One of the standard views on plural quantification is that its use commits one to the existence of abstract objects-sets. On this view claims like 'some logicians admire only each other' involve ineliminable quantification over subsets of a salient domain. The main motivation for this view is that plural quantification has to be given some sort of semantics, and among the two main candidates-substitutional and set-theoretic-only the latter can provide the language of plurals with the desired expressive power (given that the nominalist seems committed to the assumption that there can be at most countably many names). To counter this approach I develop a modal-substitutional semantics of plural quantification (on which plural variables, roughly speaking, range over ways names could be) and argue for its nominalistic acceptability.

%B Synthese %V 191 %P 1605–1626 %G eng %R http://dx.doi.org/10.1007/s11229-013-0354-5 %0 Journal Article %J History and Philosophy of Logic %D 2014 %T S\lupecki's generalized mereology and its flaws %A Urbaniak, Rafal %XOne of the streams in the early development of set theory was an attempt to use mereology, a formal theory of parthood, as a foundational tool. The first such attempt is due to a Polish logician, Stanis\unmatched{0142}aw Leśniewski (1886–1939). The attempt failed, but there is another, prima facie more promising attempt by Jerzy S\unmatched{0142}upecki (1904–1987), who employed his generalized mereology to build mereological foundations for type theory. In this paper I (1) situate Leśniewski's attempt in the development of set theory, (2) describe and evaluate Leśniewski's approach, (3) describe S\unmatched{0142}upecki's strategy without unnecessary technical details, and (4) evaluate it with a rather negative outcome. The issues discussed go beyond merely historical interests due to the current popularity of mereology and because they are related to nominalistic attempts to understand mathematics in general. The introduction describes very briefly the situation in which mereology entered the scene of foundations of mathematics –- it can be safely skipped by anyone familiar with the early development of set theory. Section 2 describes and evaluates Leśniewski's attempt to use mereology as a foundational tool. In Section 3, I describe an attempt by S\unmatched{0142}upecki to improve on Leśniewski's work, which resulted in a system called generalized mereology. In Section 4, I point out the reasons why this attempt is still not successful. Section 5 contains an explanation of why Leśniewski's use of Ontology in developing arithmetic also is not nominalistically satisfactory.

%B History and Philosophy of Logic %V 35 %P 289–300 %G eng %R http://dx.doi.org/10.1080/01445340.2014.917837 %0 Journal Article %J European review %D 2014 %T Stanislaw Leśniewski: rethinking the philosophy of mathematics %A Urbaniak, Rafal %XNear the end of the XIXth century part of mathematical research was focused on unification: the goal was to find ˝one sort of thing˝ that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterpize hit a rough patch with Frege's failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. Quite parallel to this development was the work of Stanislaw Le{\textbackslash}'sniewski (1886-1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Lesniewski attempted to find a nominalistically acceptable replacement for set theory in the foundations of mathematics. His candidate was Mereology - a theory which instead of sets and elements spoke of wholes and parts. The goal of this paper will be to present Mereology in this context, to evaluate the feasibility of Lesniewski's project and to briefly comment on its contemporary relevance.

%B European review %V 23 %P 125–138 %G eng %0 Journal Article %J Journal of philosophical logic %D 2013 %T Gödelizing the Yablo sequence %A Cieśliński, Cezary %A Urbaniak, Rafal %XWe investigate what happens when 'truth' is replaced with 'provability' in Yablo's paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Godel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo paradox. We also look at a formulation which employs Rosser's provability predicate.

%B Journal of philosophical logic %V 42 %P 679–695 %G eng %U http://dx.doi.org/10.1007/s10992-012-9244-4 %0 Journal Article %J Foundations of science %D 2013 %T Induction from a single instance: Incomplete frames %A Urbaniak, Rafal %A Van De Putte, Frederik %XIn this paper we argue that an existing theory of concepts called dynamic frame theory, although not developed with that purpose in mind, allows for the precise formulation of a number of problems associated with induction from a single instance. A key role is played by the distinction we introduce between complete and incomplete dynamic frames, for incomplete frames seem to be very elegant candidates for the format of the background knowledge used in induction from a single instance. Furthermore, we show how dynamic frame theory provides the terminology to discuss the justification and the fallibility of incomplete frames. In the Appendix, we give a formal account of incomplete frames and the way these lead to induction from a single instance.

%B Foundations of science %V 18 %P 641–653 %G eng %R http://dx.doi.org/10.1007/s10699-012-9295-6 %0 Journal Article %J History and philosophy of logic %D 2012 %T Busting a myth about Leśniewski and definitions %A Urbaniak, Rafal %A Hämäri, K Severi %XA theory of definitions which places the eliminability and conservativeness requirements on definitions is usually called the standard theory. We examine a persistent myth which credits this theory to Lesniewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Lesniewski's published or unpublished work is known where the standard conditions are discussed. Second, Lesniewski's own logical theories allow for creative definitions. Third, Lesniewski's celebrated 'rules of definition' lay merely syntactical restrictions on the form of definitions: they do not provide definitions with such meta-theoretical requirements as eliminability or conservativeness. On the positive side, we point out that among the Polish logicians, in the 1920s and 1930s, a study of these meta-theoretical conditions is more readily found in the works of Lukasiewicz and Ajdukiewicz.

