### Wittgenstein's Tractatus: Now With Examples

Ludwig Wittgenstein’s *Tractatus Logico-Philosophicus *is
a century-old this year. It deserves its reputation as one of the most
difficult books in modern philosophy.

Tractatus Logico-Philosophicus |

Part of the difficulty is down to its subject matter. The book is about how language represents reality: how sentences like ‘The cat is on the mat’ manage to tell us something about the world. Exactly because we’re so accustomed to using these kinds of sentences, it can be hard to grasp just what Wittgenstein’s worry is. We’re like fish reading a book about the nature of water.

Another part of the difficulty can be chalked up to self-reference. Wittgenstein uses language to conduct his investigation into the nature of language. The book is thus a kind of ouroboros eating its own tail, with all of the trouble that involves.

A third part of the difficulty stems from the book’s length. It’s just 80 pages of cryptic, tweet-length remarks – sometimes precise, sometimes vague – covering everything from metaphysics to logic to ethics to theology.

But at least part of the difficulty is down to
Wittgenstein’s stubborn refusal to give any *examples *of the kinds of
things he’s talking about: names, objects, elementary propositions, atomic
facts. His reluctance is understandable in a way. Wittgenstein claimed to
establish the existence of these things on *a
priori *grounds – via argument,
rather than observation – and he thought that any examples he could give would
be misleading. But his reluctance is peculiar in another way. As we’ll see, Wittgenstein
is willing to write things that aren’t strictly true in order to help us
understand him, so it’s strange that he didn’t extend this willingness to
concrete examples and illustrations.

Ludwig Wittgenstein |

In this post, I’m
going to do what Wittgenstein didn’t. I’m going to give a quick overview of the
*Tractatus *that’s chock-full of examples and analogies.
My exposition will be rough at some points and simplistic at others. Readers
looking for maximum accuracy should consult the academic secondary literature.
I’m writing this post because even the so-called ‘Introductions’ to the *Tractatus *can be hard-going.

Let’s begin. Wittgenstein’s
main concern in the *Tractatus *is with *propositions*. Propositions
are those things that can be true or false, the kinds of things that we express
with declarative sentences. ‘The cat is on the mat’, for example, is a
declarative sentence expressing the proposition that the cat is on the mat.

Some propositions have *sense*. Roughly, they tell us
something about the world. More precisely, whether they’re true or false
depends on how the world is arranged. ‘The cat is on the mat’ expresses a
proposition with sense. It’s true if the cat is on the mat and false otherwise.
Other propositions are *senseless*. They’re true no matter how the world
is arranged, so they tell us nothing about the world. ‘Either it’s raining or
it’s not raining’ expresses a senseless proposition. Still other propositions
are *nonsense*. They also tell us nothing about the world, but not because
they’re true no matter what. Instead, they tell us nothing because they’re neither
true nor false. ‘Colourless green ideas sleep furiously’, for instance,
expresses a nonsense proposition. Or, to take one of the rare examples in the *Tractatus*,
‘The good is more identical than the beautiful’ expresses a nonsense
proposition.

These cases are pretty clear-cut, but others are less
certain. Does the proposition expressed by ‘Murder is wrong’ have sense? Does
it tell us anything about the world? How about the proposition expressed by
‘Time isn’t real’? Does that have sense? We can argue about each of these
propositions individually, but that can be a long and tedious affair. It would
be convenient if we had some *formula* to tell us which propositions are
sensical, which are senseless, and which are nonsensical.

The *Tractatus *gives the outlines of such a formula. It
presents ‘the general form of proposition’: a recipe for constructing
propositions with sense. Wittgenstein’s claim is that all sensical propositions
are of this same general form. Any proposition *not* of this form is
nonsense. Wittgenstein is thus trying to do philosophers a huge favour. In
philosophy, talking nonsense is an occupational hazard. You might well come to the end of a long
career only to discover you were talking nonsense the entire time. The *Tractatus*
is supposed to help us avoid that
fate.

