Logic, Language, Mathematics

Átírás

1 Logic, Language, Mathematics A Philosophy Conference in Memory of Imre Ruzsa September 1719, 2009 Budapest Abstracts Eötvös University of Budapest, Institute of Philosophy Budapest VIII. Múzeum krt. 4/i

2 Contents Keynote Lecture Quantiers and Admissible Propositions 1 Robert Goldblatt Plenary Lectures Natural Logic, Medieval Logic and Formal Semantics 2 Gyula Klíma Whose Logic is Three-valued Logic? 3 Ferenc Csaba Modal Constructions in Sociological Arguments 4 László Pólos Analogy in Semantics 5 László Kálmán Certain Verbs Are Syntactically Explicit Quantiers 6 Anna Szabolcsi The Treatment of Ordinary Quantication in English Proper 8 András Kornai Exporting Methods from the Foundation of Mathematics to the Foundation of Relativity Theory 9 Hajnal Andréka and István Németi In Defense of Hermeneutic Fictionalism 10 Gábor Forrai Relativity and Modal Logic 11 Robin Hirsch Tasks and Ultra-tasks 12 Zoltán Szabó Gendler Neo-Fregeanism: Revising Frege's Notion of Identity in the Philosophy of Language and Mathematics 13 Mihály Makkai Many-Dimensional Modal Logics 14 Ági Kurucz English Sessions Logic and Language of Relativity Theories 15 Gergely Székely Visualizations of Relativity, Relativistic Hypercomputing 16 Renáta Tordai i

3 Comparing Relativistic and Newtonian Dynamics in First Order Logic 17 Judit X. Madarász On Field's Nominalization of Physical Theories 18 Máté Szabó Plural Grundgesetze 19 Francesca Boccuni The Reference of Numerals in Frege 20 Edward Kanterian Grasping the Conceptual Dierence between János Bolyai's and Lobachevskii's Notions of Non-Euclidean Parallelism 21 János Tanács Prior and the Limits of de Re Temporal Possibility 22 Márta Ujvári The Indispensability of Logic 23 Nenad Miscevic Names are Not Rigid 24 Hanoch Ben-Yami Premise Semantics and Possible Worlds Semantics for Counterfactuals 25 Vladan Djordjevic Fitch's Paradox and Natural Deduction System for Modal Logic 26 Edi Pavlovic Counterfactuals, Context, and Knowledge 27 Jelena Ostojic Aristotle's Wheel and Galileo's Mistake 28 Nenad Filipovic, Una Stojnic & Vladan Djordjevic On the So-Called Dependent (Embedded) Questions 29 Anna Bro»ek Partiality and Tich y's Transparent Intensional Logic: Solutions to Selected Issues 30 Ji í Raclavský `Upgrades' and `Updates': from Degrees of Belief to the Dynamics of Epistemic Logic 32 András Benedek Ruzsa on Quine's Argument against Modal Logic 33 Zsóa Zvolenszky Denite Descriptions in Dynamic Predicate Logic 34 Péter Mekis ii

4 Hungarian Sessions A matematikai tudás eukleidészi modelljének kritikája Lakatos Imre lozóájában 35 Golden Dániel Tarski és a deácionizmus 36 Kocsis László Szemantikai értékrés Cantor mennyországának égboltján avagy mi az, amit megmentett Hilbert? 37 Geier János Kontextuális kétdimenziós szemantika 39 Kovács János A logika iskolai tanulásának els lépései 40 Kiss Olga Munkácsy Katalin Az empirikus tudományok teoretizálási törekvéseir l 41 Madaras Lászlóné Kísérlet a tulajdonnevek vizsgálatára a különböz hipertextnarratívák esetében 42 Szopos András iii

5 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 1 Quantiers and Admissible Propositions Rob Goldblatt rob.goldblatt@msor.vuw.ac.nz Victoria University, Wellington There are many quantied modal logics that cannot be characterised by validity in Kripke models, even though their propositional fragments have a complete Kripke semantics. This talk will describe a way of giving complete semantics to all quantied modal logics by taking seriously the view that only certain admissible sets of worlds should count as propositions. The challenge in such an approach comes in using the class of admissible propositions to interpret the quantiers in a validity preserving manner.

6 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 2 Natural Logic, Medieval Logic and Formal Semantics Gyula Klíma klima@fordham.edu Fordham University Recent investigations in natural logic, the logic actually encoded in natural language usage, as opposed to the formal semantic and deductive systems presented by contemporary professional logicians (see, e.g. Sanchez, V., Studies on Natural Logic and Categorial Grammar, Doctoral dissertation, University of Amsterdam, 1991; LF and Natural Logic, in Preyer, G. and G. Peter, G. (eds.), 2002, Logical Form and Language. Oxford: OUP), have sparked some interest in medieval logic, as providing both a description of a natural logic and a regimentation of an actual natural language, namely, the technical Latin of scholastic philosophy. This paper, through an analysis of John Buridan's (ca ) nominalist approach to logical semantics, will argue that in our contemporary enterprise we may in fact be able to utilize a great deal from these medieval ideas, provided we keep these ideas in proper perspective, keeping always in mind what they were meant to be used for, and what they were not (without implying, though, that we cannot use them for something else, with the relevant provisos in place).

