Wyniki wyszukiwania - Biblioteka Nauki

1

Extensions of Intuitionistic Logic Without the Deduction Theorem: Some Simple Examples

100%

Humberstone L.

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tom Nr 40

45-82

EN

We provide some illustrations of consequence relations extending that associated with intuitionistic propositional logic but lacking the Deduction Theorem, together with a discussion of issues of some interest in their own right raised by these examples. There are two main examples, with some minor variations: one in which the language of intuitionistic logic is retained, and one in which this language is expanded.

2

An alternative intuitionistic version of Mally's deontic logic

63%

Lokhorst G. J. C.

Reports on Mathematical Logic

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2016

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tom Vol. 51

35--41

EN

Some years ago, Lokhorst proposed an intuitionistic reformulation of Mally's deontic logic (1926). This reformulation was unsatisfactory, because it provided a striking theorem that Mally himself did not mention. In this paper, we present an alter- native reformulation of Mally's deontic logic that does not provide this theorem.

3

An Alternative Natural Deduction for the Intuitionistic Propositional Logic

63%

Ilić M.

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tom 45

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nr 1

EN

A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.

4

Explicit Constructive Logic ECL : a New Representation of Construction and Selection of Logical Information by an Epistemic Agent

51%

Gentilini P. , Martelli M. , Rosolini G.

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tom Vol. 140, nr 3/4

357--372

EN

One of the main goals of Explicit Constructive Logic (ECL) is to provide a constructive formulation of Full (Classical) Higher Order Logic LKω that can be seen as a foundation for knowledge representation. ECL is introduced as a subsystem Zω of LKω. The first order case Z1 and the propositional case Z0 of ECL are examined as well. A comparison of constructivism from the point of view of ECL and of the corresponding features of Intuitionistic Logic, and Constructive Paraconsistent Logic is proposed.

5

Negacja określona i logika intuicjonistyczna

44%

Wojciechowski E.

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nr 1-4

301-311

EN

The philosophy of logic distinguishes between the ontological research attitude and the epistemic research attitude. On the other hand, there is the distinction between two types of negation: the classical / external / indefinite (~) and the non-classical / internal / definite (¬). The paper presents a propositional calculus with two types of negation (~,¬), which includes both the classical and the intuitionistic propositional calculus. We associate classical negation (~) with the ontological research attitude and definite negation (¬) with the epistemic one. The last and the richest construction introduced here is thus accompanied by the ontological-epistemic research attitude.

6

Co-constructive Logics for Proofs and Refutations

44%

Trafford J.

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nr 4

22-40

EN

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov's logic of problems.

7

An Investigation into Intuitionistic Logic with Identity

44%

Chlebowski S. , Leszczyńska-Jasion D.

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nr 4

EN

We define Kripke semantics for propositional intuitionistic logic with Suszko’s identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko’s propositional identity connective.

8

Multi-Agent Dialogues and Dialogue Sequents for Proof Search and Scheduling in Intuitionistic Logic and the Modal Logic S4

38%

Sticht M.

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tom Vol. 161, nr 1/2

191--218

EN

Dialogical games as introduced by Lorenzen and Lorenz describe a reasoning technique for intuitionistic and classical predicate logic: two players (proponent and opponent) argue about the validity of a given formula according to predefined rules. If the proponent has a winning strategy then the formula is proven to be valid. The underlying game rules can be modified to have an impact on proof search strategies and increase the efficiency of such a searching process. In this paper, a multi-agent version of dialogical logic is introduced that corresponds more to multiconclusion sequent calculi for propositional intuitionistic logic rather than single-conclusion ones which are more related to two-player dialogues. We also consider an extension for the normal modal logic S4. The rules lead us to a normalization of a proof, let us focus on the proponents' relevant decisions, and therefore give explicit directives that increase compactness of the proofsearching process. This allows us to perform parts of the proof in a parallel way. We prove soundness and completeness of these multi-agent systems.

9

Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate

38%

Maffezioli P. , Orlandelli E.

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nr 2

137-158

EN

In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maehara's lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and (partly) overcome the failure of interpolation for the implication-free fragment.

10

Teoria kategorii i niektóre jej logiczne aspekty

38%

Stopa M.

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nr 64

7-58

PL

This article is intended for philosophers and logicians as a short partial introduction to category theory (CT) and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an intuitionistic logic. We next present two families of toposes whose tautologies are identical with those of classical propositional logic. The relatively extensive bibliography is given in order to support further studies.

11

CZYM JEST PLURALIZM LOGICZNY? (STANOWISKO J.C. BEALLA I GREGA RESTALLA)

26%

Czernecka-Rej B.

Roczniki Filozoficzne

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2013

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tom 61

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nr 1

5-22

EN

C. Beall and Greg Restall are advocates of a comprehensive pluralist approach to logic, which they call Logical Pluralism (LP). According to LP, there is not one correct logic, but many equally acceptable logical systems. The authors share Tarski’s conviction and follow the mainstream in thinking about logic as the discipline that investigates the notion of logical consequence. LP is the pluralism about logical consequence – a pluralist maintains that there is more than one relation of logical consequence. According to LP, classical, intuitionistic and relevant logics are not rivals, but they all are equally correct, they all count as genuine logics. The purpose of this paper is to present some remarks concerning J.C. Beall’s and Greg Restall’s exposition of LP. At the beginning, the definition of the relation of logical consequence, which is central to their proposal, is shown. According to Beall and Restall, argument is valid if, and only if, in every case when the premisses are true, then the conclusion is, too. They argue that by considering different types of cases the logical pluralist obtains different logics. The paper — apart from presenting LP — also gives a critical discussion of this approach. It seems, that the thesis of LP is far from being clear. It is even unclear what exactly LP is and where is stops. It is unclear what “equally good”, “equally correct”, “equally true” mean. It is not clear, how to explain, in scope of logic, that the system of logic, is a model of real logical connections.