In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

We recall one Tarski's result about the existence of a non-zero additive measure defined on power set of an infinite set vanishing on finite subsets. This rather surprising result goes back to 1930 and it allows to introduce a non-trivial linear functional invariant under changes of finitely many inputs.

This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

We provide for the first time an exact translation into English of the Polish version of Alfred Tarski's classic 1936 paper, whose title we translate as ?On the Concept of Following Logically?. We also provide in footnotes an exact translation of all respects in which the German version, used as the basis of the previously published and rather inexact English translation, differs from the Polish. Although the two versions are basically identical, to an extent that is even uncanny, we note (...) more than 400 differences. Several dozen of these are substantive differences due to revisions by Tarski to the Polish version which he did not incorporate in the German version. With respect to these revisions the Polish version should be regarded as more authoritative than the German. Hence scholars limited to an English translation should use ours. (shrink)

This book is well known for its proof that many mathematical systems - including lattice theory and closure algebras - are undecidable. It consists of three treatises from one of the greatest logicians of all time: "A General Method in Proofs of Undecidability," "Undecidability and Essential Undecidability in Mathematics," and "Undecidability of the Elementary Theory of Groups.".

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.

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

We provide for the first time an exact translation into English of the Polish version of Alfred Tarski's classic 1936 paper, whose title we translate as ‘On the Concept of Following Logically’. We also provide in footnotes an exact translation of all respects in which the German version, used as the basis of the previously published and rather inexact English translation, differs from the Polish. Although the two versions are basically identical, to an extent that is even uncanny, we note (...) more than 400 differences. Several dozen of these are substantive differences due to revisions by Tarski to the Polish version which he did not incorporate in the German version. With respect to these revisions the Polish version should be regarded as more authoritative than the German. Hence scholars limited to an English translation should use ours. (shrink)

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.

Editor’s Introduction: We recall that Alfred Tarski arrived in the USA from Poland in September 1939. The present series of letters starts not quite three years after his arrival; this span of time allowed him to adapt himself tentatively to his situation, and to shift much of his attention from the problems of the Old World to those of his immediate surroundings.

Cette Introduction à la logique constitue le cours de base des étudiants en sciences et en philosophie. La présentation de la logique formelle dans son contexte historique est accompagnée d'un exposé très clair des notions syntaxiques et sémantiques nécessaires au calcul des énoncés et au calcul des prédicats du premier ordre... Les méthodes formelles les plus classiques (tables de vérité, arbres de consistance, dérivations) sont explicitées au travers de nombreux exercices d'application accompagnés de leurs corrigés. La seconde partie de l'ouvrage (...) fournit au lecteur les contenus théoriques nécessaires à la compréhension des débats qui ont animés la philosophie de la logique et son dialogue avec les mathématiques depuis la fin du XIX, siècle. L'axiomatisation du calcul des énoncés (présentation de So), les notions de complétude sémantique et syntaxique comme le concept de décidabilité sont introduits. Un chapitre est consacré à l'axiomatisation du calcul des prédicats du premier ordre et permet de familiariser le lecteur à la métathéorie. La présentation des résultats fondateurs (théorème de consistance, théorème de Tarski, théorème de complétude) jette les bases de la théorie des modèles. Enfin, l'ouvrage s'achève par les fondations de la logique mathématique (système Se de la théorie des ensembles et système Sn de la théorie des nombres naturels) et par l'exposé des célèbres théorèmes de Gôdel... (shrink)

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

What is a logical constant? The question is addressed in the tradition of Tarski's definition of logical operations as operations which are invariant under permutation. The paper introduces a general setting in which invariance criteria for logical operations can be compared and argues for invariance under potential isomorphism as the most natural characterization of logical operations.

