Robert Fogelin claimed there was an error in the logic of the Tractatus. I first cover his point here before going on to show that any error in this area derived from an even more fundamental one. Correcting that further error, moreover, does more than correct the logic of the Tractatus : it has repercussions for the metaphysics and theory of value found there, in line with later developments in Wittgenstein’s philosophy. In what follows I use the Tractarian numbers to (...) indicate the paragraphs spoken about. (shrink)

It is shown that there are categorical differences between sentences and statements, which have the consequence in particular that there are no paradoxical cases of self-reference with the latter as there are with the former. The point corrects an extensive train of thought that Graham Priest has pursued over recent years, but also a much wider tradition in logic and the foundations of mathematics that has been dominant for over a century. That tradition might be broadly characterized as Formalist, or (...) Nominalist, and the improved understanding of statements leads us instead into a more Realist approach and thereby contentful logic and mathematics. (shrink)

Modern Logic has generated a lot of problems for itself through inattention to natural forms of speech. In particular it has had difficulties with a large group of ‘logical paradoxes’ through its preoccupation with the Predicate Calculus and related structures to the exclusion of other formal structures that represent natural language more fully, and thereby escape these paradoxes. In natural speech the unrecognized forms involved are principally individual referring terms with a non-specific or fictional reference. For, under the influence of (...) the Logical Empiricists, the focus in formal logic has been on individual terms with a specific, factual reference. But a similar point arises with certain predicates, and indeed even the proper concept of a predicate has been lost, producing the most notorious of the standard puzzles: Russell's Paradox. The paper inevitably makes reference to contemporary logical symbolism in places, but its purpose is to show that the only way forward is to use a symbolism that is abbreviatory only, i.e. maps directly onto natural speech. (shrink)

Despite the volume of discussion on the Liar Paradox recently, there is one stream of largely British thought on the matter which is hardly represented in the wider literature. This paper points out salient aspects of the history of this tradition, from its origin in forms of propositional quantification found in Ramsey, through to more precise symbolisations which have emerged more recently. But its purpose is to exposit, with respect to a number of contested cases, the ensuing results. Thus it (...) goes on to apply the analysis to several other well known paradoxes, including one rarely discussed, which reveals more fully the consequent consistency and completeness of natural language. (shrink)

Such a misconception of grammar characterises a very popular approach to indexicality which has been current since the 1970s, stemming from the work of Casteñeda, and Kaplan. Gareth Evans was inclined to allow, for instance, that one could say ‘“To the left (I am hot)” is true, as uttered by x at t iff there is someone moderately near to the left of x such that, if he were to utter the sentence “I am hot” at t, what he would (...) thereby say is true’ (Evans 1985: 358). But not only does this disturb the proper relation between direct and indirect speech, it continues a Fregean tradition which these very cases show to be quite mistaken about the logic of intensions. (shrink)

There were already confusions in the Middle Ages with the reading of Aristotle on negative terms, and removing these confusions shows that the four traditional Syllogistic forms of statement can be readily generalised not only to handle polyadic relations (for long a source of difficulty), but even other, more measured quantifiers than just ‘all’, ‘some’, and ‘no’. But these historic confusions merely supplement the main confusions, which arose in more modern times, regarding the logic of singular statements. These main confusions (...) originate in the inability of the mainline modern tradition to supply the ‘logically proper names’ which alone have the right to replace individual variables; an inability which has resulted in the widespread, but erroneous replacement of individual variables with ordinary proper names, i.e. names for contingent beings, in many if not most contemporary logic texts. The paper includes the exhibition and grammatical characterisation of the logically proper names that are required instead, specifying just how they differ syntactically from ordinary proper names. It also shows how ontologically significant is the distinction, since not only do logically proper names refer to necessarily existent objects (showing there are no ‘empty domains’ for Classical Logic to fail to apply to), but also thereby central features of Realism become considerably clarified. (shrink)

It is clear that Tarski's T‐scheme, ‘T”p” if and only if p’, does not hold universally. It is not expected to hold, for instance, with ambiguous sentences, or with indexical sentences. Making explicit the circumstances where it does hold, however, is not just logical housekeeping: it turns out to have more radical consequences for the whole approach to Semantics associated with Tarski than has previously been envisaged.

