2007: Volume 5

The McKinsey–Lemmon logic is barely canonical
Robert Goldblatt and Ian Hodkinson19 pages. Published November 6, 2007
We study a canonical modal logic introduced by Lemmon, and axiomatised by an inŽnite sequence of axioms generalising McKinsey’s formula. We prove that the class of all frames for this logic is not closed under elementary equivalence, and so is nonelementary. We also show that any axiomatisation of the logic involves infinitely many noncanonical formulas.

Reduction in firstorder logic compared with reduction in implicational logic
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Tigran M. Galoyan
8 pages. Published, November 8, 2007In this paper we discuss strong normalization for natural deduction in the →∀fragment of firstorder logic. The method of collapsing types is used to transfer the result (concerning strong normalization) from implicational logic to firstorder logic. The result is improved by a complement, which states that the length of any reduction sequence of derivation term r in firstorder logic is equal to the length of the corresponding reduction sequence of its collapse term r^{c} in implicational logic.

Four Variables Suffice
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Alasdair Urquhart
8 pages. Published November 11, 2007What I wish to propose in the present paper is a new form of “career induction” for ambitious young logicians. The basic problem is this: if we look at the nvariable fragments of relevant propositional logics, at what point does undecidability begin? Focus, to be definite, on the logic R. John Slaney showed that the 0variable fragment of R (where we allow the sentential con stants t and f) contains exactly 3088 nonequivalent propositions, and so is clearly decidable. In the opposite direction, I claimed in my paper of 1984 that the five variable fragment of R is undecidable. The proof given there was sketchy (to put the matter charitably), and a close examination reveals that although the result claimed is true, the proof given is incorrect. In the present paper, I give a detailed and (I hope) correct proof that the four variable fragments of the principal relevant logics are undecidable. This leaves open the question of the decidability of the nvariable fragments for n = 1, 2, 3. At what point does undecidability set in?

Not so deep inconsistency: a reply to Eklund
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JC Beall and Graham Priest
11 pages. Published November 11, 2007In his “Deep Inconsistency?” Eklund attacks arguments to the effect that some contradictions are true, and especially those based on the liar paradox, to be found in Priest’ In Contradiction. The point of this paper is to evaluate his case.

An atomic theory with no prime models
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Tarek Sayed Ahmed
4 pages. November 19, 2007We construct an atomic uncountable theory with no prime models. This contrasts with the countable case.

60% Proof: Lakatos, Proof, and Paraconsistency
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Graham Priest and Neil Thomason
11 pages. November 28, 2007Imre Lakatos’ Proofs and Refutations is a book well known to those who work in the philosophy of mathematics, though it is perhaps not widely referred to. Its general thrust is out of tenor with the foundationalist perspective that has dominated work in the philosophy of mathematics since the early years of the 20th century. It seems to us, though, that the book contains striking insights into the nature of proof, and the purpose of this paper is to explore and apply some of these.
Forcing with Nonwellfounded Models
Paul Corazza
38 pages. Published, November 8, 2007
We develop the machinery for performing forcing over an arbitrary (possibly nonwellfounded) model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is (countable) transitive. However, for establishing properties of a given (possibly nonwellfounded) model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling standard steps of presentation in Kunen and Jech.
Copyright © 2007, School of Philosophy, University of Melbourne.
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