II. THE EXISTENTIAL IMPORT OF ASSERTORIC PROPOSITIONS

Avicenna presents different analyses of the assertoric propositions in his works. In al-ʿIbāra (the correspondent of De interpretatione), he presents all the traditional oppositions and defines them by considering the truth conditions of the propositions. In his view, the couples A / O and E / I are contradictories, A and E are contraries, I and O are subcontraries, I is the subaltern of A, whereas O is the subaltern of E. Subalternation is called tadākhul and is defined by means of the truth values of the propositions concerned. These truth values, which depend on the matters of the propositions, lead to a square of oppositions, for they make all the relations of the square valid.Footnote ⁵ They are presented both in al-ʿIbāra Footnote ⁶ and al-Ishārāt wa-al-tanbīhāt. Footnote ⁷

Now what about the existential import of all these propositions? Since Avicenna defines the propositions and their relations by considering their truth conditions, we propose the following definition to existential import:

Given this definition, what is the import of the different propositions? In al-Maqūlāt, Avicenna evokes explicitly non-existent objects and claims that the singular propositions whose subject does not have an existent referent are false when they are affirmative and true when they are negative. The examples he gives are the following: “Zayd who does not exist [al-maʿdūm] is seeing” and “Zayd who does not exist is blind”; these sentences are both false, but the first one contradicts “Zayd who does not exist is not seeing”,Footnote ⁸ which is true. Thus, in case the subject's referent is non-existent, both affirmatives are false while the negative sentence is true. So the negative sentence does not have an import while the affirmative one has an import because it cannot be true when the subject's referent does not exist.

We can also note that Avicenna compares between the sentences whose subject's referent is non-existent and those whose matter is impossible, for he gives the following example: “Stones (or Zayd who does not exist) are seeing” and “Stones (or Zayd who does not exist) are not seeing”Footnote ⁹ and says that these sentences are contradictory, which means that “Stones are seeing” is false while “Stones are not seeing” is true. Elsewhere,Footnote ¹⁰ he also says that both “Stones are sick” and “Stones are healthy” are false, while “Stones are not healthy” is true. Given these examples and the comparison between both kinds of sentences, we can see that the impossible sentences behave like those whose subject's referent is non-existent with regard to their truth values. This shows that what holds for the singular propositions holds also for the indefinite ones. Since the subject of the indefinites is general, they can be quantified, which means that what holds for them, also holds for the quantified propositions. So the quantified sentences have an import when they are affirmative and do not have an import when they are negative.

This same opinion is hold in other treatises, such as al-Najāt, where Avicenna says the following: “… For the negation is true of a non-existent subject, while the affirmation, whether indefinite (= metathetic) or definite, is true only of an existent subject, so it is true to say that ‘The griffin is not seeing’ [= ‘It is not the case that the griffin is seeing’], while it is not true to say that ‘the griffin is not-seeing’”.Footnote ¹¹ So, in al-Najāt too, he defends the same opinion. Note that the subject ‘griffin’ is not a proper name, so this opinion does not only hold for the singular sentences; it is more general.

This opinion is even set up as a principle by Wilfrid Hodges, who says in a recent article:

The import is then associated with the quality of the proposition, as it is in Apuleius’, Boethius’, Abelard's and Buridan's theories, although no formal expression of the lack of import, similar to that of Buridan, is provided by Avicenna.

However, Avicenna's text is not always clear, for he sometimes expresses O as follows: “Not all men are writers, but rather some of them”,Footnote ¹³ which gives it an existential import, because of the affirmative side of the sentence, and corresponds more to what Blanché calls the Y vertex (that is, ‘I ∧ O’) than to the standard O.

Furthermore, Avicenna gives other interpretations of the assertoric propositions in al-Qiyās (the correspondent of Prior Analytics), for he analyzes these propositions by introducing temporal considerations and other conditions. These analyses show that these various propositions have different truth conditions and that their behavior with regard to the import is also different.

In al-Qiyās and al-Ishārāt, Avicenna says first that some assertorics may be expressed thus: “Every S is P as long as it exists”, or thus: “Every man is laughing (at some times)”. The latter are called “General absolutes”. Other absolutes may contain “at some times but not always”; they are called “Special absolutes” or “existential”.Footnote ¹⁴ Other sentences have the following structures: ‘S is P as long as it is S’, or ‘S is P as long as it is P’.

