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Modal Logics in the Vicinity of S1
By Brian F. Chellas & Krister Segerberg
We define pre-normal modal logics and show that S1, S10, S0.9, and S0.90 are Lewis versions of certain pre-normal logics, determination and decidability for which are immediate. At the end we characterize Cresswell logics and ponder C. I. Lewis's idea of strict implication in S1.
Source: Notre Dame J. Formal Logic 37, no. 1 (1996), 1–24
Linear Kripke Frames and Gödel Logics
By Arnold Beckmann & Norbert Preining
We investigate the relation between logics of countable linear Kripke frames with constant domains and Gödel logics. We show that for any such Kripke frame there is a Gödel logic which coincides with the logic of his Kripke frame and vice versa. This allows us to transfer several recent results on Gödel logics to the logics of countable linear Kripke frames with constant domains.
Diagonalization and Self-Reference
By Richard Heck
It is often said that diagonalization allows one to construct sentences that are self-referential. This paper investigates the sense in which that is true. I argue first that, in the standard language of arithmetic, in which we have only the symbols 0, S, +, and ?, truly self-referential sentences cannot be constructed. This is shown by considering sentences like The right-hand side of this biconditional is false iff its left-hand side is true. This sentence is intuitively inconsistent, but the sentence constructed by using diagonalization in the usual way is true and, in fact, provable in Q. This problem can be resolved by expanding the language to include function-symbols for all primitive recursive functions. It can also be resolved by proving a stronger form of the diagonal lemma that I call the structural diagonal lemma. At the end of the paper, it is suggested, however, that there are some contexts in which even these methods are insufficient.
Source: PH Online, Online Papers in Philosophy
Beliefs Review
[In Spanish]
By Eduardo Fermé
In the present essay we present the AGM belief review theory: its origins, axioms, semantics and different methods of constructing change-functions. We show the relation between the AGM model and conditional logic.
Resumen en Castellano:
En el presente ensayo presentamos la teoria de cambio de creencias AGM: sus orígenes, su axiomática, su semántica y diferentes métodos para construir funciones de cambio. Mostramos la relación entre el modelo AGM y la lógica condicional.
Source: CLE
A note on Kit Fine's Rigidity Axiom
Les Axiomes de Tarski
Jean-Yves Béziau
A la fin des années 1920, Tarski développe une théorie connue aujourd’hui sous le nom de théorie de l’opérateur des conséquence: il présente des axiomes pour un opérateur Cn qui à chaque ensemble d’objets X associe un autre ensemble d’objets Cn(X) de même nature, appelés conséquences de X. Il s’agit d’une théorie très abstraite puisque la nature des objets sur lesquels porte cet opérateur n’est pas spécifiée outre mesure.
Cette théorie en beaucoup de sens est extraordinaire et il semblerait que malgré le récent regain d’intérêt à son régard, sa valeur, sa signification et sa portée n’ont pas encore été pleinement comprises. En particulier on n’a pas encore réalisé combien cette théorie était en avance sur son temps, comment elle marque un tournant dans l’histoire de la logique moderne, en libérant la logique du carcan formaliste et en la projetant dans la sphère de la plus haute abstraction.
Le but de cet article n’est pas de présenter et de discuter de façon systématique l’origine et le développement de la théorie de l’opérateur de conséquence— il faudrait pour cela un volume suffisamment épais pour servir de banc— mais de discuter seulement d’un de ses aspects: ses axiomes. Dans d’autres articles nous avons déjà discuté ou nous discuterons d’autres aspects de cette théorie. Le présent article n’est donc qu’un parmi d’autres dont la somme pourrait finir par constituer le dit banc.
On the Storeyed Revenge of Strengthened Liars
By Jordan Howard SobelThe Strengthened Liar observes that if we follow a partiality theorist and declare the Liar sentence neither true nor false (or failing to express a proposition, or suffering from some sort of grave semantic defect), then the paradox is only pushed back. For we can go on to conclude that whatever this status may be, it implies that the Liar sentence is not true. This claim is true, but it is just the Liar sentence again. We are back in paradox.(Glanzberg 2002, p. 468; 2004, p. 29; 2001, p. 222, “We are back in our contradiction.”)
The Meta-Fibring Environment: Preservation of meta-properties by fibring
By Marcelo E. Coniglio
In this paper the categories Mcon and Seq of multiple-conclusion consequence relations and sequent calculi, respectively, are introduced. The main feature of these categories is the preservation, by morphisms, of meta-properties of the consequence relations. This feature is obtained by changing the usual concept of morphism between logics (that is, a signature morphism preserving deductions) by a stronger one (a signature morphism preserving meta-implications between deductions). This allow us to obtain better results by fibring objects in Mcon and in Seq than using the usual notion of morphism between deduction systems:
In fact, meta-fibring (that is, fibring in the proposed categories) avoids the phenomenon of fibring that we call anti-collapsing problem, as opposite to the well-known collapsing problem of fibring. Additionally, a general semantics for objects in Seq (and, in particular, for objects in Mcon) is proposed, obtaining a category of logic systems called Log. A general theorem of preservation of completeness by fibring in Log is also obtained.
