Thomas Ferguson

While participating in a symposium on Dave Ripley’s forthcoming book Uncut, I had proposed that employing a strict-tolerant interpretation of the weak Kleene matrices provided a content-theoretical conception of the bounds of conversational norms that enjoyed advantages over Ripley’s use of the strong Kleene matrices. During discussion, I used the case of sentences that are taken to be out-of-bounds for being secrets as an example of a case in which the setting of conversational bounds in practice diverged from the account championed by Ripley. In this paper, I consider an objection that my treatment of quantifiers was mistaken insofar as the confidentiality of a sentence ϕ may not lift to the sentence ∃xϕ and draw from this objection that neither the strong nor the weak Kleene interpretation of quantifiers suffices, but that a novel interpretation may do so.

Many analyses of notion of _metainferences_ in the non-transitive logic ST have tackled the question of whether ST can be identified with classical logic. In this paper, we argue that the primary analyses are overly restrictive of the notion of metainference. We offer a more elegant and tractable semantics for the strict-tolerant hierarchy based on the three-valued function for the LP material conditional. This semantics can be shown to easily handle the introduction of _mixed_ inferences, _i.e._, inferences involving objects belonging to more than one (meta)inferential level and solves several other limitations of the ST hierarchies introduced by Barrio, Pailos, and Szmuc.

Logics based on weak Kleene algebra (WKA) and related structures have been recently proposed as a tool for reasoning about flaws in computer programs. The key element of this proposal is the presence, in WKA and related structures, of a non-classical truth-value that is “contaminating” in the sense that whenever the value is assigned to a formula ϕ, any complex formula in which ϕ appears is assigned that value as well. Under such interpretations, the contaminating states represent occurrences of a flaw. However, since different programs and machines can interact with (or be nested into) one another, we need to account for different kind of errors, and this calls for an evaluation of systems with multiple contaminating values. In this paper, we make steps toward these evaluation systems by considering two logics, HYB1 and HYB2, whose semantic interpretations account for two contaminating values beside classical values 0 and 1. In particular, we provide two main formal contributions. First, we give a characterization of their relations of (multiple-conclusion) logical consequence—that is, necessary and sufficient conditions for a set Δ of formulas to logically follow from a set Γ of formulas in HYB1 or HYB2. Second, we provide sound and complete sequent calculi for the two logics.

In this article we revisit a number of disputes regarding significance logics---i.e., inferential frameworks capable of handling meaningless, although grammatical, sentences---that took place in a series of articles most of which appeared in the Australasian Journal of Philosophy between 1966 and 1978. These debates concern (i) the way in which logical consequence ought to be approached in the context of a significance logic, and (ii) the way in which the logical vocabulary has to be modified (either by restricting some notions, or by adding some vocabulary) to keep as much of Classical Logic as possible. Our aim is to show that the divisions arising from these disputes can be dissolved in the context of a novel and intuitive proposal that we put forward.

Graham Priest has frequently employed a construction in which a classical first-order model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {A}$$\end{document} may be collapsed into a three-valued model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {A}^{\sim }$$\end{document} suitable for interpretations in Priest’s logic of paradox. The source of this construction’s utility is Priest’s Collapsing Lemma, which guarantees that a formula true in the model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {A}$$\end{document} will continue to be true in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {A}^{\sim }$$\end{document}. In light of the utility and elegance of the Collapsing Lemma, extending variations of the lemma to other deductive calculi becomes very attractive. The aim of this paper is to map out some of the frontiers of the Collapsing Lemma by describing the types of expansions or revisions to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {LP}$$\end{document} for which the Collapsing Lemma continues to hold and a number of cases in which the lemma cannot be salvaged. Among what is shown is that the lemma holds for a strictly more expressive form of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {LP}$$\end{document} including nullary truth and falsity constants, that any conditional connective that can be added to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {LP}$$\end{document} without inhibiting the lemma must be theoremhood-preserving, and that the Collapsing Lemma extends to the paraconsistent weak Kleene logic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PWK}$$\end{document} as well.

