14 research outputs found
Interpolation in 16-Valued Trilattice Logics
In a recent paper we have defined an analytic tableau calculus (Formula presented.) for a functionally complete extension of Shramko and Wansingâs logic based on the trilattice (Formula presented.). This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logicâsuch as the relations [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], and [InlineEquation not available: see fulltext.] that each correspond to a lattice order in (Formula presented.); and [InlineEquation not available: see fulltext.], the intersection of [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. It turns out that our method of characterising these semantic relationsâas intersections of auxiliary relations that can be captured with the help of a single calculusâlends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that [InlineEquation not available: see fulltext.], when restricted to (Formula presented.), the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano
The Proscriptive Principle and Logics of Analytic Implication
The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the analysis of concepts, a term that presupposesâthrough the root áźÎ˝ÎŹ + ÎťĎĎ âa mereological background.
In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parryâs original system AI was later expanded to the system PAI. The hallmark of Parryâs systemsâand of what may be thought of as containment logics or Parry systems in generalâis a strong relevance property called the âProscriptive Principleâ (PP) described by Parry as the thesis that: No formula with analytic implication as main relation holds universally if it has a free variable occurring in the consequent but not the antecedent.
This type of proscription is on its face justified, as the presence of a novel parameter in the consequent corresponds to the introduction of new subject matter. The plausibility of the thesis that the content of a statement is related to its subject matter thus appears also to support the validity of the formal principle.
Primarily due to the perception that Parryâs formal systems were intended to accurately model Kantâs notion of an analytic judgment, Parryâs deductive systemsâand the suitability of the Proscriptive Principle in generalâwere met with severe criticism. While Anderson and Belnap argued that Parryâs criterion failed to account for a number of prima facie analytic judgments, othersâsuch as Sylvan and Bradyâargued that the utility of the criterion was impeded by its reliance on a âsyntacticalâ device.
But these arguments are restricted to Parryâs work qua exegesis of Kant and fail to take into account the breadth of applications in which the Proscriptive Principle emerges. It is the goal of the present work to explore themes related to deductive systems satisfying one form of the Proscriptive Principle or other, with a special emphasis placed on the rehabilitation of their study to some degree. The structure of the dissertation is as follows: In Chapter 2, we identify and develop the relationship between Parry-type deductive systems and the field of âlogics of nonsense.â Of particular importance is Dmitri Bochvarâs âinternalâ nonsense logic ÎŁ0, and we observe that two â˘-Parry subsystems of ÎŁ0 (Harry Deutschâs Sfde and Frederick Johnsonâs RC) can be considered to be the products of particular âstrategiesâ of eliminating problematic inferences from Bochvarâs system. The material of Chapter 3 considers Kit Fineâs program of state space semantics in the context of Parry logics. Recently, Fineâwho had already provided the first intuitive semantics for Parryâs PAIâhas offered a formal model of truthmaking (and falsemaking) that provides one of the first natural semantics for Richard B. Angellâs logic of analytic containment AC, itself a â˘-Parry system. After discussing the relationship between state space semantics and nonsense, we observe that Fabrice Correiaâs weaker frameworkâintroduced as a semantics for a containment logic weaker than ACâtacitly endorses an implausible feature of allowing hypernonsensical statements. By modelling Correiaâs containment logic within the stronger setting of Fineâs semantics, we are able to retain Correiaâs intuitions about factual equivalence without such a commitment. As a further application, we observe that Fineâs setting can resolve some ambiguities in Greg Restallâs own truthmaker semantics. In Chapter 4, we consider interpretations of disjunction that accord with the characteristic failure of Addition in which the evaluation of a disjunction A ⨠B requires not only the truth of one disjunct, but also that both disjuncts satisfy some further property. In the setting of computation, such an analysis requires the existence of some procedure tasked with ensuring the satisfaction of this property by both disjuncts. This observation leads to a computational analysis of the relationship between Parry logics and logics of nonsense in which the semantic category of ânonsenseâ is associated with catastrophic faults in computer programs. In this spirit, we examine semantics for several â˘-Parry logics in terms of the successful execution of certain types of programs and the consequences of extending this analysis to dynamic logic and constructive logic. Chapter 5 considers these faults in the particular case in which Nuel Belnapâs âartificial reasonerâ is unable to retrieve the value assigned to a variable. This leads not only to a natural interpretation of Graham Priestâs semantics for the â˘-Parry system Sâfde but also a novel, many-valued semantics for Angellâs AC, completeness of which is proven by establishing a correspondence with Correiaâs semantics for AC. These many-valued semantics have the additional benefit of allowing us to apply the material in Chapter 2 