We show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig's Interpolation Theorem holds but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi [12].

Here we investigate the classes RCA $^\uparrow_\alpha$ of representable directed cylindric algebras of dimension α introduced by Nemeti[12]. RCA $^\uparrow_\alpha$ can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, "purely cylindric algebraic" proof for the following theorems of Nemeti: (i) RCA $^\uparrow_\alpha$ is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain (...) a strong representation theorem for RCA $^\uparrow_\alpha$ if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories. (shrink)

Let be a countable first order structure and endow the universe of with the discrete topology. Then the automorphism group of becomes a topological group. A tuple of automorphisms is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ‐orbit of each is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of (...) view of finite combinatorics.The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in implies the existence of weakly generic tuples in for all natural number. Finally, in Theorem 3.2 we show that if is a countable model of an ℵ0‐categorical, simple theory in which all types over the empty set are stationary and has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length.At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures. (shrink)

We will study the class RSA of -dimensional representable substitution algebras. RSA is a sub-reduct of the class of representable cylindric: algebras, and it was an open problem in Andréka [1] that whether RSA can be finitely axiomatized. We will show, that the answer is positive. More concretely, we will prove, that RSA is a finitely axiomatizable quasi-variety. The generated variety is also described. We note that RSA is the algebraic counterpart of a certain proportional multimodal logic and it is (...) related to a natural fragment of first order logic, as well. (shrink)

The notion of ultratopologies was introduced in [6] motivated by the model theory of first and higher order logics. In [6] we established some model theoretical applications of ultratopologies, for example, we provided a purely set theoretical characterization for classes de.nable by second order existential formulas. The present note deals with topological properties of ultratopologies, like density and compactness.

Which ultraproducts preserve the validity of formulas of higher order logics? To answer this question, we will introduce natural topologies on ultraproducts. We will show, that ultraproducts preserving certain higher order formulas can be characterized in terms of these topologies. As an application of the above results, we provide a constructive, purely model theoretic characterization for classes definable by second order existential formulas.

Let be a countable ℵ0‐homogeneous structure. The primary motivation of this work is to study different amenability properties of (subgroups of) the automorphism group of ; the secondary motivation is to study the existence of weakly generic automorphisms of. Among others, we present sufficient conditions implying the existence of automorphism invariant probability measures on certain subsets of A and of ; we also present sufficient conditions implying that the theory of is amenable. More concretely, we show that if the set (...) of locally finite automorphisms of is dense (in particular, if has weakly generic tuples of automorphisms of arbitrary finite length), then there exists a finitely additive probability measure μ on the subsets of definable with parameters such that μ is invariant under. Moreover, if is saturated and the set of its locally finite automorphisms is dense (in particular, if is saturated and has weak generics), then the theory of is amenable. (shrink)

In [3] a certain family of topological spaces was introduced on ultraproducts. These spaces have been called ultratopologies and their definition was motivated by model theory of higher order logics. Ultratopologies provide a natural extra topological structure for ultraproducts. Using this extra structure in [3] some preservation and characterization theorems were obtained for higher order logics. The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [2], where some questions remained open. (...) Here we present the solutions of two such problems. More concretely we show 1. that there are sequences of finite sets of pairwise different cardinalities such that in their certain ultraproducts there are homeomorphic ultratopologies and 2. if A is an infinite ultraproduct of finite sets, then every ultratopology on A contains a dense subset D such that |D| < |A|. (shrink)

By a celebrated theorem of Morley, a theory T is ℵ1‐categorical if and only if it is κ‐categorical for all uncountable κ. In this paper we are taking the first steps towards extending Morley's categoricity theorem “to the finite”. In more detail, we are presenting conditions, implying that certain finite subsets of certain ℵ1‐categorical T have at most one n‐element model for each natural number (counting up to isomorphism, of course).

Among others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEA ω 's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schema-axiomatizable, as well). We will also show that the complexity of the equational theory of RPEA ω is also extremely high in the recursion theoretic (...) sense. Finally, comparing the present negative results with the positive results of Ildiko Sain and Viktor Gyuris [12], the following methodological conclusions will be drawn: The negative properties of polyadic (equality) algebras can be removed by switching from what we call the "polyadic algebraic paradigm" to the "cylindric algebraic paradigm". (shrink)

Among others we will prove that the equational theory of $\omega$ dimensional representable polyadic equality algebras is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable. We will also show that the complexity of the equational theory of RPEA$_\omega$ is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildiko Sain and Viktor Gyuris (...) [12], the following methodological conclusions will be drawn: The negative properties of polyadic algebras can be removed by switching from what we call the "polyadic algebraic paradigm" to the "cylindric algebraic paradigm". (shrink)