Avicenna also gives precisions on the subject, when he says the following:

This reading is supposed to be the most general one, that is, to include the most cases, whatever additional condition they contain. It gives an import to A, for its subject is presupposed to be existent. But when the subject is a space-temporal being, such as men or animals, it does not exist eternally, this is why Avicenna adds “always or not always” and the condition “as long as it exists”Footnote ¹⁶ which specifies the time of its existence. The existence is not only space-temporal too, for the object could be abstract, as is the case with the icosahedron given as an example in al-Qiyās (p. 21) and elsewhere.Footnote ¹⁷ Its existence is said to be “in the mind”, that is, the object is considered as existent by the human mind, although it is not perceived, but only conceived. This precision is also given in al-Ishārāt, where Avicenna analyses the A proposition as follows:

In his commentary of this quotation, Paul Thom notes that Avicenna “rejects the idea that the subject-term of an absolute or modal proposition applies just to what actually exists”.Footnote ¹⁹ So the existence is not just space-temporal, it applies also to the mathematical and other abstract objects. Thom formalizes the “mental assumption” by means of what he calls the “simple de re reading of absolute and modal propositions, according to which the subject-term is ampliated to cover whatever can fall under that term”,Footnote ²⁰ which is rendered symbolically in the following way: “J_M”,Footnote ²¹ where ‘M’ means possible, and the whole expression means ‘the possible J’. A possible subject might be seen as a conceivable or a possibly existent subject, which means that the proposition has an existential import, whenever its subject is possible. It lacks import only when the subject is not possibly existent, as is the case with the griffin.

About this kind of existence (in the mind), Avicenna says what follows:

This means that the object must be considered as possibly existent in reality, even if it is conceived by the mind at the moment, for what the mind conceives is not just a mental entity, i.e. something that could exist only in the mind; rather it is an entity that could be real or that is possibly satisfied in the real world. So, what the affirmative proposition says is the following: the object existing in the mind satisfies P, not because it is mental but because it would satisfy P if it were existent in the real world. Only in that case, the affirmative proposition could be true of it. Thus the affirmation assumes it as existent, because the inexistence of the objects makes it impossible for them to possess properties. This is perhaps the difference between the icosahedron taken as an example and the griffin mentioned later. The icosahedron is conceived not just as existing in the mind, i.e. as a psychological entity; rather it is a mathematical entity that the mind conceives as such, and it is possibly existent in the real world, while the griffin could never exist in the real world, so that no affirmative proposition having the griffin as a subject could be true as Avicenna says in the following quotation:

To make sense of the difference between the mathematical entities and the imaginary ones, we could say that the griffin could never be other than imaginary, while the mathematical objects could not be seen as pure illusions. However, the mathematical (and imaginary) objects are more clearly analyzed if one considers the distinction between the “externalist” and the “essentialist”Footnote ²⁴ readings of the subject term made by the post-Avicennan logicians of the 12^th and 13^th century. We will return to this analysis in the last section.

In Manṭiq al-mashriqiyyīn, however, Avicenna seems to express things in a different way, for he writes:

This quotation is puzzling, because of the words ‘impossibly existent’ and ‘non-existent’ which seem to suggest that the subject of an affirmative sentence could be non-existent. But when one scrutinizes it more carefully, one finds that it does not really change Avicenna's position about the import because he does not say that the affirmative sentence, in the cases given, is true. And this is important because the import of the affirmative sentence is defined by the fact that the sentence is not true when its subject does not exist. So we cannot say that Avicenna changed his mind with regard to the import of the affirmative sentences.

What about the general absolute universal propositions? These propositions are said to be not contrary by Avicenna, for they may be true together. For instance, “Every man is laughing (at some times)” and “Every man is not laughing (at some times)” are both true. The latter is explained as follows by Avicenna: “the predicate ‘is laughing’ can be negated from every man, at some times”.Footnote ²⁶ Such a reading gives it an existential import, for ‘at some times’ means ‘at some particular time in his life (wujūdihi)’.Footnote ²⁷ Hence the sentence cannot be true if men are not existent.

Likewise, O general absolute and A general absolute, that is, “Some men do not laugh, at some times” and “Every man is laughing at some times” as well as I general absolute, e.g. “Some men are laughing at some times” and E general absolute, e.g. “Every man is not laughing at some times” seem to have an existential import, for a non-existent thing cannot be so and so at some times, given its non existence in space and time. Their contradictories, according to Avicenna, are the perpetual sentences, which are sentences containing either ‘always’ or ‘never’. In Avicenna's text, the couples of contradictories are stated as follows:

He also states the following two propositions and their contradictories:

Among these propositions, those that contain ‘at some times’ have an import, even when they are negative, given the way they are analyzed by Avicenna; consequently, their contradictories should not have an import, for “Contradiction is valid only if both propositions have a different import”.Footnote ³¹ As a matter of fact, the propositions containing ‘never’ do not have an import, since they are negative. However, the import of those containing ‘always’ is not clear in Avicenna's theory, for they contain a temporal operator, which means that what the proposition says holds in a space-temporal context. Because of this space-temporal character, it is hard to imagine that the proposition is true if its subject's referent is non-existent. But they are nevertheless the contradictories of propositions which are said to have an import, so according to the condition above, they should be without import, or if they have an import, their contradictories, that is, the propositions containing ‘sometimes not’ should be without import. But Avicenna seems to give an import both to the propositions containing ‘always’ and to those containing ‘sometimes not’; this shows that there is something wrong or at least unclear in the theory.

