Dashtakī's Solution to the Liar Paradox: A Synthesis of the Earlier Solutions Proposed by Ṭūsī and Samarqandī

AbstractṢadr al-Dīn al-Dashtakī (d. 1498) has proposed a solution to the liar paradox according to which the liar sentence is a self-referential sentence in which the predicate ‘false’ is iterated. Discussing the conditions for the truth-aptness of the sentences with nested and iterated instances of the predicates ‘true’ and/or ‘false’, Dashtakī argued that the liar sentence is not truth-apt at all. In the tradition of Arabic logic, the central elements of Dashtakī's solution—the self-referentiality of the liar sentence and the implicit iteration of the predicate ‘false’—were initially highlighted in two earlier solutions proposed by Naṣīr al-Dīn al-Ṭūsī (d. 1274) and Shams al-Dīn al-Samarqandī (d. 1322), respectively. Here I investigate all three solutions and show that Dashtakī's solution can be taken as a synthesis of the other two. None of these solutions seems to be convincing at the end of the day. Nevertheless, all of them include significant logical and philosophical insights. In particular, although Dashtakī's solution is not itself compelling, it is only a few steps away from a promising solution. The appendix to this paper includes translations of the relevant passages.Keywords: The liar paradoxArabic logical-Dashtakīal-Ṭusīal-Samarqandī AcknowledgementsI am thankful to Reza Pourjavady for insightful discussions we had about al-Shīrāzī, Ibn Kammuna, and Dashtakī, and to Stephen Read for his extremely helpful comments on an earlier draft of this paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 For a classification of different families of solutions to the liar paradox in the tradition of Arabic logic, see Zarepour Citation2021, secs. 1–2.2 To the best of my knowledge, the earliest study of Dashtakī's solution in the secondary literature is provided by Miller Citation1989. A more detailed analysis of this solution has recently been offered by El-Rouayheb Citation2020.3 It is more precise if we talk about ‘the liar sentences’ instead of ‘the liar sentence’. However, for the sake of simplicity I use only the latter. Almost all the solutions that are studied in this paper are concerned, in the first place, with the liar sentence ‘every sentence I say at this moment is false’. The contextual assumption which guarantees that the latter sentence is indeed a liar sentence is that that sentence is said by someone who does not say any other sentence at the moment of saying that sentence.4 Abharī's solution is analysed and reconstructed by Zarepour Citation2021. A discussion of Ṭūsī's commentary on Abharī's solution can be found in Alwishah and Sanson Citation2009, sec. 3.5 A truth-apt sentence is capable of having a truth value. It can be either true or false. Consequently, if a sentence is not truth-apt, it has no truth-value. It can be neither true nor false.6 al-Ṭūsī Citation1974, 236, ll. 3–4.7 al-Ṭūsī Citation1974, 237, l. 4.8 In contemporary versions of the correspondence theory of truth, correspondence is usually understood as a relation between a sentence and a fact (or state of affairs) in the world that is described by the sentence. To use Ṭūsī's terminology, a declarative sentence is true if and only if what is declared (i.e. the content of declaration) is realized in the world. By contrast, Tūsī introduces correspondence as a relation between a declarative sentence and the particular thing about which something is declared (i.e. the object of declaration). So, Ṭūsī's understanding of this notion is entirely different from that of contemporary philosophers. Nevertheless, going into the details of this difference is beyond the scope of the present paper. We will shortly see that Dashtakī's understanding of the notion of correspondence is strikingly similar to the modern understanding of this notion.9 al-Ṭūsī Citation1974, 237, ll. 9–10.10 al-Ṭūsī Citation1974, 237, ll. 12–4.11 al-Ṭūsī Citation1974, 237, ll. 18–9. The original Arabic of T5 in the edition by Mohaghegh and Izutsu is disordered and hardly makes any sense. It seems that the editors were not themselves satisfied with the edition either. That is probably why they have left a question mark in parentheses at the end of the paragraph. Fortunately, the correct version of the Arabic text can be found in Ḥillī's discussion of the liar paradox (al-Ḥillī Citation2009, 223, ll. 15–6). Accordingly, both here and in the appendix, my translation of the last paragraph of Ṭūsī's discussion is based on what is quoted by Ḥillī. The