Les arguments de Maddy avancés en 1990 contre la théorie des agrégats se trouvent affaiblis par le retournement qu’elle opère en 1997. La présente communication examine cette théorie à la lumière de ce retournement ainsi que des récentes recherches sur les “Nouveaux axiomes pour les mathématiques”. Si la théorie des ensembles est la théorie de la partie–tout des singletons, identifier les singletons à leurs membres singuliers ramène la théorie des ensembles à la théorie des agrégats. Toutefois si les singletons ne (...) sont pas identiques à leurs membres singuliers, ce ne sont donc pas des objets extensionnels et ils ne peuvent former une base pour la Science. Dans tous les cas, l’hypothèse d’un continuum n’a aucun intérêt sur le plan physique. Je montre d’abord que, parce qu’il y a des prédicats non-sortaux, la tentative de Frege de faire dériver l’arithmétique de la logique bute dès ses premiers pas. Car il y a des propriétés sans nombre et la contingence de cette condition signifie que la définition du zéro donnée par Frege ne peut s’obtenir à partir de la logique. Ce résultat révèle le besoin de considérer davantage les propriétés sans nombre et donc de générer une théorie des continuums en se basant sur les agrégats méréologiques plutôt que sur des ensembles contenant des nombres de choses. (shrink)

There is a little known paradox the solution to which is a guide to a much more thoroughgoing solution to a whole range of classic paradoxes. This is shown in this paper with respect to Berry's Paradox, Heterologicality, Russell's Paradox, and the Paradox of Predication, also the Liar and the Strengthened Liar, using primarily the epsilon calculus. The solutions, however, show not only that the first-order predicate calculus derived from Frege is inadequate as a basis for a clear science, and (...) should be replaced with Hilbert and Bernays' conservative extension. Standard second-order logic, and quantified propositional logic also must be substantially modified, to incorporate, in the first place, nominalizations of predicates, and whole sentences. And further modifications must be made, so as to insist that predicates are parts of sentences rather than forms of them, and that truth is a property of propositions rather than their sentential expressions. In all, a thorough reworking of what has been called 'logic' in recent years must be undertaken, to make it more fit for use. (shrink)

The formal structure of Frege’s ‘concept script’ has been widely adopted in logic text books since his time, even though its rather elaborate symbols have been abandoned for more convenient ones. But there are major difficulties with its formalisation of pronouns, predicates, and propositions, which infect the whole of the tradition which has followed Frege. It is shown first in this paper that these difficulties are what has led to many of the most notable paradoxes associated with this tradition; the (...) paper then goes on to indicate the lines on which formal logic—and also the lambda calculus and set theory—needs to be restructured, to remove the difficulties. Throughout the study of what have come to be known as first-, second-, and higher-order languages, what has been primarily overlooked is that these languages are abstractions. Many well known paradoxes, we shall see, arose because of the elementary level of simplification which has been involved in the abstract languages studied. Straightforward resolutions of the paradoxes immediately appear merely through attention to languages of greater sophistication, notably natural language, of course. The basic problem has been exclusive attention to a theory in place of what it is a theory of, leading to a focus on mathematical manipulation, which ‘brackets off’ any natural language reading. (shrink)

Maddy's (1990) arguments against Aggregate Theory were undermined by the shift in her position in 1997. The present paper considers Aggregate Theory in the light of this, and the recent search for `New Axioms for Mathematics'. If Set Theory is the part-whole theory of singletons, then identifying singletons with their single members collapses Set Theory into Aggregate Theory. But if singletons are not identical to their single members, then they are not extensional objects and so are not a basis for (...) Science. Either way, the Continuum Hypothesis has no physical interest. (shrink)

A logic focusing on the analytic a priori and explicitly rejecting the synthetic a priori developed in the early decades of the 20th century, largely through the efforts of the Logical Empiricists. This group was very influenced by Wittgenstein's early work Tractatus Logico-Philosophicus. But Wittgenstein himself, later on, departed from the Tractatus in significant ways that the Logical Empiricists did not follow. Wittgenstein came later to accept the synthetic a priori, and out of this insight comes a non-analytic logic that (...) differs from standard 20th century logic in many distinct ways. This paper details these differences. (shrink)

