Dynamical infomorphism: form of endo-perspective

2019

This paper provides a proof that Tennant's logical system entails a paradox that is called Core logic paradox, in reference to the new name given by Tennant to his intuitionistic relevant logic.

Studia Logica

Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.

Natural Language Engineering, 1997

Logic and Information is an accessible introduction to situation theory and an insightful extension to Barwise and Perry's (1983) original work. It aims to develop the mathematics required for a science of information. So, this is indeed mostly a mathematical logic text, but an interdisciplinary one. There are topics that appeal to philosophers, linguists, computer scientists and cognitive scientists. The clear and illustrative style of the book makes it understandable both to students and more advanced researchers taking new interdisciplinary directions. Devlin states that the basic ideas of situation theory have not changed much since the first hardback edition of the book in 1991. (This paperback edition is the same as the original except for correction of typos.) However, he suggests various additional reading paths for those who want to pursue the recent mathematical developments in the field. Chapter 1 starts with an examination of how one can go about understanding information in the '' Age of Information ''. Devlin emphasizes an empirical approach, developing the mathematical devices when needed as opposed to adopting a form of logic top-down for the same purpose. A historical and critical account of classical logic, based on truth, is given from this point of view. Classical first-order predicate logic is foreseen as a constrained version of the logic of information to be developed. Two important concepts of the theory are introduced : Infons, discrete items of information ; and situations, some limited portion of the whole world linked with constraints. Analogies with other branches of mathematics and science are made liberally throughout the book. An account of cognition in terms of information is given in chapter 2, where the basic ontology of the theory is introduced. This chapter forms, if you like, a requirements specification of the theory being designed. The basic capability of agents to individuate (or discriminate) objects in their environment leads to the definition of a unit of information (an infon) as a tuple that shows whether a number of objects do or do not stand as arguments to a relation. An item of information can be true of a situation, hence is said to be supported by it. Individuation of a situation does not require its exact description to be known, however. Various theory-internal abstractions are underlined in this chapter. Taking representationconstraint pairs related with information as equivalence classes that denote an infon is one such point. Models (in this case set-theoretic) are not to be confused with the real world. More complex concepts of the theory are introduced in chapter 3. One such concept is the introduction of types as a way of detecting higher-order uniformities across situations and other objects of the theory. Related with this are parameters to denote arbitrary objects of a given type, and anchors to assign values to parameters. Examples of parameters in natural language are given in the form of pronouns or proper names. With the introduction of typeabstraction, a situation supporting a set of infons is shown to be a proposition : a claim about the world that a certain object is of a certain type. Also introduced is the concept of an oracle : a special situation that concerns a certain object within a set of related parametric infons. Between accounts of types and oracles, a rather lengthy (and probably redundant) digression is made to question and explain the nature of abstraction and formalism in mathematics.

We develop a model–theoretic analysis of the contradictions developed in the logic that do no satisfy the Non Contradiction, Excluded Middle and Philonian Principles.

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 presupposes—through the root ἀνά + λύω —a 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. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s systems—and of what may be thought of as containment logics or Parry systems in general—is a strong relevance property called the ‘Proscriptive Principle’ (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 Parry’s formal systems were intended to accurately model Kant’s notion of an analytic judgment, Parry’s deductive systems—and the suitability of the Proscriptive Principle in general—were met with severe criticism. While Anderson and Belnap argued that Parry’s criterion failed to account for a number of prima facie analytic judgments, others—such as Sylvan and Brady—argued that the utility of the criterion was impeded by its reliance on a ‘syntactical’ device. But these arguments are restricted to Parry’s 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 ‘logics of nonsense.’ Of particular importance is Dmitri Bochvar’s ‘internal’ nonsense logic Σ0, and we observe that two ⊢-Parry subsystems of Σ0 (Harry Deutsch’s Sfde and Frederick Johnson’s RC) can be considered to be the products of particular ‘strategies’ of eliminating problematic inferences from Bochvar’s system. The material of Chapter 3 considers Kit Fine’s program of state space semantics in the context of Parry logics. Recently, Fine—who had already provided the first intuitive semantics for Parry’s PAI—has offered a formal model of truthmaking (and falsemaking) that provides one of the first natural semantics for Richard B. Angell’s logic of analytic containment AC, itself a ⊢-Parry system. After discussing the relationship between state space semantics and nonsense, we observe that Fabrice Correia’s weaker framework—introduced as a semantics for a containment logic weaker than AC—tacitly endorses an implausible feature of allowing hypernonsensical statements. By modelling Correia’s containment logic within the stronger setting of Fine’s semantics, we are able to retain Correia’s intuitions about factual equivalence without such a commitment. As a further application, we observe that Fine’s setting can resolve some ambiguities in Greg Restall’s 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 ‘nonsense’ is associated with catastrophic faults in computer programs. In this spirit, we examine semantics for several ⊢-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 Belnap’s ‘artificial reasoner’ is unable to retrieve the value assigned to a variable. This leads not only to a natural interpretation of Graham Priest’s semantics for the ⊢-Parry system S⋆fde but also a novel, many-valued semantics for Angell’s AC, completeness of which is proven by establishing a correspondence with Correia’s 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 Parry’s PAI. One particular instance of the type of disjunction central to Chapter 4 is Melvin Fitting’s cut-down disjunction. Chapter 6 examines cut-down operations in more detail and provides bilattice and trilattice semantics for the ⊢-Parry systems Sfde and AC in the style of Ofer Arieli and Arnon Avron’s 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 Correia’s 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 Angell’s AC∗. Finally, we consider whether this correspondence aids in providing an interpretation to Correia’s first semantics for AC.