As to the special assertoric propositions and their contradictories, they are stated as follows in al-Ishārāt, for instance:

The two special assertorics A and I propositions have an import because of the affirmative part of each conjunction, but their contradictories do not have import, given that the disjunctions are true if at least one disjunct is true, and we find in each of these disjunctions a negative part which does not have an import.

However, although A special assertoric's contradictory is expressed in a right way, I special assertoric's contradictory is not rightly stated, for it should rather be the following: ‘Every S is either always P or never P’. This proposition is not equivalent to ‘Either every S is always P or no S is ever P’ (stated by Avicenna) because in modern logic (x)[Ax ⊃ (Bx ∨ ~Cx)] differs from (x)(Ax ⊃ Bx) ∨ (x)(Ax ⊃ ~Cx). Note that this error is not related to the Arabic formulation; it is rather a logical error, for Avicenna's formulation separates explicitly the two parts of the disjunction, but this separation is wrong.

Now what could we say about the two special E and O assertoric propositions and their contradictories? Are these propositions and their contradictories rightly stated? It seems that they are not, for given that the special assertorics E and O are themselves negative, and given that their negation is internal and puts on the whole combination “sometimes but not always” in both cases, they can be formalized in the following way: E special assertoric: No S is P (sometimes but not always) = (x)[Sx ⊃ ~(x is sometimes P ∧ x is sometimes ~P)] = (x)[Sx ⊃ [~(x is sometimes P) ∨ ~(x is sometimes ~P)] (according to the De Morgan law: “~(p ∧ q) ≡ (~p ∨ ~q)”), which in turn leads to the following formula: (x)[Sx ⊃ (x is never P ∨ x is always P)], given that: ‘never’ = ‘not sometimes’, and ‘always’ = ‘not sometimes not’.

Consequently, it is E itself that contains a disjunction, while its right contradictory should have the following structure: ~(x)[Sx ⊃ (x is never P ∨ x is always P)], which is equivalent (by the definition of the quantifiers) to: (∃x)[Sx ∧ ~(x is never P ∨ x is always P)], itself equivalent to: (∃x)[Sx ∧ [~(x is never P) ∧ ~(x is always P)] (by the other De Morgan law, that is, “~(p ∨ q) ≡ (~p ∧ ~q)”). This leads to the following formula: (∃x)[Sx ∧ (x is sometimes P ∧ x is sometimes ~P)] (which is the formal counterpart of I special assertoric above) given that ‘not never’ = ‘sometimes’, and ‘not always’ = ‘sometimes not’. In ordinary language, ~E special assertoric says simply the following: Some Ss are sometimes P and sometimes not P. Here, ~E = I.

The same could be said about O special assertoric, which is stated as follows: “Some Ss are not P, sometimes but not always”. If we formalize it, we obtain the following: (∃x)[Sx ∧ ~(x is sometimes P ∧ x is sometimes ~P)], which is equivalent to: (∃x)[Sx ∧ [~(x is sometimes P) ∨ ~(x is sometimes ~P)], itself equivalent to: (∃x)[Sx ∧ (x is never P ∨ x is always P)], that is, “Some Ss are either never P or always P”.

Its contradictory is thus the following: ~(∃x)[Sx ∧ (x is never P ∨ x is always P)], that is, (x)[Sx ⊃ ~(x is never P ∨ x is always P)], itself equivalent to: (x)[Sx ⊃ ~(x is never P) ∧ ~(x is always P)], thus in turn to: (x)[Sx ⊃ (x is sometimes P ∧ x is sometimes not P)]. In ordinary language, this is expressed as follows: “Every S is sometimes P and sometimes not P”, which corresponds to A special assertoric above, and seems very natural since ~O = A.

Consequently, E and O special assertorics, when they are rightly formulated, do not have an import, given that they are expressed by disjunctions containing at least one negative element. By contrast, their contradictories have an import, for they are expressed by conjunctions where at least one element is affirmative. Therefore, unlike what Avicenna says, E and O special assertorics are just the contradictories of I and A special assertorics respectively.

Now, what does Avicenna say about these special assertorics in al-Qiyās? In page 49, he evokes the particular (I and O) special assertorics as follows:

This is not clearly expressed because Avicenna does not say precisely what are the logical relations between the components of the contradictory propositions (are they conjunctions or disjunctions?). As we noted above, I special assertoric's contradictory (i.e. E special assertoric) is a disjunction, while O special assertoric's contradictory (i.e. A special assertoric) is a conjunction. But this difference is not specified by Avicenna himself, which makes his account insufficient, and does not help determine the import of these propositions in his theory.

