Al-Ḫūnaǧī on essentialist and externalist propositions and inferences from the impossible

ABSTRACT

Afḍal al-Dīn al-Ḫūnaǧī (d. 1248) is one of the most influential Arabic logicians who departed from and argued against Avicenna in various places in his logical works. This paper is about al-Ḫūnaǧī’s account of inferences from the impossible. In this paper, we will overview his formulation of inferences about three main occurrences of impossibility in logic and language: propositions with impossible subjects, syllogism about impossible situations, and implications from a contradictory pair. All these are based on a distinction between two ways of reading the subject term of a proposition which al-Ḫūnaǧī borrowed from Fakhr al-Dīn al-Rāzī, namely the essentialist and the externalist. Later Arabic logicians raised a crucial objection to al-Ḫūnaǧī’s account of the essentialist reading. They argued that all universal propositions are false in this reading. If that is true, many of al-Ḫūnaǧī’s proofs will be trivially valid and redundant. I will argue that the falsity of these propositions does not preclude their truth.

KEYWORDS:

• Arabic logic
• al-Ḫūnaǧī
• impossibility
• syllogism
• paraconsistency

1. Introduction

Afḍal al-Dīn al-Ḫūnaǧī (d. 1248) may be considered the second greatest Arabic logician after Avicenna. He departed from Avicenna in various topics, never shied away from criticizing Avicenna, and formulated numerous new ideas opposing the mainstream Avecennian logic. One of the cornerstones of his logic is the essentialist (bi-ḥasabi l-ḥaqīqa) and externalist (bi-ḥasabi l-Ḫāriǧ) distinction between two ways of interpreting the subject terms of propositions. This distinction was first introduced by Fakhr al-Dīn al-Rāzī (d. 1210), but found its applications in the hands of Ḫūnaǧī in many aspects of the science of logic. Roughly speaking, an externalist proposition is the one whose subject term denotes only those individuals that exist in the external world, i.e. among concrete individuals, and an essentialist proposition is the one whose subject term denotes both (externally) existent individuals and those individuals that do not exist in the external world. For Ḫūnaǧī, the subjects of essentialist propositions include not only possible individuals but also impossible individuals. Consequently, one of the main contributions of Ḫūnaǧī was expanding Avecennian logic by making more room for inferences about the impossible.

In this paper, I will give a brief overview of Ḫūnaǧī’s account of inferences from the impossible. More importantly, I try to explore an issue that Street (2014) points out without finding an answer to it; Arabic logicians after Ḫūnaǧī criticized him by arguing that all universal propositions in the essentialist reading are false and all particular propositions in the essentialist reading are true and regarded that as his egregious error.¹ If true, many aspects of Ḫūnaǧī’s logic, especially those concerning the impossible, seem to collapse.² Street (2014, 464) expresses his doubts about whether or not these objections work, but he does not go further. I try to discuss this issue in this paper. Based on Ṭūsī’s remarks in his Taʿdīl al-miʿyār, I will explain two different ways of interpreting the essence of impossible subjects of essentialist propositions and evaluate them. Then, I will argue that universal and particular essentialist propositions in Ḫūnaǧī’s logic are both true and false.

Thus, in the next section, we will examine the distinction between the essentialist and the externalist reading of the subject term of a proposition.

In the third section, we will have an overview of Ḫūnaǧī’s account of inferences from the impossible. I have divided this section into three parts: First, immediate implications between propositions whose subjects are impossible individuals. Second, propositional syllogisms consist of at least one conditional whose antecedent is impossible. For example, syllogisms with premises that are about an impossible situation in which a donkey is a horse. Third, immediate implications from a contradictory pair. This part is about Ḫūnaǧī’s rejection of the principles of connexivity according to which no premise implies a contradictory pair. He argues that a contradictory pair implies a contradictory pair. These are, very roughly, the three main ways in which impossibility occurs in logic and language.

In the fourth section, we will discuss the objections that later Arabic logicians raised against Ḫūnaǧī’s account of essentialist propositions. Based on Ṭūsī’s objection to Abharī’s objection to Ḫūnaǧī, I will present an interpretation that shows that universal propositions are both true and false according to Ḫūnaǧī’s account of essentialist propositions. Ṭūsī points out a distinction between two ways of interpreting the essence of an impossible, let us say, J; it either has the essence of the possible J (among other things) or it does not. I will evaluate both interpretations and argue that the former implies that both particular and universal propositions in the essentialist reading are both true and false. This implies that the truth of these propositions prevents Ḫūnaǧī’s account of conversion and contraposition from being trivially valid. Finally, I will end the paper with some concluding remarks.

2. Essentialist and externalist readings of the subject term of a proposition

Most of Ḫūnaǧī’s logical contributions, including his account of inferences about the impossible, rest on the essentialist/externalist distinction. Let us first look at what this distinction is.

This distinction was first made by Fakhr al-Dīn al-Rāzī (d. 1210). Rāzī’s main motivation for making such a distinction was to answer the question of what is the subject of an absolute proposition such as ‘Every J is B’.³ In other words, it answers the question of what are the Js over which ‘every’ quantifies. To understand why Rāzī introduced a new distinction between two readings of a proposition to answer this question, we should first see how his predecessors answered this question. Al-Fārābī considered the subject of such propositions as possible:⁴

[Text 1] And absolute is that which has the nature of the possible [and includes] that which has become existent after it was possible to be existent, and that which does not exist and may not exist in the future.

