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On Descriptional Propositions in Ibn Sīnā: Elements for a Logical Analysis

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Mathematics, Logic, and their Philosophies

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 49))

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Abstract

Employing Constructive Type Theory (CTT), we provide a logical analysis of Ibn Sīnā’s descriptional propositions. Compared to its rivals, our analysis is more faithful to the grammatical subject-predicate structure of propositions and can better reflect the morphological features of the verbs (and descriptions) that extend time to intervals (or spans of times). We also study briefly the logical structure of some fallacious inferences that are discussed by Ibn Sīnā. The CTT-framework makes the fallacious nature of these inferences apparent.

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Notes

  1. 1.

    Hasnawi and Hodges (2017, p. 61) have correctly pointed out that ‘substantial’ is not Ibn Sīnā’s own term. Indeed, as Strobino and Thom (2017, p. 345) have mentioned, it is only in the later stage of the tradition of Arabic logic that the terminology of ‘substantial’ and ‘descriptional’ became mainstream. Some of the other names which have been employed to refer to the distinction under discussion will be mentioned later in the chapter.

  2. 2.

    Street (2002, Sect. 1.1) and Strobino and Thom (2017, Sect. 14.2.1) emphasize that for Ibn Sīnā all propositions have either temporal or alethic modality. Absolute propositions are implicitly modal and all other propositions are explicitly modal. Lagerlund (2009, p. 233) highlights that even the absolute propositions can be taken to be descriptional.

  3. 3.

    Strictly speaking, there is an important difference between a sentence and the proposition expressed by it. Accordingly, it is a sentence (rather than a proposition) which can be read in different ways. So what a substantial (respectively, descriptional) reading of a sentence expresses is a substantial (respectively, descriptional) proposition. Nonetheless, such a clear difference between sentence and proposition cannot be detected either in Ibn Sīnā’s own discussion of the substantial–descriptional distinction or in the secondary literature on this issue. So to remain more focused on the main points we would like to make—and of course for the sake of simplicity—we do not make the sentence–proposition distinction bold.

  4. 4.

    See Hodges and Johnston (2017, p. 1057).

  5. 5.

    These examples are adopted from El-Rouayheb (2019, p. 24).

  6. 6.

    See al-Qiyās (1964, Chapter III.1, p. 126) in which Ibn Sīnā complains that previous philosophers have not paid enough attention to this distinction.

  7. 7.

    A famous passage in which Ibn Sīnā discusses this distinction can be found in the logic part of al-Išārāt (1983, Chap. 4.2, pp. 264–266). For translations of this passage see Street (2005, pp. 259–260) and Ibn Sīnā (1984, Chap. 4.2, p. 92). In the logic part of al-Naǧāt (1985, pp. 34–37)—whose translation can be found in Ahmed (2011, Sect. 48)—Ibn Sīnā proposes six different readings of necessary propositions. The second and the third readings include respectively substantial and descriptional necessities. This distinction is discussed also in al-Qiyās (1964) and Manṭiq al-Mašriqīyīn (1910). Translations of some relevant passages from these two works are provided by Hodges and Johnston (2017, Appendix A.2). They discuss a distinction between ḍarūrī and lāzim propositions in passages from Manṭiq al-Mašriqīyīn that is tantamount to the distinction between substantial and descriptional readings of propositions.

  8. 8.

    See El-Rouayheb (2017, pp. 72 & 81).

  9. 9.

    See Street (2002, p. 133).

  10. 10.

    See, among others, Rescher and vander Nat (1974), Hodges and Johnston, and Chatti (2019a, 2019b).

  11. 11.

    See Martin-Löf (1984). In what follows, a basic familiarity with CTT is assumed. All the background requirements can be found in Rahman et al. (2018, Chap. 2).

  12. 12.

    See Almog (1991, 1996).

  13. 13.

