Wilfrid Hodges; VIII*—Truth in a Structure, Proceedings of the Aristotelian Society, Volume 86, Issue 1, 1 June 1986, Pages 135–152, https://doi.org/10.1093/ari.

Tarski has exerted enormous influence not only on the development of mathematical logic, but on twentieth-century philosophy and philosophical analysis. This influence has been twofold, with the two components pulling in a sense in opposite directions. A comparison with the influence of the Vienna Circle provides an instructive vantage point in viewing Tarski’s influence. On the one hand, Tarski has provided powerful tools for logical analysis in philosophy. His first and most important contribution was to show that — and how (...) — the crucial semantical concept of truth can be defined for various formal languages. They include the most central types of language used and studied by twentieth century logicians and philosophers. The subsequent work by Tarski and his school in creating contemporary model theory in its narrow sense as a branch of technical logic has likewise enriched tremendously the conceptual arsenal of philosophers. The extent and depth of Tarski’s influence is in fact easily underestimated. In comparison, Carnap’s well-known books in logical syntax and logical semantics did not in the same way contribute to the logical techniques that philosophers might find useful. A large part of Tarski’s influence lies in facilitating a switch of emphasis from purely syntactical to model-theoretical, semantical methods. It was Gödel’s and Tarski’s early work that forced logicians and philosophers for the first time to take seriously the distinction between semantical and syntactical concepts and methods. The general theoretical implications of the distinction were not realized at once. In the light of hindsight, it is for instance quite striking that Tarski still in his classical monograph repeatedly assimilates to each other the task of defining truth for some part of mathematics or logic and axiomatizing its truths in a sense that involves explicit syntactical rules of inference. It was nevertheless Tarski more than anyone else who began to develop and exploit systematically model-theoretical concepts in logical theory. In comparison, Gödel never developed systematically the model-theoretical aspects of logic. Others, for instance Quine, stubbornly remained within the syntactical tradition. Carnap, who did switch his attention from the syntactical to a semantical approach, was not powerful enough a logician to create a sufficiently rich store of model-theoretical techniques and ideas to be equally helpful in philosophical analysis. In contrast, several major developments in analytic philosophy have had their starting points in Tarski’s ideas. Well-known cases in point are Davidson’s truth-conditional semantics and the possible worlds semantics developed by Tarski’s favorite former student Richard Montague. (shrink)

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? (...) What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski. In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable. (shrink)

The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation ∼. Then in a Hilbert space ∼ turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann "quantum logic" (...) can be interpreted by taking their "disjunction" to be ¬(∼A & ∼ B). Their logic can thus be mapped into a Boolean structure to which an additional operator ∼ has been added. (shrink)

In game-theoretical semantics, perfectlyclassical rules yield a strong negation thatviolates tertium non datur when informationalindependence is allowed. Contradictorynegation can be introduced only by a metalogicalstipulation, not by game rules. Accordingly, it mayoccur (without further stipulations) onlysentence-initially. The resulting logic (extendedindependence-friendly logic) explains several regularitiesin natural languages, e.g., why contradictory negation is abarrier to anaphase. In natural language, contradictory negationsometimes occurs nevertheless witin the scope of aquantifier. Such sentences require a secondary interpretationresembling the so-called substitutionalinterpretation of quantifiers.This interpretation is sometimes impossible,and (...) it means a step beyond thenormal first-order semantics, not an alternative to it. (shrink)