7 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 3 Whose Logic is Three-valued Logic? Ferenc Csaba csferenc@yahoo.com Eötvös University of Budapest One of Imre Ruzsa's most important achievements in philosophical logic is his system of intensional logic with semantic value gaps. These gaps are a means of handling the diculties which are caused by partial predicates, or descriptions which do not or do not uniquely denote, or variables denoting an object which is not an element of the appropriate domain. In the case of sentences, the semantic value gap truth value gap is not a genuine truth value, only a lack of such, and the presence of truth-value gaps is perfectly reconcilable with a realist attitude to semantic issues, and can serve as a means for a logic of empirical investigation. The question then arises: what happens if there are genuine truth-value gaps, e.g. sentences which are meaningful but undecidable in the strongest sense: even God does not know whether they are true or not. It would have consequences not only to our logic but for the divine logic, too. Michael Dummett has argued that the latter must be a kind of three-valued logic, while the former must be intuitionist logic. In my paper I will investigate what type of sentences could have the chance of being undecidable in the strong sense. Of course, if a sentence is strongly undecidable, we will never know that it is so; my chances therefore are very but not innitely limited.

8 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 4 Modal Constructions in Sociological Arguments László Pólos laszlo.polos@durham.ac.uk Durham Business School We rst provide an overview of the formal properties of this family of models and outline key dierences with classical rst-order logic. We then build a model to represent processes of perception and belief core to social theories. To do this, we dene our multi-modal language and then add substantive constraints that specify the inferential behavior of modalities for perception, default, and belief. We illustrate the deployment of this language to the theory of legitimation proposed by Hannan, Pólos, and Carroll (2007). This paper aims to call attention to the potential benets of modal logics for theory building in sociology.

9 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 5 Analogy in Semantics László Kálmán kalman@nytud.hu Eötvös University of Budapest / Hungarian Academy of Sciences The principle of compositionality may seem perfectly trivial. However, depending on what one means by meaning, one could argue that it imposes no substantive constraint or, to the contrary, that it cannot be obeyed at all. On the other hand, we could view this principle as a denition of meaning (or a component of such a denition), in which case it would yield a very abstract concept of meaning, one very far from empirically testable reality. In my paper, I will propose a holistic approach instead of the traditional, analytic/atomistic one. Instead of insisting on cutting forms and meanings into pieces (or building them up from primitive and complex building blocks), I will emphasise the global features of signs. I will introduce the principle of generalized compositionality, which is based on the concept of similarities between forms and meanings. (The similarity of two forms or meanings is often related to their recognizable component parts, but the relationship is more complicated and indirect than the one inherent in the traditional concept of compositionality). My generalized compositionality principle states that we interpret and produce complex signs by analogy, relying on our earlier experience on similar complex signs and their interpretation. This approach, I believe, has several attractive consequences. First, it predicts that interpretation will be subject to various frequency eects and other psychological factors (just like phonological or morphological phenomena in analogy-based models). This clearly means that we aim at a cognitively more realistic model, with a possibility of individual dierences in interpretation and a clear-cut concept of where so-called pragmatic factors enter interpretation. Second, on this approach, the dubious distinction between literal and non-literal interpretations no longer make sense: the mechanism of gurative interpretation does not dier in any way from literal interpretation.

10 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 6 Certain Verbs Are Syntactically Explicit Quantiers Anna Szabolcsi anna.szabolcsi@nyu.edu New York University Schlenker (Mind and Language, 2006) observes that there are pervasive similarities both in the logical properties of quantication over individuals, world, and times and in the linguistic devices (quantiers, denite descriptions, pronouns, demonstratives) that pertain to them. Yet their treatment has not been uniform in philosophical logic. In particular, quantication over individuals is typically executed in a syntactically explicit manner, using variables ranging over the whole universe, whereas quantication over times and worlds is typically executed using non-variablebinding operators of a much more limited power, such as the and operators of modal logic and Montague's, the abstractor over indices of worlds. Ontological symmetry could be achieved if individuals, times, and worlds were treated alike. Indeed, both in philosophical logic and in linguistics there have been signi- cant precedents for deviation from the typical strategy. Quine (1960) recasts quantication over individuals along the lines of modal propositional logic, and Ben-Shalom (1996) makes the approach linguistically more relevant by presenting the nominal restriction of determiners as the accessibility relation associated with modal operators. From the other end, Groenendijk and Stokhof's (1984) theory of questions is among the rst to demonstrate a need to quantify over worlds explicitly. Cresswell (1990), Iatridou (1994), Percus (2000), Schlenker (1999, 2004), Pratt and Francez (2001), Kusumoto (2005), Lechner (2007), and von Stechow (to appear) are among the growing number of authors who have proposed to treat certain cases of time and world quantication in a syntactically explicit manner. The primary diagnostics for explicit quantication include the existence of variable-like pronouns referring to the syntactically represented argument, the fact that the argument is not evaluated with respect to a single index, and the fact that the argument need not be linked to the closest suitable operator. A related but distinct question is the following: Among the linguistic operators with quanticational content, which ones are explicit quantiers? The existence of an explicitly quantiable argument does not make it necessary for all operators pertaining to it to be explicit quantiers. This paper examines so-called raising verbs in Shupamem (a Grasseld Bantu language), Dutch, and English. Raising verbs are non-agentive verbs whose surface subjects can be thought of as originating in the verb's innitival complement. Relevant examp-

