Abstract
One of the striking differences between natural languages, both conversational and scientific, and the extensional languages constructed by logicians is that most conditional statements, i.e., statements of the form ‘if p, then q’, of a natural language are not truth-functional. A statement compounded out of simpler statements is truth-functional if its truth-value is uniquely determined by the truth-values of the component statements. The symbolic expression of this idea of truth-functionality, as given inPrincipia Mathematica, is p ≡ q ⊃ (f (p) ≡ f (q)). That is, if ‘f (p)’ is any truth-function of ‘p’, and ‘q’ has the same truth-value as ‘p’, however widely it may differ in meaning, then ‘f (q)’ has the same truth-value as ‘f (p)’ Clearly, if I am given just the truth-values of‘p’ and ‘q’, not their meanings, I cannot deduce the truth-value of ‘ifp, then q’—with a single exception: if ‘p’ is given as true and ‘q’ as false, it follows that ‘if p, then q’ is false, provided it has a truth-value at all. On the contrary, the knowledge that matters for determination of the truth-value of a ‘natural’ conditional—let us call them henceforth ‘natural implications’, in contrast to those truth-functional statements which logicians call ‘material conditionals’ or ‘material implications’—is rather knowledge of themeanings of the component statements. In the case of simple analytic implications like ‘if A has a niece, then A is not an only child’ such knowledge of meanings is even sufficient for knowledge of the truth of the implication; at any rate knowledge of the truth-value of antecedent and consequent is irrelevant. In the case of those synthetic natural implications which assert causal connections, knowledge of meanings is not, indeed, sufficient, but it is necessary, and knowledge of the truth-values of the component statements is not presupposed by knowledge of the truth-value of the implication.’ Consider the conditional (which may or may not be ‘contrary-to-fact’): if I pull the trigger, the gun will fire. It would be sad if belief in such an implication were warranted only by knowledge of the truth of antecedent and consequent separately, for in that case it would be impossible for man to acquire the power of even limited control over the course of events by acquiring warranted beliefs about causal connections. Notice further that frequently presumed knowledge of a causal implication is a means to knowledge of the truth, or at least probability, of the antecedent; if this is an acid then it will turn blue litmus paper red; the reaction occurred; thus the hypothesis is confirmed. Knowledge of the consequences of suppositions is independent of knowledge of the truth-values of the suppositions, no matter whether the consequences be logical or causal.
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Notes
In this context ‘knowledge’ is used in the weak sense in which ‘pis known to be true’ entails that there is evidence making it highly probable that p, not the stronger claim that there is evidence making it certainthat p
I have slightly changed Carnap’s way of putting the counterintuitive consequence, in accordance with my using Dx, t’instead of ‘Dx’.
There seems to be fairly universal agreement now among philosophers of science that the simple kind of explicit definition of disposition concepts in terms of material implication is inadequate, precisely because we want to be able to say of an object which is not subjected to the test operation by which a disposition Dis defined that it does nothave DOne exception to this trend might, however, be noted: Gustav Bergmann maintains (’Cornments on Professor Hempel’s "The Concept of Cognitive Significance,"’ Proceedings of the American Academy of Arts and Sciences, July 1951, pp. 78-86) that such explicit definitions nevertheless provide adequate analyses of the disposition concepts—in a sense of ’adequate analysis’ which is obscure to me. Referring to Carnap’s example of the match which is burned up before ever being immersed in water and therefore would be soluble by the criticized definition of ’soluble’, he says ‘I propose to analyze the particularsentence ’the aforementioned match is (was) not soluble’ by means of two sentences of the ideal schema, the first corresponding to ‘This match is (was) wooden,’ the second to the law’No wooden object is soluble. — In what sense do these two sentences provide an analysisof ’soluble’? Bergmann is simply deducing ‘the match is not soluble’ from two well-confirmed premises, and is therefore perhaps giving a correct explanationof the fact described by the sentence, but since ’soluble’ reappears in the major premise—as it must if the syllogism is to be valid!—its meaning has not been analyzed at all. It is one thing to give grounds for an assertion, another thing to analyze the asserted proposition.
