| Logic Postgraduate School Budapest
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I. General outline II. List of course titles |
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III. Course descriptions
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I. General outline The Logic Postgraduate School (simply school from now on) is a joint enterprise of Eötvös Loránd University Budapest and the Mathematical Institute of the Hungarian Academy of Sciences. Scope: Nowadays, symbolic or formal logic is often devided to three parts usually referred to as Philosophical Logic, Computer Science Logic(s), and Mathematical Logic. The school intends to put equal emphasis on all three of these (for the scope of the first one see Handbook of Philosophical Logic Vol's I-IV). We do not consider these parts as three disjoint fields, rather we do the opposite, namely we treat them as three ways of looking at the same integrated subject, Logic. However, if the student wishes to do so, he/she is permitted to specialize to any one of these three general directions of Logic. In general, the student may eventually specialize to any branch of Logic represented in this part of Central Europe, let it be Set Theory; Algebraic Logic; Model-theoretic (or formal) semantics of Natural Language; Temporal Logic or Dynamic Logic in Natural Language or in Computer Science; Modal Logic; Partial Intensional Logic; Category Theoretic approaches; interaction of Logic with some fixed branch of Mathematics (like e.g. algebra, topology, combinatorics) etc. Some general views on the present situation in Logic (These views are not cornpulsory either for students or for teachers) A number of researchers in the interdisciplinary field spanning logic, linguistics, and computer science have come to feel that the time is ripe to build a more visible and permanent organizational basis for training graduate students and young scientists, who would like to join them (on a professional level) in their activities. In fact, we intend to teach a relatively new field which falls in the intersection shared by such larger disciplines as linguistics, mathematics, computer science, and psychology. The field in question has come into existence as a result, at least in part, of the fact that over the last decade or so, logic has proved applicable in many new areas, especially computer science, artificial intelligence, and cognitive science. Many issues have been clarified and new perspectives opened up by the application of logical techniques. In this process, the very conception of what logic is has been changing, often radically, so that the subject should be taken as a quite general discipline, not identifiable with any single successful application, be it to mathematics or to linguistics. Thus, the "red thread" through most of our activities is the use of logical notions and techniques, taken in this wider sense. |
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III. Course descriptions Classical Extensional Logic(an introductory course) Content (aims & scope) Pure syntactic description of classical first order logic (QC); proofs of relatively important meta-theorems. The classical sentence-calculus (PC) as a fragment of QC. Semantics of extensional languages (based on set theory). Basic and functor categories, semantical values. Extensional type theory. Interpretations. Truth functions. Variables and operators bounding variables. Formal description of the structure of extensional language. Classical first order semantics. Introduction of semantic cpncepts (interpretation, valuation of variables; model of aset of formulas, satisfiability of a set of formulas; semantic consequence relation). Soundness of QC with respect to the semantics. Completeness of QC. QC-complete sets of formulas and their satisfiability: Henkin-type proof of completeness. Elimination of constant symbols, unicity formulas. QC-saturated sets of formulas. Analitic sequences and tables, tree structures. Possibility of concrete demonstration of the semantic consequence relation. Decidability of monadic formulas. Completeness of PC with respect to the truth-value semantics. The reconstruction of QC in the framework of the (Gentzen-type) natural deduction. Short overview: intuitionistic and minimal calculi. Type theoretic extensional !languages. Grammar and semantics. Semantical metatheorems. Defining classical logical constants. The second order