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In today's mathematics, we have many variants of logic (propositional, first order, higher order, fuzzy logic, etc.). These are all self-consistent formal systems that are based on some set of axioms.

In Wittgenstein's Tractatus, he demands a logical structure for thoughts and basically says that only sentences that mirror such thoughts are valid.

However, it seems to me that he assumes there is one and only one real logic after which our thoughts can be ordered. On the other hand, in principle, the different logics could have been available to Wittgenstein before 1920.

I don't know if his considerations survive to the present day. I wonder if these problems have been discussed by other philosophers, and maybe even resolved. In particular, I'm curious what the modern relevance is of Wittgenstein's early claim regarding the usefulness of discussions about the supernatural.

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These seem to me two different questions: (1) Universalist conception of logic vs. logical pluralism (2) Relevance of TLP to today's antimetaphysics. (Did I understand you correctly?) Quick comment regarding the first question: The existence of different logical systems does not force you to embrace logical pluralism. Quine's stance regarding the primacy of classical first order logic is a good example for this. –  DBK Apr 1 '12 at 20:14
    
@DBK: (I signed in from another account when I asked this) Yes, that's true, these are actually two issures. But the second would be to broad if I wouldn't explain where my motivations to ask it come from. I'm interested in both subjects, but I really only ask the second question with respect to the problems of the third. Btw. ich bin auch Wiener ;) –  NiftyKitty95 Apr 2 '12 at 8:33
    
@CodyGray: Okay, thanks Gray. Although sadly, in a physicists community it's not always a good move to disclose that you read books on philosophy. –  NiftyKitty95 Apr 17 '12 at 7:14
    
Really? How weird. :-( Sorry if that's not what you wanted then. I can delete your account on this particular site if you want to keep things hush-hush... –  Cody Gray Apr 17 '12 at 7:16
    
@CodyGray: Haha, no it's good. I don't identify with internet accounts. And well, there are some people who think philosophy is useless, since I guess they espect a direct problem solving approach from all fields. But Wittgenstein is probably on the edge. –  NiftyKitty95 Apr 17 '12 at 7:20
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3 Answers

In today's mathematics, we have many variants of logic (propositional, first order, higher order, fuzzy logic, etc.). These are all self-consistent formal systems that are based on some set of axioms.

True. Furthermore, some logics are classical, some are non-classical, and some are deviant-- but let's not let that detain us here.

In Wittgenstein's Tractatus, he demands a logical structure for thoughts and basically says that only sentences that mirror such thoughts are valid.

And, in the Philosophical Investigations he famously walked back from the absurdity of this claim, pointing to all kinds of thoughts and sentences that do not take the form of propositions. This does not mean that we necessarily need to follow the late Wittgenstein in rejecting the early Wittgenstein, but we should at least be aware of the critique, and be prepared to respond to it.

However, it seems to me that he assumes there is one and only one real logic after which our thoughts can be ordered. On the other hand, in principle, the different logics could have been available to Wittgenstein before 1920.

I don't recall him addressing this at all, nor do I see what the substitution of a different logical formalism would have on his project. Do you have a concrete example where the use of a different logic would result in a significant change to the system of the Tractatus? Or is this merely idle speculation?

I don't know if his considerations survive to the present day. I wonder if these problems have been discussed by other philosophers, and maybe even resolved. In particular, I'm curious what the modern relevance is of Wittgenstein's early claim regarding the usefulness of discussions about the supernatural.

Wait, what? How does "the supernatural" come into play here at all? At the moment, this appears to be a gross non sequitur. If you think there is an argument to be made which would link the substitution of non-classical or deviant logics into the schema of the Tractatus and "the supernatural", go ahead and make the argument-- I don't think you can expect us to connect those particular dots without a lot more to go on.

