Wittgenstein's Tractarian Apprenticeship

Landini, Gregory

doi:10.15173/russell.v23i2.2045

Cited by 8 publications

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“…This is born out in Section 5, where we provide tableau calculi for Tractarian logic that employ a remarkably small number of simple inference rules. See Soames (1983), Sundholm (1992), Varga von Kibéd (1993), Miller (1995), Cheung (2000), Jacquette (2001), Floyd (2002), McGray (2006), and Landini (2007). thus is licensed by 3.315 and 5.501. In languages with infinitely many predicates, this exceeds the expressive resources of first-order logic; however, in the absence of any comprehension axioms whatsoever, this is still weaker than even predicative second-order logic.…”

Section: The Fogelin-geach Debatementioning

confidence: 99%

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

Rogers

Wehmeier

2012

The Review of Symbolic Logic

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AbstractIn theTractatus, 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 theTractatus, 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 theTractatuswhen 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.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.5.503

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Section: The Fogelin-geach Debatementioning

confidence: 99%

“…A significant literature has emerged in response to this dispute, nearly uniformly in opposition to Fogelin's contention. See Soames (1983), Sundholm (1992), Varga von Kibéd (1993), Miller (1995), Cheung (2000), Jacquette (2001), Floyd (2002), McGray (2006), and Landini (2007).…”

Section: The Fogelin-geach Debatementioning

confidence: 99%

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

Rogers

Wehmeier

2012

The Review of Symbolic Logic

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“…Shosky finds the latter "evidently familiar" to Whitehead and Russell "as early as 1910, if not before" [129, p. 18] [87, p. 162] as well as [129] and [8]. Landini seems unaware that [87] was updated to McGuinness [86] two years before Landini's [77] appeared. But it is not important to us here who was the first to use truth table technique, use a truth table, publish a truth table or truth table technique, or even see that truth tables explain extensional logical relevance ("following from") as truth-ground containment.…”

Section: Diagrams Of Valid Arguments Show the Premisses Contain The Cmentioning

confidence: 99%

The Concept of Relevance and the Logic Diagram Tradition

Dejnožka

2010

Log. Univers.

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What is logical relevance? Anderson and Belnap say that the "modern classical tradition [,] stemming from Frege and WhiteheadRussell, gave no consideration whatsoever to the classical notion of relevance." But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson-Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein's Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains.

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“…In the meantime, Hintikka (1956) seems to have been the only serious attempt to spell out the proposal. After Hintikka's canonical paper there has been little serious discussion of the exclusive interpretation of quantifiers until the issue was taken up in Floyd (2001), Wehmeier (2004) and Landini (2007). §3 Wehmeier's Account of the Elimination of Identity Since Wittgenstein gives no explicit set of rules for handling his exclusive quantifiers and names, there is some leeway in interpreting the intended convention.…”

Section: §1 Introductionmentioning

confidence: 99%

Wittgenstein’s Elimination of Identity for Quantifier-Free Logic

Lampert¹,

Säbel²

2020

The Review of Symbolic Logic

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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 $a=b \wedge b=c \rightarrow a=c$ . 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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Wittgenstein's Tractarian Apprenticeship

Cited by 8 publications

References 0 publications

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

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

The Concept of Relevance and the Logic Diagram Tradition

Wittgenstein’s Elimination of Identity for Quantifier-Free Logic

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