Capturing Transparent Truth in LFIs

PAILOS, FEDERICO MATÍAS; BARRIO, EDUARDO ALEJANDRO; SZMUC, DAMIÁN

Buenos Aires

Workshop; 1st Workshop CLE-Buenos Aires Logic Group; 2014

Buenos Aires Logic Group-S.A.D.A.F.

The logics of formal inconsistency (LFIs) are powerful paraconsistent logics that encode classical logic and allow to fix an interesting distinction between contradictions and inconsistencies. These systems, introduced by Carnielli and Marcos [2002], internalize the metatheoretical notion of consistency, expressing it in the object language. Hence one can isolate contradictions in such a way that the application of the principle of explosion is restricted to consistent sentences only, thus avoiding triviality. This is achieved by means of adding to a collection of appropriate axioms and rules already accepted in classical propositional logic a restricted principle of explosion, that can be read as saying that explosion is valid if it is applied to consistent formulas. If the formula is not consistent, then explosion is not a safe principle. In contrast to LP, there is no commitment to the truth of the sentences that express the contradiction in the LFIs. Contradictions may well be taken as a provisional situation that can be decided later, at least in principle. In the framework of LFIs there is a place for contradictions of an epistemic character. It is really important to point out that the classical reasoning can be restored into LFIs. This is the inferential behavior in consistent fragment of the language of the LFIs is completely classical. In particular, Carnielli [2013] has shown how to adapt the incompleteness results about PA to LFIs assuming the consistency of the Gödel sentence. Thus, the classical Gödelian arguments can be restored in LFIs. In this paper, we are going to explore how to talk about truth into LFIs. We are going to explore two different systems: mbC and LPT showing that adding truth to these theories leads to trivial results. Specially, we will focus on the Strengthened Liar and Currys sentence formulated in a language that contains transparent truth and the connective of consistency. Then, we will show that one is not capable of avoiding trivialization in the presence of these sentences. In this way, one cannot express transparent truth in a language that is capable to talk about consistency and express mbC and LPTs conditional. In the second part of the paper, we will introduce a new conditional that is in principle capable to avoid triviality even in presence of transparent truth and consistency. Modifying the 3-valued matrices of MPT, we will try to get stable valuations for the Strengthened Liar and Currys sentence. The main difference with MPTs conditional is that when the antecedent receive value ½, and the consequent receives value 0, then the conditional receives (the designated value) ½. In this way, we may recover all instances of the diagonal lemma, even the ones with the truth predicate or the consistent operator. A further consequence of this replacement of conditionals is that there will be no longer possible to define with it the consistency operator, which nevertheless be introduced as primitive (and explained with a truth table). Finally, we express some concerns regarding the semantic advantages and disadvantages of the resulting theory. We will discuss to what extent it manages to represent self-reference and, therefore, to be a satisfying theory of truth.