Tarski and Primitivism About Truth
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Poznań Studies in the Philosophy of the Sciences and the Humanities), 21Pp
Forthcoming in: Uncovering Facts and Values, ed. A. Kuzniar and J. Odrowąż-Sypniewska (Poznań Studies in the Philosophy of the Sciences and the Humanities), 21pp. Abstract Alfred Tarski seems to endorse a partial conception of truth, the T-schema, which he believes might be clarified by the application of empirical methods, specifically citing the experimental results of Arne Næss (1938a). The aim of this paper is to argue that Næss’ empirical work confirmed Tarski’s semantic conception of truth, among others. In the first part, I lay out the case for believing that Tarski’s T-schema, while not the formal and generalizable Convention-T, provides a partial account of truth that may be buttressed by an examination of the ordinary person’s views of truth. Then, I address a concern raised by Tarski’s contemporaries who saw Næss’ results as refuting Tarski’s semantic conception. Following that, I summarize Næss’ results. Finally, I will contend with a few objections that suggest a strict interpretation of Næss’ results might recommend an overturning of Tarski’s theory. Keywords: truth, Alfred Tarski, Arne Næss, Vienna Circle, experimental philosophy Joseph Ulatowski ORDINARY TRUTH IN TARSKI AND NÆSS 1. Introduction Many of Alfred Tarski's better known papers on truth (e.g. 1944; 1983b), logical consequence (1983c), semantic concepts in general (1983a), or definability (1948) identify two conditions that successful definitions of “truth,” “logical consequence,” or “definition” require: formal correctness and material (or intuitive) adequacy.1 The first condition Tarski calls “formal correctness” because a definition of truth (for a given formal language) is formally correct when it is constructed in a manner that allows us to avoid both circular definition and semantic paradoxes. -
Alfred Tarski His:Bio:Tar: Sec Alfred Tarski Was Born on January 14, 1901 in Warsaw, Poland (Then Part of the Russian Empire)
bio.1 Alfred Tarski his:bio:tar: sec Alfred Tarski was born on January 14, 1901 in Warsaw, Poland (then part of the Russian Empire). Described as \Napoleonic," Tarski was boisterous, talkative, and intense. His energy was often reflected in his lectures|he once set fire to a wastebasket while disposing of a cigarette during a lecture, and was forbidden from lecturing in that building again. Tarski had a thirst for knowledge from a young age. Although later in life he would tell students that he stud- ied logic because it was the only class in which he got a B, his high school records show that he got A's across the board| even in logic. He studied at the Univer- sity of Warsaw from 1918 to 1924. Tarski Figure 1: Alfred Tarski first intended to study biology, but be- came interested in mathematics, philosophy, and logic, as the university was the center of the Warsaw School of Logic and Philosophy. Tarski earned his doctorate in 1924 under the supervision of Stanislaw Le´sniewski. Before emigrating to the United States in 1939, Tarski completed some of his most important work while working as a secondary school teacher in Warsaw. His work on logical consequence and logical truth were written during this time. In 1939, Tarski was visiting the United States for a lecture tour. During his visit, Germany invaded Poland, and because of his Jewish heritage, Tarski could not return. His wife and children remained in Poland until the end of the war, but were then able to emigrate to the United States as well. -
Alfred Tarski and a Watershed Meeting in Logic: Cornell, 1957 Solomon Feferman1
Alfred Tarski and a watershed meeting in logic: Cornell, 1957 Solomon Feferman1 For Jan Wolenski, on the occasion of his 60th birthday2 In the summer of 1957 at Cornell University the first of a cavalcade of large-scale meetings partially or completely devoted to logic took place--the five-week long Summer Institute for Symbolic Logic. That meeting turned out to be a watershed event in the development of logic: it was unique in bringing together for such an extended period researchers at every level in all parts of the subject, and the synergetic connections established there would thenceforth change the face of mathematical logic both qualitatively and quantitatively. Prior to the Cornell meeting there had been nothing remotely like it for logicians. Previously, with the growing importance in the twentieth century of their subject both in mathematics and philosophy, it had been natural for many of the broadly representative meetings of mathematicians and of philosophers to include lectures by logicians or even have special sections devoted to logic. Only with the establishment of the Association for Symbolic Logic in 1936 did logicians begin to meet regularly by themselves, but until the 1950s these occasions were usually relatively short in duration, never more than a day or two. Alfred Tarski was one of the principal organizers of the Cornell institute and of some of the major meetings to follow on its heels. Before the outbreak of World War II, outside of Poland Tarski had primarily been involved in several Unity of Science Congresses, including the first, in Paris in 1935, and the fifth, at Harvard in September, 1939. -