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(i) A different logic B would make any conclusion in his logic A, which couldn't be drawn in B not a valid in B. Therefore different underlying logics (if they really differ) mean different legal thoughts. (ii) The supernatural is related because according to Wittgenstein many concepts, which relate to e.g. god (e.g. Good and evil) are not to be talked about. The question asks to what extend it is today reasonable to discard all the things he wants to discard on the last page of the book. PS: I really like the "What would you like to know?" question on your page, given that one can't answer. –  NiftyKitty95 Apr 17 '12 at 8:49
    
1) may be true, depending on the logics, etc.-- this is why I asked the OP for an example; 2) Wittgenstein did not identify the supernatural as "that which must be passed over in silence"-- furthermore, his later work consisted largely of attempting to say precisely those things. re: P.S.--- Glad you liked it, I chose it (in part) because of the number of ways it can be read. –  Michael Dorfman Apr 17 '12 at 9:13
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Visual images and gesture extraction isn't the point-- it is that the gesture does not translate simply to "disgust". Furthermore, meaning is always understood within a context, and the context is never saturated. There is more to the world than mathematics or computation. and many things which resist formalization. Finally, as far as I recall, Wittgenstein was not by any means lacking mathematical training. –  Michael Dorfman Apr 18 '12 at 11:25
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Also - even if 'something so simple looking as predicate calculus can express all of knowledge in the first place' were true (which is dubious as Michael notes), are we assuming there is something defective in natural language? That the formal expression is the correct one? Just what is it about the formal expression that makes it better than the original, informal expression? 'I asked him for a breadknife, and he gives me a razorblade because it is sharper'. –  adrianos Apr 20 '12 at 19:44
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@RonMaimon Can you explain why formal and precise is "better"? Wittgenstein's point about the knife/razor shows that 'better' is purpose relative. Similarly, what exactly is imprecise about 'this JPG is of my grandma'? Isn't the point of saying it just to say who the image is of, or perhaps to make a joke? How does your formal language do that better than natural langauge? –  adrianos Apr 23 '12 at 12:48
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The existence of different logics is interesting, but not particularly relevant, because of Gödel/Turing universality. If you have a statement about possibilities, a statement in modal logic, you can encode it in a semantics about possible worlds, in a first order logical form about an expanded universe of discourse (see Kripke semantics). Further, if you have a "fuzzy logic", you can speak about it in an appropriate axiomatic mathematical system which includes real numbers, and makes a map between propositions and fuzzy values.

There are also Bayesian probability calculi which can be thought of as a different logic, some people regard quantum amplitudes as a logic, while other schemes regard permission as in the logic. What you put in the logic and what you put in the axioms is largely up to you.

But the main point is that the ordinary first order logic is complete, it will produce any logical consequence of any axioms, and this was proved by Gödel. This means that if you have a mathematically precise description of some other logic, you can always talk about this logic in terms of first order logic, and consider the other logic as axioms on top of first order logic. This is not a natural point of view, but it is a possible point of view.

The universality of first order logic is the logical analog of Turing universality--- that a finite complexity computer with unbounded memory can simulate any other computer with suitable programming. The formalism of logic is like the instruction set, the axioms are like the instructions of the program, and the deductions are the running of the program. A Turing machine can do Gödel deduction (you can program a computer to deduce in first order logic) and Gödel deduction can describe a computer, so the two results are essentially equivalent, and anything that can be stated in any coherent logical system is something that is meaningful for a computer to analyze and interpret.

So there really is only one type of logic, and Wittgenstein is mostly right. Although it is a mistake to attribute this to Wittgenstein, who was not as mathematically or logically precise at the mathematicians and logicians of the early 20th century in whose footsteps he followed, not as precise, nor as iconoclastic, as Russell, and did not contribute commensurately to the formal developments as Russell did. Further, one may say that Wittgenstein's ideas have a sonority and a lack of mathematical precision that leads those in technical fields to perhaps use the phrase "running one's mouth". The proper attribution of first order logic, the recognition of its importance in philosophy, and the associated logical positivism, better belongs to Hilbert, Frege, Boole, Quine, Gödel, Turing, Russel, Whitehead and others who come before. It continued with those who built the Vienna school, including Carnap, which made logical positivism the ascendant philosophy until the 1970s.

The use of predicate language to remove ambiguity, and the thesis that all statements should be formulated in some predicate language about precise observable criteria, is the central tenet of logical positivism, which flourished in the mid-20th century, but took a beating in the 1970-90s.

