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Moore's Paradox - Contradiction and Philosophy

How is sense distinguished from nonsense in the discussion of philosophical problems? That is the question logic of language asks. It is philosophy's first question.

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"Contradiction isn't the unique thing philosophers think it is"

... it makes no sense to assert 'p is the case and I don't believe that p is the case'. This assertion has to be ruled out ... just as a contradiction is. And this shows that ... contradiction isn't the unique thing [logicians] think it is. (Letter to Moore, October 1944 [M.42], q.v. for the discussion)

Any complex proposition of the form (p and not-p) is necessarily false. That is logic's "Principle of Contradiction". A contradiction is unique in that it is the only necessarily false complex proposition, just as a tautology (p or not-p) is the only necessarily true complex proposition. (Wittgenstein's sample tautology: 'If I know either it is raining or it is not raining, I know nothing about the weather' (TLP 4.461).)

The remaining types of proposition (whether simple or complex) are either true or false ([possibly] true if not contradicted, false if contradicted): "subject to contradiction" was Pascal's characterization.

... in this way confounding the definitions they call nominal, which ... are arbitrary ... with those they call real, which are really propositions by no means arbitrary but subject to contradiction ... (Blaise Pascal, "On the Geometrical Mind")

Contradiction and tautology have an important place in Wittgenstein's Tractatus Logico-Philosophicus [as marking the limits: here are the two extremes], and this was why he was excited by what he called Moore's Paradox (CV p. 76), an example of which would be: 'There is a fire in this room, and I don't believe there is a fire in this room.'

When he wrote to Moore in 1944 Wittgenstein was, I think, looking backward (to the old way of thinking) rather than forward with his comments to Moore. "Analytical philosophers" were investigating linguistic forms (Russell's "Theory of Descriptions" is an example), and what Wittgenstein said to Moore was relevant to those investigations. But Wittgenstein's own investigations were taking another direction at that time.

'It's impossible, but I believe it'

The proposition 'It's false, but I believe it' is like the proposition 'It's true, but I don't believe it'.

"One can't believe impossible things," Alice says, but the White Queen replies: "Try again: draw a long breath, and shut your eyes" (Through the Looking Glass, v). But can you believe what is impossible? The word 'can' asks for definition or explanation of meaning or rule of grammar (Wittgenstein's revised concept 'grammar' and its identification with logic).

Nothing is impossible in logic except nonsense (i.e. undefined words or combinations of words; undefined language cannot state a logical possibility because "mere sound without sense" does not state anything), and you cannot believe nonsense however much you may write down or speak the nonsense sound or "sign without a use in the language" (The word 'sign' in Wittgenstein's jargon means: the physical aspect only of language, in contrast to its use or meaning; a sign in itself is simply noise).

But by 'impossible' the White Queen means: what is highly improbable. Someone might believe that water in his tea kettle will turn to ice if placed over a fire, although what believing that amounts to is hard to say (cf. OC § 338). Or you might believe in the miracles of sacred texts. Such events are not "real possibilities" because they are never seen to happen; but what can be described ("logical possibility") can be believed too, and some ways of life are visibly founded on improbable beliefs.


Contradiction as proof of falsehood

Part of the oldest way of thinking in logic-philosophy is the assertion that, solely on the basis of form, a contradiction (p and not-p) is necessarily false; and that if we construct arguments that contain no contradictions our arguments will be valid (or at least not necessarily invalid): "There is a contradiction in your reasoning: therefore, you have reasoned wrongly (your conclusion doesn't stand to reason)." Whereas no other standard of falseness or invalidity might be agreed about, the Principle of Contradiction has always stood.

The contradiction that refutes

And it still stands if there is a distinction between contradictions in form (syntax) and contradictions in sense (semantic): Uncovering the latter by cross-questioning a thesis (claim to knowledge) is one method of refutation in the Socratic method (the other being to show that the thesis is unclear in meaning). Contradiction as the test of truth, as the test of knowledge, for of course what we know is truth, not falsity.

The "refutation" alluded to in Sophist 216b (cf. Euthyphro 6e-7e) is the Socratic elenchus, the point in the discussion where the contradiction in Socrates' companion's thinking is shown to him, thus allowing the companion to realize that he does not know what he thought he knew, which is the only path to Socratic ignorance (in contrast to conceited, presumptuous ignorance).

But in Moore's case there is something (p and not-q) that is also necessarily false -- but its form does not show you this; you have to look to its meaning to see that (p and not-q) is necessarily false in this instance. (Of course, there are difficulties with the notion of "necessarily false": (1) combinations of words that have no use in the language are nonsense and therefore can be neither true nor false. Further, (2) we do sometimes use "necessarily false" propositions, as e.g. when faced with a fact we can't deny, we nonetheless say, "I don't (or, I can't) believe it!" "Necessarily false" propositions sometimes do have a use in the language: although they are contradictions in sense, we are not uttering nonsense when we do use them.)

