Abstract-Propositional Gödel logic G extends intuitionistic logic with the non-constructive principle of linearity (A → B) ∨ (B → A). We introduce a Curry-Howard correspondence for G and show that a simple natural deduction calculus can be used as a typing system. The resulting functional language extends the simply typed λ-calculus via a synchronous communication mechanism between parallel processes, which increases its expressive power. The normalization proof employs original termination arguments and proof transformations implementing forms of code mobility. Our results provide a computational interpretation of G, thus proving A. Avron's 1991 thesis.

Abstract. Starting with the deontic principles in Mīmām . sā texts we introduce a new deontic logic. We use general proof-theoretic methods to obtain a cut-free sequent calculus for this logic, resulting in decidability, complexity results and neighbourhood semantics. The latter is used to analyse a well known example of conflicting obligations from the Vedas.

The Mīmāṃsā school of Indian philosophy elaborated complex ways of interpreting the prescriptive portions of the Vedic sacred texts. The present article is the result of the collaboration of a group of scholars of logic, computer science, European philosophy and Indian philosophy and aims at the individuation and analysis of the deontic system which is applied but never explicitly discussed in Mīmāṃsā texts. The article outlines the basic distinction between three sorts of principles-hermeneutic, linguistic and deontic. It proposes a mathematical formalization of the deontic principles and uses it to discuss a well-known example of seemingly conflicting statements, namely the prescription to undertake the malefic Śyena sacrifice and the prohibition to perform any harm.

This paper studies the notions of conceptual grounding and conceptual explanation (which includes the notion of mathematical explanation), with an aim of clarifying the links between them. On the one hand, it analyses complex examples of these two notions that bring to the fore features that are easily overlooked otherwise. On the other hand, it provides a formal framework for modeling both conceptual grounding and conceptual explanation, based on the concept of proof. Inspiration and analogies are drawn with the recent research in metaphysics on the pair metaphysical grounding-metaphysical explanation, and especially with the literature in philosophy of science on the pair causality-causal explanation.

Along the lines of the Abramsky "Proofs-as-Processes" program, we present an interpretation of multiplicative linear logic as typing system for concurrent functional programming. In particular, we study a linear multiple-conclusion natural deduction system and show it is isomorphic to a simple and natural extension of λ-calculus with parallelism and communication primitives, called λ`. We shall prove that λ`satisfies all the desirable properties for a typed programming language: subject reduction, progress, strong normalization and confluence. * Funded by FWF grant P32080-N31. † Funded by ANR JCJC project Intuitions Bolzaniennes. linking proofs to programs, propositions to types, proof normalization to computation. Soon after, Wadler [32] introduced the session typed π-calculus CP, shown to tightly correspond to classical linear logic. LimitationsThis important research notwithstanding, it appears that there is still ground to cover toward canonical and firm foundations for concurrent computation. First of all, asynchronous communication does not appear to rest on solid foundations. Yet this communication style is easier to implement and more practical than the purely synchronous paradigm, so it is widespread and asynchronous typed process calculi have been already investigated (see [13]). Unfortunately, linear logic has not so far provided via Curry-Howard a logical account of asynchronous communication. The reason is that there is a glaring discrepancy between full cut-elimination and π-calculus reduction which has not so far been addressed. On one hand, as we shall see, linear logic does support asynchronous communication, but only through the full process of cut-elimination, which indeed makes essential use of asynchronous communication. On the other hand, the π-calculus of [9] only mimics a partial cut-elimination process that only eliminates top-level cuts. Indeed, by comparison with its type system, the π-calculus lacks some necessary computational reduction rules. Some of the missing reductions, corresponding to commuting conversions, were provided in Wadler's CP. The congruence rules that allow the extra reductions to mirror full cut-elimination, however, were rejected: in Wadler's [32] own words, "such rules do not correspond well to our notion of computation on processes, so we omit them". A set of reductions similar to that rejected by Wadler is regarded in [27] as sound relatively to a notion of "typed context bisimilarity". The notion, however, only ensures that two "bisimilar" processes have the same input/output behaviour along their main communication channel; the internal synchronization among the parallel components of the two processes is not captured and may differ significantly. Thus, the extensional flavor of the bisimilarity notion prevents it to detect the intensionally different behaviour of the related processes, that is, how differently they communicate and compute.