Logics Of Formal Inconsistency Research Articles

Paraconsistent Arithmetic with a Local Consistency Operator and Global Selfreference

The Provability Logic and Proof-Theory of the system of Paraconsistent Arithmetic PRACI are presented. PRACI is based on the paraconsistent predicate calculus CI corresponding to the C-system Ci introduced by Carnielli et al. [W.A. Carnielli, M.E. Coniglio, J. Marcos, Logics of formal inconsistency, in D. Gabbay, F. Guenthner eds., Handbook of Philosophical Logic, volume 14, Kluwer Academic Publishers, 2005]. PRACI can support an infinity of contradictions B∧¬B without trivializing, but reject identifications between different numbers such as 0=1. In PRACI a new propositional connective °(.) is added, so that °A can be read as “A is consistent”. We obtain a system with a local selfreference, based on the local consistency assertions °A, and a global selfeference, based statements involving PrPRACI(.). The fundamental relationPrT(#°B)→¬PrT(#B) betweeen local and global consistency is investigated. It states that in a paraconsistent setting, the provability of the non-trivialty of Arithmetic could be reduced to that of some suitable local consistency assertions, so that we can speak of a possible weakened Hilbert's program.

Formal inconsistency and evolutionary databases

This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning.

Le logica impecabile del irrational

This paper is written in INTERLINGUA2, a form of modern Latin without declensions whose use in science was initiated by G. Peano, preceded by Descartes and Leibniz. I am following here the IALA conventions for INTERLINGUA of 1952. To the best of my knowledge, this is the first text in logic or philosophy ever written in INTERLINGUA. The paper offers an introduction to some philosophical and logical questions concerned with the problem of the contradictory in logic, traditionally seen as some form of irrationality, as well as a comparison between some distinct positions, their logical approaches and reciprocal criticisms. A brief account of the history of the subject is also sketched. In particular, some recent results about the logics of formal inconsistency (LFIs), the society semantics and its general form, the possible-translations semantics, are emphasized here not only as a new method for combining logics, but also as an impeccable foundation to what is taken to be as the irrational. These syntactical and semantical tools have the double intention of, on the one hand, to systematize and to precisely define an ample class of logic systems, and on the other hand to offer alternative semantic interpretations to certain less studied non-classical logics, while making possible to combine simple logics so as to obtain other logics with a richer structure. We try to assess here the interest, the degree of success and the capability the LFIs and of the possible-translations semantics (as well as its associate, the society semantics) as conceptual contrivances to overcome the irrational.