Despite a renewed interest in Richard Angell’s logic of analytic containment ), the first semantics for \ introduced by Fabrice Correia has remained largely unexamined. This paper describes a reasonable approach to Correia semantics by means of a correspondence with a nine-valued semantics for \. The present inquiry employs this correspondence to provide characterizations of a number of propositional logics intermediate between \ and classical logic. In particular, we examine Correia’s purported characterization of classical logic with respect to his semantics, showing the condition Correia cites in fact characterizes the “logic of paradox” \ and provide a correct characterization. Finally, we consider some remarks on related matters, such as the applicability of the present correspondence to the analysis of the system \ and an intriguing relationship between Correia’s models and articular models for first degree entailment

Recent scholarship in intellectual humility (IH) has attempted to provide deeper understanding of the virtue as personality trait and its impact on an individual's thoughts, beliefs, and actions. A limitations-owning perspective of IH focuses on a proper recognition of the impact of intellectual limitations and a motivation to overcome them, placing it as the mean between intellectual arrogance and intellectual servility. We developed the Limitations-Owning Intellectual Humility Scale to assess this conception of IH with related personality constructs. In Studies 1 (n= 386) and 2 (n = 296), principal factor and confirmatory factor analyses revealed a three-factor model – owning one's intellectual limitations, appropriate discomfort with intellectual limitations, and love of learning. Study 3 (n = 322) demonstrated strong test-retest reliability of the measure over 5 months, while Study 4 (n = 612) revealed limitations-owning IH correlated negatively with dogmatism, closed-mindedness, and hubristic pride and positively with openness, assertiveness, authentic pride. It also predicted openness and closed-mindedness over and above education, social desirability, and other measures of IH. The limitations-owning understanding of IH and scale allow for a more nuanced, spectrum interpretation and measurement of the virtue, which directs future study inside and outside of psychology.

In this paper, we consider some contributions to the model theory of the logic of formal inconsistency $\mathsf{QmbC}$ as a reply to Walter Carnielli, Marcelo Coniglio, Rodrigo Podiacki and Tarcísio Rodrigues’ call for a ‘wider model theory.’ This call demands that we align the practices and techniques of model theory for logics of formal inconsistency as closely as possible with those employed in classical model theory. The key result is a proof that the Keisler–Shelah isomorphism theorem holds for $\mathsf{QmbC}$, i.e. that the strong elementary equivalence of two $\mathsf{QmbC}$ models $\mathfrak{A}$ and $\mathfrak{B}$ is equivalent to them having strongly isomorphic ultrapowers. As intermediate steps, we introduce some notions of model-theoretic equivalence between $\mathsf{QmbC}$ models, explicitly prove Łoś’ theorem and introduce a useful technique of model-theoretic ‘atomization’ in which the satisfaction sets of non-deterministically evaluated formulae are associated with new predicates. Finally, we consider some of the extensions of $\mathsf{QmbC}$, explicitly showing that Keisler–Shelah holds for $\mathsf{QCi}$ and suggesting that it holds of extensions like $\mathsf{QCila}$ and $\mathsf{QCia}$ as well.

This book aids in the rehabilitation of the wrongfully deprecated work of William Parry, and is the only full-length investigation into Parry-type propositional logics. A central tenet of the monograph is that the sheer diversity of the contexts in which the mereological analogy emerges – its effervescence with respect to fields ranging from metaphysics to computer programming – provides compelling evidence that the study of logics of analytic implication can be instrumental in identifying connections between topics that would otherwise remain hidden. More concretely, the book identifies and discusses a host of cases in which analytic implication can play an important role in revealing distinct problems to be facets of a larger, cross-disciplinary problem. It introduces an element of constancy and cohesion that has previously been absent in a regrettably fractured field, shoring up those who are sympathetic to the worth of mereological analogy. Moreover, it generates new interest in the field by illustrating a wide range of interesting features present in such logics – and highlighting these features to appeal to researchers in many fields.

Of the interpretations of disjunction in which Addition fails discussed in Chap. 4, Melvin Fitting’s cut-down disjunction stands out as an interpretation with a plausible explanation for the failure. This chapter examines cut-down operations in more detail and with more rigor. In particular, we consider bilattice and trilattice semantics for the $$\vdash $$ ⊢ -Parry systems $$\mathsf {S}_{\mathtt {fde}}$$ S fde and $$\mathsf {AC}$$ AC —both of which have been claimed as ‘rivals’ to $$\mathsf {E}_{\mathtt {fde}}$$ E fde —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.

