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While reading Logicomix, I came across a scene that I don't quite understand.

Russell: ...Logicians are creating elaborate ways to "say the same things in different words"...this "everything is a tautology" stuff smells of metaphysical bosh!

logicomix image

This scene seems to imply that Russell didn't view logic as tautologies. But then, what could logic be, if it is not a series of tautologies?

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Wittgenstein was reviving Kant's old view that logical deduction only brings out what is implicitly thought in the premises. Of course, Kant had in mind Aristotle's term logic, which is roughly equivalent to the logic of one place predicates (monadic predicate calculus) in modern terms, and Wittgenstein had in mind Boole's logic of propositions (classes), which is where the idea of logic as tautological comes from. Those two are equivalent in logical strength, which is too say, they both have very little.

It was discovered already in the 19th century, largely by Peirce and Frege, that generalizations about logic based on those two examples completely fail once even two place predicates (relations) are taken into account, let alone polyadic predicates. We now know, for example, that while monadic calculus and propositional logic are decidable (there is an algorithm based on truth tables for deciding whether a given formula is a tautology) polyadic calculus is not. Some polyadic "tautologies" we may never know because we are not smart or lucky enough to discover them.

Russell stressed the import of this already in his Principles of Mathematics, p.32 back in 1903:

"The calculus of relations is a more modern subject than the calculus of classes. Although a few hints for it are to be found in De Morgan, the subject was first developed by C. S. Peirce. A careful analysis of mathematical reasoning shows (as we shall find in the course of the present work) that types of relations are the true subject-matter discussed, however a bad phraseology may disguise this fact; hence the logic of relations has a more immediate bearing on mathematics than that of classes or propositions, and any theoretically correct and adequate expression of mathematical truths is only possible by its means."

Peirce himself distinguished two types of logical reasoning, which he called corollarial and theorematic. The former corresponds to Kant's "analytic" reasoning and Wittgenstein's "tautologies", the latter to non-trivial reasoning that we see on display already in Euclidean geometry and arithmetic. Hintikka later elaborated on what goes on in non-"tautological" logic much further, see What is the difference between depth and surface information?

To put it more starkly, if Kant and Wittgenstein were right about logic then all of Euclid's theorems, and the Last Fermat Theorem too, would be trivialities "tautologically contained" in the axioms of geometry and arithmetic. Unlike Kant, who thought that some extra-logical "synthetic a priori" means are employed in mathematics, Wittgenstein was fully aware that not even all of arithmetic will be in the offering on his conception of logic. But... he did not care. Friedman explains in Logical Truth and Analyticity in Carnap's "Logical Syntax of Language":

"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".

We can now understand why Logicomix has Russell call his "everything is a tautology" stuff "metaphysical bosh".

For more on the nuances of early Wittgenstein's conception see Was Wittgenstein anticipating Gödel? (he wasn't). On how late Wittgenstein radically changed his ways see Koshkin, Wittgenstein, Peirce, and paradoxes of mathematical proof. For more on Peirce's view of logic that Russell was channelling see Chevalier, Peirce’s relativization of the analytic vs. synthetic dichotomy. Here is from Peirce himself, who was anticipating Gödel's undecidability results, in a way:

"Since Kant, especially, it has been customary to say that deduction only elicits what was implicitly thought in the premisses; and the famous distinction of analytical and synthetical judgments is based upon that notion. But the logic of relatives shows that this is not the case in any other sense than one which reduces it to an empty form of words. Matter entirely foreign to the premisses may appear in the conclusion [...] But neither Kant nor the scholastics provide for the fact that an indefinitely complicated proposition, very far from obvious, may often be deduced by mathematical reasoning, or necessary deduction, by the logic of relatives, from a definition of the utmost simplicity, without assuming any hypothesis whatever (indeed, such assumption could only render the proposition deduced simpler); and this may contain many notions not explicit in the definition." [CP 3.641 and 2.361]

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    Thank you for the informative answer. So as far as I understood, there are two types of logic: propositional and non-propositional. When Wittgenstein said "All logical statements are tautologies", he was referring to the former. When he said "Logic cannot be speak of, but can only be shown", he was referring to the latter, and was directly attacking Russell's attempt to systemize logic with ramified type theory. In conclusion, he argued that logic should be delimited to propositional calculus, and any attempt to go further than that should be abandoned. Is my understanding correct?
    – Dimen
    Jan 20 at 3:43
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    @Dimen Wittgenstein did not distinguish between those two, and his "logic must take care of itself" applied to all of logic (that he was willing to recognize). Only looking from outside in can we say that he was basically discarding polyadic calculus, to him that was not even logic. Over and above that, he was also discarding meta-logic, discourse about logic. To him there was a universal language and no "outside" means to talk about it, hence it can be "shown but not said". But on this Russell was in agreement with him, he just had a broader logic within that universal language.
    – Conifold
    Jan 20 at 5:19
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As a matter of terminology, some logicians use 'tautology' as a synonym for a logical truth, while others restrict it to logical truths of the propositional calculus. I shall use the more general term logical truth.

