57 research outputs found
From Analogical Proportion to Logical Proportions
International audienceGiven a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators ( aâ§b and aÂŻâ§bÂŻ), or dissimilarity indicators ( aâ§bÂŻ and aÂŻâ§b) pertaining to the pair (a, b), to the ones associated with the pair (c, d). There are 120 semantically distinct logical proportions. One of them models the analogical proportion which corresponds to a statement of the form âa is to b as c is to dâ. The paper inventories the whole set of logical proportions by dividing it into five subfamilies according to what they express, and then identifies the proportions that satisfy noticeable properties such as full identity (the pair of equivalences defining the proportion hold as true for the 4-tuple (a, a, a, a)), symmetry (if the proportion holds for (a, b, c, d), it also holds for (c, d, a, b)), or code independency (if the proportion holds for (a, b, c, d), it also holds for their negations (aÂŻ,bÂŻ,cÂŻ,dÂŻ)). It appears that only four proportions (including analogical proportion) are homogeneous in the sense that they use only one type of indicator (either similarity or dissimilarity) in their definition. Due to their specific patterns, they have a particular cognitive appeal, and as such are studied in greater details. Finally, the paper provides a discussion of the other existing works on analogical proportions
Modality, Potentiality and Contradiction in Quantum Mechanics
In [11], Newton da Costa together with the author of this paper argued in
favor of the possibility to consider quantum superpositions in terms of a
paraconsistent approach. We claimed that, even though most interpretations of
quantum mechanics (QM) attempt to escape contradictions, there are many hints
that indicate it could be worth while to engage in a research of this kind.
Recently, Arenhart and Krause [1, 2, 3] have raised several arguments against
this approach and claimed that, taking into account the square of opposition,
quantum superpositions are better understood in terms of contrariety
propositions rather than contradictory propositions. In [17] we defended the
Paraconsistent Approach to Quantum Superpositions (PAQS) and provided arguments
in favor of its development. In the present paper we attempt to analyze the
meanings of modality, potentiality and contradiction in QM, and provide further
arguments of why the PAQS is better suited, than the Contrariety Approach to
Quantum Superpositions (CAQS) proposed by Arenhart and Krause, to face the
interpretational questions that quantum technology is forcing us to consider.Comment: Published in: New Directions in Paraconsistent Logic, J-Y B\'eziau M.
Chakraborty & S. Dutta (Eds.), Springer, in press. arXiv admin note: text
overlap with arXiv:1404.518
A modal theorem-preserving translation of a class of three-valued logics of incomplete information
International audienceThere are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, áŽukasiewicz and Nelson logics. We show that Priestâs logic of paradox, closely connected to Kleeneâs, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management
Fuzzy Extensions of Conceptual Structures of Comparison
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