation of the validity of (Pu) is to be foundin the intuitively acceptable character of the various principlesthat it implies. We note first that it does not conflict with anyof the previous valid or invalid statement forms involving ' s'.Also, (Pu) leads to the following transitivity relations whichseem intuitively correct:

(P12) p s q . q s r : D . p s r ,(P13) p s q . q c r : 3 . p s r .

Further consequences will be developed later.We saw before (2.2) that most ordinary uses of causal implica-

tion are elliptical. (Pu) implies that the same must be true ofcontrary-to-fact implication. As before, it is impossible todecide exactly what the meaning of an elliptical sentence is, buta plausible translation of such a sentence as ' If this object hadbeen released then and there (a) it would have fallen' is ' If thisobject had been released then and there and had been heavierthan air it would have fallen ' [' HaRs . s Fa '] . Anotherpossible translation is " Ra ' and some of the true propositionsabout the particular situation (a) imply ' Fa ' in a specialcontrary-to-fad sense', where the italicized phrase is defined asfollows. An antecedent implies a consequent in this specialcontrary-to-fact sense when (1) the antecedent is false, (2) theantecedent causally implies (c) the consequent, (3) the ante-cedent does not strictly imply the consequent (cf. 2.5), and (4)the antecedent is causally possible and the consequent is notcausally necessary {cf. 2.5).1

A similar problem of translation obtains with respect to sub-junctive sentences of the form ' If f had been the case, g wouldhave been the case anyway ' or ' Even if f had been the case gwould have been the case ' ; e.g., ' Even if this object had beenblue at a it would have fallen'. Such a sentence is assertedwhen it is believed that sufficient conditions would have beenfulfilled to cause the phenomenon even if the one in questionhad been different. Thus a plausible translation of it is ' If thisreleased object had been heavier than air and blue it would havefallen' [' HaRaBa . s Fa ']. Note that this follows from' HaRa. s Fa ', for (Pu) and (P8) imply by ordinary logical rules

(PM) p s q . 3 : p r . « q .1 In this connexion see Nelson Goodman, " The Problem of Counter-

factual Conditionals ", The Journal of Philosophy, vol. 44 (1947), pp. 113-128, part II .

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Other possible translations are ' HaBaBa . s Fa : Fa ~ Ba ' and' ' - ' (Ba a ~ Fa). ~ Ba'. The first of these, together with theassertion that ' HaBaBa' is causally possible (c/. 2.5), impliesthe second (by the calculus of section 3). It should be empha-sized that we are suggesting only plausible translations of theEnglish sentences under consideration, and are not attemptingto give a precise analysis of them. ^Precise analysis is notrequired for our purpose, since the analyses we ghcg. are suffi-ciently close to correct usage to satisfy the continuity require-ment stated in section 1.

2.4. Causal Implication and Material ImplicationA very common use of a causal universal is to make predictions

from it. For example,(x)(ExcDx), Ea .-. Da

and(x)(ExcDx), ~ D a .-. ~ E a

are valid arguments. And of course these are valid whether' a ' refers to the future, as in the case of a prediction, or to thepast. When a prediction is disconfinned the causal universal isthereby refuted:

Ea, ~ D a .-. ~(x)(ExcDx).These arguments can all be validated by the principle that acausal implication implies a material implication:

(P16) p c q . a . p a q .This implies

(P16) (x)(fxcgx)3(x)(fx3gx);in our terminology : a causal universal implies the correspondingmaterial universal.

It is natural to ask: Do the converses of (P1S) and (P18) hold ?Note that if they did, a causal implication would be equivalentto a material implication and a causal universal would be equiva-lent to the corresponding material universal, with the resultthat the logic of causal propositions would reduce in immediateand simple fashion to an extensional logic (cf. section 1). Goodarguments can be advanced, however, to show that these equiva-lences are fallacious:

(F5) p c q . = . p 3 q,(F6) (x)(fxcgx) = (x)(fXDgx).

This is the case even though these distinctions are not reflectedby ordinary grammatical form. Thus the English 'if . . .then . . .' may be used indifferently for causal implication and

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material implication (as well as for strict implication); theof implication involved is determined from the content of thesentence or its context.