For a given logic, such as classical logic, a logical truth is a proposition that comes out true under all circumstances, or all interpretations, or all uniform substitutions of its non-logical terms, or all (insert your preferred account of validity here). It is also correct to say that all logical truths are logically equivalent to one other, since logical equivalence is itself a relation that holds between propositions that have the same truth value under all circumstances. So we can agree with Wittgenstein that all theorems of logic are logical truths and all are logically equivalent.

But this is far from saying that all theorems of logic say the same thing in different words. Still less does it mean that logic is somehow redundant, or can be disposed of, because its propositions are mere logical truths or tautologies.

  1. For one thing, 'meaning' is a more fine-grained concept than truth conditions. Hence, to say of two propositions that they are logically equivalent, or even necessarily equivalent, falls short of saying that they say the same thing, or mean the same thing. For more on this, see the SEP article on hyperintensionality.

  2. Logic has epistemological value. Nobody is logically omniscient, so the expression of a logical truth, or the exhibition of a logical proof, can provide information that a person previously lacked. Logic can be particularly useful in combining information from different sources and revealing something previously unknown.

  3. Logic can be useful in axiomatizing a body of information. Although this does not present any new information, it has value by showing how the comparatively less obvious theorems of a theory are deducible from the comparatively more obvious axioms. This can also be extended to axiomatizing the logic itself, which may help to provide an epistemological justification of the logic. This relates to what Russell was aiming to do in building the foundations of logic and mathematics.

  4. A logical truth or proof may serve to provide an algorithm for computing a result. Logic is closely related to computation, via the Curry-Howard correspondence. So, a logical proposition or proof is not redundant or merely 'tautological' in some trivial sense just because it expresses a logical truth.

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This scene seems to imply that Russell didn't view logic as tautologies.

Correct. Wittgenstein's view about "logic=tautologies" was grounded on propositional logic and truth table. Unfortunately, truth table is not applicable to predicate logic and thus valid predicate logic formulas are not tautologies in the propositional sense.

In addition, Russel (and Whithead) were porponents of Logicism philosophy of logic and mathematics, according to which all mathematics can be derived from "purely" logical axioms; thus, it is not reasonable to assert that all mathematics is only "saying the same things in different words" (see Logicomix above: "Whitehead and I spent over a thousand pages to build foundations for logic and ... [mathematics]").

Consider Principia's Axiom of Infinity (but Multiplicative Axiom as well): according to Russell it is a logical axiom but we cannot describe it as a tautology.


But it is also fair to say that W's Tractarian views on mathematics are not easily reducible to the slogan: "all mathematics is tautology".

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There's an element of truth and of dramatic license in the graphic novel. It is, after all, a work of literature rather than reportage. Russell had lost the Christian faith he was brought up in as a young man and displayed all the evangelical fervour of a man converted to a new creed by writing his book, Why I am not a Christian. As metaphysics is closely connected to theology, this explains the disparaging reference to 'metaphysical boosh' in the extract above. But he wasn't so disparaging about Wittgenstein's intense engagement with logic, after all, Russell was a logician himself. Nevertheless, he was taken aback by Wittgenstein's presumption that he had dissolved the 'problems of life' by showing logic could answer all questions that could be meaningfully asked and what was left, the problems of life, were not questions at all. It wasn't Wittgenstein's early philosophy that he had an issue with, but his later philosophy where he ranged further than the limited horizons of his early work (though he said, those limits were the limits of the world).

While Russell was serving four months in Brixton prison in 1918 under The Defense of The Realm Act for a mocking reference to the American Army he wrote a book on mathematical philosophy where he wrote:

The importance of 'tautology' for a definition of mathematics was pointed out to me by my former pupil, Ludwig Wittgenstein, who was working on the problem. I do not know whether he has solved it. Or whether he is dead or alive.

Wittgenstein was indeed working on this problem and had written a small book on this called The Tractatus Logico-Philosophicus whilst a soldier in the Austrian army. When he returned to Cambridge, Russell offered to write an introduction and it was this friendly gesture on Russell's part that prompted the publishing of the book - it had Russell's stamp of approval on it. However, when Wittgenstein recieved Russell's introduction on his Tractatus on 9th March 1920, he was disappointed and later said:

There's was so much of it that I'm not quite in agreement with - both where you are critical of me and also where you are simply trying to elucidate my point of view. But that doesn't matter. The future will pass judgement on us ...

Well, one philosopher did in 2016, Mario Bunge, an Argentinian-Canadian philosopher came to the conclusion referred to in the extract above as he recounted in his autobiography, Between Two Worlds: Memoirs of a Philosopher-Scientist. This is because of Wittgenstein's view of mathematics as simply a collection of tautologies given an axiomatic system. Bunge saw this as being obviously wrong.

