The logic of paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, (...) concluding that it will be extremely difficult to appeal to second-order LP for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made. (shrink)

The present paper concerns a technical study of PRIEST'S logic of paradox [Pri 79], We prove that this logic has no proper paraconsistent strengthening. It is also proved that the mentioned logic is the largest paraconsistent one satisfaying TARSKI'S conditions for the classical conjunction and disjunction together with DE MORGAN'S laws for negation. Finally, we obtain for the logic of paradox an algebraic completeness result related to Kleene lattices.

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

Counterexamples to reassurance relative to a relation between models of the logic of paradox are provided. Another relation, designed to fix the problem in logic without equality, is introduced and discussed in connection with the issue of classical recapture.

B. Kooi and A. Tamminga present a correspondence analysis for extensions of G. Priest’s logic of paradox. Each unary or binary extension is characterizable by a special operator and analyzable via a sound and complete natural deduction system. The present paper develops a sound and complete proof searching technique for the binary extensions of the logic of paradox.

In this article I should like briefly to ground two mutually determining propositions:1. The philosophical logic that has emerged in the twentieth century and that corresponds to contemporary culture is a logic of paradox.

In this paper, we aim to devise a logic that can deal with both the paradoxes that motivate dialetheism and the paradoxes related to the irrelevance of material implication. We propose the semantics and the sequent calculus of a relevant logic inspired by difference-making accounts of causation and arguably true to Graham Priest’s Logic of Paradox \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document}: a relevant logic that validates those and only those (...) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document}-consequences that are not irrelevant. The new logic’s relevant implication is then added to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document} to obtain a logic with the logical vocabulary required to cope with the mentioned paradoxes. (shrink)

Many have claimed that legitimate constitutional democracy is either conceptually or practically impossible, given infinite regress paradoxes deriving from the requirement of simultaneously democratic and constitutional origins for legitimate government. This paper first critically investigates prominent conceptual and practical bootstrapping objections advanced by Barnett and Michelman. It then argues that the real conceptual root of such bootstrapping objections is not any specific substantive account of legitimacy makers, such as consent or democratic endorsement, but a particular conception of the logic (...) of normative standards—the determinate threshold conception—that the critic attributes to the putatively undermined account of legitimacy. The paper further claims that when we abandon threshold conceptions of the logic of legitimacy in favor of regulative-ideal conceptions, then the objections, from bootstrapping paradoxes to the very idea of constitutional democracy, disappear. It concludes with considerations in favor of adopting a more demanding conception of the regulative ideal of constitutional democracy, advanced by Habermas, focusing on potentials for developmental learning. (shrink)

In this paper we explain that the paraconsistent logic LP promoted by Graham Priest can only be supported by trivial dialetheists, i.e., those who believe that all sentences are dialetheias.

The author once again offers a definition of paradox that fundamentally differs from a Hegelian notion of contradiction. In a paradox, thought should logically identify and comprehend the mental re...

In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of (...) paradox. In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule. (shrink)

Taking our inspiration from modal correspondence theory, we present the idea of correspondence analysis for many-valued logics. As a benchmark case, we study truth-functional extensions of the Logic of Paradox (LP). First, we characterize each of the possible truth table entries for unary and binary operators that could be added to LP by an inference scheme. Second, we define a class of natural deduction systems on the basis of these characterizing inference schemes and a natural deduction system for (...) LP. Third, we show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics. (shrink)

We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\varphi$ and $(\varphi\odot\psi)$ also satisfies $\psi$ , except trivially. We derive this as a corollary of a more general result concerning variable sharing.

