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Published online by Cambridge University Press: 05 May 2010
The Tractatus first appeared in 1921, the same year that Post's “Introduction to a General Theory of Elementary Propositions” appeared in the American Journal of Mathematics. As the latter is the first piece clearly to present and exploit the distinction between a deductive system and a truth-functional interpretation of such a system, we may conclude that Wittgenstein's views had been arrived at somewhat before a variety of logical concepts had received the clarification and refinement incipient on the now taken-for-granted distinction between proof and model theory. One such concept, of considerable interest to Wittgenstein, was that of inference. The following constitutes an attempt to explicate his notion. In particular, I shall attempt to show that his repudiation of “laws of inference” is closely tied to his rejection of logical constants; and that both can be seen as the product of what might be termed a “metaphysics of completeness”—before, of course, any (presently recognizable) notion of completeness had achieved a measure of precision or currency.
1 Vol. 43, 163–185.
2 Ludwig Wittgenstein und der Wiener Kreis: Gesprache, aufgezeichnet von Friedrich Waismann, ed. McGuinnes, B. F. (Oxford: Basil Blackwell, 1967), 68–69Google Scholar (“Zu Heidegger”).
3 “What the Tortoise Said to Achilles”, Mind 4 (NS) (1895), 278–280.Google Scholar
4 “The Runabout Inference-Ticket”, Analysis 21 (1960), 38–39.CrossRefGoogle Scholar
5 Prior himself does not discuss justification per se. He speaks of inferring a Statement “in an analytically valid way” from another statement. While I don't wish to suggest that “analytic validity” (whatever it is) is the same thing as justification, I think it is nonetheless clear that Prior's discussion would have no force if the “analytic validity” of an inference did not justify it.
6 Stevenson, J. T., “Roundabout the Runabout Inference-Ticket”, Analysis 21 (1961), 124–128CrossRefGoogle Scholar. Belnap, N. D., “Tonk, Plonk and Plink”, Analysis 22 (1962), 130–134.CrossRefGoogle Scholar
7 Belnap refers his development of the notion to Post.
8 Ibid., 132.
9 Wittgenstein (Harmondsworth: Penguin, 1973), 46.Google Scholar
10 This claim is never stated explicitly. The two explicit claims are that internal relations cannot be represented (4.122), and that no proposition can represent general logical form (4.121). But note that prima facie there would be no problem in representing internal relations unless their representation (tantamount to the representation of the proposition's specific logical form) required the representation of general logical form. For although on Wittgenstein's theory of representation, no proposition could represent its own logical form, presumably some other proposition, with a different logical form, could. It would only be impossible for any proposition to represent a given set of internal relations if the representation of those internal relations presupposed the representation of the one thing no proposition could represent, viz., general logical form.
11 The most concise version of this problem with which I am familiar is one proposed by Hans Herzberger in a set of seminar notes on Kripke languages, distributed at the University of Toronto in March 1979. English is there demonstrated to have Carnap closure, where the latter is defined as follows: A language, L, is Carnap-closed iff for each class G which is definable in L there is a sentence k which affirms its own membership in that class, so that k is true iff k ε G. It follows that no Carnap-closed language can define its own class of non-true sentences; and hence, since English is Carnap-closed, English is incapable of expressing its own concept of non-truth. Yet clearly, even as speakers of no other language, at an intuitive level we have a concept of non-truth for English. That is, we appear to possess an inexpressable concept. For a fuller appreciation of the problems involved, and the difficulties in producing coherent resolutions, see Herzberger, H. G., “Truth and Modality in Semantically Closed Languages”, in Martin, R. L., ed., The Paradox of the Liar (Yale, 1970)Google Scholar; “new Paradoxes for Old” (in manuscript); Smullyan, R. M., Theory of Formal Systems (Princeton, 1961).Google Scholar
12 It would also seem, given Wittgenstein's view of the way negation operates, that the negative claim is itself nonsense as it presupposes the positive claim. Presumably, because couched in the negative, it is somehow less psychologically pernicious. But it is undoubtedly a major rung in the ladder we must eventually throw away. For a less breezy trip through the intricacies of the semantic paradoxes and related problems of expressibility, see, for example, Thomson, James, “On Some Paradoxes”, in Butler, R. J., ed., Analytical Philosophy (Oxford: Basil Blackwell, 1962), 104–119Google Scholar; Herzberger, Hans G., “Paradoxes of Grounding in Semantics”, Journal of Philosophy 67 (1970), 145–167CrossRefGoogle Scholar; Parsons, Charles, “The Liar Paradox”, Journal of Philosophical Logic 3 (1974), 381–412.CrossRefGoogle Scholar
