We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two kinds of import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle's system appears then as a fragment of a broader system which can be developed by using FOL. 1 The case of Frege is more complex; in his Begriffsschrift, he presents a square he assumes to be valid; but he formalizes the propositions in exactly the same way as Russell, and these formalizations do invalidate the square. 2 See for instance Quine 1950, Chapter 15, French translation (p. 96) and Russell 1959, French translation, Chapter VI (p. 83)where both authors assume that the universals do not have an import, when translated by conditionals in the modern symbolism.
In this paper, I raise the following problem: How does Avicenna define modalities? What oppositional relations are there between modal propositions, whether quantified or not? After giving Avicenna's definitions of possibility, necessity and impossibility, I analyze the modal oppositions as they are stated by him. This leads to the following results:(1) The relations between the singular modal propositions may be represented by means of a hexagon. Those between the quantified propositions may be represented by means of two hexagons that one could relate to each other.(2) This is so because the exact negation of the bilateral possible, i.e. 'necessary or impossible' is given and applied to the quantified possible propositions. (3) Avicenna distinguishes between the scopes of modality which can be either external (de dicto) or internal (de re). His formulations are external unlike al-Fārābī's ones.However his treatment of modal oppositions remains incomplete because not all the relations between the modal propositions are stated explicitly. A complete analysis is provided in this paper that fills the gaps of the theory and represents the relations by means of a complex figure containing 12 vertices and several squares and hexagons.
In this paper, I raise the following problem: what propositions have an import in Avicenna's modal logic? Which ones do not? Starting from the assumption that the singular and quantified propositions have an import if they require the existence of their subject's referent(s) to be true, I first discuss the import of the absolute propositions then I analyze the import of the modal propositions by considering Avicenna's definitions and the relations between these propositions. This leads to the following results: Avicenna's general opinion is that the affirmatives, be they assertoric or modal, have an import while the negatives do not. The possible affirmative propositions are given an import both in the externalist and the internalist post-Avicennan readings, provided that the subject is not impossible. However, the theory is not always clear, for the propositions containing ‘sometimes not’ are given an import, together with the negative necessaries containing ‘as long as it is P’, despite their negative character; the necessary affirmative propositions containing ‘as long as it is P’ are given an import, although they do not require it. In addition, Avicenna's analysis of the special assertorics E and O (containing the internal conditions ‘at some times but not always’) and their contradictories is erroneous, which does not help determine their import. But when correctly analyzed, these special E and O do not have an import, while their contradictories – I and A special assertorics respectively – have an import.
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