Source: CLE
Minimalists about Truth can (and should) be Epistemicists, and it helps if they are revision theorists too
By Greg Restall
Minimalists about truth say that the important properties of the truth predicate are revealed in the class of T-biconditionals. Most minimalists demur from taking all of the T-biconditionals of the form 'p' is true if and only if p, to be true, because to do so leads to paradox. But exactly which biconditionals turn out to be true? I take a leaf out of the epistemic account of vagueness to show how the minimalist can avoid giving a comprehensive answer to that question. I also show that this response is entailed by taking minimalism seriously, and that objections to this position may be usefully aided and abetted by Gupta and Belnap’s revision theory of truth.
Polynomial Ring Calculus for Logical Inference
By Walter Carnielli
This paper proposes a new "all-purpose" algebraic proof method applicable to general truth-functional sentential logics and to some non-truth-functional logics. The method, based on reducing polynomials over finite fields, is particularly apt for finitely-many-valued logics (and for classical propositional logic PC ). It can be extended to certain non-finitely valued logics and non-truth-functional logics as well, provided they can be characterized by two-valued dyadic semantics. The resulting mechanizable proof method introduced here is of interest for automatic proof theory, and seems also to be appropriate for investigating questions on complexity.
Source: CLE
Possible worlds semantics for credulous and contraction inference
By Alexander Bochman
A possible worlds semantics is suggested for a broad class of nonmonotonic inference relations, including not only traditional skeptical ones, but also credulous and contraction inference. The semantics could be used to provide a canonical framework for studying and comparing different kinds of nonmonotonic inference.
Intertranslating Counterfactuals and Updates
By Mark Ryan & Pierre-Yves Schobbens
We recall that the Ramsey Rule can be seen as axiomatising the relationship of inverse accessibility relations which exists between the notions of update and counterfactual conditional. We use this fact to translate between postulates for updates and postulates for counterfactuals. Thus, Katsuno/Mendelzon?s postulates U1{U8 are translated into counterfactual postulates C1{C8 (theorem 6), and many of the familar counter-factual postulates are translated into postulates for updates (theorem 7). Our conclusions are summarised in table 5.
Causation and Counterfactuals
A recommended book
Edited by John Collins, Ned Hall and L. A. Paul, Published by Bradford Books
One philosophical approach to causation sees counterfactual dependence as the key to the explanation of causal facts: for example, events c (the cause) and e (the effect) both occur, but had c not occurred, e would not have occurred either. The counterfactual analysis of causation became a focus of philosophical debate after the 1973 publication of the late David Lewis's groundbreaking paper, "Causation," which argues against the previously accepted "regularity" analysis and in favor of what he called the "promising alternative" of the counterfactual analysis. Thirty years after Lewis's paper, this book brings together some of the most important recent work connecting--or, in some cases, disputing the connection between--counterfactuals and causation, including the complete version of Lewis's Whitehead lectures, "Causation as Influence," a major reworking of his original paper. Also included is a more recent essay by Lewis, "Void and Object," on causation by omission. Several of the essays first appeared in a special issue of the Journal of Philosophy, but most, including the unabridged version of "Causation as Influence," are published for the first time or in updated forms.
Other topics considered include the "trumping" of one event over another in determining causation; de facto dependence; challenges to the transitivity of causation; the possibility that entities other than events are the fundamental causal relata; the distinction between dependence and production in accounts of causation; the distinction between causation and causal explanation; the context-dependence of causation; probabilistic analyses of causation; and a singularist theory of causation.
Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity [le] [omega]
By Richard Zach
Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers [forall] p,[exist] p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most [omega] , the resulting logics are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some G?del-Dummett logics with quantifiers over propositions.
Source: Journal of Philosophical Logic
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Yablo's paradox rides again: a reply to Ketland
By Otavio Bueno & Mark Colyvan
Yablo ' s paradox is generated by the following (infinite) list of sentences (called Yablo's list):(S1) For all k> 1, Sk is not true.
(S2) For all k> 2, Sk is not true.
(S3) For all k> 3, Sk is not true.
...
(Sn) For all k>n, Sk is not true.
A little reflection reveals that this list is paradoxical. The source and nature of the paradox has been the focus of a fascinating debate. The crucial issue, of course, is whether Yablo's paradox involves circularity. Stephen Yablo (1993), Roy Sorensen (1998), and Bueno and Colyvan (2003b) have argued that the Yablo list generates a liar-like paradox without circularity. In the other camp are Graham Priest (1997) and JC Beall (2001), who argue that the paradox involves a fixed-point construction and therefore is circular. In Bueno and Colyvan (2003a), we respond by showing that there is a way of deriving a contradiction from the Yablo list without invoking any fixed-point construction and so, it would seem, the paradox does not essentially involve circularity.
In a recent paper, Jeffrey Ketland (2004) argues that our response is incorrect, and claims that the derivation presented in our paper is invalid.
(...)
