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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)On truth-schemes for intensional logics
By Janusz Czelakowski and Wieslaw Dziobiak
Scope Dominance with Upward Monotone Quantifiers
By Alon Altman, Ya'acov Peterzil & Yoad WinterWe 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)
Evidentiality, Modality and Probability
By Eric McCready & Norry OgataWe show in this paper that some expressions indicating source of evidence are part of propositional content and are best analyzed as a special kind of epistemic modal. Our evidence comes from the Japanese evidential system. We consider six evidentials in Japanese, showing that they can be embedded in conditionals and under modals and that their properties with respect to modal subordination are similar to those of ordinary modals. We show that these facts are difficult for existing theories of evidentials, which assign evidentials necessarily widest scope, to explain. We then provide an analysis using a logical system designed to account for evidential reasoning; this logic is the first developed system of probabilistic dynamic predicate logic. This analysis is shown to account for the data we provide that is problematic for other theories.
Keywords: evidentiality, modality, probability, Japanese, dynamic semantics
Source: Semantics Archive
Essence and Modality
By Edward N. ZaltaRecently, K. Fine raised counterexamples to the the traditional definition of essential property in terms of modality. On the traditional definition, ‘property F is essential to object x' is defined in terms of the modal claim ‘necessarily, if x exists, then x is F'. The definiens, it is argued, is not a sufficient condition for the definiendum. One counterexample, which assumes modal set theory, is that (a) necessarily, if Socrates exists, then he has the property being a member of {Socrates}, but (b) the property being a member of {Socrates} is not essential to Socrates. Another counterexample (which assumes the existence of an object not identical to Socrates (e.g., the Eiffel Tower). Fine suggests that (a) necessarily, if Socrates exists, then he has the property of being distinct from the Eiffel Tower, but (b) the property being distinct from the Eiffel Tower is not essential to Socrates, since "nothing in Socrates' nature connects him in any special way to the Eiffel Tower".
In this paper, I analyze the relationship between essence and modality and reconsider the above counterexamples in light of the logic and theory of abstract objects. This axiomatic theory offers a foundational metaphysics and yields a clear analysis of the nature of abstract objects in general and mathematical objects such as {Socrates}. The theory is consistent with our intuitions about what ordinary objects there are, and the underlying logic offers a new understanding of the properties essential to ordinary objects. The analysis of mathematical and other abstract objects offers a more refined view of their essential properties than that offered by modal set theory.
In the paper,, the claim ‘x has F necessarily' becomes ambiguous in its application to abstract objects. In the case of ordinary objects, the definition of ‘F is essential to x' be reconstructed in several ways. The conclusion is that the traditional definition of essential property for abstract objects in terms of modal notions is not correct, but not because of Fine's first counterexample. Moreover, in the case of ordinary objects, the relationship between essential properties and modality, once properly understood, can handle the second counterexample.
Published in Mind, Volume 115/Issue 459 (July 2006): 659-693
ON THE DA COSTA, DUBIKAJTIS AND KOTAS' SYSTEM OF THE DISCURSIVE LOGIC, D*2
By Janusz CiuciuraIn the late forties, Stanislaw Jaskowski published two papers on the discursive (or discussive) sentential calculus, D2. He provided a definition of it by an interpretation in the language of S5 of Lewis. The known axiomatization of D2 with discursive connectives as primitives was introduced by da Costa, Dubikajtis and Kotas in 1977. It turns out, however, that one of the axioms they used is not a thesis of the real Ja?›kowski's calculus. In fact, they built a new system, D*2 for short, that differs from D2 in many respects. The aim of this paper is to introduce a direct Kripke-type semantics for the system, axiomatize it in a new way and prove soundness and completeness theorems. Additionally, we present labeled tableaux for D*2.
Keywords: discursive (discussive) logic, D2, paraconsistent logic, labelled tableaux.
Published in Logic and Logical Philosophy, Volume 14 (2005), 235-252
The complexity of theorem-proving procedures
By Stephen A. CookIt is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be "reduced" to the problem of determining whether a given propositional formula is a tautology. Here "reduced" means, roughly speaking, that the first problem can be solved deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of determining tautologyhood has the same polynomial degree as the problem of determining whether the first of two given graphs is isomorphic to a subgraph of the second. Other examples are discussed. A method of measuring the complexity of proof procedures for the predicate calculus is introduced and discussed.
In Proceedings of the 3rd ACM Symposium on Theory of Computing, pages 151--158, 1971
Author's link (a compressed pdf file of a scanned version)
Other link
The Concept of Mathematical Elucidation:
theory and problems
By José SeoaneThere is a contrast between concepts which may be treated in accordance with the criteria of mathematical rigor and concepts which are not susceptible of such a treatment. We will call theoretical concepts to the former and pre-theoretical to the latter. In mathematical world, sentences which relate theoretical and pre-theoretical concepts (in a determinated way) are denominated thesis; sentences which relate only theoretical concepts (in a determinated way) are denominated theorems. Elucidatory processes in mathematics have thesis as their principal output. I intend to establish in this paper that the introduction of the concept of mathematical elucidation has an important theoretical value to the effects of studying a certain special type of intended conceptual relation and a certain type of justificatory argumentation for it. The analysis of the contrast between thesis and theorems will allow us to construct a context where the interest for those conceptual relations and their supporting justificatory mechanisms arises naturally. Then, I will attempt to offer some structural features of mathematical elucidation qua conceptual relation and their impact on the strategies of elucidatory justification. This is what I grandiloquently call theory in the heading. I will suggest also a classification of the problems which a theoretical reflexion as the one proposed may contribute to clarify and I will make some brief observations on some paradigmatic examples of each one of the categories of the classification constructed. This work should be considered as a modest definition of a sort of research program.