11 les in English are aspectual begin (as in The paint began to dry ), seem, and threaten (as in The barn threatened to collapse ). I will suggest that scope interaction with an appropriate subject indicates that such verbs are syntactically explicit quantiers over times and worlds, and moreover ones that acquire scope in the same manner as expressions quantifying over individuals (by quantier raising and scope reconstruction). I thus add a new diagnostic for syntactically explicit quantication. 7

12 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 8 The Treatment of Ordinary Quantication in English Proper András Kornai kornai@mokk.bme.hu Budapest University of Technology and Economics We bring together some well-known lines of criticism directed at Montague Grammar, such as (i) taking a stilted, highly regulated variety of language as the object of inquiry; (ii) ignoring the meaning of content words; and (iii) the failure to treat hyperintensionals; and oer a coherent, and we believe much simpler, alternative using an algebraic variety of model structures.

13 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 9 Exporting Methods from the Foundation of Mathematics to the Foundation of Relativity Theory Hajnal Andréka and István Németi andreka@math-inst.hu / nemeti@math-inst.hu Rényi Institute of Mathematics, Budapest We use experience gained during the success story of the foundation of mathematics to serve as guideline for elaborating foundation for natural sciences. Since spacetime is the arena in which the processes of physics and indeed most of natural sciences unfold, it seems to be reasonable to start with elaborating a logic based foundation for spacetime. For this, Tarski's work, in particular his rst-order logic axiomatization and analysis of geometry, is a good starting point. Goldblatt's book on spacetime geometry already made progress in this direction. We report on progress made in this direction in our school in the last 10 years. In particular, we will show how one can build up relativity theories (including general relativity and cosmology and Einstein's E = mc 2 ) purely within logic, as theories in the sense of logic, and with no other prerequisites than some familiarity with the basics of logic. This will provide, as a byproduct, a logic based foundation for relativity (in analogy with the foundation of mathematics) as well as a conceptual analysis for relativity theories. Further, it will provide a gentle (and streamlined) introduction to relativity for the questioning mind or for the logically minded. We touch upon connections with the logical theory of denability (Reichenbach, Tarski, Beth, Makkai). Instead of putting the emphasis on a particular formulation of relativity theory, we put the emphasis on the connections (interpretations) between dierent theories leading up to logical dynamics, the technical counterpart of which is known as algebraic logic.

14 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 10 In Defense of Hermeneutic Fictionalism Gábor Forrai Department of Philosophy, University of Miskolc Hermeneutic ctionalism about mathematics maintains that mathematics is not committed to the existence of abstract objects such as numbers are supposed to be. Mathematical sentences are true, but they should not construed literally. Numbers are just ctions in terms of which we can conveniently describe things which exist. The paper defends Stephen Yablo's hermeneutic ctionalism against an objection proposed by John Burgess and Gideon Rosen. The objection, directed against all forms of nominalism, goes as follows. Nominalism can take either a hermeneutic form and claim that mathematics, when rightly understood, is not committed to existence of abstract objects, or a revolutionary form and claim that mathematics is to be understood literally but is false. The hermeneutic version is said to be untenable because there is no philosophically unbiased linguistic argument to show that mathematics should not be understood literally. Against this I argue that it is wrong to demand that hermeneutic ctionalism should be established solely on the basis of linguistic evidence. In addition, there are reasons to think that hermeneutic ctionalism cannot even be defeated by linguistic arguments alone.

15 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 11 Relativity and Modal Logic Robin Hirsch Department of Computer Science, University College London There are two funny things about the special theory of relativity: (i) the speed of light is constant and (ii) all observers can themselves by observed. Relativity theory encourages us to abandon any absolute frame of reference. It discourages us from making statements such as the length of this rod is x, but prefers the length of this rod is x in frame of reference F. It is therefore natural to use modal logic to describe relativity theory. In this talk I'll review a number of modal logics that attempt to describe aspects of relativity theory. In particular we will see how property (ii) above has to be handled carefully, if we are to restrict to standard Kripke semantics for modal logics.

16 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 12 Tasks and Ultra-tasks Zoltán Szabó Gendler Yale University Can we count the primes? There is a near unanimous consensus that in principle we can. I believe the near-consensus rests on a mistake: we tend to confuse counting the primes with counting each prime. To count the primes, I suggest, is to come up with an answer to the question How many primes are there? because of counting each prime. This, in turn requires some sort of dependence of outcome on process. Building on some ideas from Max Black, I argue that barring very odd laws of nature such dependence cannot obtain.

