Coherence of proof-net categories

Abstract

The notion of proof-net category defined in this paper is closely related to graphs implicit in proof nets for the multiplicative fragment without constant propositions of linear logic. Analogous graphs occur in Kelly's and Mac Lane's coherence theorem for symmetric monoidal closed categories. A coherence theorem with respect to these graphs is proved for proof-net categories. Such a coherence theorem is also proved in the presence of arrows corresponding to the mix principle of linear logic. The notion of proof-net category catches the unit free fragment of the notion of star-autonomous category, a special kind of symmetric monoidal closed category.

Open Problems in Proof-Theoretic Semantics

“…We identify a particular path that has not yet been explored, and we call this path 'intensional' or 'strong' harmony. The need to consider such a notion on the background of the discussion, initiated by Prawitz [45], on the identity of proofs, in particular in the context of Kosta Došen's work (see Došen [6,8,9], Došen and Petrić [12]), was raised by Luca Tranchini. 4 As the background to this issue we first present two conceptions of harmony, which are not reliant on the notion of identity of proofs.…”

“…This is because disjunction, which is used to express the introduction meaning for multiple introduction rules, is translated into disjunctionfree second-order logic in a way that makes its introduction meaning identical to its elimination meaning. 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.…”

“…Nevertheless, I should mention that the proper recognition of categorial logic within proof-theoretic semantics is still a desideratum. For the topic of categorial proof theory the reader is referred to Došen's work, in particular to his programmatic statement of 1995 [6], his contribution to this volume [11] and the detailed expositions in two monographs [7,12].…”

On the Paths of Categories

“…Implication is again characterized through adjunction in the categorial setting for the proof theory of linear propositional logic without modalities, which is investigated in [10].…”

What Is the Problem with Proof Nets for Classical Logic?

“…What has to be dropped is the property of being closed, i.e., there is no longer a bijection between the proofs of A ⊢ B ⇒ C and A ∧ B ⊢ C , which means we lose currying. This approach is studied in [14].…”