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Logic, Language, Mathematics
A Philosophy Conference in Memory of Imre
Ruzsa
September 17�19, 2009
Budapest
Abstracts
Eötvös University of Budapest, Institute of PhilosophyBudapest VIII. Múzeum krt. 4/ihttp://phil.elte.hu/ruzsaconf
Contents
Keynote Lecture
Quanti�ers and Admissible Propositions 1Robert Goldblatt
Plenary Lectures
Natural Logic, Medieval Logic and Formal Semantics 2Gyula Klíma
Whose Logic is Three-valued Logic? 3Ferenc Csaba
Modal Constructions in Sociological Arguments 4László Pólos
Analogy in Semantics 5László Kálmán
Certain Verbs Are Syntactically Explicit Quanti�ers 6Anna Szabolcsi
The Treatment of Ordinary Quanti�cation in English Proper 8András Kornai
Exporting Methods from the Foundation of Mathematicsto the Foundation of Relativity Theory 9Hajnal Andréka and István Németi
In Defense of Hermeneutic Fictionalism 10Gábor Forrai
Relativity and Modal Logic 11Robin Hirsch
Tasks and Ultra-tasks 12Zoltán Szabó Gendler
Neo-Fregeanism: Revising Frege's Notion of Identityin the Philosophy of Language and Mathematics 13Mihály Makkai
Many-Dimensional Modal Logics 14Ági Kurucz
English Sessions
Logic and Language of Relativity Theories 15Gergely Székely
Visualizations of Relativity, Relativistic Hypercomputing 16Renáta Tordai
i
Comparing Relativistic and Newtonian Dynamicsin First Order Logic 17Judit X. Madarász
On Field's Nominalization of Physical Theories 18Máté Szabó
Plural Grundgesetze 19Francesca Boccuni
The Reference of Numerals in Frege 20Edward Kanterian
Grasping the Conceptual Di�erence between János Bolyai'sand Lobachevskii's Notions of Non-Euclidean Parallelism 21János Tanács
Prior and the Limits of de Re Temporal Possibility 22Márta Ujvári
The Indispensability of Logic 23Nenad Miscevic
Names are Not Rigid 24Hanoch Ben-Yami
Premise Semantics and Possible Worlds Semanticsfor Counterfactuals 25Vladan Djordjevic
Fitch's Paradox and Natural Deduction System for Modal Logic 26Edi Pavlovic
Counterfactuals, Context, and Knowledge 27Jelena Ostojic
Aristotle's Wheel and Galileo's Mistake 28Nenad Filipovic, Una Stojnic & Vladan Djordjevic
On the So-Called Dependent (Embedded) Questions 29Anna Bro»ek
Partiality and Tich y's Transparent Intensional Logic:Solutions to Selected Issues 30Ji°í Raclavský
`Upgrades' and `Updates': from Degrees of Belief tothe Dynamics of Epistemic Logic 32András Benedek
Ruzsa on Quine's Argument against Modal Logic 33Zsó�a Zvolenszky
De�nite Descriptions in Dynamic Predicate Logic 34Péter Mekis
ii
Hungarian Sessions
A matematikai tudás eukleidészi modelljének kritikájaLakatos Imre �lozó�ájában 35Golden Dániel
Tarski és a de�ácionizmus 36Kocsis László
Szemantikai értékrés Cantor mennyországának égboltján� avagy mi az, amit megmentett Hilbert? 37Geier János
Kontextuális kétdimenziós szemantika 39Kovács János
A logika iskolai tanulásának els® lépései 40Kiss Olga � Munkácsy Katalin
Az empirikus tudományok teoretizálási törekvéseir®l 41Madaras Lászlóné
Kísérlet a tulajdonnevek vizsgálatára a különböz®hipertextnarratívák esetében 42Szopos András
iii
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 1
Quanti�ers and Admissible Propositions
Rob Goldblatt
Victoria University, Wellington
There are many quanti�ed 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 quanti�ed 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
quanti�ers in a validity preserving manner.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 2
Natural Logic, Medieval Logic and Formal
Semantics
Gyula Klíma
Fordham University
Recent investigations in �natural logic�, the logic actually encoded in naturallanguage usage, as opposed to the formal semantic and deductive systems pre-sented by contemporary professional logicians (see, e.g. Sanchez, V., Studieson Natural Logic and Categorial Grammar, Doctoral dissertation, Universityof Amsterdam, 1991; �LF and Natural Logic�, in Preyer, G. and G. Peter, G.(eds.), 2002, Logical Form and Language. Oxford: OUP), have sparked someinterest in medieval logic, as providing both a description of a �natural logic�and a �regimentation� of an actual natural language, namely, the technical La-tin of scholastic philosophy. This paper, through an analysis of John Buridan's(ca. 1300-1362) nominalist approach to logical semantics, will argue that in ourcontemporary enterprise we may in fact be able to utilize a great deal from the-se medieval ideas, provided we keep these ideas in proper perspective, keepingalways 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, withthe relevant provisos in place).
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 3
Whose Logic is Three-valued Logic?
