Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valuedlogic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their (...) originals. This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valuedLogic into S5. (shrink)
I apply Kooi and Tamminga's (2012) idea of correspondence analysis for many-valued logics to strong three-valuedlogic (K3). First, I characterize each possible single entry in the truth-table of a unary or a binary truth-functional operator that could be added to K3 by a basic inference scheme. Second, I define a class of natural deduction systems on the basis of these characterizing basic inference schemes and a natural deduction system for K3. Third, I show that each (...) of the resulting natural deduction systems is sound and complete with respect to its particular semantics. Among other things, I thus obtain a new proof system for Lukasiewicz's three-valuedlogic. (shrink)
Kleene’s strong three-valuedlogic extends naturally to a four-valuedlogic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it four-valued analogs of Kleene’s weak three-valuedlogic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for bilinear (...) bilattices the extensions do not produce any new equivalences. (shrink)
A three-valued propositional logic is presented, within which the three values are read as ?true?, ?false? and ?nonsense?. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.
Our aim is to model the behaviour of a cognitive agent trying to solve a complex problem by dividing it into sub-problems, but failing to solve some of these sub-problems. We use the powerful framework of erotetic search scenarios combined with Kleene’s strong three-valuedlogic. ESS, defined on the grounds of Inferential Erotetic Logic, has appeared to be a useful logical tool for modelling cognitive goal-directed processes. Using the logical tools of ESS and the three- (...) class='Hi'>valuedlogic, we will show how an agent could solve the initial problem despite the fact that the sub-problems remain unsolved. Thus our model not only indicates missing information but also specifies the contexts in which the problem-solving process may end in success despite the lack of information. We will also show that this model of problem solving may find use in an analysis of natural language dialogues. (shrink)
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 -language) by the addition of unary and binary connectives. Tamminga has also applied this (...) technique to the paracomplete analogue of LP, strong Kleene logic \. In this paper, we generalize these results for the negative fragments of LP and \, respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \, 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 \ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic \ and its dual \ and LP). (shrink)
The paper consists of two parts. The first part begins with the problem of whether the original three-valued calculus, invented by J. Łukasiewicz, really conforms to his philosophical and semantic intuitions. I claim that one of the basic semantic assumptions underlying Łukasiewicz's three-valuedlogic should be that if under any possible circumstances a sentence of the form "X will be the case at time t" is true (resp. false) at time t, then this sentence must (...) be already true (resp. false) at present. However, it is easy to see that this principle is violated in Lukasiewicz's original calculus (as the cases of the law of excluded middle and the law of contradiction show). Nevertheless it is possible to construct (either with the help of the notion of "supervaluation", or purely algebraically) a different three-valued, semi-classical sentential calculus, which would properly incorporate Łukasiewicz's initial intuitions. Algebraically, this calculus has the ordinary Boolean structure, and therefore it retains all classically valid formulas. Yet because possible valuations are no longer represented by ultrafilters, but by filters (not necessarily maximal), the new calculus displays certain non-classical metalogical features (like, for example, nonextensionality and the lack of the metalogical rule enabling one to derive "p is true or q is true" from" 'p ∨ q' is true"). The second part analyses whether the proposed calculus could be useful in formalizing inferences in situations, when for some reason (epistemological or ontological) our knowledge of certain facts is subject to limitation. Special attention should be paid to the possibility of employing this calculus to the case of quantum mechanics. I am going to compare it with standard non-Boolean quantum logic (in the Jauch-Piron approach), and to show that certain shortcomings of the latter can be avoided in the former. For example, I will argue that in order to properly account for quantum features of microphysics, we do not need to drop the law of distributivity. Also the idea of "reading off" the logical structure of propositions from the structure of Hilbert space leads to some conceptual troubles, which I am going to point out. The thesis of the paper is that all we need to speak about quantum reality can be acquired by dropping the principle of bivalence and extensionality, while accepting all classically valid formulas. (shrink)
Four known three-valued logics are formulated axiomatically and several completeness theorems with respect to nonstandard intuitive semantics, connected with the notions of information, contrariety and subcontrariety is given.
