Gabriel Sandu

Frege has one magnificent achievement to his credit, viz. the creation of modern formal logic. As a philosopher and as a theoretical logician, he was nevertheless as parochial as he was, geographically speaking. Hence Frege’s concepts and problems offer singularly unfortunate starting points for constructive work in the foundations of logic and mathematics. Even if he is right in some of his views, they depend on severely restrictive assumptions that have to be noted and eliminated. These restrictive assumptions have not been sufficiently acknowledged in most of the recent discussions of Frege’s ideas. This has given rise to wild overstatements of Frege’s achievements. To give only two examples, Frege is sometimes said to have had within his reach “the concepts necessary to frame the notion of the completeness of a formalization of logic as well as soundness.” Or, he is credited with the creation of mathematical logic “issuing in a formal system logic incorporating propositional calculus, first- and second-order quantification theory, and a theory of sets developed within second-order quantification theory.” We will soon see that both these claims are not only severely exaggerated but that they miss some of the most characteristic features of Frege’s entire way of thinking about logical and semantical matters.

This chapter explores logical semantics, that is, the structural meaning of logical expressions like connectives, quantifiers, and modalities. It focuses on truth-theoretical semantics for formalized languages, a tradition emerging from Carnap's and Tarski's work in the first half of the last century that specifies the meaning of these expressions in terms of the truth-conditions of the sentences in which they occur. It considers Tarski-style definitions of the semantics of a given language in a stronger metalanguage, Tarski's impossibility results, and attempts to overcome them in the post-Tarskian tradition.

The so-called New Theory of Reference (Marcus, Kripke etc.) is inspired by the insight that in modal and intensional contexts quantifiers presuppose nondescriptive unanalyzable identity criteria which do not reduce to any descriptive conditions. From this valid insight the New Theorists fallaciously move to the idea that free singular terms can exhibit a built-in direct reference and that there is even a special class of singular terms (proper names) necessarily exhibiting direct reference. This fallacious move has been encouraged by a mistaken belief in the substitutional interpretation of quantifiers, by the myth of thede re reference, and a mistaken assimilation of direct reference to ostensive (perspectival) identification. Thede dicto vs.de re contrast does not involve direct reference, being merely a matter of rule-ordering (scope).The New Theorists' thesis of the necessity of identities of directly referred-to individuals is a consequence of an unmotivated and arbitrary restriction they tacitly impose on the identification of individuals.

Independence‐Friendly logic (IF‐logic) introduced by Hintikka and Sandu (1989) studies patterns of dependence and independence of quantifiers which exceed those found in ordinary first‐order logic. The present survey focuses on the game‐theoretical interpretation of IF‐logic, including connections to solution concepts (equilibria in mixed strategies) in classical game theory, but we shall also present its compositional interpretation together with its connections to notions of dependence and dependence between terms.

entities in mathematics There is a line of argument which keeps ontological commitments to the minimum by making use of conservativity results. The argument goes back to Hilbert who set its general frame. Hilbert’s concern was with certain abstract (ideal) entities in mathematics but the argument has been applied without discrimination to avoid ontological commitment to mathematical entities in physics (Field) or to avoid an ontological commitment to substantial properties in the case of truth (Horwich, Field, Williams).

Logics in which a relation R is semantically incomplete in a particular universe E, i.e. the union of the extension of R with its anti-extension does not exhaust the whole universe E, have been studied quite extensively in the last years. (Cf. van Benthem (1985), Blamey (1986), and Langholm (1988), for partial predicate logic; Muskens (1996), for the applications of partial predicates to formal semantics, and Doherty (1996) for applications to modal logic.) This is not so with semantically incomplete generalized quantifiers which constitute the subject of the present paper. The only systematic study of these quantifiers from a purely logical point of view, is, to the best of my knowledge, that by van Eijck (1995). We shall take here a different approach than that of van Eijck and mention some of the abstract properties of the resulting logic. Finally we shall prove that the two approaches are interdefinable

In this paper I am going to inquire to what extent the main requirements of a minimalist theory of truth and falsity (as formulated, for example, by Horwich and Field) can be consistently implemented in a formal theory. I will discuss several of the existing logical theories of truth, including Tarski-type (un)definability results, Kripke’s partial interpretation of truth and falsity, Barwise and Moss’ theory based upon non-well-founded sets, McGee’s treatment of truth as a vague predicate, and Hintikka’s languages of imperfect information, to see which axioms of the minimalist theory they satisfy or fail to satisfy. Finally, I will discuss the relation between the minimalist program and compositionality.