Some Remarks on Proof-Theoretic Semantics

Abstract

This is a tripartite work. The first part is a brief discussion of what it is to be a logical constant, rejecting a view that allows a particular self-referential "constant" • to be such a thing in favour of a view that leads to strong normalisation results. The second part is a commentary on the flattened version of Modus Ponens, and its relationship with rules of type theory. The third part is a commentary on work (joint with Nissim Francez) on "general elimination rules" and harmony, with a retraction of one of the main ideas of that work, i.e. the use of "flattened" general elimination rules for situations with discharge of assumptions. We begin with some general background on general elimination rules.

Open Problems in Proof-Theoretic Semantics

“…This second proposal would require the revision of basic tenets of proof-theoretic semantics, because ever since the work of von Kutschera [36], Prawitz [47] and Schroeder-Heister [53] on general constants, and since the work on general elimination rules, especially for implication, by Tennant [69,70], López-Escobar [37] and von Plato [43], 12 the idea of the indirect elimination rules as the basic form of elimination rules for all constants has been considered a great achievement. That said, the abandonment of projection-based conjunction and modus-ponens-based implication has received some criticism (Dyckhoff [15], Schroeder-Heister [66], Sect. 15.8).…”

Proof‐theoretic semantics of natural deduction based on inversion

“…Recently, there are doubts concerning general elimination rules formulated by Dyckhoff, 2016, even disadvantages such as: ‘too many deductions’, ‘disharmonious mess’ (p. 81), and others.…”

Bilateral Rules as Complex Rules

“…7 Nonetheless, Rumfitt's bilateral system is a development of this early tradition, and the author is explicitly skeptical about meaning-theoretical usages of sequent calculus. 8 Moreover, none of the criticisms that we will consider about this system crosses the limits of this traditional approach. Hence, later alternative approaches to proof-theoretic semantics can be overlooked in what follows.…”

“…See[28] (general proof theory) and[6].2 See[6] p. 258. This criterion has been criticized in[8], where the author proposes a new criterion.3[27], p. 33. For a historical account of the development of this principle, see[24].…”