Plato and Parmenides

Contradiction in sense rather than in form is the Socratic elenchus, as indeed it must be (because meaning is use in the language, not language form), Parmenides notwithstanding. Parmenides: "what is, is" and so it is a contradiction and therefore false to say that change occurs -- because change would be for "what is" to be "what is not", and nothing can be both "what is" and "what is not". Parmenides interests Plato rather than Socrates, so much so that Plato invents logical form in order to avoid saying "what is not is", as in Russell's example 'There is a gold mountain'.

G.E. Moore, Wittgenstein, and "the form of words"

Wittgenstein was in a struggle between the notions that meaning is determined by linguistic form and that meaning is determined by use, so that Moore's having found a contradiction based on meaning that did not have the form of a contradiction was important to Wittgenstein: it "showed that [formal] contradiction isn't the unique thing philosophers think it is".

If I had to say what is the main mistake made by philosophers of the present generation, including Moore, I would say that it is that when language is looked at, what is looked at is the form of words and not the use made of the form of words. (LC p. 2)


Contradictions in Form and Sense

We might distinguish between a contradiction in form and a contradiction in sense -- but we must not misunderstand the expression 'contradiction in sense' to mean that "the sense of the expression is senseless". Instead, we might apply it to e.g. orders with contradictory instructions, e.g. 'Come in and don't come in!' because we normally know what to do with each instruction taken separately ('Come in!', 'Don't come in!') but not what to do with their combination.

It might also be noted that a contradiction in sense need not be a contradiction in form, e.g. 'Come here and go away!' That command -- i.e. combination of words -- is nonsense. But then some philosophers would claim that its "logical form" -- by which they mean its "true" form -- is really 'Come and do not come' or (p and not-p), so attached are those fellows to the Principle of Contradiction. (Plato invented the notion of logical form, precisely in order to avoid the contradiction 'A is and A is not'.)

The proposition 'If A is like B, then A is not like B' is also a contradiction -- but it has a use in our language. And there is no need (What need would there be?) to recast that proposition in a different form. Cf. Wittgenstein's own example: 'This is beautiful and this is not beautiful' (RPP i § 37).

Query: one who does not know but knows.

The proposition 'one who knows and doesn't know': (p and ~p) is its form, which is a contradiction, but it is neither nonsense nor false in all cases.

Socrates: 'Man's wisdom it to know that he is without wisdom' is 'Socratic ignorance'. A paradox is an example of a contradiction that is not nonsense: 'I know that I do not know.'

How does 'The king is dead. Long live the king! [= 'The king is not dead']' [= 'The king is dead, and the king is not dead'] differ from 'This is beautiful, and this is not'? In both instances, 'king' is a common name, not a proper name; it is not 'King Anselmus is dead! Long live King Ethelblau!' (In this context the grammar of a common name is more like the grammar of a demonstrative pronoun than like the grammar of a proper name.) Thus, it seems that 'The king is dead. Long live the king!' is a contradiction (although it hasn't the surface grammar or form of one), but a contradiction with a use in the language.

'Nothing is impossible' contradiction

Why? Because if nothing is impossible, then something is impossible, namely, for something to be impossible. And so the proposition 'Nothing is impossible' appears to imply a contradiction. Nevertheless that proposition has a use in our language. Why?

Wittgenstein: teaching philosophy is teaching students who have brought with them "a mass of false and far too simple ideas" about the logic of our language. (Logic misconceptions point to false paths in philosophy, especially to false grammatical analogies.) Wittgenstein shows the student a non-naive -- because the distinction between language-with-meaning and language-without-meaning (nonsense) is asked about rather than simply assumed -- view of how our language works (of what gives language meaning).

As warning sign posts against false paths, Wittgenstein points out that propositions have varied uses in our language, not only that of statement of fact. Note that only statements of fact are necessarily put to the test of contradiction -- or to the test of experience -- which the proposition 'Nothing is possible' clearly is not.

The proposition 'Nothing is impossible' is an exhortation to focus on why something is possible rather than on why it is not -- to presume that one can realize one's dream, although at the risk of over-reaching oneself (for some things may in fact be impossible). An exhortation is not a statement of fact. The "general form [i.e. essence] of a proposition" (TLP 4.5) is not "This is how things stand if the proposition is true", because there are many types of propositions (proposition-types), not just one, and exhortation is an example.


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