Wittgenstein Vs. Frege and Russell

teorema Vol. XL/2, 2021, pp. 45-61 ISNN 0210-1602 [BIBLID 0210-1602 (2021) 40:2; pp. 45-61] Are Rules of Inference Superfluous? Wittgenstein vs. Frege and Russell Gilad Nir RESUMEN En el Tractatus 5.132 Wittgenstein argumenta que la justificación inferencial depen- de solo de la comprensión de las premisas y la conclusión y no está mediada por ningún otro acto adicional. Tomando lo anterior como base, Wittgenstein defiende que las reglas de inferencia de Frege y Russell “carecen de sentido” y son “superfluas”. Esta línea de argumento es problemática, puesto que no está claro que pueda haber una explicación viable de la inferencia de acuerdo con la cual no haya tal mediación. Muestro que el re- chazo por parte de Wittgenstein de las reglas de inferencia puede estar motivado si se tie- ne en cuenta su interpretación holista de la relación entre inferencia y comprensión. PALABRAS CLAVE: Wittgenstein, Frege, Russell, inferencia, holismo. ABSTRACT In Tractatus 5.132 Wittgenstein argues that inferential justification depends solely on the understanding of the premises and conclusion, and is not mediated by any further act. On this basis he argues that Frege’s and Russell’s rules of inference are “senseless” and “superfluous”. This line of argument is puzzling, since it is unclear that there could be any viable account of inference according to which no such mediation takes place. I show that Wittgenstein’s rejection of rules of inference can be motivated by taking account of his holistic construal of the relation between inference and understanding. KEYWORDS: Wittgenstein, Frege, Russell, Inference, Holism. In paragraph 5.132 of the Tractatus Logico-Philosophicus Wittgenstein argues that inferential justification depends solely on the understanding of the premises and conclusions, and is not mediated by any further act or state [Wittgenstein (1960), henceforth TLP]. On this basis he goes on to reject what he calls Frege’s and Russell’s “laws of inference” (Schlussgesetze), a term which, as I will argue below, refers to the rules of inference of their systems. These, he says, “are senseless and would be superfluous”. This line of argument is puzzling, however, for it is far from clear that there 45 46 Gilad Nir could be any viable account of the justificatory nature of inference in which no such mediation takes place. What seems to make inference into a form of justification (by contrast to the mere association of thoughts) is that we draw the conclusion in light of our recognition of the goodness of the inferential connection. Indeed, Frege acknowledges this in saying that to infer is “to make a judgment because we are cognisant of other truths as providing a justification for it” [Frege (1979), p. 3] and Russell acknowledges this in saying that in inferring, the applicability of the rule “must be simply perceived.” [Russell (1903), p. 41].1 Both Frege and Russell construe the logical mediation of premises and conclusion in terms of the application of rules of inference. Further- more, in spelling out the role of such rules Frege and Russell are careful to distinguish it from the contribution made by adding a further premise to the inference. Frege says that rules such as modus ponens cannot be ex- pressed in his logical system “because they form its basis” [Frege (1972), p. 136], and Russell says that the rule of inference “eludes formal statement” [Russell (1903), p. 34]. They thus seem to treat the rules of inference in terms of what we might nowadays call the meta-language of their systems; and the advantage in doing so is that it allows them to avoid the regress which, as pointed out by Lewis Carroll, threatens the idea that inferential justification depends on the appreciation of validity [Carroll (1895)]. To simplify somewhat, Carroll shows that if the appreciation of validity had to be codified as a premise that is added to the inference – if the inference would not be fully justified otherwise – the form of the original inference would thereby be altered, and a further demand for justification would then arise with respect to the new inferential sequence. Moreover, if the appreciation of the validity of the new inference had to be codified as a premise as well, a further, as yet unjustified inferential sequence would emerge, and there would be no stopping this regress. But since rules of inference in Fre- ge’s and Russell’s accounts capture the appreciation of validity without adding such idle premises, they seem to account for inferential justification without giving rise to the regress. Indeed this is why Frege’s and Russell’s inexpressible rules of inference do not seem to be senseless and superfluous. What, then, is the point of Wittgenstein’s critique? As I understand him, and