To see how Wittgenstein unearths ‘the general form of proposition’,
let’s start with a particular example. Take the proposition expressed by ‘One
of the Beatles was a bachelor.’ This proposition is sensical. It tells us
something about the world. This proposition can also be *analysed*, by
which I mean it can be broken down into parts. ‘One of the Beatles was a
bachelor’ can be analysed into four propositions, connected by the word ‘or’:
(1) John was a bachelor, or (2) Paul was a bachelor, or (3) George was a
bachelor, or (4) Ringo was a bachelor. The symbol ‘bachelor’ can also be
analysed, this time into two symbols connected by the word ‘and’. A bachelor is
(1) unmarried, and (2) a man.

The Beatles |

This process of analysis might well continue, but it can’t
go on forever. Eventually, we’ll reach what Wittgenstein called *elementary
propositions*: propositions that can’t be broken down. These elementary
propositions are combinations of *names*: symbols that can’t be broken
down.

Since our example proposition – One of the Beatles was a
bachelor – has sense, each of its elementary propositions must
also have sense. For these elementary propositions to have sense, each of the
names in these elementary propositions must *refer to something*. The
proposition expressed by ‘London is in England’ has sense because each of its
terms refers to something. The proposition expressed by ‘London is in
Flobertness’ is nonsensical, because ‘Flobertness’ doesn’t refer to anything.
Wittgenstein thus concludes that *names* must refer to *simple objects*.
These objects can’t be broken down. If they could be broken down, they might
cease to exist. Then the corresponding name would have no reference, and any
elementary proposition featuring that name would be nonsense. Wittgenstein’s
enquiry into the nature of language has thus led him to a conclusion about the
nature of reality: there must be simple objects forming the substance of the
world.

Wittgenstein gives us no examples of objects, names, or
elementary propositions. He establishes their existence through argument rather
than observation, and so is happy to let others do the work of identifying
them. He thinks that he can know that objects, names, and elementary
propositions exist, even in the absence of examples. Think of it this way. I
can know that you have an ancestor who spent their entire life underwater, even
though I can’t point out any such ancestor. Take your parents and ask, ‘Did one
of them spend their entire life underwater?’. If the answer is – as I suspect –
‘No,’ then take your parents’ parents: did one of them spend their entire life
underwater? That’ll also be a ‘No,’ so take your parents’ parents’ parents, and
so on. Eventually, the answer must be ‘Yes,’ because we all descended from sea
creatures. I can thus know that you have at least one entirely aquatic ancestor,
even though I know of no examples. Wittgenstein’s argument is analogous: if a
proposition has sense, its analysis *must* end with elementary
propositions made up of names referring to simple objects. Never mind that
examples are hard to come by.

All that said, here’s one way of illustrating Wittgenstein’s
view. Let the fundamental particles of matter – quarks, leptons, and the like –
be our *objects*. Each such object is assigned a number, which serves as
its *name*. Elementary propositions are then combinations of numbers. We
might express them with lists, like ‘13, 28, 567, 435.’

(This illustration isn’t entirely faithful to the theory
expressed in the *Tractatus*. Wittgenstein’s objects can’t be divided.
They’re ‘unalterable and subsistent,’ and they exist in every world that we can
imagine. Fundamental particles might lack some of these properties.)

Note that elementary propositions are *just *combinations
of names, like ‘13, 28, 567, 435.’ They *don’t* say ‘28 is between 13 and
567’ or ‘567 crashes into 435’ or anything like that. How then can elementary
propositions have sense? How can a mere combination of names tell us something
about the world?

Wittgenstein’s answer is given by his famous *picture
theory of meaning*. It’s an idea that’s said to have occurred to him in a
Paris traffic-court, where he saw an accident reconstructed with toys. Those
toys stood as proxies for the people and cars involved in the accident, and the
way that the toys were arranged showed the way that the people and cars were
arranged at the time of the accident. The toys together constituted a *picture*:
‘a model of reality.’ Wittgenstein claims that elementary propositions are also
a kind of picture: an elementary proposition is a picture of an *atomic fact*.
An atomic fact is a combination of objects, an elementary proposition is a
combination of names, and the way that the names are combined in an elementary
proposition *shows* how the corresponding objects are combined in an
atomic fact.

