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Published online by Cambridge University Press: 01 January 2020
The goal of standard semantics is to provide truth conditions for the sentences of a given language. Probabilistic Semantics does not share this aim; it might be said instead, if rather cryptically, that Probabilistic Semantics aims to provide belief conditions.
The central and guiding idea of Probabilistic Semantics is that each rational individual has ‘within’ him or her a personal subjective probability function. The output of the function when given a certain sentence as input represents the degree of likelihood which the individual would assign to that sentence. One can characterize these functions via a set of axioms, and in the terms of this defined structure develop probabilistic analogues of all important semantical notions (e.g. validity, entailment). Then, dealing with ‘being given probability 1’ instead of ‘truth,’ one can proceed to give a completely adequate semantics for the particular language under consideration. The axioms delimiting the class of probability functions are, in fact, chosen with this goal and language in mind.
1 Popper, Karl The Logic of Scientific Discovery (New York: Basic Books 1959)Google Scholar
2 Field, Hartry H. ‘Logic, Meaning and Conceptual Role,’ Journal of Philosophy, 74 (1977) 379–409Google Scholar. The groundwork laid out in this section comes mostly from Field and two papers by Hugues Leblanc cited below. It should also be noted that William Harper anticipated Field's work in his dissertation, and his work included the identity predicate.
3 Leblanc, H. ‘Probabilistic Semantics for First-Order Logic,’ Zeitschrift fuer Mathematische Logik und Grundlagen der Mathematik, 25 (1979) 497–509CrossRefGoogle Scholar; ‘Probabilistic Semantics: An Overview,’ mimeographed.
4 Leblanc, ‘Probabilistic Semantics for First-Order Logic.’ This result is his lemma 1. (y).
5 The following remarks stem from comments on an earlier version of this paper by Bas van Fraassen. The ‘venn-diagram’ example given below is more or less directly taken from these comments.
