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)

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)

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.

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 implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the “logical” content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system (...) for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice and automated theorem proving of Reverse Mathematics results. (shrink)

In this paper I want to show that a main theme of Neurath’s philosophical work was the formulation of a radically empiricist theory of science. His approach, dubbed "encyclopedism", can be characterized by the following five theses: scientific knowledge is (1) fallible, (2) pluralistic, (3) holistic, (4) can be systematized only locally, and (5) does not give us a faithful description of the real world. (4) is to be considered as the most original thesis of encyclopedism and is discussed in (...) detail. (shrink)

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)

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)

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.

Between 1940 and 1945, while still a student of theoretical physics and without any contact with the history of science, Thomas S. Kuhn developed a general outline of a theory of the role of belief in science. This theory was well rooted in the philosophical tradition of Emerson Hall, Harvard, and particularly in H. M. Sheffer’s and C. I. Lewis’s logico-philosophical works—Kuhn was, actually, a graduate student of the former in 1945. In this paper I reconstruct the development of that (...) general outline after Kuhn’s first years at Harvard. I examine his works on moral and aesthetic issues—where he displayed an already ‘anti-Whig’ stance concerning historiography—as well as his first ‘Humean’ approach to science and realism, where his earliest concern with belief is evident. Then I scrutinise his graduate work to show how his first account of the role of belief was developed. The main aim of this paper is to show that the history of science illustrated for Kuhn the epistemic role and effects of belief he had already been theorising about since around 1941.Keywords: Intension; Belief; Causality; Thomas Kuhn; Clarence Irving Lewis; Henry Sheffer. (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)

The article ‘A Note on Strict Implication’ was submitted for publication by C.I. Lewis and C.H. Langford but withdrawn in proof. The paper is, according to notes and letters by both Lewis and Langford, largely by Lewis. It constitutes an early attempt by Lewis to give meanings for the modal connectives using abstract objects. To be necessary, for example, is for a statement to have the same intension as a truth-functional tautology. This theory prefigures the view of Lewis's 1946 book, (...) Analysis of Knowledge and Valuation. This introduction to the article looks at the historical background of the piece in addition to its argument and examines why it was withdrawn. (shrink)

This paper sets out a semantics for C.I. Lewis's logic S2 based on the ontology of his 1923 paper ‘Facts, Systems, and the Unity of the World’. In that article, worlds are taken to be maximal consistent systems. A system, moreover, is a collection of facts that is closed under logical entailment and conjunction. In this paper, instead of defining systems in terms of logical entailment, I use certain ideas in Lewis's epistemology and philosophy of logic to define a class (...) of models in which systems are taken to be primitive elements but bear certain relations to one another. I prove soundness and completeness for S2 over this class of models and argue that this semantics makes sense of at least a substantial fragment of Lewis's logical theory. (shrink)

The calculus of relations was created and developed in the second half of the nineteenth century by Augustus De Morgan, Charles Sanders Peirce, and Ernst Schröder. In 1940 Alfred Tarski proposed an axiomatization for a large part of the calculus of relations. In the next decade Tarski's axiomatization led to the creation of the theory of relation algebras, and was shown to be incomplete by Roger Lyndon's discovery of nonrepresentable relation algebras. This paper introduces the calculus of relations and the (...) theory of relation algebras through a review of these historical developments. (shrink)

H. B. Smith, Professor of Philosophy at the influential 'Pennsylvania School' was (roughly) a contemporary of C. I. Lewis who was similarly interested in a proper account of 'implication'. His research also led him into the study of modal logic but in a different direction than Lewis was led. His account of modal logic does not lend itself as readily as Lewis' to the received 'possible worlds' semantics, so that the Smith approach was a casualty rather than a beneficiary of (...) the renewed interest in modality. In this essay we present some of the main points of the Smith approach, in a new guise. (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)

We give a reading of binary necessity statements of the form “ϕ is necessary for ψ” in terms of proofs. This reading is based on the idea of interpreting such statements as “Every proof of ψ uses ϕ”.

Translator's summary The translated paper is an extract, published in 1945, of an unpublished thesis, of both historical and technical import, dealing with notions of definability and their relation to invariance under automorphisms. The author develops a metamathematical Galois theory, and discusses and anticipates some aspects of higher-order model theory in an informal but conceptually rich manner.

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)

The paper is a brief survey of the most important semantic constructions founded on the concept of possible world. It is impossible to capture in one short paper the whole variety of the problems connected with manifold applications of possible worlds. Hence, after a brief explanation of some philosophical matters I take a look at possible worlds from rather technical standpoint of logic and focus on the applications in formal semantics. In particular, I would like to focus on the fruitful (...) marriage of possible world semantics and algebra and its evolution leading to very general construction of Wójcicki called referential semantics and some of its refinements. The presentation is informal and sketchy; the main purpose is to put in one place a short, and readable I hope, description of the most important constructions and to point out the main sources of these solutions. (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)

In 1880, when Oliver Wendell Holmes (later to be a Justice of the U.S. Supreme Court) criticized the logical theology of law articulated by Christopher Columbus Langdell (the first Dean of Harvard Law School), neither Holmes nor Langdell was aware of the revolution in logic that had begun, the year before, with Frege's Begriffsschrift. But there is an important element of truth in Holmes's insistence that a legal system cannot be adequately understood as a system of axioms and corollaries; and (...) this element of truth is not obviated by the more powerful logical techniques that are now available. (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)

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)

I argue that recent defenses of the view that in 1936 Tarski required all interpretations of a language to share one same domain of quantification are based on misinterpretations of Tarski’s texts. In particular, I rebut some criticisms of my earlier attack on the fixed-domain exegesis and I offer a more detailed report of the textual evidence on the issue than in my earlier work. I also offer new considerations on subsisting issues of interpretation concerning Tarski’s views on the logical (...) correctness of certain omega-arguments, on the Tarskian proof that Etchemendy took to be modal and fallacious, and on Tarski’s appeals to the “common concept of consequence”. (shrink)