Abstract
We investigate incomplete symbols, i.e. definite descriptions with scope-operators. Russell famously introduced definite descriptions by contextual definitions; in this article definite descriptions are introduced by rules in a specific calculus that is very well suited for proof-theoretic investigations. That is to say, the phrase ‘incomplete symbols’ is formally interpreted as to the existence of an elimination procedure. The last section offers semantical tools for interpreting the phrase ‘no meaning in isolation’ in a formal way.
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Notes
Especially On Denoting (1905), OD for short.
That so-and-so can be a complex expression seems plausible from Russells considerations in OD, p.479: “[...] ... a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the present King of France, the centre of mass of the solar system at the first instance of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form.
As it will be seen later ‘incomplete symbol’ will be interpreted syntactically and ‘no meaning in isolation’ semantically in our approach.
In [24] speaking of a secondary respectively primary occurrence of a definite description.
(10) and (11) are equivalent under the condition that ιxA(x) exists.
The review in question is in [26].
Definitions 3, 4, and 5 are all taken from [3], p.1ff - with minor modifications.
cf. [3], p.2f.
The proofs of both lemmata are as in Buchholz (2002/03).
More detailed proofs of the cut-lemma, and Cut-elimination are stated below.
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Acknowledgments
The author is indebted to J. Czermak, G. Dorn, B. Fitelson, H. Leitgeb, O. Hjortland, P. Oppenheim, G. Sauermoser, and an anonymous referee.
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Gratzl, N. Incomplete Symbols — Definite Descriptions Revisited. J Philos Logic 44, 489–506 (2015). https://doi.org/10.1007/s10992-014-9339-1
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DOI: https://doi.org/10.1007/s10992-014-9339-1