Ferenc Csaba
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 di�culties 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 sen-
se: even God does not know whether they are true or not. It would have con-
sequences 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 in�nitely � limited.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 4
Modal Constructions in Sociological Arguments
László Pólos
Durham Business School
We �rst provide an overview of the formal properties of this family of modelsand outline key di�erences with classical �rst-order logic. We then build a modelto represent processes of perception and belief core to social theories. To dothis, we de�ne our multi-modal language and then add substantive constraintsthat specify the inferential behavior of modalities for perception, default, andbelief. We illustrate the deployment of this language to the theory of legitimationproposed by Hannan, Pólos, and Carroll (2007). This paper aims to call attentionto the potential bene�ts of modal logics for theory building in sociology.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 5
Analogy in Semantics
László Kálmán
Eötvös University of Budapest / Hungarian Academy of Sciences
The principle of compositionality may seem perfectly trivial. However, depen-ding on what one means by �meaning�, one could argue that it imposes nosubstantive constraint or, to the contrary, that it cannot be obeyed at all. Onthe other hand, we could view this principle as a de�nition of �meaning� (or acomponent of such a de�nition), in which case it would yield a very abstractconcept 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 intopieces (or building them up from primitive and complex building blocks), I willemphasise the global features of signs. I will introduce the principle of generali-zed compositionality, which is based on the concept of similarities between formsand meanings. (The similarity of two forms or meanings is often related to the-ir recognizable component parts, but the relationship is more complicated andindirect than the one inherent in the traditional concept of compositionality).My generalized compositionality principle states that we interpret and producecomplex signs by analogy, relying on our earlier experience on similar complexsigns and their interpretation.
This approach, I believe, has several attractive consequences. First, it pre-dicts that interpretation will be subject to various frequency e�ects and otherpsychological factors (just like phonological or morphological phenomena inanalogy-based models). This clearly means that we aim at a cognitively mo-re realistic model, with a possibility of individual di�erences in interpretationand a clear-cut concept of where so-called �pragmatic� factors enter interpre-tation. Second, on this approach, the dubious distinction between �literal� and�non-literal� interpretations no longer make sense: the mechanism of ��gurative�interpretation does not di�er in any way from �literal� interpretation.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 6
Certain Verbs Are Syntactically Explicit
Quanti�ers
Anna Szabolcsi
New York University
Schlenker (Mind and Language, 2006) observes that there are pervasive similari-ties both in the logical properties of quanti�cation over individuals, world, andtimes and in the linguistic devices (quanti�ers, de�nite descriptions, pronouns,demonstratives) that pertain to them. Yet their treatment has not been uniformin philosophical logic. In particular, quanti�cation over individuals is typicallyexecuted in a syntactically explicit manner, using variables ranging over thewhole universe, whereas quanti�cation over times and worlds is typically execu-ted using non-variablebinding operators of a much more limited power, such asthe � and ♦ operators of modal logic and Montague's � , the abstractor overindices 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) recastsquanti�cation over individuals along the lines of modal propositional logic, andBen-Shalom (1996) makes the approach linguistically more relevant by presen-ting the nominal restriction of determiners as the accessibility relation associatedwith modal operators. From the other end, Groenendijk and Stokhof's (1984)theory of questions is among the �rst to demonstrate a need to quantify overworlds explicitly. Cresswell (1990), Iatridou (1994), Percus (2000), Schlenker(1999, 2004), Pratt and Francez (2001), Kusumoto (2005), Lechner (2007), andvon Stechow (to appear) are among the growing number of authors who haveproposed to treat certain cases of time and world quanti�cation in a syntacti-cally explicit manner. The primary diagnostics for explicit quanti�cation includethe existence of variable-like pronouns referring to the syntactically representedargument, the fact that the argument is not evaluated with respect to a singleindex, and the fact that the argument need not be linked to the closest suitableoperator.
A related but distinct question is the following: Among the linguistic ope-rators with quanti�cational content, which ones are explicit quanti�ers? Theexistence of an explicitly quanti�able argument does not make it necessary forall operators pertaining to it to be explicit quanti�ers. This paper examinesso-called raising verbs in Shupamem (a Grass�eld Bantu language), Dutch, andEnglish. Raising verbs are non-agentive verbs whose surface subjects can bethought of as originating in the verb's in�nitival complement. Relevant examp-
les in English are aspectual begin (as in The paint began to dry), seem, andthreaten (as in The barn threatened to collapse). I will suggest that scope inter-action with an appropriate subject indicates that such verbs are syntacticallyexplicit quanti�ers over times and worlds, and moreover ones that acquire scopein the same manner as expressions quantifying over individuals (by �quanti�erraising� and �scope reconstruction�). I thus add a new diagnostic for syntacti-cally explicit quanti�cation.
7
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 8
The Treatment of Ordinary Quanti�cation in
English Proper
András Kornai
Budapest University of Technology and Economics
We bring together some well-known lines of criticism directed at MontagueGrammar, such as
(i) taking a stilted, highly regulated variety of language as the object of in-quiry;
(ii) ignoring the meaning of content words; and
(iii) the failure to treat hyperintensionals;
and o�er a coherent, and we believe much simpler, alternative using an algebraicvariety of model structures.
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
[email protected] / [email protected]
Rényi Institute of Mathematics, Budapest
We use experience gained during the success story of the foundation of mat-hematics to serve as guideline for elaborating foundation for natural sciences.Since spacetime is the arena in which the processes of physics and indeed mostof natural sciences unfold, it seems to be reasonable to start with elaboratinga logic based foundation for spacetime. For this, Tarski's work, in particularhis �rst-order logic axiomatization and analysis of geometry, is a good startingpoint. Goldblatt's book on spacetime geometry already made progress in thisdirection. We report on progress made in this direction in our school in the last10 years.