What about the propositions containing ‘as long as it is S’, called “descriptional” par some commentators,Footnote ³⁷ that is, Aw, Ew, Iw and Ow?

The affirmatives may be true, but they would be false if the subject does not refer to any existing thing. They seem, then, to have an import. This import is suggested explicitly in al-ʿIbāra where Avicenna claims: “there is no affirmation at all unless the subject is [taken to be] as we said”.Footnote ³⁸ This opinion, however, can be challenged when the descriptional sentences are concerned, for instance in the modern account of the universals.

As to the negatives, they would have an import if they were false and would be without import if they were true, whenever the subject's referent is non-existent. Consequently, they should not have an import if the negation is external. For instance, a sentence like “The griffin is not a cat as long as it is a griffin” is true despite the non existence of griffinsFootnote ³⁹ when the negation is external, i.e. when the whole sentence is interpreted as “It is not the case that the griffin is a cat as long as it is a griffin”.

Those which say: “S is P as long as it is P” are true since they could be interpreted as “Every S is P iff it is P” and “Some Ss are Ps iff they are Ps”, in which case their consequent is always true. However, the negative propositions ‘No S is P as long as it is P’ and ‘Some Ss are not P as long as they are P’ are both false if their antecedents are true, given that their consequents are both always false.

The two affirmatives should not have an import, given that they are always true, but we do not find in Avicenna's text anything that suggests that they are without import. On the contrary, he seems to give them an import, by saying “All what is described as B, is necessarily and always described as A, as long as its essence exists, as long as it is A, and A is not the predicate but rather a part of the predicate”.Footnote ⁴⁰

As to the negatives, they would have an import if they are false, and would be without import if they are true. As a matter of fact, Avicenna holds them false for he says: “[As to] ‘Not every B is A as long as it is A’, this negative [sentence] is never true”.Footnote ⁴¹ The word ‘never’ suggests that they are even always false; therefore they have an import in his theory. Let us now turn to the import of the modal propositions in Avicenna's theory. In order to clarify it, we will first present the definitions of the modalities provided by Avicenna.

III. AVICENNA'S DEFINITIONS OF NECESSITY AND POSSIBILITY

Avicenna talks about several kinds of possibility, which are the unilateral possible, that is “possibly α”, the bilateral possible, i.e. “possibly α and possibly ~α”, and what he calls the “narrowest-possible” (translation Hodges 2010Footnote ⁴²), i.e. what is neither actual nor necessary, nor impossible, which he explains as follows: “the third meaning is that which is not actual and is not necessary (ḍarūrī) in the future”.Footnote ⁴³

Necessity may be absolute, as in (1) “God is necessarily existent”, or conditional when a condition is added. The conditions are the following: (2) “as long as S exists”, (3) “as long as it is S”, (4) “when it is P”, (5) “In some determined times”, or (6) “at regular times” not determined precisely, e.g. in “Every man is breathing”. These conditions are stated in al-Qiyās as well as al-Najāt. Footnote ⁴⁴ In Manṭiq al-mashriqiyyīn, they are slightly different.

The sentences containing (2), (5) and (6) have an import, for they contain the existential condition or temporal considerations. As to (3) and (4), if we consider Avicenna's examples, we would say that they have an import too, although some passages of al-Qiyās and al-Ishārāt Footnote ⁴⁵ seem to suggest that they may be without import. We will return to these passages in section 4.

In addition, Avicenna distinguishes in al-ʿIbāra between two interpretations of possible propositions, which are the following:

1. (1) “Every man is possibly a writer”

2. (2) “It is possible that every man is a writer”Footnote ⁴⁶

We will call them internal when the modality is in front of the predicate as in (1) and external when the modality is in front of the whole sentence as in (2). This distinction is not equivalent to the medieval de re / de dicto distinction, which is more complex; it simply involves the scope of the modal operator.

The contradictions between modal quantified propositions are stated very clearly in al-Qiyās (pp. 49–50), when the modality is external.Footnote ⁴⁷ The simple (de dicto) ones are the following: □A / ◊O; □E / ◊I; □I / ◊E; □O / ◊A.Footnote ⁴⁸

IV. THE EXISTENTIAL IMPORT OF MODAL PROPOSITIONS

Now, what is the import of these modal propositions? For the singular proposition “Zayd is a writer” (formalized here by α), Avicenna gives the following truth values: “Possibly α” and “Possibly ~α” are both true, while “Not possibly α” is false; “Necessarily α” and “Necessarily not α” are both false while “Not necessarily α” is true.Footnote ⁴⁹ These values correspond to the case where Zayd exists, for in that case, the two possible propositions are both true, while the two necessary ones are both false. Avicenna does not seem to consider the case where Zayd does not exist in his evaluations, for it is when Zayd exists that either α is true or ~α is true and consequently, ◊α and ◊~α are both true, while □α and □~α are both false.

