The Other Ways of Paradox 
By William G. Lycan 

In "The Ways of Paradox" (1966), Quine offered his classic characterization of the notion of paradox, a taxonomy for paradoxical arguments, and some vocabulary for discussing them. In this paper I shall generalize Quine's taxonomy and defend a simpler characterization. My characterization will have the virtue or the flaw (as might be) of making paradox a matter of degree. 

Posted by Tony Marmo at 22:40 GMT

Updated: Thursday, 7 January 2010 23:32 GMT

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The Modal Logic of Agreement and Noncontingency

Lloyd Humberstone

The formula DA (it is noncontingent whether A) is true at a point in a Kripke model just in case all points accessible to that point agree on the truth-value of A. We can think of D-based modal logic as a special case of what we call the general modal logic of agreement, interpreted with the aid of models supporting a ternary relation, S, say, with OA (which we write instead of DA to emphasize the generalization involved) true at a point w just in case for all points x, y, with Swxy, x and y agree on the truth-value of A. The noncontingency interpretation is the special case in which Swxy if and only if Rwx and Rwy, where R is a traditional binary accessibility relation. Another application, related to work of Lewis and von Kutschera, allows us to think of OA as saying that A is entirely about a certain subject matter.

Keywords: modal logic; contingency; noncontingency; subject matters; supervenience

 

Source: Notre Dame J. Formal Logic Volume 43, Number 2 (2002), 95-127.

Posted by Tony Marmo at 01:13 BST

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A propositional logic for Tarski's consequence operator

By Hércules de Araújo Feitosa, Mauri Cunha do Nascimento & Maria Claudia Cabrini Grácio

This paper presents the TK-algebras associated to Tarski's consequence operator and introduces the TK Logic. So it shows the adequacy (soundness and completeness) of TK Logic relative to the algebraic model given by TK-algebras.

Source: CLE e-prints Vol. 7(1), 2007

Posted by Tony Marmo at 15:19 BST

Updated: Monday, 4 June 2007 15:37 BST

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Truth-Definitions and Definitional Truth

By Douglas Patterson 

Putnam, Etchemendy, Heck and others have criticized Tarski’s definitions of truth on the grounds that they turn what ought to be contingent truths about the truth conditions of sentences into logical, mathematical or necessary truths. I argue that this criticism rests on the misguided assumption that substitution in accord with a good definition preserves logical, mathematical or necessary truth. I give a number of examples intended to show that substitution in accord with good definitions need preserve none of these. The paper should be of interest not only to students of Tarski, but to anyone interested in definition and analyticity, and it includes some discussion of the contingent a priori, logicism, the nature of applied mathematics, and early Wittgensteinian doctrines about showing and saying.

Source: Online Papers in Philosophy 

Posted by Tony Marmo at 08:33 BST

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On truth-schemes for intensional logics

By Janusz Czelakowski and Wieslaw Dziobiak

The paper is concerned with the question of definability of truth-conditions for the connectives of intensional logics. A certain general solution of the problem is proposed for the class of self-extensional logics. The paper develops some ideas initiated by Suszko and Wojcicki in the seventies.

Source: Reports on Mathematical Logic 41 (2006)

Posted by Tony Marmo at 20:30 GMT

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Scope Dominance with Upward Monotone Quantifiers

By Alon Altman, Ya'acov Peterzil & Yoad Winter

We give a complete characterization of the class of upward monotone generalized quantifiers Q₁ and Q₂ over countable domains that satisfy the scheme Q₁xQ₂yφ→Q₂yQ₁xφ.
This generalizes the characterization of such quantifiers over finite domains, according to which the scheme holds iff Q₁ is ∃ or Q₂ is ∀ (excluding trivial cases). Our result shows that in innite domains, there are more general types of quantifiers that support these entailments.