    As pointed out by Ranta (1994, p. 55), “the most serious criticism against the type-theoretical analysis of everyday language comes from intuitionistic thinking” (i.e., from the very same framework within which CTT is developed). The concern is that although intuitionistic logic is an appropriate tool for mathematical reasoning, its application outside mathematics is inappropriate. This is mainly because, by contrast with mathematical reasoning in which objects are almost always fully presented, everyday reasoning is usually based on an incomplete presentation of objects. For example, although a natural number can be fully presented by its canonical expression, giving a full presentation of a continent seems to be extremely difficult, if not impossible. Stated differently, the presentations of continents (like many other things) in the natural language is usually incomplete in the sense that they are usually referred to by expressions which only partially determine what a continent is. There seems to be no canonical expression of the non-mathematical objects like continents, humans, trees, etc. One possible way to deal with this concern, as Ranta (1994, pp. 55–56) suggests, is “to study delimited models of language use, ‘language games’. Such a ‘game’ shows, in an isolated form, some particular aspect of the use of language, without any pretention to covering all aspects.” For example, a term like ‘human’, depending on the context, can be partially modelled by the set of canonical names of the people who are referred to by the term ‘human’ in that specific context. Accordingly, a set like {John, Mary, Jones, Madeline} can be considered as the interpretation of the term ‘human’ in a certain context. The elements of such a set are fully represented by the canonical names ‘John’, ‘Mary’, etc. Although we are still far from the full presentation of humans in flesh and blood, we have a model which enables us to formalize certain fragments of language in which talking about humans is nothing but talking about those four persons. By developing such models, we can formalize larger fragments of language. An alternative dialogical approach for dealing with this concern is put forward by Rahman et al. (2018, Sect. 10.4). This dialogical alternative is inspired by Martin-Löf (2014).

  14. 14.

    This picture needs to be refined. As we will shortly see, even in the descriptional reading the whatness of the objects of the subject term is mentioned, albeit only implicitly.

  15. 15.

    In CTT, the well-formation is not only syntactic but also semantic. Consider, for example, the predicate Hungry. The well-formedness of this predicate can be expressed by ‘Hungry(x): prop (x: Animal)’, which reveals not only the correct syntactical use of that predicate but also the semantic domain of the objects of which that predicate can be true.

  16. 16.

    In their thorough and meticulous discussion of Plato’s Cratylus, Lorenz and Mittelstrass (1967) highlight the distinction between naming (ὀνομάζειν)—as establishing what something is—and stating (λέγειν)—as establishing how something is. They (1967, p. 6) point out that “[t]he subject has to be effectively determined, i.e., it must be a thing correctly named, before one is going to state something about it”.

  17. 17.

    In CTT, the judgment that the proposition B(a) is true is usually represented by ‘B(a) true’. But as long as we are considering a proposition itself (without making any judgment that it is true) we do not really need to add ‘true’.

  18. 18.

    Ibn Sīnā proposes this example in the logic part of al-Išārāt (1983, Chap. 10.1, pp. 501–502). We are grateful to Alexander Lamprakis for drawing our attention to this example.

  19. 19.

    That Imraʾa al-Qays belongs to the category Human is not explicitly mentioned in (6) and (7). But it is necessary be added to the picture. See the next section for more details on this issue.

  20. 20.

    Notice that if we simply take Good-Poet(x) as a predicate, then from Good-Poet(a) we cannot infer either that a is good or that a is a poet.

  21. 21.

    The example is borrowed from the logic part of al-Išārāt (1983, Chap. 4.2, p. 265).

  22. 22.

    On how and why the bearer of the subject of descriptional propositions is concealed see Schöck (2008, pp. 350–351).

  23. 23.

    A truth-maker is in fact a rudimentary form of what is called proof-object in CTT. See Ranta (1994, p. 54). However, in the context of this chapter, we assume ‘truth-maker’ and ‘proof-object’ to be synonymous terms. It is also worth mentioning that in the CTT-framework one and the same true proposition has according to the rule more than one object which makes it true.