main idea of the paragraph in question is that although the liar sentence seems to be of the same species as other declarative sentences, it is not subject to truth and falsity. And if one thinks that truth and falsity are applicable to the liar sentence just because it looks similar to declarative sentences to which truth and falsity are unproblematically applicable, one has made a mispredication.12 Wittgenstein Citation1974, 3.332. Miller (Citation1989, 177, n. 19) has highlighted the parallel between this phrase of Tractatus and part of the view defended by Jalāl al-Dīn al-Dawānī (d. 1502) regarding the ill-formedness of the liar sentence. However, it seems that Dawānī's view is itself inspired by Ṭūsī's idea.13 Ibn Kammūna Citation2008, 75, ll. 6–11. Ibn Kammūna completed this work, i.e. Al-Kāshif, in 1278, four years after Ṭūsī's death.14 The principal source of this distinction for the majority of Muslim logicians is Avicenna's al-‘Ibāra. For Avicenna's views regarding this distinction, see Hodges Citation2012 and Kaukua Citation2020. For many Muslim logicians, the most crucial difference between these two types of propositions is that, contrary to simple negative propositions, metathetic affirmative propositions have an existential import. Unlike (2), (1) can be true even if X does not exist.15 al-Ḥillī Citation2009, 223, l. 17.16 al-Dashtakī Citation2007, 9, ll. 2–6. The phrase ‘nafs al-amr’ literally means the thing itself. However, its rich technical content is hard to capture in translation. That is why, both in the body of the paper and in the appendix, I use the original Arabic term instead of its translation. Roughly speaking, when it is said that P is the case in nafs al-amr it means that the essences and true natures of things are such that they make P the case. For two recent studies of the notion of nafs al-amr in the Arabic tradition, see Kaş Citation2021 and Spiker Citation2021.17 al-Dawānī Citation2007, 116, ll. 5–7.18 al-Samarqandī Citation2020, 645, l. 21–645, l. 7. All of my references to Qisṭās al-afkār are to the edition by A. Fallahi. However, in the specific section on the liar paradox, there is no important difference between this edition and the edition by Pehlivan (i.e., al-Samarqandī 2014).19 The main problem with the liar sentence is that it refers to itself and if it refers to something other than itself the paradox will disappear. Samarqandī has apparently noticed that if the liar sentence refers to something other than itself, it does not matter whether it refers to only one other sentence or to more than one sentence. There is no paradox in any of these cases. That is perhaps why he goes with the simpler case and addresses the liar sentence when it refers to only one sentence distinct from itself.20 Unfortunately, it is not clear at all why Samarqandī considers only two possible candidates for the reference of ‘every sentence I say’. For example, it is not clear why he does not address the case in which the declarer intends that phrase to refer to the liar sentence and nothing else. Is this because he thinks that the declarer cannot entertain such an intention? If so, for what reason? Unfortunately, the text remains silent on these questions.21 This is, indeed, in the same spirit as the construction proposed by El-Rouayheb Citation2020, 255.22 al-Dashtakī Citation2007, 8, ll. 3–8.23 The most important work in which this distinction is discussed is perhaps Kripke Citation1977.24 al-Dawānī Citation2007, 113, ll. 4–11.25 al-Dawānī Citation2007, 115, ll. 2–3.26 al-Dashtakī Citation2007, 14, ll. 7–10.27 Admittedly, my formulation of the Principle of Compositionality suffers from some imprecisions and ambiguities. However, my aim was just to highlight the overall similarity between the grounding element of Dashtakī's argument and the general idea behind the Principle of Compositionality. This aim can be achieved by mentioning even a rough formulation of this principle, or so I hope. For a meticulous discussion of the Principle of Compositionality, see Pelletier Citation1994.28 al-Dashtakī Citation2007, from 14, l. 11 to 15, l. 4. Unsurprisingly, no trace of quotation marks can be found in the original Arabic manuscripts of the works that we discuss here. Following El-Rouayheb Citation2020, I employ the modern convention of quotation marks to prevent possible confusion in reading the sentences under discussion. More generally, I use quotation marks to distinguish references to linguistic strings (e.g. words and sentences) from references to their semantic content.29 