This paper is an historical study of Tarski's methodology of deductive sciences (in which a logic S is identified with an operator Cn S, called the consequence operator, on a given set of expressions), from its appearance in 1930 to the end of the 1970s, focusing on the work done in the field by Roberto Magari, Piero Mangani and by some of their pupils between 1965 and 1974, and comparing it with the results achieved by Tarski and the Polish school (...) (Łoś, Suszko, Słupecki, Pogorzelski, Wójcicki). In the last section of the paper we will then compare these works with some recent developments in algebraic logic: this will lead to a better understanding of the results of the methodology of deductive science, but at the same time will show some intrinsic limits to such an approach to logic. Even if Magari's work on diagonizable algebras and universal algebra and Mangani's axiomatization of MV-algebras and results in model theory are rather famous, the articles on closure operators, published in the 1960s, are almost totally unknown outside Italy (mainly because of a linguistic limitation, the papers we analyse having been written and published in Italian). This paper aims to fill the gap in the literature and to enable the international community to get acquainted with this part of Italian logic. The same applies to some works published in Barcelona (in Catalan) at the end of the 1970s, analysed in the last section. (shrink)

This volume is presented in honour of Heinz Post, who founded a distinc tive and distinguished school of philosophy of science at Chelsea College, University of London. The 'Chelsea tradition' in philosophy of science takes the content of science seriously, as exemplified by the papers presented here. The unifying theme of this work is that of 'Correspondence, Invariance and Heuristics', after the title of a classic and seminal paper by Heinz Post, published in 1971, which is reproduced in this volume (...) with the kind permission of the editors and publishers of Studies in History and Philosophy of Science. Described by Paul Feyerabend in Against Method as "brilliant" and " . . . a partial antidote against the view which I try to defend" (1975, p. 61, fn. 17), this paper, peppered with illustrative examples from the history of science, brings to the fore some of Heinz Post's central concerns: the heuristic criteria used by scientists in constructing their theories, the intertheoretic relationships which these criteria reflect and, in particular, the nature of the correspondence that holds between a theory and its predecessors (and its suc cessors). The appearance of this volume more than twenty years later is an indica tion of the fruitfulness of Post's contribution: philosophers of science continue to explore the issues raised in his 1971 paper. (shrink)

The dual character of invariance under transformations and definability by some operations has been used in classical works by, for example, Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this article, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in L∞∞ so as to cover the cases that are of interest in the logicality debates, getting McGee’s theorem about quantifiers (...) invariant under all permutations and definability in pure L∞∞ as a particular case. We also prove some optimality results along the way, regarding the kinds of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms. (shrink)

The failed criterion of logical truth proposed by Carnap in the Logical Syntax of Language was based on the determinateness of all logical and mathematical statements. It is related to a conception which is independent of the specifics of the system of the Syntax, hints of which occur elsewhere in Carnap’s writings, and those of others. What is essential is the idea that the logical terms are invariant under reinterpretation of the empirical terms, and are therefore semantically determinate. A certain (...) objection to Carnap’s version of the invariance conception has been repeated several times in the literature. It is based on Gödel incompleteness, which is puzzling, since Carnap’s Syntax is otherwise quite careful to take account of Gödel. We show here that, in fact, the objection is invalid and is based on a confusion about determinacy. Sorting this out is worthwhile not only for the purpose of better understanding Carnap’s thinking in the Syntax, though. The invariance conception is also related to recent work in the philosophy of logic regarding “logicality”—the characterization of logical concepts—following a proposal of Tarski’s. It is even connected to some very recent developments in the foundations of mathematics. (shrink)

In Carnap’s autobiography, he tells the story how one night in January 1931, “the whole theory of language structure” in all its ramifications “came to [him] like a vision”. The shorthand manuscript he produced immediately thereafter, he says, “was the first version” of Logical Syntax of Language. This document, which has never been examined since Carnap’s death, turns out not to resemble Logical Syntax at all, at least on the surface. Wherein, then, did the momentous insight of 21 January 1931 (...) consist? We seek to answer this question by placing Carnap’s shorthand manuscript in the context of his previous efforts to accommodate scientific theories and metalinguistic claims within Wittgenstein’s Tractatus theory of meaning. The breakthrough of January 1931 consists, from this viewpoint, in the rejection of the Tractatus theory in favor of the meta-mathematical perspective of Hilbert, Gödel, and Tarski. This was not yet the standpoint of the published Logical Syntax, as we show, but led naturally to the “principle of tolerance” and thus to Carnap’s mature philosophy, in which the inconsistencies between this first view and the principle of tolerance, which survived into the published Syntax, were overcome. (shrink)