In particular, we will show how one can build up relativity theories (includinggeneral relativity and cosmology and Einstein's E = mc2) purely within logic,as theories in the sense of logic, and with no other prerequisites than somefamiliarity with the basics of logic. This will provide, as a byproduct, a logicbased foundation for relativity (in analogy with the foundation of mathematics)as well as a conceptual analysis for relativity theories. Further, it will provide agentle (and streamlined) introduction to relativity for the questioning mind orfor the logically minded.
We touch upon connections with the logical theory of de�nability (Reichen-bach, Tarski, Beth, Makkai). Instead of putting the emphasis on a particularformulation of relativity theory, we put the emphasis on the connections (in-terpretations) between di�erent theories leading up to logical dynamics, thetechnical counterpart of which is known as algebraic logic.
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 aga-
inst 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 understo-
od, 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 her-
meneutic version is said to be untenable because there is no philosophically
unbiased linguistic argument to show that mathematics should not be under-
stood 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.
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. Itdiscourages 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 thereforenatural to use modal logic to describe relativity theory.
In this talk I'll review a number of modal logics that attempt to describeaspects of relativity theory. In particular we will see how property (ii) above hasto be handled carefully, if we are to restrict to standard Kripke semantics formodal logics.
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.
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 founda-tions of mathematics, one that is based on a new version of type theory called"First Order Logic with Dependent Sorts" (FOLDS). This is an extension tothe classical Russell-Ramsey type theory, and it has the ambition of servingas the logical basis of a fully comprehensive foundational system, in the spritof Frege's Grundgesetze der Arithmetic. The novelty of the approach lies in anew systematic and �relativistic� conception of identity. In this, identity is nolonger a primitive as it is in Frege; rather, it is de�ned on the basis of the logicof FOLDS. Identity becomes type-dependent; it becomes meaningless to askif entities of di�erent types are identical (equal) or not. Category theory, andits extension to higher dimensional categories, a currently emerging branch ofabstract mathematics, is a natural environment for the FOLDS language andthe FOLDS identity concept. In the talk, I will make an attempt to relate mymathematical work to the philosophy of mathematics of Frege and of subsequentthinkers.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 14
Many-Dimensional Modal Logics
King's College London
Many-dimensional propositional modal logics (multi-modal logics having many-dimensional Kripke frames among their frames) have been studied both in puremodal logic and in various computer science and arti�cial intelligence applica-tions. 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 theknown results and open questions on the axiomatisation and decision problemsof many-dimensional modal logics.
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 anew idea at all, among others, it goes back to such leading mathematiciansand philosophers as Hilbert, Reichenbach, Carnap, Gödel, Tarski, Suppes andFriedman.
There are many examples showing the bene�ts of using axiomatic methodin the foundations of mathematics. That motivates the Hungarian school led byHajnal Andréka and István Németi to apply this method in the foundations ofrelativity 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 assumpt-ions (axioms); and to prove the surprising predictions (theorems) of relativitytheories using a minimal number of convincing axioms.
Via a sample of results in the application of axiomatic method to specialand general relativity theories, we try to show that their application to physicsis a promisingly fruitful research area.
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 Ist-ván Németi at the Rényi Institute of Mathematics, see the abstracts of IstvánNémeti, Hajnal Andréka, Judit X. Madarász and Gergely Székely. We will pre-sent visualizations of relativity. For example, we will present a movie showingwhat an astronaut would see while �ying through a huge Kerr-Newmann worm-hole or any other kind of wormhole. We will also outline the ideas of relativistichypercomputing, i.e., how Malament-Hogarth spacetimes can be used for de-signing arti�cial systems computing beyond the Turing barrier. Any spacetimeadmitting a CTC (closed timelike curve) is suitable for constructing such a hy-percomputer, but the existence of CTC's is not really needed for this. A muchmilder condition called Malament-Hogarth property is su�cient. We refer to[1], [2], and [3] for more detail. (The most satisfactory solution to the so calledblue-shift problem is available in [4].)
References
[1 ] Dávid, Gy., Németi, I., Relativistic computers and the Turing barrier.Applied Mathematics and Computation 178 (2006). http://www.math-inst.hu/pub/algebraic-logic/beyondturing.pdf
[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).http://www.math-inst.hu/pub/algebraic-logic/turing.html
[4 ] Andréka, H., Németi, I., Németi, P., Presentation in The science and
philosophy of unconventional compuing SPUC 2009, Cambridge, March2009
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 the-
ories of �rst-order logic. To illustrate the similarities between Newtonian and
relativistic dynamics we axiomatize them such that they di�er in one axiom
only. This one axiom di�erence, however, leads to radical di�erences in the pre-
dictions of the two theories. One of their major di�erences manifests itself in
the relation between relativistic and rest masses.
The statement that the centerlines of a system of point masses viewed from
two di�erent reference frames are related exactly by the coordinate transforma-
tion 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.
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 onto-logically committed to mathematical objects, because of the indispensability ofmathematics in our best scienti�c theories. Indispensability means that physicaltheories refer to and quantify over mathematical entities such as sets, numbersand functions. In his famous book Science Without Numbers' Hartry Field ar-gues that this is not the case. We can �nominalize� our physical theories, thatis we can reformulate them in such a way that (1) the new version preservesthe attractivity of the theory, and (2) the nominalized theory does not containquanti�cations over mathematical entities.
I'm going to reconsider Field's nominalization procedure for a toy physictheory formulated in a �rst order language, in order to make a clear distinctionbetween the following three steps:
- the physical theory in terms of empirical observations;
- the standard physical theory, which contains quanti�cation over mathe-matical entities, as usual;
- the nominalized version of the theory without any reference to mathema-tical entities.
