Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...) computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some (...) directions we tried to give an overview of the most recent results and research trends, too. (shrink)

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)

SC , CA , QA and QEA stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension , respectively. Generalizing a result of Németi on cylindric algebras, we show that for K {SC, CA, QA, QEA} and ordinals , the class Nr K of -dimensional neat reducts of -dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if (...) and only if > 1.From this it easily follows that for 1 , the operation of forming -neat reducts of algebras in K does not commute with forming subalgebras, a notion to be made precise. (shrink)

In this paper we try to initiate a search for an explicite and direct denition of ultraproducts in categories which would share some of the attractive properties of products, coproducts, limits, and related category theoretic notions. Consider products as a motivating example.

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of (...) accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that~the Twin Paradox becomes provable in AccRel, but it is not provable without IND. (shrink)

Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity--including such revolutionary areas as black hole physics, relativistic computers, new cosmology--are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction.

We investigate Kerr–Newman black holes in which a rotating charged ring-shaped singularity induces a region which contains closed timelike curves (CTCs). Contrary to popular belief, it turns out that the time orientation of the CTC is oppo- site to the direction in which the singularity or the ergosphere rotates. In this sense, CTCs “counter-rotate” against the rotating black hole. We have similar results for all spacetimes sufficiently familiar to us in which rotation induces CTCs. This motivates our conjecture that perhaps (...) this counter-rotation is not an accidental oddity particular to Kerr–Newman spacetimes, but instead there may be a general and intuitively comprehensible reason for this. (shrink)

We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.

What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has been endorsed by many authors (cf. van Benthem 1991). We will investigate these fragments, and find that, illuminating and interesting though they are, they lack the required nice behaviour in our sense. (Several new negative results support this claim.) (...) As a counterproposal, then, we define a large fragment of predicate logic characterized by its use of only bounded quantification. This so-called guarded fragment enjoys the above nice properties, including decidability, through an effectively bounded finite model property. (These are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like modal operators. This observation inspires a generalized semantics for first-order predicate logic with accessibility constraints on available assignments (cf. N´emeti 1986, 1992) which moves the earlier quantifier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candidates sharing various desirable features with basic modal logic – in particular, its decidability. (shrink)

In this paper, we introduce a general technology, calledtaming, for finding well-behaved versions of well-investigated logics. Further, we state completeness, decidability, definability and interpolation results for a multimodal logic, calledarrow logic, with additional operators such as thedifference operator, andgraded modalities. Finally, we give a completeness proof for a strong version of arrow logic.

We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order (...) logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic. (shrink)

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, (...) Glymour and Halvorson. (shrink)

A series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

We investigate in an algebraic setting the question of which logical languages can express the properties integral, permutational, and rigid for algebras of relations.

We shall show that certain natural and interesting intervals in the lattice of varieties of representable relation algebras embed the lattice of all subsets of the natural numbers, and therefore must have a very complicated lattice-theoretic structure.

This is an abstract of our paper \A characterisation of Floyd-provable programs" submitted to Theoretical Computer Science. ! denotes the set of natural numbers. Y =d fyi : i 2 !g is the set of variable symbols. L denotes the set of classical rst order formulas of type t possibly with free variables , where t is the similarity type of arithmetic, i.e. it consists of \+; ; 0; 1" with arities \2; 2; 0; 0".

Theorem 1 states a negative result about the classical semantics j= ! of program schemes. Theorem 2 investigates the reason for this. We conclude that Theorem 2 justies the Henkin-type semantics j= for which the opposite of the present Theorem 1 was proved in [1]{[3] and also in a dierent form in part III of [5]. The strongest positive result on j= is Corollary 6 in [3].

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.

We give two theories, Th1 and Th2, which are explicitly definable over each other , but are not definitionally equivalent. The languages of the two theories are disjoint.

A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc 2. The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.

Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result (...) to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic, equality-free, quasi-polyadic algebras with rectangular atoms. The error consists in the implicit assumption of a property that does not, in general, hold. We then give a correct proof of their theorem. Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of finite dimension satisfying certain special identities is representable. We introduce a modification of the notion of a rich algebra that, in our opinion, renders it more natural. In particular, under this modification richness becomes a density notion. Moreover, our notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. We show that a finite dimensional algebra is rich iff it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density . As a consequence, every finite dimensional rich algebra of logic is representable. We do not have to assume the validity of any special identities to establish this representability. Not only does this give an improvement of the Henkin-Tarski representation theorem for rich cylindric algebras, it solves positively an open problem in op. cit. concerning the representability of finite dimensional rich quasi-polyadic algebras without equality. (shrink)

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, (...) is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski’s other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski’s axiom system. (shrink)

This book features more than 20 papers that celebrate the work of Hajnal Andréka and István Németi. It illustrates an interaction between developing and applying mathematical logic. The papers offer new results as well as surveys in areas influenced by these two outstanding researchers. They also provide details on the after-life of some of their initiatives. Computer science connects the papers in the first part of the book. The second part concentrates on algebraic logic. It features a range of papers (...) that hint at the intricate many-way connections between logic, algebra, and geometry. The third part explores novel applications of logic in relativity theory, philosophy of logic, philosophy of physics and spacetime, and methodology of science. They include such exciting subjects as time travelling in emergent spacetime. The short autobiographies of Hajnal Andréka and István Németi at the end of the book describe an adventurous journey from electric engineering and Maxwell’s equations to a complex system of computer programs for designing Hungary’s electric power system, to exploring and contributing deep results to Tarskian algebraic logic as the deepest core theory of such questions, then on to applications of the results in such exciting new areas as relativity theory in order to rejuvenate logic itself. (shrink)

Hilbert’s axiomatic approach to the sciences was characterized by a dynamic methodology tied to scientific and mathematical fields under investigation. In particular, it is an analytic art for choosing axioms but, at the same time, it has to include dynamically synthetic procedures and meta-theoretical reflections. Axioms have to be useful, or capture something, or help as part of explanations. The Andréka-Németi group use several formal axiomatic theories together to re-capture, predict, recover or explain the phenomena of special relativity, general relativity (...) and classical kinematics. Their scientific methodology is close to Hilbert’s conceptions of how science should be done ideally. In this paper, we compare Formica’s reading of Hilbert’s axiomatic method to the method used by the Andréka-Németi group. (shrink)

"Cet ouvrage collectif présente le dialogue entre de jeunes chercheurs en phénoménologie et Alexander Schnell, phénoménologue contemporain ; à partir de son ouvrage paru en 2020, Le Clignotement de l'être. Il défend la vivacité de la phénoménologie transcendantale aujourd'hui, en tant que méthode philosophique qui interroge les secrets de la subjectivité et son rapport à la réalité, au fil conducteur des concepts d'intentionnalité, de réduction, de générativité, d'absolu, et d'histoire. La question directrice qui rassemble les contributions réunies ici est la (...) suivante : quand le phénoménologue effectue la réduction phénoménologique et fait l'expérience du transcendantal comme tel, à quelle réalité fait-il face?"--Page 4 of cover. (shrink)

Ordinary people tend to be realists regarding perceptual experience, that is, they take perceiving the environment as a direct, unmediated, straightforward access to a mindindependent reality. Not so for (ordinary) philosophers. The empiricist influence on the philosophy of perception, in analytic philosophy at least, made the problem of perception synonymous with the view that realism is untenable. Admitting the problem (and trying to offer a view on it) is tantamount to rejecting ordinary people’s implicit realist assumptions as naive. So what (...) exactly is the problem? We can approach it via one of the central arguments against realism – the argument from hallucination. The argument is intended as a proof that in ordinary, veridical cases of perception, perceivers do not have an unmediated perceptual access to the world. There are many versions of it; I propose the following1: 1. Hallucinations that are subjectively indistinguishable from veridical perceptions are possible. 2. If two subjective states are indistinguishable, then they have a common nature. 3. The contents of hallucinations are mental images, not concrete external objects. 4. Therefore, the contents of veridical perceptions are mental images rather than concrete external objects. The key move is, I believe, from the fact that hallucinations that are subjectively indistinguishable from cases of veridical perception are possible to an alleged common element, factor, or nature, in the form of a mental state, in the two cases – that is, premise 2. Disjunctivism, at its core, can be taken as simply denying this move, and arguing that all that follows from the premise stating the possibility of hallucinations that are subjectively indistinguishable from cases veridical perception is that there is a broader category, that of “experience as of...”, which encompasses both cases.. (shrink)