As anyone who has talked with a language-learner knows, syntactically incorrect sentences often succeed in expressing a proposition. This is true even in the case of formal languages.
Formal semantics, say of the Tarski sort, has difficulties with syntactically incorrect sentences. One approach to saving the formal semantics is as follows: Given a syntactically incorrect sentence, we find a contextually appropriate syntactically correct sentence in the vicinity (and what counts as vicinity depends on the pattern of errors made by the language user), and apply the formal semantics to that. For instance, if someone says “The sky are blue”, we replace it with “The sky is blue” in typical contexts and “The skies are blue” in some atypical contexts (e.g., discussion of multiple planets), and then apply formal semantics to that.
Sometimes this is what we actually do when communicating with someone who makes grammatical errors. But typically we don’t bother to translate to a correct sentence: we can just tell what is meant. In fact, in some cases, we might not even ourselves know how to translate to a correct sentence, because the proposition being expressed is such that it is very difficult even for a native speaker to get the grammar right.
There can even be cases where there is no grammatically correct sentence that expresses the exact idea. For instance, English has a simple present and a present continuous, while many other languages have just one present tense. In those languages, we sometimes cannot produce an exact grammatically correct translation of an English sentence. One can use some explicit markers to compensate for the lack of, say, a present continuous, but the semantic value of a sentence using these markers is unlikely to correspond exactly to the meaning of the present continuous (the markers may have a more determinate semantics than the present continuous). But we can imagine a speaker of such a language who imitates the English present continuous by a literal word-by-word translation of “I am” followed by the other language’s closest equivalent to a gerund, even when such translation is grammatically incorrect. In such a case, assuming the listener knows English, the meaning may be grasped, but nobody is capable of expressing the exact meaning in a syntactically correct way. (One might object that one can just express the meaning in English. But that need not be true. The verb in question may be one that does not have a precise equivalent in English.)
Thus we cannot account for the semantics of syntactically incorrect sentences by applying semantics to a syntactically corrected version. We need a semantics that works directly for syntactically incorrect sentences. This suggests that formal semantics are necessarily mere approximate models.
Similar issues, of course, arise with poetry.
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