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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

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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

[email protected]

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.


Natural Logic, Medieval Logic and Formal

Semantics

Gyula Klíma

[email protected]

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).


Whose Logic is Three-valued Logic?

Ferenc Csaba

[email protected]

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.


Modal Constructions in Sociological Arguments

László Pólos

[email protected]

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.


Analogy in Semantics

László Kálmán

[email protected]

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.


Certain Verbs Are Syntactically Explicit

Quanti�ers

Anna Szabolcsi

[email protected]

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


The Treatment of Ordinary Quanti�cation in

English Proper

András Kornai

[email protected]

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.


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.


In Defense of Hermeneutic Fictionalism

Gábor Forrai

[email protected]

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.


Relativity and Modal Logic

Robin Hirsch

[email protected]

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.


Tasks and Ultra-tasks

Zoltán Szabó Gendler

[email protected]

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.


Neo-Fregeanism: Revising Frege's Notion of

Identity in the Philosophy of Language and

Mathematics

Mihály Makkai

[email protected]

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.


Many-Dimensional Modal Logics

Ági [email protected]

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 and Language of Relativity Theories

Gergely Székely

[email protected]

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.


Visualizations of Relativity, Relativistic

Hypercomputing

Renáta Tordai

[email protected]

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


Comparing Relativistic and Newtonian

Dynamics in First Order Logic

Judit X. Madarász

[email protected]

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.


On Field's Nominalization of Physical Theories

Máté Szabó

[email protected]

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.


Plural Grundgesetze

Francesca Boccuni

[email protected]

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.


The Reference of Numerals in Frege

Edward Kanterian

[email protected]

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


Grasping the Conceptual Di�erence between

János Bolyai's and Lobachevskii's Notions of

Non-Euclidean Parallelism

János Tanács

[email protected]

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.


Prior and the Limits of de Re Temporal

Possibility

Márta Ujvári

[email protected]

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.


The Indispensability of Logic

Nenad Miscevic

[email protected]

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.


Names are Not Rigid

Hanoch Ben-Yami

[email protected]

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.


Premise Semantics and Possible Worlds

Semantics for Counterfactuals

Vladan Djordjevic

[email protected]

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.


Fitch's Paradox and Natural Deduction System

for Modal Logic

Edi Pavlovic

[email protected]

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.


Counterfactuals, Context, and Knowledge

Jelena Ostojic

[email protected]


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.


Aristotle's Wheel and Galileo's Mistake

Nenad Filipovic, Una Stojnic & Vladan Djordjevic

[email protected] /



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.


On the So-Called Dependent (Embedded)

Questions

Anna Bro»ek

[email protected]

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.


Partiality and Tich y's Transparent Intensional

Logic: Solutions to Selected Issues

Ji°í Raclavský

[email protected]

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


Ùpgrades' and Ùpdates': from Degrees of

Belief to the Dynamics of Epistemic Logic

András Benedek

[email protected]

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 ofùpdates' of epistemic states and ùpgrades' 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 ùpdate' and ùpgrade' 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.


Ruzsa on Quine's Argument against Modal

Logic

Zsó�a Zvolenszky

[email protected]

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.


De�nite Descriptions in Dynamic Predicate

Logic

Péter Mekis

[email protected]

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)


A matematikai tudás eukleidészi modelljének

kritikája Lakatos Imre �lozó�ájában

Golden Dániel

[email protected]

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.


Tarski és a de�ácionizmus

Kocsis László

[email protected]

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.


Szemantikai értékrés Cantor mennyországának

égboltján � avagy mi az, amit megmentett

Hilbert?

Geier János

[email protected]

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


Kontextuális kétdimenziós szemantika

Kovács János

[email protected]

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.


A logika iskolai tanulásának els® lépései

Kiss Olga � Munkácsy Katalin


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?


Az empirikus tudományok teoretizálási

törekvéseir®l

Madaras Lászlóné

[email protected]

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.


Kísérlet a tulajdonnevek vizsgálatára a

különböz® hipertextnarratívák esetében

Szopos András

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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.

logic, language, mathematicsphil.elte.hu/ruzsaconf/ruzsaconf_abstracts.pdf · logic, language,...

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