This chapter continues the consideration of the potential for interpreting ‘nonsense’ values as catastrophic faults in computational processes, focusing on the particular case in which Nuel Belnap’s ‘artificial reasoner’ is unable to retrieve the semantic value assigned to a variable. This leads not only to a natural interpretation of Graham Priest’s semantics for the $$\vdash $$ ⊢ -Parry system $$\mathsf {S}^{\star }_{\mathtt {fde}}$$ S fde ⋆ but also a novel, many-valued semantics for Angell’s $$\mathsf {AC}$$ AC, completeness of which is proven by establishing a correspondence with Correia’s semantics for $$\mathsf {AC}$$ AC. These many-valued semantics have the additional benefit of allowing the application the material in Chap. 2 to the case of $$\mathsf {AC}$$ AC, thereby defining natural intensional extensions of $$\mathsf {AC}$$ AC in the spirit of Parry’s $$\mathsf {PAI}$$ PAI.

This chapter identifies as develops a few salient facets of the relationship between Parry-type deductive systems and the field of ‘logics of nonsense.’ Of particular importance is Dmitri Bochvar’s ‘internal’ nonsense logic $$\mathsf {\Sigma }_{0}$$ Σ 0, establishing a strong connection between such logics of nonsense and Parry systems more generally. We observe that two $$\vdash $$ ⊢ -Parry subsystems of $$\mathsf {\Sigma }_{0}$$ Σ 0 —Harry Deutsch’s $$\mathsf {S}_{\mathtt {fde}}$$ S fde and Frederick Johnson’s $$\mathsf {RC}$$ RC —can be considered to be the products of two particular ‘strategies’ of eliminating problematic inferences from Bochvar’s system. For example, this chapter shows that Deutsch’s $$\mathsf {S}_{\mathtt {fde}}$$ S fde and Johnson’s $$\mathsf {RC}$$ RC can be interpreted as the Parry-type subsystems of Bochvar’s nonsense logic generated by taking paraconsistent and connexive fragments of $$\mathsf {\Sigma }_{0}$$ Σ 0, respectively.

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.

In the field of logics of formal inconsistency, the notion of “consistency” is frequently too broad to draw decisive conclusions with respect to the validity of many theses involving the consistency connective. In this paper, we consider the matter of the axiom 0—i.e., the schema ◦ ◦ϕ—by considering its interpretation in contexts in which “consistency” is understood as a type of verifiability. This paper suggests that such an interpretation is implicit in two extracanonical LFIs—Sören Halldén’s nonsense-logic C and Graham Priest’s cointuitionistic logic daC—drawing some interesting conclusions concerning the status of 0. Initially, we discuss Halldén’s skepticism of this axiom and provide a plausible counterexample to its validity. We then discuss the interpretation of the operator in Priest’s daC and show the equivalence of 0 to the intuitionistic principle of testability. These observations suggest that it may be fruitful for members of the LFI community to look outside the canon for evidence concerning the adoption of principles like 0.

An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯, the closure of the Peano axioms in Wansing’s connexive logic C. Namely, the paper conjectured that C♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and Nelson’s paraconsistent N4, this also supplements Nelson’s own proof of the Post consistency of N4♯. Insofar as the present technique allows infinite models, this resolves Nelson’s concern that N4♯ is of interest only to those accepting that there are finitely many natural numbers.

The Routley star, an involutive function between possible worlds or set-ups against which negation is evaluated, is a hallmark feature of Richard Sylvan and Val Plumwood's set-up semantics for the logic of first-degree entailment. Less frequently acknowledged is the weaker mate function described by Sylvan and his collaborators, which results from stripping the requirement of involutivity from the Routley star. Between the mate function and the Routley star, however, lies an broad field of intermediate semantical conditions characterizing an infinite number of consequence relations closely related to first-degree entailment. In this paper, we consider the semantics and proof theory for deductive systems corresponding to set-up models in which the mate function is cyclical. We describe modifications to Anderson and Belnap's consecution calculus LE_fde2 that correspond to these constraints, for which we prove soundness and completeness with respect to the set-up semantics. Finally, we show that a number of familiar metalogical properties are coordinated with the parity of a mate function's period, including refined versions of the variable-sharing property and the property of gentle explosiveness.

