The book provides a historical and systematic exposition of the semantic theory of truth formulated by Alfred Tarski in the 1930s. This theory became famous very soon and inspired logicians and philosophers. It has two different, but interconnected aspects: formal-logical and philosophical. The book deals with both, but it is intended mostly as a philosophical monograph. It explains Tarski’s motivation and presents discussions about his ideas as well as points out various applications of the semantic theory of truth to philosophical (...) problems. (shrink)

This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to understand (...) Tarski’s and Gödel’s work, explaining why the problems they discussed are still unsolved. Finally, the book reports on some of the most influential positions in contemporary philosophy of mathematics, i.e., Maddy’s mathematical naturalism and Shapiro’s mathematical structuralism. Last but not least, the book introduces Biancani’s Aristotelian philosophy of mathematics as this is considered important to understand current philosophical issue in the applications of mathematics. One of the main purposes of the book is to stimulate readers to reconsider the Aristotelian position, which disappeared almost completely from the scene in logic and mathematics in the early twentieth century. (shrink)

We can distinguish two non-equivalent ways in which a natural language argument can be valid: it can be interpretationally or representationally valid. However, there is just one notion of classical first-order validity for formal languages: truth-preservation in all classical first-order models. To ease the tension, Baumgartner suggests that we should understand interpretational and representational validity as imposing different adequacy conditions on formalizations of natural language arguments. I argue against this proposal. To that end, I first show that Baumgartner’s definition of (...) representational validity is extensionally inadequate. I present a number of natural language arguments that we pre-theoretically hold to be representationally valid, but are not representationally valid according to Baumgartner’s definition. I then point to two further untenable features of Baumgartner’s definitions: according to Baumgartner’s definition of a representationally correct formalization, we cannot arrive at formalizations in a recursive way, and Baumgartner’s definition of representational validity is non-monotonic. I conclude that interpretational and representational validity cannot be understood as merely imposing different adequacy conditions on formalizations. If we want to capture our interpretational and representational intuitions, we need two different formal definitions of validity. (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)

A theory of definitions which places the eliminability and conservativeness requirements on definitions is usually called the standard theory. We examine a persistent myth which credits this theory to Leśniewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Leśniewski's published or unpublished work is known where the standard conditions are discussed. Second, Leśniewski's own logical theories allow for creative definitions. Third, Leśniewski's celebrated ‘rules of definition’ lay (...) merely syntactical restrictions on the form of definitions: they do not provide definitions with such meta-theoretical requirements as eliminability or conservativeness. On the positive side, we point out that among the Polish logicians, in the 1920s and 1930s, a study of these meta-theoretical conditions is more readily found in the works of Łukasiewicz and Ajdukiewicz. (shrink)

Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural assumptions regarding (...) what an acceptable neo-logicistic theory of value-ranges might look like, successfully implementing this alternative approach is impossible. (shrink)

Fine and Antonelli introduce two generalizations of permutation invariance — internal invariance and simple/double invariance respectively. After sketching reasons why a solution to the Bad Company problem might require that abstraction principles be invariant in one or both senses, I identify the most fine-grained abstraction principle that is invariant in each sense. Hume’s Principle is the most fine-grained abstraction principle invariant in both senses. I conclude by suggesting that this partially explains the success of Hume’s Principle, and the comparative lack (...) of success in reconstructing areas of mathematics other than arithmetic based on non-invariant abstraction principles. (shrink)

Maria Kokoszyńska-Lutmanowa was one of the most outstanding female representatives of the Lvov-Warsaw School. After achieving her PhD in philosophy under Kazimierz Twardowski’s supervision, she was Kazimierz Ajdukiewicz’s assistant. She was also influenced by Alfred Tarski whose results in semantics she analyzed and popularized. After World War II, she got the chair of logic in University of Wrocław and she organized studies in logic in this academic center. In the 1930s, Kokoszyńska kept in contact with members of the Vienna Circle (...) and became a kind of connecting factor between Polish logicians and the Viennese group. In Poland, she presented the views of members of the Vienna Circle. In Vienna, she emphasized the results of her Polish colleagues. In the present paper, some of Kokoszyńska’s results connected with the matters discussed in the Vienna Circle are presented, namely with the problem of metaphysics, the status of logic and the idea of unity of science. (shrink)

