Abstract
In The Logical Foundations of Probability 1 Carnap develops a theory of probability based on classical predicate logic in which probability is treated as a generalization of logic. In this respect Carnap’s theory is in the tradition of the classical probability theorists, notably La Place and Hume,2 for in the classical theorie probability is considered to be the proportion of possible cases in which a proposition is true, and is thus a generalization of the logical concept of validity, or truth in all possible cases, and consistency, or truth in some possible case. Hume’s account differs slightly but importantly: in it probability is related to inference and the logical concept of implication is generalized. On this theory the probability of a proposition relative to a set of premises is the proportion of extensions of the premises in which the conclusion holds. Carnap’s confirmation functions are defined in this way too, where the requisite logical concepts are those of predicate logic, and his account differs from the classical theorists - apart from the additional precision available through the use of advanced logical techniques - mainly in the role played by the principle of indifference. This principle asserts that distinct atomic cases are equiprobable. The classical theorists accepted it3 and Carnap makes use of it for a priori measures only in restricted ways.
I am indebted to Patrick Suppes for several conversations which helped clarify many of the questions with which this paper is concerned. Work on this paper was supported in part by NSF Grant # Gs-2099.
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References
Second edition, Chicago, 1962.
P. S. La Place, A Philosophical Essay On Probabilities, New York 1951. D. Hume, A Treatise of Human Nature (ed. by L. A. Selby-Bigge), Oxford 1888, esp. Sections 11 and 13 of Book I Part III.
Cf. e. g. Hume’s discussion, op. cit, pp. 128f.
As evidenced in his ‘The Aim of Inductive Logic’ in Logic, Methodology and Philosophy of Science (ed. by E. Nagel, P. Suppes and A. Tarski), Stanford 1962.
F. P. Ramsey, ‘Truth and Probability’ in The Foundations of Mathematics, London 1931. B. De Finetti, ‘La prevision, ses lois logiques, ses sources subjectives’, Annales de L’Institut Henri Poincaré, 7 (1937).
For a guide to the considerable literature on this subject see the introduction to H. Kyburg, and H. Smokier, Studies in Subjective Probability, New York 1964. This book includes Ramsey’s article mentioned above as well as a translation of De Finetti’s monograph.
Cf. my article ‘Some Remarks on Coherence and Subjective Probability’, Philosophy of Science, 32 (January 1965) 4, where this is done rather clumsily.
‘The Aim of Inductive Logic. ’
The situation remains as far as I know as it was described by Carnap in ‘On the Application of Inductive Logic’, Philosophy and Phenomenological Research, 8 (September, 1947), 133–148. Cf. R. C. Jeffrey, ‘Goodman’s Query’, Journal of Philosophy 63 (May 26, 1966) 11, 281–288 for a proposal and my article ‘Characteristics of Projectible Predicates’, Journal of Philosophy, 64 (May, 1967), 280–286 in which the proposal is shown to be inadequate.
In ‘Judgment and Belief, in The Logical Way of Doing Things (ed. by K. Lambert), New York, 1969, I associate degree of transparency of belief with the extent to which it is behavioristic as distinct from mentalistic. That paper is concerned with the relation between mentalistic and behavioristic belief, and ignores - except for a few casual comments - the question of the nature of the theory of probability associated with this view of belief. Both there and in the present essay belief is taken to be relative to a logic, and both papers are attempts at accounting for features of this relativity.
The full condition is: p is a coherent betting function on X just in case for every pair X1, X2 of disjoint subsets of X, there is some conjunctively consistent Y such that Y implies n 1 propositions in X 1, n 2 propositions in X2 and \((n{}_1 - \sum\limits_{A \in X{}_2} {p(A)) \geqslant (n{}_2 - \sum\limits_{A \in X{}_1} {P(A)).} } \)
We think of the agent betting on propositions in X 1 and against those in X 2. C 1 is a special case of this.
‘Truth and Probability’, p. 189.
He does say that probability is a generalization of formal logic (ibid., p. 186) but he does not appear to think of different probability theories as based upon different logics.
Church’s Thesis is that all and only general recursive functions are effectively computable. It was first proposed by Alonzo Church in ‘An Unsolvable Problem of Elementary Number Theory’, American Journal of Mathematics 58 (1936) 345–363. Cf. S. C. Kleene’s discussion in Introduction to Metamathematics, Princeton 1952, p. 130 ff.
See S. C. Kleene and R. E. Vesley, The Foundations of Intuitionistic Mathematics, Amsterdam 1965.
See, for example, B. van Fraassen, ‘Singular Terms, Truth Value Gaps, and Free Logic’, Journal of Philosophy 63 (1966) 481–495.
In, for example, Knowledge and Belief, Cornell 1962.
P. Suppes, ‘The Philosophical Relevance of Decision Theory’, Journal of Philosophy 58 (1961) 605–614.
H. Gaifman, ‘Concerning Measures on First Order Calculi’, Israel Journal of Mathematics 2, 1–18. D. Scott and P. Krauss, ‘Assigning Probabilities to Logical Formulas’, in Aspects of inductive Logic (ed. by J. Hintikka and P. Suppes), Amsterdam 1966, pp.219–264.
I use A, B, C,… to range over propositions, X, Y, Z,… and P, Q,… to range over sets of propositions. P always refers to a set of propositions which is properly covered by the family {P i }. In referring to the unit set {A}, set brackets are frequently omitted.
Cf. reference 11 for the generalized form. The proof does not differ essentially.
Cf. e.g. Probability and the Logic of Rational Belief, Middletown 1961.
In R. C. Stalnaker, ‘Probability and Conditionals’, mimeographed, Yale Univ., this relation is explored in some depth.
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Vickers, J.M. (1970). Probability and Non Standard Logics. In: Lambert, K. (eds) Philosophical Problems in Logic. Synthese Library, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-3272-8_5
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