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ABSTRACT
Jay Newhard [2021. “Alethic Undecidability and Alethic Indeterminacy.” Synthese 199: 2563–2574. doi:10.1007/s11229-020-02900-z] proposes a novel solution to the Liar Paradox, which he calls the alethic indeterminacy solution to the Liar Paradox. Bradley Armour-Garb [2021. “Troubles for Alethic Indeterminacy.” Inquiry. doi:10.1080/0020174X.2021.1932166] raises a pair of objections to the alethic indeterminacy solution. Both objections are based upon the alethic indeterminacy solution’s alleged commitment that the truth conditions for a Liar Sentence are indeterminate, and therefore not true. In this paper, this alleged commitment is shown to be mistaken. The alethic indeterminacy solution is compatible with maintaining that the truth conditions for a Liar Sentence are a non-truth-functional biconditional which is true, and not indeterminate. Two independent accounts of non-truth-functional biconditionals are defended. The first account is based on the claim that indicative conditionals express a non-truth-functional dependency relation between consequent and antecedent. The second account is based upon connected sentences having interdependent truth conditions, which can occur with any polyadic connective. These accounts are fully general, not ad hoc measures. Each account solves both of Armour-Garb’s objections. With both objections met, the alethic indeterminacy solution is shown to be a viable solution to the Liar Paradox worth further development.
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Notes
1 Newhard Citation2021, 2567. In this passage, Newhard discusses a truth-teller sentence, T, as well as a Liar Sentence, L. The edits and ellipses mark homologous remarks about T.
2 Armour-Garb points out that ‘on Kleene logic, whether weak or strong, if both sides of a biconditional have value ‘i’, then the biconditional has that value, too … .’ (Armour-Garb Citation2021, 5) Newhard notes this as well in the passage quoted at the close of Section 2.
3 Of course, a biconditional is standardly understood as a conjunction of two conditionals. For now, we can leave open whether conjunction in the truth conditions for a Liar Sentence is truth-functional or non-truth-functional. It is certainly not controversial to treat a conjunction as truth-functional, and there is no obvious reason why it would be problematic for a conjunction of two non-truth-functional conditionals to be truth-functional. On the other hand, it might turn out that the same or similar reasons for thinking that an indicative conditional is non-truth-functional apply to some conjunctions as well. See Section 5.
4 The sentence in the text immediately preceding this one is: ‘Thus, if the truth-conditions for a sentence, S, obtain, then that sentence is true.’ (Armour-Garb Citation2021, 8) That sentence is true. However, the sentence ‘If (3) were true, and if (3) specified the truth-conditions for (L), then it would follow that (L) would be true.’ is not an instance of that schema, because the schema refers to S, which is the sentence for which truth conditions are given, and (3) is the biconditional sentence stating truth conditions for L.
Armour-Garb begins that sentence, ‘Thus, if the truth-conditions for a sentence, S, obtain, then that sentence is true.’ (Armour-Garb Citation2021, 8; emphasis added) It is somewhat peculiar to draw this as a conclusion, since that is just what it is for the truth conditions of a sentence to be met. Also, there is a conflation here between truth conditions, which are a state of affairs which might or might not obtain, in the preceding sentence, and truth conditions which are a biconditional sentence whose right side states the truth conditions (state of affairs) which need to obtain in order for the sentence to be true and whose left side attributes truth to the proposition or sentence type whose truth depends on that state of affairs obtaining, in the subsequent sentence.
5 Armour-Garb does not state explicitly why ‘it would follow that L would be true.’ (Armour-Garb Citation2021, 8) It might be that what Armour-Garb has in mind is that for a biconditional to be true, both sentences on either side of the biconditional must be true; hence, if (3) is true, ‘L is true’ is true; therefore, L is true. Although a biconditional is true when both sentences on either side are false as well as when both are true, it might be that he ruled out the case where both sides are false, presuming that one of the sides is true, and concluded that both sides must be true. Against this, it should be pointed out that it is circular to presume that (the left side) L is true to conclude that L is true. Also, to presume that the right side is true begs the question against the alethic indeterminacy solution. Further, the presumption that a biconditional is true when both sentences on either side of the biconditional are true holds only if (3) is truth-functional, whereas it is assumed explicitly for this case that (3) is non-truth-functional.
6 Armour-Garb also argues that (3) does not meet either of two sufficient conditions for non-truth-functionality. It is shown in Section 5 that the argument against one of these sufficient conditions is mistaken, and that the other sufficient condition is in fact a sufficient condition for alethic indeterminacy, rather than non-truth-functionality. Of course, these two sufficient conditions might not exhaust the sufficient conditions for non-truth-functionality.