This book presents the state of the art in the fields of formal logic pioneered by Graham Priest. It includes advanced technical work on the model and proof theories of paraconsistent logic, in contributions from top scholars in the field. Graham Priest’s research has had a considerable influence on the field of philosophical logic, especially with respect to the themes of dialetheism—the thesis that there exist true but inconsistent sentences—and paraconsistency—an account of deduction in which contradictory premises do not entail the truth of arbitrary sentences. Priest’s work has regularly challenged researchers to reappraise many assumptions about rationality, ontology, and truth. This book collects original research by some of the most esteemed scholars working in philosophical logic, whose contributions explore and appraise Priest’s work on logical approaches to problems in philosophy, linguistics, computation, and mathematics. They provide fresh analyses, critiques, and applications of Priest’s work and attest to its continued relevance and topicality. The book also includes Priest’s responses to the contributors, providing a further layer to the development of these themes.

Yehoshua Bar-Hillel and Rudolph Carnap’s classical theory of semantic information entails the counterintuitive feature that inconsistent statements convey maximal information. Theories preserving Bar-Hillel and Carnap’s modal intuitions while imposing a veridicality requirement on which statements convey information—such as the theories of Fred Dretske or Luciano Floridi—avoid this commitment, as inconsistent statements are deemed not information-conveying by fiat. This paper produces a pair of paradoxical statements that such “veridical-modal” theories must evaluate as both conveying and not conveying information, although Bar-Hillel and Carnap’s theory accommodates these statements without inconsistency. Moreover, the paradoxes are independently interesting as the mode in which they self-refer bears on their evaluation

In this paper, we look at applying the techniques from analyzing superintuitionistic logics to extensions of the cointuitionistic Priest-da Costa logic daC (introduced by Graham Priest as “da Costa logic”). The relationship between the superintuitionistic axioms- definable in daC- and extensions of Priest-da Costa logic (sdc-logics) is analyzed and applied to exploring the gap between the maximal si-logic SmL and classical logic in the class of sdc-logics. A sequence of strengthenings of Priest-da Costa logic is examined and employed to pinpoint the maximal non-classical extension of both daC and Heyting-Brouwer logic HB . Finally, the relationship between daC and Logics of Formal Inconsistency is examined

This paper offers a framework for extending Arnon Avron and Iddo Lev’s non-deterministic semantics to quantified predicate logic with the intent of resolving several problems and limitations of Avron and Anna Zamansky’s approach. By employing a broadly Fregean picture of logic, the framework described in this paper has the benefits of permitting quantifiers more general than Walter Carnielli’s distribution quantifiers and yielding a well-behaved model theory. This approach is purely objectual and yields the semantical equivalence of both α-equivalent formulae and formulae differing only by codenotative terms. Finally, we make a brief excursion into non-deterministic model theory, proving a strong Łoś’ Theorem and compactness for all finitely-valued, non-deterministic logics whose quantifiers have intensions describable in a first order metalanguage

The hallmark of the deductive systems known as ‘conceptivist’ or ‘containment’ logics is that for all theorems of the form , all atomic formulae appearing in also appear in . Significantly, as a consequence, the principle of Addition fails. While often billed as a formalisation of Kantian analytic judgements, once semantics were discovered for these systems, the approach was largely discounted as merely the imposition of a syntactic filter on unrelated systems. In this paper, we examine a number of prima facie unrelated deductive contexts in which Addition fails and attempt to harmonise them by developing a computational interpretation of conceptivist logics

This paper examines the relationships between the many-valued logics G~ and Gn~ of Esteva, Godo, Hajek, and Navara, i.e., Godel logic G enriched with Łukasiewicz negation, and neighbors of intuitionistic logic. The popular fragments of Rauszer's Heyting-Brouwer logic HB admit many-valued extensions similar to G which may likewise be enriched with Łukasiewicz negation; the fuzzy extensions of these logics, including HB, are equivalent to G ~, as are their n-valued extensions equivalent to Gn~ for any n ≥ 2. These enriched systems extend Wansing's logic I4C4, showing that Łukasiewicz negation is a species of Nelson's negation of constructible falsity and yielding a Kripke-style semantics for G~ and Gn~ to complement the many-valued semantics.