17 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 13 Neo-Fregeanism: Revising Frege's Notion of Identity in the Philosophy of Language and Mathematics Mihály Makkai McGill University Since the middle 1990's, I have been working on a new approach to the foundations of mathematics, one that is based on a new version of type theory called "First Order Logic with Dependent Sorts" (F OLDS). This is an extension to the classical Russell-Ramsey type theory, and it has the ambition of serving as the logical basis of a fully comprehensive foundational system, in the sprit of Frege's Grundgesetze der Arithmetic. The novelty of the approach lies in a new systematic and relativistic conception of identity. In this, identity is no longer a primitive as it is in Frege; rather, it is dened on the basis of the logic of F OLDS. Identity becomes type-dependent; it becomes meaningless to ask if entities of dierent types are identical (equal) or not. Category theory, and its extension to higher dimensional categories, a currently emerging branch of abstract mathematics, is a natural environment for the F OLDS language and the F OLDS identity concept. In the talk, I will make an attempt to relate my mathematical work to the philosophy of mathematics of Frege and of subsequent thinkers.

18 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 14 Many-Dimensional Modal Logics Ági Kurucz King's College London Many-dimensional propositional modal logics (multi-modal logics having manydimensional Kripke frames among their frames) have been studied both in pure modal logic and in various computer science and articial intelligence applications. They are also connected to algebras of relations in algebraic logic, and to nite variable fragments of modal, intuitionistic and classical predicate logics. In this talk we discuss some of these connections. We also give a survey of the known results and open questions on the axiomatisation and decision problems of many-dimensional modal logics.

19 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 15 Logic and Language of Relativity Theories Gergely Székely Rényi Institute of Mathematics, Budapest Applying mathematical logic in the foundations of relativity theories is not a new idea at all, among others, it goes back to such leading mathematicians and philosophers as Hilbert, Reichenbach, Carnap, Gödel, Tarski, Suppes and Friedman. There are many examples showing the benets of using axiomatic method in the foundations of mathematics. That motivates the Hungarian school led by Hajnal Andréka and István Németi to apply this method in the foundations of relativity theories. This talk is based on the research of this school. Our school's general aims are to axiomatize relativity theories within pure rst-order logic using simple, comprehensible and transparent basic assumptions (axioms); and to prove the surprising predictions (theorems) of relativity theories using a minimal number of convincing axioms. Via a sample of results in the application of axiomatic method to special and general relativity theories, we try to show that their application to physics is a promisingly fruitful research area.

20 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 16 Visualizations of Relativity, Relativistic Hypercomputing Renáta Tordai Rényi Institute of Mathematics This talk is strongly related to the school directed by Hajnal Andréka and István Németi at the Rényi Institute of Mathematics, see the abstracts of István Németi, Hajnal Andréka, Judit X. Madarász and Gergely Székely. We will present visualizations of relativity. For example, we will present a movie showing what an astronaut would see while ying through a huge Kerr-Newmann wormhole or any other kind of wormhole. We will also outline the ideas of relativistic hypercomputing, i.e., how Malament-Hogarth spacetimes can be used for designing articial systems computing beyond the Turing barrier. Any spacetime admitting a CT C (closed timelike curve) is suitable for constructing such a hypercomputer, but the existence of CT C's is not really needed for this. A much milder condition called Malament-Hogarth property is sucient. We refer to [1], [2], and [3] for more detail. (The most satisfactory solution to the so called blue-shift problem is available in [4].) References [1 ] Dávid, Gy., Németi, I., Relativistic computers and the Turing barrier. Applied Mathematics and Computation 178 (2006). [2 ] Andréka, H., Németi, I., Németi, P., General relativistic hypercomputing and foundation of mathematics. Natural Computing, to appear. [3 ] Etesi, G., Németi, I., Non-Turing computations via Malament-Hogarth space-times. International Journal of Theoretical Physics 41,2 (2002). [4 ] Andréka, H., Németi, I., Németi, P., Presentation in The science and philosophy of unconventional compuing SPUC 2009, Cambridge, March 2009

21 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 17 Comparing Relativistic and Newtonian Dynamics in First Order Logic Judit X. Madarász Rényi Institute of Mathematics This talk is strongly related to the talks of Hajnal Andréka, István Németi and Gergely Székely. We introduce and compare Newtonian and relativistic dynamics as two theories of rst-order logic. To illustrate the similarities between Newtonian and relativistic dynamics we axiomatize them such that they dier in one axiom only. This one axiom dierence, however, leads to radical dierences in the predictions of the two theories. One of their major dierences manifests itself in the relation between relativistic and rest masses. The statement that the centerlines of a system of point masses viewed from two dierent reference frames are related exactly by the coordinate transformation between them seems to be a natural and harmless assumption; and it is natural and harmless in Newtonian dynamics. However, in relativistic dynamics it leads to a contradiction. We are going to present a siple geometric proof for this surprising fact.