fragment. Soundness of the EC inference system. Meta-theorems concerning EC; PC, and QC as parts of EC. The generalized semantics. Completeness of EC with respect to the generalized semantics. Lindström's theorem. Naturalness of the generalized semantics of EC: Sain-Sacks theorem. KPU-absolute semantics' and logics. Perhaps: directions in refining the generalized semantics. Literature - Ruzsa Imre: Klasszikus modális és intenzionális logika, Akadémiai Kiadó, 1983. - Ruzsa Imre: Logikai szintaxis és szemantika L, IL, Akadémiai Kiadó, 1989., 1990. - Handbook of Philosophical Logic I, II, III, Reidel, 1985-1989. - H. B. Enderton: A Mathematical Introduction to Logic, Academic Press, 1972. - H. Andréka, T. Gergely, and I. Németi: Easily Comprehensible Mathematical Logic and its Model Theory, KFKI, 1975. Potential lecturers Imre Ruzsa; László Pólos |
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Model Theory Prerequisites Naive set theory. Familiarity with some of the basic ideas of logic would be useful but is not necessary. Content (aims & scope) General theory of (model theoretic) languages (syntax, semantics, validity), intuitive notions of algorithm, decidability, and enumerability notions of compactness and complete calculi in this general setting (which is the simplest form of abstract model theory). The Galois connection induced by the validity relation, axiomatizable hull, theory generated by a set of formulae. Model theory of propositional calculus as an example. The (model theoretic) language of the Tarski-Givant book as a still very simple example halfway between propositional logic and predicate logic. Model theory of predicate calculus: Referential transparency. Examples of undistinguishable non-isomorphic models expressible and non-expressible properties in first order logic. Product reduced product, ultraproduct. Los theorem, compactness theorem. Characterization of axiomatizable hulls, finite axiomatizability. Saturated (rich) and atomic (poor) rnodels, the Löwenheim-Skolem theorems. Mostowski-Ehrenfeucht game. Applications. Ultraproduct methods, saturated ultrapowers good ultrafilters, Keisler-Shelah theorem, Mod Th(K) = Ur Up K (characterizations of elementary hulls in terms of ultraproducts). Fragments of first order logic: equational logic. Subalgebras. Homomorphic images. Free algebras. Birkhoff characterizations of equationally definable classes (varieties). Analogous theorems for other fragments, e.g; for the universal Horn or implicational fragment (quasivarieties), the universally quantified fragment. Congruence lattice, subdirect irreducible and simple algebras, subdirect products. Discriminator algebras and discriminator varieties. Basic category theoretic properties of equational- and quasi-equational classes; reflective subcategories and implicational ones. Literature J. L. Bell and A. B. Slomson, Models and Ultraproducts. North-Holland, 1969. C. C. Chang and H. J. Keisler, Model Theory. North-Holland, 1977. R. McKenzie, G. McNulty, and W. Taylor, Algebras, Lattices, Varieties. Wadsworth & Brooks/Cole Advanced books & Software, 1987. Chapter 0. "General Algebra" of L. Henkin, J. D. Monk, and A. Tarski, Cylindric Algebras, North-Holland, 1985. A. Tarski and S. Givant, A formalization of set theory without variables. AMS Colloquium Publisher volume 41, 1987. Potential lecturers István Németi |
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Algebraic logic Prerequisites (This can be changed depending on `demand'. We are very flexible concerning prerequisites.) Familiarity with the basic ideas of universal algebra. Naive set theory. Some vague ideas concerning what logic is about. Content (aims & scope) The backbone of the course would be the following Lecture Notes Volume of the present Logic Graduate School: H. Andréka - I. Németi - I. Sain: Algebraic Logic vi+103 pp. (1991). A less complete version of this appeared in Studia Logica (special volume dedicated to Algebraic Logic, eds: W. J. Blok and D. J. Pigozzi, 1991) in the form of Németi: Algebraizations of quantifier logics, an introductory overview. (The title of the latter is slightly misleading in the present context, namely the course will not be restricted to quantifier logics a great emphasis will be on algebraic treatment of non-classical propositional logics too.) Aspects of the course (1) Algebraic approaches to non-classical logics, multimodal algebras, Boolean algebras with operators (BAO's) and their representations via Kripke models. (2) Expanding distributive lattices and Boolean algebras from unary relations to n-ary relations. Algebraic theories of relations of higher ranks (relation algebras, cylindric-, polyadic-, other algebras, JOnsson's clones of relations, abstract model theoretic connections etc.). (3) Algebraizations of quantifier logics. (4) The higher level logical structure consisting of theories (perhaps logics), and interpretations between them. (Relevant to Goguen-Burstall's Institutions (used in computer science).) Categories of theories. (5) Universal algebraic logic (algebraization of a general theory of logics or (generalized) abstract model theory), e.g., the equivalence of completeness theorems in logics with representations theorems in algebra; or the connection between deduction theorems in logic and Equationally Definable Principal Congruences in algebra. (6) Connections between algebraic logic and `pure' logic. The above listed aspects do not necessarily appear as separate topics (that would be taught at different times), because, e.g.,the algebras listed in (2) are all special cases of BAO's in (1) which turn out to be a key tool in treating the whole of (1) and large portion of (5). So these items are really only aspects of a single coherent topic. Literature The main reference will be the above mentioned Lecture Notes. - A. Tarski and S. Givant, A formalization of set theory without variables. AMS Coll. Pub. Vol. 41, 1987. - L. Henkin, J. D. Monk, and A.Tarski, Cylindric Algebras PartIL North-Holland, 1985. - I. Németi, "Algebraizations of quantifier logics, an introductory overview", in: Studia Logica 1990. Some of the additional, optional literature made available. - W. J. Blok and D. Pigozzi, Algebraizable logics. Memoires of AMS, 1989. - B. Jónsson, "The theory of binary relations", in: Algebraic Logic, Colloq. Math. Soc. J.Bolya Vol. 54, North-Holland, 1991. - R. I. Goldblatt, "Varieties of complex algebras", Annals of Pure & Applied Logic 1989. - R. I. Goldblatt, "Mathematics of modal logics", PartsI-II, in: Reports on Math. - H. Andréka, I. Németi, and LSain, Universal algebraic logic. Preprint 1988, 80 pages. Potential lecturers H. Andréka, I. Németi, I. Sain. |
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Metalogic Prerequisites Naive set theory and some axiomatic set theory, e.g. independence results (familiarity with independence of CH). Classical logic (first-order formulas, models, recursive enumerability, Gödel's completeness theorem). Content (aims & scope) Set theoretical foundations of logic and logical foundations of set theory. Resolution of the "chicken-egg" problem i.e. the vicious circularity in this. On the one side: The precise, true meaning of set theoretic independence results in view of all this. On the other side: If everything reduces to such formulas of LÎ which are not mathematical objects, then what are the models of first order logic? What is a symbol on the most basic level where we didn't yet started building math? What is a natural number? Do we know anything about the natural numbers when we start building our very basic "meta-language" or "background language" LÎ? If we succeeded in proving something about our "object language" (e.g. about the set theoretically coded version of the "metalanguage") then what did we learn about the original meta language this way? (Tools available for making such inferences.) How much a priori mathematics do we have to presuppose in order to build up our meta language LÎwhich will serve as a foundation for building up math? (The ideal situation would be zero but a careful analysis yielding nonzero but small seems to be preferable to a careless zero. A rigorous and careful proof of the following train of thought. If there is a formula jCH of standard second order logic L2 such that Ţ2jCH ifl' the Continuum Hypothesis is true (where Ţ2 denotes validity in all standard second order models); then there cannot exist a complete and sound effective inference system for L2. If there is time left then: A relatively easy variant of Gödel's incompleteness theorem, either Kalmár's diagonalization proof or the set theoretic version where coding (Gödel numbering) is practically trivial (transparent). Literature - J. D. Monk: Mathematical Logic. - I. Ruzsa: Logikai szintaxis és szemantika. Akadémiai Kiadó, Budapest, 1988. - J. Barwise: Admissible sets and structures. - M. Manzano: Higher order logic (in preparation). - I. Sain: There are general rules for specifying semantics. - J. L. Bell and M. Machover: A course in mathematical logic. pp.497-508, especially section 10.5. - H. B. Enderton: Mathematical Introduction to Logic. Much more should be covered than Ebbinghaus et al.: "Math. Logic, section VIL3-4, pp.108-114", which however is a reasonable starting point if the exercises are elaborated in detail (though this book does not really address the central questions to be discussed in the course). |
