# Ludwig the Thick-Headed Logician^{seuss}

The problem with geniuses is that there's always the possibility that they're having a joke at our expense; or that their seemingly profound sayings are actually the gropings of a dimwit upon which we project our conceptions of what a genius would say. More than occasionally it has seemed to me that Ludwig Wittgenstein, one of the most respected, or at least discussed, philosophers of the twentieth century, may fall into the latter category.

Wittgenstein was canonized early for his work *Tractatus Logico-Philosophicus*. The Vienna Circle worshiped him, even though he refused to meet with them or acknowledge their existence. He was a protegé of Bertrand Russell, who brought him to Cambridge, where he reigned as the respected Delphic Oracle for the rest of his life, even though he published nothing. Many of his notebooks were published after his death, the most notable being *Philosophical Investigations*.

The *Tractatus*^{bib} is mainly renowned for its oracular (the word is inescapable) pronouncements, such as

6.52 | We feel that when |

Oooh. Actually, I rather like that one.

And of course, the immortal

7 | What we cannot speak about we must consign to silence. |

Although a better translation might be, "About that whereof you cannot speak you must remain silent." Yes, all the paragraphs are numbered, to make them easier to quote and/or seem more profound. There's a Dewey Decimal-kind of system, so that 6.52 is a "comment" on 6.5. So we have

5.6 | The … |

5.63 | I am my world. (The microcosm) |

5.631 | There is no such thing as the subject that thinks or entertains ideas If I wrote a book called |

Whaddya think?

The parts of the *Tractatus* that *aren't* quoted so much are about logic. These parts form most of the *Tractatus*, and what they have to say about logic is mostly wrong, which raises the question why we should attach much significance to the profundities that supposedly follow from the logic.

Wittgenstein supposedly introduced truth tables in this work, so he gets credit for that, although the uses he puts them to are now generally forgotten. (He had an argument with Frege about whether T and F were objects (Frege) or just entries in truth tables (Wittgenstein). Nowadays we let them play both roles, so Frege won that one.)

But here are some of the mistakes he made: He repeatedly asserts things like, "When the truth of one proposition follows from the truth of others, we can see this from the structure of the propositions. [5.13] … 'Laws of inference,' which are supposed to justify inferences, as in the works of Frege and Russell, have no sense and would be superfluous. [5.132]" His alternative to rules of inference is to imagine all the elementary propositions about the world being lined up, after which all formulas about them could be constructed in advance. Even without complexity theory, this is a hard idea to swallow.^{semantics}

How about (5.1): "Truth-functions [truth tables] can be arranged in series. That is the foundation of the theory of probability." I don't think anyone ever followed up on *this* bizarre suggestion.

Then there's the convoluted technique of 6.1203 for checking whether a formula is a tautology. It involves drawing lines between formulas and truth values, but I couldn't get it to work; for any nontrivial formula the links rapidly congeal into an unreadable mess. Now, it is probable that this paragraph has been hailed as the seed from which has sprung tableau-based methods for theorem proving, but it is a flimsy foundation for the later announcements that "… in a suitable notation we can in fact recognize the formal properties of propositions by the mere inspection of the propositions themselves [6.122].… there can never be surprises in logic [6.1251]. … Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases [6.1262]."

It's odd that someone so aware of Russell and Whitehead's work in *Principia Mathematica* would think that proving theorems in logic was so straightforward (granted that there is a lot more to *PM* than proving logic theorems), especially considering that Wittgenstein includes first-order logic in his claims. He explicitly grants at one point that there might well be an infinity of elementary propositions, as well as of the "names" of objects that the propositions are about. So: "Mathematics is a method of logic [6.234]. … Indeed, it is a consequence of this method [equation solving] that every proposition of mathematics must be obviously true [6.2341]."

But the biggest blunder in the *Tractatus* is its treatment of equality.