See D. J. O’Connor, ’The Analysis of Conditional Sentences,’ Mind, July 1951, p. 354. Also, the Finnish philosopher E. Kaila once attempted to escape from Carnap’s conclusion that disposition concepts are not explicitly definable by proposing that ‘Dx’be taken as neither true nor false in case x is not subjected to O(which proposal, incidentally, is consonant with Carnap’s proposal of introducing dispositional predicates by reduction sentences, as we shall see later): ’Wenn-So,’ Theoria, 1945, Part II.
This has been called the ’paradox of confirmation.’ See C. G. Hempel, ’Studies in the Logic of Confirmation,Mind, January, April 1945; and R. Carnap,Logical Foundations of Probability, Section 87 ( Chicago: Univ. of Chicago Press, 1950 ).
For a lucid warning against this confusion, see R. Carnap, ‘Truth and Confirmation,’ in H. Feigl and W. Sellars (eds.),Readings in Philosophical Analysis ( New York: AppletonCentury-Crofts, 1949 ).
Indirect confirmation of a conditional is distinguished from (a) direct confirmation, consisting in the verification of the conjunction of antecedent and consequent, (b) vacuous confirmation, consisting in the verification of the negation of the antecedent.
D is here assumed to be an intrinsic disposition in the sense explained above. The above schema is, with a slight alteration, copied from Anders Wedberg’s ’The Logical Construction of the World’ (Theoria, 1944, Part III, p. 237), who cites it for purposes of criticism from Kaila’s Den mänskliga kunskapenA variant of this definition schema has more recently been proposed by Thomas Storer: ’On Defining "Soluble",’ Analysis, June 1951.
The latter restriction has been suggested to me by Michael Scriven.
For example, the painstaking attempt made by B. J. Diggs, in ‘Counterfactual Conditionals’ (Mind, October 1952), to achieve an extensional analysis of the counterfactual conditional is guided by this idea.
One might, though, take the more moderate view that warranted assertion of counterfactual conditionals merely involves statistical determinism, i.e., belief in the existence of a statistical law relative to which the consequent is inferable from the antecedent with a probability sufficiently high to warrant practical reliance on the conditional. But on either view singular counterfactual conditionals derive their warrant from a law, whether causal or statistical.
Notice that while ‘A says"p"’ does not entail, but at best confers a high probability upon’A believes thatp ,’ the latter proposition is entailed by’A asserts thatp ,’ according to my usage of ’assert’ as an intentional verb. I am not denying, of course, that there may be a proper purely behavioristic sense of ’assert’; nor do I deny that’A asserts thatp’ may properly be so used that it is compatible with’A does not believe thatp.’ My usage may be explicated as follows: A believes thatp and utters a sentence expressing the proposition thatp.
This seems to be overlooked by O’Connor who, following Broad, concludes his analysis of conditional sentences (loc. cit.)with the claim that "a particular contrary-to-fact conditional has exactly the same meaning as the corresponding universal indicative statement." The examples given by him indicate that by the universal statement correspondingto the ‘particular’ contrary-to-fact conditional he means the universal conditional of which the latter is a substitution instance. Obviously, it might be true to say ’if the trigger of the gun had been pulled, the gun would have fired’ though there are exceptions to the generalization ’any gun fires if its trigger is pulled.’ The singular conditional is elliptical; in asserting it one presupposes the presence in the particular situation of various causal conditions which the antecedent does not explicitly mention. (See, on this point, my article ’Philosophical Analysis, Translation Schemas and the Regularity Theory of Causation,’ Journal of Philosophy, October 9, 1952, and my book Analytische Erkenntnistheorie, Chapter IV A (Vienna: Springer Verlag, 1955); also R. Chisholm, ’Law Statements and Counterfactual Inference,’ Analysis, April 1955). "
It might be objected that the law of inertia can be formulated in such a way that it is not contrary to fact: if no unbalancedforces act on a body, then it is at rest or in uniform motion relative to the fixed stars. But when the law is used for the derivation of the orbit of a body moving under the influence of a central force, it is used in the contrary-to-fact formulation since the tangential velocities are computed by making a thought experiment: how would the body move at this moment if the central force ceased to act on it and it moved solely under the influence of its inertia?
