Truthmaker gaps and the no-no paradox



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8. The strengthened no-no paradox.
Consider the following sentences:
The neighbouring sentence is not truth-made.

The neighbouring sentence is not truth-made.
(where a sentence is truth-made iff there exists something which makes it true). Call these sentences the SNN sentences. Symmetry considerations dictate that the SNN sentences must both possess the same truth-value. Suppose they are both true. Given what they say, and given Tarski’s truth-schema, they are both not truth-made. Given TM1, it follows that they are both not true. Contradiction! Conclude: they are not both true. Suppose they are both false. Given what they say, and given Tarski’s falsity-schema, they are both truth-made. Given TM2, it follows that they are both true. Contradiction! Conclude: they are not both false. Thus, despite their symmetry, the SNN sentences must differ in truth-value. Such is the strengthened no-no paradox.12

If one accepts TM1 and TM2 then there is no extensional difference between the no-no sentences and the SNN sentences. Unlike the dunno-dunno paradox and the no-no paradox, however, TM1 is used to derive the contradiction. Given that Sorensen has already rejected TM1 then this strengthened paradox appears to present no additional problem—he can simply treat the proof as a reductio of TM1. However, a problem remains.

Suppose the SNN sentences are false. Given what they say, and given Tarski’s falsity-schema, they are both truth-made. Given TM2, they are both true, and so they are both not false. Contradiction! The SNN sentences thus take one of the three assignments: T-T, T-F, F-T. But now there seem to be no grounds to determine which is the correct assignment. So, the SNN sentences exhibit the second dimension of paradoxicality and so, for Sorensen, they have groundless truth-values. However, now the asymmetrical assignments are ruled out since these assignments entail that one of the sentences is truth-made. Hence, the SNN sentences must be both true. But now the SNN sentences are not truth-teller like at all and Sorensen loses his explanation of why they have groundless truth-values! Much worse, we have proved that they are both true. Given knowledge of a truthmaker–truthmaking gap solution to the truth-teller, and closure, the SNN sentences are both known to be true. Given K2, the SNN sentences are truth-made. Given TM2 they are both true. Given what they say, and given Tarski’s truth-schema, they are both not truth-made. Contradiction!

Just as with the dunno-dunno paradox it seems that the only option for Sorensen, if he is to retain K2/K1, is to take the SNN sentences to be akin to the liar.


9. Armour-Garb and Woodbridge on the strengthened no-no paradox.
Armour-Garb and Woodbridge (2005), hereafter ‘AGW’, briefly consider whether the strengthened no-no paradox is a problem for Sorensen. As it turns out, they underestimate how problematic the strengthened no-no paradox proves to be. Their presentation is as follows (where I follow their numbering):
(7) (8) has no truthmaker.

(8) (7) has no truthmaker.


AGW (2005) say:
If (7) and (8) were both false then each would have a truthmaker and, thus, would be true. So, to maintain consistency, they cannot both be false. Ascribing divergent truth-values staves off inconsistency, but […] there are two ways of doing this, with nothing favouring one over the other. The problem for Sorensen is that if one of these sentences is true and the other is false, then the true one—whichever it is—has a truthmaker. But it is utterly indeterminate which of these sentences is the true one. Of course, Sorensen can consistently claim that both (7) and (8) are true—and, as such, without truthmakers—but this appears to undermine the motivation for positing truthmaker gaps in the first place, since they were introduced as a means for resolving, consistently, the indeterminacy of both the open-pair and the sorites.
(where the name ‘open-pair’ is the name AGW use for the no-no sentences). As I read them, AGW are posing Sorensen two problems. The first is that (7) and (8) are epistemically indeterminate in that they have unknowable truth-values (that’s what AGW mean when they say “But it is utterly indeterminate which of these sentences is the true one.”). Yet, if (7) and (8) take different truth-values, then one sentence is true and so has a truthmaker. But then how can Sorensen explain why the true sentence of the pair has an unknowable truth-value if it has a truthmaker? The second is that it would be illegitimate to accept that T-T is the only remaining assignment on the grounds that a truthmaker–truthmaking gap theory cannot properly account for the epistemic indeterminacy exhibited by (7) and (8) when these sentences have divergent truth-values.

The second problem only emerges if the first problem proves to be intractable. As it turns out, Sorensen (2005b) thinks that AGW “only cleanly exclude the F-F assignment”. Is he right? With respect to the first problem, Sorensen thinks that (7) and (8) are epistemically symmetrical: they are both known/knowable or both not known/knowable. But he also thinks that epistemic symmetry does not entail truth-value or truthmaker symmetry. In fact he conjectures that (7) and (8) do take different truth-values, such that one is truth-made the other is not:


But having a truthmaker is only a necessary condition for knowability. If (8) is unknowable (because it has no truthmaker) then its perfect resemblance to (7) renders (7) unknowable. Having planted the seed of absolute unknowability through a truthmaker gap, we can grow further unknowables amongst neighbouring truths that do have truthmakers. Ignorance can spread without perfect symmetry (2005b).13
One way to unpack what Sorensen is really after here is to make use of a version of the following safety principle on knowledge: one knows that p only if one couldn’t easily have been wrong.14 In more detail:
(SAF) If the method one has used to form one’s true belief that p could easily have produced a false belief then one’s true belief that p is not knowledge.
Suppose that (7) is true and (8) is false. Suppose one forms a belief, via method M, that (7) is true. Thus one’s belief is both true and is made true (since what (8) says is not the case). Since method M could just have easily have produced the (different but symmetrical) belief that (8) is true, and since (8) is false, then one could easily have formed a false belief. Hence, given SAF, one’s belief that (7) is true is not knowledge, despite the fact that (7) has a truthmaker. Furthermore, since no method could distinguish between the truth-status of (7) and (8) then one’s belief that (7) is true cannot be knowledge—again, despite the fact that (7) has a truthmaker. Thus, AGW’s first problem has been answered and so the second problem does not arise.15

Still, both AGW and Sorensen fail to spot that the asymmetric assignments are ruled out on different grounds. As we have seen, if the assignments T-F and F-T are both consistent, and if a truthmaker–truthmaking gap solution to the truth-teller is correct, then (7) and (8) must lack truthmakers. But if (8) is false, then (7) has a truthmaker. Contradiction. Inconsistency is a good reason to show that the only remaining assignment is T-T. But, as we have also seen, we have proved this to be the case on (putatively) known assumptions. So, via closure, (7) and (8) are known, and so, via K2, are truth-made. But if they are truth-made then, via TM2, they are true. Given that they say, and given Tarski’s truth-schema, they are not truth-made. Contradiction!

As it turns out, Sorensen is wary that (7) and (8) may, after all, yield a contradiction:
Suppose Woodbridge and Armour-Garb came up with a formal refutation of the T-F assignments. That very refutation would become a premise for the back-up position of declaring the T-T assignment correct by a process of elimination.

I even have a back-up to this back-up. Suppose Woodbridge and Armour-Garb somehow go on to eliminate the T-T assignment. (A T-T assignment can be eliminated by adding “and the other statement is false” to (7) and (8).) That refutation would give me the second premise needed to activate my last resort: declaring (7) and (8) meaningless. After all, if the pair is meaningful, there is some way to consistently assign them truth-values. If there is no such way, then these sentences must get the same “last ditch” treatment as the liar paradox (2005b).


It’s worth noting that the augmented sentences Sorensen alludes to above do not give rise to a strengthened paradox. Consider these sentences:



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