4.241 | When I use two signs with one and the same meaning, I express this by putting the sign '=' between them. So ' … |

4.242 | Expressions of the form ' |

The problem with this superficially plausible idea appears as soon as you consider a formula such as ∀*x*(*f*(*x*) ⊃*x=a*), which Wittgenstein does in paragraph 5.5301. For here *x* is not a "sign" whose "meaning" is the same as that of *a*, but just a placeholder for some arbitrary object, which or might not happen to be *a*. The formula says that all objects that have property *f* "are *a*," i.e., equal to *a*. Or, in his words: "What this proposition says is simply that *only a * satisfies the function *f*, and not that only things that have a certain relation to *a* satisfy the function *f*." But the clause after the comma in the quoted sentence is simply wrong. As Kripke says somewhere, the "certain relation" in question is merely the smallest reflexive relation, a perfectly well defined entity. It may sound useless, but Wittgenstein has provided the perfect example of its usefulness. We're all familiar with others, such as specifying of a sequence *S* that *S _{i}*=

*S*⊃

_{j}*i=j*, which is the standard way to say that the elements of the sequence are distinct.

(One way to see the point is to invert the implications, so that we get formulas like ∀*x*(*x≠a* ⊃¬*f*(*x*)). Now "≠" seems like a perfectly good relation, or at least it's not obvious what's wrong with it. This formula says the same thing as before, only in the form, "Anything but *a* does not have property *f*." Wittgenstein's comeback would probably be that all such syntactic transformations do not change the underlying proposition, which is "obviously" always the same.)

The *reductio ad absurdum* of Wittgenstein's position on equality is where it gets him with existential quantifiers. In what follows I use a more modern notation than his for the formulas:

5.532 | … I do not write '∃ (So Russell's '∃ x,y(f(x,y))) ∨ (∃x(f(x,x)))'.) … |

5.5321 | Thus, for example, instead of '∀ And the proposition, ' |

Like many of Wittgenstein's proposals in the *Tractatus,* this one sounds superficially plausible, but is completely unworkable. The problem is that it makes substitution into contexts with bound variables almost impossible. There is a useful little lemma in formal logic that states that if *Q* does not contain a free occurrence of *x*, then '∃*x*(*P* ∨ *Q*)' is equivalent to '(∃*xP*) ∨ *Q*'. But this is not true in Wittgenstein's world. Suppose *P* is just '*P*(*x*)' and *Q* is '*N*(*a*) ⊃ ∃*y*(*R*(*a,y*)'. In the version with wide quantification over *x*, the second quantifier is inside the first, and therefore (I guess, who really knows) it must become

*x*(

*P*(

*x*) ∨ (

*N*(

*a*) ⊃ (

*R*(

*a,x*) ∨ ∃

*y*(

*R*(

*a,y*)))))

if it is to be equivalent to the version where the first quantifier has narrow scope.

There's a less obvious philosophical problem, too. We've gone to all this trouble to avoid having to mention the harmless little predicate '=', but we end up having to think about equality constantly. In fact, basic predicate calculus does not treat equality as a built-in constant. You must define it by adding axioms. That raises the question whether equality *can* be defined by axioms. That is, in all models of these axioms does '=' refer to a predicate recognizable as equality? If you *do* want to think of '=' as a logical symbol (akin to '∀' or '∨'), then you can use "first-order logic with equality," and simply stipulate that '=' refers to the relation {⟨*ο,ο*⟩ | for any *ο*}. Wittgenstein's approach makes all these issues impossible to avoid, or to deal with explicitly.

In his defense, when he wrote *Tractatus Logico-Philosophicus*, nobody really understood these issues very well. Tarski's theory of truth was still ten years in the future, although Skolem had already published what is now known as the Skolem-Lowenheim Theorem.

But Wittgenstein fled from formal logic. He repudiated the whole idea that if you expressed everything in a perfect language you could avoid mistakes. Well, who can blame him for *that* insight, whether or not you find his later philosophy as profound as most philosophers think. Perhaps, though, he took a look at how technical formal logic was getting and decided he just couldn't deal with it. He decamped so speedily and completely that no one seems to think of his early work in the context of logic at all.

### Notes

Note **seuss** With apologies to Dr. Seuss. If this note is puzzling, you are among the benighted many who believe that his work began with *The Cat in the Hat*, which was actually about where it came to an end.

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Note **bib** All quotes from the translation by D.F. Pears and B.F. McGuinness, 1961, Routledge and Kegan Paul. The original German version was published in 1921, and had a perfectly ordinary German title, which could be translated, "A Logico-Philosophical Treatise." The Latin translation must have been inspired by the feeling that we had another Newton or Russell on our hands.

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Note **semantics** Okay, I guess you could say that Wittgenstein was groping toward a semantic, as oppose to proof-theoretic, approach to logic, which makes him, again, a pioneer. It's his insistence that there's only one, obvious approach that seems so strange.

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