Having Field's nominalization procedure reconstructed, it will be clear thatthere is no di�erence between the original and the nominalized versions of thetheory, at least, there is no di�erence from a formalist point of view. It is becau-se the only di�erence would come from the di�erent �meanings� of the variablesover which the quanti�cations are running. The formalist philosophy of mathe-matics, however, denies that the variables have meanings at all. So, the formalsystems as abstract mathematical entities are still included in physical theories;and this fact is highly enough for the structural platonist or immanent realistto apply the Quine-Putnam argument.
Finally, therefore, I will suggest a completely di�erent way for the objectionto the Quine-Putnam argument.
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 mathema-tics exposed in Grundgesetze der Arithmetik is inconsistent. The contradictionis derived from the infamous Basic Law V. This principle is crucial to Frege'slogicism as it embeds the tenet that tightly connects natural numbers, conceivedas equivalence classes, to concepts. As far as it is currently known, moreover,the so far provided consistent subsystems of Grundgesetze displaying some vers-ion of Basic Law V cannot interpret second-order Peano arithmetic. This seemsto show that Frege's programme could not be completely recovered, after all.Secondly, these subsystems may be challenged with respect to the issue of towhat extent they actually capture Frege's notion of concept. In particular, boththese subsystems are based on a more or less radical limitation of the universeof Fregean concepts, which seems to be incompatible with Frege's spirit.
The aim of this article is to present a consistent predicative second-ordersystem with plural comprehension and Basic Law V, Plural Grundgesetze (PG),which is capable of deriving second-order Peano axioms. The main features ofPG are plural quanti�cation, which will guarantee the power of full second-orderlogic to PG, and predicative comprehension for concepts. I will also analyse theissue regarding predicativism from a Fregean perspective.
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 doesnot commit him to the view that prior to this analysis numerals already refer-red to particular objects, numbers; the requirements for a faithful de�nition ofnumber did not involve for him criteria for the preservation of sense or referencein the transition from pre-systematic uses of numerals and number statementsto their use in his formal system. For the pre-systematic use is too vacillatingand 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; logicdescribes pre-existing relations between Platonic objects (thoughts), and his ac-count 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 senseor reference of ordinary uses of number. His revisionism is limited to predica-tive/attributive uses considered irrelevant for scienti�c purposes (FA 57, 60).Without some preservation of sense and reference the point and nature of thetransition 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 whomthe sense and reference of numerals and number statements is a system-internalfeature. But it is demonstrated that this misses not only Frege's Platonism,but also his insistence on the applicability of arithmetic. Finally, it is arguedthat while it is correct to stress, as Weiner does, that Frege's logicism had anepistemological agenda (to prove the analyticity of arithmetical truths), thischaracterisation must be supplemented by the ontological aspect of his project,which is to prove that numbers are objects and thus that arithmetic is a sciencewith a proper subject matter.
References
• Joan Weiner, What's in a Numeral? Frege's Answer, in: Mind, 116: 2007
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 21
Grasping the Conceptual Di�erence between
János Bolyai's and Lobachevskii's Notions of
Non-Euclidean Parallelism
János Tanács
Budapest University of Technology and Economics, Department ofPhilosophy and History of Science
The presentation is going examine the di�erence 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 Un-
tersuchungen is elaborated in order to see how Bolyai's and Lobachevskii's no-
tions of parallelism di�er. The careful examination of the Commentary reveals
a seeming incoherence of Bolyai's translation, and �nally the explanation of this
incoherence o�ered by the received view and that of the novel position will be
compared and assessed.
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 onwhether radical coming-into-being is a genuine de re possibility of individuals.He considers it by the putative complete property swap of two individuals, JuliusCaesar and Mark Antony, through worlds.
Prior's original solution to the dilemma of the Leibnizian vs haecceistic po-sition with respect to property-indiscernible worlds consists in pointing out thatthe property swap must necessarily stop at the property of origin. However, thepossibility he denies is temporal and not logical; for, when we ask, `when was itpossible', it is easy to see that `after his birth . . . it was clearly too late for himto have had di�erent parents.' And as to the de re possibility of having di�erentparents `before Caesar existed' the obvious retort is there would seem to havebeen no individual identi�able as Caesar . . . who could have been the subject ofthis possibility'.
This sounds fairly trivial. But by parity of reasoning we can get an uncom-fortable consequence; for, if Caesar (or any other actual individual) could nothave been the subject, before his birth, of the (later) unrealized possibility, equ-ally, 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 thesubject of a de re possibility of being (going to be) born � i.e., at least not beforeour conception. This amounts to saying that what is once actual is preceded bywhat is non-possible, contravening thus the logic of propositional modalities.
The air of paradox can be dissolved by denying, with Prior, that the possi-bility of origin is a genuine de re possibility. As a possibility it is general or dedicto: it is possible that someone be born to such and such parents, but it is notpossible of someone that he should be born to these or other parents.
