Wittgenstein’s N-operator is a ‘primitive sign’ which shows every complex proposition is the result of the truth-functional combination of a finite number of component propositions, and thus provid...

Wittgenstein's conception of the general form of a truth function given in thesis 6 can be presented as a sort of a trade-off: the author of the Tractatus is unable to reconcile the simplicity of his original idea of a series of forms with the simplicity of his generalisation of Sheffer's stroke; therefore, he is forced to sacrifice one of them. As we argue in this paper, the choice he makes – to weaken the logical constraints put on the concept (...) of a series of forms, thus effectively metaphorising that concept for the sake of upholding the N-operation's role of generating the series – is unfortunate. An actual expansion of a series of truth functions as defined in 6 would require either making decisions at each step (Anscombe) or outwardly rejecting the concept of a series (Sundholm). However, neither is faithful to Wittgenstein's own fundamental intuitions regarding the nature of logic. For this reason, a different trade-off that prioritises upholding the basic features of a series of forms over the simplicity of the operation that generates that series seems to be much more reasonable. We offer such a trade-off by developing the schema already present in the Tractatus (5.101). The key element of our alternative solution is the construction of an operation that can perform the task of producing all consecutive truth functions of a given collection of atomic propositions as an invariant difference between any base and its successor throughout the series. We define it as a composition of three functions (in an algebraic sense): the first turns a symbol of a given truth function into a binary number, the second increments that number, the third turns the result back into a symbol of another truth function. We also show how the operation can be defined by means of a different notation without invoking binary arithmetic. (shrink)

Wittgenstein's Tractatus carefully distinguished the concept all from\nthe notion of a truth-function, and thereby from the quantifiers.\nI argue that Wittgenstein's rationale for this distinction is lost\nunless propositional functions are understood within the context\nof his picture theory of the proposition. Using a model Tractatus\nlanguage, I show how there are two distinct forms of generality implicit\nin quantified Tractatus propositions. Although the explanation given\nin the Tractatus for this distinction is ultimately flawed, the distinction\nitself is a genuine one, and the forms of generality that (...) Wittgenstein\nindicated can be seen in the quantified sentences of contemporary\nlogic. (shrink)

In his Tractatus, Wittgenstein maintained that arithmetic consists of equations arrived at by the practice of calculating outcomes of operations$\Omega ^{n}(\bar {\xi })$defined with the help of numeral exponents. Since$Num$(x) and quantification over numbers seem ill-formed, Ramsey wrote that the approach is faced with “insuperable difficulties.” This paper takes Wittgenstein to have assumed that his audience would have an understanding of the implicit general rules governing his operations. By employing the Tractarian logicist interpretation that theN-operator$N(\bar {\xi })$and recursively defined arithmetic (...) operators$\Omega ^{n}(\bar {\xi })$are not different in kind, we can address Ramsey’s problem. Moreover, we can take important steps toward better understanding how Wittgenstein might have imagined emulating proof by mathematical induction. (shrink)

Wittgenstein's "Tractatus" claims the possibility to represent quantificational logic by means of a single basic operation. Fogelin regards the "Tractatus" system as seriously flawed and thus not suited for this aim, while Geach and Soames propose extensions and corrections of the Tractarian notational system. If the unusual Tractarian view -- every variable can be conceived as a propositional variable -- is taken full account of, we find out, that the "Tractatus" system is well suited to deal even with the problematic (...) cases of mixed quantifiers with many-place predicates. By expanding the view of variables as certain two-dimensional lists of propositions we find adequate representations of formulas like fxy in the logic of the "Tractatus". Surprisingly we don't need anything which corresponds to formulas with two or more free variables like fxy, and still get the full power of ordinary quantificational logic. (shrink)

The power of Wittgenstein's N operator described in the Tractatus is that every proposition which can be expressed in the Russellian variant of the predicate calculus familiar to him has an equivalent proposition in an extended variant of his N operator notation. This remains true if the bound variables are understood in the usual inclusive sense or in Wittgenstein's restrictive exclusive sense. The problematic limit of Wittgenstein's N operator comes from his claim that symbols alone reveal the logical status of (...) propositions. This would require knowledge of the size of the Tractarian domain of unchangeable, simple objects, and this, Wittgenstein maintained, cannot be known. (shrink)

The present text discusses whether there is a tension between aphorisms 6.1-6.13 of the Tractatus and the Church-Turing theorem about the decidability of predicate logic. We attempt to establish the following points: (i) Aphorisms 6.1-6.13 are not consistent with the Church-Turing theorem. (ii) The logical symbolism of the Tractatus, built from the N-operator, can (and should) be interpreted as expressively complete with respect to first-order formulas. (iii) Wittgenstein’s reasons for believing that Logic is decidable were purely philosophical and the undecidability (...) result shows that there are aspects of his criticisms of Frege and Russell that become unjustified in light of these results. (shrink)

In this paper, I provide a new reading of Wittgenstein’s N operator, and of its significance within his early logical philosophy. I thereby aim to resolve a longstanding scholarly controversy concerning the expressive completeness of N. Within the debate between Fogelin and Geach in particular, an apparent dilemma emerged to the effect that we must either concede Fogelin’s claim that N is expressively incomplete, or reject certain fundamental tenets within Wittgenstein’s logical philosophy. Despite their various points of disagreement, however, Fogelin (...) and Geach nevertheless share several common and problematic assumptions regarding Wittgenstein’s logical philosophy, and it is these mistaken assumptions which are the source of the dilemma. Once we recognize and correct these, and other, associated expository errors, it will become clear how to reconcile the expressive completeness of Wittgenstein’s N operator, with several commonly recognized features of, and fundamental theses within, the Tractarian logical system. (shrink)

The purpose of this paper is threefold. First, I visit the Fogelin–Geach-dispute, criticizeMiller''s interpretation of the Geachian notationN(x:N(fx)) and conclude that Fogelin''s argumentagainst the expressive completeness of the Tractariansystem of logic is unacceptable and that the adoptionof the Geachian notation N(x:fx) would not violate TLP5.32. Second, I prove that a system of quantificationtheory with finite domains and with N as the solefundamental operation is expressively complete. Lastly, I argue that the Tractarian system is apredicate-eliminated many-sorted theory (withoutidentity) with finite domains (...) and with N as the solefundamental operation, and thus is expressivelycomplete. (shrink)