In this article the author first described the developments which brought to focus the importance of consistency proofs for mathematics, and which led Hilbert to promote the science of metamathemat-ics. Further comments and remarks concern the (partly analogous) beginnings of the work on the decision problem, Gödel?s theorems and related matters, and general metamathematics. An appendix summarizes a text by the author on completeness and categoricity.

Alfred Tarski’s theory of truth, to which we will also refer as the classical theory,1 has a conspicuous place in mathematics as well as in general philosophy. The place in philosophy appears the more prominent of the two, although it is still somewhat unsettled, and perhaps even controversial.

The relevance of metamathematical researches for philosophy of math- ematics is an indubitable matter. In the paper I shall speak about impli- cations of metamathematics for general philosophy, especially for classical epistemological problems. Let us start with a historical observation con- cerning Hilbert's programme, the rst research programme in metamathe- matics as a separate study of formal systems. This programme was strongly in uence by epistemological considerations. In fact, Hilbert wanted to se- cure all classical mathematics against inconsistencies and this (...) aim had be achieved with the help of nitary consistency proof. Hilbert's claim is like the Cartesian project of reduction of all concepts to clarae et distinctae ideae. The epistemological commitment of original Hilbert's programme may be treated as a prelim- inary heuristic motivation for looking for epistemological implications of metamathematical results. It seems reasonable to examine in this respect three so-called limitative theorems: the rst Godel's incompleteness theo- rem , the second Godel's incom- pleteness theorem consistency of S, cannot be proved in S, providing that S is consistent) and Tarski's undenability theorem ; \S" stands for formal system containing elementary number theory. From the above-mentioned theorems we obtain an important, from philosophical point of view, conclusion: semantics of S cannot be expressed in S. Now, I shall present three applications of limitative theorems to the analysis of classical epistemological problems. (shrink)

The nominalistic ontology of Kotarbinski, Slupecki and Tarski does not provide any direct interpretations of the sets of higher types which play important roles in type theory and in set theory. For this and other reasons I will interpret those theories as descriptions of some finite structures which are actually constructed in human imaginations and stored in their memories. Those structures will be described in this lecture. They are hinted by the idea of Skolem functions and Hilbert's -symbols, and (...) they constitute a finitistic modification of Tarski's concept of a model. They suggest also a form of the evolutionary process which leads to the development of human intelligence and language. (shrink)

The purpose of this article is to highlight a selected few of Alfred Tarski's career achievements. The choice of these achievements is subjective. Section 1 is a general sketch of his life and work, emphasizing his role as researcher, teacher, organizer and founder of a scientific school. Section 2 discusses his contributions to set theory. Section 3 discusses his contributions to the foundations of geometry and to measure theory. Section 4 looks at his metamathematical work, and especially the decision (...) problem for formalized theories. Section 5 is a selected bibliography to illustrate Sects. 1–4. (shrink)

The larger part of Yearbook 6 of the Institute Vienna Circle constitutes the proceedings of a symposium on Alfred Tarski and his influence on and interchanges with the Vienna Circle, especially those on and with Rudolf Carnap and Kurt Gödel. It is the first time that this topic has been treated on such a scale and in such depth. Attention is mainly paid to the origins, development and subsequent role of Tarski's definition of truth. Some contributions are primarily (...) historical, others analyze logical aspects of the concept of truth. Contributors include Anita and Saul Feferman, Jan Wolenski, Jan Tarski and Hans Sluga. Several Polish logicians contributed: Gzegorczyk, Wójcicki, Murawski and Rojszczak. The volume presents entirely new biographical material on Tarski, both from his Polish period and on his influential career in the United States: at Harvard, in Princeton, at Hunter, and at the University of California at Berkeley. The high point of the analysis involves Tarski's influence on Carnap's evolution from a narrow syntactical view of language, to the ontologically more sophisticated but more controversial semantical view. Another highlight involves the interchange between Tarski and Gödel on the connection between truth and proof and on the nature of metalanguages. The concluding part of Yearbook 6 includes documentation, book reviews and a summary of current activities of the Institute Vienna Circle. Jan Tarski introduces letters written by his father to Gödel; Paolo Parrini reports on the Vienna Circle's influence in Italy; several reviews cover recent books on logical empiricism, on Gödel, on cosmology, on holistic approaches in Germany, and on Mauthner. (shrink)

This paper presents the life and work of Alfred Tarski, one of the most distinguished and influential logicians in the entire history.