22 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 18 On Field's Nominalization of Physical Theories Máté Szabó Eötvös University of Budapest Quine and Putnam's Indispensability Argument claims that we must be ontologically committed to mathematical objects, because of the indispensability of mathematics in our best scientic theories. Indispensability means that physical theories refer to and quantify over mathematical entities such as sets, numbers and functions. In his famous book Science Without Numbers' Hartry Field argues that this is not the case. We can nominalize our physical theories, that is we can reformulate them in such a way that (1) the new version preserves the attractivity of the theory, and (2) the nominalized theory does not contain quantications over mathematical entities. I'm going to reconsider Field's nominalization procedure for a toy physic theory formulated in a rst order language, in order to make a clear distinction between the following three steps: - the physical theory in terms of empirical observations; - the standard physical theory, which contains quantication over mathematical entities, as usual; - the nominalized version of the theory without any reference to mathematical entities. Having Field's nominalization procedure reconstructed, it will be clear that there is no dierence between the original and the nominalized versions of the theory, at least, there is no dierence from a formalist point of view. It is because the only dierence would come from the dierent meanings of the variables over which the quantications are running. The formalist philosophy of mathematics, however, denies that the variables have meanings at all. So, the formal systems as abstract mathematical entities are still included in physical theories; and this fact is highly enough for the structural platonist or immanent realist to apply the Quine-Putnam argument. Finally, therefore, I will suggest a completely dierent way for the objection to the Quine-Putnam argument.

23 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 19 Plural Grundgesetze Francesca Boccuni University of Padua It is well-known that the logical system for the logicist foundation of mathematics exposed in Grundgesetze der Arithmetik is inconsistent. The contradiction is derived from the infamous Basic Law V. This principle is crucial to Frege's logicism as it embeds the tenet that tightly connects natural numbers, conceived as equivalence classes, to concepts. As far as it is currently known, moreover, the so far provided consistent subsystems of Grundgesetze displaying some version of Basic Law V cannot interpret second-order Peano arithmetic. This seems to show that Frege's programme could not be completely recovered, after all. Secondly, these subsystems may be challenged with respect to the issue of to what extent they actually capture Frege's notion of concept. In particular, both these subsystems are based on a more or less radical limitation of the universe of Fregean concepts, which seems to be incompatible with Frege's spirit. The aim of this article is to present a consistent predicative second-order system with plural comprehension and Basic Law V, Plural Grundgesetze (P G), which is capable of deriving second-order Peano axioms. The main features of P G are plural quantication, which will guarantee the power of full second-order logic to P G, and predicative comprehension for concepts. I will also analyse the issue regarding predicativism from a Fregean perspective.

24 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 20 The Reference of Numerals in Frege Edward Kanterian Trinity College, Oxford Joan Weiner has recently (2007) argued that Frege's analysis of numerals does not commit him to the view that prior to this analysis numerals already referred to particular objects, numbers; the requirements for a faithful denition of number did not involve for him criteria for the preservation of sense or reference in the transition from pre-systematic uses of numerals and number statements to their use in his formal system. For the pre-systematic use is too vacillating and indeterminate for pure science. I demonstrate that her account faces both exegetical and substantive dif- culties. It ignores Frege's robust realism in both logic and arithmetic; logic describes pre-existing relations between Platonic objects (thoughts), and his account of number and arithmetical truth in general is subservient to this realism. It is also not true that Frege does not ask for the preservation of any sense or reference of ordinary uses of number. His revisionism is limited to predicative/attributive uses considered irrelevant for scientic purposes ( F A 57, 60). Without some preservation of sense and reference the point and nature of the transition from pre-systematic to systematic arithmetic would be left wanting. In fact, as I show, on Weiner's account Frege turns into a formalist for whom the sense and reference of numerals and number statements is a system-internal feature. But it is demonstrated that this misses not only Frege's Platonism, but also his insistence on the applicability of arithmetic. Finally, it is argued that while it is correct to stress, as Weiner does, that Frege's logicism had an epistemological agenda (to prove the analyticity of arithmetical truths), this characterisation must be supplemented by the ontological aspect of his project, which is to prove that numbers are objects and thus that arithmetic is a science with a proper subject matter. References Joan Weiner, What's in a Numeral? Frege's Answer, in: Mind, 116: 2007

25 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 21 Grasping the Conceptual Dierence between János Bolyai's and Lobachevskii's Notions of Non-Euclidean Parallelism János Tanács Budapest University of Technology and Economics, Department of Philosophy and History of Science The presentation is going examine the dierence between János Bolyai's and Lobachevskii's notions of non-euclidean parallelism. The examination starts with the summary of a widespread view of historians of mathematics on János Bolyai's notion of non-euclidean parallelism used in the rst paragraph of his Appendix. After this a novel position of the location and meaning of Bolyai's term parallela in his Appendix is put forward. Subsequently János Bolyai's Hungarian manuscript, the Commentary on Lobachevskii's Geometrische Untersuchungen is elaborated in order to see how Bolyai's and Lobachevskii's notions of parallelism dier. The careful examination of the Commentary reveals a seeming incoherence of Bolyai's translation, and nally the explanation of this incoherence oered by the received view and that of the novel position will be compared and assessed.

