全面解析the paradox of the liarfrom the different components of modern logic to the ancient greek philosophical thinking
It s worth being clear that there is no reductio ad absurdum escape from the Liar. For, what unrecognized, implicit assumption have we made such that we can take the Liar s
double-contradictory conclusion to constitute a reductio ad absurdum proof for its falsehood?
Well, to begin, we ve assumed that there is a liar sentence. So, perhaps the Liar proves there is no such sentence. No, this is ridiculous! We didn t assume that a liar sentence exists. The sentence is right there on page 1! (You don t need to turn back; the rumored-to-be-non-existent sentence will occur in my next sentence.)
We ve also implicitly assumed that the sentence, This sentence is false actually says something, i.e., that the sentence is meaningful, it expresses a proposition. Maybe the Liar proves that that is a perfectly grammatical sentence that expresses no proposition, is
meaningless.5 This suggestion fails for two reasons.
First Reason. If the sentence This sentence is false expresses no proposition, then it follows that it is meaningless. Therefore, that sentence is neither true nor false, and, so, we seem to have escaped. But, consider the sentence:
This sentence is not true.
This new sentence is also a paradoxical liar sentence.
Hmwk. 2. Prove it.
Now, according to the proposed solution, the new sentence is also meaningless, and,
therefore, neither true nor false, and, thus, not true. But, the new sentence just says that it is not true. Therefore, it is true (for what it says is the case)! But if the sentence is true, then it cannot be meaningless. Consequently, the proposal does not resolve the paradox.
Second Reason. The proposal only ever had a chance of resolving versions 1a & 1b in which a sentence directly predicated falsehood of itself. In versions 2a & 2b, where a sentence indirectly predicates falsehood to itself by ascribing a truth-value to another sentence, it s clear that the sentence is perfectly meaningful because you effortlessly
understood it. After all, it was only as a result of reading it that you went on to consider precisely the sentence to which it referred.
Perhaps another example shows the point even more clearly:
The sentence printed in the box on p. 160 of Howard DeLong’s A Profile of Mathematical Logic (Addison-Wesley, 1970) is true.
Go ahead; look it up. (I ll wait.)
Finally, we ve implicitly assumed the : (i) there are only two truth-values, viz., the true and the false, and (ii) every sentence must have one, and only one, of these. Therefore, we ve implicitly assumed that our liar sentences have to be either true or false. Perhaps the Liar shows that some meaningful sentences have no truth-value. This is a 5 The Stoic logician Chrysippus suggested this resolution.