The larger part of Yearbook 6 of the Institute Vienna Circle constitutes the proceedings of a symposium on Alfred Tarski and his influence on and interchanges with the Vienna Circle, especially those on and with Rudolf Carnap and Kurt Gödel. It is the first time that this topic has been treated on such a scale and in such depth. Attention is mainly paid to the origins, development and subsequent role of Tarski's definition of truth. Some contributions are primarily (...) historical, others analyze logical aspects of the concept of truth. Contributors include Anita and Saul Feferman, Jan Wolenski, Jan Tarski and Hans Sluga. Several Polish logicians contributed: Gzegorczyk, Wójcicki, Murawski and Rojszczak. The volume presents entirely new biographical material on Tarski, both from his Polish period and on his influential career in the United States: at Harvard, in Princeton, at Hunter, and at the University of California at Berkeley. The high point of the analysis involves Tarski's influence on Carnap's evolution from a narrow syntactical view of language, to the ontologically more sophisticated but more controversial semantical view. Another highlight involves the interchange between Tarski and Gödel on the connection between truth and proof and on the nature of metalanguages. The concluding part of Yearbook 6 includes documentation, book reviews and a summary of current activities of the Institute Vienna Circle. Jan Tarski introduces letters written by his father to Gödel; Paolo Parrini reports on the Vienna Circle's influence in Italy; several reviews cover recent books on logical empiricism, on Gödel, on cosmology, on holistic approaches in Germany, and on Mauthner. (shrink)

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)

According to a common opinion, the word ‘semantics’ , derived from the Greek word semantikos , appeared for the first time, at least in modern times, in the book Essai de semantique, science de significations by M. J. A. Bréal . However, Quine says in his lectures on Carnap:As used by C. S. Peirce, “semantic” is the study of the modes of denotation of signs: whether a sign denotes its object through causal or symptomatic connection, or through imagery, or through (...) arbitrary convention and so on. This sense of semantic, namely a theory of meaning, is used also in empirical philology: empirical semantic is the study of historical changes of meanings of words.1For Bréal, semantics was a branch of general linguistics. In particular, semantics was occupied with so-called lexical meaning and its changes through time. Thus, semantics in this sense belonged to what was called “the diachronic treatment of language”. This tradition is fairly alive in contemporary linguistic theory. Quine’s description of the word ‘semantic’ in Peirce corresponds, which Quine explicitly states, to its use in philology. However, some linguists ascribe a more theoretical role to lingustic semantics. Karl Bühler is an example. In his Sprachtheorie he says that a theory of semantic functions of language is a part of theory of language.2 This account is to be found also among philosophers. It is also rather obvious that Peirce did not limit his semantic only to empirical studies. Linguists also use the word ‘semasiology’ instead of ‘semantics’; Bühler proposed the term ‘sematology’ for a general theory of symbols. (shrink)

The paper examines the interplays between logic and politics in the Polish School of Logic starting from 1914. The Polish School of Logic flourished between 1920 and 1939. Philosophically, it was influenced by Kazimierz Twardowski (1866–1938). For Twardowski logic is fundamental for every kind of human activity, professional and private and this means that every argument should be formulated and proceed by correct inferential rules. These rules involve semiotics, formal logic and methodology of science. The paper shows how this position (...) was shared by Twardowski’s students, including Jan Łukasiewicz, Stanisław Leśniewski, Kazimierz Ajdukiewicz and Tadeusz Kotrabiński, as well as by the next generation of logicians and philosophers and especially by Alfred Tarski. These authors considered logic, philosophy and science as completely neutral with respect to politics and ideology but they also considered logical skills as indispensable in political activities. (shrink)

The observation that the standard solution of the paradox of the Liar is not satisfactory as a pragmatic solution of a semantic problem restores its former status as a semantic antinomy. Since the antinomy originates from Tarski's T scheme, a conservative modification of the standard semantics is looked for, which would prevent applying the scheme T to anomalous statements. Two such modifications are considered. The first is simpler and implies Kleene's weak tables for three-valued logic. The second, more complex (...) but also more intuitive, agrees with Kleene's strong tables. This undermines the seemingly natural intuition that a sentence and the sentence attributing truth to it have the same truth conditions, which is compatible only with the weak tables. (shrink)

Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing exists. The (...) result can then be strengthened to the conclusion that infinitely many things exist. Those things are not just all Gödel codes of sentences but rather all natural numbers. Against this background inclusive negative free logic collapses into noninclusive free logic, which collapses into classical logic. The consequences for ontological deflationism with respect to truth are discussed. (shrink)