Many of Tarski’s better known papers are either about or include lengthy discussions of how to properly define various concepts: truth, logical consequence, semantic concepts, or definability. In general, these papers identify two primary conditions for successful definitions: formal correctness and material adequacy. Material adequacy requires that the concept expressed by the formal definition capture the intuitive content of truth. Our primary interest in this paper is to better understand Tarski’s thinking about material adequacy, and whether components of his view (...) developed over time. More precisely, we are concerned with how Tarski’s understanding of the content of the common-sense, every-day usage of truth may have developed over time. We distinguish this concern from the character of the extensional criterion of adequacy Tarski proposes: that a materially adequate definition must entail all instances of Convention T. We will develop our reading of Tarski as follows: first, we will review the “Polemical Remarks,” focusing primarily on §§14 and 17, and Tarski’s references to Naess’ empirical research. Next, we will provide a summary and discussion of Naess’ work, especially his findings with respect to Tarski’s definition of truth and his research that suggests there is no single common or everyday concept of truth. Third, we will consider several possible objections to our interpretation of the Tarski–Naess dialectic. We will conclude that Tarski’s conception of material adequacy developed over time, potentially because of what he had learned through his interactions with Naess. (shrink)

This volume portrays the Polish or Lvov-Warsaw School, one of the most influential schools in analytic philosophy, which, as discussed in the thorough introduction, presented an alternative working picture of the unity of science.

Narrowly speaking, the correspondence theory of truth is the view that truth is correspondence to a fact -- a view that was advocated by Russell and Moore early in the 20 th century. But the label is usually applied much more broadly to any view explicitly embracing the idea that truth consists in a relation to reality, i.e., that truth is a relational property involving a characteristic relation (to be specified) to some portion of reality (to be specified). During the (...) last 2300 years this basic idea has been expressed in many ways, resulting in a rather extended family of views, theories, and theory sketches. The members of the family employ various concepts for the relevant relation (correspondence, conformity, congruence, agreement, accordance, copying, picturing, signification, representation, reference, satisfaction) and/or various concepts for the relevant portion of reality (facts, states of affairs, situations, events, objects, sequences of objects, sets, properties, tropes). The resulting multiplicity of versions and reformulations of the theory is due to a blend of substantive and terminological differences. (shrink)

A good argument is one whose conclusions follow from its premises; its conclusions are consequences of its premises. But in what sense do conclusions follow from premises? What is it for a conclusion to be a consequence of premises? Those questions, in many respects, are at the heart of logic (as a philosophical discipline). Consider the following argument: 1. If we charge high fees for university, only the rich will enroll. We charge high fees for university. Therefore, only the rich (...) will enroll. There are many different things one can say about this argument, but many agree that if we do not equivocate (if the terms mean the same thing in the premises and the conclusion) then the argument is valid, that is, the conclusion follows deductively from the premises. This does not mean that the conclusion is true. Perhaps the premises are not true. However, if the premises are true, then the conclusion is also true, as a matter of logic. This entry is about the relation between premises and conclusions in valid arguments. (shrink)

This paper is a critique of coercive theories of meaning, that is, theories (or criteria) of meaning designed to do down ones opponents by representing their views as meaningless or unintelligible. Many philosophers from Hobbes through Berkeley and Hume to the pragmatists, the logical positivists and (above all) Wittgenstein have devised such theories and criteria in order to discredit their opponents. I argue 1) that such theories and criteria are morally obnoxious, a) because they smack of the totalitarian linguistic tactics (...) of the Party in Orwell’s 1984 and b) because they dehumanize the opposition by portraying them as mere spouters of gibberish; 2) that they are profoundly illiberal since if true, they would undermine Mill’s arguments for free speech; 3) that such theories are prone to self-contradiction, pragmatic and otherwise; 4) that they often turn against their creators including what they were meant to exclude and excluding what they were meant to include; 5) that such theories are susceptible to a modus tollens pioneered by Richard Price in his Review Concerning the Principle Questions of Morals(1758); and 6) that such theories are prima facie false since they fail to ‘predict’ the data that is it their business to explain (or, in the case of criteria, fail to capture the concept that they allegedly represent). The butcher’s bill is quite considerable: some of Hobbes, a fair bit of Locke, half of Berkeley, large chunks of Hume, Russell's Theory of Types, verificationism in its positivist and Dummettian variants, much of pragmatism and most of Wittgenstein - all these have to be sacrificed if we are to save our souls as philosophic liberals. (shrink)

This special issue collects together nine new essays on logical consequence :the relation obtaining between the premises and the conclusion of a logically valid argument. The present paper is a partial, and opinionated,introduction to the contemporary debate on the topic. We focus on two influential accounts of consequence, the model-theoretic and the proof-theoretic, and on the seeming platitude that valid arguments necessarilypreserve truth. We briefly discuss the main objections these accounts face, as well as Hartry Field’s contention that such objections (...) show consequenceto be a primitive, indefinable notion, and that we must reject the claim that valid arguments necessarily preserve truth. We suggest that the accountsin question have the resources to meet the objections standardly thought to herald their demise and make two main claims: (i) that consequence, as opposed to logical consequence, is the epistemologically significant relation philosophers should be mainly interested in; and (ii) that consequence is a paradoxical notion if truth is. (shrink)