The argument to show that (F5) is invalid is simple. For if(F5) were valid a false proposition would causally imply anyproposition, and hence by (P9) counterfactually imply any pro-position. E.g., ' This glass was not released ' would imply both'(Releasing this glass) c (it exploded)' and '(Releasing this glass)c (it didn't explode)', and hence both ' If this glass had beenreleased it would have exploded ' and ' If this glass had beenreleased it would not have exploded '. A separate argument isrequired to establish the invalidity of (F6). For one could holdthat a causal universal is equivalent to the corresponding materialuniversal, but still hold that something more is meant by acausal implication than a mere material implication. E.g., onecould hold that ' Ea c Da ' is true if and only if ' (x)(Ex 3 Dx)'is true. Such a position appears logically arbitrary, to be sure,but it is a possible one. To show that (Fe) is invalid we pointout two unacceptable consequences which follow from theassumption of its validity.

(1) Consider the following true proposition: ' All the bookson this desk are in German'. It is clearly a mere summary offacts, a universal proposition true by accident, not by law, sowe would naturally symbolize it by ' (x)(Bx 3 Gx)' rather thanby ' (x)(Bx c Gx)'. Moreover, we would say that since it is trueby accident and not by law the corresponding contingent universal' (x)(Bx3Gx).~(x)(BxcGx)' is true. But on the theorythat a causal universal is equivalent to the corresponding materialuniversal the quoted statement is self-contradictory!

(2) Suppose next that ' Russell's Principles is not a book onthis desk ' [' <•»•' Ba '] is also true. Under the proposed equiva-lence we could validly conclude ' If Russell's Principles were onthis desk it would be in German' [' Ba s Ga']. For we wouldbe given ' (x)(Bx 3 Gx)', and that it is equivalent to' (x)(Bx c Gx)', so we could infer by ordinary logical rules' (x)(Bx c Gx)'. We could specify this to get ' Ba c Ga ' whichwe could combine with ' ~ Ba ' to get' Ba c Ga . ~ Ba '. Thenby (PB) we could infer ' Ba s Ga '. Hence if (F6) were valid wewould be unable to distinguish those cases where we can validlydeduce contrary-to-fact implications (cf. 2.3) from those wheresuch a deduction cannot be validly made.

There is one feature of the examples just employed in (1) and(2) which merits attention. Consider the sentence 'All thebooks on this desk are in German'. It was translated as

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' (x)(Bx 3 Gx)', where the predicate ' B ' means ' being a bookon this desk now'. This predicate makes reference to a par-ticular place and time, and in this respect is to be contrastedwith such a predicate as ' E ' [' being an electron beam '] whichneed not include in its definition any restriction to a particularspatiotemporal region. On this basis we call ' B ' an indexicalpredicate and ' E ' a symbolic predicate.1 The relevance, of thisdistinction to the matter at hand is that one might argue thatthough (F6) is a fallacy in general, it is a valid principle in alanguage which contains no indexical predicates. (This wouldmean that it would be a valid principle in a scientific language,for since science investigates only repeatable qualities a scien-tific language need contain no indexical predicates.)

We can, however, construct examples to show that (F6) is afallacy even when the predicates involved are non-indexical.Thus we could modify the above example by replacing theindexical predicate ' B ' by a complicated symbolic predicate' B ' such as ' being a book on a desk which is of such a size andshape, which is five feet from an individual with such and such ahistory, . . . , . . . ' . By making this description sufficientlycomplicated we could make it highly probable that it was uniqueto this particular desk. Then ' (x)(Bx 3 Gx)' would veryprobably be true, while ' (x)(Bx c Gx)' would very probably befalse. The fact that we could not be certain of the uniquenessof the property defined does not destroy the example, becausethe truth of either a causal universal or a material universal isnever known with more than probability.

2.5. Causal Implication and Strict ImplicationA causal implication implies a material implication (P15) but

not vice versa (F5); in contrast, a strict implication implies acausal implication but not vice versa.2 These facts are expressedby the principle

(FIT) p -S q - 3 - p c q,and the fallacy

(F7) p -3 q . a . p c q.It should be noted that because of (P17), ' c ' differs in meaning

1 For an elaboration of this distinction see A. W. Burks, " Icon, Index,and Symbol ", Philosophy and Phenomenological Research, vol. 9 (1948-49),pp. 673-689.