Basic Concepts of Set Theory, Functions and Relations 1. Basic
Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6 -
Alfred Tarski, Friend and Daemon Benjamin Wells
Alfred Tarski, Friend and Daemon Benjamin Wells Engaging Tarski When I had passed doctoral exams in 1963 and Alfred Tarski’s name stayed with me after I read sounded Bob Vaught and John Addison on their about the Banach-Tarski paradox in [3] during availability for supervising my research, both said high school. I then discovered logic (and Tarski’s that the department recognized me as Tarski’s definition of truth) in the last year of college but student. News to me, despite the second opinion. still considered myself to be a topologist, not from So I called him to learn more about my fate. He love but from intimate contact in four courses as acknowledged the claim, “Good,” and invited me an undergraduate. So before arriving at UC Berke- for the first serious nighttime talk. ley in fall 1962 for graduate work, I thought only Source of the Title vaguely about taking a course with him. But my inclinations were shifting: dutifully registering Throughout cooperation and separation, Alfred for fall classes in topological groups and algebraic and I were friends, cordial and personable, but geometry, I added metamathematics and Tarski’s not really personal. He also established another general algebraic systems. The next fall, Tarski of- role, that of daemon in the sense of [4]: a leonine fered set theory. These two courses were my only externalized conscience, at least. The scope of Tarski lectures, but there were numerous seminars. this conscience was foremost mathematical, with a hope for political, a goal of cultural, a reserva- Visitors to Berkeley constantly identified Tarski tion on philosophical and moral, and a hint of with whatever topic occupied him so fruitfully and spiritual. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
Definitions and Nondefinability in Geometry 475 2
Definitions and Nondefinability in Geometry1 James T. Smith Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which han- dled some basic concepts awkwardly and imprecisely. They would introduce precision re- quired for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geo- metric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By fol- lowing this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics. 1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for general- ization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the ques- tion, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal. -
Lecture 1: Tarski on Truth Philosophy of Logic and Language — HT 2016-17
Lecture 1: Tarski on Truth Philosophy of Logic and Language — HT 2016-17 Jonny McIntosh [email protected] Alfred Tarski (1901-1983) was a Polish (and later, American) mathematician, logician, and philosopher.1 In the 1930s, he published two classic papers: ‘The Concept of Truth in Formalized Languages’ (1933) and ‘On the Concept of Logical Consequence’ (1936). He gives a definition of truth for formal languages of logic and mathematics in the first paper, and the essentials of the model-theoretic definition of logical consequence in the second. Over the course of the next few lectures, we’ll look at each of these in turn. 1 Background The notion of truth seems to lie at the centre of a range of other notions that are central to theorising about the formal languages of logic and mathematics: e.g. validity, con- sistency, and completeness. But the notion of truth seems to give rise to contradiction: Let sentence (1) = ‘sentence (1) is not true’. Then: 1. ‘sentence (1) is not true’ is true IFF sentence (1) is not true 2. sentence (1) = ‘sentence (1) is not true’ 3. So, sentence (1) is true IFF sentence (1) is not true 4. So, sentence (1) is true and sentence (1) is not true Tarski is worried that, unless the paradox can be resolved, metatheoretical results in- voking the notion of truth or other notions that depend on it will be remain suspect. But how can it be resolved? The second premise is undeniable, and the first premise is an instance of a schema that seems central to the concept of truth, namely the following: ‘S’ is true IFF S, 1The eventful story of Tarski’s life is told by Anita and Solomon Feferman in their wonderful biography, Alfred Tarski: Life and Logic (CUP, 2004). -
Carnap's Contribution to Tarski's Truth