For further elaboration of this argument against the extensional interpretation of laws, see my article ‘Reduction Sentences and Disposition Concepts,’ in P. A. Schilpp (ed.).The Philosophy of Rudolf Carnap ( La Salle, Ill.: Open Court, 1963 ).
Notice that a property may fail to be purely general in this sense even if it is not defined in terms of a particular object, e.g., ’being the highest mountain.’
This criticism applies to C. G. Hempel and P. Oppenheim’s explication of ’law’ relative to a simplified extensional language system, in ‘Studies in the Logic of Explanation,’ Philosophy of Science, April 1948; reprinted in H. Feigl and M. Brodbeck (eds.),Readings in the Philosophy of Science (New York: Appleton-Century-Crofts, 1953 ). R. Chisholm makes the same criticism, in ’Law Statements and Counterfactual Inference,’ Analysis, April 1955.
A Note on Natural Laws and So-Called ’Contrary-to-Fact Conditionals, Mind, January 1949.
He could hardly mean it just in the sense of ’(3y) (y e A • x = y)’, for this says nothing else than ‘x e A’, and so does not amount to one of alternative interpretations of ‘x e A’.
Introduction to Logical Theory(London: Methuen; New York: Wiley, 1952), p. 199.
A lawlike statement is a statement which expresses a law if it is true.
Scientific Explanation (Cambridge: Cambridge Univ. Press, 1953), Chapter 9.
The truth-condition of a sentence is that state of affairs whose existence is the necessary and sufficient condition for the truth of the sentence. One might instead speak simply of the propositionexpressed by a sentence, if it were not for the purpose of emphasizing the connection between the concepts of truth and of semantic meaning (reference).
Strictly speaking, they are incompatible only if the antecedent describes a logical possibility. But contrary-to-fact conditionals with self-contradictory antecedents are analytic, and we are here concerned only with conditional sentences that express empirical propositions. Cf. the following statement by H. Reichenbach, in Nomological Statements and Admissible Operations(Amsterdam: North-Holland Publishing Company, 1953). ’Introduction’: "Assume we say ’if ahad happened, then bwould have happened.’ If this is to be a reasonable implication, it should be required that the contrary implication‘if ahad happened, then not-b would have happened’ be not true."
A predicate ’P’is time-dependent if only statements of the form ‘x is Pat time t; not statements of the form ’x is P, ’are complete.
Contrary-to-Fact Conditionals,’ Journal of Philosophy, January 1951.
By the truth-condition of an assertion I mean of course the truth-condition of the asserted sentence.
I deliberately use ‘unusual’ tenseless sentences in order not to contaminate propositions with pragmatic properties of assertion events, such as their temporal relation to the asserted facts.
Philosophy of Science, October 1936, p. 445.
A more detailed discussion of reduction sentences, especially in connection with the analytic-synthetic dualism that has been branded a ’dogma of empiricism,’ is contained in my articles ’Reduction Sentences and Open Concepts,’ Methodos, Vol. 5, No. 17, and ’Reduction Sentences and Disposition Concepts,’ in P. A. Schilpp (ed.), The Philosophy of Rudolf Carnap, op. citSee also similar comments by Hempel, in ’The Concept of Cognitive Significance: A Reconsideration’ (American Academy of Arts and Sciences, July 1951; and ’A Logical Appraisal of Operationism’ (Scientific Monthly, October 1954).
See A. W. Burks, ‘The Logic of Causal Propositions,’ Mind, July 1951, for an axiomatic definition of causal implication.
In ’On the Presuppositions of Induction; Review of Metaphysics, June 1955, A. W. Burks formulates a ’uniformity’ axiom to the effect that a universal causal implication is logically equivalent to any substitution instance of itself; or, equivalently, that any two substitution instances of a universal causal implication are equivalent. This is another way of characterizing causal implication, since obviously ‘Fa D Ga’does not entail ‘(x)(Fx DGx)’.
Cf. A. W. Burks,’Dispositional Statements,’Philosophy of Science, July 1955. The same non-extensional analysis has been proposed by W. Sellars, in the course of discussions held at the Minnesota Center for Philosophy of Science.