Fifteen essays are contained in this collection, all relating to Heinz Post ’ s article ‘ Correspondence, Invariance and Heuristics ’, also reprinted. In this article, written in the heyday of the post - positivist movement, Post aims to convince his fellowphilosophers of science to bring the issue of heuristics back to the philosophical stage. Examining a wealth of theories and models from the physics and chemistry of the last 300 years, Post extracts several strategies of theory construction of which (...) he considers the General Correspondence Principle to be the most important. According to this principle, any acceptable newtheory should explain the well - confirmed part of its predecessor. Later Post states the General Correspondence Principle more precisely and uses it with what he considers its de facto validity to argue against incommensurability, Kuhn - losses, 1 and relativism. Post himself seems to support a kind of convergent realism which is most notably expressed in his credo that science progresses linearly. (shrink)

This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language L∞,∞. It continues with a review of critiques of the Tarski-Sher thesis, and a proposal in response to one of those critiques via homomorphism invariance. That has quite divergent characterization results depending (...) on its formulation, one in terms of FOL, the other by Bonnay in terms of L∞,∞, both without equality. From that we move on to a survey of Bonnay’s work on similarity relations between structures and his results that single out invariance with respect to potential isomorphism among all such. Turning to the critique that calls for sameness of meaning of a logical operation across domains, the paper continues with a result showing that the isomorphism invariant operations that are absolutely definable with respect to KPU−Inf are exactly those definable in full FOL; this makes use of an old theorem of Manders. The concluding section is devoted to a critical discussion of the arguments for set-theoretical criteria for logicality. (shrink)

For a complete first order theory of Boolean algebras T which has nonisomorphic countable models, we determine the first limit ordinal α = α(T) such that We show that for some and for all other T‘s, A nonprincipal ideal I of B is almost principal, if a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has complete extensions and characterize them by invariants (...) similar to the Tarski’s invariants. (shrink)

With assets of over US$1.0 trillion and growing, public pension funds in the United States have become a major force in the private sector through their holding of equity positions in large publicly traded corporations. More recently, these funds have been expanding their investment strategy by considering a corporation's long-term risks on issues such as environmental protection, sustainability, and good corporate citizenship, and how these factors impact a company's long-term performance. Conventional wisdom argues that the fiduciary responsibility of the pension (...) funds' trustees must be solely focused on their beneficiaries and, therefore, their investment criteria must be based strictly on narrowly defined financial measures. It is also asserted that well-established financial measurements of corporate performance already include long-term risk assessment through discounted present value of future flow of earnings. Consequently, all other criteria are contrary to the best interest of the pension funds' beneficiaries. In this paper, we assert that, contrary to conventional wisdom, pension funds, and for that matter other mutual funds, must be concerned with the long-term survival and growth of corporations. These measures are generally referred to "socially responsible investing" and when applied to corporations, it is termed "socially responsible corporate conduct." We demonstrate that current measurement of future risk assessment invariably understates, and quite often completely overlooks, these long-term risks because of the inherent bias towards short-run on the part of financial intermediaries whose compensation depends greatly on short-term results. Furthermore, there is ample evidence to suggest that these intermediaries have been engaging in self-serving practices and thus failing in their duties to serve their clients', i.e. pension funds', best interests. Because of their large holdings in the total market as well as individual companies, these funds cannot easily divest from poorly performing companies without destabilizing the companies' stock and overall markets. Hence, they must opt for a strategy of emphasizing investment criteria that encourage companies to take into account long-term aspects of their operations in terms of their impact on environment, sustainability, and community welfare, to name a few. We argue that an exclusionary, and even a primary, focus on short-term financial criteria is no longer a viable option. It also calls for the pension funds to encourage greater transparency and accountability of the entire corporate sector through improved corporate governance. Thus socially responsible investing practices are not merely discretionary and desirable activities; they are a necessary imperative, which both the corporations and public pension funds, and other large institutional holders, will ignore at serious peril to themselves. Finally, the paper considers some of the recent developments where corporations have been responding to these challenges and how their actions might be strengthened through greater disclosure and transparency of corporate activities. It also makes recommendations for the pension funds to support further research in creating new measurement standards that further refine the concept of socially responsible investing as a necessary ingredient of long-term corporate survival and growth in the context of a changing economic, environmental and socio-political dynamic. (shrink)

