2008: Volume 6

Categorical Abstract Algebraic Logic: Equivalential πInstitutions
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George Voutsadakis
24 pages. Published August 4, 2008
The theory of equivalential deductive systems, as introduced by Prucnal and Wrónski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as πinstitutions. More precisely, the notion of an Nequivalence system for a given πinstitution is introduced. A characterization theorem for Nequivalence systems, previously proven for Nparameterized equivalence systems, is revisited and a “transfer theorem” for Nequivalence systems is proven. For a πinstitution I having an Nequivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those Nprotoalgebraic πinstitutions having an Nequivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of πinstitutions, as they relate to the existence of Nequivalence systems.

Bayesians sometimes cannot ignore even very implausible theories
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(even ones that have not yet been thought of)
Branden Fitelson and Neil Thomason
12 pages. Published August 4, 2008
In applying Bayes’s theorem to the history of science, Bayesians sometimes assume – often without argument – that they can safely ignore very implausible theories. This assumption is false, both in that it can seriously distort the history of science as well as the mathematics and the applicability of Bayes’s theorem. There are intuitively very plausible counterexamples. In fact, one can ignore very implausible or unknown theories only if at least one of two conditions is satisfied: (i) one is certain that there are no unknown theories which explain the phenomenon in question, or (ii) the likelihood of at least one of the known theories used in the calculation of the posterior is reasonably large. Often in the history of science, a very surprising phenomenon is observed, and neither of these criteria is satisfied.

ChurchRosser property and intersection types
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George Koletsos and George Stavrinos
18 pages. Published August 4, 2008
We give a proof via reducibility of the ChurchRosser property for the system D of λcalculus with intersection types. As a consequence we can get the confluence property for developments directly, without making use of the strong normalization property for developments, by using only the typability in D and a suitable embedding of developments in this system. As an application we get a proof of the ChurchRosser theorem for the untyped λcalculus.

A Rejection System for the FirstDegree Formulae of some Relevant Logics
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Ross T. Brady
15 pages. Published August 4, 2008
The standard Hilbertstyle of axiomatic system yields the assertion of axioms and, via the use of rules, the assertion of theorems. However, there has been little work done on the corresponding axiomatic rejection of nontheorems. Such Hilbertstyle rejection would be achieved by the inclusion of certain rejectionaxioms (raxioms) and, by use of rejectionrules (rrules), the establishment of rejectiontheorems (rtheorems). We will call such a proof a rejectionproof (rproof). The ideal to aim for would be for the theorems and rtheorems to bemutually exclusive and exhaustive. That is, if a formula A is a theorem then it is not an rtheorem, and if A is a nontheorem then it is an rtheorem. In this paper, I present a rejecion system for the firstdegree formulae of a large number of relevant logics.

Modal Formulas True at Some Point in Every Model
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Lloyd Humberstone
13 pages. Published August 4, 2008
In a paper on the logical work of the Jains, Graham Priest considers a consequence relation, semantically characterized, which has a natural analogue in modal logic. Here we give a syntactic/axiomatic description of the modal formulas which are consequences of the empty set by this relation, which is to say: those formulas which are, for every model, true at some point in that model.

Review: Eckart MenzlerTrott’s — Logic’s Lost Genius: The Life of Gerhard Gentzen
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John N. Crossley
5 pages. September 15, 2008

Mathematical and Physical Continuity
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Mark Colyvan and Kenny Easwaran
7 pages. Published September 16, 2008
There is general agreement in mathematics about what continuity is. In this paper we examine how well the mathematical definition lines up with common sense notions. We use a recent paper by Hud Hudson as a point of departure. Hudson argues that two objects moving continuously can coincide for all but the last moment of their histories and yet be separated in space at the end of this last moment. It turns out that Hudson’s construction does not deliver mathematically continuous motion, but the natural question then is whether there is any merit in the alternative definition of continuity that he implicitly invokes.

Reply to Beall and Priest
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Matti Eklund
13 pages. Published November 7, 2008
In my “Deep Inconsistency” (Australasian Journal of Philosophy, 2002), I compared my meaninginconsistency view on the liar with Graham Priest’s dialetheist view, using my view to help cast doubt on Priest’s arguments for his view. Jc Beall and Priest have recently published a reply to my article (Australasian Journal of Logic, 2007). I here respond to their criticisms. In addition, I compare the meaning–inconsistency view with Anil Gupta and Nuel Belnap’s revision theory of truth, and discuss how best to deal with the strengthened liar.

