Is the principle of contradiction a consequence of x^2 = x?

Is the principle of contradiction a consequence of x^2 = x

Logic may be simple. Logic, congruence and algebra

This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The notion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacement theorem and algebraization of logic are studied (including MacLane-Curry’s theorem and a discussion about Curry’s algebras). We also examine how these concepts are related to such notions as semantics, truth-functionality and bivalence. We argue that a logic, which is simple, can deserve the name logic and that the opposite view is connected with a reductionist perspective (reduction of logic to algebra)

The paraconsistent logic Z. A possible solution to Jaśkowski’s problem

We present a paraconsistent logic, called Z, based on an intuitive possible worlds semantics, in which the replacement theorem holds. We show how to axiomatize this logic and prove the completeness theorem

Encoding hybridised institutions into first order logic

"First published online: 12 November 2014"A ‘hybridization’ of a logic, referred to as the base logic, consists of developing the characteristic features of hybrid logic on top of the respective base logic, both at the level of syntax (i.e. modalities, nominals, etc.) and of the semantics (i.e. possible worlds). By ‘hybridized institutions’ we mean the result of this process when logics are treated abstractly as institutions (in the sense of the institution theory of Goguen and Burstall). This work develops encodings of hybridized institutions into (many-sorted) first order logic (abbreviated FOL) as a ‘hybridization’ process of abstract encodings of institutions into FOL, which may be seen as an abstraction of the well known standard translation of modal logic into first order logic. The concept of encoding employed by our work is that of comorphism from institution theory, which is a rather comprehensive concept of encoding as it features encodings both of the syntax and of the semantics of logics/institutions. Moreover we consider the so-called theoroidal version of comorphisms that encode signatures to theories, a feature that accommodates a wide range of concrete applications. Our theory is also general enough to accomodate various constraints on the possible worlds semantics as well a wide variety of quantifications. We also provide pragmatic sufficient conditions for the conservativity of the encodings to be preserved through the hybridization process, which provides the possibility to shift a formal verification process from the hybridized institution to FOL.We thank both Till Mossakowski and Andrzej Tarlecki for the technical suggestion of using the predicates D. The work of the first author has been supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0439. The work of the second author was funded by the European Regional Development Fund through the COMPETE Programme, and by the Portuguese Foundation for Science and Technology through the projects FCOMP-01-0124-FEDER-028923 and NORTE-01-0124-FEDER-000060

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 ( a∧b and a¯∧b¯), or dissimilarity indicators ( a∧b¯ and a¯∧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 “a is to b as c is to d”. 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 (a¯,b¯,c¯,d¯)). 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