Perhaps because Wittgenstein himself in various places renounced the
Tractatus Logico-Philosophicus, his interpreters seem to feel a need to locate the proof which could definitively dissuade a Tractarian believer. I am suspicious of all such proofs by whomever they may be proffered. Anscombe, for instance, claims that the Tractarian claim that the truths of logic are tautological falls to Church's proof "that multiple quantification theory has no decision procedure; that is, that there cannot be a method by which one could settle, concerning any well-formed formula of that theory, whether it was a theorem or not." But a proof could easily be given in turn that the propositions of the Tractatus itself have no "decision procedure," since they all present themselves under the force of the final retraction which denies them the status of propositions. In order for a proof to demonstrate the
falsehood of any claim in the Tractatus, it would first have to regard the claim as something which might be either true or false, and in this way it would fail to receive their meaning and wind up having no relevance to the Tractatus whatsoever.
Here is a typical statement (from Culture and Value) of Wittgenstein renouncing the Tractatus:
I might say: if the place I want to get to could only be reached by way of a ladder, I would give up trying to get there. For the place I really have to get to is a place I must already be at now.
Anything that I might reach by climbing a ladder does not interest me.
What Wittgenstein renounces in the Tractatus is its own manner of renunciation--it "throws away the ladder" only after having used it to get somewhere. "It is a great temptation," says another remark in Culture and Value, "to try to make the spirit explicit." If the temptation
tries to mitigate itself by intending subsequently to put this explicitation under erasure, that only makes the temptation all the more insidious, because it betrays the spirit while pretending to piety. It is Wittgenstein's succumbing to this temptation in the Tractatus which calls for its renunciation.
In regard to propositional logic, the tautological view holds--if it's tautological (valid, really, rather than true), the system can prove it. So if Witt. meant that he was correct (I believe).
ReplyDeleteIn regard to first order logic as a whole, it's bit more difficult--. First order logic is complete (any valid deduction can be shown to be valid...), yet there may be undecidable arguments (ie, if it's not valid, the system may not show it). There may be invalid arguments which the proof method (usually a reductio ad absurdum of some type) does not prove: that's Church/Turing's Halting Problem (and really, Goedelian incompleteness, eventually, in regards to natural numbers--disproving Frege/Russell, along with the purely tautological view). In that sense, Abscombe was correct.
I consider Wittgenstein somewhat of a pragmatist, however, and his point was that the methods of formal logic were limited. "Merely tautologies" is what he wanted to suggest, even if the great mirror of logic had a certain formal ...oh je ne sais quoi. With Phil. Investigations, St. Ludwig went native, he's discussing ordinary language, family resemblance, meaning as use (tho' still somewhat pragmatist)--the PI's nearly anthropology, IMHO.
J,
ReplyDeleteYou've put the case maybe a little too telegraphically for someone (such as me) not fluent in "first order logic" to follow. What is the justification of your substitution of validity for truth? Isn't validity reducible to truth anyway? I would take such a reduction to be the precondition of a science that claimed to be able to say that "it is valid to argue" in one way or another, a statement which would have to be true or false. Or if there is no such science, by what sort of power do you claim to be able to "show" validity?
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ReplyDeleteTho' at times I am not sure Witt. used "tautology" in the ordinary sense (or I misinterpret it). The law of the excluded middle is a basic tautology: "Either Jr's in the GOP, OR he isn't." But I don't think Witt. always used it in that sense, as in true in all cases, vacuously true. Or a biconditional (I found me TLP! yay).
ReplyDeleteSo, CLARIFICATION: in the restricted propositional sense, Truth does apply to the tautological statement: ie true in all possible worlds. But that's NOT what Miss Anscombe was getting at when she's discussing the Halting Problem, which has to do with the proofs, completeness of the entire system of prop, OR F-O logic...so, she could be in error.
Right. Wittgenstein doesn't use tautology "in the ordinary sense." And this is because there is no such thing as the ordinary sense of tautology. But maybe you can explain that ordinary sense to me. "True in all cases, vacuously true?" According to this formulation, something can be true that is not "the case." Is this not a violation of anything that could be called an "ordinary sense" of truth? Or perhaps the ordinary sense of tautology presupposes a developed sense of truth? I do not think that is so, but I do have a lot to learn about logic.
ReplyDeleteWittgenstein doesn't use tautology "in the ordinary sense." And this is because there is no such thing as the ordinary sense of tautology.
ReplyDeleteWell, tautology has a fairly precise definition, apart from the TLP--even in ordinary language. I think you are correct that Wittgenstein wants to do more than "mere logic" (and "mere tautologies"--but he does include truth tables, including one for the biconditional, which is ...tautologous--True, for all values of the argument)
As with the law of the excluded middle---when someone says "either OJ killed her, or he didn't,", or "either your computer's on, or it isn't"-- it's fairly clear what he means, but vaucuous in the sense that it doesn't say much (or anything, really). Witt.says that somewhere as well--tautologies don't really have "sense." It's merely form....yet one could extrapolate metaphysics from those "givens", I suppose. Baboons don't know what the law of the excluded middle is (do they)---
Some logician types have criticized tautology but that seems to me about like criticizing the moves of a knight in chess (or any piece). That's how the game's played.
You are correct that Witt. rejected the TLP. Yet--he was the one with the great interest in formal logic as a youth (a very rich and privileged youth), and he met Frege, who sent him to Cambridge, and to the dastardly Russell (and his cronies). Oh well--blame Frege, or better blame LW.
At times (from reading Witt's Poker, and other things), Witt. comes off as an incredibly pampered baby (or...mentally ill). His vati was one of the wealthiest men in europe. He could have done anything, even after the TLP. Popper considered Witt. pretty much marzipan, or something...
What role do tautologies have in the game? That is, what do you do with them?
ReplyDelete