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 elementa-ry logical-inferential practices by their unavoidability and global indispensabilityfor all our cognitive e�orts. It starts by agreeing with prominent apriorists abouttheir attempt to justify such beliefs either from naturalistic computationalistconsiderations of unavoidability (inevitability) (Horwich) or from constitutive-ness (Boghossian) or from global indispensability Argument (C. Wright), andthen proceeds to argue that unavoidable and indispensable tools provide entitle-ment/justi�cation for projects if projects are themselves meaningful. However,we are justi�ed to think that our most general cognitive project is meaningful,and justi�ed partly of the basis of its up to date success; and this basis is a
posteriori. Therefore, the whole re�ective justi�cation from compellingness andunavoidability is a posteriori. This suggests that the justi�cation of our intui-tional 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 thata priori/ a posteriori distinction is useful and to the point. What is needed isre�nement and respect for structure, not rejection of the distinction.
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 di�erent 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 di�culty
� 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.
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 (PWS) for counterfactuals is modal logicwith the addition of the so-called selection function, whose role is to somehowseparate important from unimportant worlds. A counterfactual A > C is true i�C holds at the important A-worlds. The older, premise semantics (PS) says thatA > C is true i� A, together with some further true premises B1, B2, . . . , entailsC. The main problem for PWS is to explain which worlds are important, and forPS it is to specify which truths are to be included among the B's. That problemis very di�cult to be solved in general, but in particular cases we often do haveclear intuitions about the importance of worlds and about the B's. I argue thatour intuitions used in PS are more basic, since in testing our selection functionwe use our intuitions from PS, rather than the other way around, that is, we saythat the important worlds are those where the B's hold, and we do not explainthe B's in terms of important worlds. Although PWS is a much more powerfullogical tool, if what I said is correct, we still need to investigate the relationbetween the two semantics. That explains the motive behind the two results Iwill defend. The �rst says that the standard interpretation of Goodman's PSis not correct since it validates conditional excluded middle, which Goodmanrejects, and, second, that Lewis' notion of cotenability, which allegedly capturesthe intentions of the premise semanticists, fails to do so, and that this is aproblem for Lewis' and not for the premise semantics.
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.
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, butthere are di�erent views about the way context in�uences the truth-conditionsfor counterfactuals. Di�erent theories explain the context dependency of coun-terfactuals in di�erent ways. For example, the so-called standard theories (Stal-naker, Lewis), and the so-called pragmatic theories or strict implication analysis
of counterfactuals (Warmbrod, von Fintel et al.) o�er explanations that are di�e-rent in many important respects. I will argue that the pragmatic theories give anexplanation that better �ts our language practice. I will conclude by pointing towhat I see as another advantage of the pragmatic theories: in applying counter-factuals to epistemology (like Nozick, DeRose and others who de�ne knowledgein terms of counterfactuals), the standard view of the truth-conditions leadsto denying the closure principle, and a speci�c version of the pragmatic view,which I will de�ne, leads to epistemic contextualism and enables us to keep theclosure principle.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 28
Aristotle's Wheel and Galileo's Mistake
Nenad Filipovic, Una Stojnic & Vladan Djordjevic
[email protected] / [email protected]
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 in�nite division of a continuum is impossible: by banding the straight
line into the circle, one can obtain in�nitely many parts, or sides, because, as
Galileo believed, circle is a polygon with in�nitely 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 bet-
ween 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
o�ering a solution to the paradox based on contemporary theories of continu-
um, we will present Galileo's putative solution, and point to its signi�cance 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 criti-
que, 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
de�nition which Galileo might reject, we believe that our second argument does
not presuppose anything external to Galileo's theory.
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 byanalysis of dependent questions, i.e. of expressions which
(i) are parts of compound questions and
(ii) are isomorphic with some independent questions (scil. questions sensustricto).
One may meet the tendency to explicate the sense of independent questions bythe 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 const-ructing settheoretical or possible-worlds semantics for independent questions.In my opinion, these tendencies are abortive and lead to irrelevant explicationsof the sense of questions sensu stricto.
But on the other hand, semiotic functions of the so-called dependent qu-estions as parts of compound expressions require deeper analysis. My papercontains a proposal of such an analysis.
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 ar-guments) is frustrating: classical logical laws (e.g., De Morgan law for exchangeof quanti�ers) designed for total functions usually (if not ever) collapse. To in-corporate partial functions, Tichý suggested the modi�cation of (the naturaldeduction for) the logic of simple theory of types mainly by the correction ofthe rule of β-reduction (because of partiality, β-reduction is not a rule equiva-lent to β-expansion). As it is apparent from Tichýh's collected papers and hismonograph, Tichý's transparent intensional logic, treating both modal and tem-poral variability, is a powerful logical system for logical analysis (explication)of natural language meaning.1 The present author shows how to de�ne withinTichý's system �3-valued connectives� which get a value even when an �inputproposition� is gappy (e.g., �exclusion negation� or �totalizing true-predicate�).Another contribution is made by the correct formulation of the extensionalityprinciple for partial functions. Another contribution is made by correct for-mulation of the notion �complementary function�, i.e. a function non-F havingextensions which are �complementary� to extensions of the function F (not onlytwo intuitively plausible explications, but rather partial classes complicate thematter).
References
(1) Tichý, P., The Foundations of Partial Type Theory. Reports on Mathema-tical 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 OtagoPress, 2004
(4) Raclavský, J., De�ning Basic Kinds of Properties, in: T. Marvan, M. Zo-uhar (eds.), The World of Language and the World beyond Language, Fi-lozo�cký ústav SAV, 2007. [The text includes a rigorous classi�cation of
1The adoption of partial functions for logical analysis of natural language was stressed alsoby Imre Ruzsa (e.g., An Approach to Intensional Logic, Studia Logica 40 (1981)).
properties (as functions from possible worlds to classes of individuals) suchas �being a non-F � within Tychý's system; it can be easily generalized toclassi�cation of all intensions or rather all functions.]
