@article {Pawlowski2017, title = {Paradoxes of informal provability and many-valued indeterministic provability logic}, year = {Submitted}, author = {Urbaniak, Rafal and Pawlowski, Pawel} } @incollection {2045, title = {Logics of (formal and informal) provability}, booktitle = {Handbook of Formal Philosophy}, year = {In Press}, publisher = {Springer}, author = {Urbaniak, Rafal and Pawlowski, Pawel}, editor = {Hansson, Sven Ove and Hendricks, Vincent F.} } @article {Pawlowski2017d, title = {Many-valued logic of informal provability: a non-deterministic strategy}, journal = {The Review of Symbolic Logic}, year = {In Press}, author = {Pawlowski, Pawel and Urbaniak, Rafal} } @incollection {RaD:i, title = {Induction}, booktitle = {Handbook of Formal Philosophy}, year = {2018}, publisher = {Springer}, keywords = {induction, logic, probability}, isbn = {978-3-319-77433-6}, doi = {10.1007/978-3-319-77434-3}, author = {Urbaniak, Rafal and Batens, Diderik}, editor = {Hansson, Sven Ove and Hendricks, Vincent F.} } @incollection {D:Abd, title = {Abduction Logics Illustrating Pitfalls Of Defeasible Methods}, booktitle = {Applications of formal philosophy: the road less travelled}, series = {Logic argumentation \& reasoning}, volume = {14}, year = {2017}, pages = {169{\textendash}193}, publisher = {Springer}, address = {Berlin}, abstract = {On the one hand this paper offers an introduction to adaptive logics, focussing on properties that are imposed upon adaptive logics by the fact that they explicate defeasible reasoning. On the other hand new adaptive logics of abduction are presented and employed to illustrate those properties. These logics were developed in view of the criticism to existing adaptive logics of abduction.}, isbn = {978-3-319-58507-9,331958507X,978-3-319-58505-5}, doi = {10.1007/978-3-319-58507-9_8}, author = {Batens, Diderik}, editor = {Rafa{\l} Urbaniak and Gillman Payette} } @incollection {Pawlowski2017a, title = {Philosophical Aspects of an Alleged Connection Between the Axiom of Choice and Predicting the Future}, booktitle = {Applications of Formal Philosophy The Road Less Travelled}, year = {2017}, pages = {213-220}, publisher = {Springer}, author = {Pawlowski, Pawel}, editor = {Urbaniak, Rafal and Gillman, Payette} } @article {4443770, title = {Plural quantifiers: a modal interpretation}, journal = {Synthese}, volume = {191}, number = {7}, year = {2014}, pages = {1605{\textendash}1626}, abstract = {

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 {\textquoteright}some logicians admire only each other{\textquoteright} 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.

}, issn = {0039-7857}, doi = {http://dx.doi.org/10.1007/s11229-013-0354-5}, author = {Urbaniak, Rafal} } @incollection {4090467, title = {Pluralists about pluralism? Different versions of explanatory pluralism in psychiatry}, booktitle = {New directions in the philosophy of science}, year = {2014}, pages = {104{\textendash}119}, publisher = {Springer}, abstract = {

In this contribution, I comment on Raffaella Campaner{\textquoteright}s defense of explanatory pluralism in psychiatry (in this volume). In her paper, Campaner focuses primarily on explanatory pluralism in contrast to explanatory reductionism. Furthermore, she distinguishes between pluralists who consider pluralism to be a temporary state on the one hand and pluralists who consider it to be a persisting state on the other hand. I suggest that it would be helpful to distinguish more than those two versions of pluralism {\textendash} different understandings of explanatory pluralism both within philosophy of science and psychiatry {\textendash} namely moderate/temporary pluralism, anything goes pluralism, isolationist pluralism, integrative pluralism and interactive pluralism. Next, I discuss the pros and cons of these different understandings of explanatory pluralism. Finally, I raise the question of how to implement or operationalize explanatory pluralism in scientific practice; how to structure the {\textquotedblleft}genuine dialogue{\textquotedblright} or shape {\textquotedblleft}the pluralistic attitude{\textquotedblright} Campaner is referring to. As tentative answers, I explore a question-based framework for explanatory pluralism as well as social-epistemological procedures for interaction among competing approaches and explanations.