In the first half of the fifteenth century, the Italian logician, natural philosopher, and doctor of medicine Cajetan of Thiene wrote a commentary on William Heytesbury’s Regulae solvendi sophismata, which later became a part of the printed edition of Heytesbury’s treatises. Several late fifteenth century reprints sustained its circulation and further influence. Following Heytesbury, Cajetan listed four alternative treatments of paradoxes, where the first three were formulated in general logico-semantic terms and the last one in terms of obligationes. The present (...) analysis reconstructs the first three positions in terms of the theories of logical operators endorsed as part of the solution to paradoxes. This reconstruction uncovers different underlying views of operators, namely context-sensitive, value-functional, and supervaluationist.Priore dimidia parte saeculi 15 Caietanus de Thiena, logicus, physicus et medicus, commentarium super G. Hentisberi Regulis solvendi sophismata conscripsit, quod posterius una cum Hentisberi tractatibus typis impressum est. Cuius commentarii notitiam auctoritatemque continuam iteratae nonnullae eius editiones in fi ne 15 saeculi factae sustinebant. Caietanus quattuor vias tractandi insolubilia distinxit, quarum tres primae conceptibus generalibus logico-semanticis, quarta doctrina de obligationibus innixae sunt. In analysi hic proposita auctor primas tres vias reconstruit, doctrinas varias de logicis coniunctionibus vel notis reserans, super quibus illae viae solvendi paradoxa fundantur. Quarum prima vim notarum a contextu sermonis dependentem facit. Altera notas pure “compositionaliter” tractat. Tertia iuxta modum doctrinae de “supervaluatione” omnes formales tautologias servat, aliis considerationibus neglectis. (shrink)

The main aim of this paper is to elucidate, from a logical point of view, the phenomenon of Simpson reversal — the paradox of a statistical reasoning. We define a binary relation of supporting in the following way: a sentence A supports a sentence B if and only if the probability of B is higher when A is true, than when A is false. It appears that a statistical argument occurring in Simpson paradox cannot be formalized by means (...) of a binary relation. We generalize the relation of support introducing the third parameter. Then we argue that it properly mirrors main features of the statistical argument occurring in Simpson paradox. (shrink)

There are three distinct questions associated with Simpson’s paradox. Why or in what sense is Simpson’s paradox a paradox? What is the proper analysis of the paradox? How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal account of the first two (...) questions. We argue that the “how to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”. (shrink)

This paper provides a defense of the full strength of classical logic, in a certain form, against those who would appeal to semantic paradox or vagueness in an argument for a weaker logic. I will not argue that these paradoxes are based on mistaken principles; the approach I recommend will extend a familiar formulation of classical logic by including a fully transparent truth predicate and fully tolerant vague predicates. It has been claimed that these principles are (...) not compatible with classical logic; I will argue, by both drawing on previous work and presenting new work in the same vein, that this is not so. We can combine classical logic with these intuitive principles, so long as we allow the result to be nontransitive. In the end, I hope the paper will help us to handle familiar paradoxes within classical logic; along the way, I hope to shed some light on what classical logic might be for. (shrink)

This paper presents a model for the structure of universal frameworks in logic, mathematics, and physics that are closed to logical conclusion by the mechanism of paradox across a dualism of elements. The prohibition takes different forms defined by the framework of observation inherent to the structure. Forms include either prohibition to conclusion on the logical relationship of internal elements or prohibition to conclusion based on the existence of an element not included in the framework of a first (...) element. The model is applied to logical arguments in philosophy, mathematics, and physics and is initially a geometrical analysis of quantum theory and its application in experiment. Conclusion from the analysis is extended to give insight into the complexity of infinities above the two-dimensional boundary of the model. (shrink)

Classical logic, of first or higher order, is extended with sentential operators and quantifiers, interpreted substitutionally over unrestricted substitution class. Operators mark a single layered, consistent metalanguage. Self-reference, arising from substitutional quantification over sentences, allows to express paradoxes which, unlike contradictions, do not lead to explosion. Semantics of the resulting language, using semi-kernels of digraphs, is non-explosive yet two-valued and has classical semantics as a special case for clasically consistent theories. A complete reasoning is obtained by extending LK with (...) two rules for sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. (shrink)

