Classical Negation Strikes Back: Why Priest’s Attack on Classical Negation Can’t Succeed

In his theory of natural laws David Lewis rejects the authenticity of impossible worlds on the grounds that the contradiction contained within his modifier "in (the world) w" is tantamount to a contradiction in the whole theory, which seems unacceptable. At the same time, in philosophical discourse very often researchers use counterfactual situations and thought experiments with impossible events and objects. There is a need to apply the theory of worlds to genuine, concrete, but impossible worlds. One way to do this is to reject Lewis's classical negation on the grounds that it leads to problems of completeness and inconsistency inside the worlds. The proposed extension for impossibility is compatible with Lewis's extensional metaphysics, although it leads to some loss for description completeness in semantics.

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IF Modal Logic and Classical Negation

Studia Logica ◽

10.1007/s11225-012-9462-3 ◽

2013 ◽

Vol 102 (1) ◽

pp. 41-66

Author(s):

Tero Tulenheimo

Keyword(s):

Modal Logic ◽

Classical Negation

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Paraconsistency, self-extensionality, modality

Logic Journal of IGPL ◽

10.1093/jigpal/jzy064 ◽

2018 ◽

Vol 28 (5) ◽

pp. 851-880

Author(s):

Arnon Avron ◽

Anna Zamansky

Keyword(s):

Intuitionistic Logic ◽

Modal Logics ◽

The Other ◽

Paraconsistent Logics ◽

Classical Negation ◽

Inconsistent Theories ◽

Replacement Property ◽

General Method

Abstract Paraconsistent logics are logics that, in contrast to classical and intuitionistic logic, do not trivialize inconsistent theories. In this paper we take a paraconsistent view on two famous modal logics: B and S5. We use for this a well-known general method for turning modal logics to paraconsistent logics by defining a new (paraconsistent) negation as $\neg \varphi =_{Def} \sim \Box \varphi$ (where $\sim$ is the classical negation). We show that while that makes both B and S5 members of the well-studied family of paraconsistent C-systems, they differ from most other C-systems in having the important replacement property (which means that equivalence of formulas implies their congruence). We further show that B is a very robust C-system in the sense that almost any axiom which has been considered in the context of C-systems is either already a theorem of B or its addition to B leads to a logic that is no longer paraconsistent. There is exactly one notable exception, and the result of adding this exception to B leads to the other logic studied here, S5.

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An Algorithm for Producing Fuzzy Negations via Conical Sections

Algorithms ◽

10.3390/a12050089 ◽

2019 ◽

Vol 12 (5) ◽

pp. 89 ◽

Cited By ~ 3

Author(s):

Souliotis ◽

Papadopoulos

Keyword(s):

Artificial Intelligence ◽

Neural Networks ◽

Structural Element ◽

Specific Field ◽

New Class ◽

Fuzzy Implications ◽

Classical Negation ◽

Geometrical Concepts ◽

Computational Processes

In this paper we introduced a new class of strong negations, which were generated via conical sections. This paper focuses on the fact that simple mathematical and computational processes generate new strong fuzzy negations, through purely geometrical concepts such as the ellipse and the hyperbola. Well-known negations like the classical negation, Sugeno negation, etc., were produced via the suggested conical sections. The strong negations were a structural element in the production of fuzzy implications. Thus, we have a machine for producing fuzzy implications, which can be useful in many areas, as in artificial intelligence, neural networks, etc. Strong Fuzzy Negations refers to the discrepancy between the degree of difficulty of the effort and the significance of its results. Innovative results may, therefore, derive for use in literature in the specific field of mathematics. These data are, moreover, generated in an effortless, concise, as well as self-evident manner.

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Revisiting Explicit Negation in Answer Set Programming

Theory and Practice of Logic Programming ◽

10.1017/s1471068419000267 ◽

2019 ◽

Vol 19 (5-6) ◽

pp. 908-924

Author(s):

FELICIDAD AGUADO ◽

PEDRO CABALAR ◽

JORGE FANDINNO ◽

DAVID PEARCE ◽

GILBERTO PÉREZ ◽

...

Keyword(s):

Answer Set Programming ◽

Regular Operator ◽

Strong Negation ◽

Classical Negation ◽

Explicit Negation ◽

Answer Set

AbstractA common feature in Answer Set Programming is the use of a second negation, stronger than default negation and sometimes called explicit, strong or classical negation. This explicit negation is normally used in front of atoms, rather than allowing its use as a regular operator. In this paper we consider the arbitrary combination of explicit negation with nested expressions, as those defined by Lifschitz, Tang and Turner. We extend the concept of reduct for this new syntax and then prove that it can be captured by an extension of Equilibrium Logic with this second negation. We study some properties of this variant and compare to the already known combination of Equilibrium Logic with Nelson’s strong negation.

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