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Logic and Natural Language, Formal Semantics According to the general purpose of the Budapest Postgraduate School in Logic we would like to of@er a global view about the various apphcations of logical formalisms and methods outside mathematics. One of the most important fields where logic is used is linguistic theory and first of all the semantics of natural langage(s). The main goal of this course is to facilitate integrated research between linguists and logicians on this field. Prerequisites Knowledge of standard presentations of first-order logic including cylindric algebras. Higher-orer logic, G-models (i.e. Henkin-type "General models" ), G-completness. Basic modal logic with Kripke-models. Elements of universal algebra including many-sorted algebras and algebraic logic, theory of formal languages and computability (algorithmic decidability) theory. Some basic facts about the l-calculus. Non-compulsory, but useful: basics of abstract model theory or general theory of logics. Schedule, Content (aims & scope) 1. The fields of linguistcs (Uniformity and diversity of natural languages. Levels of linguistic description. Comparison with formal languages. Cornpetence and performance. A short survey of the history of linguistic theory. Structural linguistics.) 2. The methods of linguistics (Linguistic facts, linguistic explanation. The use of linguistic data: minimal pairs, crucial examples, etc. Restrictivity vs. universality. The relationship between syntax and semantics.) 3. Formal languages and natural language syntax (Finite state automata, context-free and context sensitive grammars. Complexity of natural language. Transformation rules. Filters and other reduction possibilities. First two are presupposed, will not be discussed, only mentioned and then used.) 4. Unification grammars (The mathematical theory of unification. Standard unification grammars. BUG (Budapest Unification Grammar): a demonstration.) 5. Between syntax and semantics (l-calculus, combinators as syntactic tools. Directionality, monotonicity. (l-calculus is presupposed. )) 6. Generalised quantifier theory (Relational view of quantifiers. Generalised quantifiers and model theory. Semantic properties of natural la guage quantifiers. Branching quantification. Semantic automata.) 7. Alge raic semantics (Many-sorte algebras for syntax. Ambiguity and term algebras. Safe derivers for manysorted algeb as. Montague grammars. (Many-sorted algebras and term algebras (i.e. free algebras) are presupposed.)) 8. Montague grammar (Intensional 1ogic. The PTQ-fragment: syntactic rules, translation rules and reduction rules.) 9. Plurals and mass-terms (The proble of non-Boolean conjunction. Set-theoretic perspective: Scha's solution. Algebraic perspective: Link's solution. Hydras. Group-readings. Generalised quantifiers and plurals.) 10. Anaphoric relations (What is anaphoric relationship`? Donkey sentences. File-change semantics and DRT: a comparision. Problems.) 11. Dynamic theories (Dynamic Pr dicate Logic. Dynamic Montague Grammar. General theory of dynamic interpretation.) 12. Partial models (The sources of partiality. Partial models. Non-monoton logics. The problem of questions. Data semantics and its applications.) 13. Presupposition (What is presupposed? Factive verbs, posessives, definite descriptions etc. Classical theories of presupposition. The use of many-valued logics. The Stalnaker-Heim approach. Dynamic and partial t eory of presupposition. Literature - Barwise, . 