Rational Reasons, Counterfactual Statements and “Impossible Worlds” in the Philosophical Justifications of Thought Experiments

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.

SOME MODEL-THEORETIC RESULTS ON THE 3-VALUED PARACONSISTENT FIRST-ORDER LOGIC QCIORE

The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs.

Revisiting Explicit Negation in Answer Set Programming

A 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.

An Algorithm for Producing Fuzzy Negations via Conical Sections

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.

Paraconsistency, self-extensionality, modality

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.

Logical rules and the determinacy of meaning

The use of conventional logical connectives either in logic, in mathematics, or in both cannot determine the meanings of those connectives. This is because every model of full conventional set theory can be extended conservatively to a model of intuitionistic set plus class theory, a model in which the meanings of the connectives are decidedly intuitionistic and nonconventional. The reasoning for this conclusion is acceptable to both intuitionistic and classical mathematicians. En route, I take a detour to prove that, given strictly intuitionistic principles, classical negation cannot exist.