Tarski On Logical Notions

We try to explain Tarski's conception of logical notions, as it emerges from alecture of his, delivered in 1966 and published posthumously in 1986 (Historyand Philosophy of Logic 7, 143–154), a conception based on the idea ofinvariance. The evaluation of Tarski's proposal leads us to consider an interesting(and neglected) reply to Skolem in which Tarski hints at his own point of view onthe foundations of set theory. Then, comparing the lecture of 1966 with Tarski'slast work and with an earlier paper written with Lindenbaum, it is shown thatTarski's conception of logical notions, with its essentially type-theoretic character,did not undergo any significant modifications throughout his life. A remark onTarski's prudential attitude on the topic in the famous paper on the concept oflogical consequence (and elsewhere) concludes our paper.     

Ray on Tarski on Logical Consequence

In Logical consequence: A defense of Tarski (Journal of Philosophical Logic, vol. 25, 1996, pp. 617–677), Greg Ray defends Tarski"s account of logical consequence against the criticisms of John Etchemendy. While Ray"s defense of Tarski is largely successful, his attempt to give a general proof that Tarskian consequence preserves truth fails. Analysis of this failure shows that de facto truth preservation is a very weak criterion of adequacy for a theory of logical consequence and should be replaced by a stronger absence-of-counterexamples criterion. It is argued that the latter criterion reflects the modal character of our intuitive concept of logical consequence, and it is shown that Tarskian consequence can be proved to satisfy this criterion for certain choices of logical constants. Finally, an apparent inconsistency in Ray"s interpretation of Tarski"s position on the modal status of the consequence relation is noted.     

Logical matrices and non-structural consequence operators

In the present paper, we study some properties of matrices for non-structural consequence operators. These matrices were introduced in a former work (see [3]). In sections 1. and 2., general definitions and theorems are recalled; in section 3. a correspondence is studied, among our matrices and Wójcicki's ones for structural operators. In section 4. a theorem is given about operators, induced by submatrices or epimorphic images, or quotient matrices of a given one. Such matrices are used to characterize lattices of non-structural consequence operators, by constructing lattices, antiisomorphic to them (see section 5.). In the last section, a sufficient condition is given for a non-structural operator to be finite.     

塔斯基：语义性真理论与符合论

塔斯基语义性真理论是否为符合论是一个争论的热点。目前有两种观点认为它是符合论,其一是根据对象语言与元语言的区分,到目前为止此观点已受到激烈批评,其二是根据“满足”概念的递归定义,此观点到目前为止还没有被人详细地阐明。本文认为塔斯基的语义性真理论是符合论,为此将首先对第一种观点及其反对意见进行深入分析,并得出结论认为该观点不能成立,继而详细阐明第二种观点,表明它是塔斯基本人的意图。在讨论中本文将对一系列的逻辑哲学问题进行探讨。     

A note on three-valued logic and Tarski theorem on truth definitions

We introduce a notion of semantical closure for theories by formalizing Nepeivoda notion of truth. [10]. Tarski theorem on truth definitions is discussed in the light of Kleene's three valued logic (here treated with a formal reinterpretation of logical constants). Connections with Definability Theory are also established.A first version was prepared for sec. 5-Gnsaga-CNR and presented at the Logic Seminar, Institute of Philosophy, University of Firenze, in the academic year 1978–79. The author would like to thank the referee for his criticism and the Alexander von Humboldt-Stiftung for the support during the revision of the paper.     

Tarski on the Necessity Reading of Convention T

Tarski’s Convention T is often taken to claim that it is both sufficient and necessary for adequacy in a definition of truth that it imply instances of the T-schema where the embedded sentence translates the mentioned sentence. However, arguments against the necessity claim have recently appeared, and, furthermore, the necessity claim is actually not required for the indefinability results for which Tarski is justly famous; indeed, Tarski’s own presentation of the results in the later Undecidable Theories makes no mention of an assumption to the effect that the definition of truth implies the biconditionals. This raises a question: was Tarski in fact committed to the necessity claim in the important papers of the 1930s and 40s? I argue that he was not. The discussion of this apparently esoteric interpretive issue in fact gets to the heart of many important questions about truth, and in the final sections of the paper I discuss the importance of the T-biconditionals in the theory of meaning and the relation of deflationary and inflationary theories of truth to the semantic paradoxes.     

