This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much more, (...) I argue. "If you love me, kiss me", a conditional imperative, mixes a declarative antecedent ("you love me") with an imperative consequent ("kiss me"); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three -valued imperatives which mixes declaratives with imperatives. I develop such a logic. (shrink)

The mereological predicate ‘is part of’ can be used to define the predicate ‘is identical with’. I argue that this entails that mereological theories can be ideologically simpler than nihilistic theories that do not use the notion of parthood—contrary to what has been argued by Ted Sider. Moreover, if one accepts an extensional mereology, there are good philosophical reasons apart from ideological simplicity to give a mereological definition of identity.

The text of Sophist 237-9 is aporetic and shares with many other dialogues this structure: A question is asked and an answer, given in a single sentence, is reached and accepted by the interlocutor. The the interlocutor is examined further and his assent undermined. I argue that the Stranger does not share Theaetetus' perplexity and holds the rejected answer. I explain the Stranger's behavior by appealing to his pedagogy.

Il y a un dilemme de la philosophie scientifique, limitée ici pour simplifier à la philosophie scientifique des sciences, assez évident dans lequel Carnap se trouve pris, mais auquel ni lui ni, sauf erreur de ma part, ses successeurs et commentateurs n’ont cru devoir s’arrêter : Ou bien la philosophie en question est scientifique au même sens que les autres sciences, c’est-à-dire ici qu’elle est elle-même l’une des sciences qu’elle a pour objets. On se demande alors à quoi pourrait ressembler (...) plus précisément une telle philosophie scientifique, à supposer qu’elle fût possible. Ou bien la philosophie en question n’est pas scientifique au sens des autres sciences, c’est-à-dire ici qu’elle est la seule qui puisse les prendre toutes pour objets, mais la seule aussi qu’elle ne puisse prendre pour objet. La prétendue philosophie scientifique des sciences n’est alors pas la philosophie de toutes les sciences, c’est seulement une philo­sophie des sciences non philosophiques. Ce dilemme vaut pour la philosophie scientifique en général et pour celle de Carnap en particulier, pour laquelle il prend une forme assez tranchante pour qu’on puisse en discuter sérieusement, surtout si l’on s’en tient à la philosophie scientifique des sciences identifiée par Carnap, à l’époque de la Syntax, à l’étude syntaxique, ou syntaxe, du Langage de la science. La réponse négative, savante et incontestée qu’il conviendrait alors d’apporter à la question cruciale de la première branche est connue : non, une syntaxe adéquate du Langage de la science dans ce Langage même est impossible, à moins que celui-ci ne soit inconsistant. Dans le présent article, on propose une formulation et une esquisse de démonstration inhabituelles de cette impossibilité, ne mobilisant aucun codage, gödélien ou autre, de la syntaxe élémentaire du Langage de la science dans ce Langage même ; et l’on remet en cause le critère d’adéquation impliqué dans la démonstration de l’impossibilité en question, rouvrant ainsi la possibilité de sortir par le haut d’un dilemme dont Carnap s’est cru obligé, ou du moins s’est conduit comme s’il était obligé, de sortir par le bas. (shrink)

Il y a un dilemme de la philosophie scientifique, limitée ici pour simplifier à la philosophie scientifique des sciences, assez évident dans lequel Carnap se trouve pris, mais auquel ni lui ni, sauf erreur de ma part, ses successeurs et commentateurs n’ont cru devoir s’arrêter : Ou bien la philosophie en question est scientifique au même sens que les autres sciences, c’est-à-dire ici qu’elle est elle-même l’une des sciences qu’elle a pour objets. On se demande alors à quoi pourrait ressembler (...) plus précisément une telle philosophie scientifique, à supposer qu’elle fût possible. Ou bien la philosophie en question n’est pas scientifique au sens des autres sciences, c’est-à-dire ici qu’elle est la seule qui puisse les prendre toutes pour objets, mais la seule aussi qu’elle ne puisse prendre pour objet. La prétendue philosophie scientifique des sciences n’est alors pas la philosophie de toutes les sciences, c’est seulement une philo­sophie des sciences non philosophiques. Ce dilemme vaut pour la philosophie scientifique en général et pour celle de Carnap en particulier, pour laquelle il prend une forme assez tranchante pour qu’on puisse en discuter sérieusement, surtout si l’on s’en tient à la philosophie scientifique des sciences identifiée par Carnap, à l’époque de la Syntax, à l’étude syntaxique, ou syntaxe, du Langage de la science. La réponse négative, savante et incontestée qu’il conviendrait alors d’apporter à la question cruciale de la première branche est connue : non, une syntaxe adéquate du Langage de la science dans ce Langage même est impossible, à moins que celui-ci ne soit inconsistant. Dans le présent article, on propose une formulation et une esquisse de démonstration inhabituelles de cette impossibilité, ne mobilisant aucun codage, gödélien ou autre, de la syntaxe élémentaire du Langage de la science dans ce Langage même ; et l’on remet en cause le critère d’adéquation impliqué dans la démonstration de l’impossibilité en question, rouvrant ainsi la possibilité de sortir par le haut (en restant sur la première branche) d’un dilemme dont Carnap s’est cru obligé, ou du moins s’est conduit comme s’il était obligé, de sortir par le bas (en se repliant sur la seconde). (shrink)

