全面解析the paradox of the liarfrom the different components of modern logic to the ancient greek philosophical thinking
as ungrammatical and, therefore, meaningless. Consequently, no sentence in any of the languages could directly say anything at all about itself.
Hmwk. 3. Prove that version 1a of the Liar cannot be constructed in Tarski s hierarchy.
Of course, one could utter or write the words This sentence is false or I am lying, but these putative self-referential constructions would be utterly ungrammatical—like the
English construction Shoes pointedly my sleep eat —and, hence, not sentences; therefore, not meaningful; and, so, without truth-values.
Equally, indirectly self-referring sentences would be impossible to construct.
Hmwk. 4. Prove that version 2a of the Liar cannot be constructed in Tarski s hierarchy.
Therefore, no sentence in any language in the series could (directly or indirectly) predicate falsity of itself. Consequently, no liar sentence could be constructed in any language in the series, and the paradox would be resolved.
Despite the fact that Tarski s proposal would, if carried out, quash the liar paradox, the proposal has few adherents. (Yet, it s impossible to overstate the historical importance of Tarski s work.) The problem is that Tarski s proposal is too crude, for it banishes not only liar sentences but all (directly or indirectly) self-referring sentences. Consequently, it is too extreme, for, again, the vast majority of self-referring sentences are perfectly banal, and some are practically irreplaceable—consider the indirectly self-referring sentence, All English sentences begin with a capital letter. There is no good reason to proscribe all of these sentences and very good reasons not to proscribe some of them.
It is also interesting to note that successfully carrying out Tarski s proposal makes that proposal impossible. If Tarski s stratification were carried out, the description of the hierarchy of languages would have to be impossible, for the sentences of that description would have to belong to some finite level of the hierarchy in order to be meaningful, but in order to describe the whole hierarchy they couldn t belong to any of its levels.