(5) Raclavský, J., Explications of Being Truth [in Czech, expanded Englishversion is in preparation], SPFFBU B 53 (2008) [Three kinds of truth pre-dicate are explicated by means of Pavel Tychý's transparent intensionallogic. The �rst predicate applies to propositions; the second applies toso-called constructions (some of them construct propositions); the thirdapplies to expressions (usually expressing constructions). Since mappingsmay be partial and constructions may be abortive, a partial and a totalvariant correspond to each kind. To the second and the third kind it cor-responds also a partial-total variant (which is the most natural one), and apartial-partial variant too (for the last kind they exist two combinations ofthe two preceding versions). The truth of expressions is language-relative.]
(6) Raclavský, J., Semantic Concept of Existential Presupposition. [Just be-fore submission. The explication of the semantic concept of existentialpresupposition in the connection with deriving of existential statements,distinguishing their de dicto / de re (in a rather generalized, Tichý's, sen-se) variants.]
31
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, HungarianAcademy of Science
While standard epistemic logic described agents' knowledge states in some �xedsituation, the Dynamic Turn' (Van Benthem) in the 1980s which also showed inAI and in linguistics, turned to belief revision theories and dynamic semantics,considering what holds, or what is known at di�erent points of time. Along thelines of Ruzsa's intensional logic we should make a distinction between changeof 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 lightof recent results in Dynamic Epistemic Logics we characterize epistemic updatesas dynamic models of change in epistemic states as a result of epistemic actions(observation, learning, communication), and doxastic upgrades as changes inreasoning, (e.g., algorithms in game theoretic settings, revisions of plausibilityor preference change), and argue for an extended framework and interpreta-tion of multi-agent dynamic modal logics. The moral of the review of variousapproaches to `update' and `upgrade' logics is: the dynamic representation ofagents' epistemic possibilities over factual changes remains a crucial question ofthe semantics of knowledge.
For Imre Ruzsa the semantic analysis of `knowledge' was a major motivati-on for the development of modal logic. As one of the path-�nders of probabilitylogic, he was also interested in measures of belief in addition to formal repre-sentations of objective probability. He was aware of the limitations of Hintikka'sepistemic logic that modeled static situations. Reconsidering some historicalapproaches to model epistemic events in probability logic, game theory and inlinear-time temporal logics, I show that Ruzsa's ideas can be considered as for-erunners of some important independent developments in modal and measuretheoretic representations of epistemic concepts.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 33
Ruzsa on Quine's Argument against Modal
Logic
Zsó�a Zvolenszky
Department of Logic, Eötvös University of Budapest
Through the 1970s and 1980s�the days when ELTE Philosophy was namedMarxism-Leninism�Imre Ruzsa prepared logic books and articles with sharp,comprehensive, up-to-date surveys of the most recent international develop-ments in logic and the philosophy of language. For decades to come, the chap-ters of his Classical, Modal and Intensional Logic would be just about the onlyHungarian-language sources available on W. V. O. Quine's famous argumentagainst modal logic, on Saul Kripke's modal semantics that seemed to bypassthe Quinean objections, and on Kripke's arguments about the semantics of natu-ral language: that proper names are rigid designators. My talk will explore thesechapters of Ruzsa's book, showing just how much of the Quinean argument Ru-zsa 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 newlight on connections not so much between Quine's argument and Kripke's for-mal work (as Ruzsa and others had thought), but instead between the Quineanargument and Kripke's thesis about proper names being rigid designators.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 34
De�nite 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 (DPL, see Gro-enendijk&Stokhof [1991]) enriched with the iota operator (a.k.a. descriptor) asa framework to model the dynamics of de�nite descriptions. The dynamic be-havior of descriptions was put forward by David Lewis (Lewis [1979]). It can beillustrated by the following discourse:
(1) �A man walks in the park. He meets a woman. The man hugs her. Aman watches from a distance. He walks a dog. The dog sni�s. The man isjealous.�
In this example, various occurrences of de�nite descriptions are used to referto the most salient individual at a given point of the discourse, instead of theone and only individual that satis�es the condition set up in the description.The referent is identi�ed via a special kind of discourse information that Lewiscalls salience ranking. With a technical implementation of salience ranking into�rst-order semantics, our version of DPL is capable to model the dynamicsof descriptions in a fully compositional way. It is a highly unusual feature ofthe system that not only formulas but also terms are evaluated in a dynamicfashion, and thus are capable of updating discourse information.
References
(1) Groenendijk, J. & Stokhof, M., Dyamic predicate logic. Linguistics andPhilosophy 14 (1990)
(2) Lewis, D., Score-keeping in a language-game, Journal of Pjilosophical Lo-gic 8 (1979)
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 35
A matematikai tudás eukleidészi modelljének
kritikája Lakatos Imre �lozó�ájában
Golden Dániel
MTA Filozó�ai Kutatóintézet
Lakatos Imre az In�nite regress and foundations of mathematics cím¶ írásá-ban úgy határozza meg saját kontribúcióját, mint �annak megmutatását, hogya modern matematika�lozó�a mélyen az általános episztemológiába ágyazódik,s csak ennek kontextusában értheto meg�. Ennek megfelel®en a matematikaitudás problémáját abban az általános keretben helyezi el, ahol a két véglet aszkepticizmus és a dogmatizmus pozíciója. A szkeptikus támadás el®li meneküléssorán a tudás racionális megalapozásával próbálkozhatunk, amelynek a mate-matika területén Lakatos három történeti kísérletét különíti el: az eukleidészi
programot, az empiricista programot és az induktivista programot.
