Compositional Anomalies in the Semantics of Evidence

Semantic theory plays a central role in the normative and descriptive theory of deductive inference, but its role in the study of inductive inference has been much less prominent. The disparity is both odd and understandable. It is odd because deductive and inductive reasoning both rely heavily on linguistic representations, and semantic theory is the natural tool for investigating inference within propositional structures. It is understandable because the logical approach to induction, as championed by Lambert, Keynes, and Carnap, to name only a few, was eclipsed by important developments in the theory of subjective probability, following the work of De Finetti, Savage, and others. From a semantical perspective, the basis for subjective probability theory is very elementary, namely, that there exist Boolean algebras of events that are ordered by how strongly one believes that they are true (the belief strength ordering). The axiomatic theory of subjective probability specifies further properties of belief strength that characterize so-called coherent beliefs; if strength of belief is coherent in this sense, then there exist numerical probabilities that represent the belief strength ordering and satisfy the mathematical laws of probability (Fine, 1973; Krantz, Luce, Suppes, & Tversky, 1971)1. Historically, the study of natural language semantics has been closely linked to theories of deductive inference because deductive relations among propositions serve as clues to semantic structure (Davidson & Harman, 1975a; Quine 1960). A theory of semantics relates three aspects of language: the syntactic structure of propositions, i.e., a specification of how complex propositions are built from simpler parts; the semantic structure of propositions, i.e., a specification of the relation between propositional structure, reference, and truth values; and inference rules that define inferential relations in terms of syntactic and semantic structure. Together these three aspects constitute a compositional theory of inference, a theory of the relationship between deductive inference and the compositional structure of propositions. A classical example of this line of analysis is the inference rule called modus ponens according to which the truth of a proposition Q may be inferred, given that if P, then Q and P are true. The truth table for the material conditional (the ifthen statement of the propositional calculus) is a semantic hypothesis concerning the meaning of conditional statements. It serves as part of the explanation for why inferences of the form of modus ponens are valid. Of course, this hypothesis is open to debate--the truth table for the material conditional is widely accepted as a semantic analysis of if-then statements in mathematical proofs, but it is quite debatable as an analysis of conditionals in ordinary (non-mathematical) discourse (Traugott, ter Meulen, Reilly, & Ferguson, 1986). Our point is simply that deductive relations