1979, On branching quantifiers in English, Journal of Philosophical Logic. - Barwise, . and R. Cooper, 1981, Generalised quantifiers and natural language, in: Linguistics and Philosophy. - Benthem, J.F.A.K. van, 1981, Why is semantics what?, in: J. Groenendijk, T. Janssen and M. Stokhof (eds.) Formal methods in the study of language.. Proceedings of the Amsterd m conference on Montague Grammar and related topics, MC-Tracts 135. & 136., Mat ematisch Centrum, Amsterdam. - Benthem, J.F.A.K. van, 1987, Categorial grammar and type theory, ITLI, Amsterdam. . Buszkowski, W. 1990, Classical categorial grammars, in: L. Kálmán and 1. Pólos (eds.) Papers fr m the second symposium on logic and language, Akadémiai Kiadó, Budapest. - Gallin, 1975, Intensional and higher-order modal logic,Mathemat,ics Studies 17., North-Ho land, Amsterdam. - Groenend'jk, J. and M. Stokhof 1989, Dynamic Predicate Logic, ITLI, Amsterdam. - Groenend' jk, J. and M. Stokhof 1990, Dynamic Montague Grammar, in: L. Kálmán and l. Pólos eds.) Papers from the second symposium on logic and language, Akadémiai Kiadó, B dapest. - Heim, I. 983, File change semantics and the familiarity theory of definiteness, in: R. Bauerle, Ch. Schwarze and A.v. Stechow (eds.) Meaning, use, and interpretation of language, Walter de Gruyter, Berlin - New York. - Kamp, H. 1981, A theoty of truth and semantic representation, in: J. Groenendijk, T. Janssen a d M. Stokhof (eds.) Formal methods ín the study of language. Proceedings of the Amst rdam conference on Montague Grammar and related topics, MC-Tracts 135. & 136., athematisch Centrum, Amsterdam. - Krifka, M 1990, Boolean and non-Boolean conjunction, in: L. Kálmán and 1. Pólos (eds.) Papers fr m the se.cond symposium on logic and language, Akadémiai Kiadó, Budapest. - Link, G. 1983, The logical analysis of plurals and mass-terms: a lattice-theoretical approach, in: R. Bauerle, Ch. Schwarze and A.v. Stechow (eds.) Meaning, use, and interpretation of language, Walter de Gruyter, Berlin - New York. - Montague, R. 1970, Universal grammar, repr. in: R.H.Thomason (ed.) Formal Philosophy. Selected Papers of Richard Montague,(1974) Yale University Press, New Haven. - Montague, R. 1973, The proper treatment of quantification in ordinary English, repr. in: R.H.Thomason (ed.) Formal Philosophy. Selected Papers of Richard Montague, (1974) Yale University Press, New Haven. - Soames, S. 1988, Presupposition, in: D. Gabbay and F. Guenthner (eds.) Handbook of Philosophical Logic , Vol IV. - Westerstahl, D. 1988, Quantifiers in formal and natural languages, in: D. Gabbay and F. Guenthner (eds.) Handbook of Philosophical Logic , Vol IV. Potential lecturers András Máté; László Pólos |
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Logics of programs Prerequisites - Basics of axiomatic set theory, especially independence results, and models of ZFC. (There is no need for technical knowledge like forcing. Understanding and insights are needed.) - Basics of classical and nonclassical logic, Kripke models. (Some algebraic logic would be most useful but is not indispensable.) - First order logic, model theory, Gödel's incompleteness theorem, basics of Peano's arithmetic. - Basics of Turing computability (or any equivalent notion like recursive enumerability). - Absolute logics (in set theoretic sense), the effect of the choice of our metamathematical tools on logic (this is not absolutely indispensable, but would be most useful; this is another course offered [is strongly related to set the,oretic model theory like KPU-absolute logics)). Content (aims & scope) Temporal logics for cornputer science, temporal logic representation of concurrency, nondeterminism etc., temporal expression and provability of termination, eventualities, invariance properties of programs etc. Completeness issues of temporal logics; computably tractable inference systems in completeness issues. Dynamic logics, Floyd-Hoare logic. Comparative study of the proof theoretic powers of the various logics of programs and program verification methods. Applications of nonstandard models in the latter as well as in characterizing or calibrating the proof theoretical powers of various (effective or tractable) inference systems. Nonstandard logics of programs. Dynamic algebras and temporal algebras (as special Boolean algebras with operators), a deeper study of propositional dynamic- and temporal logics. Basic ideas of nonmonotonic logic. Literature - R. Goldblatt: Logics of time and computation. Center for the Study of Language and Information, Lecture Notes Number 7, 1987. - I. Sain: Dynamic logic with nonstandard model theory. Dissertation, Hungarian Academy of Sciences, Budapest (in Hungarian). 