contrary to the existing literature on the topic, Wittgenstein does not mean to reiterate Carroll’s objection, but he does mean to criti- cize Frege’s and Russell’s appeal to rules of inference, for different reasons. Wittgenstein rejects the idea that the goodness of an inference is some content which is to be cognized or a fact which must be perceived. As he sees it, Frege and Russell take the rules of inference to justify in- teorema XL/2, 2021, pp. 45-61 Are Rules of Inference Superfluous? Wittgenstein vs. Frege and Russell 47 ferences by bringing such substantive content to bear on the premises and conclusion — content that would not otherwise be there. But Witt- genstein rejects the idea that logic is a source of substantive content, let alone inexpressible content; moreover he rejects the idea that the com- ponents of inference – the premises and conclusion – are discretely indi- viduated atoms of thought, which only form logical connections with one another when some such additional content is appealed to. Instead, he locates inference within the holistic context of meaning, understanding and reasoning: the appreciation of logical relations is internal to the signifi- cant use of signs, and does not need to be introduced by means of a sepa- rate act, be it the act of adding a premise or the act of applying a rule. The structure of the paper is as follows. I begin by clearing up the target of Wittgenstein’s critique, showing that it is the rules of inference of Frege’s and Russell’s logical systems that TLP 5.132 rejects. In Section II I go on to explain what precisely Wittgenstein finds to be problematic in Frege’s and Russell’s articulations of the role of rules of inference. In Section III I discuss Wittgenstein’s approach to reasoning, and the holistic context in which he locates inference, understanding and meaning. In Section IV I consider how Wittgenstein’s rejection of rules of inference reflects his general understanding of the relation between logic and thought, and in particular how it relates to his claim that inferential relations show themselves in propositions, but cannot be said by them. I argue that this claim does not commit Wittgenstein to the idea of inexpressible content which he criticizes Frege and Russell for endorsing. I. WHAT ARE SCHLUSSGESETZE? To begin with, let us consider the precise shape taken by Wittgen- stein’s critique of Frege’s and Russell’s conceptions of inference: If p follows from q, I can infer from q to p; derive p from q. The mode of inference is to be understood from the two propositions alone. Only they themselves can justify the inference. ‘Laws of inference,’ [‘Schlußgesetze’] which—as in the works of Frege and Russell—are supposed to justify inferences, are senseless, and would be superfluous. [TLP 5.132, translation emended]. Wittgenstein starts from the premise that all we need in order to justify an inference is an understanding of the premises and conclusion, and proceeds to the claim that the appeal to “laws of inference” would be teorema XL/2, 2021, pp. 45-61 48 Gilad Nir “senseless” and “superfluous.” But the term “Laws of inference” (Schlußge- setze) is not used as a technical term by either Frege or Russell. It might be taken to refer either to what Frege and Russell call the “basic laws” and “principles” (i.e. the axioms) of their systems, or to what they call the “rules” that govern proofs conducted in those systems. Russell, in particular, often fails to maintain a clear terminological distinction with respect to these two notions, and Frege is not always careful in this regard either. One reason to read ‘Schlußgesetze’ as specifically denoting rules, however, is that the second sentence of 5.132 employs a distinctively Fregean term – ‘mode of inference’ (Art des Schlusses) – which Frege uses in Be- griffsschrift to refer to the single rule of inference of that system, modus ponens. Indeed, a common reading of 5.132, which I will henceforth refer to as the standard interpretation, takes Frege’s and Russell’s rules of inference to be Wittgenstein’s target. Many standard interpreters of 5.132 take Wittgenstein’s reasons for objecting to their rules of inference to co- incide with the worry raised by Lewis Carroll [e.g. Mounce (1981), p. 44f., Kenny (2006), p. 78, Baker (1988), p. 130f.] But it is, as I pointed out, far from clear that Frege’s and Russell’s construals of inference are vulnerable to Carroll’s regress. Since Frege and Russell deny that the rules can be encoded as propositions of the language that they govern, appealing to such rules does not result in adding premises to the inference, and so it does not lead to the regress.3 These standard interpreters thus have Wittgenstein attacking a straw man.

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