37 research outputs found
Three-valued logics, uncertainty management and rough sets
This paper is a survey of the connections between three-valued logics and rough sets from the point of view of incomplete information management. Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such as rough sets. We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an equivalence relation is very limited, thus casting doubts on the practical relevance of truth-functional logical renderings of rough sets
Potential infinity, abstraction principles and arithmetic (Leniewski Style)
This paper starts with an explanation of how the logicist research program can be approached within the framework of LeĆniewskiâs systems. One nice feature of the system is that Humeâs Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of LeĆniewskiâs own arithmetic and a brief investigation into its properties
Constructive Fuzzy Logics
We generalise Kripkeâs semantics for Intuitionistic logic to Hajekâs BL and consider the constructive subsystems of GBLewf and Intuitionistic Affine logic or ALi. The genesis of our semantics is the Poset Product construction for GBL-algebras elucidated in a series of papers by Peter Jipsen, Simone Bova, and Franco Montagna. We present natural deduction systems for all of these systems and corresponding deduction theorems for these same. We present the algebraic semantics for each of the logics under consideration, demonstrate their soundness and completeness with respect to these algebraic semantics. We also show how the classical Kripke semantics for Intuitionistic logic can be recast in terms of Poset Products. We then proceed to the main results, showing how a very natural generalisation of the Kripke semantics holds for each of GBLewf , ALi and Hajekâs BL based on the embedding results of Jipsen and Montagna and the decidability results of Bova and Montagna. We demonstrate soundness and completeness of the logics under our semantics in each case, with the exception of ALi, whose robust completeness with respect to the intended models (relational models with frames valued in involutive pocrims) we leave as an open problem for the ambitious reader
Achieving while maintaining:A logic of knowing how with intermediate constraints
In this paper, we propose a ternary knowing how operator to express that the
agent knows how to achieve given while maintaining
in-between. It generalizes the logic of goal-directed knowing how proposed by
Yanjing Wang 2015 'A logic of knowing how'. We give a sound and complete
axiomatization of this logic.Comment: appear in Proceedings of ICLA 201
Implementing Theorem Provers in Logic Programming
Logic programming languages have many characteristics that indicate that they should serve as good implementation languages for theorem provers. For example, they are based on search and unification which are also fundamental to theorem proving. We show how an extended logic programming language can be used to implement theorem provers and other aspects of proof systems for a variety of logics. In this language first-order terms are replaced with simply-typed λ-terms, and thus unification becomes higher-order unification. Also, implication and universal quantification are allowed in goals. We illustrate that inference rules can be very naturally specified, and that the primitive search operations of this language correspond to those needed for searching for proofs. We argue on several levels that this extended logic programming language provides a very suitable environment for implementing tactic style theorem provers. Such theorem provers provide extensive capabilities for integrating techniques for automated theorem proving into an interactive proof environment. We are also concerned with representing proofs as objects. We illustrate how such objects can be constructed and manipulated in the logic programming setting. Finally, we propose extensions to tactic style theorem provers in working toward the goal of developing an interactive theorem proving environment that provides a user with many tools and techniques for building and manipulating proofs, and that integrates sophisticated capabilities for automated proof discovery. Many of the theorem provers we present have been implemented in the higher-order logic programming language λProlog
Parikh and Wittgenstein
A survey of Parikhâs philosophical appropriations of Wittgensteinian themes, placed into historical context against the backdrop of Turingâs famous paper, âOn computable numbers, with an application to the Entscheidungsproblemâ (Turing in Proc Lond Math Soc 2(42): 230â265, 1936/1937) and its connections with Wittgenstein and the foundations of mathematics. Characterizing Parikhâs contributions to the interaction between logic and philosophy at its foundations, we argue that his work gives the lie to recent presentations of Wittgensteinâs so-called metaphilosophy (e.g., Horwich in Wittgensteinâs metaphilosophy. Oxford University Press, Oxford, 2012) as a kind of âdead endâ quietism. From early work on the idea of a feasibility in arithmetic (Parikh in J Symb Log 36(3):494â508, 1971) and vagueness (Parikh in Logic, language and method. Reidel, Boston, pp 241â261, 1983) to his more recent program in social software (Parikh in Advances in modal logic, vol 2. CSLI Publications, Stanford, pp 381â400, 2001a), Parikhâs work encompasses and touches upon many foundational issues in epistemology, philosophy of logic, philosophy of language, and value theory. But it expresses a unified philosophical point of view. In his most recent work, questions about public and private languages, opportunity spaces, strategic voting, non-monotonic inference and knowledge in literature provide a remarkable series of suggestions about how to present issues of fundamental importance in theoretical computer science as serious philosophical issues