>With assets of over US$1.0 trillion and growing, public pension funds in the United States have become a major force in the private sector through their holding of equity positions in large publicly traded corporations. More recently, these funds have been expanding their investment strategy by considering a corporations long-term risks on issues such as environmental protection, sustainability, and good corporate citizenship, and how these factors impact a companys long-term performance. Conventional wisdom argues that the fiduciary responsibility of the pension (...) funds trustees must be solely focused on their beneficiaries and, therefore, their investment criteria must be based strictly on narrowly defined financial measures. It is also asserted that well-established financial measurements of corporate performance already include long-term risk assessment through discounted present value of future flow of earnings. Consequently, all other criteria are contrary to the best interest of the pension funds beneficiaries. In this paper, we assert that, contrary to conventional wisdom, pension funds, and for that matter other mutual funds, must be concerned with the long-term survival and growth of corporations. These measures are generally referred to socially responsible investing (SRI) and when applied to corporations, it is termed socially responsible corporate conduct (SRCC). We demonstrate that current measurement of future risk assessment invariably understates, and quite often completely overlooks, these long-term risks because of the inherent bias towards short-run on the part of financial intermediaries whose compensation depends greatly on short-term results. Furthermore, there is ample evidence to suggest that these intermediaries have been engaging in self-serving practices and thus failing in their duties to serve their clients, i.e. pension funds, best interests. Because of their large holdings in the total market as well as individual companies, these funds cannot easily divest from poorly performing companies without destabilizing the companies stock and overall markets. Hence, they must opt for a strategy of emphasizing investment criteria that encourage companies to take into account long-term aspects of their operations in terms of their impact on environment, sustainability, and community welfare, to name a few. We argue that an exclusionary, and even a primary, focus on short-term financial criteria is no longer a viable option. It also calls for the pension funds to encourage greater transparency and accountability of the entire corporate sector through improved corporate governance. Thus socially responsible investing practices are not merely discretionary and desirable activities; they are a necessary imperative, which both the corporations and public pension funds, and other large institutional holders, will ignore at serious peril to themselves. Finally, the paper considers some of the recent developments where corporations have been responding to these challenges and how their actions might be strengthened through greater disclosure and transparency of corporate activities. It also makes recommendations for the pension funds to support further research in creating new measurement standards that further refine the concept of socially responsible investing as a necessary ingredient of long-term corporate survival and growth in the context of a changing economic, environmental and socio-political dynamic. (shrink)

Being building block of data sciences, link prediction plays a vital role in revealing the hidden mechanisms that lead the networking dynamics. Since many techniques depending in vertex similarity and edge features were put forward to rule out many well-known link prediction challenges, many problems are still there just because of unique formulation characteristics of sparse networks. In this study, we applied some graph transformations and several inequalities to determine the greatest value of first and second Zagreb invariant, S K (...) and S K 1 invariants, for acyclic connected structures of given order, diameter, and pendant vertices. Also, we determined the corresponding extremal acyclic connected structures for these topological indices and provide an ordering giving a sequence of acyclic connected structures having these indices from greatest in decreasing order. (shrink)