26 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 22 Prior and the Limits of de Re Temporal Possibility Márta Ujvári Department of Philosophy, Corvinus University In chapter VIII of Papers on Time and Tense Prior elaborates his polemic on whether radical coming-into-being is a genuine de re possibility of individuals. He considers it by the putative complete property swap of two individuals, Julius Caesar and Mark Antony, through worlds. Prior's original solution to the dilemma of the Leibnizian vs haecceistic position with respect to property-indiscernible worlds consists in pointing out that the property swap must necessarily stop at the property of origin. However, the possibility he denies is temporal and not logical; for, when we ask, `when was it possible', it is easy to see that `after his birth... it was clearly too late for him to have had dierent parents.' And as to the de re possibility of having dierent parents `before Caesar existed' the obvious retort is there would seem to have been no individual identiable as Caesar... who could have been the subject of this possibility'. This sounds fairly trivial. But by parity of reasoning we can get an uncomfortable consequence; for, if Caesar (or any other actual individual) could not have been the subject, before his birth, of the (later) unrealized possibility, equally, he could not have been the subject of the later realized possibility either. Which means, that none of us who was going to be born could have been the subject of a de re possibility of being (going to be) born i.e., at least not before our conception. This amounts to saying that what is once actual is preceded by what is non-possible, contravening thus the logic of propositional modalities. The air of paradox can be dissolved by denying, with Prior, that the possibility of origin is a genuine de re possibility. As a possibility it is general or de dicto: it is possible that someone be born to such and such parents, but it is not possible of someone that he should be born to these or other parents.

27 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 23 The Indispensability of Logic Nenad Miscevic Central European University, University of Maribor The paper discusses the currently prominent strategy of justifying our elementary logical-inferential practices by their unavoidability and global indispensability for all our cognitive eorts. It starts by agreeing with prominent apriorists about their attempt to justify such beliefs either from naturalistic computationalist considerations of unavoidability (inevitability) (Horwich) or from constitutiveness (Boghossian) or from global indispensability Argument (C. Wright), and then proceeds to argue that unavoidable and indispensable tools provide entitlement/justication for projects if projects are themselves meaningful. However, we are justied to think that our most general cognitive project is meaningful, and justied partly of the basis of its up to date success; and this basis is a posteriori. Therefore, the whole reective justication from compellingness and unavoidability is a posteriori. This suggests that the justication of our intuitional armchair beliefs and practices in general is plural and structured, with a priori and a posteriori elements combined in a complex way. It seems thus that a priori/ a posteriori distinction is useful and to the point. What is needed is renement and respect for structure, not rejection of the distinction.

28 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 24 Names are Not Rigid Hanoch Ben-Yami Central European University This claim is established by relying on the trivial observation that the same name may be used to name dierent people. The problem this creates has been noticed by Kripke, and he tried to reply to it in the 1980 Preface to Naming and Necessity, but his explanation fails. Other attempts to overcome the diculty by indexing names, by individuating names according to their reference, and more are examined and rejected as well. It is doubtful whether the concept of rigidity should play any role in describing our modal discourse.

29 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 25 Premise Semantics and Possible Worlds Semantics for Counterfactuals Vladan Djordjevic University of Belgrade A typical possible worlds semantics (P W S) for counterfactuals is modal logic with the addition of the so-called selection function, whose role is to somehow separate important from unimportant worlds. A counterfactual A > C is true i C holds at the important A-worlds. The older, premise semantics (P S) says that A > C is true i A, together with some further true premises B 1, B 2,..., entails C. The main problem for P W S is to explain which worlds are important, and for P S it is to specify which truths are to be included among the B's. That problem is very dicult to be solved in general, but in particular cases we often do have clear intuitions about the importance of worlds and about the B's. I argue that our intuitions used in P S are more basic, since in testing our selection function we use our intuitions from P S, rather than the other way around, that is, we say that the important worlds are those where the B's hold, and we do not explain the B's in terms of important worlds. Although P W S is a much more powerful logical tool, if what I said is correct, we still need to investigate the relation between the two semantics. That explains the motive behind the two results I will defend. The rst says that the standard interpretation of Goodman's P S is not correct since it validates conditional excluded middle, which Goodman rejects, and, second, that Lewis' notion of cotenability, which allegedly captures the intentions of the premise semanticists, fails to do so, and that this is a problem for Lewis' and not for the premise semantics.

30 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 26 Fitch's Paradox and Natural Deduction System for Modal Logic Edi Pavlovic University of Rijeka I use Basin-Matthews-Vigano's labeled deduction system for modal logic to reformulate Fitch's paradox. In the paper some possible solutions are discussed in this new formal framework.

