Functional Interpretations of Intuitionistic Linear Logic

We present three different functional interpretations of intuitionistic linear logic ILL and show how these correspond to well-known functional interpretations of intuitionistic logic IL via embeddings of IL into ILL. The main difference from previous work of the second author is that in intuitionistic linear logic (as opposed to classical linear logic) the interpretations of !A are simpler and simultaneous quantifiers are no longer needed for the characterisation of the interpretations. We then compare our approach in developing these three proof interpretations with the one of de Paiva around the Dialectica category model of linear logic

Computational Interpretations of Classical Linear Logic

Abstract. We survey several computational interpretations of classical linear logic based on two-player one-move games. The moves of the games are higher-order functionals in the language of finite types. All interpretations discussed treat the exponential-free fragment of linear logic in a common way. They only differ in how much advantage one of the players has in the exponentials games. We dis-cuss how the several choices for the interpretation of the modalities correspond to various well-known functional interpretations of intuitionistic logic, including Gödel’s Dialectica interpretation and Kreisel’s modified realizability.

Unifying Functional Interpretations: Past and Future

This article surveys work done in the last six years on the unification of various functional interpretations including G\"odel's dialectica interpretation, its Diller-Nahm variant, Kreisel modified realizability, Stein's family of functional interpretations, functional interpretations "with truth", and bounded functional interpretations. Our goal in the present paper is twofold: (1) to look back and single out the main lessons learnt so far, and (2) to look forward and list several open questions and possible directions for further research.Comment: 18 page

On an Intuitionistic Logic for Pragmatics

We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justications and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, dened as a hypothesis that in some situation the truth of p is epistemically necessary

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

From IF to BI: a tale of dependence and separation

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Vaananen, and their compositional semantics due to Hodges. We show how Hodges' semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural role, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio

A Galois connection between classical and intuitionistic logics. II: Semantics

Three classes of models of QHC, the joint logic of problems and propositions, are constructed, including a class of subset/sheaf-valued models that is related to solutions of some actual problems (such as solutions of algebraic equations) and combines the familiar Leibniz-Euler-Venn semantics of classical logic with a BHK-type semantics of intuitionistic logic. To test the models, we consider a number of principles and rules, which empirically appear to cover all "sufficiently simple" natural conjectures about the behaviour of the operators ! and ?, and include two hypotheses put forward by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these turns out to be either derivable in QHC or equivalent to one of only 13 principles and 1 rule, of which 10 principles and 1 rule are conservative over classical and intuitionistic logics. The three classes of models together suffice to confirm the independence of these 10 principles and 1 rule, and to determine the full lattice of implications between them, apart from one potential implication.Comment: 35 pages. v4: Section 4.6 "Summary" is added at the end of the paper. v3: Major revision of a half of v2. The results are improved and rewritten in terms of the meta-logic. The other half of v2 (Euclid's Elements as a theory over QHC) is expected to make part III after a revisio