This book examines philosophical approaches to linguistic vagueness, a puzzling feature of natural language that gives rise to the ancient Sorites paradox and challenges classical logic and semantics. -/- The Sorites, or Paradox of the Heap, consists in three claims: (1) One grain of sand does not make a heap. (2) One billion grains of sand do make a heap. (3) For any two amounts of sand differing by at most one grain: either both are heaps of (...) sand, or neither one is. The third claim is rendered plausible by an initial conviction that vague predicates like ‘heap’ tolerate small changes. However, the repeated application of a tolerance principle to the second claim yields the further proposition that one grain of sand does make a heap – which contradicts claim number one. Consequently, many philosophers reject or modify tolerance principles for vague predicates. -/- Inga Bones reassesses prominent responses to the Sorites and defends a Wittgensteinian dissolution of the paradox. She argues that vague predicates are, indeed, tolerant and discusses how this finding relates to the paradox itself, to the notion of validity and to the concept of a borderline case . (shrink)

Paradoxes have played an important role both in philosophy and in mathematics and paradox resolution is an important topic in both fields. Paradox resolution is deeply important because if such resolution cannot be achieved, we are threatened with the charge of debilitating irrationality. This is supposed to be the case for the following reason. Paradoxes consist of jointly contradictory sets of statements that are individually plausible or believable. These facts about paradoxes then give rise to a deeply troubling (...) epistemic problem. Specifically, if one believes all of the constitutive propositions that make up a paradox, then one is apparently committed to belief in every proposition. This is the result of the principle of classical logical known as ex contradictione (sequitur) quodlibetthat anything and everything follows from a contradiction, and the plausible idea that belief is closed under logical or material implication (i.e. the epistemic closure principle). But, it is manifestly and profoundly irrational to believe every proposition and so the presence of even one contradiction in one’s doxa appears to result in what seems to be total irrationality. This problem is the problem of paradox-induced explosion. In this paper it will be argued that in many cases this problem can plausibly be avoided in a purely epistemic manner, without having either to resort to non-classical logics for belief (e.g. paraconsistent logics) or to the denial of the standard closure principle for beliefs. The manner in which this result can be achieved depends on drawing an important distinction between the propositional attitude of belief and the weaker attitude of acceptance such that paradox constituting propositions are accepted but not believed. Paradox-induced explosion is then avoided by noting that while belief may well be closed under material implication or even under logical implication, these sorts of weaker commitments are not subject to closure principles of those sorts. So, this possibility provides us with a less radical way to deal with the existence of paradoxes and it preserves the idea that intelligent agents can actually entertain paradoxes. (shrink)

http://dx.doi.org/10.5007/1808-1711.2008v12n2p203 The preface paradox is the apparent pragmatic inconsistency that occurs when the author of a book declares in its preface that despite believing that it is highly probable that everything the book maintains is true it is also highly probable that the book contains at least some errors. The preface paradox has often been presented as an example of a logically inconsistent belief that it is nevertheless rational to accept, supporting the suggestion that rationality has nothing immediately (...) to do with avoiding formal logical inconsistency in one’s beliefs. In contrast with complicated received solutions, I explain a simple method of avoiding inconsistency in preface paradox assertions by indexing the two claims to distinct probability or justification sources, referencing in the first case the conscientious efforts to eliminate errors and in the second case the chagrin of past experience in discovering previously unsuspected errors in published writings. The solution thereby helps preserve the concept of rationality as logical consistency of belief, and avoids the conclusion that the preface paradox proves that it is sometimes rational to accept logically inconsistent beliefs. (shrink)