1986. - I. Sain: Comparing and characterizing the powers of established program verification methods. In: Proc. Conf. "Many-sorted logic and its applications in Computer Science" (Leeds, U.K.), 1988, to appear. - H. Andréka, I. Németi, and I. Sain: Survey of effective completeness methods in first order temporal logic in "Temporal logics in Computer Science" (eds: L. Bolc, A. Szalas), Academic Press, to appear. - D. Gabbay and F. Günther: Handbook of Philosophical Logic Vol II, Reidel 1984. - I. Németi: Dynamic algebras in FCT 1981 (Springer Lecture Notes in Computer Science) - V. R. Pratt: Dynamic algebras. In: Studia Logica, 1990, to appear. - J. van Leeuven (ed.): Handbook of Theoretical Computer Science. North Holland 1989; within this: D. Kozen - J. Tiuryn: Logics of Programs. - T.Gergely and L.Ury: First order programming theories, in preparation. Potential lecturers Ildikó Sain, István Németi |
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Tarski and Trends in 20th Century Philosophical Logic (History of Logic) Content (aims & scope) Russell's importance for the philosophy of logic. Developments of logical techniques, new notational convention. Results in philosophical logic, theory of definite descriptions. The early days of logical positivism, the Vienna Circle. Carnap's syntactic period. The beginning of the area of logical semantics. Polish tradition. The philosophical and logical position of the Lvov-Warsaw school. Twardowski, Lukasiewicz, Lesniewski. The need of model theoretic-semantical foundations of deductive systems against Hilbertism. Tarski's field of interest and his results. His investigation in set theory, foundation of infinite notions. Results in metalogic (metamathematics), and in algebra. The definition of the consequence relation. The consequences of K. Gödel's (1931) result, and Tarski's epoch-marking work, the introduction of the notion of truth into formalized language. The philosophical (adequateness) and formal aspects of the semantic conception of truth. The ehmination of the paradox of the liar, and the object language - meta language construction. The construction of the definition of the notion "true sentence" in the language of class calculus. The creations of the Tarskian semantic concept of truth and the development of logical semantics. Carnap's semantic period. The definition of existential semantic system. Publications of Meaning and Necessity, and intensional semantic theory on the basis of the Fregean tradition. The importance of Meaning and Necessity. Farther developments in the Frege-Tarski type truth-value semantics: Carnap, Kripke, Prior, von Wright, Hintikka type modal theories, possible world semantics. Montague's intensional logic and formal pragmatics. Rival (anti-Tarskian) semantic theories, and between them, game theoretic semantics and dialogue games. Literature A. J. Ayer: Logical Positivism, New York, 1959. D. Gabbay - F. Guenthner (eds. ), Handbook of Philosophical Logic I-III. Reidel, Dortrecht, 1983. R. Carnap: The Logical Syntax of Language, 1937. R. Carnap: Introduction to Semantics, 1942. R. Carnap: Meaning and Necessity, 1958. J. Hintikka - Kulas: The Game of the Language, Reidel, Dortrecht, 1983. B. Saarinen (ed.): Game Theoretical Semantics, Reidel, Dortrecht, 1979. A. Tarski: Logic, Semantics and Metamathematics, Oxford, 1956. A. Tarski: Bizonyítás és igazság, Budapest, 1990. L. Henkin (ed.): Proceedings of the Tarski symposium, American Mathematical Society, Providence, 1974. R. Thomason (ed.): Formal Philosophy: Selected Papers of Richard Montague, New Haven, 1974. J. Wolenski: Logic and Philosophy in the Lvov-Warsaw School, Kluwer Academic Publishers, 1989. Potential lecturers Anna Madarász |
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History of Logic I-II. The course focuses on the connection of logic with philosophy in general and on some interrelations between mathematical and philosophical thinking conveyed by logic. The first semester presupposes only minimal knowledge of logic; the second one bases on the fundamental classical metatheorems of logic. Some familiarity with the eleatic philosophy, Plato, Artistotel, the scholastic philosophy, Leibniz and Wittgenstein would be useful. Each class will be based on short fragments taken from the works indicated as sources below. First semester 1. The rise of systematic reasoning in the Greek philosophy and mathematics. The importance of indirect reasoning. Sources: The fragments of Parmenides and Zeno. The Elements of Euclid. 