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Thanks for clearing up some of the logic points. The other (main) question remains to be answered. And I have yet to understand how first order logic encompasses the others... I'll look into it. And it's funny, I really only wrote the "On the other hand, in principle the different logics could have been available to Wittgenstein before 1920." line only because it seems to me that he basically tries to translate Gödels semantic considerations to truth of statements in common language. In any case, I don't value truth as much as many others, so I regard many philosophical questions as sensible. –  NiftyKitty95 Apr 2 '12 at 8:44
    
How does ordinary first-order logic deal with paraconsistent logics such as dialetheism? –  Michael Dorfman Apr 17 '12 at 8:37
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@MichaelDorfman: You write down the axioms of Peano Arithmetic, or some set theory, say, ZF, in first order logic, so that you can talk about mathematics. Then you define the statements of dialethistic logic as text strings (integers), and the deduction rules as manipulations of these (recursive functions, i.e. computations). If the dialethistic logic is well defined, it will have a well defined computer program to do deductions, and this means it can be modelled in PA or ZF or any other reasonable axiom system. Now any statement about the other logic is a first order statement in PA. –  Ron Maimon Apr 18 '12 at 3:27
    
@RonMaimon "it will have a well defined computer program to do deductions" This is a property of logics known as computability and closely related to decidability. I'm not sure what you mean by "well defined" here, but since neither first order logic nor ZF is decidable, I don't know why an imbedded dialetheistic logic should be presumed to be decidable. Do you have any reference to a place where a dialetheistic logic (such as Priest's Logic of Paradox) is proven to be decidable? I would be very interested to see such a proof. Thanks! –  Dennis Jan 7 '13 at 18:06
    
@Dennis: The only point is that if your logic makes sense as a system, then it is embeddable inside a first order system, by just formalizing the terms in ZF. It's like universality, once you have one computer with one instruction set, you can simulate any other computer, and that computer can simulate yours back. Likewise, once you have one logic, the standard choice is first order logic, you can simulate any other, by modeling it with axioms, and vice versa, the other logic can include first order logic, or at least should. –  Ron Maimon Jan 22 '13 at 20:53
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However, it seems to me that he assumes there is one and only one real logic after which our thoughts can be ordered. On the other hand, in principle, the different logics could have been available to Wittgenstein before 1920.

Wittgenstein's early and later view of logic entails that there cannot be multiple logics. A calculus is a means of transforming symbols, and there can be as many of these as one has transformation rules. Logic however, in Wittgenstein's sense of the word, is different. He regarded propositions as bipolar (Tractatus) and, later, bivalent (Investigations). He rejected Brouwer's view that there are propositions with no truth value and Lukasiewicz's multi-valued logics (in the 1932-1935 Cambridge lectures). Their systems are calculi which specify alternative transformation rules, but these are not the rules of language and not, therefore, the rules of logic in the required sense. A language can be governed by these rules only by altering what it means to be a proposition, and therefore what it means to have a language. Logic, proposition and language are all internally related concepts.

I don't know if his considerations survive to the present day. I wonder if these problems have been discussed by other philosophers, and maybe even resolved.

These considerations have very much survived. Michael Dummett in particular has argued that alternative logics (such as dropping LOM or multiple truth values) entail alternative metaphysics. This has led to much discussion about how we could build a 'theory of meaning' for language such that the answers to metaphysical questions would all be determined by the right meaning theory. This actually goes against both the early and later view of Wittgenstein that logic does not entail any metaphysics at all, for the propositions of logic do not correspond to any state of affairs. They are rules which are tautologies (in the Tractatus) or grammatical rules (Investigations).

There is a modern resurgence of interest in metaphysics that is profoundly un-Wittgensteinian. The problems have not so much been resolved as exacerbated, and a return to Wittgenstein's penetrating solutions (in the later philosophy) would be very helpful.

In particular, I'm curious what the modern relevance is of Wittgenstein's early claim regarding the usefulness of discussions about the supernatural.

There are a number of comments on the 'mystical' in the Tractatus which are related to the doctrine of showing what cannot be said, but this doctrine was abandoned by the later Wittgenstein as confused. There is no such thing as ineffable metaphysics. Those modern philosophers however, like Dummett and Davidson, who have revived the old Tractatus approach to philosophy, could well be said to be committed to mystical, or supernatural views, about the nature of the world, reality etc. I do not see, though, that they have solved any 'problems' in these areas any more than Plato, Aristotle, Aquinas or Descartes solved them.

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