Exactly true and non-falsity logics meeting infectious ones

Logic Journal of the IGPL, 2018

Infectious logics are systems which have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated (i) as a way to treat different pathological sentences (like the Liar and the Truth-Teller) differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps, and (ii) as a way to treat the semantic pathology suffered by at least some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof-theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems.

1995

We note rst that the rst degree entailment of Lukasiewicz's 3valued logic and a 3-valued logic that is extracted of Belnap's 4-valued logic is the same. Then, we give an axiomatization of that entailment as the calculus E fde +A^:A!B _:B, where E fde is the rst degree entailment of Anderson-Belnap's logic E of relevance and necessity.

This short and informal article briefly introduces a four-valued logic.

Journal of Applied Non-Classical Logics, 2014

Three-valued logics belong to a family of nonclassical logics that started to flourish in the 1920s and 1930s, following the work of (Lukasiewicz, 1920), and earlier insights coming from Frege and Peirce (see (Frege, 1879), (Frege, 1892), (Fisch and Turquette, 1966)). All of them were moved by the idea that not all sentences need be True or False, but that some sentences can be indeterminate in truth value. In his pioneering paper, Lukasiewicz writes: "Three-valued logic is a system of non-Aristotelian logic, since it assumes that in addition to true and false propositions there also are propositions that are neither true nor false, and hence, that there exists a third logical value."

South American Journal of Logic, 2017

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions, we argue that a proper account of conditional can be obtained by extending the logical notation of Frege's ideography. From a dialogical redefinition of truth-values as moves in a game, it becomes possible to characterize the logical meaning of " If " , and only " If ". That is: by getting rid of the paradoxes of material implication, whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections. Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed in this algebraic and game-theoretical light.

Bulletin of the Section of Logic, 2007

It has been emphasized by Hiroakira Ono, Petr Hájek, and other logi-cians that there exists a close relationship between substructural and many-valued logics, see, for example, [11], [13], [18]. This relationship has many aspects, and in the present paper, we take the ...

Notre Dame Journal of Formal Logic, 2002

In a finitary closure space, irreducible sets behave like two-valued models, with membership playing the role of satisfaction. If f is a function on such a space and the membership of $fx_1 ,\ldots, x_n$ in an irreducible set is determined by the presence or absence of the inputs $x_1 ,\ldots, x_n$ in that set, then f is a kind of truth function. The existence of some of these truth functions is enough to guarantee that every irreducible set is maximally consistent. The closure space is then said to be expressive. This paper identifies the two-valued truth functional conditions that guarantee expressiveness.

Journal of Logic, Language and Information, 2006

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of "revenge Liar" arguments, also higherorder combinations of generalized truth values have been suggested to account for so-called hypercontradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's "useful four-valued logic", one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a "bi-and-a-half" lattice which determines seven-valued logics different from Priest's Logic of Paradox.

In: Peter Schroeder-Heister on Proof-Theoretic Semantics, Springer, 2021

In some presentations of classical and intuitionistic logics, the object-language is assumed to contain (two) truth-value constants: verum and falsum, that are, respectively, true and false under every bivalent valuation. We are interested to de ne and study analogical constants that in an arbitrary multi-valued logic over truth-values have the truth-value v_i under every (multi-valued) valuation. We Definition ne such constants proof-theoretically via their associated I/E-rules in a natural-deduction proof system. As is well known, the absence or presence of such constants has a significant deductive impact on the logics studied. In particular, we propose a generalization of the notions of contradiction and explosiveness of a logic to the context of multi-valued logics.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1987