JOURNAL FOR THE HISTORY OF ANALYTICAL PHILOSOPHY CARNAP’S CONTRIBUTION TO TARSKI’S TRUTH VOLUME 3, NUMBER 10 MONIKA GrUBER EDITOR IN CHIEF KEVIN C. KLEMENt, UnIVERSITY OF MASSACHUSETTS In his seminal work “The Concept of Truth in Formalized Lan- guages” (1933), Alfred Tarski showed how to construct a for- EDITORIAL BOARD mally correct and materially adequate definition of true sentence GaRY EBBS, INDIANA UnIVERSITY BLOOMINGTON for certain formalized languages. These results have, eventu- GrEG FROSt-ARNOLD, HOBART AND WILLIAM SMITH COLLEGES ally, been accepted and applauded by philosophers and logi- HENRY JACKMAN, YORK UnIVERSITY cians nearly in unison. Its Postscript, written two years later, SANDRA LaPOINte, MCMASTER UnIVERSITY however, has given rise to a considerable amount of contro- LyDIA PATTON, VIRGINIA TECH versy. There is an ongoing debate on what Tarski really said MARCUS ROSSBERG, UnIVERSITY OF CONNECTICUT in the postscript. These discussions often regard Tarski as pu- MARK TEXTOR, KING’S COLLEGE LonDON tatively changing his logical framework from type theory to set AUDREY YAP, UnIVERSITY OF VICTORIA theory. RICHARD ZACH, UnIVERSITY OF CALGARY In what follows, we will compare the original results with those REVIEW EDITORS presented two years later. After a brief outline of Carnap’s pro- JULIET FLOYD, BOSTON UnIVERSITY gram in The Logical Syntax of Language we will determine its sig- CHRIS PINCOCK, OHIO STATE UnIVERSITY nificance for Tarski’s final results. ASSISTANT REVIEW EDITOR SEAN MORRIS, METROPOLITAN STATE UnIVERSITY OF DenVER DESIGN DaNIEL HARRIS, HUNTER COLLEGE C 2015 MONIKA GrUBER CARNAP’S CONTRIBUTION TO TARSKI’S TRUTH couple of years ago. In particular, in addition to the previously studied languages, he decides to investigate the languages the MONIKA GrUBER structure of which cannot be brought into harmony with the principles of the theory of semantical categories. -
The History of Logic
c Peter King & Stewart Shapiro, The Oxford Companion to Philosophy (OUP 1995), 496–500. THE HISTORY OF LOGIC Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and nega- tion in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (uni- versal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’). -
Taboo Versus Axiom
TABOO VERSUS AXIOM KATUZI ONO Some important formal systems are really developable from a finite number of axioms in the lower classical predicate logic LK or in the intuitionistic predicate logic LJ. Any system of this kind can be developed in LK (or in LJ) from the single conjunction of all the axioms of the system. I have been intending to develop usual formal systems starting from TABOOS and standing on the primitive logic LO at first, since the logic too could be brought up to the usual logics by means of TABOOS in LOυ. In developing any formal system from TABOOS, we are forced to assume unwillingly that all the TABOOS are mutually equivalent, as far as we adopt more than one TABOOS for the system. If we could develop formal systems from single- TABOO TABOO-systems, we could get rid of this unwilling assumption. It would be well expected that any formal theory developable from a finite number of axioms in LK or in LJ would be developable from a single TABOO.2) In the present paper, I will prove that this is really the case. Before stating the theorem, let us define the δ-TRANSFORM Spy] of any sentence ® with respect to an h-ary relation £y (λ>0). SΓ&3 is introduced by a structural recursive definition as follows: @C8Qs (r)((©->g(r))->#(r)) for any elementary sentence ©, Received January 20, 1966. *> As for this plan, see Ono [1]. LO is called PRIMITIVE SYSTEM OF POSITIVE LOGIC in Ono [1]. The terminology PRIMITIVE LOGIC together with its reference notation LO has been introduced in Ono [2], 2) In my work [1], I had to assume this to bring up logics by means of TABOO systems. -
Biographies.Pdf
Chapter udf Biographies bio.1 Georg Cantor his:bio:can: An early biography of Georg Cantor sec (gay-org kahn-tor) claimed that he was born and found on a ship that was sailing for Saint Petersburg, Russia, and that his parents were unknown. This, however, is not true; although he was born in Saint Petersburg in 1845. Cantor received his doctorate in mathematics at the University of Berlin in 1867. He is known for his work in set theory, and is credited with founding set theory as a distinctive research discipline. He was the first to prove that there are infinite sets of different sizes. His theo- ries, and especially his theory of infini- ties, caused much debate among mathe- maticians at the time, and his work was controversial. Cantor's religious beliefs and his Figure bio.1: Georg Cantor mathematical work were inextricably tied; he even claimed that the theory of transfinite numbers had been communicated to him directly by God. Inlater life, Cantor suffered from mental illness. Beginning in 1894, and more fre- quently towards his later years, Cantor was hospitalized. The heavy criticism of his work, including a falling out with the mathematician Leopold Kronecker, led to depression and a lack of interest in mathematics. During depressive episodes, Cantor would turn to philosophy and literature, and even published a theory that Francis Bacon was the author of Shakespeare's plays. Cantor died on January 6, 1918, in a sanatorium in Halle. 1 Further Reading For full biographies of Cantor, see Dauben(1990) and Grattan-Guinness(1971).