to the case of AC to define intensional extensions of AC in the spirit of Parryâs PAI. One particular instance of the type of disjunction central to Chapter 4 is Melvin Fittingâs cut-down disjunction. Chapter 6 examines cut-down operations in more detail and provides bilattice and trilattice semantics for the â˘-Parry systems Sfde and AC in the style of Ofer Arieli and Arnon Avronâs logical bilattices. The elegant connection between these systems and logical multilattices supports the fundamentality and naturalness of these logics and, additionally, allows us to extend epistemic interpretation of bilattices in the tradition of artificial intelligence to these systems. Finally, the correspondence between the present many-valued semantics for AC and those of Correia is revisited in Chapter 7. The technique that plays an essential role in Chapter 4 is used to characterize a wide class of first-degree calculi intermediate between AC and classical logic in Correiaâs setting. This correspondence allows the correction of an incorrect characterization of classical logic given by Correia and leads to the question of how to characterize hybrid systems extending Angellâs ACâ. Finally, we consider whether this correspondence aids in providing an interpretation to Correiaâs first semantics for AC
SEQUENTIAL CALCULI FOR MANY-VALUED LOGICS WITH EQUALITY DETERMINANT
Abstract We propose a general method of constructing sequential calculi with cut elimination property for propositional finitely-valued logics with equality determinant. We then prove the non-algebraizability of the consequence operations of cut-free versions of such sequential calculi. Key words and phrases: many-valued logic, equality determinant, sequential calculus, cut elimination, algebraizable sequential consequence operation. One of the main issues concerning many-valued logics is to find their appropriate useful axiomatizations. Since the development of the formalism of many-place sequents in [13] which enabled one to axiomatize arbitrary finitely-valued logics, the main emphasis within the topic has been laid on developing generic approaches dealing with variations of the approac
Sweet SIXTEEN: Automation via Embedding into Classical Higher-Order Logic
An embedding of many-valued logics based on SIXTEEN in classical higher-order logic is presented. SIXTEEN generalizes the four-valued set of truth degrees of Dunn/Belnapâs system to a lattice of sixteen truth degrees with multiple distinct ordering relations between them. The theoretical motivation is to demonstrate that many-valued logics, like other non-classical logics, can be elegantly modeled (and even combined) as fragments of classical higher-order logic. Equally relevant are the pragmatic aspects of the presented approach: interactive and automated reasoning in many-valued logics, which have broad applications in computer science, artificial intelligence, linguistics, philosophy and mathematics, become readily enabled with state of the art reasoning tools for classical higher-order logic
Relevant generalization starts here (and here = 2)
There is a productive and suggestive approach in philosophical logic based on the idea of generalized truth values. This idea, which stems essentially from the pioneering works by J.M. Dunn, N. Belnap, and which has recently been developed further by Y. Shramko and H. Wansing, is closely connected to the power-setting formation on the base of some initial truth values. Having a set of generalized truth values, one can introduce fundamental logical notions, more specifically, the ones of logical operations and logical entailment. This can be done in two different ways. According to the first one, advanced by M. Dunn, N. Belnap, Y. Shramko and H. Wansing, one defines on the given set of generalized truth values a specific ordering relation (or even several such relations) called the logical order(s), and then interprets logical connectives as well as the entailment relation(s) via this ordering(s). In particular, the negation connective is determined then by the inversion of the logical order. But there is also another method grounded on the notion of a quasi-field of sets, considered by BiaĹynicki-Birula and Rasiowa. The key point of this approach consists in defining an operation of quasi-complement via the very specific function g and then interpreting entailment just through the relation of set-inclusion between generalized truth values.In this paper, we will give a constructive proof of the claim that, for any finite set V with cardinality greater or equal 2, there exists a representation of a quasi-field of sets <P(V ), âŞ, âŠ, â> isomorphic to de Morgan lattice. In particular, it means that we offer a special procedure, which allows to make our negation de Morgan and our logic relevant
On a multilattice analogue of a hypersequent S5 calculus
In this paper, we present a logic MMLS5n which is a combination of multilattice logic and modal logic S5. MMLS5n is an extension of Kamide and Shramkoâs modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MMLS5n in the spirit of Restallâs one for S5 and develop a Kripke semantics for MMLS5n, following Kamide and Shramkoâs approach. Moreover, we prove theorems for embedding MMLS5n into S5 and vice versa. As a result, we obtain completeness, cut elimination, decidability, and interpolation theorems for MMLS5n. Besides, we show the duality principle for MMLS5n. Additionally, we introduce a modification of Kamide and Shramkoâs sequent calculus for their multilattice version of S4 which (in contrast to Kamide and Shramkoâs original one) proves the interdefinability of necessity and possibility operators. Last, but not least, we present Hilbert-style calculi for all the logics in question as well as for a larger class of modal multilattice logics