}, isbn = {9783319043814}, doi = {10.1007/978-3-319-04382-1_8}, author = {Van Bouwel, Jeroen}, editor = {Galavotti, Maria Carla and Dieks, Dennis and Gonzalez, Wenceslao J. and Hartmann, Stephan and Uebel, Thomas and Weber, Marcel} } @incollection {de2014proof, title = {The Proof Is in the Process: A Preamble for a Philosophy of Computer-Assisted Mathematics}, booktitle = {New Directions in the Philosophy of Science}, series = {The Philosophy of Science in a European Perspective}, volume = {5}, year = {2014}, pages = {15{\textendash}33}, publisher = {Springer}, abstract = {

According to some well-known mathematicians well-versed in computer-assisted mathematics (CaM), “Computers are changing the way we are doing mathematics”. To what extent this is really true is still an open question. Indeed, even though some philosophers of math have taken up the challenge to think about CaM, it is unclear in what sense exactly a machine (can) affect(s) the so-called “queen of the sciences”. In fact, some have concluded that issues raised by the use of the computer in mathematics are not specific to the use of the computer per se. However, such findings seem precarious since a systematic study of computer-assisted mathematics is still lacking. In this paper I argue that in order to understand the impact of CaM, it is necessary to take more seriously the computer itself and how it is actually used in the process of doing mathematics. Within such an approach, one searches for characteristics that are specific to the use of the computer in mathematics. I will focus on a feature that is beyond any doubt inherently connected to the use of computing machinery, viz. mathematician-computer interactions. I will show how such interactions are fundamentally different from the usual interactions between mathematicians and non-human aids (a piece of paper, a blackboard etc) and how such interactions determine at least two more characteristics of CaM, viz. the significance of time and processes and the steady process of internalization of mathematical tools and knowledge into the machine. I will restrict myself to the use of the computer within so-called experimental mathematics since this is the main object of CaM within the philosophical literature.

}, doi = {10.1007/978-3-319-04382-1_2}, author = {De Mol, Liesbeth}, editor = {Galavotti, Maria Carla and Dieks, Dennis and Gonzalez, Wenceslao J. and Hartmann, Stephan and Uebel, Thomas and Weber, Marcel} } @article {4443783, title = {S\lupecki{\textquoteright}s generalized mereology and its flaws}, journal = {History and Philosophy of Logic}, volume = {35}, number = {3}, year = {2014}, pages = {289{\textendash}300}, abstract = {

One 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 Le{\'s}niewski (1886{\textendash}1939). The attempt failed, but there is another, prima facie more promising attempt by Jerzy S\unmatched{0142}upecki (1904{\textendash}1987), who employed his generalized mereology to build mereological foundations for type theory. In this paper I (1) situate Le{\'s}niewski{\textquoteright}s attempt in the development of set theory, (2) describe and evaluate Le{\'s}niewski{\textquoteright}s approach, (3) describe S\unmatched{0142}upecki{\textquoteright}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 {\textendash}- it can be safely skipped by anyone familiar with the early development of set theory. Section 2 describes and evaluates Le{\'s}niewski{\textquoteright}s attempt to use mereology as a foundational tool. In Section 3, I describe an attempt by S\unmatched{0142}upecki to improve on Le{\'s}niewski{\textquoteright}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 Le{\'s}niewski{\textquoteright}s use of Ontology in developing arithmetic also is not nominalistically satisfactory.

}, doi = {http://dx.doi.org/10.1080/01445340.2014.917837}, author = {Urbaniak, Rafal} } @article {4443772, title = {Stanislaw {L}e{\'s}niewski: rethinking the philosophy of mathematics}, journal = {European review}, volume = {23}, number = {1}, year = {2014}, pages = {125{\textendash}138}, abstract = {

Near the end of the XIXth century part of mathematical research was focused on unification: the goal was to find {\textacutedbl}one sort of thing{\textacutedbl} 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{\textquoteright}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}{\textquoteright}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{\textquoteright}s project and to briefly comment on its contemporary relevance.

}, author = {Urbaniak, Rafal} } @article {4443766, title = {G{\"o}delizing the Yablo sequence}, journal = {Journal of philosophical logic}, volume = {42}, number = {5}, year = {2013}, pages = {679{\textendash}695}, abstract = {

We investigate what happens when {\textquoteright}truth{\textquoteright} is replaced with {\textquoteright}provability{\textquoteright} in Yablo{\textquoteright}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{\textquoteright}s provability predicate.

}, issn = {0022-3611}, url = {http://dx.doi.org/10.1007/s10992-012-9244-4}, author = {Cie{\'s}li{\'n}ski, Cezary and Urbaniak, Rafal} } @article {4443727, title = {Induction from a single instance: {I}ncomplete frames}, journal = {Foundations of science}, volume = {18}, number = {4}, year = {2013}, pages = {641{\textendash}653}, abstract = {

In 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.

}, issn = {1233-1821}, doi = {http://dx.doi.org/10.1007/s10699-012-9295-6}, author = {Urbaniak, Rafal and Van De Putte, Frederik} } @article {3148393, title = {Busting a myth about Le{\'s}niewski and definitions}, journal = {History and philosophy of logic}, volume = {33}, number = {2}, year = {2012}, pages = {159{\textendash}189}, abstract = {

A 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{\textquoteright}s published or unpublished work is known where the standard conditions are discussed. Second, Lesniewski{\textquoteright}s own logical theories allow for creative definitions. Third, Lesniewski{\textquoteright}s celebrated {\textquoteright}rules of definition{\textquoteright} 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.