%B History and philosophy of logic %V 33 %P 159–189 %G eng %R http://dx.doi.org/10.1080/01445340.2011.583771 %0 Journal Article %J ERKENNTNIS %D 2012 %T Numbers and propositions versus nominalists: yellow cards for Salmon & Soames %A Urbaniak, Rafal %XSalmon and Soames argue against nominalism about numbers and sentence types. They employ (respectively) higher-order and first-order logic to model certain natural language inferences and claim that the natural language conclusions carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that. I argue that this strategy fails because the nominalist can accept those natural language consequences, provide them with plausible and non-committing truth conditions and account for the inferences made without committing themselves to abstract objects. I sketch a modal account of higher-order quantification, on which instead of ranging over sets, higher order quantifiers are used to make (logical) possibility claims about which predicate tokens can be introduced. This approach provides an alternative account of truth conditions for natural language sentences which seem to employ higher-order quantification, thus allowing the nominalist to evade Salmon's argument. I also show how the nominalist can account for the occurrence of apparently singular abstract terms in certain true statements. I argue that the nominalist can achieve this by, first, dividing singular terms into real singular terms (referring to concrete objects) and only apparent singular terms (called onomatoids), introduced for the sake of brevity and simplicity, and then providing an account of nominalistically acceptable truth conditions of sentences containing onomatoids. I develop such an account in terms of modally interpreted abstraction principles and argue that applying this strategy to Soames's argument allows the nominalists to defend themselves. One would hope and perhaps conjecture that the whole general set theory, however beautiful it is, will in the future disappear. With the higher types Platonism begins. The tendencies of Chwistek and others ('Nominalism') of speaking only of what can be named are healthy. [Alfred Tarski](1)

%B ERKENNTNIS %V 77 %P 381–397 %G eng %R http://dx.doi.org/10.1007/s10670-012-9402-7 %0 Journal Article %J Synthese %D 2012 %T 'Platonic' thought experiments: how on earth %A Urbaniak, Rafal %XBrown (The laboratory of the mind. Thought experiments in the natural science, 1991a, 1991b; Contemporary debates in philosophy of science, 2004; Thought experiments, 2008) argues that thought experiments (TE) in science cannot be arguments and cannot even be represented by arguments. He rest his case on examples of TEs which proceed through a contradiction to reach a positive resolution (Brown calls such TEs "platonic"). This, supposedly, makes it impossible to represent them as arguments for logical reasons: there is no logic that can adequately model such phenomena. (Brown further argues that this being the case, "platonic" TEs provide us with irreducible insight into the abstract realm of laws of nature). I argue against this approach by describing how "platonic" TEs can be modeled within the logical framework of adaptive proofs for prioritized consequence operations. To show how this mundane apparatus works, I use it to reconstruct one of the key examples used by Brown, Galileo's TE involving falling bodies.

%B Synthese %V 187 %P 731–752 %G eng %R http://dx.doi.org/10.1007/s11229-011-0008-4 %0 Journal Article %J Philosophia Mathematica %D 2011 %T How not to use the Church-Turing thesis against platonism %A Urbaniak, Rafal %XOlszewski claims that the Church-Turing thesis can be used in an argument against platonism in philosophy of mathematics. The key step of his argument employs an example of a supposedly effectively computable but not Turing-computable function. I argue that the process he describes is not an effective computation, and that the argument relies on the illegitimate conflation of effective computability with there being a way to find out.

%B Philosophia Mathematica %V 19 %P 74–89 %G eng %R http://dx.doi.org/10.1093/philmat/nkr001 %0 Journal Article %J Studia Logica %D 2010 %T Neologicist nominalism %A Urbaniak, Rafal %XThe goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those difficulties while still being able to put abstraction principles to a foundational use.

%B Studia Logica %V 96 %P 149-173 %G eng %R 10.1007/s11225-010-9279-x %0 Journal Article %J Reports on Mathematical Logic %D 2010 %T Response to a Critic (Definability and Ontology) %A Urbaniak, Rafal %B Reports on Mathematical Logic %V 45 %P 255-259 %G eng %0 Journal Article %J The Reasoner %D 2009 %T Bogus singular terms and substitution salva denotatione %A Urbaniak, Rafal %B The Reasoner %V 3 %P 4-5 %8 June %G eng %0 Journal Article %J Logic Journal of IGPL %D 2009 %T Capturing dynamic conceptual frames %A Urbaniak, Rafal %XThe main focus of this paper is to develop an adaptive formal apparatus capable of capturing arguments conducted within a conceptual framework. I first explain one of the most recent theories of concepts developed by cognitivists, in which a crucial part is played by the notion of a *dynamic frame*. Next, I describe how a dynamic frame may be captured by a finite set of formulas and how a formalized adaptive framework for reasoning within a dynamic frame can be developed.

