On the logic of nonmonotonic conditionals and conditional probabilities: Predicate logic
This very rich paper explores the possibilities of construing conditionals as semantic relations on an object language. It extends the author's work of ``On the logic of nonmonotonic conditionals and conditional probabilities'' [J. Philos. Logic 25, 185-218 (1996; Zbl 0854.03019)] to the case of predicate logic. The general idea is to take Popper functions (conditional probability functions) as the primitives of a semantic theory. A (nonmonotonic) conditional \(\rightarrow\) is interpreted as a set of ordered pairs of sentences in a first order language. \(R\) is a set of pairs that counts as an ``EQ-model class'' just in case every \(\rightarrow\) in \(R\) satisfies a certain set of conditions. The idea is that ``associated with each nonmonotonic conditional, \(\rightarrow\), is a way of assigning meanings to sentences and a way of measuring classes of possible worlds; and, relative to its associated meanings and measures, a conditional assertion \(C \rightarrow B\) holds \textit{just in case} \(B\) is true in almost all possible worlds in which \(C\) is true.'' The treatment of quantification, although it is essentially substitutional, is based on ``name-extensions'' of a given language: for example, given an objectual interpretation of a language \(L\) with a finite number of individual names that makes \((\exists x)Fx\) true, despite the fact that \(Fc\) is false for every name \(c\), we can construct a name-extension \(L^+\) containing a new name \(b\) that satisfies \(F\). Thus we get what amounts to an objectual semantics for quantification (complete with compactness), without (quite) going beyond a substitutional interpretation of the quantifies. This is what makes it possible to give a unified quite general theory of nonmonotonic conditionals in terms of Popper functions.
scientific article; zbMATH DE number 1156515