Tractarian First-Order Logic: Identity and the N-Operator

Abstract

<jats:title>Abstract</jats:title><jats:p>In the<jats:italic>Tractatus</jats:italic>, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of the<jats:italic>Tractatus</jats:italic>, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in the<jats:italic>Tractatus</jats:italic>when taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.</jats:p><jats:p><jats:disp-quote><jats:p>As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.</jats:p><jats:attrib>5.503</jats:attrib></jats:disp-quote></jats:p>

On Wittgenstein's transcendental deductions

“…The lacuna has to do with the fact that, in the Tractatus, Wittgenstein speaks of 'infinity' as if it were one number and neither identifies nor discusses any additional, transfinite numbers. Yet, since selections of elementary propositions stand in one-to-many relations both to their truth-possibilities, and truth-functions, then if there exist infinite totalities of elementary propositions (as was presumed above in our explication 7 Geach (1981Geach ( , 1982, as well as Rogers and Wehmeier (2012), have proposed notational alternatives designed to circumvent these and other issues identified by Fogelin with regards to the expressive completeness of Wittgenstein's N operator. For a more detailed critical consideration of these alternatives, see Connelly (2017).…”

Wittgenstein's Philosophy of Mathematics

“…This makes it difficult to translate his notation into ours. After a decades-long dispute, Rogers and Wehmeier (2012) settled that Operator N can express first-order logic with identity. However, Wittgenstein's logic is more powerful than this.…”

Incompatibility, inconsistency, and logical analysis in Tractatus Logico-Philosophicus

“…For a contemporary account, see Frascolla (2007). Rogers and Wehmeier (2012) show that the primary notational constraints laid down in Tractatus -that compound formulas involve only the N-operator, and that identity be expressed by sameness of name rather than via an equation -are jointly satisfiable by a formal language equal in expressive power to first-order logic. Miller (1995) supplies a Tractarian situation semantics with a truth definition materially equivalent to Tarski-style truth definitions.…”