2-3. Plato. The method of mathematics and its relation to the philosophy. Logical prineiples as philosophical arguments. The theory of Forms as logical semantics. Sources: Meno, Phaedo, The Republic., The Sophist. 4-5. Aristotle. Theory of substance and logic. Categorical propositions and syllogisms: the first coherent theory of logic. Problems of modality in logic and ontology. Paradoxes. Aristotel about mathematics. Sources: Categories, Hermeneutics, First Analytics, Metaphysics. 6-7. The Stoic logic. The natural deduction system of propositional logic. The Stoic semantics. Sources in: Hülser, K.-H.: Die Fragmente der Dialektik 8. Aristotelianism in the late antiquity. Discussions with the Stoics, attempts to the synthesis. Sources: Galen: Institutio Logica, Alexander of Aphrodisias: In Aristotelís Analyticam Prioram... 9. Between antiquity and scholasticism: Poxphyry, Boéthius, Avicenna. Source: Porphyry: Institutio Logica. 10-12. The logic of the schlasticism, the golden era of logical discussions. Proprietates terminorum, suppositio-theory, consequentiae: different views and their motivations. Sources: Abélard: Dialectica, Willam of Shyreswood: Introductiones in Logicam, Buridan: Sophismata, Pseudo-Scotus: In Universam Logicam Quaestiones, Ockham: Summa Logicae. Second semester 1. Lebniz: the first attempt for the mathamatization of the logic. The ideas of lingua characterica universalis and calculus ratiocinator; the relations between mathematics, logic and philosophy. Sources: Opuscules et fragments inédits (ed. Couturat); Generales Inquisitiones.... 2. Kant and the German philosophical tradition of transcendental logic. Source: Kritík der reinen Vernunft. 3. Bolzano's critique of Kant based on logical Platonism. His metalogical theory of implication. The foundational problems of mathematical analysis. Sources: Wissenschaftslehre; Paradoxien des Unendlichen. 4. The algebraic way to the mathematization of logic: early attempts, de Morgan, Boole, Schröder, Peirce. Source: Boole: An Investigation of the Laws of Thought... 5. Mathematical and philosophical motivations for a foundation of mathamatics. Cantor's set theory. Sources in: Cantor: Gesammelte Werke (ed. Zermelo). 6-8. Frege. The first modern system of mathematical logic, based on the concept of function. The Sinn-Bedeutung theory and its motivations. Logic and philosophy of mathematics; the first fall of logicism. Sources: Translations from the Philosophical Writings of Gottlob Frege; The Foundations of Arithmetic. 9. Criticism of Frege in the Tractatus; logical platonism and conservativism. Source: Wittgenstein: Tractatus logic.o philosophicus. 10. Type-theoretical logic: an attempt to the `correction' of logicism. Sources: Russell: `Mathematical logic based on the theory of types', Ramsey: `The foundations of mathematics'. 11. The logic in the philosophy of the Vienna Circle and the metalogical results of the thirties. (The second fall of logicism.) Sources: Writings by Carnap and Reichenbach. 12. The influence of logic in contemporary philosophy. Quine and the third life of logicism. Source: Quine: From a Logícal Point of View. Potential lecturers András Máté |
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Artifacts in logic Prerequisites - The "philosophy" of axiomatic set theory with an emphasis of independence results. Models of ZFC (only nontechnical knowledge is needed, there is no need for technical knowledge like forcing; understanding and insights are needed). - Gödel's incompleteness theorem, basics of Turing computability or any equivalent notion (existence of enumerable but not decidable sets). Content (aims & scope) The reasons why the standard version of higher order logic is not absolute. Consequences of non-absoluteness of the definition of a logic to mathematical results concerning that logic (e.g. incompleteness). Methodology of mathematical modelling of extra-mathematical phenomena elaborated for logic (especially philosophical and computer science logics). When do the results reflect only on the way the model was built instead of on the original phenomenon (logical system) under study. (The analogon of Heisenberg's uncertainty principle in formal logic.) How Henkin and his followers constructed absolute versions of higher order logics. Absoluteness w.r.t. weak systems of set theory, KPU-absoluteness. KPU-absolute versionsrof logics of programs, first order temporal logics etc. General methodology of constructing absolute (KPU-absolute) versions of (new) logics. Literature - (Parts of ) J. Barwise: Admissible Sets and Structures. Springer-Verlag, 1975. - K. L. Manders: First-order logical systems and set-theoretic definability. Preprint, Univ. of Pittsburgh, 1983. - I. Sain: There are general rules for specifying semantics: Observations on Abstract Model Theory. CL& CL (Computational Linguistics and Computer Languages) Vol XIII, 1979, pp. 195-250. - J. Vaananen: Set theoretic definability of logics. In: Handbook of Model Theoretic Logics (eds: J. Barwise and S. Feferman), Springer-Verlag, 1985. - A. Pasztor: Recursive programs and denotational semantics in absolute logics of programs. Theoretical Computer Science, to appear. Potential lecturers István Németi, Ildikó Sain, Hajnal Andréka |
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Logic and Cosmology Prerequisites Axiomatic set theory (independence methods, models of set theory), classical logic. The course can be given without these prerequisites but that needs a special arrangement and efforts on both sides. Content (aims & scope) One of the starting points of this course is the interplay of ideas between Einstein and Gödel which was started when they were both in Princeton; Gödel's work on relativity and its connections with later works e.g. by S. Hawking. We will derive the basic qualitative postulates of relativity theory by using only tools of logic and without using the usual mathematics of relativity like analysis and differential geometry. The emphasis is on the qualitative aspects of (the frontier areas of -, and paradoxical looking parts of ) relativity theory and cosmology (black holes, wormholes etc. , and on the possibility of approaching these by the machinery of modern logic (especially by using nonstandard models of set theory as an analogy). Connections with logics of time (systems related to temporal logic). Literature - Kurt Gödel: A remark about the realtionship between Relativity theory and idealistic philosophy. In: Albert Einstein: Philosopher-Scientist (ed.: P. A. Schilpp) Harper and Raw (Evanston, Illinois), (1949), pp. 557-562. - L. C. Epstein: Relativity visualized. - S. Hawking: A brief history of time. - R. Goldblatt: Diodorean modality in Minkowski Spacetime. Studia Logica 39(1980) 219-236. - R. Rucker: Infinity and the mind. - M. S. Morris, K. S. Thorne, and U. Yurtsever: "Wormholes, FTL, Time machines", Physics review letters 61, 1446(1988). - Michael Morris: Ph.D.Dissertation, Caltech. kb. 1986. - J. L. Bell: Toposes and local set theories. An introduction. Oxford Univ. Press, 1988. Chapter 8: (2) Some analogies with the theory of relativity. - Kompaneec: Fizika. MIR, Tankönyv. kb. 1960-62. - Kunen: Set theory. Potential lecturers István Németi, Hajnal Andréka |
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Category Theoretic Approaches to Algebraic Logic Content (aims & scope) Basic ideas of algebraic logic. Categories, enriched categories, algebraic theories (in Lawvere's sense), the Makkai et al. approach, enriched algebraic theories, pretoposes, toposes etc. Category of theories and interpretations (acting as theory morphisms). Institutions (by Goguen and Burstall). Connections between category theoretic logic and ``pure logic". The model theoretic aspects of this connection. Makkai's ultracategories (as a vehicle for a very strong kind of a completeness theorem or Stone representation theorem). Connections between category theoretic logic and Tarskian algebraic logic. The injectivity approach to categorical logic. Literature - M. Makkai and R. Paré: Accessible categories: The foundations of categorical model theory. Contemporary Math., 1989. - J. Adámek, H. Herrlich, and G. Strecker: Abstract and concrete categories, Wiley 1990. - A. Daigneault: Lawvere's elementary theories and polyadic and cylindric algebras. Fund. Math. Vol. 66 (1969) 307-328. - J. D. Monk: Review on the book: First order categorical logic, by M. Makkai and G. Reyes. Bull. Amer. Math. Soc. Vol. 84,6 (1978) 1378-1380. - M. Makkai: Stone duality for first order logic. Adv. Math. 65 (1987) 97-170. - H. Andréka and I. Németi: A general axiomatizability theorem formulated in terms of cone-injective subcategories. In: Universal Algebra, Proc. Coll. Esztergom, Colloq. Math. Soc. J. Bolyai 29, 1981. 13-35. - R. Guitart and C. Lair: Calcul syntaxique des modeles et calcul des formules internes. Diagrammes, Vo1.4, Dec. 1980. 106pp. |
