Abstract
The recent, short debate over the alethic undecidability of a Liar Sentence between Stephen Barker (Barker 2012, 2014) and Mark Jago (Jago 2016) is revisited. It is argued that Jago’s objections succeed in refuting Barker’s alethic undecidability solution to the Liar Paradox, but that, nevertheless, this approach may be revived as the alethic indeterminacy solution to the Liar Paradox. According to the alethic indeterminacy solution, there is genuine metaphysical indeterminacy as to whether a Liar Sentence bears an alethic property, whether truth or falsity. While the alethic indeterminacy solution is presented here, and some revenge cases are considered and addressed, the primary aim of this paper is to revive and defend this underexplored and auspicious approach to solving the Liar Paradox.
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Notes
I follow Barker and Jago in taking sentences to be bearers of truth without commitment to whether sentences are the primary bearers of truth. Nothing in this discussion turns on that issue.
For Barker, a sentence whose truth conditions specify no non-alethic fact as a potential grounding fact is ground-unspecifiable. (2014, p. 203) Barker argues that “Ground-unspecifiable sentences are undecidable with respect to their grounding status.” (2014, p. 206) That is, for Barker, any ground-unspecifiable sentence is ground-undecidable.
A more detailed dependency graph would indicate the truth of L depending reciprocally on the truth of ~ L, but that does not affect Jago’s objection.
Although Barker introduces TM as “the principle of truth-maker maximalism” (2014, p. 201) comparison with the standard formulation of truthmaker maximalism, TMM (MacBride 2020, Sect. 2.1) reveals two significant differences.
- TMM:
-
For every truth, there must be something in the world that makes it true.
One is that TMM is not concerned with false sentences, since truthmaking concerns entities related to true sentences. The second is that TMM does not require that the something in the world be a non-alethic fact. The misleading introduction of and name for TM does not bear crucially on Barker’s arguments, nor on Jago’s arguments.
I take it that metaphysical indeterminacy occurs when there is no fact of the matter as to whether an object bears a property. Here, cases such as the indeterminacy in the alethic status of L and T arise where there is no fact of the matter as to whether L, or T, bears truth, or falsity. There is metaphysical indeterminacy for both objects bearing both properties. For an endorsement of the claim that it can be metaphysically indeterminate what properties an object has, see Eklund 2011 at Sect. 4. For a general discussion of metaphysical indeterminacy, see Williams 2008.
It is hoped that this characterization of metaphysical indeterminacy is sufficiently neutral that no more detailed commitment to a particular theory of metaphysical indeterminacy is required. Nevertheless, it must be noted that at least one popular theory of metaphysical indeterminacy is not compatible with the alethic indeterminacy solution; specifically, the supervaluational theory of Barnes and Williams (2011). Their theory is incompatible with the alethic indeterminacy solution because it is “fully classical and bivalent” (2011, Part II); that is, according to their theory, there is a determinate fact of the matter in every possible world as to whether an object in that possible world bears a property. To emphasize the difference, I describe the alethic indeterminacy solution as attributing genuine metaphysical indeterminacy to L and T. For similar reasons, Bradford Skow introduces the term ‘deep metaphysical indeterminacy’ for “reality [which] cannot be completely precisified” (2010, p. 852; italics removed) such as that posited by the orthodox interpretation of quantum mechanics.
The alethic indeterminacy of T and L follows from their truth conditions. It also follows from their truth conditions that they are ungrounded. Unlike the alethic undecidabilty solution, according to the alethic indeterminacy solution, the alethic status of a sentence does not depend on its grounding status.
According to Cobreros et al., “A transparent truth predicate T is one that, paired with some quotation device \(\langle\,\rangle\), allows, for any wff A, for the claim \(T\langle A\rangle\) to be substituted for A or vice versa, in all extensional contexts in all arguments without change in validity.” (2013, p. 841, italics in original; see also Beall 2009) Permissible substitution need not preserve meaning, but must preserve alethic status. In a language with a transparent truth predicate, contradiction is derived as follows: Assume L is true, conclude L (by transparency), conclude L is not true (by identity); next, assume L is not true, conclude L (by identity), conclude L is true (by transparency). Assuming that L is neither true nor false also leads to contradiction. However, these three cases are not exhaustive, since there remains the case of L having an incomplete alethic evaluation. If that case is assumed, no contradiction results. I thank an anonymous reviewer for this journal for pressing me to consider this alternate derivation.
