Tractatus 6 Reconsidered: An Algorithmic Alternative to Wittgenstein's Trade-Off

Abstract

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.

Keywords:

• Ludwig Wittgenstein
• Tractatus Logico-Philosophicus
• truth functions
• series of forms
• N-operation
• N-operator

Notes

1 In contemporary logic, this functor (NOR) is called the Peirce's arrow, while the term ‘Sheffer stroke’ is reserved for the NAND operator. When referring to Wittgenstein's N-operation, we will use the former. For historical reasons, the term ‘Sheffer stroke’ can be applied to both.

2 All references to Wittgenstein's Tractatus Logico-Philosophicus will be given using the numbering of its theses or, in the case of references to the ‘Author's Preface ’, a page number. References are to the translation by D. F. Pears and B. F. McGuinness (Wittgenstein 1965).

3 It is worth noting that Tractatus does not dismiss the existence of such abstract entities as redness, loudness and the horse (as a species), though it does not explicitly acknowledge their existence, either: it remains neutral regarding the ontological status of first-order properties. For discussion of this issue, see, for example: Anscombe 1965, p. 97; Anscombe 1966; Stenius 1960, p. 61n; Copi 1966; Allaire 1966; Wedin 1990.

4 The actual symbol used in 5.5 is ‘(- - - - - T) (ξ,…)’ because Wittgenstein writes ‘ –’ instead of ‘F’ and does not care about the ratio between the numbers of signs in both brackets.

5 Fogelin (1975) noted that Tractarian notation based on the N-operation is insufficient because it does not allow for the construction of such mixed multiply general propositions as $\mathrm{\forall }x\mathrm{\exists y}\mathrm{fxy}$. Fogelin's criticism sparked a debate involving Peter Geach (1981) and Scott Soames (1983) who defended Wittgenstein's account by introducing basically equivalent simple notational devices that allow one to specify the variable ranges. As has been subsequently pointed out by Fogelin (1982), the price to be paid for this solution is the endorsement of the undecidability of some logical propositions because Geach's notational devices violate the Tractarian demands for the finiteness and successiveness of the operation (cf. 5.32). Geach responded in turn that ‘Fogelin still appears …to confuse the performance of one operation on a (possibly) infinite class of operands with the performance of an infinite number of operations’ (Geach 1982). Sundholm summed up the debate by pointing out that ‘both sides of the dispute are right and that the error lies in a fundamental inconsistency embedded in the Tractatus itself’ (Sundholm 1992, p. 64).

6 Note that in Wittgenstein's notation the definiens is to the left of the equal sign.

7 We shall not discuss the exact meaning of this expression, of which Sundholm has already given a detailed exposition (Sundholm 1992).

8 Anscombe makes here a mistake claiming that this formula is equivalent to q when, in fact, the term generated is equivalent to $q\wedge \sim p$. She is probably treating the formula as the inference scheme ‘if $p\vee q$ holds true, and if $\sim p$ holds true, then q holds true ’, and not as a formula of propositional logic. However, the error does not undermine her method.

9 It should be mentioned here that the first interpreter ready to accept this price was Bertrand Russell, who explained in his Introduction to the Tractatus (Russell 1922, p. xv) that the symbol in thesis 6 ‘means whatever can be obtained by taking any selection of atomic propositions, negating them all, then taking any selection of the set of propositions now obtained, together with any originals – and so on indefinitely’. Symptomatically, although he was generally dissatisfied with Russell's text, Wittgenstein did not oppose this explanation.

10 Assuming that there is a countable number of predicates and a countable number of atomic propositions, it is possible to sequentially generate all possible first-order logic formulas. The idea is very simple: we divide all well-formed formulas into subsets based on the length of the formula, the maximum index of the predicate used, and the maximum index of the atomic proposition used. For each of these subsets, we generate a three-dimensional matrix and exhaust it with a known diagonal traversal algorithm, so that each formula will be generated after a finite number of steps. Although the idea is relatively simple, its implementation is technically quite complicated and should be the subject of a separate article.