Notes to the Tortoise (excerpt)

Harold Brown

[. . . .] And, more significantly, Carroll was not only guilty of a logical contradiction in a specific argument, but he was also guilty of a much more serious existential contradiction for, believing that he had doomed logic, he continued to attempt to contribute to logic so that at his death in 1898, three years after the publication of the dialogue in question, he was still at work on the projected second and third parts of his Symbolic Logic. [5] These reflections suggest that Carroll was not attempting to destroy logic after all, but had an entirely different point in mind. Indeed, Carroll’s point is quite clear in the dialogue itself, for the problem of the infinite progress is not generated by the bare attempt to draw a logical inference, but by the refusal to make use of any rule of inference that has not been explicitly stated.

The way out of the infinite progress, then, would be to reject the legitimacy of the tortoise’s claim that any logical rules that we use might just as well be stated in the argument, Rather, it would seem, in order to actually draw an inference we must make use of rules that are not mentioned and the point of the dialogue becomes one about the way in which logic plays its role in arguments: It functions by remaining as a context or medium within which argument takes place, not as an explicit part of an argument. This does not mean that we never state or analyze laws of logic; we obviously can and do. But it does mean that in order to actually draw an inference we must make use of rules of inference which are not, at that moment, explicitly stated.

But even if this interpretation of Carroll’s intent is correct, many philosophers would still draw the conclusion that scepticism has triumphed, for if every logical inference requires that we make use of rules of inference which are not stated, then we can never be completely certain that our rules of inference are really legitimate. Certainty is only possible here, it will be maintained, after all of our logical rules have been stated and subjected to analysis. This is an argument to which we will return in the final section of this paper, but before we consider it I want to try to show that the tortoise’s discovery is only one specific instance of a more general and not totally unfamiliar phenomenon: the conceptual tools which we use in order to carry out a process of reasoning or of analysis cannot, at the same time, be the objects of that process, just as the physical tools that we use to build or tear down a structure cannot, at the same time, be the objects that are being constructed or destroyed.

II

All contemporary students of philosophy have already encountered this phenomenon in their study of symbolic logic. It is now fashionable to present logic as a formal, uninterpreted calculus and then to offer an interpretation of this calculus as a system of logic. But we do not begin to teach logic by walking silently into the classroom and writing strings of sign-designs on the blackboard. Rather, we begin by talking about what we are going to do and even by explicitly distinguishing between object language and metalanguage, between the language which we are constructing and the language in which we are going to talk about the language that we are constructing. The distinction itself, I take it, is made in a meta-meta-language and we can immediately see the possibility of the same sort of infinite progress that the tortoise talked about beginning again. If we insist on using no language which is not first mentioned we can never begin to speak―at some point a language must be presupposed in order to get discourse started. And not only must we presuppose some language in which to discuss logic, but we must also obey the laws of logic in that language; we must not, for example, make contradictory claims about our object language. We even go so far, in modern logic, as to prove theorems about our calculus and the proof of these theorems presupposes a logic, usually the very same logic that we are attempting to embody in our calculus. Thus we find that before we can state any laws of logic at all, we must first accept both a logic and a language which have never been subjected to analysis.

It is worth noting here that these conclusions are completely parallel to the result of Gödel’s demonstration that it is impossible to prove the absolute consistency of arithmetic (as well as of most formal systems). The difficulty in producing such a proof arises from the need to carry out the proof of the consistency of one system within the context of another, more powerful, system whose own consistency has not been proven. Gödel originally set out to get around this difficulty by means of Gödel numbering, i.e., by developing a means of mapping the meta-language into the object language. The results of this attempt, the theorems on completeness and consistency, are well known, and it might not be excessively far fetched to suggest that Carroll’s 1895 paper can be viewed as an informal precursor of Gödel’s theorems.

The same phenomenon pervades all of meta-philosophy. Meta-philosophy is the discipline which chooses to talk about talk, but uses a language in order to do this. Thus the meta-philosopher must remember, whenever he points out an ambiguity or unclarity in someone else’s talk, that he is presupposing the consistency and clarity of his own talk. It is always possible that meta-meta-analysis will reveal that there was, for example, no ambiguity in the talk in question, but only an appearance of an ambiguity generated by an ambiguity or lack of clarity in the meta-language. Any philosopher who seriously asks, “What do you mean by such and such a term” both assumes that he and the person he is questioning can use some language meaningfully and at the same time opens himself up to the legitimate retort, “What do you mean by ‘mean?’” Once again, many philosophers might view this argument as an attack on the legitimacy of all analysis, but I do not think that this conclusion necessarily follows from the argument, although I will reserve my reasons for making this claim until the final section of this paper. For the moment we are merely noting a fact about the way in which language carries out its function that is analogous to the tortoise’s point about the way in which logic carries out its function. For just as logic is the medium in which all valid inferring is done, so language is the medium in which all talking is done, Even when we are talking about language we do so in a language and no matter how many facts about the structure and function of language we make explicit, we still do so within a language―and the language with which we make things explicit cannot itself simultaneously be the explicit object of that language.

SOURCE: Brown, Harold [I.] “Notes to the Tortoise,” The Personalist, vol. 53, 1972, pp. 104-109; this excerpt, pp. 105-106.

Note: This excerpt is preceded by an account of Lewis Carroll’s What the Tortoise Said to Achilles (1895), and is followed by examples of laboratory apparatus viz. fundamental theory change in physics.  Implications for logic, language, and science include rejection of Hume’s skepticism and recognition that all knowledge endeavors “take place within a medium of a presupposed body of concepts and techniques.”

Dialectic by Harold I. Brown

Harold I. Brown on presuppostions &smp; scientific theory change

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Vienna Circle, Karl Popper, Frankfurt School, Marxism, McCarthyism & American Philosophy:
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“What is the Relationship Between Logic and Reality?”
by R. Dumain

History, Sociology, & Scope of Logic: Select Bibliography

Historical & Philosophical Perspectives on Mathematics & Mathematical Logic:
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American Philosophy Study Guide

Offsite:

Harold I. Brown @ academia.edu