John Buridan's Sophismata

The no-no paradox (Week 14, 13 February)

Greenough, Truth-maker gaps and the no-no paradox

  1. Sorensen holds that the no-no sentences are indeterminate. What exactly does he mean by that?

    His view is that the sentences have a truth-value, but that there is nothing in virtue of which they are true/false, i.e. they lack truth/falsity-makers. Hence having a truthmaker is sufficient but not necessary for truth, a relatively controversial claim. A different way of putting the truthmaking claim is in terms of supervenience; that the no-no sentences have such and such truth-values does not supervene on what there is or how things are. That is, there may be indiscernible worlds w and v such that a no-no sentence is true at one world but false at the other. If the supervenience principle held, it would have to be true/false at one world iff true/false at the other.

    Since we cannot know the truth-values of the no-no pair, the sort of indeterminacy posited by Sorensen is epistemic, a position defended in more generality by Timothy Williamson, and one that many find implausible, at least for the sort of indeterminacy of sentences like 'This stick is red', for a borderline red stick.

  2. Greenough says that a general explanation of what is paradoxical about the no-no paradox and its kin must appeal to the second, not first, dimension of paradoxicality. Do you agree?

    His claim rests on a paradox of p. 551. The pair is not symmetrical and yet they seem paradoxical since they have two distinct consistent truth-value assignments. He is right that the pair are not symmetrical since (2) is not equivalent to "(1) is false"; if it were, we would have a mere syntactic variation of the no-no. But there is still something more puzzling (dare we say 'paradoxical') about the no-no, since there is a sense in which their symmetry rules out any consistent assignment of truth-values. The same is not true of the paradox on p. 551.

  3. One of Greenough's main objections to Sorensen's view comes from the following passage: "Only one assignment remains: the dunno-dunno sentences must both be true. But then the dunno-dunno sentences are not truth-teller like at all and Sorensen loses his explanation as to why these sentences have groundless truth-values" [554]. Do you agree?

    Why do the dunno-dunno pair have groundless truth-values? Have we not just proved that they must be given the symmetrical assignment T-T? (Recall we are rejecting the Symmetry Thesis.) Doesn't that proof ground the fact that they are both true? Since they can only given one assignment, they obviously cannot be paradoxical along the second dimension, so they are not paradoxical for the same reason the truth-teller is. But one might question why they are paradoxical at all.

    By the same token, we can prove that 'The barber of Seville doesn't exist' can only receive the value T. Doesn't that proof ground that's sentence's being true?

  4. What is an example of a "strengthened" no-no paradox? Why does it pose a problem for Sorensen's view?

    One is given by Armour-Garb and Woodbridge and consists of the following pair:

    1. (ii) has no truthmaker;
    2. (i) has no truthmaker.
    Sorensen thinks that they differ in truth-value even though they are symmetric, because are epistemicallysymmetric in the sense the sense they are either both knowable or both not. Indeed, he even thinks one of them could even have a truthmaker without being knowable. This claim strikes me as reasonable but not for reasons that would help Sorensen. (E.g. suppose ZFC is objectively the best theory of sets (modulo logical equivalence). Then since the continuum hypothesis is neither provable nor disprovable in ZFC, and since proof of it is our only means of knowing it, we cannot know it. But if the universe of sets exists, then clearly it is a truth-/falsity-maker for CH.) For what could serve as a truthmaker for one without serving as a truthmaker for the other? There seems to be no plausible answer forthcoming.

    It is clear why each of the pair cannot be false: then both have a truthmaker and so they do not exhibit a truthmaking gap. But as Greenough notes and Sorensen doesn't, if they receive either a T-F or F-T assignment then one of them has a truthmaker, contrary to the truthmaker-truthmakign gap view defended by Sorensen. But via this reasoning we have shown that the only consistent assignment (given Sorensen's view) is T-T, and so both have truthmakers and are hence true, contrary to what they say.

Sorensen, A definite no-no

  1. What is the definite no-no paradox, and how does it relate to the no-no?

    The definite no-no consists of the pair:

    1. (ii) is not definitely true;
    2. (i) is not definitely true.
    The sentences are similar in that that they cannot both be consistently assigned falsity, but they can be both consistently assigned differently truth-values. (We are of course not assuming the Symmetry Thesis.)

    However, one difference between the no-no and definite no-no is that the definite no-no cannot be solved in the way that Buridan proposes to solve the no-no. If (i) is true, then the conjunction of it and (ii) would only imply that (i) is an indefinite truth, and not a contradiction, as Sorensen notes. However, I don't think what Sorensen says about truthmaking (i.e. that what makes p true need not make p definitely true) undercuts the problem posed by the definite no-no. If they receive different truth-values, as Sorensen's view predicts, then one of them is definitely true. How could it lack a truthmaker? Indeed, the most we can prove is that one of them is definitely true. But the only way we could come to know their truth-values is via proof. And if we have no proof that one is definitely true or a proof that the other is, that undermines the proof that at least one of them is definitely true!

  2. Sorensen says that the epistemicist (e.g. him) can admit that "proof is a good way of making propositions knowable while denying that it is universally effective [...] [s]ome proofs are too complicated or too long or otherwise inaccessible to yield knowledge of their conclusions" [p. 229]. Even if this were true, how does it help Sorensen?

    First, why isn't the opponent simply free to formulate the paradox in terms of a notion of definiteness for which proof suffices for definiteness and knowledge? Second, the proof we have concerning the correct assignment to the definite no-no does not at all seem to exhibit this sort of character: i.e. the proof is not so complicated or long or other inaccessible that it would not constitute knowledge of its conclusion (given knowledge of its premises).