6 research outputs found
Relevant Logics Obeying Component Homogeneity
This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues
Relevant Logics Obeying Component Homogeneity
This paper discusses three relevant logics (S*fde , dS*fde , crossS*fde) that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde , dS*fde , crossS*fde. Second, the paper establishes complete sequent calculi for S*fde , dS*fde , crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define (a given family of) containment logics, we explore the single-premise/single-conclusion fragment of S*fde , dS*fde , crossS*fde and the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues
Semantical analysis of weak Kleene logics
This paper presents a semantical analysis of the Weak Kleene Logics K3W and PWK from the tradition of Bochvar and HalldĂŠn. These are three-valued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical consequence in PWKâthat is, we individuate necessary and sufficient conditions for a set Î of formulas to follow from a set Î in PWKâand a characterisation result for logical consequence in K3W. The paper also investigates two subsystems of K3W and PWK and discusses the relevance of the results against existing background. Finally, the paper discusses some issues related to Weak Kleene Logicsâin particular, their philosophical interpretation and the reading of conjunction and disjunctionâand points at some open issues
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