  24. 24.

    To put it in more technical language, if in the proposition ‘every B, as long as it exists, is C’, the bearers of the description B are of the type A, then first(z): A must be understood as what Sundholm (1989, p. 10) calls ‘A-injection’.

  25. 25.

    See Chatti (2019b, pp. 113–114). Since Ibn Sīnā considers existential import for A-form propositions, Chatti emphasizes that the above formulas must include the conjunct ‘(∃x)S(x)’.

  26. 26.

    Hasnawi and Hodges (2017, p. 61) label such propositions as ‘(a-)’ which can be considered as an abbreviation for ‘A-form lāzim’ propositions.

  27. 27.

    This formalization is in accordance with what Hodges and Johnston (2017, p. 1061) put forward following Rescher and vander Nat (1974). The conjunct ‘(∃t)(∃x)Moving(t,x)’ is added to guarantee the existential import of the proposition.

  28. 28.

    For a detailed technical definition of time scales, see Ranta (1994, Sect. 5.1).

  29. 29.

    For a detailed technical definition of time spans, see Ranta (1994, p. 115).

  30. 30.

    See Ranta (1994, Sect. 5.4).

  31. 31.

    This terminology is borrowed from Recanati (2007a, 2007b).

  32. 32.

    Recall that as pointed out before, it is assumed that a proposition has different truth-makers (or proofs or justifications). In the present context this amounts to the assumption that a proposition has different truth-makers during different time spans. That a proposition is true in a specific time span is equivalent to that one of its truth-makers is obtained in that time span.

  33. 33.

    See Ranta (1994, p. 108).

  34. 34.

    In order to avoid notational complexity we omitted one variable within the timing function. Indeed, strictly speaking, the correct formalization must be τ(x, b(x)) = d: span(T) (x: Human, b(x): Running(x)).

  35. 35.

    Hasnawi and Hodges (2017, p. 61) label such propositions as ‘(a-m)’, which can be considered as an abbreviation for ‘A-form muwāfiq’ propositions.

  36. 36.

    This formalization is suggested by Hodges and Johnston (2017, p. 1061), following Rescher and vander Nat(1974). Again, the conjunct ‘(∃t)(∃x)S(t,x)’ is added to preserve the existential import.

  37. 37.

    For Ibn Sīnā time is the number or magnitude of motion. Although he does not explicitly talk about events, his definition of time shows that he does not consider an independent existence for it. This suffices to convince us that the enrichment approach is preferable to the saturation approach. For a detailed discussion on Ibn Sīnā’s view regarding time, see Lammer (2018, Chap. 6).

  38. 38.

    See notes 25, 27, and 36.

  39. 39.

    The existence of the subject matter of non-mathematical propositions can be presented either by ‘logical games’ or by dialogical verification procedures. See note 13.

  40. 40.

    An alternative approach is based on the introduction of the notion of ontological dependence. See Rahman and Redmond (2015).

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Acknowledgements

We are thankful to Leone Gazziero (STL) , Laurent Cesalli (Genève), and Tony Street (Cambridge), leaders of the ERC-Generator project “Logic in Reverse: Fallacies in the Latin and the Islamic traditions,” and to Claudio Majolino (STL) , associated researcher to that project, for fostering the research leading to the present study. We should also thank Vincent Wistrand (UMR: 8163, STL) and Alexis Lamprakis (München) for many fruitful discussions from which we have benefited a lot. The present paper was written while Mohammad Saleh Zarepour was a Humboldt Research Fellow at LMU Munich. We are thankful to Alexander von Humboldt Foundation for their support.

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Rahman, S., Zarepour, M.S. (2021). On Descriptional Propositions in Ibn Sīnā: Elements for a Logical Analysis. In: Mojtahedi, M., Rahman, S., Zarepour, M.S. (eds) Mathematics, Logic, and their Philosophies. Logic, Epistemology, and the Unity of Science, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-53654-1_16

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