Truthmaker gaps and the no-no paradox



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TRUTHMAKER GAPS AND THE NO-NO PARADOX

Patrick Greenough

St. Andrews / University of Sydney

March 2nd 2009

1. The no-no paradox.
Consider the following sentences:
The neighbouring sentence is not true.

The neighbouring sentence is not true.
Call these the no-no sentences. Symmetry considerations dictate that the no-no sentences must both possess the same truth-value. Suppose they are both true. Given Tarski’s truth-schema—if a sentence S says that p then S is true iff p—and given what they say, they are both not true. Contradiction! Conclude: they are not both true. Suppose they are both false. Given Tarski’s falsity-schema—if a sentence S says that p then S is false iff not-p—and given what they say, they are both true, and so not false. Contradiction! Conclude: they are not both false. Thus, despite their symmetry, the no-no sentences must differ in truth-value. Such is the no-no paradox.1

Sorensen (2001, 2005a, 2005b) has argued that: (1) The no-no paradox is not a version of the liar but rather a cousin of the truth-teller paradox. (2) Even so, the no-no paradox is more paradoxical than the truth-teller. (3) The no-no and truth-teller sentences have groundless truth-values—they are bivalent but give rise to “truthmaker gaps”. (4) It is metaphysically impossible to know these truth-values. (5) A truthmaker gap response to the no-no paradox provides reason to accept a version of epistemicism.

In this paper it is shown that a truthmaker gap solution to the no-no and truth-teller paradoxes runs afoul of the dunno-dunno paradox, the strengthened no-no paradox, and the strengthened truth-teller paradox. In consequence, the no-no paradox is best seen as a form of the liar paradox. As such, it cannot provide a case for epistemicism.

2. Two dimensions of paradoxicality.
The no-no paradox is generated from the following theses: (1) The no-no sentences both express propositions (in the same context of utterance), (2) Classical logic and classical semantics are both valid. (3) The no-no sentences are (relevantly) symmetrical, and (4) If the no-no sentences are symmetrical then they must possess the same truth-value. Call this last thesis “The Symmetry Thesis”. Sorensen assumes that the (1) – (3) are valid but that the Symmetry Thesis fails. Without this thesis, the contradiction does not arise and so Sorensen does not take the no-no paradox to be a version of the liar but rather a version of the truth-teller.

The truth-teller sentence “This sentence is true” has two pathological features.2 Firstly, unlike the liar sentence, it can consistently possess either truth-value and yet there seem to be no grounds which determine that it has one truth-value rather than the other. Secondly, there seems to be no way of finding out the truth-value of the truth-teller sentence—either via proof or empirical investigation. Although Sorenson assumes that the no-no paradox is akin to the truth-teller, he takes the latter to be less paradoxical:


In the standard liar paradox, the problem is that there is no consistent assignment of truth-values. In the truth-teller paradox, the problem is that there are too many consistent assignments. […] The no-no paradox shares this feature but poses the further problem of assigning asymmetrical truth-values to symmetrical sentences. The no-no paradox has two dimensions of arbitrariness (2001, p.167).
And so we have two dimensions of paradoxicality to contend with:
Dimension One: A pair of sentences is paradoxical if the sentences are (relevantly) symmetrical but differ in truth-value (relative to the same circumstances of evaluation).
Dimension Two: A sentence (or sentence pair) is paradoxical if there are too many consistent assignments of truth-value (relative to the same circumstances of evaluation) whereby there is nothing to determine which is the correct assignment.

It’s worth noting at the outset that it is a mistake to take the no-no sentences to be more paradoxical than the truth-teller. Consider the following sentence tokens:





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