Relative Entailment Among Probabilistic Implications

We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective; the semantics of this sort of implication is defined in terms of a threshold on a conditional probability of the consequent, given the antecedent: we are dealing with what the data analysis community calls confidence of partial implications or association rules. Existing studies of redundancy among these partial implications have characterized so far only entailment from one premise and entailment from two premises, both in the stand-alone case and in the case of presence of additional classical implications (this is what we call "relative entailment"). By exploiting a previously noted alternative view of the entailment in terms of linear programming duality, we characterize exactly the cases of entailment from arbitrary numbers of premises, again both in the stand-alone case and in the case of presence of additional classical implications. As a result, we obtain decision algorithms of better complexity; additionally, for each potential case of entailment, we identify a critical confidence threshold and show that it is, actually, intrinsic to each set of premises and antecedent of the conclusion

Linear Types, Protocols, and Processes in Classical F°

Session types and typestate both promise a type system that can reason about protocol adherence. The complexity budgets of most programming languages, however, do not allow for new forms of types aimed at specific problem domains--even domains as broad as these. Classical F◦ --read F-pop --is a typed λ-calculus based on classical (i.e., full) linear logic, wherein session types arise naturally from the interaction between the usual sums, products, and implications of linear logic and a simple process model, with the dualizing negation of classical logic naturally accounting for how a protocol is seen by each of a channel\u27s endpoints. Classical F◦ expressions evaluate to processes, reminiscent of those in the π-calculus, that communicate over channels, but source expressions, rather than including processes and channels, employ only two novel control operators that account for process creation and communication. F◦ is introduced by way of its intuitionistic fragment, which even on its own can account for typestate: the combination of linearity and polymorphism leads to natural encodings of many programmer-specified protocols. In fact, any protocol expressible as a regular language can be encoded in an intuitionistic F◦ type. F◦ distinguishes between linear and unrestricted types by using kinds together with a notion of subkinding, avoiding the pitfalls of approaches based on type qualifiers or modalities; kinds are related by a subkinding order that allows unrestricted types to be treated as though they were linear. Soundness for intuitionistic and classical F◦ is proved both in the standard operational sense of preservation and progress and for an augmented semantics that shows more directly that the expected properties of linearity are preserved. This establishes the absence of deadlocks in closed, well-typed F◦ programs; it also guarantees that such programs will not leak processes as long as their result types are unrestricted

Classical BI: Its Semantics and Proof Theory

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus \`a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.Comment: 42 pages, 8 figure

Non-normal modalities in variants of Linear Logic

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of Linear Logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic Linear Logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of Kripke resource models extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatization and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the logical framework to use by instantiating it as logics of agency. In particular, we propose a logic to reason about the resource-sensitive use of artefacts and illustrate it with a variety of examples

On Various Negative Translations

Several proof translations of classical mathematics into intuitionistic mathematics have been proposed in the literature over the past century. These are normally referred to as negative translations or double-negation translations. Among those, the most commonly cited are translations due to Kolmogorov, Godel, Gentzen, Kuroda and Krivine (in chronological order). In this paper we propose a framework for explaining how these different translations are related to each other. More precisely, we define a notion of a (modular) simplification starting from Kolmogorov translation, which leads to a partial order between different negative translations. In this derived ordering, Kuroda and Krivine are minimal elements. Two new minimal translations are introduced, with Godel and Gentzen translations sitting in between Kolmogorov and one of these new translations.Comment: In Proceedings CL&C 2010, arXiv:1101.520

Logical operators for ontological modeling

We show that logic has more to offer to ontologists than standard first order and modal operators. We first describe some operators of linear logic which we believe are particularly suitable for ontological modeling, and suggest how to interpret them within an ontological framework. After showing how they can coexist with those of classical logic, we analyze three notions of artifact from the literature to conclude that these linear operators allow for reducing the ontological commitment needed for their formalization, and even simplify their logical formulation

Buying Logical Principles with Ontological Coin: The Metaphysical Lessons of Adding epsilon to Intuitionistic Logic

We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where one can get an analogue of Diaconescu’s result, but also can disentangle the roles of certain other assumptions that are hidden in mathematical presentations. It is our view that these results have not received the attention they deserve: logicians are unlikely to read a discussion because the results considered are “already well known,” while the results are simultaneously unknown to philosophers who do not specialize in what most philosophers will regard as esoteric logics. This is a problem, since these results have important implications for and promise signif i cant illumination of contem- porary debates in metaphysics. The point of this paper is to make the nature of the results clear in a way accessible to philosophers who do not specialize in logic, and in a way that makes clear their implications for contemporary philo- sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism. Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett, realism, antirealis

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