If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature β become small. Thus, microscopic particles like electrons must (...) remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2πħ)3 are reduced to points in the phase space, the partition functionTr{exp(−βĤ) for N identical bosons (fermions) approaches (2πħ)−3N(N!) ∫ ... ∫ d3r1 d3p1 ... d3rN d3pN exp(−βH). The two factors, (2πħ)−3N and (N!)−1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles. (shrink)

Seven hundred adolescents with mean age of 15.3 years. Multigroup confirmatory factor analysis was performed to assess measurement invariance of the MSPSS scale across Chinese and South Asian ethnic minority samples. Results show that the original three-factor structure of the MSPSS was supported in both samples. Measurement invariance was supported in terms of configural, metric, and partial scalar invariance. Given partial scalar invariance was achieved, the latent mean differences were compared across samples. Chinese adolescents had higher levels of all three (...) types of social support when compared with their South Asian ethnic minority counterparts. The present study attempts to assess the measurement invariance of the MSPSS across ethnic groups in Asian society, which sheds light on future research that involves perceived social support among adolescents in Chinese culture contexts. (shrink)

Learning to recognize visual objects from examples requires the ability to find meaningful patterns in spaces of very high dimensionality. We present a method for dimensionality reduction which effectively biases the learning system by combining multiple constraints via the use of class labels. The use of extensive class labels steers the resulting lowdimensional representation to become invariant to those directions of variation in the input space that are irrelevant to classification; this is done merely by making class labels independent of (...) these directions. We also show that prior knowledge of the proper dimensionality of the target representation can be imposed by training a multi-layer bottleneck network. Computational experiments involving non-trivial categorization of parameterized fractal images and of human faces indicate that the low-dimensional representation extracted by our method leads to improved generalization in the learned tasks and is likely to preserve the topology of the original space. (shrink)

Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal (...) result to key developments and concepts in the history of model theory, notably ultraproducts and preservation results. (shrink)

A universal property of visual word identification is position-invariant letter identification, such that the letter is coded in the same way in CAT and ACT. This should provide a fundamental constraint on theories of word identification, and, indeed, it inspired some of the theories that Frost has criticized. I show how the spatial coding scheme of Colin Davis can, in principle, account for contrasting transposed letter priming effects, and at the same time, position-invariant letter identification.

Tout s’est passé très vite, prenant l’Église au dépourvu. Le mouvement pour la reconnaissance des mariages et des unions civiles entre personnes du même sexe a connu un succès extraordinaire. La hiérarchie de l’Église catholique oppose une fin de non-recevoir à toute reconnaissance positive des couples homosexuels, en invoquant une morale sexuelle qu’on dit ancienne et invariable,..

In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings of matter, (...) gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics. (shrink)

In Pure Inductive Logic, the rational principle of Predicate Exchangeability states that permuting the predicates in a given language L and replacing each occurrence of a predicate in an L-sentence phi according to this permutation should not change our belief in the truth of phi. In this paper we study when a prior probability function w on a purely unary language L satisfying Predicate Exchangeability also satisfies the principle of Unary Language Invariance.

Supercommentary on Tattvacintāmaṇidīdhiti, 16th century commentary, by Raghunātha Śiromaṇi on Tattvacintāmaṇi, 13th century basic work of the neo-Nyāya school in Hindu philosophy, by Gaṅgeśa; portion dealing with definition of invariable concomitance (vyāpti).

Commentary on Pañcalakṣaṇī of Mathurānātha Tarkavāgīśa, 1600-1675, work dealing with definition of invariable concomitance (vyāpti) with reference to Nyaya philosophy.

Commentaries on Pañcalakṣaṇī of Mathurānātha Tarkavāgīśa, 1600-1675, work dealing with definition of invariable concomitance (vyāpti) with reference to Nyaya philosophy ; with supercommentary on Māthurīvyāptipañcakaśatakoṭi.