2010: Volume 9

Review: Vagueness and Degrees of Truth
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Christian G. Fermüller
9 pages. Published November 2, 2010
A Review of Nicholas J.J. Smith, Vagueness and Degrees of Truth, Oxford University Press, 2008.

Cantor’s Proof in the Full Definable Universe
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Laureano Luna and William Taylor
16 pages. Published November 3, 2010
Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the scope of quantifiers reveals a natural way out.

ComplementTopoi and Dual Intuitionistic Logic
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Luis EstradaGonzález
19 pages. Published November 26, 2010
Mortensen studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complementtopos logic, which throws some light on the problem of giving a dualization for LJ.

AJL COMMENT
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One Philosopher is Correct (Maybe)
Paul Skokowski
3 pages. Published December 1, 2010
It is argued that there may be one philosopher who is correct.
2010: Volume 8
In Memory of Professor Robert K. Meyer

The DCompleteness of T_{→}
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R. K. Meyer and M. W. Bunder
8 pages. Published September 22, 2010
A Hilbertstyle version of an implicational logic can be represented by a set of axiom schemes and modus ponens or by the corresponding axioms, modus ponens and substitution. Certain logics, for example the intuitionistic implicational logic, can also be represented by axioms and the rule of condensed detachment, which combines modus ponens with a minimal form of substitution. Such logics, for example intuitionistic implicational logic, are said to be Dcomplete. For certain weaker logics, the version based on condensed detachment and axioms (the condensed version of the logic) is weaker than the original. In this paper we prove that the relevant logic T[→], and any logic of which this is a sublogic, is Dcomplete.

Extending Metacompleteness to Systems with Classical Formulae
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Ross T. Brady
22 pages. Published September 22, 2010
In honour of Bob Meyer, the paper extends the use of his concept of metacompleteness to include various classical systems, as much as we are able. To do this for the classical sentential calculus, we add extra axioms so as to treat the variables like constants. Further, we use a onesorted and a twosorted approach to add classical sentential constants to the logic DJ of my book, Universal Logic. It is appropriate to use rejection to represent classicality in the onesorted case. We then extend these methods to the quantified logics, but we use a finite domain of individual constants to do this.

Boolean Conservative Extension Results for some Modal Relevant Logics
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Edwin D. Mares and Koji Tanaka
19 pages. Published September 22, 2010
This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean negation.

Logics without the contraction rule and residuated lattices
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Hiroakira Ono
32 pages. Published September 22, 2010
In this paper, we will develop an algebraic study of substructural propositional logics over FL_{ew}, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCKlogics, Lukasiewicz’s manyvalued logics and fuzzy logics, within a uniform framework.

Models for Substructural Arithmetics
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Greg Restall
18 pages. Published September 22, 2010
This paper explores models for arithmetic in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R.

A Logic for Vagueness
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John Slaney
35 pages. Published November 2, 2010
This paper presents F, substructural logic designed to treat vagueness. Weaker than Lukasiewicz’s infinitely valued logic, it is presented first in a natural deduction system, then given a Kripke semantics in the manner of Routley and Meyer's ternary relational semantics for R and related systems, but in this case, the points are motivated as degrees to which the truth could be stretched. Soundness and completeness are proved, not only for the propositional system, but also for its extension with firstorder quantifiers. The firstorder models allow not only objects with vague properties, but also objects whose very existence is a matter of degree.
Copyright © 2010, Philosophy Department, University of Melbourne.
Individual papers are copyright their authors.