But what happens if Zayd does not exist?

In this case, only ~α is true, hence ◊~α is true and □α is false, being contradictories. Thus ◊~α does not have an import, while □α has an import.

What about ◊α and □~α? Given that α is false, ◊α is indeterminate: it could be true or false. If it is true, it does not have an import, but if it is false, it has an import. The same may be said about □~α.

Thus, absolutely speaking, we may have the following situations:

1. ◊α : with import and □~α: without import

or

2. ◊α : without import and □~α: with import

Both cases validate the modal square, but which one is hold by Avicenna? To answer this question, we have to take into account the following facts:

1. Avicenna compares between ‘stones are men’, i.e. the affirmative sentences whose matter is impossible, and ‘Zayd who does not exist is seeing’, i.e. the affirmative sentences where the subject is empty; since the sentences whose matter is impossible are always false when they are affirmative, we would say that the affirmative sentences whose subject is empty are always false too; consequently α is always false, and ◊α is false too when Zayd is non-existent; if so, ◊α has an import, since it could not be true when the subject is empty.

2. ‘never’ and ‘□~’ are very close, even if they are not equivalent; since ‘never’ does not have an import, ‘□~’ should not have an import too. In addition □~ = ~◊, so if “◊α” has an import, its contradictory “~◊α” does not have an import by the above theorem (section 2).

The above interpretation is corroborated by the following passage:

Given this treatment of non-existent objects, we may say that the possible affirmative propositions have an import, while the negative necessary ones do not have an import, only because they are negative. This opinion may be generalized to the quantified propositions, for the possible A-proposition (1) ‘Every man is possibly a writer’ (A◊) is hold true, because the subject “man” is not empty, while the possibility A-proposition (2) ‘Possibly Every man is a writer’ (◊A) is said to be “doubtful” [“qad yushakku fīhi”]Footnote ⁵¹ only because the possibility operator puts on the quantifier, that is, only because it is not very likely that every existent man is a writer as appears in what follows: “According to some people, it is impossible that all [existent] men are writers”.Footnote ⁵² This means that (2) presupposes the existence of men. Thus the second proposition has an import and the first one too, because it is affirmative and because the possibility affects only the predicate.

The import of the substantive necessity Ai.e. “Necessarily every horse is an animal (as long as it exists)” (= □A) and the necessary A “Every horse is necessarily an animal (as long as it exists)” (= A□) is even more obvious for they include explicitly the clause of existence.

However, in al-Qiyās, Avicenna raises a question close to the problem of existential import by saying:

His answer is:

If so, A◊ applies also to future [i.e. non-existent yet] persons. Does this mean that this proposition does not have an import, after all, i.e. may be true even when its subject is non-existent? This raises a problem in the theory, because it would contradict what is said in al-ʿIbāra to the effect that the affirmative propositions cannot be true in case the subject's referent is not existent.

Furthermore, in al-Qiyās, Avicenna says that “Possibly every animal is a person” is true,

The example given in that passage suggests the narrowest-possible, for it may be read as follows: ~(every animal is a person) ∧ ◊(every animal is a person) ∧ ◊~(every animal is person), and formalized as: O ∧ ◊A ∧ ◊O. We may think that this proposition does not have an import, but if we take into account the formula which expresses it, it would have an import for if ◊A has an import, both “◊A ∧ ◊O” and “O ∧ ◊A ∧ ◊O” have an import too, because if ◊A is false in case the subject's referents are non-existent, the two conjunctions will be false too; they would also require the existence of the subject's referents to be true, and this is exactly what it means for a proposition to have an import.

This analysis refutes what Allan Bäck says about the import of the possible propositions in Avicenna's theory, for according to this author, the general possible has an import while the narrow-possible does not have an import. This opinion is hold in the following quotation:

By saying that, A. Bäck refers to al-Qiyās (pp. 34–6), where Avicenna analyzes the example “Zayd is sitting”. However, what Avicenna says is: “This sitting is not perpetually existent nor perpetually not existent as long as the essence of the subject exists” and he adds “The thing can never sit at all” (al-Qiyās, pp. 34–5); so unlike what A. Bäck says, it is the sitting that need not exist at all, not Zayd; consequently the analysis Avicenna provides in this text does not prove that the narrow-possible does not have import; on the contrary, Avicenna insists on the fact that the subject (Zayd) exists and says that what may be not existent at all is the sitting. So the narrow-possible has an import as we noted above, unlike what A. Bäck claims. In addition, when S's referent does not exist, ‘S is not P’ is true; it is not false, as A. Bäck says.