The coherence of Omniscience: A defense

International Journal for Philosophy of Religion -     

Logical inference in English: A preliminary analysis

The perfect fit of syntactic derivability and logical consequence in first-order logic is one of the most celebrated facts of modern logic. In the present flurry of attention given to the semantics of natural language, surprisingly little effort has been focused on the problem of logical inference in natural language and the possibility of its completeness. Even the traditional theory of the syllogism does not give a thorough analysis of the restricted syntax it uses.My objective is to show how a theory of inference may be formulated for a fragment of English that includes a good deal more than the classical syllogism. The syntax and semantics are made as formal and as explicit as is customary for artificial formal languages. The fragment chosen is not maximal but is restricted severely in order to provide a clear overview of the method without the cluttering details that seem to be an inevitable part of any grammar covering a substantial fragment of a natural language. (Some readers may feel the details given here are too onerous.)I am especially concerned with quantifier words in both object and subject position, with negation, and with possession. I do not consider propositional attitudes or the modalities of possibility and necessity, although the model-theoretic semantics I use has a standard version to deal with such intensional contexts.An important point of methodology stressed in earlier publications (Suppes, 1976; Suppes & Macken, 1978; Suppes, 1979) is that the semantic representation of the English sentences in the fragment uses neither quantifiers nor variables, but only constants denoting given sets and relations, and operations on sets and relations.In the first section, I rapidly sketch the formal framework of generative syntax and model-theoretic semantics, with special attention to extended relation algebras. The second section states the grammar and semantics of the fragment of English considered. The next section is concerned with developing some of the rules of inference. The results given are quite incomplete. The final section raises problems of extension. Classical logic is a poor guide for dealing with inferences involving high-frequency function words such as of, to, a, in, for, with, as, on, at, and by. Indeed, the line between logical and nonlogical inference in English seems to be nonexistent or, if made, highly arbitrary in character-much more so than has been claimed by those critical of the traditional analyticsynthetic tradition.No theorems on soundness or completeness are considered because of the highly tentative and incomplete character of the rules of inference proposed. However, because of the variable-free semantics used, soundness is easy to establish for the rules given.The research reported here has been supported in part by National Science Foundation Grant No. SED77-09698.     

Supervaluationism and Logical Consequence: A Third Way

It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness. This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion.     

The Definition of Moral Dilemmas: A Logical Problem

This paper concerns one of the undecided disputes of modern moral philosophy: the possibility of moral dilemmas. Whereas proponents of the possibility of moral dilemmas often appeal to moral experience, many opponents refer to ethical theory and deontic logic. My aim in this paper is to clarify some of the tension between moral experience and ethical theory with respect to moral dilemmas. In Part One I try to show that a number of logical arguments against the possibility of moral dilemmas, though apparently very different, turn out to be basically the same, as they are all based on the following concept of ought: if A ought to be done, doing B is impermissible and doing A itself is permissible. In Part Two I present an overview of several definitions of moral dilemmas that have been given by proponents of moral dilemmas: definitions that define moral dilemmas in terms of oughts and definitions that define them in terms of reasons. I conclude that, while reason is to weak, ought is too strong a concept to define moral dilemmas with. In this way, the arguments from Part One create a logical problem for proponents of the possibility of moral dilemmas to define moral dilemmas.     

A Logical Theory of Localization

A central problem in applying logical knowledge representation formalisms to traditional robotics is that the treatment of belief change is categorical in the former, while probabilistic in the latter. A typical example is the fundamental capability of localization where a robot uses its noisy sensors to situate itself in a dynamic world. Domain designers are then left with the rather unfortunate task of abstracting probabilistic sensors in terms of categorical ones, or more drastically, completely abandoning the inner workings of sensors to black-box probabilistic tools and then interpreting their outputs in an abstract way. Building on a first-principles approach by Bacchus, Halpern and Levesque, and a recent continuous extension to it by Belle and Levesque, we provide an axiomatization that shows how localization can be realized wrt a basic action theory, thereby demonstrating how such capabilities can be enabled in a single logical framework. We then show how the framework can also enable localization for multiple agents, where an agent can appeal to the sensing already performed by another agent and the knowledge of their relative positions to localize itself.