Published in Journal of Logic, Language and Information, Volume 14, Number 4, October 2005, pp. 445-455(11)

Link to the article in the Journal
Author's Link

Posted by Tony Marmo at 00:26 GMT

Updated: Saturday, 25 November 2006 19:51 GMT

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Worlds and Times

By Ulrich Meyer

There are many parallels between the role of possible worlds in modal logic and that of times in tense logic. But the similarities only go so far, and it is important to note where the two come apart. This paper argues that even though worlds and times play similar roles in the model theories of modal and tense logic, there is no tense analogue of the possible-worlds analysis of modal operators. An important corollary of this result is that presentism cannot be the tense analogue of actualism.

Keywords: tense logic; modal logic; times; possible worlds; actuality operator; presentism; actualism

Published in The Notre Dame Journal of Formal Logic
An unpublished version may be downloaded from the Author's page

Posted by Tony Marmo at 19:09 BST

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Logic Inference in Polynomial Format

By Walter Carnielli

The methods described in this paper have a promising potential to any truth-functional multi-valued logic: there is an exciting area of research in designing new proof theory techniques for such logics, and simplifying applications to multi-valued logics in decision tables and discovering patterns, as in several other fields (it is well-known that multi-valued logics find applications in artificial intelligence, database theory and data mining, modeling reasoning and model checking, for instance). It is important to emphasize that the method is also plainly applicable to non-finite valued logics, and also to represent binary semantics for many-valued logics5 (cf. [13]) and even to quantum circuits and quantum gates (cf. [1]). The arguments advanced here try to conceptualize this approach, in particular when extended to quantification and non-finite valued logics, as inheritance of an admirable legacy in the mathematical thinking, which may have been disregarded by logicians.

Source: CLE e-prints

Posted by Tony Marmo at 02:41 BST

Updated: Friday, 30 June 2006 02:42 BST

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Modal Deduction in Second-Order Logic and Set Theory-I

By Johan van Benthem, Giovanna D'Agostino, Angelo Montanari, Alberto Policriti

We investigate modal deduction through translation into standard logic and set theory. Derivability in the minimal modal logic is captured precisely by translation into a weak, computationally attractive set theory \Omega. This approach is shown equivalent to working with standard first-order translations of modal formulas in a theory of general frames. Next, deduction in a more powerful second?order logic of general frames is shown equivalent with set?theoretic derivability in an `admissible variant' of \Omega. Our methods are mainly model?theoretic and set?theoretic, and they admit extension to richer languages than that of basic modal logic. Read more [1] [2]; [3]

Appeared in Journal of Logic and Computation 1997 7(2):251-265

Modal Deduction in Second-Order Logic and Set Theory - II

By Johan van Benthem, Giovanna D'Agostino, Angelo Montanari, Alberto Policriti

In this paper, we generalize the set-theoretic translation method for poly-modal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor such a translation to work with specific cases of extended modal logics.

Keywords: Modal Logic; Modal Deduction; Translation Methods; Second-Order Logic; Set Theory
Appeared in Studia Logica, Volume 60, Number 3, May 1998, pp. 387-420(34)

Modal Logic and Set Theory: a Set-Theoretic Interpretation of Modal Logic

Giovanna D’Agostino, Angelo Montanari, and Alberto Policriti

In this paper, we describe a novel set-theoretic interpretation of modal logic and show how it allows us to build promising bridges between modal deduction and set -theoretic reasoning. More specifically, we describe a translation technique that maps modal formulae into set-theoretic terms, thus making it possible to successfully exploit derivability in first-order set theories to implement derivability in modal logic.

Appeared in JFAK, a collection of essays dedicated to Johan van Benthem on the occasion of his 50th birthday, on June 12, 1999 (Edited by Jelle Gerbrandy et al.)

Posted by Tony Marmo at 00:01 BST

Updated: Saturday, 10 June 2006 03:38 BST

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A Short Note on Gentzen's LJ and NJ Systems Isomorphism

By Wagner Sanz

We state a new intuitionistic sequent calculus and use it to clarify Gentzen's NJ and LJ isomorphism, it contains new negation rules which are immediate readings of what seems to be good and sound natural deduction rules.
Keywords: Sequent Calculus, Natural Deduction, Intuitionism, Negation.