Lakatos cambridge-i doktori értekezésének eredeti befejezése (amely szinténaz összegy¶jtött írások második kötetében jelent meg) az eukleidészi programheurisztikájaként mutatja be az �analízis és szintézis módszerét�, amelyet Pappusleírása nyomán ismertet. A Bizonyítások és cáfolatok egyik lábjegyzetében pedigazt mondja Lakatos, hogy a matematikai felfedezésnek ezt a módszerét váltottafel a XVII. századot követ®en a szerencsére és/vagy a zseni megérzéseire történ®hivatkozás. Ezt az irracionális fordulatot kívánja Lakatos megel®zni azzal, hogya matematikai (és tágabban a tudományos) tudás kvázi-empirikus modelljére
tesz javaslatot.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 36
Tarski és a de�ácionizmus
Kocsis László
PTE Filozó�a Tanszék
Tarski az igazság szemantikai koncepciójának kidolgozásakor alapvet®en az igaz-ság korrespondencia-elméleti meghatározásának pontosabb kifejtésére törekszik,miszerint az igazság nem más, mint a valóságnak való megfelelés. Tarski ezzela lépéssel úgy t¶nik, hogy elkötelez®dik egy olyan elmélet lehet®sége mellett,amely az igazság természetét egy explicit de�níció segítségével kívánja megha-tározni, és amely ennél fogva elismeri azt, hogy az igazság egy lényeges, nemprimitív, de�niálható természettel rendelkez® fogalom. Érdekes módon egy ilyenprojekt lehetetlensége mellett érvelnek a magukat de�ácionistáknak tartó �lozó-fusok, miközben elméleteik alapjait nagyrészt Tarski igazságról vallott nézetei-ben vélik felfedezni. Tehát a de�ácionisták, akik szerint az igazságnak nem ad-ható explicit de�niciója, Tarskit megpróbálják de�ácionistaként értelmezni, mégha ez Tarski eredeti szándékának ellentmondani is látszik. El®adásomban arraa vitára szeretnék re�ektálni, amely a Tarski által nyújtott igazság-koncepcióde�ációs jellegével kapcsolatban robbant ki, miközben mindjobban tisztázni sze-retném Tarski elméletének helyét a kortárs igazságelméletek között.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 37
Szemantikai értékrés Cantor mennyországának
égboltján � avagy mi az, amit megmentett
Hilbert?
Geier János
Stereo Vision LTD, Budapest
Közismert, hogy a halmazelmélet axiomatizálásának vezéralakja Hilbert volt.Ruzsa [1, p176] szerint �Hilbert semmiképen sem akart lemondani Cantor transz-�nit matematikájáról. . . � Másutt Ruzsa [1, p183]: �A halmazelmélet axiomati-zálásának természetes célja, hogy az antinómiák kiküszöbölése mellett a naivhalmazelmélet értékes részéb®l minél többet megmentsen.� Nyilvánvaló, hogyaz �értékes rész� magja nem más, mint a Cantor-féle átlós eljárás és az azon ala-puló hatványhalmaz-tétel (CHT ). Megmenteni csak azt lehet, ami el®tte márlétezett, így jogosan vethet® fel a kérdés: az ún. �naiv halmazelmélet� kereteinbelül � azaz a 19. sz. végére kialakult (és napjaink �hétköznapi matematikusai�által is rendszeresen használt) tiszta, világos, természetes matematikai gondolko-dásmód (TMG) szerint � hibátlan-e a CHT bizonyítása? Itt arra a gondolatme-netre utalok, amit minden, e témával foglalkozó tankönyvben megtalálhatunk;például Ruzsa [1, p147].
El®adásomban virtuális id®utazásra invitálok az 1890-es évekbe, amikor meg-jelentek a halmazelméleti antinómiák éppen a nevezett gondolatmenet parafrá-zisaiként, és még nem volt se ZFC, se NBG, de volt egy egységes konszenzusa TMG-r®l. Ennek fényében kimutatni szándékozom: a CHT bizonyításánaktankönyvi, �naiv� gondolatmenete hibás, mert az indirekt levezetésnek egy adottpontján nem veszi �gyelembe az ott fellép® szemantikai értékrést. A hiba kimu-tatásának alapja szintén megtalálható Ruzsa [1, p178]-ban, amikor arról beszél,hogy �. . . bizonyos dolgok között egy kétváltozós F (a, b) reláció van adva olymódon, hogy . . .minden d-re F (d, d) és F (d, s) közül pontosan az egyik telje-sül, és . . . ez a speciális s elem is a számításba jöhet® dolgok közé tartozik.�Ugyanakkor elfogadom, hogy a CHT a ZFC-nek tétele.
Következmények:
(1) Hilbert nem mentett meg semmit, ellenben (tévedésb®l?, Zermeloval, Fra-enkellel és másokkal együtt) �. . . egy új, más világot teremtett�.
(2) A �Russell-antinómia� nem antinómia.
Hivatkozások
[1 ] Ruzsa Imre (1966) A matematika néhány �lozó�ai problémájáról. In:Világnézeti nevelésünk természettudományos alapjai IV., Tankönyvkiadó,Budapest.