In addition, the fact that the possibility is de dicto does not necessarily mean that the subject is not satisfied. For it is clear that what is meant by Avicenna's example is that the hypothesis that in some (future) world, ‘all animals would be human beings’ is conceivable, in case all the other animals disappear in that particular time. But this does not mean that the extension of the subject “animal” is empty. Rather it illustrates a particular situation where the extension of the word “animal” is drastically reduced so that it becomes equivalent to that of “man”.

In the passage of Avicenna's text cited above, Avicenna does not evoke necessity at all. But he does evoke necessity elsewhere in al-Qiyās, while talking about the same kind of situation, when he says:

This passage suggests a “combined de dicto / de re sense”Footnote ⁵⁸ as Paul Thom calls it. For the de dicto meaning is related to the quantifier, while the de re meaning is related to the matter of the proposition, i.e. the relation between the subject (animal) and the predicate (man), which is essential. This de dicto possibility could be interpreted as being ‘truth is some possible circumstance (or world)’ other that the actual one. But this ‘world’ should not be interpreted as some ‘possible alternative’ of the actual world; rather it is some ‘future’ possible world, which means that the de dicto modality is temporal rather than alethic, unlike the de re necessity, which seems to be alethic, its temporal character being simply a consequence of its necessity.

As to the possible affirmative particulars, they have an import, for all the reasons given above and because ◊I is equivalent to ~□E; consequently if □E does not have an import (because it is negative), then its contradictory has an import. The same may be said about I◊, i.e. ~(E□).

Furthermore, if □E and E□ do not have an import, then ◊E and E◊ should not have an import too, given that the possible propositions are implied by the necessary ones in Avicenna's frame, so if the necessary propositions are true when the subject's referents do not exist, then the possible propositions will be true too, and would not have an import for that reason. The same may be said about ◊O and O◊, which are the contradictories of □A and A□ respectively and are implied by O; so given that O may be true when the subject is empty, ◊O and O◊ may also be true; hence they do not have an import for that reason and because they contradict the necessary universal affirmatives which do have an import.

As shown in “Avicenna on possibility and necessity”,Footnote ⁵⁹ the narrow-possibility propositions, are expressed as follows:

which is equivalent to

The same may be said about the universal negative, that is:

which is equivalent to:

Both are formalized as: ◊E ∧ ◊I (= ◊I ∧ ◊E).

These propositions have an import because one of their conjuncts has an import. Their contradictories are respectively: □O ∨ □A and □E ∨ □I.

Both disjunctions do not have an import because □O and □E do not have an import.

As to A narrow-possible, that is: “Every S is ◊P ∧ ◊~P = Every S is ◊P ∧ Every S is ◊~P = A◊ ∧ E◊”,Footnote ⁶⁰ it has an import because A◊ has an import. Its contradictory, i.e. ~(A◊ ∧ E◊) = ~A◊ ∨ ~E◊ = O□ ∨ I□ does not have an import because O□ does not have an import. In the same way, I narrow-possible is formalized as: Some Ss are ◊P and ◊~P. Its contradictory is expressed by: Every S is □~P or □P.Footnote ⁶¹

If the affirmatives have an import, their contradictories should not have an import because they contain a negative predicate and a disjunction; therefore they will be true even in case the Ss do not exist, given that when □~P is true, the disjunction, that is, the consequent of the conditional, is true.

The narrowest-possibility propositions, that is:

seem to have an import, for in each conjunction there is at least one conjunct which has an import, namely the affirmative possible proposition in each conjunction. Their negations do not have an import, because in each disjunction, there is at least one disjunct without import, namely the negative one(s). These negations are respectively the following: “I ∨ □E ∨ □I”,“E ∨ □E ∨ □I”, “A ∨ □A ∨ □O” and “O ∨ □A ∨ □O”.Footnote ⁶³

As to the necessity “descriptional” propositions, that is, □Aw, □Ew, □Iw and □Ow and their correspondent possibility ones, that is, ◊Aw, ◊Ew, ◊Iw, ◊Ow, we have to examine them closely to determine their import. The logical structure of the universals involve a conditional, for ‘Every S is P as long as it is S’ says what follows: ‘Every S is P if it is S’ and may be formalized as ‘(x)[Sx ⊃ (Sx ⊃ Px)]’ which is equivalent to ‘(x)(Sx ⊃ Px)’. One can add the modalities afterwards.

If S's referents do not exist, can they be true? According to H. Lagerlund, the condition “as long as it is S” is added to the existential condition, for he says:

However, the text considered is far from clear, for Avicenna says the following:

What he uses in front of “as long as its essence is existent” is a real negation expressed by “nor does it mean”; so, it seems that unless an “essence” as essence cannot be non-existent, the import is not required, for the text itself does not say that the import is required for this kind of propositions. But we may interpret the negation as denying the fact that the subject always exists. In that case, the proposition has an import, for the subject S is presupposed existent in the time where it is S.