Any language $$\mathcal {L}$$ L of classical logic, of first- or higher-order, is expanded with sentential quantifiers and operators. The resulting language $$\mathcal {L}^+\!$$ L +, capable of self-reference without arithmetic or syntax encoding, can serve as its own metalanguage. The syntax of $$\mathcal {L}^+$$ L + is represented by directed graphs, and its semantics, which coincides with the classical one on $$\mathcal {L}$$ L, uses the graph-theoretic concepts of kernels and semikernels. Kernels provide an explosive semantics, while semikernels (...) generalize this to situations where paradoxes do not lead to explosion, thus distinguishing them from contradictions. Paradoxes arise only at the metalevel due to specific interpretations of the operators, but they can be avoided: $$\mathcal {L}^+$$ L + can express paradoxes but remains free from them. For an expansion $$\mathcal {L}^+$$ L + of any FOL language $$\mathcal {L}$$ L, with the non-explosive semantics, a complete reasoning system is obtained by extending Gentzen’s classical sequent calculus with two rules for the sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. The novel semantics and self-referential capabilities seem promising for a further extension of classical logic towards one capable also of consistently expressing its own syntax and truth theory. (shrink)

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical (...) means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem. (shrink)

The operation of developing a concept is a common procedure in mathematics and in natural science, but has traditionally seemed much less possible to philosophers and, especially, logicians. Meir Buzaglo's innovative study proposes a way of expanding logic to include the stretching of concepts, while modifying the principles which block this possibility. He offers stimulating discussions of the idea of conceptual expansion as a normative process, and of the relation of conceptual expansion to truth, meaning, reference, ontology and (...) class='Hi'>paradox, and analyzes the views of Kant, Wittgenstein, Godel, and others, paying especially close attention to Frege. His book will be of interest to a wide range of readers, from philosophers to logicians, mathematicians, linguists, and cognitive scientists. (shrink)

Twisted Logic: Navigating Life's Puzzles, Paradoxes, and Big Questions draws upon an array of popular and novel puzzles and paradoxes to help us understand and navigate our everyday world, as well as engaging with the big questions beyond. It will appeal to all those interested in learning about twisted logic and the ways in which intuition and common sense can sometimes lead us astray. The book is designed for everyone and is accessible to the layman and student alike. (...) No prior mathematical skill or technical knowledge is required. (shrink)

This review is mainly expository. At one place, Following a suggestion of carnap's, It is suggested that advances in brain physiology might enable us to revive the notion of conventional truth.

I argue here that Kierkegaardian faith is essentially, albeit paradoxically, worldly---that Kierkegaardian faith is a form of world-affirmation. A correlate of this claim is that faithlessness of any kind is ultimately a form of aesthetic resignation grounded in a deep seated world-alienation. The paradox of faith’s worldliness is found in the fact that, for Kierkegaard, faith both excludes and includes resignation in itself. I make sense of this paradox by appealing to Kierkegaard’s idea of “an annulled possibility,” and (...) conclude that faith’s love of the world is an affirmation via a double negation. (shrink)

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for (...) the logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution. (shrink)

We extend the ordinary logic of knowledge based on the operator K and the system of axioms S₅ by adding a new operator Uφ, standing for "the agent utters φ", and certain axioms and a rule for U, forming thus a new system KU. The main advantage of KU is that we can express in it intentions of the speaker concerning the truth or falsehood of the claims he utters and analyze them logically. Specifically we can express in the (...) new language various notions of lying, as well as of telling the truth. Consequently, as long as lying or telling the truth about a fact is an intentional mode of the speaker, we can resolve the Liar paradox, or at least some of its variants, turning it into an ordinary (false or true) sentence. Also, using Kripke structures analogous to those employed by S. Kraus and D. Lehmann in [3] for modelling the logic of knowledge and belief, we offer a sound and complete semantics for KU. (shrink)

I consider a puzzling case presented by Jose Benardete, and by appeal to this case develop a paradox involving counterfactual conditionals. I then show that this paradox may be leveraged to argue for certain non-obvious claims concerning the logic of counterfactuals.

Awarded the 1988 Johnsonian Prize in Philosophy. Published with the aid of a grant from the National Endowment for the Humanities.

