Theno–no paradox is a distinctiveparadox belonging to the family of thesemantic paradoxes (like theLiar paradox). It derives its name from the fact that it consists of two sentences each simply denying what the other says.
A variation on the paradox occurs already inThomas Bradwardine’sInsolubilia.[1] The paradox itself appears as the eighth sophism of chapter 8 ofJohn Buridan’sSophismata.[2] Although the paradox has gone largely unnoticed even in the course of the 20th-century revival of the semantic paradoxes, it has recently been rediscovered (and dubbed with its current name) by the US philosopherRoy Sorensen,[3] and is now appreciated for the distinctive difficulties it presents.[4]
The notion oftruth seems to be governed by the naive schema:
(where we use single quotes to refer to the linguistic expression inside the quotes). Consider however the two sentences:
Reasoning inclassical logic, there are four possibilities concerning (N1) and (N2):
Yet, possibilities 1. and 2. are ruled out by the instances of (T) for (N1) and (N2). To wit, possibility 1. is ruled out because, if (N1) is true, then, by (T), (N2) is not true; possibility 2. is ruled out because, if (N1) is not true, then, by (T), (N2) is true. It would then seem that either of possibilities 3. and 4. should obtain. Yet, both of those possibilities would also seem repugnant, as, on each of them, two perfectly symmetrical sentences would mysteriously diverge intruth value.
Generally speaking, the paradox instantiates the problem of determining the status of ungrounded sentences that are notinconsistent.[5] More in particular, the paradox presents the challenge of expanding one’s favourite theory of truth with further principles which either express the symmetry intuition against possibilities 3. and 4.[6] or make them acceptable in spite of their intuitive repugnancy.[7] Because (N1) and (N2) do not lead to inconsistency, a certain strand in the discussion of the paradox has been willing to assume both the relevant instances of (T) and classical logic, thereby deriving the conclusion that either possibility 3. or possibility 4. holds.[8] Such conclusion has in turn been taken to have momentous consequences for certain influential philosophical theses. Consider, for example, the thesis oftruthmaker maximalism:
If, as per possibilities 3. and 4., one of (N1) or (N2) is true and the other one is not true, then, given the symmetry between the two sentences, it might seem that there is nothing that makes true whichever of the two is in fact true. If so, (TM) would fail.[10] These and similar conclusions have however been contested by other philosophers on the grounds that, as evidenced byCurry's paradox, joint reliance on (T) and classical logic might be problematic even when it does not lead to inconsistency.[11]