In al-Qiyās, Avicenna claims:

Here too, as in al-Ishārāt, the durable existence of the subject's referents is not explicitly required. But the ‘moving’ and the ‘changing’ are not just concomitant, for the first implies the second: by definition a moving thing necessarily changes. If so, the logical relation between the subject and the predicate is an implication, not a conjunction, and if one considers the modern interpretation of the implication (which is material and philonian), one would say that the import is not required, because this implication is true when its antecedent is false. But the philonian implication is not hold by Avicenna. On the contrary, the syllogisms he holds such as Darapti, for instance, require the conditional to have a true antecedent, because the particular conclusion in that mood could not be true if the antecedents of its two universal premises are false. This being so, the proposition should have an import, even when interpreted as an implication.

As to the propositions where the condition is related to P, they are expressed thus:

• – Every B is □A as long as it is A

• – Some Bs are □As as long as they are As

These should be true, whatever the truth value of the antecedent is, because ‘As long as it is A’ may be formalized as ‘Ax ≡ Ax’, which is always true.

But the negative sentences of this kind, which are expressed as follows:

• – No B is □A as long as it is A

• – Some Bs are □ not As as long as they are As

contain ‘~Ax ≡ Ax’ as a consequent; since this formula is always false, the whole proposition would be false if Bx is true, i.e. if there are Bs and it would be true if there are no Bs, that is if Bx is false.

Given that the affirmatives are always true, they should not have an import. But Avicenna says:

Hence, he gives them an import quite explicitly.

As to the negatives, Avicenna considers the assertoric propositions of that kind as false and even always false as we saw above (section 2). As a consequence, the necessary ones are false too. But saying that these propositions are never true means that they have an import, for if they are always false, their antecedent is always true, hence it is always satisfied, and consequently never empty.

As to the possible propositions of that kind, they are expressed as follows:

• – Every B is ◊A as long as it is A

• – Some Bs are ◊As as long as they are As

• – Every B is ◊ not A as long as it is A

• – Some Bs are ◊ not As as long as they are As.

The affirmatives are true; so they should not have an import. But if the condition of existence is required in this kind of sentences too, they would have an import. Note that if possibility is interpreted in the narrow meaning, we would have in all cases a negative part in the consequent, which would make it false, because if we say “Every B is ◊A and ◊~A as long as it is A”, the second conjunct is contradictory, in which case the whole proposition would be false if the antecedent is true, and true if the antecedent is false. The negative assertorics are hold false by Avicenna, so the possible ones are false too; consequently the possible universals and particulars seem to have an import.

V. EVALUATION OF THE THEORY

Avicenna's position is not always clear although, generally speaking, he defends the opinion that the affirmatives have an import, while the negatives do not.

We may try to clarify Avicenna's position with regard to the subject-term, and consequently the import of the propositions in the light of the distinction between the essentialist and the externalist readings of the subject-term made by the post-Avicennan logicians.

This distinction is first made by Fakhr al-Dīn al-Rāzī who expresses it as follows (as reported by Tony Street):

Here is how Rāzī first made the distinction, for the a-proposition (“Every J is B”):

The first reading is the essentialist, while the second is the externalist. The essentialist reading gives an import to all the true affirmative propositions whose subject is a conceptual (for instance, a mathematical) entity, such as the icosahedron (Avicenna's example), and the triangle given here as an example. The proposition presupposes the existence of these subjects, (for he says “were it to exist”) even if this existence is not external or actual. The second reading – the externalist one – does not give the same truth conditions to the proposition. For the truth of the proposition depends on the external existence of the things that satisfy the subject-term. This is why Rāzī considers the proposition “every figure is a triangle” as true in this reading only “if the only figures existent externally were triangles”: in this particular situation of the external world, the proposition would be true and would have an import. As to the proposition “every septagon is a figure”, it is false in the externalist reading, if there are no septagons in the external world. Given that it is false in that external reading because the subject is non-existent, it has an import, by our definition above, for this means that the proposition, in that reading, requires the existence of the subject to be true. The same proposition is considered by Rāzī as false in the essentialist reading, which probably means that he takes the septagon to be a non-existent object. If so, the proposition has an import too in the essentialist reading, since it requires the existence of the subject to be true.