al-Dashtakī Citation2007, 15, ll. 5–11.30 al-Dashtakī Citation2007, 15, ll. 14–17.31 This solution is strikingly similar to a solution proposed by John Dumbleton (d. ca. 1349). On his solution(s) to the liar paradox, see Read Citation2021, sec. 3.6, and Citation2022, sec. 6, and Bartocci and Read Citation2022.32 Another translation of this passage is provided by Alwishah and Sanson Citation2009, 125–7.33 Quṭb al-Dīn al-Shīrāzī (d. 1311) has mentioned a faithful Persian translation of this passage in his Durrat al-tāj (al-Shīrāzī Citation1990, 460–1), albeit without disclosing that he has borrowed it from Ibn Kammūna. It is now well-known that Ibn Kammūna's Al-Kāshif has been one of the major sources of al-Shīrāzī's Durrat al-tāj. See Pourjavady and Schmidtke Citation2004.34 This presentation of the liar paradox is probably borrowed from Abharī's Kashf al-ḥaqā’iq. See al-Abharī Citation1998, 217, ll. 4–6.35 This presentation of the liar paradox is problematic. If the liar sentence is false, then some of the sentences that the liar has said in this house are false. Since the liar sentence is the only sentence that the liar has said in this house, the only candidate for being true is itself. So, the liar sentence must be both false and true. But it is not clear why Ibn Kammūna does not follow this simple line of argument and instead considers the possibility that the liar might have said sentences other than the liar sentence. He then discusses two cases: (1) The liar sentence itself is one of the liar's true sentences. (2) The liar sentence is false, but all other sentences of the liar are true. He seems to miss the third possibility: (3) The liar sentence and some other sentences of the liar are false, but there are still some other true sentences that are said by the liar. Ibn Kammūna fails to establish that in case (2) the liar sentence is both true and false. If ‘every sentence I say in this house is false’ is not the only sentence that is said by the liar in this house, the mere falsity of that sentence cannot make what it expresses the case. So, if this sentence is false and all other sentences said by the liar in this house are true, then what the liar sentence expresses is not the case. So, we cannot conclude its truth. Moreover, case (3) is not discussed at all. Therefore, we are justified in concluding that Ibn Kammūna fails to show that the particular version of the liar sentence he discusses is actually paradoxical.36 This paragraph must be seen as a commentary on Ṭūsī's solution. It is interesting that although Ṭūsī himself sees his own solution as incompatible with that of Abharī, Ibn Kammūna seems to endorse both solutions at the same time. The following paragraph includes a brief presentation of Abharī's solution. 37 This latter line of argument seems to be unsound, too. Even if we take the sentence ‘every sentence I say in this house is false’ as a khārijī proposition, we cannot conclude that it is false because its subject has no instance in the external world. This sentence is itself said by the liar. So, it can be considered a khārijī object of itself.38 These square brackets have been added by the editor of the Arabic text.39 This sentence is quoted from al-Ṭūsī Citation1974, 237, l. 5.40 This phrase seems to be quoted from al-Ṭūsī Citation1974, 237, ll. 18–19. See also note 1.41 All the headings in square brackets, in sections (E)-(G), have been added by the editor of the Arabic texts.42 The phrase in the square brackets has been added by the editor of the Arabic text.43 The editor of the Arabic text has correctly mentioned that the whole paragraph is quoted from Sharḥ al-qisṭās. Nevertheless, it is not clear why he has put the whole paragraph in square brackets.44 These square brackets have been added by the editor of the Arabic text.45 The phrase in the square brackets has been added by the editor of the Arabic text.46 Dawānī attributes the quoted claim to Samarqandī. However, Dashtakī mentions it as if it is his own view, or at least this is how things are reflected in the Arabic edition I consulted.47 The square brackets have been added by the editor of the Arabic text.48 The square brackets have been added by the editor of the Arabic text. This is because the third disjunct is not mentioned in some manuscripts.49 This title, added by the editor of the Arabic text, does not fit what follows in the paragraph below it. Dawānī agrees with Dashtakī that Tūsī's solution is not convincing. Dawānī merely confirms Dashtakī's conclusion by providing some additional examples and observations.