Structuralism has recently moved center stage in philosophy of mathematics. One of the issues discussed is the underlying logic of mathematical structuralism. In this paper, I want to look at the dual question, namely the underlying structures of logic. Indeed, from a mathematical structuralist standpoint, it makes perfect sense to try to identify the abstract structures underlying logic. We claim that one answer to this question is provided by categorical logic. In fact, we claim that the latter can be seen—and (...) probably should be seen—as being a structuralist approach to logic and it is from this angle that categorical logic is best understood. (shrink)

In the Böhm theorem workshop on Crete, Zoran Petric called Statman’s “Typical Ambiguity theorem” the typed Böhm theorem. Moreover, he gave a new proof of the theorem based on set-theoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Böhm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment of intuitionistic linear logic without the multiplicative unit the weak typed Böhm theorem holds. The system IMLL (...) exactly corresponds to the linear lambda calculus with multiplicative pairing. The system IMLL also exactly corresponds to the free symmetric monoidal closed category without the unit object. As far as we know, our separation result is the first one with regard to these systems in a purely syntactical manner. (shrink)

Monoidal logics were introduced as a foundational framework to analyse the proof theory of deontic logic. Building on Lambek’s work in categorical logic, logical systems are defined as deductive systems, that is, as collections of equivalence classes of proofs satisfying specific rules and axiom schemata. This approach enables the classification of deductive systems with respect to their categorical structure. When looking at their proof theory, however, one can see that there are similarities between monoidal and substructural logics. The purpose of (...) the present paper is to address this issue and highlight the differences between these two approaches. We argue that monoidal logics provide a more flexible foundational framework that enables a finer analysis of the relationship between negation and other logical connectives. We show that the elimination of double negation is independent from the de Morgan dualities, that monoidal deductive systems are not necessarily weakly distributive and that deduc... (shrink)

In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic , which is the linear fragment of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories / of an ambient *-autonomous category . Similar structures were first introduced by M. Barr in the late 1970’s in abstract duality theory and more recently in work on game semantics for linear logic. The paper has two goals: to discuss concrete models and (...) to present various completeness theorems.As concrete examples, we present a hypercoherence model, using Ehrhard’s hereditary/anti-hereditary objects, a Chu-space model, a double gluing model over our categorical framework, and a model based on iterated double gluing over a *-autonomous category.For the multiplicative fragment of. (shrink)

We study the coherence, that is the equality of canonical natural transformations in non-free symmetric monoidal closed categories . To this aim, we use proof theory for intuitionistic multiplicative linear logic with unit. The study of coherence in non-free smccs is reduced to the study of equivalences on terms in the free category, which include the equivalences induced by the smcc structure. The free category is reformulated as the sequent calculus for imll with unit so that only equivalences on derivations (...) in this system are to be considered. We establish that any equivalence induced by the equality of canonical natural transformations over a model can be axiomatized by some set of “critical” pairs of derivations. From this, we derive certain sufficient conditions for full coherence, and establish that the system of identities defining smccs is not Post-complete: extending this system with an identity that does not hold in the free smcc does not in general cause the free smcc to collapse into a preorder. In order to give a larger context to these results, we study the equality of canonical morphisms in non-free symmetric monoidal categories, and establish that w.r.t. a broad subclass of smccs, the equivalences induced by the equality of canonical natural transformations over a model coincide with the equivalences induced by the equality of canonical morphisms for all interpretations in that model. (shrink)

Coherence with respect to Kelly–Mac Lane graphs is proved for categories that correspond to the multiplicative fragment without constant propositions of classical linear first-order predicate logic without or with mix. To obtain this result, coherence is first established for categories that correspond to the multiplicative conjunction–disjunction fragment with first-order quantifiers of classical linear logic, a fragment lacking negation. These results extend results of [K. Došen, Z. Petrić, Proof-Theoretical Coherence, KCL Publications , London, 2004 ; K. Došen, Z. Petrić, Proof-Net Categories, (...) Polimetrica, Monza, 2007 ], where coherence was established for categories of the corresponding fragments of propositional classical linear logic, which are related to proof nets, and which could be described as star-autonomous categories without unit objects. (shrink)