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 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

Dualities for Plonka sums

Plonka sums consist of an algebraic construction similar, in some sense to direct limits, which allows to represent classes of algebras defined by means of regular identities (namely those equations where the same set of variables appears on both sides). Recently, Plonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Plonka sum representation in terms of dualisable algebras.Comment: 12 pages; the paper was awarded with the "SILFS Logic Prize" and is appearing on "Logica Universalis

Adding 4.0241 to TLP

Tractatus 4.024 inspired the dominant semantics of our time: truth-conditional semantics. Such semantics is focused on possible worlds: the content of p is the set of worlds where p is true. It has become increasingly clear that such an account is, at best, defective: we need an ‘independent factor in meaning, constrained but not determined by truth-conditions’ (Yablo 2014, p. 2), because sentences can be differently true at the same possible worlds. I suggest a missing comment which, had it been included in the Tractatus, would have helped semantics get this right from the start. This is my 4.0241: ‘Knowing what is the case if a sentence is true is knowing its ways of being true’: knowing a sentence’s truth possibilities and what we now call its topic, or subject matter. I show that the famous ‘fundamental thought’ that ‘the “logical constants” do not represent’ (4.0312) can be understood in terms of ways-based views of meaning. Such views also help with puzzling claims like 5.122: ‘If p follows from q, the sense of “p” is contained in the that of “q”’, which are compatible with a conception of entailment combining truth-preservation with the preservation of topicality, or of ways of being true

Immune Logics

This article is concerned with an exploration of a family of systems—called immune logics—that arise from certain dualizations of the well-known family of infectious logics. The distinctive feature of the semantic of infectious logics is the presence of a certain “infectious” semantic value, by which two different though equivalent things are meant. On the one hand, it is meant that these values are zero elements for all the operations in the underlying algebraic structure. On the other hand, it is meant that these values behave in a value-in-value-out fashion for all the operations in the underlying algebraic structure. Thus, in a rather informal manner, we will refer to immune logics as those systems whose underlying semantics count with a certain “immune” semantic value behaving in a way that is somewhat dual to that of the infectious values. In a more formal manner, carrying out this dualization will prove to be not as straightforward as one could imagine, since the two characterizations of infectiousness discussed above lead to two different outcomes when one tries to conduct them. We explore these alternatives and provide technical results regarding them, and the various logical systems defined using such semantics

Conjunction and Disjunction in Infectious Logics

In this paper we discuss the extent to which conjunction and disjunction can be rightfully regarded as such, in the context of infectious logics. Infectious logics are peculiar many-valued logics whose underlying algebra has an absorbing or infectious element, which is assigned to a compound formula whenever it is assigned to one of its components. To discuss these matters, we review the philosophical motivations for infectious logics due to Bochvar, Halldén, Fitting, Ferguson and Beall, noticing that none of them discusses our main question. This is why we finally turn to the analysis of the truth-conditions for conjunction and disjunction in infectious logics, employing the framework of plurivalent logics, as discussed by Priest. In doing so, we arrive at the interesting conclusion that —in the context of infectious logics— conjunction is conjunction, whereas disjunction is not disjunction

Containment Logics: Algebraic Completeness and Axiomatization

The paper studies the containment companion (or, right variable inclusion companion) of a logic ⊢. This consists of the consequence relation ⊢ r which satisfies all the inferences of ⊢ , where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. In accordance with the work started in [10], we show that a different generalization of the Płonka sum construction, adapted from algebras to logical matrices, allows to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization

Pure Refined Variable Inclusion Logics

In this article, we explore the semantic characterization of the (right) pure refined variable inclusion companion of all logics, which is a further refinement of the nowadays well-studied pure right variable inclusion logics. In particular, we will focus on giving a characterization of these fragments via a single logical matrix, when possible, and via a class of finite matrices, otherwise. In order to achieve this, we will rely on extending the semantics of the logics whose companions we will be discussing with infectious values in direct and in more subtle ways. This further establishes the connection between infectious logics and variable inclusion logics