Sunday, February 12, 2006

liar liar

Some thoughts on the Liar paradox...

Let Z be a sentence, A be a name for Z and B be an unassigned name. Assume that there is a language that follows the following principles:

Principle T: A is true <-> Z

Principle S: (A = B) -> (A is true <-> B is true)

Principle B: ~(A is true & A is not true) & (A is true or A is not true)

Now let L be a sentence which says of itself that it is not true:

The Liar Sentence (LS): L = ‘L is not true’

The Liar reasoning proceeds by first assuming that L is not true. Then by Principle T, it follows that ‘L is not true’ is true. Then by Principle S and the Liar sentence it follows that L is true. There is no problem yet. The problem arises by adjunction of the assumption and L is true. This violates Principle B.

The Liar Reasoning:

1. L is not true {Assumption}
2. L is not true -> ‘L is not true’ is true {Principle T}
3. ‘L is not true’ is true {1, 2, MP}
4. (‘L is not true’ = L) -> (‘L is not true’ is true <-> L is true) {Principle S}
5. ‘L is not true’ is true <-> L is true {LS, 4, MP}
6. L is true {3, 5, MP}
7. L is true & L is not true {1, 6, ADJ}
8. ~(L is true & L is not true) {Principle B}

Contextualists and hierarchical theorists take issue with Principle T. They argue that in one way or other Principle T is significantly more complex than our na├»ve construal would have it and due to this certain instances of it do not hold.[1] Others have taken issue with Principle S. The worry is roughly that two sentences ‘A’ and ‘A is true’ do not always have the same truth-value.[2] Dialetheists, who contend that there are true contradictions, have challenged the left conjunct of Principle B,[3] while others who appeal to truth-values gaps have taken issue with the right conjunct of Principle B.[4] Still others maintain that every principle is faithful to our language and therefore our folk theory is inconsistent. [5] In every case except the latter, the purposed solutions can be seen as attempts to unmask the hidden mistake in our reasoning. Giving up Principle S is very unattractive and counterintuitive. Challenging principle B, means giving up classical logic. Therefore, if there is some explanation of why Principle T does not always hold it would be the path of least resistance. It seems that the contextualist has the best shot at providing a "happy-face solution" to the Liar paradox, but I think they have some huge problems (see Schiffer 2003 for happy-face/unhappy-face distinction).

...and the suggestion that the Liar sentence is meaningless or doesn't express a propostion doesn't solve anything.

Consider this strengthened Liar sentence:

l = ‘l does not express a true proposition’

Let us first assume that l does expresses a proposition; that is let us try to assign a truth-value to l. Suppose first that l is true, then ‘l does not express a true proposition’ is true, so l is not true. If l is not true, then ‘l does not express a true proposition’ is not true, so l does express a true proposition and l is true. Contradiction. This is the familiar paradox.

However, when faced with this paradoxical reasoning one may be tempted to claim that what is really going on is that l does not express a proposition at all, e.g. l is not “truth-evaluable” or l is “meaningless”. Let us follow this suggestion and assume that l does not express a proposition. If l does not express a proposition at all, then it surely follows that l does not express a true proposition. Yet, this is precisely what l says, so l must be true after all and therefore l expresses a proposition. Again we get a contradiction. Now what?

[1]See Parsons (1974), Burge (1979), (1981) Simmons (1993), Glanzberg (2001), (2004), Gauker, (forthcoming).
[2] See Skyrms (1970)
[3] See Priest (1987).
[4] See Kripke (1975).
[5] See Tarski (1956), Chiraha (1979), Eklund (2002a), (2002b). Inconsistency views are of course unhappy-face solutions.

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