This essay offers a conception of logic by which logic may be considered to be exceptional among the sciences on the backdrop of a naturalistic outlook. The conception of logic focused on emphasises the traditional role of logic as a methodology for the sciences, which distinguishes it from other sciences that are not methodological. On the proposed conception, the methodological aims of logic drive its definitions and principles, rather than the description of scientific phenomena. The notion of a methodological discipline (...) is explained as a relation between disciplines or practices. Logic serves as a methodological discipline with respect to any theoretical practice, and this generality, as well as logic’s reflexive nature, distinguish it from other methodological disciplines. Finally, the evolution of model theory is taken as a case study, with a focus on its methodological role. Following recent work by John Baldwin and Juliette Kennedy, we look at model theory from its inception in the mid-twentieth century as a foundational endeavour until developments at the end of the century, where the classification of theories has taken centre-stage. (shrink)

In this paper I discuss a prevailing view by which logical terms determine forms of sentences and arguments and therefore the logical validity of arguments. This view is common to those who hold that there is a principled distinction between logical and nonlogical terms and those holding relativistic accounts. I adopt the Tarskian tradition by which logical validity is determined by form, but reject the centrality of logical terms. I propose an alternative framework for logic where logical terms no longer (...) play a distinctive role. This account employs a new notion of semantic. (shrink)

This is a comprehensive study of the Derveni Papyrus. The papyrus, found in 1962 near Thessaloniki, is not only one of the oldest surviving Greek papyri but is also considered by scholars as a document of primary importance for a better understanding of the religious and philosophical developments in the fifth and fourth centuries BC. Gábor Betegh aims to reconstruct and systematically analyse the different strata of the text and their interrelation by exploring the archaeological context; the interpretation of (...) rituals in the first columns of the text; the Orphic poem commented on by the author of the papyrus; and the cosmological and theological doctrines which emerge from the Derveni author's exegesis of the poem. Betegh discusses the place of the text in the context of late Presocratic philosophy and offers an important preliminary edition of the text of the papyrus with critical apparatus and English translation. (shrink)

In standard model-theoretic semantics, the meaning of logical terms is said to be fixed in the system while that of nonlogical terms remains variable. Much effort has been devoted to characterizing logical terms, those terms that should be fixed, but little has been said on their role in logical systems: on what fixing their meaning precisely amounts to. My proposal is that when a term is considered logical in model theory, what gets fixed is its intension rather than its extension. (...) I provide a rigorous way of spelling out this idea, and show that it leads to a graded account of logicality: the less structure a term requires in order for its intension to be fixed, the more logical it is. Finally, I focus on the class of terms that are invariant under isomorphisms, as they render themselves more easily to mathematical treatment. I propose a mathematical measure for the logicality of such terms based on their associated Löwenheim numbers. (shrink)

This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical (...) consequence. Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro's blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed. (shrink)

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension—it is a logic of cardinality. In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, (...) the more coarse-grained and less “intensional” it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things “in the world”. Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus’s principles of explicit and implicit extensionality, we are able to characterize purely logical languages as “sub-extensional”, namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality. (shrink)

This book outlines the broad spectrum of answers to important questions presented in Jewish literature, covering theological issues bearing on the meaning of the Torah and of revelation, as well as hermeneutical questions regarding understanding of the halakhic text.

This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in on (...) the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike. (shrink)

We explain the general significance of integer-based descriptors for natural shapes and show that the evolution of two such descriptors, called mechanical descriptors (the number _N_(_t_) of static balance points and the Morse–Smale graph associated with the scalar distance function measured from the center of mass) appear to capture (unlike classical geophysical shape descriptors) one of our most fundamental intuitions about natural abrasion: shapes get monotonically _simplified_ in this process. Thus mechanical descriptors help to establish a correlation between subjective and (...) objective descriptors of perceived objects. (shrink)

A Critical Notice of Thomas KjellerJohansen, Plato’s Natural Philosophy. A Study of the Timaeus-Critias.

ישעיהו ליבוביץ, מגדולי האישים שקמו לעם ישראל בדורות האחרונים, השפיע על ההגות היהודית ועל החברה הישראלית בדרכים שונות. דבריו עוררו תמיד דיון ותגובה. הספר כולל מאמרים המתארים והמנתחים את הגותו היהודית והכללית, השקפותיו בתחומי הפילוסופיה ובשאלות תרבות וחברה, לאומיות וציוניות.

