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At 6.1 in the Tractatus, Wittgenstein says, "The propositions of logic are tautologies."

When he says this, is he referring to the fact that the axioms of propositional logic as presented, for example, in the Principia Mathematica (published before the Tractatus) or Hilbert & Ackermann (published after) are only tautologies, and therefore any statements that can be proven therefrom are also tautologies?

Or does he mean something more mystical, such as, "the propositions of logic can tell us nothing meaningful about the world because meaning is transcendental," or somesuch (I'm still trying to understand the work as a whole, this is probably not the correct way to phrase it, but I hope my meaning is understood)?

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I would take this completely literally.

We feel as though logic and formal mathematics really produce or contain meaning, but in fact they just rearrange the meaning we put into them. They are nothing but extremely sophisticated tautologies, and once we have seen the path through them, and gotten the clarity they can help us achieve, we are left with no more meaning than we brought to the table to begin with.

Any meaning they 'produce' in us is not really in them, it is in us. Mathematics and logic are a place where Plato's notion of anamnesis is almost literally correct.

I am not sure that is about any transcendental nature of truth. But it is in its own way deep. It is about the fact that formal language does not contain or confer meaning, it only transfers a point of view from one person to another. This can be useful, but it cannot introduce meaning, it can only point at it.

In his later philosophy, this kind of thinking leads to the characterization of the use of language, especially its formal aspects of grammar and reference, as a 'game' we play in order to communicate, and distinct from the actual act of communication itself.

From my own pet 'neo-Intuitionist' position on mathematics, it confirms that the meaning behind mathematics or logic is not in the formal results but in the intuitions that are already accessible to us, which are used and strengthened by going through the process. Anything you can accomplish by knowing mathematical or logical results is merely a shortcut or an increase in leverage, and is something completely different from what you accomplish by doing mathematics or logic.

  • Another nice answer. Would you say that our mathematical derivations do add something to our understanding, as opposed to adding meaning. For example, a derived theorem adds to both our understanding of our assumptions and to our understanding of how our assumptions relate to our derivation. – Nick R Aug 15 '15 at 17:31
  • I don't know. The derivation itself explores human psychology in a productive way, and clearly adds something to our understanding. The result itself is of more questionable value -- it really is a complicated tautology, and it really does only focus attention or eliminate steps... These are useful things, but do they improve understanding? – jobermark Aug 16 '15 at 0:06
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The claim that propositions of logic, and analytic truths in general, are tautologies was a consensus view before Frege, and can be found in Locke, Hume and Kant, see Was Locke right that analytic knowledge is vacuous?. This is because the only logic known to them was the Aristotelian one, which only allows syllogisms. Frege famously expanded the scope of logic to include not only predicates and quantifiers beyond that, but also arithmetic and much of set theory. However, Frege's logic led to paradoxes (the set of all sets not containing themselves being the most famous one), and Russell had to limit it in Principia Mathematica by imposing various syntactic rules, often unintuitive and convoluted, such as hierarchy of types.

To Wittgenstein this type of syntactic largesse was unacceptable. As he says in the Tractatus "clearly the laws of logic cannot in their turn be subject to laws of logic" and "Frege says that any legitimately constructed proposition must have a sense. I say that any proposition is legitimately constructed". So Wittgenstein rejects the expansion of logic and the Frege-Russell formalization of it, although he does not quite go back to syllogistics. But all he allows are features common to any language: substitution (of objects into predicates, etc.), and iterative combination of simple expressions using connectives.

In Friedman's assessment, these "do not lead us to the rich higher-order principles of classical analysis and set theory. As Godel's arithmetization of syntax again decisively shows, all that is forthcoming is primitive recursive arithmetic". That is propositional logic and quantifier-free theory of natural numbers, tautologies indeed. "Of course, the Tractatus is itself quite clear on the restricted scope of its conception of logic and mathematics in comparison with Frege's (and Russell's) conception. Wittgenstein's response to this difficulty is also all too clear: so much the worse for classical mathematics and set theory."

See more in Was Wittgenstein anticipating Gödel?

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