Abstract
I discuss various attempts to emulate standard set theory within the framework of Leśniewski’s system: the Leśniewski–Sobociński strategy of distinguishing between collective and distributive totalities (a heap of stones is a collective totality; a set of chairs is not) I show the strategy, as used by Sobociński, leads to a few paradoxes. Further, I argue that using the means available within Ontology itself does not result in a theory strong enough to mimic set theory. I then describe Słupecki’s generalized mereology and argue that it is too different from set theory to be able to play a foundational role.
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- 1.
It is important to emphasize that the question I am concerned with is not whether there is a way of introducing into the language of Ontology a formal counterpart of the expression ‘the class of’ which would mimic correctly Leśniewski’s understanding of this term (which was quite unusual), but rather how the expression ‘the class of’ in the nowadays most common, set-theoretic use can be imitated in the language of Leśniewski’s systems. Thus, even if Leśniewski would not share the intuitions behind the examples above, it does not matter. The point is that the sense of ‘the class of’ in which the above examples come out true seems to be a legitimate and widespread understanding of the expression in question and Mereology falls short of accounting for this plausibility.
- 2.
There is an intriguing comment on Leśniewski’s philosophical method made by Twardowski: “In general, those who follow Leśniewski, very arbitrarily demand an analysis where they find it convenient; however, whenever someone demand an analysis where it is inconvenient, they refer to intuition. And when the opponent in the discussion tries at some point to refer to intuition as well, they respond: “We cannot understand what you claim to be intuitively given”. (K. Twardowski’s Diary, ms. 2407/3)”. [The quote comes from Kazimerz Twardowski’s archive, located in the library of the Institute of Philosophy and Sociology of the Polish Academy of Sciences, Warsaw. The reference is to signatures in this collection.]
- 3.
The first four theorems in this chapter have been proven in a rather tedious manner by Sobociński (who credits Leśniewski with the proofs). I give simplified and streamlined proof sketches based on Sobociński’s proofs.
- 4.
In the original notation of the papers at issue parentheses surrounding arguments of different categories were of different shapes. For the sake of convenience I follow this aspect of the original notation, as well as I keep using \(\phi \) as a functor variable. There is no obvious assignment of shapes of brackets to semantic categories, I will just arbitrarily pick different shapes for different categories for the purposes of this chapter.
- 5.
Most of theorems and proof sketches about higher-order epsilon operators have been developed by the author.
- 6.
What makes a connective a higher-order epsilon is an interesting question. The intuition is that \(f\) is an epsilon if the truth of \(f(\delta , \gamma )\) requires some sort of uniqueness of the referent(s) of \(\delta \) (presumably, up to coextensiveness), and some sort of inclusion between the referent(s) of \(\delta \) and the referent(s) of \(\gamma \). This however is far from providing a formally correct and precise definition.
- 7.
To be fair, details and weaknesses of the translations discussed in paragraph (iii) require further investigation, which is beyond the scope of this book.
- 8.
The problem with reconstructing Słupecki’s generalized mereology is that the only paper that he published which pertained to it is a four-page abstract, so quite a few details have to be filled out by the reader. What follows is my best shot and understanding what the system looks like.
- 9.
“Uogólniaj a̧ c mereologiȩ kierujȩ siȩ nastȩpuj a̧ cymi dwoma celami: (a) w systemie uogólnionym pojȩcie zbioru powinno zachować niezmieniony sens intuicyjny, w szczególności zbiory powinny być tak zdefiniowane, by były konkretnymi przedmiotami; (b) uogólniony system mereologii powinien nadawać siȩ do podbudowania matematyki w tym zakresie, w jakim służ a̧ do tego celu prosta teoria typów logicznych lub aksjomatyczna teoria mnogości. Celu tego nie spełnia pierwotny system Leśniewskiego, w którym nie może być np. zbudowala arytmetyka liczb naturalnych.”
- 10.
The background logic is weaker but the properties of the parthood relation are essentially the same.
- 11.
Basically, the uniqueness follows from extensionality.
- 12.
So Słupecki’s system, unlike Leśniewski’s, is not free of existential assumptions.
- 13.
It is a fairly awkward procedure, for the axioms of the theory no longer are theorems of system M, but the set of theorems of M is nevertheless well defined. Perhaps, there is an axiomatization of M in the language of M. This, however, lies beyond the scope of present considerations (in particular, because I think that as a replacement for set theory M is flawed anyway). This does not mean that the system is not interesting in itself.
- 14.
“St a̧ d też wynika, że system M, a tym bardziej cały system uogólnionej mereologii, wystarcza do ugruntowania matematyki, gdyż—jak wiadomo—do celu tego wystarcza system \(\mathbf{S }_{1}\). Wniosek ten może wydać siȩ trywialny z tego wzglȩdu, że pełny system prostej teorii typów może byćbez żadnych uzupełnień podbudow a̧ matematyki. W pracy podajȩ jednak argument (zdajȩ sobie sprawȩ, że argumenty te nie s a̧ całkowicie przekonywaj a̧ ce i dlatego tu ich nie powtarzam) przemawiaj a̧ ce za tym, że terminy stałe, należ a̧ ce odpowiednio do tych kategorii semantycznych do których należ a̧ zmienne \( f, \phi , \varPhi , \ldots \) nie s a̧ nazwami zbiorów ani też żadnych innych przedmiotów, lecz wyrażeniami pozbawionymi samodzielnego sensu.Terminy te nie mog a̧ też oznaczać przedmiotów konkretnych, a wiȩc pełny system prostej teorii typów nie spełnia celu (a), o którym poprzednio była mowa.”
References
Davis, C. (1975). An investigation concerning the Hilbert-Sierpiński logical form of the axiom of choice. Notre Dame Journal of Formal Logic, 16, 145–184.
Dummet, M. (1973). Frege’s way out: A footnote to a footnote. Analysis, 33, 139–140.
Geach, P. (1956). On Frege’s way out. Mind, 65, 408–408.
Henry, D. (1972). Medieval logic and metaphysics: A modern introduction. London: Hutchinson.
Landini, G. (2006). The ins and outs of Frege’s way out. Philosophia Mathematica, 14, 1–25.
Lejewski, C. (1985). Accommodating the informal notion of class within the framework of Leśniewski’s Ontology. Dialectica, 39, 217–241.
Linsky, L., & Schumm, G. (1971). Frege’s way out: A footnote. Analysis, 32, 5–7.
Quine, W. V. (1955). On Frege’s way out. Mind, 64, 145–159.
Słupecki, J. (1958). Towards a generalized Mereology of Leśniewski. Studia Logica, 8, 131–154.
Sobociński, B. (1949). L’analyse de l’antinomie Russellienne par Leśniewski. Methodos, 1–2(1, 2, 3; 6–7):94–107, 220–228, 308–316, 237–257 [translated as “Leśniewski’s analysis of Russell’s paradox” (srzednicki and Ricky, 1984, 11–44)].
Urbaniak, R. (2008). Leśniewski and Russell’s paradox: Some problems. History and Philosophy of Logic, 29(2), 115–146.
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Urbaniak, R. (2014). Sets Revisited. In: Leśniewski's Systems of Logic and Foundations of Mathematics. Trends in Logic, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-00482-2_7
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