Whether a proposed solution to the Liar Paradox which is vulnerable to revenge problems fails entirely or is merely unsatisfactory because—distinguishing paradoxes finely enough—it solves the Liar Paradox but only by engendering another paradox is a matter of some debate. (See Beall 2007a, b section 1.3.) My own view is that a genuine solution to the Liar Paradox generates no genuine problems. That is, it is reasonable to expect that a genuine solution to the Liar Paradox includes a prima facie implausible or puzzling claim; otherwise, the paradox would have been solved long ago. Nevertheless, such a claim which remains implausible beyond prima facie examination or for which there are contravening considerations is the mark of a specious solution. Of course, settling whether a certain claim is merely prima facie implausible or is thoroughgoingly implausible is often thorny. Similarly, it may be thorny to establish whether a problematic claim is independent of a particular solution. Even so, the strategy of offering a solution to the Liar Paradox at the cost of engendering another paradox is a non-starter.
Note that while the occurrence of ‘not’ in L is, strictly speaking, ambiguous between choice and exclusion negation, it is generally given as a case distinguished from a self-attribution of falsity, and in discussions of the Liar Paradox is generally understood as expressing exclusion negation.
In discussions of truth-functionality, it is common to ascribe truth-functionality both to sentences and to connectives. Strictly speaking, it is sentences which are truth-functional, since to be truth-functional is to have a truth-value which is a function solely of the truth values of its constituents, and it is sentences which have truth values, rather than logical connectives. Nevertheless, a truth-functional sentence is truth-functional in part because it is logically complex, due to containing one or more logical connectives. The evaluation schemes for the connectives (e.g., strong Kleene, Bochvar, Łukasiewicz) presume that the connectives are truth-functional. (Better: the evaluation schemes for the connectives presume that the truth conditions of the connected sentences are not interdependent; see below.) While the latter sense of ‘truth-functional’ derives from the former, both senses are legitimate.
There are other reasons for holding that some disjunctions are non-truth-functional, which are some of the same reasons for holding that some conditional sentences are non-truth-functional. For some discussion, see Hunter 1971: chapter 16, especially p 53.
More intricate cases are possible (for example, ‘Snow is orange or R5a is true or R5b is true’) but do not concern the discussion of R3.
It might be tempting to infer that positing that there are non-truth-functional disjunctions requires a commitment to ‘or’ being ambiguous between a truth-functional connective and a non-truth-functional connective. (I thank an anonymous reviewer for bringing this concern to my attention.) The alethic indeterminacy solution neither requires nor posits this ambiguity. It may be explicitly assumed that ‘or’ expresses a truth-functional connective in R5, and that both occurrences of ‘or’ do in ‘Snow is orange or R5a is true or R5b is true’. Nevertheless, where the truth conditions of the disjuncts are interdependent, the disjunction is non-truth-functional. The first ‘or’ in ‘Snow is orange or R5a is true or R5b is true’ remains truth-functional, whichever of the two potential logical structures the sentence has, since the truth conditions for the first disjunct are not related to those of the other disjuncts. In some disjunctions, the truth conditions for the disjuncts are interdependent, while in other disjunctions, the truth conditions for the disjuncts are not interdependent; the meaning of ‘or’ in both types of disjunction is the same. It is simply that the interdependence of the truth conditions of the disjuncts prevents the truth conditions of the disjunction from being truth-functional. Accordingly, it is sentences, rather than connectives, which are described in the text as non-truth-functional. These considerations apply likewise to other logically complex sentences, mutatis mutandis. Truth-functionality requires that the truth conditions of the sentences joined by the connective not be interdependent. This interdependence is a very rare occurrence, but it is pertinent here.
Note that the rule of addition (disjunction introduction) is standardly proved using truth tables, i.e., by assuming that all pertinent sentences or propositions are truth-functional; that is, by assuming that there is no interdependence among the truth conditions for any component sentences or propositions.
The gappy, glutty, revisionary, and contextual theories and their defenders are well known, and many in total, and so will not be recited here. I include among the nihilistic solutions those solutions which deny that a liar sentence expresses a proposition, or makes a statement, or has truth conditions, for example, Chisholm (1966), chapter 7, Skyrms (1970a, b), Kneale (1971); those solutions which claim that the liar proposition is inexpressible, for example Badici and Ludwig (2007) and Shaw (2013); and those solutions which deny that there are alethic properties, for example, Woodbridge (2005) and Armour-Garb and Woodbridge (2015).
The case is made clearly and forcefully by Tim Maudlin in Maudlin (2007), pp. 184–186.
I would like to thank two anonymous reviewers for this journal for their helpful comments.
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Newhard, J. Alethic undecidability and alethic indeterminacy. Synthese 199, 2563–2574 (2021). https://doi.org/10.1007/s11229-020-02900-z
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DOI: https://doi.org/10.1007/s11229-020-02900-z