In his famous paper Der Wahrheitsbegriff in den formalisierten Sprachen (Polish edition: Nakładem/Prace Towarzystwa Naukowego Warszawskiego, wydzial, III, 1933), Alfred Tarski constructs a materially adequate and formally correct definition of the term “true sentence” for certain kinds of formalised languages. In the case of other formalised languages, he shows that such a construction is impossible but that the term “true sentence” can nevertheless be consistently postulated. In the Postscript that Tarski added to a later version of this paper (Studia Philosophica, (...) 1, 1935), he does not explicitly include limits for the kinds of language for which such a construction is possible. This absence of such limits has been interpreted as an implied claim that such a definition of the term “true sentence” can be constructed for every language. This has far-reaching consequences, not least for the widely held belief that Tarski changed from an universalistic to an anti-universalistic standpoint. We will claim that the consequence of anti-universalism is unwarranted, given that it can be argued that the Postscript is not in conflict with the existence of limits outside of which a definition of “true sentence” cannot be constructed. Moreover, by a discussion of transfinite type theory, we will also be able to accommodate other of the changes made in Tarski’s Postscript within a type-theoretical framework. The awareness of transfinite type theory afforded by this discussion will lead, in turn, to an account of Tarski’s Postscript that shows a gradual change in his logical work, rather than any of the more radical transitions which the Postscript has been claimed to reflect. (shrink)

I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.

The paper [Tarski: Les fondements de la géométrie des corps, Annales de la Société Polonaise de Mathématiques, pp. 29—34, 1929] is in many ways remarkable. We address three historico-philosophical issues that force themselves upon the reader. First we argue that in this paper Tarski did not live up to his own methodological ideals, but displayed instead a much more pragmatic approach. Second we show that Leśniewski's philosophy and systems do not play the significant role that one may be tempted to (...) assign to them at first glance. Especially the role of background logic must be at least partially allocated to Russell's systems of Principia mathematica. This analysis leads us, third, to a threefold distinction of the technical ways in which the domain of discourse comes to be embodied in a theory. Having all of this in place, we discuss why we have to reject the argument in [Gruszczyński and Pietruszczak: Full development of Tarski's Geometry of Solids, The Bulletin of Symbolic Logic, vol. 4, no. 4, pp. 481—540] according to which Tarski has made a certain mistake. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style ofPrincipia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of (...) axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition (Carnap and Tarski) and two proponents of the first-order tradition (Gödel and Hilbert), we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory. (shrink)

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. -/- There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably (...) infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. -/- Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects. (shrink)

Many papers have been devoted to the semantic turn Carnap took in the late 1930s after Tarski had explained to him his method for defining truth and his work on the establishment of scientific semantics. Commentators have often argued that the major turn in Carnap’s approach to languages had already been taken in the Logical Syntax of Language, but they have usually assumed that Carnap was happy to subsequently follow Tarski and adopt Tarskian semantics. In this paper, it is argued (...) that this assumption needs to be qualified and that Carnap was actually far from following Tarski when he decided to complement his syntactic method with a semantic one. Carnap and Tarski had different goals, divergent programs, and dissenting views on truth and semantics. After exploring several possible methods for the explication of logical concepts such as L-truth and L-implication, Carnap opted for definitions based on concepts he had found in Wittgenstein’s Tractatus and Waismann’s work on a logical interpretation of probability. Carnap’s reasons for taking a semantic turn are to be understood in the context of his principle of tolerance. He first hoped he could use Tarski’s technique to recover in the semantic setting the completeness result he had tried to establish in the Logical Syntax. His eventual adoption of non-Tarskian semantics can be accounted for by his program of a logical analysis of science—including empirical science—and his ambition to elaborate a unified logical framework for deductive, inductive, and modal logic, and for explicating analyticity. (shrink)

G. Schiemer has recently ascribed to Carnap the so-called domains-as-fields conception of models, which he subsequently used to defend Carnap’s treatment of extremal axioms against J. Hintikka’s criticism that the number of tuples in a relation, and not the domain of discourse, is optimised in Carnap’s treatment. We will argue by a careful textual analysis, however, that this domains-as-fields conception cannot be applied to Carnap’s early semantics, because it includes a notion of submodel and subrelation that is not only absent (...) from Carnap’s work at that time, but even contradicts it. As a consequence, Schiemer’s defense of Carnap’s extremal axioms against Hintikka’s criticism fails. We will reconcile Carnap’s treatment of extremal axioms and Hintikka’s observation by taking into account the practice of axiomatics in the early twentieth century. If one realises that, in Carnap’s time, a predicate for the domain of discourse was often introduced in the formal theory, and that Carnap defined such predicates from the basic relations of an axiom system, the apparent disagreement between optimising relations and optimising domains disappears. (shrink)

What we call first‐order logic over fixed domain was initiated, in a certain guise, by Peirce around 1885 and championed, albeit in idiosyncratic form, by Zermelo in papers from the 1930s. We characterise such logics model‐ and proof‐theoretically and argue that they constitute exploration of a clearly circumscribed conception of domain‐dependent generality. Whereas a logic, or family of such, can be of interest for any of a variety of reasons, we suggest that one of those reasons might be that said (...) logic fosters some clarification regarding just what qualifies as a logical concept, a logical operation, or a logical law. (shrink)

True beliefs and truth-preserving inferences are, in some sense, good beliefs and good inferences. When an inference is valid though, it is not merely truth-preserving, but truth-preserving in all cases. This motivates my question: I consider a Modus Ponens inference, and I ask what its validity in particular contributes to the explanation of why the inference is, in any sense, a good inference. I consider the question under three different definitions of ‘case’, and hence of ‘validity’: the orthodox definition given (...) in terms of interpretations or models, a metaphysical definition given in terms of possible worlds, and a substitutional definition defended by Quine. I argue that the orthodox notion is poorly suited to explain what's good about a Modus Ponens inference. I argue that there is something good that is explained by a certain kind of truth across possible worlds, but the explanation is not provided by metaphysical validity in particular; nothing of value is explained by truth across all possible worlds. Finally, I argue that the substitutional notion of validity allows us to correctly explain what is good about a valid inference. (shrink)

This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ (...) in his theory of axiomatics is presented. Specifically, it is shown that Carnap’s early model theory is based on a convention to simulate domain variation that is not identical but logically comparable to the modern account. (shrink)

In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to (...) be. By providing such a reassessment, the paper contributes to a reevaluation of Carnap’s contributions to the development of modern logic. (shrink)

In the paper the following questions are discussed: What is logical consequence? What are logical constants? What is a logical system? What is logical pluralism? What is logic? In the conclusion, the main tendencies of development of modern logic are pointed out.