We develop a semantic model for evidence that deals with interactions between propositional structure, evidence, and judgments of belief. It was a lot of fun to work on this chapter with John Miyamoto. I learned a lot from him. I think this is probably one of the most important scholarly contributions I’ve been involved in.

Miyamoto, J., Gonzalez, R., & Tu, S. (1995). Compositional anomalies in the semantics of evidence. In J. R. Busemeyer, R. Hastie, and D. L. Medin (Eds.), Decision Making from a Cognitive Perspective, Volume 32 of the Psychology of Learning and Motivation series. Academic Press, 319-383.   PDF

Introduction from the Chapter

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).

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 between conditionals and other statements constitute evidence for the semantics of conditionals, and  more generally, deductive relations among diverse natural language statements constitute evidence  for theories of the semantic structure of natural language (Davidson & Harman, 1975b; McCawley,  1993).

What we hope to show in the present paper is that there are many empirical phenomena that exhibit interesting interactions between the compositional structure of propositions and inferences drawn inductively from evidence. Just as deductive relations serve as clues to the semantic structure of propositions, relations between belief strength, propositional structure, and evidence serve as further clues to semantic structure. We will begin by describing six empirical phenomena that exemplify interactions between propositional structure, evidence, and judgments of belief. We then describe the basic elements of a semantic approach to the theory of evidence. A semantic theory of evidence is a theory of how strength of belief is affected by two factors: (i) the structure of the evidence, and (ii) the compositional structure of propositions. What we present is a framework for studying the interaction between propositional structure, evidence, and belief strength. Next, we consider a variety of empirical results that illustrate concretely the relationship between strength of belief, propositional structure, and the structure of evidence. Some of these results are well-known in the literature; for example, we will discuss conjunction and disjunction errors in probability judgment from the perspective of a semantic theory of evidence. Other results are less well-known; for example, aspects of counterfactual reasoning can be treated as problems in the semantics of evidence. What we hope to show is that the concept of a semantic theory of evidence provides a unifying framework for seeking general answers to the question of how belief strength is affected by natural language representations and the structure of evidence.

Many of the phenomena discussed in this paper can be characterized as compositional anomalies–they are cases in which observed relations in belief strength conflict with the semantic structure of propositions, or at least with well-established theories of this semantic structure. The term “compositional” is deserved because the conflicts arise in the relationship between the belief strengths of complex propositions and their component propositions. One reason for thinking that semantic theories of evidence will have something new to say about language structure is that compositional anomalies demonstrate the existence of perplexing inconsistencies between inductive and deductive inference. We will return to this point after presenting examples of such inconsistencies.