This book brings together papers from a conference that took place in the city of L'Aquila, 4–6 April 2019, to commemorate the 10th anniversary of the earthquake that struck on 6 April 2009. Philosophers and scientists from diverse fields of research debated the problem that, on 6 April 1922, divided Einstein and Bergson: the nature of time. For Einstein, scientific time is the only time that matters and the only time we can rely on. Bergson, however, believes that scientific time (...) is derived by abstraction, even in the sense of extraction, from a more fundamental time. The plurality of times envisaged by the theory of Relativity does not, for him, contradict the philosophical intuition of the existence of a single time. But how do things stand today? What can we say about the relationship between the quantitative and qualitative dimensions of time in the light of contemporary science? What do quantum mechanics, biology and neuroscience teach us about the nature of time? The essays collected here take up the question that pitted Einstein against Bergson, science against philosophy, in an attempt to reverse the outcome of their monologue in two voices, with a multilogue in several voices. (shrink)

Moral obligation and permissibility are usually thought to be interdefinable. Following the pattern of the duality definitions of necessity and possibility, we have that something’s being permissible could be defined as its not being obligatory to not do it. And that something’s being obligatory could be defined as its not being permissible to not do it. In this paper, I argue that neither direction of this alleged interdefinability works. Roughly, the problem is that a claim that some act is obligatory (...) or permissible entails that there is a moral law, whereas a negative claim that some act is not obligatory or not permissible does not. Nevertheless, one direction of the interdefinability can potentially be salvaged. I argue that, if we do not require the conceptual possibility of moral dilemmas, then there is a way to plausibly define obligation in terms of permissibility. I conclude that permissibility is the only feasible deontic primitive. (shrink)

The origin of paraconsistent logic is closely related with the argument, ‘from the assertion of two mutually contradictory statements any other statement can be deduced’; this can be referred to as ex contradictione sequitur quodlibet (ECSQ). Despite its medieval origin, only by the 1930s did it become the main reason for the unfeasibility of having contradictions in a deductive system. The purpose of this article is to study what happened earlier: from Principia Mathematica to that time, when it became well (...) established. The two main historical claims that I am going to advance are the following: (1) the first explicit use of ECSQ as the main argument for supporting the necessity of excluding any contradiction from deductive systems is to be found in the first edition of the book Grundzüge der Theoretischen Logik (Hilbert, D. and Ackermann, W. 1928. Grundzüge der Theoretischen Logik. Berlin: Julius Springer Verlag); (2) Łukasiewicz's position regarding the logical constraints against contradictions varies considerably from his studies on the principle of (non-) contradiction in Aristotle, published in 1910 and what is stated in his `authorized lectured notes' on mathematical logic that appeared in 1929. The two texts are: 1) a paper in German (Łukasiewicz, J. 1910. `Über den Satz des Widerspruchs bei Aristotles'. Bulletin International de l'Académie des sciences de Cracovie, Classe d'Histoire et de Philosophie, pp. 15–38) [English translation: Łukasiewicz, J. 1971. ‘On the principle of contradiction in Aristotle’, Review of Metaphysics, XXIV, 485–509]; and 2) a book in Polish. Łukasiewicz, J. 1910. O zasadzie sprzecznosci u Aristotelesa Studium krytyczne, Warsaw: Panstwowe Wydawnictwo Naukowe [German translation: Łukasiewicz, J. 1993. Über den Satz des Widerspruchs bei Aristotles. Hildesheim: Georg Olms Verlag]. The lecture notes were then published as a book (Łukasiewicz, J. 1958. Elementy Logiki Matematycznej. Warszawa: Panstwowe Wydawnictwo Naukowe [PWN] and then translated into English (Łukasiewicz, J. 1963. Elements of Mathematical Logic. Oxford, New York: Pergamon Press/the Macmillan Company). The second half of this article will concentrate on Łukasiewicz's position on ECSQ. This will lead me to propose that to regard him as a forerunner of paraconsistent logic by virtue of those early writings is accurate only if his book published in Polish is considered but not if the analysis is restricted to the paper originally published in German (as has been the case for the principal reconstructions of the history of paraconsistent logic). Furthermore, I will stress that in the 1929 book he presented one formalization of ECSQ as an axiom for sentential calculus and, also, he used ECSQ to defend the necessity of consistency, apparently independently of Hilbert and Ackermann's book. At the end, I will suggest that the aim of twentieth century usage of ECSQ was to change from the centuries-long philosophical discussion about contradictions to a more ‘technical’ one. But with paraconsistent logic viewed as a technical solution to this restriction, then, the philosophical problem revives but having now at one's disposal an improved understanding of it. Finally, Łukasiewicz's two different positions about ECSQ open an interesting question about the history of paraconsistent logic: do we have to attempt a consistent reconstruction of it, or are we prepared to admit inconsistencies within it? (shrink)