The essentialist reading seems to apply mostly to the mathematical entities according to Ṭūsī too, who takes Avicenna's own example and says:

The icosahedron is thus taken to exist as an “intelligible reality”,Footnote ⁷⁰ which gives such propositions an existential import when interpreted in the essentialist reading. But, as we saw above with the example of the griffin, the imaginary beings could not exist in Avicenna's theory and the same may be said about the impossible entities. So all affirmative propositions whose subjects are imaginary or impossible do not have an import, whether in the externalist or in the essentialist readings. This opinion seems to be widely shared by the post-Avicennan logicians, except Khunājī (d. 1251) and Abharī (d. 1265) in some texts.Footnote ⁷¹ The latter two logicians admit the existence of impossible entities, which leads to some strange and undesirable results, such as the falsity of all E-propositions and the truth of all I-propositions.Footnote ⁷² Since Avicenna does not admit the existence of the impossible and imaginary entities, he avoids these undesirable results. His theory gives an import to the affirmative propositions, whether they are interpreted in the externalist or the essentialist reading, because both readings give the same results with regard to the import, provided that the subject-term is a possible entity.Footnote ⁷³

As to the proposition “Possibly every animal is a man” (section 4 above), which is comparable to Rāzī's proposition “every figure is a triangle”, Avicenna seems to interpret it in an externalist way, because he holds it true if the situation of the world is such that every single animal is a man. In that interpretation, it would have an import because the only reason why it is true is the fact that the only existent animals are men in that (future) situation. In that reading, the proposition has an import, because the subject is posited as existent in that particular future situation.

When interpreting the possible proposition “Every person is possibly a writer”, Avicenna says explicitly that the subject “person” should refer to all existent persons, whether in the past, the present or the future.Footnote ⁷⁴ So his reading of the subject-term seems to be externalist too, if we consider for instance, Abharī's definition of the externalist reading, which is the following:

In this case, it has an import if the future persons are said to be so-and-so (or possibly so-and-so, in Avicenna's example) in so far as they are existent, but it would not have an import, in Rāzī's account of the externalist reading above, if the future persons are not assumed to be existent (in that future situation). According to Paul Thom, this reading, as it is rendered by Abharī, extends “the ut nunc reading” of the “medieval Latin logicians”, because it talks about “all actual Js, past, present and future”.Footnote ⁷⁶ If so, there is no incoherence in Avicenna's theory, for this possible proposition has an import in that particular reading, despite the fact that it talks about future persons too, provided that these persons are posited as existent in that future situation.

We may also interpret Avicenna's example in the essentialist reading, if we consider the following part of the quotation given above (section 4): “‘every person’ means everyone who is a person”.Footnote ⁷⁷ In that case, the subject-term may be seen as the “essence” of the person, for the essentialist reading is expressed as follows by Taḥtānī (d. 1365), for instance: “‘Every J is B’ is considered at times according to the essence (whereupon it's called ‘essentialist’, as though [the subject] is an essence in a proposition used in the sciences)…”.Footnote ⁷⁸ In this reading, the proposition has an import too, because its truth does not depend on the external existence of the persons, but on the existence of the essence.

So Avicenna seems to admit both the externalist and the essentialist readings. His opinion on the essentialist reading does not extend to the impossible objects, however, unlike Khunājī's opinion. So the propositions with an impossible object are false when they are affirmative and true when they are negative. In both the externalist and the essentialist readings, the affirmative propositions have an import, while the negative ones do not have an import.

But we find some confusions in his texts especially in his analysis of some kinds of negative propositions. For he gives an import to both the propositions containing ‘sometimes not’ and those containing ‘always’, But these propositions could not have the same import, because they contradict each other. Since “sometimes not” equals “not always”, the propositions containing them, e.g. “People do not always laugh” could be analyzed as follows: “It is not the case that people always laugh”. This makes them true even if no person exists. Avicenna's analysis of this kind of negative sentences is not therefore very convincing, nor logically correct.

Another confusion is related to the sentences containing ‘as long as it is P’, which are always true when they are affirmative. Because they are always true, they should not have an import. But Avicenna seems to give them an import because of their affirmative character. This does not match with the logical structure of these propositions which are tautological, that is, true in all cases, therefore even when the subject does not exist.

The analysis of the special negative assertorics where the temporal conditions are internal is also confused, for Avicenna does not give their right structures, as we saw in section 2. Consequently, their import is not determined clearly in his theory. But when one analyzes them in the right logical way, one finds that the special negative assertorics do not have an import, while the affirmatives have an import.

As to the descriptional sentences, if we consider them for themselves, they do not require existential import, even when they are necessary, for “Every griffin is necessarily winged, as long as it is a griffin” may be hold true, even if griffins do not exist, given that the sentence says what follows: ‘Every S is □P if it is S’. This is formalized as: ‘(x) [Sx ⊃ (Sx ⊃ □Px)]’ which is equivalent to ‘(x)(Sx ⊃ □Px)’ and does not seem to require the existence of griffins to be true, because one could hold it true only on the basis of the definition of the griffin, even if that object is non-existent. We may even give descriptional sentences that are true independently of the context, for instance, the following: “Every griffin is necessarily imaginary as long as it is a griffin”. If so, the descriptional (universal) propositions do not behave as the substantive ones and do not require the existence of their subjects to be true.