A formal theory of truth, alternative to tarski's 'orthodox' theory, based on truth-value gaps, is presented. the theory is proposed as a fairly plausible model for natural language and as one which allows rigorous definitions to be given for various intuitive concepts, such as those of 'grounded' and 'paradoxical' sentences.

The liar paradox is standardly supposed to arise from three conditions: classical bivalent truth value semantics, the Tarskian truth schema, and the formal constructability of a sentence that says of itself that it is not true. Standard solutions to the paradox, beginning most notably with Tarski, try to forestall the paradox by rejecting or weakening one or more of these three conditions. It is argued that all efforts to avoid the liar paradox by watering down any of the three assumptions (...) suffers serious disadvantages that are at least as undesirable as the liar paradox itself. Instead, a new solution is proposed that admits that if the liar sentence is true then it is false, in the first paradox dilemma horn, but denies that the liar sentence is true, but asserting instead that it is false, and refuting the second paradox dilemma horn according to which it is supposed to follow that if the liar sentence is false then it is true. The reasoning for the second paradox dilemma horn is flawed, in that is not only not supported by but actually contradicted by the Tarskian truth schema. We could only infer the second dilemma horn if it were to clasically follow from the assumption that the liar sentence is false, and from the three liar paradox conditions, that therefore it is false that the liar sentence is false. This entire sentence can be shown to be false on the basis of the standard truth schema, thus blocking the paradox. Alternative formulations of the liar sentence are discussed, and the formal proofs and counterproofs for the two paradox dilemma horns, are considered along with the further philosophical implications of maintaining a resolute stance that the liar sentence is simply false. (shrink)