31 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 27 Counterfactuals, Context, and Knowledge Jelena Ostojic University of Belgrade It is common opinion that counterfactuals are highly context-dependent, but there are dierent views about the way context inuences the truth-conditions for counterfactuals. Dierent theories explain the context dependency of counterfactuals in dierent ways. For example, the so-called standard theories (Stalnaker, Lewis), and the so-called pragmatic theories or strict implication analysis of counterfactuals (Warmbrod, von Fintel et al.) oer explanations that are dierent in many important respects. I will argue that the pragmatic theories give an explanation that better ts our language practice. I will conclude by pointing to what I see as another advantage of the pragmatic theories: in applying counterfactuals to epistemology (like Nozick, DeRose and others who dene knowledge in terms of counterfactuals), the standard view of the truth-conditions leads to denying the closure principle, and a specic version of the pragmatic view, which I will dene, leads to epistemic contextualism and enables us to keep the closure principle.

32 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 28 Aristotle's Wheel and Galileo's Mistake Nenad Filipovic, Una Stojnic & Vladan Djordjevic / / University of Belgrade There are two peculiar mathematical-metaphysical thought-experiments that are crucial to Galileo's consideration of the notion of continuum. The rst one opposes an Aristotelian claim that was generally accepted at that time that an actual innite division of a continuum is impossible: by banding the straight line into the circle, one can obtain innitely many parts, or sides, because, as Galileo believed, circle is a polygon with innitely many sides. The second one applies the same conception of the circle as a key idea to the solution to an ancient paradox known as The Aristotle's Wheel. Galileo uses an analogy between circles and polygons with nitely many sides for his very original, unusual and interesting solution, and that solution is our main topic in this paper. After oering a solution to the paradox based on contemporary theories of continuum, we will present Galileo's putative solution, and point to its signicance to Galileo's theory of continuum. We will then give two arguments aimed to show a contradiction in Galileo's solution. Our intention is to suggest an inner critique, without appealing to any particular modern or old theory of continuum, and without using any claim that could not be ascribed to Galileo. Although our rst argument might fall short of our target, since it applies a Euclidean denition which Galileo might reject, we believe that our second argument does not presuppose anything external to Galileo's theory.

33 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 29 On the So-Called Dependent (Embedded) Questions Anna Bro»ek Department of Logical Semiotics, Warsaw University In general theory of questions, the more and more important role is played by analysis of dependent questions, i.e. of expressions which (i) are parts of compound questions and (ii) are isomorphic with some independent questions (scil. questions sensu stricto). One may meet the tendency to explicate the sense of independent questions by the sense of dependent ones, e.g. the sense of questions such as: (1) Where is Budapest situated? is explicated by the sense of sentences such as: (2) A knows where Budapest is situated. where (2) contains (1) as a part. The analysis of dependent questions is often the point of departure of constructing settheoretical or possible-worlds semantics for independent questions. In my opinion, these tendencies are abortive and lead to irrelevant explications of the sense of questions sensu stricto. But on the other hand, semiotic functions of the so-called dependent questions as parts of compound expressions require deeper analysis. My paper contains a proposal of such an analysis.

34 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 30 Partiality and Tich y's Transparent Intensional Logic: Solutions to Selected Issues Ji í Raclavský Department of Philosophy, Masaryk University To work with partial functions (having no value but a gap for some of their arguments) is frustrating: classical logical laws (e.g., De Morgan law for exchange of quantiers) designed for total functions usually (if not ever) collapse. To incorporate partial functions, Tichý suggested the modication of (the natural deduction for) the logic of simple theory of types mainly by the correction of the rule of β-reduction (because of partiality, β-reduction is not a rule equivalent to β-expansion). As it is apparent from Tichýh's collected papers and his monograph, Tichý's transparent intensional logic, treating both modal and temporal variability, is a powerful logical system for logical analysis (explication) of natural language meaning. 1 The present author shows how to dene within Tichý's system 3-valued connectives which get a value even when an input proposition is gappy (e.g., exclusion negation or totalizing true-predicate). Another contribution is made by the correct formulation of the extensionality principle for partial functions. Another contribution is made by correct formulation of the notion complementary function, i.e. a function non- F having extensions which are complementary to extensions of the function F (not only two intuitively plausible explications, but rather partial classes complicate the matter). References (1) Tichý, P., The Foundations of Partial Type Theory. Reports on Mathematical Logic, 14 (1982) (2) Tichý, P., The Foundations of Frege's Logic, Walter de Gruyter, 1998 (3) Tichý, P., Pavel Tychý's Collected Papers in Logic and Philosophy. V. Svoboda, B. Jespersen, C. Cheyne (eds.), Dunedin: University of Otago Press, 2004 (4) Raclavský, J., Dening Basic Kinds of Properties, in: T. Marvan, M. Zouhar (eds.), The World of Language and the World beyond Language, Filozocký ústav SAV, [The text includes a rigorous classication of 1 The adoption of partial functions for logical analysis of natural language was stressed also by Imre Ruzsa (e.g., An Approach to Intensional Logic, Studia Logica 40 (1981)).