This paper introduces and explores a conservative extension of inquisitive logic. In particular, weak negation is added to the standard propositional language of inquisitive semantics, and it is shown that, although we lose some general semantic properties of the original framework, such an enrichment enables us to model some previously inexpressible speech acts such as weak denial and ‘might’-assertions. As a result, a new modal logic emerges. For this logic, a Fitch-style system of natural deduction is formulated. The main result (...) of this paper is a theorem establishing the completeness of the system with respect to inquisitive semantics with weak negation. At the conclusion of the paper, the possibility of extending the framework to the level of first order logic is briefly discussed. (shrink)

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. (shrink)

Contrary to common misconceptions, today's logic is not devoid of existential import: the universalized conditional ∀ x [S→ P] implies its corresponding existentialized conjunction ∃ x [S & P], not in all cases, but in some. We characterize the proexamples by proving the Existential-Import Equivalence: The antecedent S of the universalized conditional alone determines whether the universalized conditional has existential import, i.e. whether it implies its corresponding existentialized conjunction.A predicate is an open formula having only x free. An existential-import predicate (...) Q is one whose existentialization, ∃ x Q, is logically true; otherwise, Q is existential-import-free or simply import-free.How abundant or widespread is existential import? How abundant or widespread are existential-import predicates in themselves or in comparison to import-free predicates? We show that existential-import predicates are quite abundant, and no less so than import-free predicates. Existential.. (shrink)

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. (shrink)

Résumé1Le rôle prlvilégié que joue l'implication « si … alors » dans la pensée donne à sa formalisation loglque une importance capitale. Mais la formalisation classique se heurte à certaines difficultés.2On montre, par la méthode L de Gentzen, que c'est la partie positive de la logique intuitionniste qui exprime au plus près l'idée intuitive de l'implication.3L'implication est liée à la négation. On est conduit à distinguer « réfutable », «absurde» et «faux».4L'analyse de ces notions peut se faire aussi par la (...) méthode N de Gentzen. Elle conduit à une certaine symétrie dans les résultats.5Un système S, dû à F. B. Fitch, conduit à une nouvelle sorte de négation.6La négation dans S peut s'interpréter comme «contraire aux faits».Zusammenfassung1Dei der besonderen Rolle der Implikation «wenn … dann »im Denken 1st deren logische Formalisierung von wesentlicher Bedeutung. Die klassische Formalisierung stösst auf gewisse Schwierigkeiten.2Gentzens Méthode L erweist, dass der positive Teil der intuitionistischen Logik der intuitiven Idee der Implikation am nächsten kommt.3Die Implikation ist an die Negation gebunden. Folgende Unterscheidungen ergeben sich: widerlegbar, unmöglich, falsch.4Die Analyse dieser Begriffe kann auch nach Gentzens Méthode N geschehen. Sie führt zu einer gewissen Symmetrie in den Ergebnissen.5Das logische System S, das F. B. Fitch aufgestellt hat, führt zu einer neuen Art der Negation.6Die Negation kann in S als «im Widerspruch zu den Tatsachen» interpretiert werden. (shrink)

One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley-Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...) a general conception of conditionality that may unify the three given conceptions. (shrink)

We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources are tightly related, (...) and can be cleared up by adopting a particular substructural logic in place of classical logic. We then argue that our perspective can be justified via an informational semantics of contraction-free substructural logics. (shrink)

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that (...) restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

I provide a tableau system and completeness proof for a revised version of Carnap's semantics for quantified modal logic. For Carnap, a sentence is possible if it is true in some first order model. However, in a similar fashion to second order logic, no sound and complete proof theory can be provided for this semantics. This factor contributed to the ultimate disappearance of Carnapian modal logic from contemporary philosophical discussion. The proof theory I discuss comes close to Carnap's semantic vision (...) and provides an interesting counterpoint to mainstream approaches to modal logic. Despite its historical origins, my intention is to demonstrate that this approach to modal logic is worthy of contemporary attention and that current debate is the poorer for its absence. (shrink)

As is known, a logic S is paraconsistent if the rule ECQ (E contradictione quodlibet) is not a rule of S. Not less well known is the fact that Lewis’ modal logics are not paraconsistent. Actually, Lewis vindicates the validity of ECQ in a famous proof currently known as the “Lewis’ proof” or “Lewis’ argument.” This proof essentially leans on the Disjunctive Syllogism as a rule of inference. The aim of this paper is to define a series of paraconsistent logics (...) included in S4 where the Disjunctive Syllogism is valid only as a rule of proof. (shrink)

Henry M. Sheffer is well known to logicians for the discovery (or rather, the rediscovery) of the ?Sheffer stroke? of propositional logic. But what else did Sheffer contribute to logic? He published very little, though he is known to have been carrying on a rather mysterious research program in logic; the only substantial result of this research was the unpublished monograph The General Theory of Notational Relativity. The main aim of this paper is to explain, as far as possible (given (...) the scanty evidence), the nature of Sheffer's program, and the reasons for its failure. The paper concludes with a discussion of Sheffer's only true logical descendant, C.H. Langford, and his contributions to model theory. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

The axiomatic presentation of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generalizations of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective.