With that last sentence, we’ve completed our descent from
the top – an ordinary proposition expressed by ‘One of the Beatles was a
bachelor’ – down to the very bottom: names, objects, elementary propositions, and
atomic facts. Now to climb back up again. Wittgenstein’s claim is that all
propositions with sense are founded on elementary propositions. More precisely,
each proposition with sense is a *truth-function *of elementary
propositions. To understand what’s meant by ‘truth-function,’ return to our
Beatles example. The proposition expressed by ‘One of the Beatles was a
bachelor’ is a truth-function of the following four propositions: (1) John was
a bachelor, (2) Paul was a bachelor, (3) George was a bachelor, (4) Ringo was a
bachelor. What that means is that the truth of ‘One of the Beatles was a
bachelor’ *depends only* on the truth of the latter four propositions. Once
you know whether each of the latter four propositions are true, you know whether ‘One of the Beatles was a bachelor’ is true. More generally for
Wittgenstein, if you know the truth of all the elementary propositions, you know everything there is to know.

That gives us Wittgenstein’s formula for determining whether
a proposition has sense. If a proposition could be true or false, depending on
the truth of elementary propositions, that proposition has *sense*. If a proposition
is true no matter which elementary propositions are true, that proposition is *senseless*.
And if a proposition is *not* a truth-function of elementary propositions
– if you couldn’t tell whether it’s true or false even if you knew all the
elementary propositions – that proposition is *nonsense*.

This formula turns out to be bad news for philosophers. In
Wittgenstein’s estimation, only scientific questions can be given answers with
sense. The propositions of ethics (like ‘Murder is wrong’), aesthetics (like
‘Paris is beautiful’) and theology (like ‘God exists’) turn out to be nonsense.
Indeed, *all* philosophical propositions turn out to be nonsense.

But hang on a minute! The *Tractatus *is filled to the
brim with philosophical propositions. Does Wittgenstein think he’s been writing
nonsense this whole time? Funnily enough, yes:

My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them—as steps—to climb up beyond them. (He must, so to speak, throw away the ladder after he has climbed up it.) He must transcend these propositions, and then he will see the world aright. – 6.54

But this answer only invites more questions: what’s the
point of reading the *Tractatus*? How can a book full of nonsense help us
‘see the world aright’?

Wittgenstein’s answer is as follows: the *Tractatus *shows
that any attempt to answer philosophical questions must be nonsense. There
simply are no sensical answers to such questions. And so, in the words of
Wittgenstein’s final proposition, ‘What we cannot speak about we must pass over
in silence.’

But here one might object. Even if Wittgenstein’s
propositions imply their own nonsensicality, that doesn’t prove that *all *philosophical
propositions are nonsense. It doesn’t even prove that *Wittgenstein’s *propositions
are nonsense. They might just be plain-old-false. Consider an analogy. The
sentence ‘Any sentence consisting of more than five words is nonsense’ implies
its own nonsensicality. But that sentence isn’t nonsense. It’s false.

Clearly, Wittgenstein didn’t think that his propositions
were false. What might justify his own, more radical reading? Here’s one answer.
Suppose that the propositions of the *Tractatus *strike us as the only
viable answers to philosophical questions *even after we recognise that these
propositions imply their own nonsensicality*. Then we might conclude that all
philosophical propositions must be nonsense.

I’ll end with a metaphor. Imagine you’re in a clearing in a South American rainforest, searching for El Dorado. Many paths are open to you, and you don’t know which will take you there. You choose a path that looks promising, but after travelling a while you realise that you’re back where you started. You resolve to try again. But even knowing that the first path leads you back to the clearing, it still strikes you as the only way to reach your destination. In that case, you might conclude, there’s no need to try another path. There is no El Dorado.