Moore’s Paradox is a test case for any formal theory of belief. In Knowledge and Belief, Hintikka developed a multimodal logic for statements that express sentences containing the epistemic notions of knowledge and belief. His account purports to offer an explanation of the paradox. In this paper I argue that Hintikka’s interpretation of one of the doxastic operators is philosophically problematic and leads to an unnecessarily strong logical system. I offer a weaker alternative that captures in a (...) more accurate way our logical intuitions about the notion of belief without sacrificing the possibility of providing an explanation for problematic cases such as Moore’s Paradox. (shrink)

The logic B+ is Routley and Meyer’s basic positive logic. Wedefine the logics BK+ and BK′+ by adding to B+ the K rule and to BK+the characteristic S4 axiom, respectively. These logics are endowed witha relatively strong non-constructive negation. We prove that all the logicsdefined lack the K axiom and the standard paradoxes of consistency.

This volume is published concurrently with the one reviewed below and together they unite a number of Quine's previously scattered papers into two compact volumes; this volume deals with his more philosophical work while the other is concerned with more purely technical logical studies. The twenty-one essays cover the period 1934-1964 and none have appeared between hard covers before. Several of the articles—"The ways of paradox," "Foundations of mathematics," "On the application of modern logic," and "Necessary truth"—are essentially (...) popular expositions. The others are generally more restricted in both scope and appeal, and deal with the ontology of the sentential calculus, truth by convention, implicit definition, modal logic, and ontological reduction in the sciences. Several articles concern the philosophy of science directly: "On simple theories of a complex world," "Posits and reality," and "The scope and language of science." This fine collection will be of significant help in presenting the work of a distinguished philosopher to a wider audience, as well as providing the professional with a source of discussion.—P. J. M. (shrink)

Alexius Meinong's Gegenstandstheorie is subject to a formal semantic paradox. The theory of defective objects originally developed by Meinong in response to Ernst Mally's paradox about self-referential thought is rejected as a general solution to paradox in the object theory. The intentionality thesis is also refuted by the counter-example of the unapprehended mountain. It is argued that despite these difficulties, an object theory is required in order to make intuitively correct sense of ontological commitment. ;A version of (...) Meinong's theory is formalized. The logic has non-standard truth value semantics and an objectual interpretation of quantifiers for a domain of existent and non-existent objects. Theories of intentional, referential, and extensional identity are offered, together with two theories of definite description, lambda abstraction, eight systems of alethic modal logic, and formal metatheory. The logic avoids the incompleteness and undecidability limitations of Godel and Church, and all logical, semantic, and set theoretical paradoxes, without type theory or Tarskian hierarchy of metalanguages. It is naively complete and mechanically decidable. ;Wittgenstein's private language argument is refuted as an obstacle to defining private mental objects in the logic. The argument is shown to be valid, but unsound. The object theory logic may therefore provide a mixed or impure private language, and its domain may contain private mental objects as well as ordinary public and non-existent objects. This means that it constitutes an object theory logic of intention, and can be used to represent the logical structure of intentional and specifically phenomenological philosophical theories that require the intelligible designation of private mental objects. ;Appendices include Henkin-type consistency, completeness, and compactness proofs ; criticism of Kazimierz Twardowski's arguments for the distinction between content and object in phenomenological psychology; translation of and commentary on Mally's essay, "Uber die Unabhangigkeit der Gegenstande vom Denken"; a discussion of the distinction between nuclear and extra-nuclear properties, and the eliminability of the modal moment; and a philosophical application of the logic to a solution of the paradox of analysis. (shrink)

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egré, Ripley and van Rooij have put forward an approach based on a non-transitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated (...) by this logic. In this paper, we show that this logic, once it is adequately understood, is weaker than classical logic. Moreover, the logic is in a way similar to the paraconsistent logic LP. (shrink)