I first show in this paper how twentieth century Set Theory got into its greatest tangle by, amongst other things, regarding relational remarks like ‘Rxy’ asbinary functions. I then show how the lack of indexicality, and of ‘that’-clauses, in Modern Logic led that subject into its intractable difficulties with the Theory of Truth. Both errors arose not only through a contempt for ordinary language, but also through the related failure to recognise that being logical is not a matter of being (...) brainy, but of being coherent. It is not a mathematical talent, but a literary one. Later in the paper I go on to demonstrate this same conclusion with respect to Modal Logic and General Intensional Logic, and in particular with respect to fictions, since these are the central items that have been misunderstood, as is witnessed in some recent writings of Graham Priest. (shrink)

This paper is concerned with locating the specific assumption that led Frege into Russell's Paradox. His understanding of reflexive pronouns was weak, for one thing, but also, by assimilating concepts to functions he was misled into thinking one could invariably replace a two-place relation with a one-place property. /// Este trabajo se ocupa de localizar el supuesto específico que llevó a Frege a la Paradoja de Russell. Por una parte, su comprensión de los pronombres reflexivos era débil pero, por otra, (...) al asimilar los conceptos a las funciones pensó equivocadamente que siempre podíamos reemplazar una relación diádica con una propiedad monádica. (shrink)

Les arguments de Maddy avancés en 1990 contre la théorie des agrégats se trouvent affaiblis par le retournement qu’elle opère en 1997. La présente communication examine cette théorie à la lumière de ce retournement ainsi que des récentes recherches sur les “Nouveaux axiomes pour les mathématiques”. Si la théorie des ensembles est la théorie de la partie–tout des singletons, identifier les singletons à leurs membres singuliers ramène la théorie des ensembles à la théorie des agrégats. Toutefois si les singletons ne (...) sont pas identiques à leurs membres singuliers, ce ne sont donc pas des objets extensionnels et ils ne peuvent former une base pour la Science. Dans tous les cas, l’hypothèse d’un continuum n’a aucun intérêt sur le plan physique. Je montre d’abord que, parce qu’il y a des prédicats non-sortaux, la tentative de Frege de faire dériver l’arithmétique de la logique bute dès ses premiers pas. Car il y a des propriétés sans nombre et la contingence de cette condition signifie que la définition du zéro donnée par Frege ne peut s’obtenir à partir de la logique. Ce résultat révèle le besoin de considérer davantage les propriétés sans nombre et donc de générer une théorie des continuums en se basant sur les agrégats méréologiques plutôt que sur des ensembles contenant des nombres de choses. (shrink)

The epsilon calculus improves upon the predicate calculus by systematically providing complete individual terms. Recent research has shown that epsilon terms are therefore the “logically proper names” Russell was not able to formalize, but their use improves upon Russell’s theory of descriptions not just in that way. This paper details relevant formal aspects of the epsilon calculus before tracing its extensive application not just to the theory of descriptions, but also to more general problems with anaphoric reference. It ends by (...) contrasting a Meinongian account of cross-reference in intensional constructions with the epsilon account. (shrink)

Epsilon Calculi are extended forms of the predicate calculus that incorporate epsilon terms. Epsilon terms are individual terms of the form ‘εxFx’, being defined for all predicates in the language. The epsilon term ‘εxFx’ denotes a chosen F, if there are any F’s, and has an arbitrary reference otherwise. Epsilon calculi were originally developed to study certain forms of Arithmetic, and Set Theory; also to prove some important meta-theorems about the predicate calculus. Later formal developments have included a variety of (...) intensional epsilon calculi, of use in the study of necessity, and more general intensional notions, like belief. An epsilon term such as ‘ εxFx’ was originally read ‘the first F’, and in arithmetical contexts ‘the least F’. More generally it can be read as the demonstrative description ‘that F’, when arising either deictically, i.e. in a pragmatic context where some F is being pointed at, or in linguistic cross-reference situations, as with, for example, ‘There is a red haired man in the room. That red haired man is Caucasian’. The application of epsilon terms to natural language shares some features with the use of iota terms within the theory of descriptions given by Bertrand Russell, but differs in formalising aspects of a slightly different theory of reference, first given by Keith Donnellan. More recently epsilon terms have been used by a number of writers to formalise cross-sentential anaphora, which would arise if ‘that red haired man’ in the linguistic case above was replaced with a pronoun such as ‘he’. There is then also the similar application in intensional cases, like ‘There is a red haired man in the room. Celia believed he was a woman.’. (shrink)