* For logics of strict implication see C. I. Lewis and C. H. Langford,Symbolic Logic ; Ruth C. Barean, " A Functional Calculus of First OrderBased on Strict Implication", The Journal of Symbolic Logic, vol. 11(1946), pp. 1-16; and Rudolf Camap, Meaning and Necessity.

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from the ordinary usage of ' cause' in that the ordinary usagesignifies that the implication is causal (in our sense of ' c') butnot strict, i.e., the ordinary meaning of ' p causally implies q 'should probably be symbolized as ' p c q . ~ (p -3 q) '• Ourreasons for introducing this principle—apart from its intrinsicplausibility—are as follows.

Adding (Pi7) to our already existing logical machinery is thesimplest way of handling those contrary-to-fact implications andnon-counterfactual uses of the subjunctive that are based onstrict implications. Thus any statement of the form ' If thepremises had been true the conclusion would have been true'follows from the validity of the argument and the falsity of thepremises. Likewise, in deducing the practical consequences ofa belief we may say ' If this were true than such and such wouldbe the case', which is probably best symbolized by means of' c '. Again, such a sentence as ' If the Bock of Gibraltar were aman it would be rational' [' G s R '] can be dealt with by meansof (P17), for it would follow from the strict implication ' 6 - j B '(assuming the traditional definition of ' man') and the factualpremise ' ~ G ' by (P17) and (Pu).

A second reason for introducing (P17) is that it takes the placeof several more specialized principles. Consider, for example,

p s q . q - 3 r : 3 . p s r ,p s q . a : p 8. pq.

These principles are intuitively valid, and by means of them wecan validate the argument: ' If Russell's Principles were placedon this desk it would not be in German. .•. If Russell's Principleswere placed on this desk it would not be the case that all thebooks on this desk are in German'

[' Ba s ~ Ga, .-. Ba s ~ (x)(Bx 3 Gx)'].For the second principle allows us to infer ' Ba s. Ba ~- Ga 'from the premise, and since

' Ba ~Ga . -3 ~(x)(Bx 3 Gx)'is true, the conclusion follows by the first principle. But thisargument can also be validated by making use of the moregeneral principles (P,7) and the first of the following two validprinciples of distribution.

(P,8) p c q . p c r : = :pc .q r ,(pi») p v q c r : = r p c r . q c r .

The line of argument is then as follows. By (P17) we infer' Ba c Ba ' from ' Ba -3 Ba '. By (Pu) we infer ' ~ Ba ' and' Ba c ~ Ga ' from the premise. ' Ba c Ba ' and ' Ba c ~ Ga '

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give ' Ba c. Ba ~ Ga' by (P18), and this with ' ~ Ba ' gives' Ba 8 . Ba ~ Ga' by (Pu). Using (P17) again we infer' Ba »—• Ga . c ~ (x)(Bx D Gx)' from the corresponding strictimplication. The conclusion follows by (Pi3).

One consequence of (P17) which must be noted is that it intro-duces causal paradoxes derivable from the so-called paradoxesof strict implication. Thus because a contradiction strictlyimplies anything ' Ba <~ Ba. -3 Ga' is true and hence by (P1T)' Ba ~ Ba . c Ga' is true; since ' ~ (Ba «•»* Ba)' is true onlogical grounds we may infer ' Ba <*•> Ba~r»43a_l_by (Pu). Simi-larly, ' Ba c. Ga v ~ Ga ' is true. These resultshSve a certainsanction in ordinary usage; e.g., ' If Russell's Principles wereplaced on this desk it would either be in German or not inGerman' does not seem an unreasonable consequence of' Russell's Principles is not on this desk'. In any event, as wehave previously noted there is considerable arbitrariness inherentin the symbolic formulation of any common-sense idea. Oncea case of vagueness is posed clearly by means of a precise sym-bolism its resolution should be determined by convenience. Itis on this basis that we adopt (P17).

There is a consequence of introducing the paradoxes of strictimplication into the logic of causal propositions which is of con-siderable importance. Because of these paradoxes statementsof the form ' p c q ' and ' p c ~ q ' are not simple contraries ;e.g., ' Ba <—- Ba . c Ga' and ' Ba ~ Ba . c ~ Ga ' are both true.Thus it is a fallacy that

(F8) p c q . 3 ~ ( p c ~ q ) .Similarly it is a fallacy that

(F,) p » q . 3 ~ ( p s ~ q ) .In order to state the circumstances under which ' p c q ' and' p c r*~ q ' (and ' p 5 q ' and ' p s ~ q') are contraries it mightseem that we should qualify the above statement forms bystipulating that they hold only if the antecedent of each causal(or contrary-to-fact) implication is logically possible (not self-contradictory) and the consequent of each such implication isnot logically necessary. This qualification would exclude thecases we have been considering so far. But there are also" paradoxical" cases to be considered which are not derivedfrom the paradoxes of strict implication though they are parallelto them.