Nondeterministic programs occurring in recently developed programming languages define nondeterminate partial functions. Formulas (Boolean expressions) of such nondeterministic languages are interpreted by a nonempty subset of {T (true), F (false), U (undefined)}. As a semantic basis for the propositional part of a corresponding nondeterministic three-valuedlogic we study the notion of a truth-function over {T, F, U} which is computable by a nondeterministic evaluation procedure. The main result is that these truth-functions are precisely the functions satisfying four (...) basic properties, called -isotonic, –-isotonic, hereditarily guarded, and hereditarily guard-using, and that a function satisfies these properties iff it is explicitly definable (in a certain normal form) from if..then..else..fi, binary choice, and constants. (shrink)
Łukasiewicz has often been criticized for his motive for inventing his three-valuedlogic, namely the avoidance of determinism. First of all, I want to show that almost all of the critcism along this line was wrong. Second I will indicate that he made mistakes, however, in constructing his system, because he had other motives at the same time. Finally I will propose some modification of his system and its interpretation which can attain his original purpose in some (...) sense. (shrink)
We introduce a notion of semantical closure for theories by formalizing Nepeivoda notion of truth. [10]. Tarski theorem on truth definitions is discussed in the light of Kleene's threevaluedlogic (here treated with a formal reinterpretation of logical constants). Connections with Definability Theory are also established.
In this paper, I consider a family of three-valued regular logics: the well-known strong and weak S.C. Kleene’s logics and two intermedi- ate logics, where one was discovered by M. Fitting and the other one by E. Komendantskaya. All these systems were originally presented in the semantical way and based on the theory of recursion. However, the proof theory of them still is not fully developed. Thus, natural deduction sys- tems are built only for strong Kleene’s logic (...) both with one (A. Urquhart, G. Priest, A. Tamminga) and two designated values (G. Priest, B. Kooi, A. Tamminga). The purpose of this paper is to provide natural deduction systems for weak and intermediate regular logics both with one and two designated values. (shrink)
The article demonstrates that to describe the property of atomicity of transactions in database systems, we need a three-valuedlogic with propositional connective characterized in the same way as Blamey’s interjunction. However, the article explains that since Blamey’s partial logic with interjunction is a logic without tautologies, it does not satisfy some salient conditions of being a logic of atomic transactions. The article introduces a logic of the considered kind, and provides an example (...) of the formal exposition of the case of an atomic transaction. Finally, the article explains the philosophical significance of the introduced logic. (shrink)
This article investigates Kripke-style semantics for two sorts of logics: pseudo-Boolean and weak-Boolean logics. As examples of the first, we introduce G3 and S53pB.G3 is the three-valued Dummett–Gödel logic; S53pB is the modal logic S5 but with its orthonegation replaced by a pB negation. Examples of wB logic are G3wB and S53wB.G3wB is G3 with a wB negation in place of its pB negation; S53wB is S5 with a wB negation replacing its orthonegation. For each (...) system, we provide a three-valued Kripke-style semantics with and without star operation . We prove soundness and completeness theorems in each case. Note that wB logics may be equivalent to logics with Baaz’s projection Δ. We finally introduce the G3 and the S53pB both with Δ and show that they are equivalent to G3wB and S53wB, respectively. (shrink)
Prior's three-valued modal logic Q was developed as a philosophically interesting modal logic. Thus, we should be able to modify Q as a temporal logic. Although a temporal version of Q was suggested by Prior, the subject has not been fully explored in the literature. In this paper, we develop a three-valued temporal logic $Q_t $ and give its axiomatization and semantics. We also argue that $Q_t $ provides a smooth solution to (...) the problem of future contingents. (shrink)
The article concerns two axiom systems of Słupecki for the functionally complete three-valued propositional logic: W1–W6 and A1–A9. The article proves that both of them are inadequate—W1–W6 is semantically incomplete, on the other hand, A1–A9 governs a functionally incomplete calculus, and thus, it cannot be a semantically complete axiom system for the functionally complete three-valuedlogic.
Recently, Bourne constructed a system of three-valuedlogic that he supposed to replace Łukasiewicz’s three-valuedlogic in view of the problems of future contingents. In this paper, I will show first that Bourne’s system makes no improvement to Łukasiewicz’s system. However, finding some good motivations and lessons in his attempt, next I will suggest a better way of achieving his original goal in some sense. The crucial part of my way lies in reconsidering the (...) significance of the intermediate truth-value so as to reconstruct Łukasiewicz’s three-valuedlogic as a kind of extensional modal logic based on partial logic. (shrink)
ABSTRACT In this paper we introduce a set of six logical values, arising in the application of three-valued logics to time intervals, find its algebraic structure, and use it to define a six-valuedlogic. We then prove, by using algebraic properties of the class of De Morgan algebras, that this semantically defined logic can be axiomatized as Belnap's ?useful? four-valuedlogic. Other directions of research suggested by the construction of this set of six (...) logical values are described. (shrink)
We show how voting rules like the simple and the absolute majority rules, unanimity, consensus, etc. can be represented as logical operators in Łukasiewicz’s three-valuedlogic. First, we prove tha...