In this paper I will do three things. One, to explain why conceptual role semantics seems an attractive theory of meaning (I). Two, to sketch a version of it which has a good chance of withstanding some of the standard objections (II-III). Three, to see what follows from this version with respect to the naturalization of meaning (IV).

In the accepted view, the basic disposition of believers is one of absolute obedience, humility, and lack of critique, doubt, or, indeed, defiance of God. Only through such a disposition do believers convey their absolute faith and establish the appropriate hierarchy between God and humans. This article challenges this view and argues that, in mainstream rabbinic tradition, the believer is not required to renounce his or her moral autonomy and certainly not his or her understanding of God and the world. (...) Indeed, faith rests on such understanding; moreover, human autonomy is the mechanism through which humans convey God’s goodness and perfection. Their questions and criticism are part of a persistent effort to close the evident gap between their assumptions about God’s goodness and the flawed imperfect reality. The analysis focuses on rabbinic tradition but its implications go beyond it, presenting a model of a life of faith that compels subjects, as believers, to preserve their constitutive foundations as rational autonomous creatures. (shrink)

The paper’s main task is to show how much Hume’s philosophy of passions is indebted to and continues the tradition of the philosophy of affects of the 17th century, in spite of the obvious fact that he departed from the main philosophical project of the 17th century, the tripartite unity of mathematics, metaphysics, and mechanical physics. A restructuring of Hume’s order of passions and its comparison to the order followed by Descartes will show up a special „cognitivist” character of Hume’s (...) thinking on emotion, even if the special experimental framework of Hume’s theory of passions prevents him from theorizing about the free will’s distinguished role in our lives or about the will’s grounding in the concept of a providentially active biblical God. (shrink)

Am Beispiel der im dritten Teil der Abhandlung über die Methode entworfenen "provisorischen Moral" soll gezeigt werden, wie Descartes den Versuch unternimmt, die zwei wichtigsten zeitgenössischen Denkanstöße, die mechanistische Philosophie und die traditionelle christliche Moral, zu Ende zu denken, ihren Gegensatz scharf hervorzuheben und sie miteinander zu versöhnen. Die Analyse des Sinns der Vorläufigkeit, des Vorwurfs des Konformismus, der traditionellen Unterscheidung von Negation und Privation - auch mit Blick auf ihre weitere Entwicklung im Briefwechsel zwischen Spinoza und Blyenbergh - führt (...) zu dem Ergebnis, daß der Versöhnungsversuch, dessen Ziel die Ausarbeitung einer "definitiven" Ethik war, am Ende gescheitert ist. Descartes leitet eine Ära ein, in der nach Maßstäben und Beweggründen nur noch innerhalb einer provisorischen Ethik - dessen wahrer Gegensatz eine providenzielle Ethik ist - Ausschau gehalten werden kann. (shrink)

This book is a first attempt to examine the thought of key contemporary Jewish thinkers on the meaning of tradition in the context of two models. The classic model assumes that tradition reflects lack of dynamism and reflectiveness, and the present’s unqualified submission to the past. This view, however, is an image that the modernist ethos has ascribed to the tradition so as to remove it from modern existence. In the alternative model, a living tradition emerges as open and dynamic, (...) developing through an ongoing dialogue between present and past. The Jewish philosophers discussed in this work—Joseph B. Soloveitchik, Yeshayahu Leibowitz, David Hartman, and Eliezer Goldman—ascribe compelling canonic status to the tradition, and the analysis of their thought discloses the tension between these two models. The book carefully traces the course they have plotted along the various interpretations of tradition through their approach to Scripture and to Halakhah. (shrink)

Continuity as appears to us immediately by intuition differs from its current formalization, the arithmetical continuum or equivalently the set of real numbers used in modern mathematical analysis. Motivated by the known mathematical and physical problems arising from this formalization of the continuum, our aim in this paper is twofold. Firstly, by interpreting Chaitin’s variant of Gödel’s first incompleteness theorem as an inherent uncertainty or fuzziness of the arithmetical continuum, a formal set-theoretic entropy is assigned to the arithmetical continuum. Secondly, (...) by analyzing Noether’s theorem on symmetries and conserved quantities, we argue that whenever the four dimensional space-time continuum containing a single, stationary, asymptotically flat black hole is modeled by the arithmetical continuum in the mathematical formulation of general relativity, the hidden set-theoretic entropy of this latter structure reveals itself as the entropy of the black hole, indicating that this apparently physical quantity might have a pure set-theoretic origin, too. (shrink)