Now the main reason Bunge was so dismissive of Wittgenstein is partly due to the conventionally held thought that tautologies don't amount to anything. That they are vacuous truths because they hold simply by logical means. This is a view that roughly came about due to the formalist perspective on axiomatic systems. However, Euclid held that his axioms for geometry were actually true. This was affirmed by Descartes who talked about axioms bring based on 'clear and distinct ideas' And whilst his theorems are in a sense tautologies, nevertheless, what Euclid was doing wasn't vacuous. He arranged a great deal of material on geometry on a systematic basis and founded them deductively and it was an enormous source of inspiration to future generations of geometers, physicists and philosophers. To Wittgenstein too - he based his Tractatus on that of Spinoza's and Spinoza's geometric method of rational theology was directly inspired by Euclid.

Rather than thinking of theorems as tautologies let us simply think about theorems. After all, formally they are equivalent! Not any old theorem will do, a good theorem must enlighten and open up new lines of enquiries. In fact, often theorem's do not exist on their own but interlock and many theorems are generalisations of a primordial theorem whose importance has been recognised by tradition and so belong to a family of theorems, one expanding upon another. Others are variations on a theme. Now, let us walk back to the tautology side of this equivalence and see what we have. Well, we have a set of axioms and rules for their 'logical' deduction. So what do we deduce? After all so many deductions are possible, indeed infinitely many. We don't know where to begin. As Euclid already understood, mathematics does not stand apart from the world but bears witness to it - or rather, the world to it. It is what is called the Correspondance Theory of Truth. Wittgenstein clearly states this as proposition Tractatus 4.25:

If an elementary proposition is true then a state of affairs exists; if an elementary proposition is false, then the state of affairs does not exist.

His 'state of affairs' is a fact in the real world and so a proposition is true iff this fact referred to by the proposition is true. He referred to this referencing as it's sinn or sense. A term he borrowed from Frege. A proposition without a sense is not actually a proposition. Thus the propositions of formal mathematics or of formal logic, lacking sense, aren't true propositions. This is why modern logicians call them simply sentences, and the logical system, a grammar, for building sentences. This is not logic as Aristotle conceived of it, as a process of truth preservation, because no truth is being asserted. Instead, modern logicians think of them as a formal language.

Thus, one might think that this has been retrograde step. Once logic was about truth but truth has been emptied out of it. This is not good news for philosophers who are in pursuit of truth - and this explains Bunge's reaction.

But this is only one half of proposition-world divide. And to be fair to Bunge, it's the only side, roughly speaking, that Wittgenstein worked on. The other side was supplied by Godel a decade after the Tractatus was published. He simply realised that we can have a formal world of reference. For example, an axiomatic theory of number, which by the preceding, is simply a syntactic theory and hence not about number, can be shown to actually be about number, by referencing the set of numbers and validating all theorems about them. This latter side is called semantics for obvious reasons and it's both together - syntax and semantics or syntactic logic and a model - that constitutes a full logic. And the theory of this is called model theory.

But this is again unfair to Wittgenstein, he also worked on the semantic side. In fact, he saw it of such cardinal importance and the natural beginning point of logic that in the first two propositions of the Tractatus, he says:

The world is all that is the case

And

The world is a totality of facts, not things.

This appears to be exactly the opposite to what we would at first say. The world is made of things and it is the human mind that distinguishes facts and this should not be confused with the world itself. But actually, Wittgenstein was creating a logical space for the facts that held of the world. This is the semantic side - in model theory - of the world. What can be said to hold of it. They are the theorems of the world, it's model or picture.

That this is so little understood has been because of a fetishisation of the formal side of Wittgenstein's work.

So Bunge is right. Wittgenstein did impoverish logic and mathematics. But he was also wrong, in that he laid a seed that later bloomed. It's in the perspective of model theory that we can subvert Wittgensteins view of mathematics as merely tautologies. Here, this is fine, because we are not interested in theorem finding in these systems but taking a bird's eye view and viewing all mathematical theories at once and it's at this new level - that of model theory - that theorems, in the traditional sense, are to be found and have been found. Itvis, in vrief, new mathematics. In a sense, it was not mathematics that Wittgenstein was referring to but meta-mathematics.

But I'll leave the last word to Wittgenstein by way of Russell:

He used to come to my rooms at midnight and for hours he would walk back and forth like a caged tiger. On arrival, he would announce that when he left my rooms he would commit suicide. So in spite of getting sleepy, I didn't like to turn him out.

On one such evening, after an hour or two of dead silence, I said to him, "Wittgenstein are you thinking of logic or your sins?"

"Both he said," and reverted to silence.

Sinful logic, now there's a thought.

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Well of COURSE logic has some part mathematical but not all like mathematical that mathematics can not reduce it in tautology as in Witgensteins LOGİC.

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