Abstract
Both Bolzano and Tarski were unsure what counts as logic. This means that Bolzano’s concept of logical analyticity, like Tarski’s of logical consequence, is not completely determinate. In a posthumously published paper, Tarski offers a proposal for demarcating the logical objects in a type-hierarchy, based on the idea of invariance under arbitrary permutations of the domain of individuals. In this paper I comment on and extend Tarski’s proposal and show how to combine it with Bolzano’s procedure of variation among concepts, to obtain a definition of logical constants in a logically significant fragment of a purported Bolzanian realm of meanings in themselves. I conclude with doubts about the propriety and utility of such a realm.
[S]ometimes it seems to me convenient to include mathematical terms, like the ∈-relation, in the class of logical ones, and sometimes I prefer to restrict myself to terms of ‘elementary logic’. Is any problem involved here?
Alfred Tarski, letter to Morton White (Tarski 1987, 29).
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References
Hodges 1983, 56. For a useful comparison cf. Berg 1962, 116–8, which also cautions against drawing the parallels too closely.5Cf. Hacking 1976, 1979, Peacocke 1976, McCarthy 1981, Barwise and Cooper 1981, Westerstdhl 1985.
See Tarski 1956 (21983) (hereafter referred to as LSM), 417 n.
On the origins of the Lw6w-Warsaw School see Skolimowski 1967 and Wolenski 1988. 6Another possible source is Benno Kerry. See Haller 1982, vii.
Bolzano’s name occurs more often than any other in Twardowski 1894.
D3.mbska 1978, 123 reports that Twardowski regularly held a course in Lw6w on “The Attempts to Reform Traditional Logic”, which dealt with the work of Bolzano, Brentano, Boole, and Schröder.
Lukasiewicz 1913a devotes a section to comparison with Bolzano. It is also possible that Lukasiewicz, like others who went to Austrian Gymnasium, first came into contact with Bolzano’s ideas in the thin disguise of Zimmermann’s Philosophische Propaedeutik. For intriguing speculations on the possible extent of this subterranean influence, cf. Sebestik 1985, 102.10See Wolenski and Simons 1989.
Both Bolzano and Twardowski argue that the occurrence of indexical expressions in sentences of ordinary language does not entail the relativity of truth, since both deny that sentences are the primary truth-bearers (for Bolzano it is propositions, for Twardowski judgements). In discussing the point, Twardowski uses an example which is also found in Bolzano’s discussion. Bolzano (WL ¡ì147) has “Der Duft dieser Blume ist angenehm”; Twardowski 1902, 416 (in German translation) has “Diese Blume riecht angenehm”, but the Polish original, “Won tego kwiatu jest przyjemna”, is closer to Bolzano.
Proposition’ translates `Satz an sich’.
Concept’ translates `Vorstellung an sich’, and `denoting’ translates `gegenständlich’. A denoting concept is one under which at least one object falls.
The requirement that the concepts substituted all be denoting results from Bolzano’s view that all propositions with a non-denoting subject concept are false. Without this restriction the theory is uninteresting. While it is worth considering relaxing this condition, the language B below in effect follows Bolzano’s lead by requiring in classical fashion that every name denotes an individual.
Square brackets are used to form names of propositions and concepts from sentences and phrases respectively. While such a notation has its problems, we cannot go into them here.
Bolzano Wissenschaftslehre, referred to throughout as WL’, ¡ì148. Reference is made by section number to enable any of the various editions and translations to be used. 17For part of this revisionist picture, see Chapter 15 below
See the note by M. White to Tarski 1987.
have heard it said that Tarski, Carnap, and Neurath disputed the question in Vienna in the early ‘thirties, but I have been unable to find a reference to such discussions. 21Tarski 1986. The same idea is found (independently) in Mautner 1946.
A permutation may be looked on as a bijective function of the domain onto itself, or expressed as a one-one relation on the domain. Tarski used the latter conception in LSM, we shall use the former. Theoretically, they are equivalent, but the functional notation is perhaps easier to calculate with.
See LSM 156 n., 189, 214, 385.
These were first considered all together by Schröder.
LSM, 387. Cf. also Westerstdhl 1985, 395.
Tarski 1986, 152.
/bid. Of course it is also possible to give a second-order set theory. The sets themselves remain individuals, so in this sense the method is still “first-order”.
Tarski’s dissertation under Legniewski (the only one the latter supervised ¡ª good quality control) is reprinted in LSM 1–23.
Cf. the remarks on their relationship in Wolenski & Simons 1989.
See Section 7 of the long truth paper, which Tarski added for the 1935 German translation.
n how to bridge the gap between these different languages, see Chapter 11. 32 LSM,384-392.
When dealing with extensional concepts, `class’ is perhaps more appropriate than `property’, since the latter is normally construed intensionally. But `class’ is reminiscent of `set’, and hence of abstract individuals ¨¤ la ZF. Neither term is optimal.
LSM 384. Tarski and Lindenbaum admit that a range of systems may count as `logic’; theirs is chosen to be representative.
See Hodges 1986, Woledski & Simons 1989.
Suggested at a lecture given at the University of Geneva in June 1987.
The category of sentence-forming functors taking expressions of category X as inputs. For the notation, see below.
Mostowski 1957. Cf. Barwise & Cooper 1981, Westerstahl 1985. 44But in agreement (for once) with Wittgenstein.
Ways of meaning are the basis of the semantics for Le¡ìniewski’s Ontology in Chapter 12. No claim is made that this concept coincides with the medieval one of this name, but the medieval notions of modus and functio are worth considering as possible starting points for modern semantic theories
As a view about Frege’s philosophy, this is controversial. It is upheld by Baker & Hacker 1984, and I think they are (for once) right. No other view properly explains Frege’s principle that sense determines reference, much as one may wish Frege had thought otherwise.
Cf. note 28. Note that the sense of `definition’ relevant here is not one to do with human linguistic conventions, but concerns the coextensionality or cointensionality of a simple concept with a complex one.
These are the true “logically proper names”. Their sense or intension consists solely in their designating the individual they do. Such atomic intensions form the basis of a possible identity criterion for intensions. Cf. Morscher 1981, 111.
See Frege’s essay “The Thought” (Der Gedanke) Frege 1967, 342–61; Frege 1984, 35272.
Cf. Morscher 1981, 125. Berg (Berg 198) disagrees with Morscher. The case of sets is admittedly the less clear of the two, but I think Morscher is essentially right.
WL, ¡ì101. It is not assumed of course that any human being is ever in mental touch with more than a few of these concepts.
WL, ¡ì148, note 1. This example is probably not representative of Bolzano’s considered view on the identity of propositions using converse relations. He does however consider active/passive sentence¡ªpairs to express the same proposition. Here we see that his work lacks explicit principles for deciding when two linguistic sentences express the same proposition.
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Simons, P. (1992). Bolzano, Tarski, and the Limits of Logic. In: Philosophy and Logic in Central Europe from Bolzano to Tarski. Nijhoff International Philosophy Series, vol 45. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8094-6_2
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