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Wittgenstein's Elimination of Identity for Quantifier-Free Logic

Lampert, Timm and Säbel, Markus (2016) Wittgenstein's Elimination of Identity for Quantifier-Free Logic. [Preprint]

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Abstract

One of the central logical ideas in Wittgenstein's Tractatus logico-philosophicus is the elimination of the identity sign in favor of the so-called "exclusive interpretation" of names and quantifiers requiring different names to refer to different objects and (roughly) different variables to take different values. In this paper, we examine a recent development of these ideas in papers by Kai Wehmeier. We diagnose two main problems of Wehmeier's account, the first concerning the treatment of individual constants, the second concerning so-called "pseudo-propositions" ("Scheinsätze") of classical logic such as a=a or a=b v b=c -> a=c. We argue that overcoming these problems requires two fairly drastic departures from Wehmeier's account: (1) Not every formula of classical first-order logic will be translatable into a single formula of Wittgenstein's exclusive notation. Instead, there will often be a multiplicity of possible translations, revealing the original "inclusive" formulas to be ambiguous. (2) Certain formulas of first-order logic such as a=a will not be translatable into Wittgenstein's notation at all, being thereby revealed as nonsensical pseudo-propositions which should be excluded from a "correct" conceptual notation. We provide translation procedures from inclusive quantifier-free logic into the exclusive notation that take these modifications into account and define a notion of logical equivalence suitable for assessing these translations.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Lampert, Timmlampertt@staff.hu-berlin.de
Säbel, Markusmarkus.saebel@hu-berlin.de
Keywords: Wittgenstein, Logic, Identity, Kai Wehmeier
Subjects: General Issues > Logical Positivism/Logical Empiricism
Specific Sciences > Mathematics
Depositing User: Markus Säbel
Date Deposited: 08 Apr 2017 13:21
Last Modified: 08 Apr 2017 13:21
Item ID: 12974
Subjects: General Issues > Logical Positivism/Logical Empiricism
Specific Sciences > Mathematics
Date: April 2016
URI: https://philsci-archive.pitt.edu/id/eprint/12974

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