In the Middle Ages, the Liar paradox (e. g., ‘‘this sentence is false’’) and its cognates were known as the insolubles (insolubilia), with some authors explicitly admitting that no solution is forthcoming. It is not known how the discussion entered Latin medieval thought, but occasional late ancient and Arabic analysis of the problems involved in self-referential paradoxes are known, although they are not at very high level of sophistication. Aristotle mentions in his Sophistical Refutations (180a27-180b7) a person who swears that he will break his oath, but it seems very improbable that this text would have been the historical origin of the medieval discussion of the paradox. It was, nevertheless, often mentioned in the discussions.
Insolubles were usually presented as sentences uttered in some supposed context, calling for an evaluation. The paradigmatic example was ‘‘Socrates says something false’’ assumed to be uttered by Socrates in a situation where he utters nothing else. other versions were also produced, and often in ways that differ in interesting logical ways. For example, imagine a situation where A says that ‘‘What B says is false,’’ while B says ‘‘What A says is true.’’
One early solution to the basic paradox was to claim that Socrates somehow fails to formulate a proposition carrying a truth value. In the version of the solution, given in the anonymous Insolubilia monacensia, it is simply claimed that despite uttering something, Socrates says nothing. The idea relies somehow on making a distinction between asserting and uttering, which are to be conceived as two elements of making a claim successfully. In the insoluble case, the two elements do not work properly together. One should thus respond ‘‘you are not saying anything.’’ The solution was called cancellation (cassatio).
According to the solution known as the theory of ‘‘transcasus’’ the insoluble sentence fails to refer to itself, and instead refers to something else. For example, the claim ‘‘Socrates says something false’’ as said by Socrates turns out to refer to something he said immediately before the sentence. An associated theory makes a distinction between exercised act and signified act, thus making it possible to undo the self-referential relation. John Duns Scotus gave this solution to the paradox.
A more general kind of solution was to prohibit self-reference from language, claiming either that no expression in a language is able to refer to itself, or more specifically that the semantic predicates ‘‘true’’ and ‘‘false’’ cannot refer to the wholes they are parts of. As an obvious objection, medieval logicians considered the above-mentioned case where A says that B speaks the truth while B says that A lies. The reference here is not self-reflexive, but circular, and it seems completely arbitrary to simply prohibit such structures. William Ockham resorts to a version of this solution in his Summa logicae.
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