7 In Section 5, a second, independent account is presented where polyadic connectives connect sentences having interdependent truth conditions, which can occur with any polyadic connective, including conjunction.
8 That is, setting aside component sentences which are alethically indeterminate. Those cases do not bear on the absurdities presented here of indicative conditionals whose component sentences are either true or false.
9 Examples 1, 6, and 10 are given by Alfred Tarski (Citation1941, 26); examples 2, 7, and 11 are given by Garlandus Compotista in the 11th or 12th century (cited by Boh Citation1993, 5); example 3 is given by Bertrand Russell (Citation1903, 34); examples similar to 9 are given by Gamut (Citation1991, 34).
10 It should be noted that relevance logics (also called ‘relevant logics’) have been developed with the aim of avoiding the paradoxes of material implication, and require that the antecedent of a material conditional be relevant to the consequent. See Mares Citation2022.
11 See also C. I. Lewis (Citation1912, 522 and Citation1917).
12 This observation is credited to Hugh MacColl (Citation1908) by Egré and Rott (Citation2021, §1).
13 It should be noted that relevance logics (also called ‘relevant logics’) have been developed with the aim of avoiding the paradoxes of material implication, and require that the antecedent of a material conditional be relevant to the consequent. See Mares Citation2022.
14 An indicative conditional materially implies the corresponding material conditional. This can be shown easily by examining the four cases.
15 As Russell and Whitehead (Citation1910, 8) write, ‘ … the particular functions of propositions which we shall have occasion to construct or to consider explicitly are all truth-functions. This fact is closely connected with a characteristic of mathematics, namely, that mathematics is always concerned with extensions rather than intentions.’
16 Russell and Whitehead (Citation1910, 7, using ‘·’ for conjunction) do this explicitly: ‘Thus “p ≡ q” stands for “(p ⊃ q) · (q ⊃ p).” It is easily seen that two propositions are equivalent when, and only when, they are both true or are both false.’
17 While I expect that the bafflement among students is common and well known, it is worth noting that Robert Farrell writes that ‘A didactic problem which inevitably confronts the teacher of elementary symbolic logic is the justification of the truth-table for p ⊃ q’ (Citation1975, 301) and that ‘Students of truth-functional logic frequently regard material implication to be patently absurd.’ (Citation1979, 383) See also Anjum Citation2009.
18 Frege’s Begriffsschrift is his first attempt ‘to demonstrate that arithmetic was reducible to logic.’ (Beaney Citation1997, 47; italics removed) In the section of the Begriffsschrift on conditionality, Frege presents conditionals as truth-functional and writes, ‘The causal link implicit in the word ‘if’, however, is not expressed by our symbols … .’ Frege’s project in the Begriffsschrift is explicitly to invent a language attaining the precision necessary for the ‘recognition of a scientific truth’, a precision lacking in ordinary language. (Cf. Frege Citation1997, 48f, especially his analogy of the relationship between the Begriffsschrift and ordinary language to the relation between a microscope and the eye on page 49.) (Frege Citation1997, 57) Russell addresses the point in several works, usually briefly. He distinguishes between material implication, a truth-functional relation which he introduces by stipulation, and formal implication, where the antecedent and consequent are related by topic. Like Frege, it is clear that Russell’s purpose is progress in mathematics and mathematical logic, work for which such stipulation is apt. Russell (Citation1903, 8) notes that ‘The connection of mathematics with logic, according to the above account [i.e., his], is exceedingly close.’ Discussing ‘implication, i.e. “p implies q,” or “if p, then q.” This is to be understood in the widest sense that will allow us to infer the truth of q if we know the truth of p. Thus we interpret it as meaning: “Unless p is false, q is true,” or “either p is false or q is true.” (The fact that “implies” is capable of other meanings does not concern us; this is the meaning which is convenient for us.)’ (Russell Citation1920, 147) Russell makes very similar remarks in Russell Citation1908: ‘That this is not the usual sense, may be admitted; all that I affirm is that it is the sense which I most often have to speak of, and therefore for me the most convenient sense.’ (Russell Citation1908, 301); and also in Russell Citation1903, 34, §37; and in Russell and Whitehead Citation1910, 7, ‘But “implies” as used here expresses nothing else than the connection between p and q also expressed by the disjunction “not-p or q.” … When it is necessary explicitly to discriminate “implication” from “formal implication,” it is called “material implication.”’
19 The connection might even be epistemic, though I will henceforth set aside that case; see Johnston Citation1996, 98 and 106, and Strang Citation1970. According to Strang Citation1970, the history and etymology of ‘if’ is closely related to doubt. This understanding of the conditional makes it fairly straightforward to extend the account to what have come to be known as Austinian conditionals, e.g., ‘There are biscuits on the sideboard if you want some.’ (cf. Austin Citation1956) as well as counterfactual conditionals and subjunctive conditionals.