Leitgeb (2002) objects against the clarity of the debate about the alleged (non-)circularity of Yablos paradox, arguing that there actually are at least two notions of self-reference and circularity at play.One, on which Yablos paradox is not circular, is defined via thereference of the constituents of a sentence, and another, on which the paradox is circular, is defined via syntactic mappings and fixedpoints. More importantly, Leitgeb argues that both definitions arent satisfactory and that before we can undertake a serious debate about the circularity of Yablos paradox we first need to clarify the notions involved. I will focus on Leitgebs criticism of the first definition^{1}and will argue that the problems arise not as much on the level of our definition of circularity as on the level of our definition of reference of sentences (aboutness). Leitgebs main worry is the failure of a requirement called Equivalence Condition, which says that if a formula is self-referential, any formula logically equivalent to it should also be self-referential. I will argue that preservation under logical equivalence is unreasonable with respect to self-reference,but is indeed needed with respect to aboutness. Since Leitgeb sown tentative notion of aboutness doesnt satisfy the requirement, I will suggest another approach which fixes this problem. I also explain why the intuitions that circularity should satisfy the equivalence condition are misled. Next, I argue that the new notion of aboutness is not susceptible to slingshot arguments. Finally, I compare it with Goodmans notion of absolute aboutness, emphasizing those features of Goodmans approach that make his notion inapplicable in the present discussion. ^{[1]I would like to express my gratitude to all the people who discussed earlier versions of this paper with me: Hannes Leitgeb, Jeffrey Ketland, Karl Georg Niebergall, Diderik Batens, Joke Meheus, Maarten Van Dyck, Stefan Wintein, Martin Bentzen, Christian Straßer, Ghent Centre for Logic and Philosophy of Science members, and the participants of PhDs in Logic workshop (Gent 2009)}

It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be dened in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics (where the variables range over all subsets of the domain) in which the identity relation is not denable. The point is that the denability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed.

%B Australasian Journal of Logic %V 7 %P 48–55 %G eng %0 Journal Article %J The Reasoner %D 2009 %T PhD's in Logic - report (with S. Wintein) %A Urbaniak, Rafal %B The Reasoner %V 3 %P 6–7 %G eng %0 Generic %D 2009 %T Reasoning with dynamic conceptual frames. %A Urbaniak, Rafal %E Weber, Erik %E Libert, Thierry %E Marage, Pierre %E Vanpaemel, Geert %B Logic, Philosophy and History of Science in Belgium. Proceedings of the Young Researchers Days 2008 %I Koninklijke Vlaamse Academie van België %C Brussel %P 84-89 %G eng %0 Journal Article %J The Reasoner %D 2009 %T Slingshot arguments: two versions %A Urbaniak, Rafal %B The Reasoner %V 3 %P 4–5 %G eng %0 Journal Article %J History and Philosophy of Logic %D 2008 %T Leśniewski and Russell's paradox: some problems %A Urbaniak, Rafal %B History and Philosophy of Logic %V 29 %P 115–146 %G eng %0 Generic %D 2008 %T Reducing sets to modalities %A Urbaniak, Rafal %E Hieke, Alexander %E Hannes, Leitgeb %B Proceedings of the 31st International Wittgenstein Symposium of the Austrian Ludwig Wittgenstein Society %I Department for Culture and Science of the Province of Lower Austria %C Kirchberg am wechsel %V XVI %P 359-361 %G eng %0 Conference Paper %B Logica 2007 Yearbook %D 2007 %T Time Travel and Conditional Logics %A Urbaniak, Rafal %B Logica 2007 Yearbook %G eng %0 Journal Article %J Reports on Mathematical Logic %D 2006 %T On Ontological Functors of Lesniewski's Elementary Ontology %A Urbaniak, Rafal %XWe present an algorithm which allows to define any possible sentence-formative functor of Le&\#347;niewski's Elemen- tary Ontology (LEO), arguments of which belong to the category of names. Other results are: a recursive method of listing possible functors, a method of indicating the number of possible n-place ontological functors, and a sketch of a proof that LEO is function- ally complete with respect to {&\#8743;,&\#172;, &\#8704;, &\#949;}

%B Reports on Mathematical Logic %V 40 %P 15–43 %G eng %0 Journal Article %J Journal of Logic and Computation %D 2006 %T On representing 2We propose an intuitive understanding of the statement: an ax-iom (or: an axiomatic basis) determines the meaning of the only specific constant occurring in it. We introduce some basic semantics for functors of the category ^{s}⁄_{n,n} of Le´sniewskis Ontology. Using these results weprove that the popular claim that the axioms of Ontology determine themeaning of the primitive constants is false.