Corresponding to the implications with logically impossibleantecedents there are implications with causally impossible ante-cedents, i.e., antecedents which can never be exemplified because

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they contradict existing laws. E.g., assuming' (x)(BxHx. c Fx) 'to be true, ' BaHa ~ Fa . 8 Wa ' [' If this released, heavier thanair, object had not fallen it would have been white'] and' RaHa ~ Fa . c Wa ' are such implications. Similarly, corre-sponding to the implications with logically necessary consequentsthere are implications with causally necessary consequents, i.e.,consequents true by virtue of a causal law. Under the sameassumption as before ' Wac:BaHa .3Fa ' ['If this object iswhite then (causally) if it is released and is heavier than air itwill fall'] is such an implication. It seems desirable to countthese statements as true, rather than false, just as we count thecorresponding strict implications true. We shall do this, thoughwe emphasize that there is e certain arbitrariness in this pro-cedure.

Thus there are " paradoxes " of causal implication analogous tothe paradoxes of strict implication : a causally impossible ante-cedent causally implies any consequent and a causally necessaryconsequent is causally implied by any antecedent. To statethese principles symbolically we introduce symbols for theconcepts of causal possibility ( ' ^ ' ) and causal necessity (' GO '),*corresponding to the concepts of logical possibility (' ^ ') and

logical necessity (' • '). The " paradoxes " of causal implicationare then

(P : :) p S . q c p .With these causally modal concepts we can also state the prin-ciples of contraries for causal and contrary-to-fact implications :

(Pis) ^ P 3 ~ ( p » q . p « ~ q ) .{It is unnecessary to require that ' p ' be logically possible aswell as causally possible, for as we shall see by (P,8) that whichis causally possible is also logically possible.)

Lewis Carroll's barber shop paradox * provides an example ofthe use of the causal modalities. The owner of the barber shoplays down the following rules : (1) ' If Carr and Allen are out,then Brown must be in ' [' CA. c ~ B ' ] , and (2) ' If Allen goesout Brown mast accompany him' [' A c B'] . Carroll's argumentis as follows : (a) the first rule implies ' C 3 . A c <•— B ' , (b) thesecond rule implies ' ~ (A c ~ B) \ and (c) by modus UAlena

1 These are what Hans Reichenbach (Elements of Symbolic Logic,p. 392) calls physical modalities. * Burks and Copi, op. tit.

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these results imply ' ~ C ' ; i.e., the given rules imply thatCarr can never go out! Step (a) is of course invalid by (F4) andstep (b) is valid only if the true premise' Q A' is assumed (c/. PK).It should be noted that ' AC', together with the two rules,implies a contradiction by (P16) and the usual rules of logic.' AC ' is clearly logically possible ; therefore itjnust be causallyimpossible. Hence we have the valid principle

Thus the troublesome feature of Carroll's example is that therules are such that the antecedent of one represents a causalimpossibility. (Of course, this is true only on the assumptionthat the rules laid down are obeyed, and hence our symboliza-tion of Carroll's rules is not completely adequate. If they doobey the rules, the barbers have certain causal dispositions—as is shown by the fact that counterfactual propositions mayunder appropriate circumstances be inferred—so that anadequate symbolization will be of the general form given.)

The concepts of causal necessity and possibility can be relatedto the concept of causal implication, just as the correspondingideas of logical possibility and necessity can be related to theconcept of strict implication. Thus we have

(Pu) GD(p 3 q) = • p c q.Causal possibility is related to causal necessity in this way :

~ GO rw

When (Pu) is applied to a causal universal the symbol forcausal necessity (' [c]') appears inside the scope of a universalquantifier in the resultant expression. E.g., l (x)(Ex c Dx)'becomes' (x) [c] (Ex 3 Dx)'. The question arises as to the properinterpretation of this result. Now to say that for every spatio-temporal region ' Ex 3 Dx' is true on causal grounds is logicallyequivalent to saying that ' (x)(Ex 3 Dx)' is true on causalgrounds. Thus we have1

(P.T) (x)[c]fx • GD(x)fx.We are now in a position to compare the properties of the

causal symbols we have introduced with those of the corre-sponding logical symbols. The first point to note is that while acausal implication or necessity is implied by the correspondingstrict implication or logical necessity, it implies the correspondingmaterial implication or assertion without a modal operator.