In this article, we consider variations of Nuel Belnap’s ‘artificial reasoner’. In particular, we examine cases in which the artificial reasoner is faulty, e.g. situations in which the reasoner is unable to calculate the value of a formula due to an inability to retrieve the values of its atoms. In the first half of the article, we consider two ways of modelling such circumstances and prove the deductive systems arising from these two types of models to be equivalent to Graham Priest’s first-degree entailment with an ‘emptiness’ value and Richard Angell’s analytic containment, making computational interpretations of these systems possible. The Belnap-type semantics for AC bring FDE φ and AC in line with other containment logics in their neighborhood. The second half of the article examines formal questions, such as whether AC admits an analysis along the lines of that given to the related system of William Parry’s system of analytic implication, as suggested by Kurt Gödel and confirmed by Kit Fine. Furthermore, a natural means of extending these systems to languages with an intensional implication connective is investigated.

In a recent paper, John Wigglesworth explicates the notion of a set's being grounded in or ontologically depending on its members by the modal statement that in any world, that a set exists in that world entails that its members exist as well. After suggesting that variable-domain S5 captures an appropriate account of metaphysical necessity, Wigglesworth purports to prove that in any set theory satisfying the axiom Extensionality this condition holds, that is, that sets ontologically depend on their members with respect to extraordinarily weak notions of set. This paper diagnoses a number of problems concerning Wigglesworth's formal argument. For one, we will show that Wigglesworth's argument is invalid as it requires an appeal to hidden, extralogical theses concerning rigid designation and the persistence of sets across possible worlds. Having demonstrated the indispensability of these principles to Wigglesworth's argument, we will then show that even granted the enthymematic premises, the argument only proves the ontological dependence of singletons on their members and does not extend to sets in general. Finally, we will consider strengthenings of Wigglesworth's reasoning and suggest that even the weakest generalization will bear undesirable consequences.

We examine the relationship between the logics of nonsense of Bochvar and Halldén and the containment logics in the neighborhood of William Parry’s A I. We detail two strategies for manufacturing containment logics from nonsense logics—taking either connexive and paraconsistent fragments of such systems—and show how systems determined by these techniques have appeared as Frederick Johnson’s R C and Carlos Oller’s A L. In particular, we prove that Johnson’s system is precisely the intersection of Bochvar’s B 3 and Graham Priest’s non-symmetrized connexive logic and that Oller’s system lies just beneath the intersection of B 3 and Priest’s paraconsistent L P. We conclude by examining Oller’s system in more depth, giving it a characterization in terms of L P and showing that it plays the same role to Harry Deutsch’s paraconsistent containment logic S that Aleksandr Zinov'ev’s S 1 plays with respect to A I

The work of Arnon Avron and Ofer Arieli has shown a deep relationship between the theory of bilattices and the Belnap-Dunn logic \. This correspondence has been interpreted as evidence that \ is “the” logic of bilattices, a consideration reinforced by the work of Yaroslav Shramko and Heinrich Wansing in which \ is shown to be similarly entrenched with respect to the theories of trilattices and, more generally, multilattices. In this paper, we export Melvin Fitting’s “cut-down” connectives—propositional connectives that “cut down” available evidence—to the case of multilattices and show that two related first degree systems—Harry Deutsch’s four-valued \ and Richard Angell’s \—emerge just as elegantly and are as intimately connected to the theories of bilattices and trilattices as the Belnap-Dunn logic.

J. Michael Dunn’s Theorem in 3-Valued Model Theory and Graham Priest’s Collapsing Lemma provide the means of constructing first-order, three-valued structures from classical models while preserving some control over the theories of the ensuing models. The present article introduces a general construction that we call a Dunn–Priest quotient, providing a more general means of constructing models for arbitrary many-valued, first-order logical systems from models of any second system. This technique not only counts Dunn’s and Priest’s techniques as special cases, but also provides a generalized Collapsing Lemma for Priest’s more recent plurivalent semantics in general. We examine when and how much control may be exerted over the resulting theories in particular cases. Finally, we expand the utility of the construction by showing that taking Dunn–Priest quotients of a family of structures commutes with taking an ultraproduct of that family, increasing the versatility of the tool.