Source: CLE e-prints
Worlds and Times
By Ulrich MeyerThere 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
Induction and Comparison
By Paul PietroskiLogical induction may be important for theoretical linguistics, even if children do not induce languages from experience. Either our human capacities for inductive reasoning lie near the heart of our capacity to generate and understand expressions of a human language, or not. If they do, then theoretically minded linguists should try to understand human inductive capacities and the kinds of understanding they make possible, independent of other cognitive capacities. If not, then we should be clear about this, and not pretend otherwise-say, by adopting semantic theories that exploit the full resources of the logic that Frege used to reduce arithmetic to Hume's Principle. But suppose our best theories of language do presuppose that speakers have inductive capacities. Then considerations of theoretical parsimony suggest that we theorists should squeeze as much as we can from our representations of human inductive capacities, before adding controversial assumptions about how speakers understand expressions. This leaves room for hypotheses according to which speakers understand certain sentences in terms of covert quantification over abstracta.
Source: Semantics Archive
Quantification and Second-Order Monadicity
By Paul M. PietroskiOnce we grant that grammatical structure can be as complicated as logical structure, and just as distant from audible features of word strings, we can approach the study of human cognition by combining the insights of modern logic (and not just its first-order fragment) and linguistics. Those deciding where to invest might want to compare this project, in terms of the results it has delivered and its potential for delivering more in the foreseeable future, with alternative projects that philosophers of language and mind have been pursuing. My bias in this regard will be evident. Though a more dispassionate assessment might lead to much the same conclusion: for now, our best hope of learning something important about the structure of thought-and giving substance to the ancient idea of language as a mirror of the mind-lies with figuring out how Frege, Chomsky, Montague, Davidson, and many others could each be importantly right, and no doubt wrong, about the same thing: namely, the shared syntactic/semantic structure of our sentences/thoughts. My suggestion is that this structure is more conjunctive, monadic, and second-order than one might think.
Source: Semantics Archive
Truth and the Unprovability of Consistency
By Hartry FieldIt might be thought that we could argue for the consistency of a mathematical theory T within T, by giving an inductive argument that all theorems of T are true and inferring consistency. By Gödel's second incompleteness theorem any such argument must break down, but just how it breaks down depends on the kind of theory of truth that is built into T. The paper surveys the possibilities, and suggests that some theories of truth give far more intuitive diagnoses of the breakdown than do others. The paper concludes with some morals about the nature of validity and about a possible alternative to the idea that mathematical theories are indefinitely extensible.
Under review
Professor Field is one of the leading advocates of the Deflationary Theory of Truth and his views are among the most challenging ones at the turn of the century. This papers tries to tackle one of the most difficult and most important Philosophical issues of the XXth century.
Semantic computations of truth, based on associations already learned
By Patrick Suppes & Jean-Yves Beziau
In this article we try to give an account of how one determines the truth or falsity of sentences like: Paris is the capital of France, Paris is not the capital of France, Rome is the capital of France. We want to describe the computations underlying the answers given, taking into account, at least in a qualitative way, the time factor what psychologists call the latency of a response. Our theory should be able to explain the data gathered by experimentation, for example, why it takes more time to give a negative answer than a positive one, be it true or false. But the important theoretical question is what is the actual method of computation, a problem not ordinarily considered in philosophical theories of truth, but also not subject to direct empirical observation.
Intensional Models for the Theory of Types
By Reinhard MuskensIn this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up.
Source: Semantics Archive, To appear in the The Journal of Symbolic Logic
A STRATEGY FOR ASSESSING CLOSURE
By Peter MurphyThis paper looks at an argument strategy for assessing the epistemic closure principle. This is the principle that says knowledge is closed under known entailment; or (roughly) if S knows p and S knows that p entails q, then S knows that q. The strategy in question looks to the individual conditions on knowledge to see if they are closed. According to one conjecture, if all the individual conditions are closed, then so too is knowledge. I give a deductive argument for this conjecture. According to a second conjecture, if one (or more) condition is not closed, then neither is knowledge. I give an inductive argument for this conjecture. In sum, I defend the strategy by defending the claim that knowledge is closed if, and only if, all the conditions on knowledge are closed. After making my case, I look at what this means for the debate over whether knowledge is closed.
Forthcoming in Erkenntnis
Scopal Independence:
A Note on Branching and Wide Scope Readings of Indefinites and Disjunctions
By Philippe Schlenker
Hintikka claimed in the 1970s that indefinites and disjunctions give rise to 'branching readings' that can only be handled by a 'game-theoretic' semantics as expressive as a logic with (a limited form of) quantification over Skolem functions. Due to empirical and methodological difficulties, the issue was left unresolved in the linguistic literature. Independently, however, it was discovered in the 1980s that, contrary to other quantifiers, indefinites may scope out of syntactic islands. We claim [here] that branching readings and the island-escaping behavior of indefinites are two sides of the same coin: when the latter problem is considered in full generality, a mechanism of 'functional quantification' (Winter 2004) must be postulated which is strictly more expressive than Hintikka's, and which predicts that his branching readings are indeed real, although his own solution was insufficiently general. Furthermore, we suggest that, as Hintikka had seen, disjunctions share the behavior of indefinites, both with respect to island-escaping behavior and (probably) branching readings. The functional analysis can thus naturally be extended to them.
Source: Institut Jean-Nicod.
To appear in Journal of Semantics.
