4 research outputs found
Subject-matter and intensional operators I:conditional-agnostic analytic implication
Although logical settings are typically concerned with tracking alethic considerations, frameworks exist in which topic-theoretic considerationsâe.g., tracking subject-matter or topicâare given equal importance. Intuitions about extending topic through a propositional language are generally straightforward for extensional cases. For a number of reasons, arriving at a compelling account of the subject-matter of intensional operatorsâsuch as intensional conditionalsâis a more difficult task. In particular, the framework of topic-sensitive intentional modals (TSIMs) championed by Francesco Berto and his collaborators leave the topics of intensional formulae undefined, which artificially constricts the expressivity of the theory. This paper proposes an approach to fill in this lacuna, emphasizing an analogous problem in Parry-style containment logics. In this setting, the approach receives a proof-of-concept through the introduction of a natural and general family of subsystems of Parryâs PAIâwith sound and complete axiomatizationsâthat allow a fine degree of control over the topics of intensional conditionals
An algebraic study of logics of variable inclusion and analytic containment
This thesis focuses on a wide family of logics whose common
feature is to admit a syntactic definition based on specific
variable inclusion principles.
This family has been divided into three main components:
logics of left variable inclusion, containment logics, and
the logic of demodalised analytic implication.
We offer a general investigation of such logics within
the framework of modern abstract algebraic logic
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