38
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 39
Kontextuális kétdimenziós szemantika
Kovács János
Szegedi Tudományegyetem
Nyelvhasználatunk egyik alapvet® sajátossága, hogy a nyelvi megnyilatkozás ke-
retéül szolgáló kontextus egy további szerepben, a megnyilatkozás tárgyaként is
el®fordulhat. Egy adott helyen beszélhetünk például más helyekr®l, egy adott
id®pillanatban más id®pillanatokról, egy adott lehetséges világban más lehetsé-
ges világokról. Véleményem szerint a kontextus e kett®s szerepének a felismerése
a kétdimenziós szemantikai elméletek kidolgozásának egyik legfontosabb indoka.
El®adásomban egy kétdimenziós kontextuális szemantika alapjait vázolom fel, és
azt vizsgálom, hogy miként lehet rekonstruálni Kripke Naming and Necessityben
kifejtett szemantikai nézeteit az általam bemutatott szemantika keretei között.
Megpróbálok továbbá választ adni Chalmers a kétdimenziós szemantika kontex-
tuális értelmezésével kapcsolatos ellenvetéseire, valamint Soames kétdmenziós
szemantikával kapcsolatos kritikájára. Végezetül igyekszem néhány összefüggést
felmutatni episztemikus és meta�zikai szükségszer¶ség között, választ keresve
a kérdésre, hogy az elgondolhatóság valóban ad-e valamiféle támpontot a világ
meta�zikai szerkezetének feltárásához.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 40
A logika iskolai tanulásának els® lépései
Kiss Olga � Munkácsy Katalin
[email protected] / [email protected]
Corvinus Egyetem / ELTE TTK Matematikatanítási ésMódszertani Csoport
Az összevont tanulócsoportos kisiskolákban folytatott matematikatanulási vizs-gálatok közben találkoztunk azzal a problémával, hogy a hátrányos helyzet¶gyerekek, az eltér® �social dialect�-et beszél®k, nem értik tanáraik hétköznapiszavait. �k maguk nem használják az ÉS-t meg a VAGY-ot, vagyis a legegy-szer¶bb logikai m¶veleteket sem, így éles eszük, jó gyakorlati problémamegoldóképességük ellenére el vannak zárva a matematikatanulás lehetoségét®l is.
Tárgyi és képi reprezentációkkal, valamint történetmeséléssel próbáltuk ahátrányok leküzdését segíteni, ezzel kapcsolatban vannak empirikus kutatásieredményeink is. Modellként a logikai áramkörök helyett folyóágakból és gá-takból álló rendszert vizsgáltunk. A problémamegoldás sikeressége felvetette amentális m¶velet és a nyelvi reprezentáció összefüggései elemzésének szükséges-ségét.
A probléma azonban általánosan is felvetheto: a logikai elemeinek milyenhasználata jellemzi e szubkultúrákat? Az érvelések milyen szisztematikus mód-ja az, amelyre a tanár építhet? Mennyire szisztematikusak ezek (azaz mindigérvényesülnek, vagy csak általában), és milyen kapcsolatban állnak azzal a logi-kával, amit a modern matematika oktatása igényel?
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 41
Az empirikus tudományok teoretizálási
törekvéseir®l
Madaras Lászlóné
Szolnoki F®iskola
A modern természettudományok egyik megteremt®jének, Galileinek alapvet®
felismerése volt, hogy: �Egyedül logikus gondolkodással semmit sem tudhatunk
meg a tapasztalati világról; a valóságra vonatkozó minden tudásunk a tapasz-
talatból indul ki és oda torkollik.� A modern tudomány a matematika és az
empirizmus összekapcsolódásából született. Mintegy három évszázaddal kés®bb
hasonló felismerés segítette egy új diszciplina, a tudományos �lozó�a megszü-
letését, amelyhez kidolgozói hasonlóan nagy reményeket f¶ztek. Az empirizmus
és a racionalizmus módszereinek összekapcsolásával azt gondolták, hogy meg-
sz¶nik az elmélet és a gyakorlat közötti éles dichotómia mint a természet tanul-
mányozására szolgáló rivális módszerek harca, és egyben megnyílik a lehet®ség
a természetr®l szerzett ismereteink szisztematikus ellen®rzésére.
A század elején felmerült törekvések az empiriára és a logikára támaszkodva
egy egységes és tökéletesített tudomány kidolgozását vetették fel. Sikerült-e,
sikerülhetett-e ez a vállalkozás? El®adásunkban az alapkutatások gyakorlatát is
vizsgálva erre keressük a választ.
Logic, Language, Mathematics: Conference in Memory of Imre Ruzsa 42
Kísérlet a tulajdonnevek vizsgálatára a
különböz® hipertextnarratívák esetében
Szopos András
Nyíregyházi F®iskola, Magyar Nyelvtudományi Osztály
A hipertext, mint a XX. század egyik jellegzetes szövegformája, lehet®séget ad aszöveg legkülönfélébb szint¶ szervez®désére, illetve magába foglalhat olyan ele-meket is, amelyek a hagyományos szövegekben egyáltalán nem vagy csak ritkánfordulhatnak el®. A tulajdonnevek meghatározása és szerepe már a hagyomá-nyos szövegek esetében is megosztotta mindazokat, akik de�niálni próbálták.El®adásomban vázolom a hipertext-speci�kus tulajdonnevek (�nick�) jelentés- ésjelöleti problémáit: van-e jelentése, jelölete a nickneveknek, illetve tekinthet®k-eezek a nevek individuumnnévnek vagy sem.