The work of Hugh MacColl (1837–1909) suffered the same fate after his death as before it:despite being vaguely alluded to and in part even commended, on the whole it has remained an unknown quantity. Even worse, those of his ideas which have played a decisive role in the history of logic have been credited to his successors; this is especially the case with the definition of strict implication and the first formal development of formal modal logic. This paper takes an (...) initial step towards a rectification of this unfortunate misrepresentation, presenting a bibliography of MacColl’s most significant publications with particular regard to their reception. (shrink)

In this work, attention is drawn to the abundant use of metaphor and analogy in works of logic. I argue that pervasiveness of figurative language is to be counted among the features that characterize logic and distinguish it from other sciences. This characteristic feature reflects the creativity that is inherent in logic and indeed has been demonstrated to be a necessary part of logic. The goal of this paper, in short, is to provide specific examples of figurative language used in (...) logic that yield insights into the nature of the subject. I encourage the reader to take metaphors seriously, and to accept that they are not mere embellishments but key elements in our understanding of logic. (shrink)

In formal logic there is a premium on clever paraphrase, for it subsumes troublesome inferences under a familiar theory. (A paradigm is Davidson's analysis 1967 of inferences like ?He buttered his toast with a knife; so, he buttered his toast?.) But the need for paraphrase in formal logic runs deeper than the odd recalcitrant inference, and thus, I shall argue, commits logicians to some interesting consequences. First, the thesis that arguments are valid in virtue of their form must be severely (...) qualified (?4). And second, it is misleading to view a formal logical theory as a standard for justifying and criticizing inference (?7). The latter point depends on the nature and role of paraphrase, which permit a range of conflicting logical theories. Conflicting logical theories arise from the conflicting goals of logical theorists and the promiscuous nature of paraphrase makes reconciliation impossible. (shrink)

Everyday reasoning is replete with arguments which, though not logically valid, nonetheless harbor a measure of credibility in their own right. Here the claim that such arguments force us to acknowledge material validity, in addition to logical validity, is advanced, and criteria that attempt to unpack this concept are examined in detail. Of special concern is the effort to model these criteria on explications of logical validity that rely on notions of substitutivity and logical form. It is argued, however, that (...) such a parallel is not easily located and that it is uncertain that a construal of material validity can be fashioned after traditional accounts of logical validity. Attention is also given to the topics of enthymemes and to the proper domain of logic. (shrink)

The standard use of the propositional calculus ('P.C.?) in analyzing the validity of inferences involving conditionals leads to fallacies, and the problem is to determine where P.C. may be ?safely? used. An alternative analysis of criteria of reasonableness of inferences in terms of conditions of justification rather than truth of statements is proposed. It is argued, under certain restrictions, that P. C. may be safely used, except in inferences whose conclusions are conditionals whose antecedents are incompatible with the premises in (...) the sense that if the antecedent became known, some of the previously asserted premises would have to be withdrawn. (shrink)

Abstract W. E. Johnson argued that by taking into account both the epistemic and constitutive conditions for using arguments in inferences one could dissolve the paradoxes of material implication. This essay argues that the same sort of consideration can be used to dissolve the paradox of ravens in confirmation theory. It is argued in particular, and in agreement with certain points raised by the Popperians, that those instances of a generalization which are verifying but apparently not confirming cannot raise the (...) posterior of that generalization. (shrink)

I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. I argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads me to select as a plausible candidate LL, a fragment of linear logic that differs from R in that it rejects both contraction and distribution.

A major question for the relevant logics has been, “Under what conditions is Ackermann's ruleγ from -A ∨B andA to inferB, admissible for one of these logics?” For a large number of logics and theories, the question has led to an affirmative answer to theγ problem itself, so that such an answer has almost come to be expected for relevant logics worth taking seriously. We exhibit here, however, another large and interesting class of logics-roughly, the Boolean extensions of theW — (...) free relevant logics (and, precisely, the well-behaved subsystems of the 4-valued logicBN4) — for which γ fails. (shrink)

The object of this paper is to show why recent research in the psychology of deductive and probabilistic reasoning does not have.

The point of the present paper is to draw attention to some interesting similarities, as well as differences, between the approaches to the logic of noncontingency of Evgeni Zolin and of Claudio Pizzi. Though neither of them refers to the work of the other, each is concerned with the definability of a (normally behaving, though not in general truth-implying) notion of necessity in terms of noncontingency, standard boolean connectives and additional but non-modal expressive resources. The notion of definability involved is (...) different in the two cases (‘external’ for Zolin, ‘internal’ for Pizzi), as are the additional resources: infinitary conjunction in the case of Zolin, and for Pizzi, first, propositional quantification, and then, later, most ingeniously, the use of a propositional constant. As well as surveying and comparing of the work of these authors, the discussion includes some some novelties, such as the confirmation of a conjecture of Zolin’s (Theorem 2.7). (shrink)

This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...) exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge. (shrink)