Philosophy 290-3

Fall 2022

Number Title Instructor Days/time Room
290-3 Graduate Seminar: Proof Theory and Proof-Theoretic Semantics Mancosu Tu 2-4 Moses 234

The most familiar semantics for formal or natural languages are those expressed in terms of Tarskian reference conditions connecting linguistic expressions (individual constants, predicates etc.) to extra-linguistic entities (objects, sets of objects etc.). This “referential” conception of semantics is captured by model-theoretic notions. An alternative type of semantics, motivated by a view of “meaning as use”, tries to account for the semantic properties of expressions by means of rules of use which govern linguistic expressions. An especially significant approach has focused on the role of inferences. In this approach, sometimes referred to as inferential semantics, the roles of inferences in the practice of inferring plays a central role in explaining the meaning of various parts of language. The logical constant “and”, for instance, is explained in terms of the inferential behavior governing it. This leads to the study of the rules of introduction and elimination for “and”, which govern the way in which we infer sentences containing “and” and draw consequences from such sentences. Proof-theory (hence the name proof-theoretic semantics) has developed formalisms that are quite suitable for the formulation of such semantics. In particular, the system of natural deduction introduced by Gentzen in the 1930s has been a major tool in proof-theoretic semantics. In this seminar we will first study some basic results in proof-theory that are needed in the study of proof-theoretic semantics (such as for instance the natural deduction formalization of logic, the normalization of proofs in natural deduction, the sub-formula property for normal proofs etc.). Then we will explore some interesting issues concerning what properties (harmony, conservativity etc.) logical constants should satisfy if the rules of introduction and elimination for such constants can be said to fix their meanings. We will also pursue the topic of “bilateralism”, i.e. whether assertion alone should be the primary modality of inferring or whether assertion and denial should both play an independent role. For the proof-theory part of the seminar we will use chapters 3 and 4 of Mancosu et al. An Introduction to Proof Theory (OUP 2021). For the second part of the seminar we will read articles by, among others, Prior, Belnap, Dummett, Prawitz, Steinberger, Read, Rumfitt, and Tennant.