This paper aims to describe and examine Alfred Tarski's famous semantic conception of truth as well as some of the critiques presented against it. The first part of this paper is divided into five segments: criteria imposed upon every adequate definition of truth are discussed in the first segment; the second is dedicated to Tarski’s Convention T; distinction between object language and metalanguage, as well as Tarski’s attitude toward formalized and colloquial languages, is described in the third (...) segment; fourth segment is devoted to Tarski’s definition of truth using the notion of “satisfaction”; the last segment is the evaluation of Tarski’s solution of paradoxes. In the second part of this paper the selected objections against Tarski’s conception are presented with emphasis on their diversity. (shrink)

Alfred Tarski frequently declared that his semantic definition of truth was inspired by Aristotle’s views. The present paper discusses this issue in the context of Marian Wesoły’s criticism of the thesis that there is an affinity between Tarski’s views and those of Aristotle. The article concludes with an inquiry into whether Aristotle’s definition of truthfulness can be identified with the correspondence theory of truth.

The Semantic Theory of Truth The semantic theory of truth was developed by Alfred Tarski in the 1930s. The theory has two separate, although interconnected, aspects. First, it is a formal mathematical theory of truth as a central concept of model theory, one of the most important branches of mathematical logic. Second, it … Continue reading Semantic Theory of Truth →.

One can often encounter an opinion that Polish scientific philosophy deserves to be much better known than actually is. This book is thought as a response to such a claim. The papers collected in this volume are divided into two parts: Background and Influence and History and Systematics. However, there is no sharp borderline between themes which are touched in both parts. Generally speaking, all papers of the first part relate the Lvov-Warsaw School to some philosophical movements external to it (...) whereas the papers collected in the second one focus on internal issues connected with the school . Since the Polish school of mathematical logic is much better known than the Polish analytic philosophy we decided to omit here any treatment of the former. Thus, this collection centers on purely philosophical matters. We projected this volume not as an exhaustive panorama of Polish analytic philosophy but rather as a series of essays on particular persons or topic. As a result one can find here papers on Twardowski. Ajdukiewicz, Kotarbinski, Tarski and Lukasiewicz as well as on ethics on science, nominalism, and the methodology of psychology. We hope that this book will contribute to a better knowledge and evaluation of Polish achievements in analytic philosophy. We would like to express our gratitude to Professor Leszek Nowak, the editor-in-chief of Poznan Studies in the Philosophy of the Sciences and the Humanities, who initiated the idea of the collection and helped in its preparation. (shrink)

The semantic theory of truth, formulated by Alfred Tarski in the 1939s, is primarily a mathematical theory. On the other hand, it also has a considerable philosophical content. This paper presents the second aspect of this theory. It can be shown that several traditional philosophical issues pertaining to the concept of truth can be illuminated by Tarski’s account of truth. It concerns, for instance, the idea of correspondence, the relation of truth and logic, the problem of the relativity/absoluteness (...) of truth, etc. (shrink)

This papers discuss the place, if any, of Convention T (the condition of material adequacy of the proper definition of truth formulated by Tarski) in the truth-makers account offered by Kevin Mulligan, Peter Simons and Barry Smith. It is argued that although Tarski’s requirement seems entirely acceptable in the frameworks of truth-makers theories for the first-sight, several doubts arise under a closer inspection. In particular, T-biconditionals have no clear meaning as sentences about truth-makers. Thus, truth-makers theory cannot be (...) considered as the semantic theory of truth enriched by metaphysical (ontological) data. The problem of truth-makers for sentences about future events is discussed at the end of the paper. (shrink)

Ancien etudiant de Brentano et de Zimmerman, Kazimierz Twardowski, apres son election a la chaire de philosophie a Lvov en 1895, crea autour de lui un cercle d'etudiants et de collaborateurs exceptionnel, connu aujourd'hui sous le nom d'Ecole de Lvov-Varsovie. A mi-chemin entre Vienne et Cambridge, c'est a Lvov, et puis partiellement a Varsovie, que Jan Lukasiewicz, Stanislaw Lesniewski, Alfred Tarski, Kazimierz Ajdukiewicz, Tadeusz Kotarbinski et bien d'autres encore, repenserent dans un esprit d'analyse les questions fondamentales de la philosophie (...) du langage, de la logique, de la philosophie des sciences et des mathematiques. Plus qu'une simple traduction, ce livre est une version revisee de la monographie desormais classique que Jan Wolenski, connu pour ses travaux non seulement en histoire de la philosophie analytique, mais aussi en epistemologie et en theorie de la verite, a consacree a l'Ecole de Lvov-Varsovie. (shrink)