For this reason I am not impressed by the defeatist argument presented by Jan Berg in ’On Defining Disposition Predicates,’ Analysis, March 1955.
It should be noted that my discussion is here restricted to what may be called ‘causal’ dispositions, in contradistinction to what may be called ’probabilistic’ dispositions.
By a time-independent property I here mean a property which a thing has either all the time or never at all. This usage should be distinguished from the usage in which a time-independent property is a property Psuch that sentences of the form ‘x has Pat time t’are meaningless (thus Carnap argues in ’Truth and Confirmation,’ op. cit, that ’true’ is time-independent in this sense). To say of a disposition that it is time-independent in the former sense is to assert an empirical law, as explained earlier.
The schema for time-dependent dispositions, like ’electrically charged’, ’elastic’, ’kindly disposed toward X’, is analogous except that all predicate constants and predicate variables carry the time variable as second argument.
’The Logic of Causal Propositions,’ op. cit, p. 364.
Scientific Explanation, op. cit, p. 296.
Following Burks, I shall here assume for the sake of simplicity that what is called the ’cause’ is a sufficient conditionIt is well known that the events which are actually identified as ’causes,’ both in everyday life and in science, are not strictly sufficient conditions for their alleged effects. For a detailed discussion of problems created by this circumstance for an adequate formulation of an analysis of causal judgments in terms of ’constant conjunction,’ see my article ’Philosophical Analysis, Translation Schemas and the Regularity Theory of Causation,’ op. cit, or my book Analytische Erkenntnistheorie, loc. cit
If we want to express the idea that cause and effect are successive states of the same thing, as in the above example, a more complicated symbolism is needed. But this would not affect the question at issue.
System of Logic, Book III, Chapter V, section 6.
C’may be a negative condition, in which case it would be more natural to speak of the necessary absence of a specified condition. For example, in order for the day-night-day sequence to continue it is necessary that no opaque body interpose itself between sun and earth in such a way that no sunlight could reach any part of the earth’s surface at any time.
The following consideration serves as a striking demonstration of the inadequacy of extensional logic for the definition of the concepts ’sufficient condition’ and ’necessary condition’ as used in everyday life and science. We often have occasion to say that a certain disjunction of conditions is a necessary condition for a given effect though neither condition by itself is necessary. Thus a college professor may announce to his class that in order to pass his course they must either write a passing term paper or pass a final examination, but that it is not necessary to write a passing term paper, nor is it necessary to pass a final examination. Now, the extensional definition of ’q’expresses a necessary condition for ‘p’ is p D qHence the professor’s statement would take the form (p D q v r)• (p D q)• (p D r)But since this conjunction entails (p’. q’ ’ r), which contradicts the first conjunct, it is self-contradictory!
Cf. ’On the Presuppositions of Induction,’Review of Metaphysics, June 1955.
This seems to be the view W. Kneale takes of laws of nature. As Kneale himself notes in Probability and Induction(Oxford: The Clarendon Press, 1949), p. 71, the view was held by Locke: though man cannot attain to certain knowledge of the laws of nature, as he can only generalize from instances, an angel who knew the ’real essences of natural kinds would see the same sort of necessary connection between causal antecedent and causal consequent as we see between ‘being a Euclidean triangle’ and ’being a triangle whose angle-sum equals 180°.’
Of course we may justifiably assert ’Pa Qa’even if no instance of Phas ever been observed, on the evidence ‘Ra • Qa • Rb • Qb’etc., where Pand Rare similar properties (e.g., let Px = xis water subjected to a temperature exceeding 150, and Rx = xis coffee subjected to a temperature exceeding 155 and Qx = xboils).
In Burks’ system the specified incompatibility, moreover, holds only if the antecedent is physicallypossible; accordingly there are paradoxes of causal implication analogous to the familiar paradoxes of material and strict implication. This additional proviso, however, has such queer consequences as that if ice were denser than water, it would not float’ and if ice were denser than water, it would still float’ are compatible.
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Pap, A. (1978). Disposition Concepts and Extensional Logic. In: Tuomela, R. (eds) Dispositions. Synthese Library, vol 113. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1282-8_3
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