Source: CLE e-prints

Posted by Tony Marmo at 16:58 BST

Updated: Tuesday, 16 May 2006 17:02 BST

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A Logic for Ambiguous Description

By Arthur Buchsbaum

A logic formalizing ambiguity, which appears both in natural language and in mathematical discourse, is presented, through a sequent calculus and a semantics, together with some elementary results.

Author Keywords: ambiguous description; designators; rigidity
Appeared in 'Electronic Notes in Theoretical Computer Science' Volume 67, October 2002, Pages 1-18 WoLLIC'2002, 9th Workhop on Logic, Language, Information and Computation

Posted by Tony Marmo at 14:44 BST

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Degrees of Truth, Degrees of Falsity

By Toby Ord

In this paper I recall the reasons in favour of extending the classical conception of truth to include degrees of truth as well as truth value gaps and gluts, then provide a sketch of a new system of logic that provides all of these simultaneously. Despite its power, the resulting system is quite simple, combining degrees of truth and degrees of falsity to provide a very flexible and elegant conception of truth value.

Posted by Tony Marmo at 13:27 GMT

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Many-valued Logics Enriched with a Stonean Negation: a Direct Proof of Representation and Completeness

By Martinivaldo Konig

This paper studies Lukasiewicz's many-valued logic enriched with a new operator: the Stonean negation. This research focuses on the class of algebras containing the algebraic counterpart of this new logic: the class of Stonean MV-algebras. A direct proof of subdirect representation Theorem is given, as well as an algebraic completeness Theorem.

Keywords: MV-algebra, Stonean MV-algebra, Stonean negation operator, Chang's subdirect representation, Chang's algebraic completeness.

Appeared in L&PS - Logic and Philosophy of Science: Vol. 1 - No.1 - 2005

Posted by Tony Marmo at 00:01 GMT

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MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

By Robert Goldblatt

Over a period of three decades or so from the early 1930’s there evolved two kinds of mathematical semantics for modal logic. Algebraic semantics interprets modal connectives as operators on Boolean algebras. Relational semantics uses relational structures often called Kripke models, whose elements are thought of variously as being possible worlds, moments of time, evidential situations, or states of a computer. The two approaches are intimately related the subsets of a relational structure form a modal algebra (Boolean algebra with operators), while conversely any modal algebra can be embedded into an algebra of subsets of a relational structure via extensions of Stone’s Boolean representation theory. Techniques from both kinds of semantics have been used to explore the nature of modal logic and to clarify its relationship to other formalisms particularly first and second order monadic predicate logic.
The aim of this article is to review these developments in a way that provides some insight into how the present came to be as it is. The pervading theme is the mathematics underlying modal logic and this has at least three dimensions. To begin with there are the new mathematical ideas, when and why they were introduced, and how they interacted and evolved. Then there is the use of method and results from other areas of mathematical logic, algebra and topology in the analysis of modal systems. Finally, there is the application of modal syntax and semantics to study notions of mathematical and computational interest.

Appeared in: Handbook of the History of Logic. Volume 6 Dov M. Gabbay and John Woods (Editors)

Posted by Tony Marmo at 00:01 GMT

Updated: Tuesday, 22 November 2005 11:35 GMT

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From Paradox to Judgment: towards a metaphysics of expression

By Mariam Thalos

The Liar sentence is a singularly important piece of philosophical evidence. It is an instrument for investigating the metaphysics of expressing truths and falsehoods. And an instrument too for investigating the varieties of conflict that can give rise to paradox. It shall serve as perhaps the most important clue to the shape of human judgment, as well as to the nature of the dependence of judgment upon language use.

Source: The Australasian Journal of Logic

Posted by Tony Marmo at 00:01 GMT

Updated: Friday, 11 November 2005 06:33 GMT

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