The Polish School of Logic flourished in the period 1920-1939. Philosophically, it was influenced by Kazimierz Twardowski, professor at the University of Lwow (now Lviv in Ukraine), who established the Lwow-Warsaw School, to which the mentioned logical group belonged. Twardowski claimed that logic is very important in every kind of human activity, professional as well as private. Hence, every argument should be clearly formulated and proceed by correct inferential rules. These postulates involved semiotics, formal logic, and methodology of science — (...) that is, logica sensu largo. This position was accepted by Twardowski’s most distinguished students, such as Jan Łukasiewicz, Stanisław Leśniewski, Kazimierz Ajdukiewicz, and Tadeusz Kotarbiński, who graduated before 1914, as well as the next generation of logicians and philosophers, particularly by Alfred Tarski. Although all these people considered logic, philosophy, and science as completely neutral with respect to politics and ideology, they treated logical skills as indispensable in political activities. In philosophical specialized terminology, Polish logicians regarded logic as a weapon against irrationalism. This position was also represented by Polish logicians who did not belong to the group of Twardowski’s students. (shrink)

In 1930 Otto Neurath asked Alfred Tarski for information about Polish philosophers of science and their work. Tarski answered on 24 April, 1930, saying in one of the first sentences that.

There are many versions of the Liar Paradox. J. Agassi names 13 of them. But the most important one is related to Tarski’s theorem that the truth predicate is non-definable for systems that are sufficient for the formalisation of elementary arithmetic of natural numbers. Let S be such a system. We assume that S is consistent and that the syntax of S has been arithmetized as understood by G¨odel. Let E be any sentence of S. E ∗ is the (...) symbol of the G¨odel number of E. If formula A belongs to the language of system S, then formula E is a constant point for formula A if and only if S ⊢ E → A. It can be proved that every formula A ∈ S has a constant point in S. (shrink)

According to the received view formal semantics applies to natural language to some extent only. It is so because natural language is inherently indefinite, in particular, its expressions are ambiguous, vague and admits departures from syntactic rule. Moreover, intensional contexts occur in ordinary language—it results in limitations of the principle of compositionality. The ordinary conversation appeals to various principles, for instance, Grice’s maxims which exceed logical formalism. Thus, ordinary language cannot be fully formalized. On the other hand, if L is (...) a formal language, its metalanguage ML, must be partially informal—for instance, it contains, terms of ordinary mathematics, especially set theory. Even, if, for instance, due to the technique of aritmetization, ML can be represented in L, such a representation is only local. In fact, this view can be derived from some Tarski’s remarks on the role played by natural language. It is usually assumed that the universality of natural language, is the source of troubles associated with antinomies. It is so this circumstance requires a solution, for example by distinguishing levels of language. However, even if antinomies are excluded, what is informal is prior with respect to what is formal. It shows that formal semantics has limitations even with respect to formalized languages. (shrink)

Roman Ingarden offered a strong criticism of the verifiability principle in his talk delivered at the 8th International Congress in Prague in 1934. Ingarden argued that this principle either violates itself or smuggles a hidden sense. In this paper I show that Ingarden-like arguments about smuggled (but this pejorative qualification is skipped) meaning apply not only to the criteria of sense, but also to other semantic assertions within language-hierarchies in Tarski’s sense.

Poland has played an enormous role in the development of mathematical logic. Leading Polish logicians, like Lesniewski, Lukasiewicz and Tarski, produced several works related to philosophical logic, a field covering different topics relevant to philosophical foundations of logic itself, as well as various individual sciences. This collection presents contemporary Polish work in philosophical logic which in many respects continue the Polish way of doing philosophical logic. This book will be of interest to logicians, mathematicians, philosophers, and linguists.

This paper comments Susan Haack’s remarks about Twardowski’s criticism of relativism in the theory of truth. The author summarizes Twardowski’s arguments for truth-absolutism and tries to show that that their presentation by Haack is incomplete. The defense of Twardowski’s position in the paper uses ideas developed by Tarski and Kokoszyñska.