13 See Notebooks, 1.6.15; 14.6.15; 17.6.15; 27.4.16; and the concluding section of the present paper.
14 For a discussion of this revision and some of its consequences, the reader is referred to Wittgenstein's disowned paper, “Some Remarks on Logical Form”. Aristotelian Society Supplementary Volume 9, Knowledge, Experience and Realism (1929), 162–171Google Scholar. See also Kenny's chap. 6, “The Dismantling of Logical Atomism”.
15 Due to Ian Hacking.
16 If elementary propositions have internal relations to one another then any case for disjoint applications of the terms “logical form” and “internal relation” fails. Granted. this as it stands does not prove they are related in the way proposed—but it deals with the strong form of the objection, viz. that logical form cannot have anything to do with inference.
17 “Two Dogmas of Empiricism”, in From a Logical Point of View (New York: Harper & Row, 1963), §6, 43.Google Scholar
18 Ibid., 42.
19 This particular way of setting the problem was proposed by J. V. Canfield.
20 As long as we demand that F and G be simple predicates. Were they not, F might be something like “is not both red all over and non-red all over at the same Time”—in which case the truth of Fa would be a logical matter.
21 Our Quinean, of course, employs modus tollens. He reasons as follows: “If Fa. ~Ga, modus tollens, and (x)(Fx ⊃ Gx) then a contradiction results. It must not be the case that a contradiction results, so it must not be the case that (all of) Fa, ~Ga, modus tollens, and (x)(Fx ⊃ Gx).” In choosing to reject modus tollens in such a circumstance, he might argue that rules of inference are different for meta-and object-languages. This does have an air of the ad hoc, however.
22 How do we know it is the real one? I don't know. Fortunately, epistemological matters are not prima facie of immediate concern. It will suffice that we allow the possibility that we could discover “the real” one.
23 An accidental counter-instance might be something of the form “We all call this an x, but John calls it a y”; but clearly this can't be what Canfield has in mind. Thus, “non-accidental” counter-instances: which, I suppose, would have to look something like “We all call this an x, but we've all got it wrong; ‘x’ doesn't pick out x”. (?)
24 Interestingly, it is not so clear what one must do over questions about the nature of inference. If we allow that certain elementary propositions do entail others, does this affect the way in which logical form determines internal relations? On the surface, it would appear to muddy up the “formal” part of the concept of logical form—a mixing of semantics with syntax which the young Wittgenstein would have found anathema. However, does muddying the concept of logical form prevent it from playing essentially the same rôle in a somewhat messier system? Again on the surface, it would appear not.
25 The diagrammatic illustration should not be read as entailing anything incompatible with later arguments denying the possibility of a private language. To some extent, Wittgenstein's solipsism is gratuitous with respect to the language-world scheme. It is not gratuitous (but by the same token, not uniquely indicated) to the extent that language was not to be thought of as a metaphysically distinct entity, outside our (one's) control.
26 Given that a number of intuitions are wavering on the relative merits of classical as opposed to intuitionist, relevance, or quantum logics, and also on the viability of standard two-valued interpretations, one might describe the current situation from the perspective of the Tractatus as follows: Logic proper is whatever underlies and is common to all these logics (and this is not the empty set). Our unwavering commitment to these most fundamental structures of thought is explained by noting that these structures are embedded in reality, and hence not subject to our control. Depending on one's choice of epistemology, and one's view of its relation to metaphysics, one can accommodate a number of compelling and often very subtle absolutist positions from the history of philosophy, psychology, and linguistics.
27 And, by the same token, a rejection of any similar commitments to a specific “type” of logic. On this view, debates about the “true logic” are baseless.