全面解析the paradox of the liarfrom the different components of modern logic to the ancient greek philosophical thinking
V. A Proposed Solution
One way to escape would be to somehow guarantee that liar sentences could never 8arise. In 1931, the logician Alfred Tarski argued that we could block the possibility of liar sentences if we:9
1. strictly distinguish object languages from metalanguages, and
2. restrict language so that it is only in the metalanguage that one can construct sentences about the sentences of an object language, and, so, predicate truth or falsity of those object-language sentences.
Satisfying these conditions would prevent a language from doing (A) and (B).
For example, the English language would become an infinite sequence of distinct languages: English1, English2, English3, …where each succeeding language would be the meta-language for the previous one.
English1 would contain almost all of the vocabulary of the English we currently speak, but it would lack all semantic vocabulary: words like sentence, proposition, assertion,
meaning, lies, true, etc. Thus, we could talk about snow (and shoes, ships, and sealing wax, etc.) in English1 by asserting the English1 sentence, Snow is white. However, English1 would lack the resources for us to be able to talk about any English1 sentences (or any other sentences) in English1.
So, if we would want to talk about the English1 sentence Snow is white in order to say, for example, that it is true, then we would have to treat English1 as the object language and speak in its metalanguage English2. Among its vocabulary, English2 has the semantic terms needed to talk about sentences in English1. Only in English2, then, could we say that Snow is white is true, which we would do by asserting the English2 sentence, “The English1 sentence Snow is white is true.”
Now, English2 only has the semantic vocabulary needed to talk about sentences in English1, not its own sentences (or those of any other language in the sequence). Similarly, then, it would be only in English3—the metalanguage of English2—that we could construct a sentence to assert the truth of that English2 sentence, viz. “The English2 sentence, The English1 sentence, “Snow is white” is true is true.” And so on through the series of English metalanguages.
As no language in the infinite series could satisfy either (A) or (B), Tarski s
stratification banishes all same-level sentence reference and, thus, all sentence self-reference, 8 Of course, banishing all self-reference would guarantee that. But, that program would be ridiculously extreme. Almost all instances of self-reference are harmless and, indeed, very useful if not practically irreplaceable. 9 In “The Concept of Truth in Formalized Languages,” but for a very readable version I recommend Tarski s “Truth and Proof” [Scientific American, vol. 194, no. 6, pp. 63-77].