20 Especially since an indicative conditional materially implies the corresponding material conditional.
21 Hunter (Citation1971) also discusses non-truth-functional disjunctions; see Hunter Citation1971, chapter 16, especially p 53 (cited by Newhard Citation2021, 2570, footnote 11).
22 If a logically complex sentence contains more than two component sentences but there is interdependence of truth conditions between fewer than all of them, the logically complex sentence is partly non-truth-functional.
23 Armour-Garb argues that (3) does not satisfy these sufficient conditions for non-truth-functionality ‘as (3) clearly fails to possess truth-conditions that are interdependent.’ (Armour-Garb Citation2021, 7; italics in original) I hope to have shown in the preceding discussion that the truth conditions for the components of (3) are interdependent. Armour-Garb also mentions a second sufficient condition for being non-truth-functional, which is having regressive truth conditions. (cf. Armour-Garb Citation2021, 7) Although having regressive truth conditions seems to suffice for an indeterminate alethic evaluation, it does not seem to be a sufficient condition for non-truth-functionality.
24 The argument for Step (b) contains another mistake as well. Note that the same argument purporting to show that (3) is not regressive can be run to show that it isn’t non-regressive either: (cf. Armour-Garb Citation2021, 5, quoted above in section 3)
Since the set of biconditionals that specify non-regressive truth conditions is a proper subset of the set of biconditionals that specify truth conditions, if the biconditional stated in (3) is in the set of biconditionals that specify non-regressive truth conditions—that is, if (3) specifies non-regressive truth conditions—then the biconditional stated in (3) is in the set of biconditionals that specify truth conditions. Thus, if the biconditional stated in (3) does not specify truth conditions, then that biconditional does not specify non-regressive truth conditions. Since I have argued that, on Newhard’s approach, the biconditional in (3) fails to specify truth conditions for (L), it follows that the biconditional stated in (3) fails to specify non-regressive truth conditions for (L).
If this argument were sound, it would show that the truth conditions are neither regressive nor non-regressive. Since that result is a reductio ad absurdum, the argument is unsound. Armour-Garb uses ‘specify’ as a success verb, such that for a biconditional to specify truth conditions entails that the right side of the biconditional describes the state of affairs on whose obtaining the truth of the sentence to which truth is attributed on the left side of the biconditional in fact depends. Only then is the set of biconditionals that specify regressive truth conditions a proper subset of the set of biconditionals that specify truth conditions. Of course, the set of biconditionals specifying non-regressive truth conditions is also a proper subset of the set of biconditionals that specify truth conditions. Either way, the conclusion which follows validly is trivial, since it is only the claim that truth conditions which are not true are not both true and (non-)regressive, i.e. ∼(A ∧ B), rather than ∼A ∧ ∼B. The phrase ‘does not specify regressive truth conditions’ is ambiguous between not both true and regressive and neither true nor regressive. The former follows trivially but is not what is needed, since the aim is to show that the truth conditions for L are not regressive, since the motivation for the alethic indeterminacy solution depends on the truth conditions being regressive. The latter is what is needed, but does not follow. That (3) is not regressive would be a very peculiar result anyway, as explained in the next paragraph in the text.
Although Armour-Garb consistently uses ‘specify’ as a success very, if ‘specify’ is not a success verb, it is more obvious that the argument is unsound, since the set of biconditionals that specify (state) regressive truth conditions is not a proper subset of the set of biconditionals that specify (state) truth conditions. In particular, the conditional claim, ‘if the biconditional stated in (3) does not specify truth conditions, then that biconditional does not specify regressive truth conditions.’ (Armour-Garb Citation2021, 5) is false, since when the antecedent is true, (3) might nevertheless be regressive.
25 See especially Gupta and Belnap Citation1993, and Simmons Citation1993 and Citation2018. According to the alethic indeterminacy solution, the alethic evaluation of L is incomplete, and there is genuine metaphysical indeterminacy as to whether L bears an alethic property, due to the regress in the truth conditions for L. On Gupta and Belnap’s revision theory of truth, the definition of the truth predicate is circular; since the definition determines the extension of the truth predicate, and the rule of revision for L is regressive, its extension is unstable. On contextual theories, the context shifts as reasoning in the alethic evaluation of L proceeds, and the alethic evaluation of L (and the extension of the truth predicate) varies with context accordingly, in a regress of contexts. On all three, the regressivity is benign. Revision and contextual theories differ from the alethic indeterminacy solution in assigning an alethic value to a Liar sentence, but concur on the regressivity.