35 properties (as functions from possible worlds to classes of individuals) such as being a non-f within Tychý's system; it can be easily generalized to classication of all intensions or rather all functions.] (5) Raclavský, J., Explications of Being Truth [in Czech, expanded English version is in preparation], SPFFBU B 53 (2008) [Three kinds of truth predicate are explicated by means of Pavel Tychý's transparent intensional logic. The rst predicate applies to propositions; the second applies to so-called constructions (some of them construct propositions); the third applies to expressions (usually expressing constructions). Since mappings may be partial and constructions may be abortive, a partial and a total variant correspond to each kind. To the second and the third kind it corresponds also a partial-total variant (which is the most natural one), and a partial-partial variant too (for the last kind they exist two combinations of the two preceding versions). The truth of expressions is language-relative.] (6) Raclavský, J., Semantic Concept of Existential Presupposition. [Just before submission. The explication of the semantic concept of existential presupposition in the connection with deriving of existential statements, distinguishing their de dicto / de re (in a rather generalized, Tichý's, sense) variants.] 31

36 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 32 `Upgrades' and `Updates': from Degrees of Belief to the Dynamics of Epistemic Logic András Benedek Institute for Institute for Philosophical Research, Hungarian Academy of Science While standard epistemic logic described agents' knowledge states in some xed situation, the Dynamic Turn' (Van Benthem) in the 1980s which also showed in AI and in linguistics, turned to belief revision theories and dynamic semantics, considering what holds, or what is known at dierent points of time. Along the lines of Ruzsa's intensional logic we should make a distinction between change of belief and change of the world, which has a consequence for the meaning of `updates' of epistemic states and `upgrades' of measures of uncertainty. In light of recent results in Dynamic Epistemic Logics we characterize epistemic updates as dynamic models of change in epistemic states as a result of epistemic actions (observation, learning, communication), and doxastic upgrades as changes in reasoning, (e.g., algorithms in game theoretic settings, revisions of plausibility or preference change), and argue for an extended framework and interpretation of multi-agent dynamic modal logics. The moral of the review of various approaches to `update' and `upgrade' logics is: the dynamic representation of agents' epistemic possibilities over factual changes remains a crucial question of the semantics of knowledge. For Imre Ruzsa the semantic analysis of `knowledge' was a major motivation for the development of modal logic. As one of the path-nders of probability logic, he was also interested in measures of belief in addition to formal representations of objective probability. He was aware of the limitations of Hintikka's epistemic logic that modeled static situations. Reconsidering some historical approaches to model epistemic events in probability logic, game theory and in linear-time temporal logics, I show that Ruzsa's ideas can be considered as forerunners of some important independent developments in modal and measure theoretic representations of epistemic concepts.

37 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 33 Ruzsa on Quine's Argument against Modal Logic Zsóa Zvolenszky Department of Logic, Eötvös University of Budapest Through the 1970s and 1980sthe days when ELTE Philosophy was named Marxism-LeninismImre Ruzsa prepared logic books and articles with sharp, comprehensive, up-to-date surveys of the most recent international developments in logic and the philosophy of language. For decades to come, the chapters of his Classical, Modal and Intensional Logic would be just about the only Hungarian-language sources available on W. V. O. Quine's famous argument against modal logic, on Saul Kripke's modal semantics that seemed to bypass the Quinean objections, and on Kripke's arguments about the semantics of natural language: that proper names are rigid designators. My talk will explore these chapters of Ruzsa's book, showing just how much of the Quinean argument Ruzsa got right, and what aspects of it he, along with nearly all his contemporaries, missed. Based primarily on John Burgess's subsequent work, we can shed new light on connections not so much between Quine's argument and Kripke's formal work (as Ruzsa and others had thought), but instead between the Quinean argument and Kripke's thesis about proper names being rigid designators.

38 Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 34 Denite Descriptions in Dynamic Predicate Logic Péter Mekis Department of Logic, Eötvös University of Budapest We are going to introduce a version of dynamic predicate logic (DP L, see Groenendijk&Stokhof [1991]) enriched with the iota operator (a.k.a. descriptor) as a framework to model the dynamics of denite descriptions. The dynamic behavior of descriptions was put forward by David Lewis (Lewis [1979]). It can be illustrated by the following discourse: (1) A man walks in the park. He meets a woman. The man hugs her. A man watches from a distance. He walks a dog. The dog snis. The man is jealous. In this example, various occurrences of denite descriptions are used to refer to the most salient individual at a given point of the discourse, instead of the one and only individual that satises the condition set up in the description. The referent is identied via a special kind of discourse information that Lewis calls salience ranking. With a technical implementation of salience ranking into rst-order semantics, our version of DP L is capable to model the dynamics of descriptions in a fully compositional way. It is a highly unusual feature of the system that not only formulas but also terms are evaluated in a dynamic fashion, and thus are capable of updating discourse information. References (1) Groenendijk, J. & Stokhof, M., Dyamic predicate logic. Linguistics and Philosophy 14 (1990) (2) Lewis, D., Score-keeping in a language-game, Journal of Pjilosophical Logic 8 (1979)