We investigate the universal fragment of intuitionistic logic focussing on equality of proofs. We give categorical models for that and prove several completeness results. One of them is a generalization of the well known Yoneda lemma and the other is an extension of Harvey Friedman's completeness result for typed lambda calculus.

We investigate the universal fragment of intuitionistic logic focussing on equality of proofs. We give categorical models for that and prove several completeness results. One of them is a generalization of the well known Yoneda lemma and the other is an extension of Harvey Friedman's completeness result for typed lambda calculus.

A categorical structure suitable for interpreting polymorphic lambda calculus (PLC) is defined, providing an algebraic semantics for PLC which is sound and complete. In fact, there is an equivalence between the theories and the categories. Also presented is a definitional extension of PLC including "subtypes", for example, equality subtypes, together with a construction providing models of the extended language, and a context for Girard's extension of the Dialectica interpretation.

First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved (...) for DFOL. The semantics is functorial in the sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra for a DFOL theory. Agreement with standard first-order semantics is established. Applications of DFOL to constructive mathematics and categorical foundations are given. A key feature is a local propositions-as-types principle. (shrink)

We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like “Set is the archetypal Cartesian Closed Category”, which means that proofs about CCCs can be done in the “archetypal language” and then lifted to proofs in the general setting. The method works even when our archetypal language (...) is diagrammatical, has potential ambiguities, is not completely formalized, and does not have semantics for all terms. We illustrate the method with an example from hyperdoctrines and another from synthetic differential geometry. (shrink)

We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin.One level is given by an intensional type theory, called Minimal type theory. This theory extends a previous version with collections.The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model.This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks (...) to the way the extensional level is linked to the intensional one which fulfills the “proofs-as-programs” paradigm and acts as a programming language. (shrink)

In this paper we introduce a novel way of building arithmetics whose background logic is R. The purpose of doing this is to point in the direction of a novel family of systems that could be candidates for being the infamous R#1/2 that Meyer suggested we look for.

There is a natural story about what logic is that sees it as tied up with two operations: a ‘throw things into a bag’ operation and a ‘closure’ operation. In a pair of recent papers, Jc Beall has fleshed out the account of logic this leaves us with in more detail. Using Beall’s exposition as a guide, this paper points out some problems with taking the second operation to be closure in the usual sense. After pointing out these problems, I (...) then turn to fixing them in a restricted case and modulo a few simplifying assumptions. In a followup paper, the simplifications and restrictions will be removed. (shrink)

We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from [8] that any Grothendieck fibration of the kind being considered can automatically be upgraded to a two-dimensional fibration, after which the invariance property is reduced to an (...) abstract theorem concerning pseudonatural transformations of morphisms into two-dimensional fibrations. (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)

We present the paradigm of categories-as-syntax. We briefly recall the even stronger paradigm categories-as-machine-language which led from -calculus to categorical combinators viewed as basic instructions of the Categorical Abstract Machine. We extend the categorical combinators so as to describe the proof theory of first order logic and higher order logic. We do not prove new results: the use of indexed categories and the description of quantifiers as adjoints goes back to Lawvere and has been developed in detail in works of (...) R. Seely. We rather propose a syntactic, equational presentation of those ideas. We sketch the (quasi-equational) categorical structures for dependent types, following ideas of J. Cartmell (contextual categories). All these theories of categorical combinators, together with the translations from -calculi into them, are introduced smoothly, thanks to the systematic use of– - an abstract variable-free notation for -calculus, going back to N. De Bruijn, – - a sequent formulation of the natural deduction. (shrink)