Paraconsistent Vagueness: Why Not?
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Dominic Hyde and Mark Colyvan
15 pages. Published November 26, 2008
The idea that the phenomenon of vagueness might be modelled by a paraconsistent logic has been little discussed in contemporary work on vagueness, just as the idea that paraconsistent logics might be fruitfully applied to the phenomenon of vagueness has been little discussed in contemporary work on paraconsistency. This is prima facie surprising given that the earliest formalisations of paraconsistent logics presented in Jáskowski and Halldén were presented as logics of vagueness. One possible explanation for this is that, despite initial advocacy by pioneers of paraconsistency, the prospects for a paraconsistent account of vagueness are so poor as to warrant little further consideration. In this paper we look at the reasons that might be offered in defence of this negative claim. As we shall show, they are far from compelling. Paraconsistent accounts of vagueness deserve further attention.

Collapsing Arguments for Facts and Propositions
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John Howard Sobel
40 pages. Published November 27, 2008
Kurt Gödel argues in “Russell’s Mathematical Logic” that on the assumption that, contrary to Russell, definite descriptions are terms, it follows given only several “apparently obvious axioms” that “all true sentences have the same signification (as well as all false ones).” Stephen Neale has written that this argument, and others by Church, Davidson, and Quine to similar conclusions, are of considerable philosophical interest. Graham Oppy, responding to this opinion, says they are of minimal interest. Falling between these is my opinion that implications of these arguments for propositions and facts are of moderate philosophical interest, and that these arguments provide occasions for reflection of possible interest on fine lines of several theories of definite descriptions and class–abstractions.

An abstract approach to reasoning about games with mistaken and changing beliefs
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Benedikt Löwe and Eric Pacuit
20 pages. Published November 28, 2008
We do not believe that logic is the sole answer to deep and intriguing questions about human behaviour, but we think that it might be a useful tool in simulating and understanding it to a certain degree and in specifically restricted areas of application. We do not aim to resolve the question of what rational behaviour in games with mistaken and changing beliefs is. Rather, we develop a formal and abstract framework that allows us to reason about behaviour in games with mistaken and changing beliefs leaving aside normative questions concerning whether the agents are behaving “rationally”; we focus on what agents do in a game. In this paper, we are not concerned with the reasoning process of the (ideal) economic agent; rather, our intended application is artificial agents, e.g., autonomous agents interacting with a human user or with each other as part of a computer game or in a virtual world. We give a story of mistaken beliefs that is a typical example of the situation in which we should want our formal setting to be applied. Then we give the definitions for our formal system and how to use this setting to get a backward induction solution. We then apply our semantics to the story related earlier and give an analysis of it. Our final section contains a discussion of related work and future projects. We discuss the advantages of our approach over existing approaches and indicate how it can be connected to the existing literature.

Logical dynamics meets logical pluralism?
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Johan van Benthem
28 pages. Published December 17, 2008
Where is logic heading today? There is a general feeling that the discipline is broadening its scope and agenda beyond classical foundational issues, and maybe even a concern that, like Stephen Leacock’s famous horseman, it is ‘riding off madly in all directions’. So, what is the resultant vector? There seem to be two broad answers in circulation today. One is logical pluralism, locating the new scope of logic in charting a wide variety of reasoning styles, often marked by nonclassical structural rules of inference. This is the new program that I subscribed to in my work on substructural logics around 1990, and it is a powerful movement today. But gradually, I have changed my mind about the crux of what logic should become. I would now say that the main issue is not variety of reasoning styles and notions of consequence, but the variety of informational tasks performed by intelligent interacting agents, of which inference is only one among many, involving observation, memory, questions and answers, dialogue, or general communication. And logical systems should deal with a wide variety of these, making informationcarrying events firstclass citizens in their setup. The purpose of this brief paper is to contrast and compare the two approaches, drawing freely on some insights from earlier published papers. In particular, I will argue that logical dynamics sets itself the more ambitious diagnostic goal of explaining why substructural phenomena occur, by ‘deconstructing’ them into classical logic plus an explicit account of the relevant informational events.

Logical Pluralism Hollandaise
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Graham Priest
5 pages. Published December 17, 2008
Johan van Benthem compares and contrasts two research programmes, which he calls logical pluralism and logical dynamics, stating his ‘preference’ for the second of these ‘alternatives’. In this note I want to put the matter into a slightly different perspective.
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