Francesco Berto has recently written “The Gödel Paradox and Wittgenstein's Reasons,” about a paradox first formulated by Graham Priest in 1971. The major reason for disagreeing with Berto's conclusions concerns his elucidation of Wittgenstein's understanding of Gödel's theorems. Seemingly, Wittgenstein was some kind of proto-paraconsistentist. Priest himself has also, though in a different way, tried to tar Wittgenstein with the same brush. But the resolution of other paradoxes is intimately linked with the resolution of the Gödel Paradox, and with understanding (...) Wittgenstein's views of Gödel's theorems. So this paper discusses some other paradoxes before looking at Wittgenstein's relevant views. (shrink)

Prawitz proved a theorem, formalising 'harmony' in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can (...) extend the theorem to the classical case. Properly arranged there is a thoroughgoing 'harmony', in the classical rules. Indeed, as we shall see, they are, all together, far more 'harmonious' in the general sense than has been commonly observed. As this paper will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented. (shrink)

Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can (...) extend the theorem to the classical case. Properly arranged there is a thoroughgoing ‘harmony’, in the classical rules. Indeed, as we shall see, they are, all together, far more ‘harmonious’ in the general sense than has been commonly observed. As this paper will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented. (shrink)

Since there are non-sortal predicates Frege’s attempt to derive Arithmetic from Logic stumbles at its very first step. There are properties without a number, so the contingency of that condition shows Frege’s definition of zero is not obtainable from Logic. But Frege made a crucial mistake about concepts more generally which must be remedied, before we can be clear about those specific concepts which are numbers.

I have written a number of articles recently that have a rather remarkable character. They all point out trivial grammatical facts that, at great cost, have not been respected in twentieth century Logic. A major continuous strand in my previous work, with this same character, I will first summarise, to locate the kind of fact that is involved. But then I shall present an overview of the more recent, and more varied points I have made, which demonstrate the far larger (...) extent of basic grammar that has been overlooked or suppressed. I end with some remarks about how this phenomenon can have arisen – principally through logicians not being attentive enough to their own language, and occupying themselves, instead, with often quite imaginary languages. (shrink)

A paradox is generally a puzzling conclusion we seem to be driven towards by our reasoning, but which is highly counterintuitive, nevertheless. There are, amongst these, a large variety of paradoxes of a logical nature which have teased even professional logicians, in some cases for several millennia. But what are now sometimes isolated as 'the logical paradoxes' are a much less heterogeneous collection: they are a group of antinomies centered on the notion of self-reference, some of which were known in (...) Classical times, but most of which became particularly prominent in the early decades of last century. Quine distinguished amongst paradoxes such antinomies. He did so by first isolating the 'veridical' and 'falsidical' paradoxes, which, although puzzling riddles, turned out to be plainly true, or plainly false, after some inspection. In addition, however, there were paradoxes which 'produce a self-contradiction by accepted ways of reasoning', and which, Quine thought, established 'that some tacit and trusted pattern of reasoning must be made explicit, and henceforward be avoided or revised' (Quine 1966, p7). We will first look, more broadly, and historically, at several of the main conundrums of a logical nature which have proved difficult, some since antiquity, before concentrating later on the more recent troubles with paradoxes of self-reference. They will all be called 'logical paradoxes'. (shrink)

I here recall Ryle's analysis of Heterologicality, but broaden the discussion to comparable analyses not only of Heterologicality but also other puzzles about self-reference. Such matters have a crucial bearing on the debate between representational and non-representational theories of mind, as will be explained.

Russell held that ‘a exists’, where ‘a’ is a logically proper name, was necessarily true. By contrast his account of ‘The K exists’ allowed this to be contingent, since, on his Theory of Descriptions, it did not assert the existence of an individual, but merely the instantiation of some uniquely identifying properties. The present paper refines Russell’s distinction in several ways, first by providing what Russell merely gestured at, namely explicit, formally defined logically proper names. But following from this it (...) is seen that Russell’s intention with regard to ‘The K exists’ is better expressed ‘A unique K exists’, leaving the former to be assimilated into the non-contingent category, through interpreting its subject phrase ‘The K’ nonattributively. The paper closes with an exhibition of similar discriminations that are available with higher-order subjects, such as properties, numbers, and facts. (shrink)