» Cf. Camap, op. tit., pp. 178-179,186.

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This is exemplified by (P1B) and (P17) and also by

(P29) QDpDp.(Note that by (P2g) the paradoxes of strict implication becomespecial cases of the paradoxes of causal implication.) In thesecond place, the deductive interrelations of ' c ' , ' \c\', and ' < ^ 'are quite similar to those of ' -3 ', ' • ', and ' ^ '. This hasalready been illustrated by the parallelism of the causal modali-ties and the logical modalities. It is further verified by thefact that this hypothesis accounts for all the rules of the presentsection (as we shall show in section 3) and for all the fallacies(in the sense that the fallacies are not theorems). Thus it wouldseem that causal propositions have a certain modal character,despite the fact that they are empirical rather than rational.This suggests that the formalized logic of causal propositionsshould be a double modal logic. We now turn to the construc-tion of such a system.

3. The Calculus of Causal PropositionsIn this section we shall develop an applied functional calculus

of first order which will formalize the results of section 2. Thiscalculus is patterned after Church's formulation of a functionalcalculus of first order and Fitch's development of a modal logicwith quantifiers.1

The limitations of our language should perhaps be commentedon. A language of first order is clearly inadequate to state thestatistical and quantitative theories of a well-developed science.We believe, however, that besides being a first step towardachieving a truly adequate language, the present formulation isof intrinsic interest because many of the questions about causalitytraditionally considered by philosophers (e.g., Bacon, Hume, Mill,and Keynes) concern universal (non-statistical), qualitative laws.

3.1. Vocabulary and Formation RulesThe extensional part of the primitive vocabulary consists of

the logical constants <~, 3 , parentheses, and the universalquantifier; and of infinite lists of prepositional constants andvariables, individual constants and variables, and functionalconstants and variables of all degrees. This is supplemented by

* Alonzo Church, Introduction to Mathematical Logic, part I, andF. B. Fitch, " Intuitionistic Modal Logic with Quantifiers ", PortugaliaeMathematka, vol. 7 (1948), pp. 113-118. Consult these works for historicalreferences to the methods employed.

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380 ARTHUR W. BURKS

the non-extensional primitive symbols • and [c]. Symbols ofthe object language will generally be used autonomously as inthe preceding sentences. Capital Roman letters and lower-caseGreek letters, with or without numerical subscripts, will be usedas variables in the syntax language, Roman letters takingformulas of the calculus as values and Greek letters takingindividual variables as values unless otherwise specified.

A formula is any finite sequence of primitive symbols. Therecursive definition of ' well-formed formula' (' w.f.f.') is :

A prepositional variable or constant alone is w.f.If /3 is a functional variable or constant of degree n and a.lt a*

. . . , a^ are individual variables or constants, j3 (ax, a i , . . . a,) is w.f.If A and B are w.f., ~ A, (A 3 B), (<x)A, DA and [c]A are w.f.No formula is w.f. unless its being so follows from these rules.

Hereafter the Roman letter syntactic variables will take asvalues only w.f.f. Parentheses will be omitted or replaced bydots whenever this simplifies the punctuation without introducingambiguity. An occurrence of an individual variable a in a w.f.f.A will be called ' a bound occurrence' if it is in a w.f. part ofthe form (a)B ; otherwise it will be called ' a free occurrence'.

It should be noted that by these formation rules formulaswith modal operators operating on modal operators are w.f.,e.g., ' [c]([Ilp 3 [Dp) '• In connexion with the interpretation ofthese formulas it should be remembered that because of (P17) and(P2B) GD is not to be interpreted »s ' it is true on causal grounds 'but rather as ' it is true on either causal or logical grounds'.Examples of sentences with causal modalities operating oncausal modalities are sentences which assert that the appearanceof one dispositional property causes the appeararce of another,or that if a certain causal law were false such-and-such would bethe case, or that it .is a causal law that a given kind of habit orresolution arises under certain circumstances.