Carnap's system of inductive logic has very often been criticized on the ground that “degree of confirmation” is defined only for languages which are extremely over-simplified. Allegedly, it would be very difficult—and perhaps impossible—to define it adequately for languages formalized within the higher predicate calculi, or languages equivalent to these in richness, and it is such languages that would be needed were we ever to formalize the language of empirical science as a whole. Thus, this criticism bears not only on (...) Carnap's work, but on all attempts to construct an exact analysis of induction from this particular standpoint; that is, from the standpoint of “logical measure functions.”. (shrink)

In this paper I revisit R. M. Martin’s logic of belief. As with much of Martin’s work, his formal studies into belief and belief reports have gone largely unnoticed. However, in my article I suggest reasons for thinking that these studies warrant revisiting. One reason is that Martin adopted an account of the notion of belief which was more comprehensive than that employed by most rival theorists. Another reason is that Martin couched his theory in a formal pragmatics which utilised (...) a pragmatical meta-language. This method resulted in a rather novel approach which may turn out to be more enlightening than many popular alternative accounts. Martin was a constructivistic nominalist whose approach to philosophical analysis was characterised by a commitment to first-order extensional systems. Yet in the statement of his theory of belief he employed a platonistic syntax simply for the reason that this kind of syntax is a little easier to implement than one which is nominalistic. However Martin also supposed that a fully nominalised version of his theory could be developed. In my article I investigate whether it can. I do this by sketching an inscriptional pragmatics, where this pragmatical system presupposes an inscriptional semantics of the kind developed by Martin himself. While further work is required, I think that a nominalisation of Martin’s logic of belief may be developed. (shrink)

Peter Geach's puzzle of intentional identity is to explain how the claim ‘Hob thinks a witch has blighted Bob's mare, and Nob wonders whether she killed Cob's sow’ is compatible with there being no such witch. I clarify the puzzle and reduce it to the familiar problem of negative existentials. That problem is a paradox of representations that seem to include denials of commitment , to carry commitment to what they deny commitment to, and to be true. The best proposed (...) solutions can be understood through this paradox; I evaluate them, and defend a new solution. (shrink)

Jim Mackenzie; Authority, Journal of Philosophy of Education, Volume 22, Issue 1, 30 May 2006, Pages 57–65, https://doi.org/10.1111/j.1467-9752.1988.tb00177.x.

In Reason and Analysis, Prof. Brand Blanshard criticises the logical empiricist view regarding necessary statements, including the laws of logic. He distinguishes four component these of this view: (I) Necessary statements--here, the laws of logic--are resolutions or reports of linguistic usage. (II) They are conventions. (III) They are analytic (tautologies). (IV) They say nothing about the world. In this paper I first show that Prof. Blanshard is essentially right in his criticisms of (I), (II), and (IV); but that he has (...) done nothing to show that (III) is incorrect with regard to the laws of logic. Then, second, I attempt to show that even though (I), (II), and (more importantly) (IV) are false, (III) is true. That is, the laws of logic are tautologies, and yet, in a sense to be distinguished, they say something about the world. (shrink)

The fact that Gödel's famous incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of the community of logicians. Indeed, almost every more or less formal treatment of the theorem makes a reference to this connection. Gödel himself remarked in the paper announcing his celebrated result :The analogy between this result and Richard's antinomy leaps to the eye;there is (...) also a close relationship with the ‘liar’ antinomy, since … we are… confronted with a proposition which asserts its own unprovability.In the light of the fact that the existence of this connection is commonplace it is all the more surprising that very little can be learnt about its exact nature except perhaps that it is some kind of similarity or analogy. There is, however, a lot more to it than that. Indeed, as we shall try to show below, the general ideas underlying the three central theorems concerning internal limitations of formal deductive systems can be taken as different ways to resolve the Liar paradox. More precisely, it will turn out that an abstract formal variant of the Liar paradox, which can almost straightforwardly inferred from its original ordinary language version, is a possible common generalization of Gödel's incompleteness theorem, the theorem of Tarski on the undefinability of truth, and that of Church concerning the undecidability of provability. (shrink)

The comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom (...) scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences. (shrink)

Theories of propositions as sets of truth-supporting circumstances are committed to the thesis that sentences or other representations true in all and only the same circumstances express the same proposition. Theories of propositions as complex, structured entities are not committed to this thesis. As a result, structured propositions can play a role in our theories of language and thought that sets of truth-supporting circumstances cannot play. To illustrate this difference, I sketch a theory of transparent, non-deflationary truth consistent with some (...) theories of structured propositions, but inconsistent with any theory of propositions as sets of truth-supporting circumstances. (shrink)

Building on recent work by Delia Graff Fara and Ora Matushansky on appellative constructions like ‘Mirka called Roger handsome’, I argue that if Millianism about proper names is true, then the quantifier ‘something’ in ‘Mirka called Roger something’ is best understood as a kind of substitutional quantifier. Any adequate semantics for such quantifiers must explain both the logical behavior of ‘Mirka called Roger something’ and the acceptability of ‘so’-anaphora in ‘Mirka called Roger something, and everyone so called is handsome’. Millianism (...) about proper names is inconsistent with such quantifiers being standard non-substitutional second-order quantifiers. But this is not the only option for Millianism: I provide two different propositional semantics for substitutional quantification, each of which is adequate in the sense above, given Millianism. One of these is based on Tobias Rosefeldt’s work on non-nominal quantification, and I identify in what way Rosefeldt’s semantics is substitutional. (shrink)

Pour le philosophe intéressé aux structures et aux fondements du savoir théorétique, à la constitution d'une « méta-théorétique «, θεωρíα., qui, mieux que les « Wissenschaftslehre » fichtéenne ou husserlienne et par-delà les débris de la métaphysique, veut dans une intention nouvelle faire la synthèse du « théorétique », la logique mathématique se révèle un objet privilégié.

Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie...

It is often assumed that two linguistic agents can come to understand one another in part because they use the same words. That is, many philosophical theories of communication posit an intersubjective same-word relation. However, giving an account of this relation is complicated by what I call “The Variation Problem”—a problem resulting from the fact that the same word can be pronounced differently. In this paper, I first argue that previous models of the same-word relation, including Kaplanian and Chomskyan models, (...) fail to escape The Variation Problem. I then propose a new model on which the same-word relation is grounded in a particular kind of social relation that holds between the speaker and the audience. On this model, using the same word requires not that agents make the same sounds, but that they coordinate their internal linguistic representations. (shrink)

Magidor argued that category mistakes are infelicitous due to presupposition failure. The case for this position is strengthened by the consideration of a previously unnoted category of data, namely multi-sentence discourses in which category mistake phenomenology arises at the end of the last sentence, but arguably due to content contained in a previous sentence. This phenomenon is analysed in terms of the previous sentence giving rise to a presupposition that is shown to be false only in the last sentence.

I want to discuss, in some detail, a short section from Quine’s Philosophy of Logic. It runs from pages 10 to 13 of the second, revised edition of the book and carries the subheading ‘Truth and semantic ascent’.1 In these two and a half pages, Quine presents his well-known account of truth as a device of disquotation, employing what I call Quine’s Ladder. The section merits scrutiny, for it has become the central document for contemporary deflationary views about truth.

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s Induction-Axiom (...) Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

In a number of essays published over the last decade or so, W. V. Quine has made some interesting suggestions concerning the ontology of theories. If I understand him correctly, one of his principal objects has been to formulate a criterion by means of which one can correctly decide what are the ontological commitments of any given theory. My aim in this paper is to reveal what I think are inadequacies in Quine's criterion and to indicate the direction in which (...) a proper formulation lies. (shrink)

Philosophical dialetheism, whose main exponent is Graham Priest, claims that some contradictions hold, are true, and it is rational to accept and assert them. Such a position is naturally portrayed as a challenge to the Law of Non-Contradiction (LNC). But all the classic formulations of the LNC are, in a sense, not questioned by a typical dialetheist, since she is (cheerfully) required to accept them by her own theory. The goal of this paper is to develop a formulation of the (...) Law which appears to be unquestionable, in the sense that the Priestian dialetheist is committed to accept it without also accepting something inconsistent with it, on pain of trivialism—that is to say, on pain of lapsing into the position according to which everything is the case. This will be achieved via (a) a discussion of Priest's dialetheic treatment of the notions of rejection and denial; and (b) the characterization of a negation via the primitive intuition of content exclusion. Such a result will not constitute a cheap victory for the friends of consistency. We may just learn that different things have been historically conflated under the label of 'Law of Non-Contradiction'; that dialetheists rightly attack some formulations of the Law, and orthodox logicians and philosophers have been mistaken in assimilating them to the indisputable one. (shrink)

Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”—making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an (...) operator allowing it, against the opinion of many critics, to rule things out and manifest disagreement: for unlike other proposed exclusion-expressing-devices (for instance, the entailment of triviality), the operator used to formulate the notion of absolute contradiction appears to be immune both from crippling expressive limitations and from revenge paradoxes—pending a rigorous nontriviality proof for a formal dialetheic theory including it. (shrink)

We are pleased to release this edition of Ralph Johnson’s The Rise of Informal Logic as Volume 2 in the series Windsor Studies in Argumentation. This edition is a reprint of the previous Vale Press edition with some typographical errors and other minor mistakes corrected. The prime motive for gathering Ralph H. Johnson’s essays under one cover is their clear articulation of the goals, concerns and problems of the discipline of informal logic. To my knowledge all of the published articles, (...) even of the 1980s, are still in print. But some are obtainable only by special request of a journal back issue. Their availability, even their existence, is not nearly widely enough known, and this volume is dedicated to remedying that disservice to those currently working in the field of informal logic, critical thinking, argumentation, and practical reasoning. Three of these sixteen pieces appear here in print for the first time. The previously published pieces have appeared from 1980 to 1992 as chapters in collective works or as articles in journals, and these in turn published in Canada, USA, The Netherlands and Belgium. It is hoped that gathering this hitherto scattered material under one cover will contribute to a greater understanding of what informal logic is, and to an enhanced sense of the impact of Johnson’s ideas. A discipline of informal logic might exist today without the writings of Johnson and his frequent co-author, J. Anthony Blair. But it would almost certainly be quite different from what it actually is. (shrink)

Starting with Frege, the semantics (and pragmatics) of quotation has received a steady flow of attention over the last one hundred years. It has not, however, been subject to the same kind of intense debate and scrutiny as, for example, both the semantics of definite descriptions and propositional attitude verbs. Many philosophers probably share Davidson's experience: ‘When I was initiated into the mysteries of logic and semantics, quotation was usually introduced as a somewhat shady device, and the introduction was accompanied (...) by a stern sermon on the sin of confusing the use and mention of expressions’ (Davidson 1979, p. 79). Those who leave it at that, however, miss out on one of the most difficult and interesting topics in the philosophy of language. (shrink)

The strategy of this paper is twofold: First, we carry out a systematic investigation of the question of what specific kind of meaning quotation marks contribute to the overall meaning of an utterance. We consider the following kinds of meaning: literal meaning (§ 2.1), conventional implicature (§ 2.2), presupposition (§ 2.3), and conversational implicature (§ 2.4). We present arguments in favor of a pragmatic analysis of quotation marks, claiming that the notion of conversational implicature seems to be the most promising (...) alternative: All general features of this kind of meaning are met by quotational constructions. Nonetheless, an approach based on conversational implicatures faces some problems when taking direct and pure quotations into account, namely effects on truth-conditions and, allegedly, on grammaticality. Thus, our second aim is to propose acceptable solutions to these criticisms in § 3. Finally, in § 4, we consider how a radical pragmatic account of quotation could be integrated into a Neo-Gricean architecture of the semantics/pragmatics-interface. (shrink)

O presente artigo busca debater a posição de Charles Sanders Peirce e dos primeiros estudantes peirceanos de Lógica (Christine Ladd e O. H. Mitchell nos Studies in Logic, 1883) dentro do debate inspirador da visão da linguagem dentro da Filosofia Analítica, conhecido como “Lingua Universalis contra Calculus Ratiocinator”, cujos primórdios podem ser traçados desde a filosofia de Gottfried Leibniz. Para isso, comparamos esse campo do pensamento peirceano com o debate crucial entre a conceitografia de Gottlob Frege (Begriffsschrift, 1879) e a (...) lógica algébrica de George Boole (An investigation of the Laws of Thought on which are founded the mathematical theories of Logic and Probabilities, 1854). O nosso objetivo principal é observar que esse momento da filosofia peirceana pode ser comparado com o pensamento wittgensteiniano, especialmente em sua nova vertente, chamada New Wittgenstein, que tenta superar a tradicional divisão entre Tractatus Logico-Philosophicus e Investigações Filosóficas. Pela comparação entre pensadores influenciados por C. S. Peirce e a filosofia de Ludwig Wittgenstein, pretendemos abrir caminhos na compreensão do conceito de jogo de linguagem, especialmente em sua gramática e em seus operadores simbólicos, através da observação das funções de verdade [truth functions] e as tabelas de verdade [truth tables] enquanto lógica algébrica. (shrink)