There are some seemingly small points to be made, first of all, about usemention confusions in Stephen Read’s paper ‘The Truth Schema and the Liar’. But underlying them is a grammatical point that has much wider repercussions. For it generates, on its own, a more straightforward way of understanding what gets people into a tangle with Liar and Strengthened Liar sentences, and that leads to a much fuller, critical assessment of the line of approach to these matters that Read derives (...) from Bradwardine. (shrink)

Tarski’s assessment that natural language is inconsistent on account of the Liar Paradox is shown to be incorrect: what Tarski’s theorem in fact shows is that Truth is not a property of sentences but of propositions. By using propositions rather than sentences as the bearers of Truth, semantic closure within the same language is easily obtained. Tarski’s contrary assessment was partly based on confusions about propositions and their grammatical expression. But more centrally it arose through blindness to pragmatic factors in (...) language — a blindness that was common in his time, and it has continued to the present day, in discussions of ‘Open Pairs’, and Yablo-type paradoxes, for instance. For completeness, it is also shown that the Fixed Point Theorem does not apply to propositions, because of categorical differences between sentences and propositions — also predicates and properties. (shrink)

Criticisms have been aired before about the fear of certain Platonic abstract objects, propositions. That criticism extends to the widespread preference for an operator analysis of expressions like ‘It is true, known, obligatory that p’ as opposed to the predicative analysis in their equivalents ‘That p is true, known, obligatory’. The criticism in the present work also concerns Prior’s attitude to Platonic entities of a certain kind: not propositions, i.e., the referents of ‘that’-clauses, but individuals, i.e., the referents of Russell’s (...) ‘logically proper names’. Prior had a close knowledge of theories of individuals, including that of Russell. However, when Prior formulated his tense logic Q he presumed a non-Russellian, pre-suppositional account of individuals, such that they might exist at one time but not at another, and when they did not exist no statement could be made about them. This paper examines the correctness of such a pre-suppositional account of individuals, showing there are features of natural language it cannot account for. It is shown that the required features can be formalised in Hilbert’s epsilon calculus, and that using that calculus substantiates a Russellian understanding of ‘logically proper names’. (shrink)

On montre que, avec la notion traditionnelle et grammaticale du prédicat comme ce qui reste de la phrase après l’enlèvement du sujet, le paradoxe de Russell, ou d’autres comparables comme le paradoxe de Grelling et le paradoxe de la prédication, ne posent aucun problème. L’interdit formel standard sur la substitution des prédicats impliquant des variables libres dans des schémas où ces variables deviendraient liées, suffit pour prévenir le développement des paradoxes standard. On discute ensuite des réarrangements requis dans les fondations (...) de la Théorie des ensembles pour intégrer cette idée, et on explique les conséquences pour les questions étroitement liées de la Diagonalisation et du Théorème de Cantor. (shrink)

On montre que, avec la notion traditionnelle et grammaticale du prédicat comme ce qui reste de la phrase après l’enlèvement du sujet, le paradoxe de Russell, ou d’autres comparables comme le paradoxe de Grelling et le paradoxe de la prédication, ne posent aucun problème. L’interdit formel standard sur la substitution des prédicats impliquant des variables libres dans des schémas où ces variables deviendraient liées, suffit pour prévenir le développement des paradoxes standard. On discute ensuite des réarrangements requis dans les fondations (...) de la Théorie des ensembles pour intégrer cette idée, et on explique les conséquences pour les questions étroitement liées de la Diagonalisation et du Théorème de Cantor. (shrink)

It is shown that the Nominalism of much of Modern Logic is what has given rise to many of its problems, especially The Liar Paradox. Shifting to a Realist Logic, in which ‘that’-clauses have a central place, overcomes these problems. The move involved, from the study of mentioned sentences to the employment of ‘that’-clauses, reveals the indexicality of referring phrases, and it is that which enables an escape from The Liar. But also it is shown that no parallel paradox is (...) obtainable with the propositions that ‘that’-clauses designate. Several theoretical issues, however, then need to be faced about what ‘that’-clauses designate, primarily what has been called the problem of ‘The Unity of the Proposition’. The term ‘proposition’, it turns out, is not always appropriate for what ‘that’-clauses designate, so a richer language is introduced to handle all cases. But the problem of the unity of what is designated arises in all cases, and is solved by noting a parallel with Russell’s treatment of individuals and descriptive phrases. (shrink)