Truth and meaning are perhaps the most important notions of logical semantics. This is why so much weight is attached to their mutual relationship. This query is considered in this article, primarily within the frame of the semantic definition of truth, formulated by Alfred Tarski in the early 1930’s ; the remarks on other concepts will be marginal.

This paper proposes a formal framework for the cognitive relation understood as an ordered pair with the cognitive subject and object of cognition as its members. The cognitive subject is represented as consisting of a language, conequence relation and a stock of accepted theories, and the object as a model of those theories. This language allows a simple formulation of the realism/anti-realism controversy. In particular, Tarski’s undefinability theorem gives a philosophical argument for realism in epistemology.

Les débats qui ont eu lieu, lors du Congrès de philosophie scientifique de 1935, sur la sémantique et sa portée philosophique présentent un grand intérêt historique pour deux raisons. Tout d’abord, on s’accorde à y reconnaître un des évènements majeurs du congrès. En second lieu, et de façon plus substantielle, ils ont joué un rôle décisif dans le développement de la sémantique comme discipline philosophique. C’est Carnap qui en a pris l’initiative en invitant Tarski à donner deux conférences. Ce (...) sont eux qui, avec Kokoszyńska, deviendront les porte-parole de la sémantique. Neurath, qui a préparé le résumé de la discussion, avait plusieurs objections contre la sémantique à la Tarski. En particulier, il craignait que la sémantique n’introduise la métaphysique dans la philosophie. La suite a montré que les partisans de la sémantique ont gagné. (shrink)

Les débats qui ont eu lieu, lors du Congrès de philosophie scientifique de 1935, sur la sémantique et sa portée philosophique présentent un grand intérêt historique pour deux raisons. Tout d’abord, on s’accorde à y reconnaître un des évènements majeurs du congrès. En second lieu, et de façon plus substantielle, ils ont joué un rôle décisif dans le développement de la sémantique comme discipline philosophique. C’est Carnap qui en a pris l’initiative en invitant Tarski à donner deux conférences. Ce (...) sont eux qui, avec Kokoszyńska, deviendront les porte-parole de la sémantique. Neurath, qui a préparé le résumé de la discussion, avait plusieurs objections contre la sémantique à la Tarski. En particulier, il craignait que la sémantique n’introduise la métaphysique (au sens que les Viennois donnaient au mot) dans la philosophie. La suite a montré que les partisans de la sémantique ont gagné. (shrink)

The main question of our article is: What is the logical form of statements containing expressions such as “… is true” and “it is true that …”? We claim that these expressions are generally not used in order to assign a certain property to sentences. We indicate that a predicative interpretation of these expressions was rejected by Frege and adherents to the prosentential conception of truth. We treat these expressions as operators. The main advantage of our operational reading is the (...) fact that it adequately represents how the words, “true” and “truth,” function in everyday speech. Our approach confirms the intuition that so-called T-equivalences are not contingent truths, and explains why they seem to be—in some sense—necessary sentences. Moreover, our operational readingof truth expressions dissolves problems arising from the belief that there is some specific property—truth. The fact that we reject that truth is a certain property does not mean that we deny that the concept of truth plays a very important role in our language, and hence in our life. We indicate that the concept of truth is inseparable from the concept of sentence and vice versa—it is impossible to explicate one of these concepts without appeal to the other. (shrink)

Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega $, of Kripke–Feferman (...) truth theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators. (shrink)

We prove that the modal logic of a crowded locally compact generalized ordered space is $\textsf {S4}$. This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space.

In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti and Reichenbach on the one hand, and by Cooper and Cantwell on the other. Here we provide the proof theory for the resulting logics DF/TT and CC/TT, using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: DF/TT allows for (...) algebraic completeness, but not for the construction of a canonical model, while CC/TT fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects. (shrink)

We develop the theory of Krull dimension forS4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for aT1-space to have a finite modal Krull dimension can be described by an appropriate generalization of the well-known concept of a nodec space. This, in turn, can (...) be described by modal formulaszemnwhich generalize the well-known Zeman formulazem. We show that the modal logicS4.Zn:=S4+ zemnis the basic modal logic ofT1-spaces of modal Krull dimension ≤n, and we construct a countable dense-in-itselfω-resolvable Tychonoff spaceZnof modal Krull dimensionnsuch thatS4.Znis complete with respect toZn. This yields a version of the McKinsey-Tarski theorem forS4.Zn. We also show that no logic in the interval [S4n+1S4.Zn) is complete with respect to any class ofT1-spaces. (shrink)