We provide a sound and complete proof system for an extension of Kleene’s ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of “is defined” is extended to terms and formulas via a straightforward recursive algorithm. The “is defined” formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the (...) formula, its negation, and the negation of its “is defined” formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions are ubiquitous in computer science and even in (in)equation solving at schools. Our work was motivated by an attempt to precisely explain, in terms of logic, typical informal methods of reasoning in such applications. (shrink)

Infectious logics are systems that 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 as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and 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. (shrink)

The recent development and exploration of mixed metainferential logics is a breakthrough in our understanding of nontransitive and nonreflexive logics. Moreover, this exploration poses a new challenge to theorists like me, who have appealed to similarities to classical logic in defending the logic ST, since some mixed metainferential logics seem to bear even more similarities to classical logic than ST does. There is a whole ST-based hierarchy, of which ST itself is only the first step, that seems to become more (...) and more classical at each level. I think this seeming is misleading: for certain purposes, anyhow, metainferential hierarchies give us no reason to move on from ST. ST is indeed only the first step on a grand metainferential adventure; but one step is enough. This paper aims to explain and defend that claim. Along the way, I take the opportunity also to develop some formal tools and results for thinking about metainferential logics more generally. (shrink)

The recent development and exploration of mixed metainferential logics is a breakthrough in our understanding of nontransitive and nonreflexive logics. Moreover, this exploration poses a new challenge to theorists like me, who have appealed to similarities to classical logic in defending the logic ST, since some mixed metainferential logics seem to bear even more similarities to classical logic than ST does. There is a whole ST-based hierarchy, of which ST itself is only the first step, that seems to become more (...) and more classical at each level. I think this seeming is misleading: for certain purposes, anyhow, metainferential hierarchies give us no reason to move on from ST. ST is indeed only the first step on a grand metainferential adventure; but one step is enough. This paper aims to explain and defend that claim. Along the way, I take the opportunity also to develop some formal tools and results for thinking about metainferential logics more generally. (shrink)

In this paper we re-assess the philosophical foundation of Exactly True Logic ($$\mathcal {ET\!L}$$ ET L ), a competing variant of First Degree Entailment ($$\mathcal {FDE}$$ FDE ). In order to do this, we first rebut an argument against it. As the argument appears in an interview with Nuel Belnap himself, one of the fathers of $$\mathcal {FDE}$$ FDE, we believe its provenance to be such that it needs to be taken seriously. We submit, however, that the argument ultimately fails, (...) and that $$\mathcal {ET\!L}$$ ET L cannot easily be dismissed. We then proceed to give an overview of the research that was inspired by this logic over the last decade, thus providing further motivation for the study of $$\mathcal {ET\!L}$$ ET L and, more generally, of $$\mathcal {FDE}$$ FDE -related logics that result from semantical analyses alternative to Belnap’s canonical one. We focus, in particular, on philosophical questions that these developments raise. (shrink)

In recent years, some theorists have argued that the clogics are not only defined by their inferences, but also by their metainferences. In this sense, logics that coincide in their inferences, but not in their metainferences were considered to be different. In this vein, some metainferential logics have been developed, as logics with metainferences of any level, built as hierarchies over known logics, such as \, and \. What is distinctive of these metainferential logics is that they are mixed, i.e. (...) the standard for the premises and the conclusion is not necessarily the same. However, so far, all of these systems have been presented following a semantical standpoint, in terms of valuations based on the Strong Kleene truth-tables. In this article, we provide sound and complete sequent-calculi for the valid inferences and the invalid inferences of the logics \ and \, and introduce an algorithm that allows obtaining sound and complete sequent-calculi for the global validities and the global invalidities of any metainferential logic of any level. (shrink)

This essay introduces a novel framework to studying many-valued logics – the movable truth value (or MTV ) approach. After setting up the framework, we will show that a vast number of many-valued logics, and in particular many-valued logics that have previously been given very different kinds of semantics, including C, K3, LP, ST, TS, RM fde, and FDE, can all be unified within the MTV -logic approach. This alone is notable, since until now RM fde in particular has resisted (...) attempts to provide it with the _same kind_ of many-valued semantics as the other logics in this list. New proofs of the duality between LP and K3, and of the self-duality of C, ST, TS, and RM fde, are presented. The essay will conclude with a discussion of directions that further research might take. (shrink)

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but also on what we (...) call Gentzen-regular connectives. For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations. (shrink)

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘non-falsity’ versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case (...) of infectious logics, namely to the case of Deutsch's logic Sfde introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are Setl and Snfl. We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between Setl, Snfl and two well-known systems, strong Kleene three-valued logic K3 and Priest's Logic of Paradox LP. This connection allows us to investigate the characterisation of the entailment relations associated with Setl and Snfl as well as to introduce the notion of ‘infectious analogue’ of a certain logic. We also study implicative extensions of Setl and Snfl and prove soundness and completeness theorems for them as well. (shrink)

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to (...) be identified with an infinite sequence of consequence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting consequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences. (shrink)

Building on recent work, I present sequent systems for the non-classical logics LP, K3, and FDE with two main virtues. First, derivations closely resemble those in standard Gentzen-style systems. Second, the systems can be obtained by reformulating a classical system using nonstandard sequent structure and simply removing certain structural rules (relatives of exchange and contraction). I clarify two senses in which these logics count as “substructural.”.