}, issn = {0144-5340}, doi = {http://dx.doi.org/10.1080/01445340.2011.583771}, author = {Urbaniak, Rafal and H{\"a}m{\"a}ri, K Severi} } @article {3148353, title = {Numbers and propositions versus nominalists: yellow cards for {S}almon \& {S}oames}, journal = {ERKENNTNIS}, volume = {77}, number = {3}, year = {2012}, pages = {381{\textendash}397}, abstract = {

Salmon 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{\textquoteright}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{\textquoteright}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 ({\textquoteright}Nominalism{\textquoteright}) of speaking only of what can be named are healthy. [Alfred Tarski](1)

}, issn = {0165-0106}, doi = {http://dx.doi.org/10.1007/s10670-012-9402-7}, author = {Urbaniak, Rafal} } @article {3148385, title = {{\textquoteright}Platonic{\textquoteright} thought experiments: how on earth}, journal = {Synthese}, volume = {187}, number = {2}, year = {2012}, pages = {731{\textendash}752}, abstract = {

Brown (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{\textquoteright}s TE involving falling bodies.

}, issn = {0039-7857}, doi = {http://dx.doi.org/10.1007/s11229-011-0008-4}, author = {Urbaniak, Rafal} } @incollection {J:flash, title = {A Formal Logic for the Abduction of Singular Hypotheses}, booktitle = {Explanation, Prediction, and Confirmation}, year = {2011}, pages = {93{\textendash}108}, publisher = {Springer}, address = {Dordrecht}, author = {Meheus, Joke}, editor = {Dieks, Dennis and Gonzalez, Wenceslao J. and Hartmann, Stephan and Uebel, Thomas and Weber, Erik} } @article {3148473, title = {How not to use the {C}hurch-{T}uring thesis against platonism}, journal = {Philosophia Mathematica}, volume = {19}, number = {1}, year = {2011}, pages = {74{\textendash}89}, abstract = {

Olszewski 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.

}, issn = {0031-8019}, doi = {http://dx.doi.org/10.1093/philmat/nkr001}, author = {Urbaniak, Rafal} } @article {Urbaniak2010, title = {Neologicist nominalism}, journal = {Studia Logica}, volume = {96}, number = {2}, year = {2010}, pages = {149-173}, abstract = {

The 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.

}, doi = {10.1007/s11225-010-9279-x}, author = {Urbaniak, Rafal} } @article {urbaniak2010response, title = {Response to a Critic (Definability and Ontology)}, journal = {Reports on Mathematical Logic}, volume = {45}, year = {2010}, pages = {255-259}, author = {Urbaniak, Rafal} } @article {urbaniak2009bogus, title = {Bogus singular terms and substitution salva denotatione}, journal = {The Reasoner}, volume = {3}, number = {6}, year = {2009}, month = {June}, pages = {4-5}, author = {Urbaniak, Rafal} } @article {urbaniak2009capturing, title = {Capturing dynamic conceptual frames}, journal = {Logic Journal of {IGPL}}, volume = {18}, number = {3}, year = {2009}, pages = {430{\textendash}455}, abstract = {

The 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.

}, author = {Urbaniak, Rafal} } @article {urbaniak2009doxastic, title = {Doxastic synonymy vs. logical equivalence}, journal = {The Reasoner}, volume = {3}, number = {5}, year = {2009}, pages = {5-6}, author = {Urbaniak, Rafal} } @article {Urbaniak2009b, title = {Leitgeb, {\textquoteleft}{\textquoteleft}about{\textquoteright}{\textquoteright}, {Y}ablo}, journal = {Logique et Analyse}, volume = {52}, number = {207}, year = {2009}, pages = {239-254}, abstract = {

Leitgeb (2002) objects against the clarity of the debate about the alleged (non-)circularity of Yablo’s paradox, arguing that there actually are at least two notions of self-reference and circularity at play.One, on which Yablo’s 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 aren’t satisfactory and that before we can undertake a serious debate about the circularity of Yablo’s paradox we first need to clarify the notions involved. I will focus on Leitgeb’s criticism of the first definition1and 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). Leitgeb’s 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 doesn’t 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 Goodman’s notion of absolute aboutness, emphasizing those features of Goodman’s 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 Stra{\ss}er, Ghent Centre for Logic and Philosophy of Science members, and the participants of PhDs in Logic workshop (Gent 2009)