Logics of formal inconsistency
Orientadores: Walter Alexandre Carnielli, Carlos M. C. L. CaleiroTexto em ingles e portuguesTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias HumanasTese (doutorado) - Universidade Tecnica de Lisboa, Instituto Superior TecnicoResumo: Segundo a pressuposição de consistĂȘncia clĂĄssica, as contradiçÔes tĂȘm um carĂĄ[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocĂnio sensato pode entĂŁo ter lugar. Uma lĂłgica Ă© paraconsistente se ela rejeita uma tal pressuposição, e aceita ao invĂ©s que algumas teorias inconsistentes conquanto nĂŁo-triviais façam perfeito sentido. A? LĂłgicas da InconsistĂȘncia Formal, LIFs, formam uma classe de lĂłgicas paraconsistentes particularmente expressivas nas quais a noção meta-teĂłnca de consistĂȘncia pode ser internalizada ao nĂvel da linguagem obje[c]to. Como consequĂȘncia, as LIFs sĂŁo capazes de recapturar o raciocĂnio consistente pelo acrĂ©scimo de assunçÔes de consistĂȘncia apropriadas. Assim, por exemplo, enquanto regras clĂĄssicas tais como o silogismo disjuntivo (de A e {nĂŁo-,4)-ou-13, infira B) estĂŁo fadadas a falhar numa lĂłgica paraconsistente (pois A e (nao-A) poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presunção de que estamos raciocinando em um ambiente consistente (neste caso, pelo acrĂ©scimo de (consistente-.A) como uma hipĂłtese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustraçÔes destas lĂłgicas e de suas propriedades, mostrando que tais lĂłgicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocĂnio consistente dentro de tais sistemas inconsistentes sĂŁo tambĂ©m ilustradas Em cada caso, interpretaçÔes em termos de semĂąnticas polivalentes, de traduçÔes possĂveis ou modais sĂŁo fornecidas, e os problemas relacionados Ă provisĂŁo de contrapartidas algĂ©bricas para tais lĂłgicas sĂŁo examinados. Uma abordagem formal abstra[cjta Ă© proposta para todas as definiçÔes relacionadas e uma extensa investigação Ă© feita sobre os princĂpios lĂłgicos e as propriedades positivas e negativas da negação.Abstract: According to the classical consistency presupposition, contradictions have an explosive character: Whenever they are present in a theory, anything goes, and no sensible reasoning can thus take place. A logic is paraconsistent if it disallows such presupposition, and allows instead for some inconsistent yet non-trivial theories to make perfect sense. The Logics of Formal Inconsistency, LFIs, form a particularly expressive class of paraconsistent logics in which the metatheoretical notion of consistency can be internalized at the object-language level. As a consequence, the LFIs are able to recapture consistent reasoning by the addition of appropriate consistency assumptions. So, for instance, while classical rules such as disjunctive syllogism (from A and (not-A)-or-B, infer B) are bound to fail in a paraconsistent logic (because A and (not-.4) could both be true for some A, independently of B), they can be recovered by an LFI if the set of premises is enlarged by the presumption that we are reasoning in a consistent environment (in this case, by the addition of (consistent-/!) as an extra hypothesis of the rule). The present monograph introduces the LFIs and provides several illustrations of them and of their properties, showing that such logics constitute in fact the majority of interesting paraconsistent systems from the literature. Several ways of performing the recapture of consistent reasoning inside such inconsistent systems are also illustrated. In each case, interpretations in terms of many-valued, possible-translations, or modal semantics are provided, and the problems related to providing algebraic counterparts to such logics are surveyed. A formal abstract approach is proposed to all related definitions and an extended investigation is carried out into the logical principles and the positive and negative properties of negation.DoutoradoFilosofiaDoutor em Filosofia e MatemĂĄtic
Tempered Pragmatism
This paper assesses the prospects of a pragmatist theory of content. I begin by criticising the theory presented in D.H. Mellorâs essay âSuccessful Semanticsâ. I then identify problems and lacunae in the pragmatist theory of meaning sketched in Chapter 13 of Dummettâs The Logical Basis of Metaphysics. The prospects are brighter, I contend, for a tempered pragmatism, in which the theory of content is permitted to draw upon irreducible notions of truth and falsity. I sketch the shape of such a theory and illustrate the role of its pragmatist elements by showing how they point towards a promising account of the truth conditions of indicative conditionals. A feature of the account is that it validates Modus Ponens whilst invalidating Modus Tollens