We present a proof of the equivalence between two deductive systems for constructive necessity, namely an axiomatic characterisation inspired by Hakli and Negri's system of derivations from assumptions for modal logic , a Hilbert-style formalism designed to ensure the validity of the deduction theorem, and the judgmental reconstruction given by Pfenning and Davies by means of a natural deduction approach that makes a distinction between valid and true formulae, constructively. Both systems and the proof of their equivalence are formally verified (...) using the state-of-the-art proof assistant Coq. The proof approach taken throughout the paper unveils the use of some alternative proof methods that allow for a smooth transition from the high-level mathematical proofs to their mechanised counterparts. (shrink)

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that (...) restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

We present a sequent calculus for the modal logic S4, and building on some relevant features of this system we show how S4 can easily be translated into full propositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities only in correspondence with S4 modalities. We discuss the complexity of the decision problem for several classes of linear formulas naturally arising from the proposed translations.

Este livro marca o início da Série Investigação Filosófica. Uma série de livros de traduções de textos de plataformas internacionalmente reconhecidas, que possa servir tanto como material didático para os professores das diferentes subáreas e níveis da Filosofia quanto como material de estudo para o desenvolvimento pesquisas relevantes na área. Nós, professores, sabemos o quão difícil é encontrar bons materiais em português para indicarmos. E há uma certa deficiência na graduação brasileira de filosofia, principalmente em localizações menos favorecidas, com relação (...) ao conhecimento de outras línguas, como o inglês e o francês. Tentamos, então, suprir essa deficiência, ao introduzirmos traduções de textos importantes ao público de língua portuguesa, sem nenhuma finalidade comercial e meramente pela glória da filosofia. O presente volume é constituído de três traduções de verbetes importantes sobre lógica, da Enciclopédia de Filosofia da Stanford: (1) A Lógica de Aristóteles, (2) Lógica Clássica, (3) Lógica Modal. (shrink)

The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 1 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right (...) of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else). This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1. These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical. (shrink)

This paper sets out a semantics for C.I. Lewis's logic S2 based on the ontology of his 1923 paper ‘Facts, Systems, and the Unity of the World’. In that article, worlds are taken to be maximal consistent systems. A system, moreover, is a collection of facts that is closed under logical entailment and conjunction. In this paper, instead of defining systems in terms of logical entailment, I use certain ideas in Lewis's epistemology and philosophy of logic to define a class (...) of models in which systems are taken to be primitive elements but bear certain relations to one another. I prove soundness and completeness for S2 over this class of models and argue that this semantics makes sense of at least a substantial fragment of Lewis's logical theory. (shrink)

In this paper, I examine Ruth Barcan Marcus's early formal work on modal systems and the deduction theorem, both for the material and the strict conditional. Marcus proved that the deduction theorem for the material conditional does not hold for system S2 but holds for S4. This last result is at odds with the recent claim that without proper restrictions the deduction theorem fails also for S4. I explain where the contrast stems from. For the strict conditional, Marcus proved the (...) deduction theorem for S4 though restricted to arguments with necessary premises. I discuss Marcus's result and analyze her philosophical position on the significance of the deduction theorem for modal systems designed to express the notion of deducibility. (shrink)

It has been argued that ethically correct robots should be able to reason about right and wrong. In order to do so, they must have a set of do’s and don’ts at their disposal. However, such a list may be inconsistent, incomplete or otherwise unsatisfactory, depending on the reasoning principles that one employs. For this reason, it might be desirable if robots were to some extent able to reason about their own reasoning—in other words, if they had some meta-ethical capacities. (...) In this paper, we sketch how one might go about designing robots that have such capacities. We show that the field of computational meta-ethics can profit from the same tools as have been used in computational metaphysics. (shrink)