}, author = {Urbaniak, Rafal} } @article {urbaniak2009kaye, title = {The Mathematics of Logic by Richard Kaye (review).}, journal = {Bulletin of Symbolic Logic}, volume = {15}, number = {02}, year = {2009}, pages = {216{\textendash}218}, author = {Urbaniak, Rafal} } @incollection {Urbaniak2009a, title = {Modal Logic and Philosophy. {A} case study.}, booktitle = {Logic, Philosophy and History of Science in Belgium. Proceedings of the Young Researchers Days 2008}, year = {2009}, pages = {79-83}, publisher = {Koninklijke Vlaamse Academie van Belgi{\"e}}, address = {Brussel}, author = {Urbaniak, Rafal and Rostalska, Agnieszka}, editor = {Weber, Erik and Libert, Thierry and Marage, Pierre and Vanpaemel, Geert} } @article {urbaniak2009note, title = {A note on identity and higher-order quantification.}, journal = {Australasian Journal of Logic}, volume = {7}, year = {2009}, pages = {48{\textendash}55}, abstract = {

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 de ned in a language without identity, as soon as we admit second-order, set-theoretically interpreted quanti ers 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 de nable. The point is that the de nability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed.

}, author = {Urbaniak, Rafal} } @article {Urbaniak, title = {PhD{\textquoteright}s in Logic - report (with S. Wintein)}, journal = {The Reasoner}, volume = {3}, number = {4}, year = {2009}, pages = {6{\textendash}7}, author = {Urbaniak, Rafal} } @incollection {Urbaniak2008, title = {Reasoning with dynamic conceptual frames.}, booktitle = {Logic, Philosophy and History of Science in Belgium. Proceedings of the Young Researchers Days 2008}, year = {2009}, pages = {84-89}, publisher = {Koninklijke Vlaamse Academie van Belgi{\"e}}, address = {Brussel}, author = {Urbaniak, Rafal}, editor = {Weber, Erik and Libert, Thierry and Marage, Pierre and Vanpaemel, Geert} } @article {Urbaniak2009, title = {Slingshot arguments: two versions}, journal = {The Reasoner}, volume = {3}, number = {4}, year = {2009}, pages = {4{\textendash}5}, author = {Urbaniak, Rafal} } @article {urbaniak2008lesniewski, title = {Le{\'s}niewski and Russell{\textquoteright}s paradox: some problems}, journal = {History and Philosophy of Logic}, volume = {29}, number = {2}, year = {2008}, pages = {115{\textendash}146}, author = {Urbaniak, Rafal} } @incollection {Urbaniak2008a, title = {Reducing sets to modalities}, booktitle = {Proceedings of the 31st International Wittgenstein Symposium of the Austrian Ludwig Wittgenstein Society}, volume = {XVI}, year = {2008}, pages = {359-361}, publisher = {Department for Culture and Science of the Province of Lower Austria}, address = {Kirchberg am wechsel}, author = {Urbaniak, Rafal}, editor = {Hieke, Alexander and Hannes, Leitgeb} } @conference {Urbaniak2007, title = {Time Travel and Conditional Logics}, booktitle = {Logica 2007 Yearbook}, year = {2007}, author = {Urbaniak, Rafal} } @article {urbaniak2006ontological, title = {On Ontological Functors of {L}esniewski{\textquoteright}s Elementary Ontology}, journal = {Reports on Mathematical Logic}, volume = {40}, year = {2006}, pages = {15{\textendash}43}, abstract = {

We present an algorithm which allows to define any possible sentence-formative functor of Le\&\#347;niewski{\textquoteright}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;}

}, author = {Urbaniak, Rafal} } @article {Urbaniak2006, title = {On representing 2After a brief presentation of Lesniewski{\textquoteright}s notation for 1- and 2-place sentential connectives of protothetic, the article discusses a method of extending this method to n \&\#8805; 3-place sentential connectives. Such a method has been hinted at by Luschei, but in fact, no general effective method of defining such functors has been clearly and explicitly given. The purpose of this article is to provide such a method.

}, author = {Urbaniak, Rafal} } @article {urbaniak2006some, title = {Some non-standard interpretations of the axiomatic basis of Le{\'s} niewski’s Ontology}, journal = {The Australasian Journal of Logic}, volume = {4}, year = {2006}, pages = {13{\textendash}46}, abstract = {

We 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{\textfractionsolidus}n,n of Le{\textasciiacute}sniewski’s Ontology. Using these results weprove that the popular claim that the axioms of Ontology determine themeaning of the primitive constants is false.

}, author = {Urbaniak, Rafal} } @article {Urbaniak2003, title = {Analysis of Kotarbinski{\textquoteright}s reistic reasonings}, journal = {{Logiczne Podstawy Rozumowa{\'n}}, volume = {3}, year = {2003}, pages = {62{\textendash}81}, author = {Urbaniak, Rafal} }