Abstract
The leading idea around which the present discussion revolves is a concept of the probability (or likelihood) of a statement. It is presupposed that a function Pr is given which assigns to each statement p at issue some real-number value, to be indicated as Pr(p), which lies in the interval between 0 and 1 (inclusive). The function Pr is intended to provide a measure of the ‘probability’ of statements in the usual sense of that term. Specifically, it is supposed that this numerical measure function comports itself in a normal, ‘well-behaved’ way in satisfying the usual rules of the theory of statement-probabilities,1 with the particular requirement that it meets inter alia the various conditions to be stipulated below.2
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References
Thus, for example, logically equivalent statements must be accorded the same probability-values.
Several methods for securing a measure of the probability of statements in this fashion have been discussed in the literature cited in the References given at the end of the chapter. Of these, the (semantically grounded) method of Rudolf Carnap’s important treatise on the Logical Foundations of Probability is the best known. However, no particular, specific method for the assignment of statement-probabilities need be assumed for our present purposes.
See P. R. Halmos, The Foundations of Probability’, American Mathematical Monthly 51 (1944) 493–510.
On this subject the reader is referred to R. Carnap’s Logical Foundations of Probability (Chicago, 1950).
It must be remembered that we are not claiming to characterize Pr at this juncture, but simply to note some of its traits which derive from its previously fixed character as a given measure over the domain D that conforms to the usual rules of a probability-calculus for statements.
This is readily shown by means of the probabilistic interpretation of modalities now to be discussed.
The present section draws upon ideas originally developed in the author’s paper, A Probabilistic Approach to Modal Logic’, Acta Philosophica Fennica,fasc. 16 (1963) 215–226.
In this rule ‘≡’ represents the familiar algebraic notation for an identity, rather than material equivalence.
In view of Rule 0, it might be thought that - rather than introducing a two-placetruth value, and then defining M-tautology in its terms - the best procedure would be to define M-tautologousness directly by Rule O. Although this tactic would indeed provide an adequate basis for the whole of the present discussion, I have chosen to adhere to two-place truth values for the following reasons. Firstly this machinery seems to me to have certain didactic advantages in keeping the notion of statement probabilities linked with the familiar resource of truth-values. Secôndly it makes semantical considerations clearer and more explicit. Finally, it makes possible an extension of the considerations of the present discussion to such existential axioms as Lewis’s B9, or
which could not be dealt with without the two-place truth-values or some similar mechanism.
See Appendix II of C.I. Lewis and C.H. Langford, Symbolic Logic (New York, 1932 ).
At this point too it is important that the possibility-space for the Pr measure is finite, since otherwise the possession by s of Pr-measure of 1 would not guarantee the Boolean identity of s with the universal element V.
Since the days of J. M. Keynes there has existed in the literature a concept of probability that is merely qualitative or comparative rather than full-bloodedly quantitative. In particular, see Leonard J. Savage, The Foundations of Statistics (John Wiley and Sons, New York and London 1954). This gives an axiomatic basis for a qualitative concept of probability. Although the present discussion has been based on a quantitative measure of probability, it has actually made use of only a few of the grosser characteristics of such a measure. Indeed, all of the results presented would still be forthcoming if this qualitative probability concept were taken for its basis, and the condition of having probability l’ were replaced by the condition having greatest possible comparative probability’. (I owe the substance of this footnote to Professor Patrick Suppes.)
See James Dugundji’s Note on a Property of Matrices for Lewis and Langford’s Calculi of Propositions’. The Journal of Symbolic Logic 5 (1940) 150–151.
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Rescher, N. (1968). Probability Logic. In: Topics in Philosophical Logic. Synthese Library, vol 17. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3546-9_11
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