Strong Kleene Logic Research Articles

Strong Kleene Logics as a Tool for Modelling Formal Epistemic Norms

In this paper, we present two ways of modelling every epistemic formal conditional commitment that involves (at most) three key epistemic attitudes: acceptance, rejection and neither acceptance nor rejection. The first one consists of adopting the plurality of every mixed Strong Kleene logic (along with an epistemic reading of the truth-values), and the second one involves the use of a unified system of six-sided inferences, named 6SK, that recovers the validities of each mixed Strong Kleene logic. We also introduce a sequent calculus that is sound and complete with respect to both approaches. We compare both accounts, and finally, we suggest that the plurality of Strong Kleene logic as well as the general framework 6SK are linked to formal epistemic norms via bridge principles.

Change of logic, without change of meaning

AbstractChange of logic is typically taken as requiring that the meanings of the connectives change too. As a result, it has been argued that legitimate rivalry between logics is under threat. This is, in a nutshell, the meaning‐variance argument, traditionally attributed to Quine. In this paper, we present a semantic framework that allows us to resist the meaning‐variance claim for an important class of systems: classical logic, the logic of paradox and strong Kleene logic. The major feature of the semantics is that the connectives have the same meanings in these systems, so that the meaning‐variance argument is straightforwardly blocked. We discuss the effects of this semantics for two uses of the argument of meaning variance, and also consider its impact on related issues.

Minimally Nonstandard K3 and FDE

Graham Priest has formulated the minimally inconsistent logic of paradox (MiLP), which is paraconsistent like Priest’s logic of paradox (LP), while staying closer to classical logic. We present logics that stand to (the propositional fragments of) strong Kleene logic (K3) and the logic of first-degree entailment (FDE) as MiLP stands to LP. That is, our logics share the paracomplete and the paraconsistent-cum-paracomplete nature of K3 and FDE, respectively, while keeping these features to a minimum in order to stay closer to classical logic. We give semantic and sequent-calculus formulations of these logics, and we highlight some reasons why these logics may be interesting in their own right.

Normalisation for Some Quite Interesting Many-Valued Logics

In this paper, we consider a set of quite interesting three- and four-valued logics and prove the normalisation theorem for their natural deduction formulations. Among the logics in question are the Logic of Paradox, First Degree Entailment, Strong Kleene logic, and some of their implicative extensions, including RM3 and RM3⊃. Also, we present a detailed version of Prawitz’s proof of Nelson’s logic N4 and its extension by intuitionist negation.

Generalized Correspondence Analysis for Three-Valued Logics

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP (which is built in the $$ \{\vee ,\wedge ,\lnot \} $$ -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete analogue of LP, strong Kleene logic $$ \mathbf K_3 $$ . In this paper, we generalize these results for the negative fragments of LP and $$ \mathbf K_3 $$ , respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or $$ \mathbf K_3 $$ , but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of $$ \mathbf K_3 $$ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic $$ \mathbf G_3 $$ and its dual $$ \mathbf DG_3 $$ (which have different interpretations of negation than $$ \mathbf K_3 $$ and LP).

Automated Proof-searching for Strong Kleene Logic and its Binary Extensions via Correspondence Analysis

Using the method of correspondence analysis, Tamminga obtains sound and complete natural deduction systems for all the unary and binary truth-functional extensions of Kleene’s strong three-valued logic K3 . In this paper, we extend Tamminga’s result by presenting an original finite, sound and complete proof-searching technique for all the truth-functional binary extensions of K3.

Interpolation Methods for Dunn Logics and Their Extensions

The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the mathbf{Dunn}calculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In terms of the mathbf{Dunn} calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics.

Skolem Functions in Non-Classical Logics

This paper shows how to conservatively extend theories formulated in non-classical logics such as the Logic of Paradox, the Strong Kleene Logic and relevant logics with Skolem functions. Translations to and from the language extended by Skolem functions into the original one are presented and shown to preserve derivability. It is also shown that one may not always substitute s=f(t) and A(t, s) even though A(x,y) determines the extension of a function and f is a Skolem function for A.

Off-Topic: A New Interpretation of Weak-Kleene Logic

This paper offers a new and very simple alternative to Bochvar's well known nonsense -- or meaninglessness -- interpretation of Weak Kleene logic. To help orient discussion I begin by reviewing the familiar Strong Kleene logic and its standard interpretation; I then review Weak Kleene logic and the standard (viz., Bochvar) interpretation. While I note a common worry about the Bochvar interpretation my aim is only to give an alternative -- and I think very elegant -- interpretation, not necessarily a replacement.

On All Strong Kleene Generalizations of Classical Logic

By using the notions of exact truth (`true and not false') and exact falsity (`false and not true'), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the $${{\bf SK}^\mathcal{P}}$$SKP and $${\bf SK}^{\mathcal{N}}$$SKN calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the $${\bf SK}^{\mathcal{P}}$$SKP and the $${\bf SK}^{\mathcal{N}}$$SKN calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the $${\bf SK}^{\mathcal{P}}$$SKP and the $${\bf SK}^{\mathcal{N}}$$SKN calculus, we also hint at its philosophical significance.

Borderline vs. unknown: comparing three-valued representations of imperfect information

In this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and González-Rodríguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness and formalisms that handle incomplete information often lead to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility–necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions.

SUPER LIARS

Kripke’s theory of truth offered a trivalent semantics for a language which, like English, contains a truth predicate and means of self-reference; but it did so by severely restricting the expressive power of the logic. In Kripke’s analysis, the Liar (e.g.,This very sentence is not true) receives the indeterminate truth value, but this fact cannot be expressed in the language; by contrast, it is straightforward to say in English thatthe Liar is something other than true. Kripke’s theory also fails to handle the Strengthened Liar, which can be expressed in English as:This very sentence is something other than true. We develop a theory which seeks to overcome these difficulties, and is based on a detailed analysis of some of the linguistic means by which the Strengthened Liar can be expressed in English. In particular, we propose to takeliterallythe quantificational form of the negative expressionsomething other than true.Like other quantifiers, it may have different implicit domain restrictions, which give rise to a variety of negations of different strengths (e.g.,something other than true among the values{0, 1}, oramong{0, 1, 2}, etc). This analysis naturally leads to a logic with as many truth values as there are ordinals—a conclusion reached independently by Cook (2008a). We develop the theory within a generalization of the Strong Kleene Logic, augmented with negations that each have a nonmonotonic semantics. We show that fixed points can be constructed for our logic, and that it enjoys a limited form of ‘expressive completeness’. Finally, we discuss the relation between our theory and various alternatives, including one in which the wordtrue(rather than negation) is semantically ambiguous, and gives rise to a hierarchy of truth predicates of increasing strength.

Axiomatizing Kripke's theory of truth

AbstractWe investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF. namely the strength of the systemramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the systemof ramified analysis up to ε0.