The definitions are:

(AvB)(A.B)(AsB)(A-SB)(AcB)(AsB)(3«)AOA

= d f

= d l

= d l

= d f

=<W

= d l

=<U

= d f

(~AsB)~(~Av~B)«A3B).(B3A))

•(AsB)GD(A 3 B)(~A.(AcB))~(«)~A~D~A

= d f /̂ / A

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3.2. Axioms and Primitive Rule of Inference.Primitive rule:

(1) From A and A 3 B to infer B (modus ponens).The axioms are determined by the following schemata and

rules:—(1)A3.B3C:3:A3B.D.A3C.(2) As.BsA.(3) ~A3~B.3.B3A.(4) (<x)(A 3 B) 3 .A 3 (a) B if a is not free in A.(5) (a)A 3 B where a is an individual variable, fl is an

individual variable or constant, no free occurrence ofa in A is in a w.f. part of A of the form (fi)C, and Bresults from the substitution of /? for all free occurrencesof a in A.

(6) DAsfJJA.(7) [c]A3A.(8) (a)(A3B)3.(a)A3(a)B.(9) n (A3B)3 .DA3DB.

(10) GD(A 3 B) 3 . GDA 3 GDB.(11) (a)DA3D(«)A.(12) (a)[c]A 3 [c](a)A.(II) If A is an axiom, so is (<x)A.

(Ill) If A is an axiom, so is QA.Axiom schemata (9) and (11) are not used in establishing the

principles of section 2, but are included for the sake of complete-ness. By means of them the strict implication analogues ofour theorems can be proved. Hence this calculus contains asystem of strict implication as well as both an extensionalsystem and a system of causal implication.

3.3 Theorems1

' Theorem ' is denned as follows. A proof of a formula B onhypotheses Ax, A ,̂ . . ., A,,, is a finite sequence of formulassuch that each member of the sequence is an axiom or is one ofthe hypotheses or follows by (I) from preceding members of thesequence, and such that the last member of the sequence is B.Any formula for which there exists a proof without hypothesesis a theorem. To asseit that there is a proof of B on hypothesesAu A* . . ., An we write Alt A* . . ., A,, h B, and to assertthat there is a proof of B without hypotheses we write |> B.

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382 ARTHUR W. BUSKS: THE LOGIC OF CAUSAL, PROPOSITIONS

All the principles of section 2 are theorems of the calculusjust constructed. We will not show this in detail but willinstead state some metatheorems by means of which thisresult can easily be established.

Note first that every theorem of the propositional calculus isa theorem of this system and that the results of making propersubstitutions of w.f.f. of the calculus of causal propositions forthe variables of the theorems of the propositional calculus arelikewise theorems of the system. Hence the machinery of thepropositional calculus can be employed in proving the principlesof section 2. Note also that the Deduction Theorem holds forthe calculus of causal propositions.1

In addition, the following metatheorems or derived rules ofinference hold for the calculus of causal propositions.

(IV) If A,, A. . . ., A,, h B, then

Proof : * Let B l t Bto . . ., Bo (where Bn is B) be the given proofof B on the given hypotheses. To make the desired proofoperate on the given proof as follows :

(a) Replace each hypothesis A< by {c\A.{;(6) Replace each axiom C by OP, \JC 3 [cJC and GDC ;(c) Replace any B< that is neither an axiom nor an hypo-

thesis by [c](B,sB,)3. 0B,3(UBj,[c]B,3[cftoand[c]B<; where B, 3 B< and Bt are the premises fromwhich B< was inferred in the given proof.

The resulting sequence is the desired proof by virtue of (I),(III), (6), and (10).

(V) If A^A,, . . ., A^hB.then

(a)Ax, (a)A» . . ., (aJA. t- (a)B.Proof : Similar to that of (IV) but using (I), (II) and (8).

(VI) If h A 3 B, then I- [c]A 3 [c]B.Proof: By means of (IV), (I), and (10).

(VII) If h A 3 B, then H (<x)A 3 (<x)B.Proof : Similar to that of (VI), but using (V) and (8) with (I).

University of Michigan.1 Cf. Church, op. cit., sections 2.4 and 1.9 respectively.* This proof is based on one of Fitch's, op. cit., p. 117.

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viii.--the logic of causal propositions

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