Logic, Semantics, Metamathematics. By A. Tarski. Translated by J. H. Woodger. Pp. 471. 60s. 1956. (Clarendon Press)

Die grundlagen der arithmetik: by G. Frege. English translation by J. L. Austin. 119 pages, 14 × 22 cm. Breslau, Verlag von Wilhelm Koebner, 1884, and New York, Philosophical Library, 1950. Price, $4.75

Bayes or Bust?-A Critical Examination of Bayesian Confirmation Theory.

Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters-one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

The Oxford handbook of philosophy of mathematics and logic

Can there be grounding without necessitation? Can a fact obtain wholly in virtue of metaphysically more fundamental facts, even though there are possible worlds at which the latter facts obtain but not the former? It is an orthodoxy in recent literature about the nature of grounding, and in first-order philosophical disputes about what grounds what, that the answer is no. I will argue that the correct answer is yes. I present two novel arguments against grounding necessitarianism, and show that grounding contingentism is fully compatible with the various explanatory roles that grounding is widely thought to play.

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Against Grounding Necessitarianism

• When we assert something about the world: (1) we make a distinction amongst ways for the world to be, and (2) we commit to one side of the distinction. Example: When we assert ‘Snow is white’: (1) we distinguish between white-snow and non-white-snow ways for the world to be, and (2) we commit the the white-snow side of this distinction. • For the assertion to be true is for the way the world actually is to be on the side of the distinction we are committed to. For the assertion to be true is for the world to lie on the white-snow side of this distinction. 2. Logical Space

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The Construction of Logical Space

Etude de la critique du consensus entre semantique du premier ordre, semantique du second ordre et theorie des ensembles chez Quine, developpee par G. Boolos a partir d'un scheme distinguant les quantificateurs monadiques et les quantificateurs polyadiques.

Nominalism through de-nominalization

Godel claimed that Zermelo-Fraenkel set theory is ‘what becomes of the theory of types if certain superfluous restrictions are removed’. The aim of this paper is to develop a clearer understanding of Godel's remark, and of the surrounding philosophical terrain. In connection with this, we discuss some technical issues concerning infinitary type theories and the programme of developing the semantics for higher-order languages in other higher-order languages.

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Hierarchies ontological and ideological

The purpose of this paper is to defend a conception of language that does not rely on linguistic meanings, and use it to address the Sorites and Liar paradoxes.

/pdf/a-plea-for-semantic-localism-4dysmdrc66.pdf

A Plea for Semantic Localism

Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, treated as variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals. Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. For the case of first-order languages, interpretations are standardly cashed out in terms of what might be called model-theoretic interpretations (or MT-interpretations). An MT-interpretation for a first-order language L is an ordered pair 〈D, F 〉. The domain D is a non-empty set, and is intended to specify the range of the variables in L. The interpretation function F is intended to specify semantic values for the variables of L (including non-logical expressions). The semantic value of an n-place predicate-letter is a set of n-tuples of individuals in D, and the semantic value of a first-order variable is an individual in D. Truth on an MT-interpretation can then be characterized as follows:

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Agustín Rayo

Papers

The Construction of Logical Space

Nominalism through de-nominalization

Hierarchies ontological and ideological

A Plea for Semantic Localism

A Completeness Theorem for Unrestricted First-Order Languages