Thus, according to al-Fārābī, the subject of ‘every human is rational’ includes all actual and possible humans. Later Arabic logicians refused to accept Fārābī’s analysis of the subject of an absolute proposition.⁵ According to Avicenna, the subject of an absolute proposition is that which actually exists, either in the external world or only in the mind. Avicenna gives us the example of ‘every white’:⁶

[Text 2] This description [of being white] is not the description of possibility or soundness. It is definitely not understood from our saying ‘every white’ all those things that being white is sound to [be ascribed to] them, rather [what is understood is] all those that are actually white […] and this actuality is not only being existent among the concrete individuals. Sometimes, the subject is not being intended with regard to its being existent among concrete individuals, […] rather [it is intended] with regard to its being rationally perceived to be actually such and such, no matter whether it existed or not.

Thus, the subject must be existent, whether in the mind or in the external world and among other concrete individuals. It is important to note that by ‘no matter whether it existed or not’ Avicenna means that it does not matter whether it existed externally among concrete individuals, as he mentions in the previous lines. For Avicenna, those things that exist in the mind but are externally non-existent are non-existent possible things. It means that impossibilities are not among the referents of J. Avicenna defines impossible as that whose nonexistence is necessary.⁷ It never exists,⁸ neither in the mind nor in the external world.⁹

Thus, it is not difficult to see the similarity between Avicenna’s analysis of the referents of ‘every J’ and that of Fārābī. For both, the universal quantifier quantifies over possible and actual things.

In this context, Rāzī came up with a distinction between two readings of a proposition, namely the externalist and the essentialist. As Rāzī put it:¹⁰

[Text 3] When we say, ‘every J’ it is sometimes used with regard to essentiality,¹¹ and sometimes it is used with regard to external existence.

Regarding the former, he explains:

[Text 4] We do not mean by this that which is described as J externally, but rather something more general, which is: if it existed externally, it would be true that it is J, whether it exists externally or not. For we can say ‘Every triangle is a figure’ even if there are no triangles existent externally. Rather, the meaning is that anything that would be a triangle if it existed, in so far as it existed, would be a figure.

Thus, ‘every heptagonal is a figure’ is true in the essential reading and false in the external reading. We can now see how the referents of J in a proposition in the form ‘Every J is B’ are determined by these two different readings:

• ‘Every J’ (essential reading): Things which, if they existed externally, would be true of them that they are J.

• ‘Every J’ (external reading): Things that exist (externally) and are J.

Rāzī’s external/essential distinction, on the one hand, provides another way to avoid Fārābī’s analysis of the subject of an absolute proposition mentioned above, and on the other hand, is not based on Avicenna’s metaphysical assumption of mental existence. Nevertheless, it seems that Rāzī does not go beyond possibilities. At least I could not find any discussion of impossible individuals in his Mantiq al-Mulaḫḫaṣ. If this is true, then for Rāzī, the subjects of categorical propositions are either possible individuals or actual individuals.

Although the external and essential distinction does not play a major role in Rāzī’s logic, it is an inseparable part of Ḫūnaǧī’s logic.¹² Ḫūnaǧī’s use of the external/essential distinction extends the domain of subjects of categorical propositions to include impossibilities.

First, let us see how Ḫūnaǧī defines the distinction, and then we will look at how he applies it to make it possible to argue from the impossible. Ḫūnaǧī repeats Rāzī’s definition,¹³ but adds an explanation about the essentialist reading of a proposition like ‘every J is B’:

[Text 5] [A]nd its meaning (i.e. the meaning of the essentialist reading of a proposition) is that whatever is implicant for J (malzūm lil-ǧīm) is implicant for B.¹⁴

In this case, he seems concerned only with the properties that are essentially implied by the subject’s essence.¹⁵ Thus, the predications that are true of a subject in an essentialist reading are those that are implied by the essence of the subject. They are either part of the essence of the subject or the implied accidents (ʿaraḍ al-lāzim).

Ḫūnaǧī tells us explicitly what kind of things are the subject of a proposition in the essentialist reading: ‘These include actual, possible, and impossible individuals’.¹⁶ Some of the examples of impossible individuals are ‘the eclipsed that is not a moon’, ‘the stone-man’, ‘the breathing non-animal’, ‘the odd two’, and ‘non-animal man’. It is obvious that these impossibilities are not logical impossibilities. Nevertheless, the logical impossibilities follow from some necessary truths, such as ‘every eclipsed is the moon’, ‘no stone is a man’, ‘every breathing is an animal’, ‘two is even and no even number is odd’, and ‘all men are animals’.

Most of Ḫūnaǧī’s contributions and objections to Avicenna are based on the essentialist/ externalist distinction.¹⁷ I will limit myself here to some of these cases of inference from the impossible.

3. Inferences from the impossible

We will examine Ḫūnaǧī’s account of inference from the impossible in three steps. First, immediate implication from propositions whose subjects are impossible individuals. These implications have been discussed mainly in his accounts of conversion and contraposition. For the sake of simplicity, I have considered only Ḫūnaǧī’s account of conversion in this paper, because his account of contraposition is so complicated that it might distract us from our main concern.¹⁸ I will discuss this section under ‘impossible individuals’. After that, we will review Ḫūnaǧī’s account of impossible situations. Here we are concerned with conditionals with impossible antecedents. Then we will see Ḫūnaǧī’s rejection of principles of connexivity where he explains to us that a contradictory pair implies a contradictory pair.

3.1. Impossible individuals

Imagine a vegetable-man; something that is both a human and a vegetable. This is an impossible individual for men are animals but vegetables are not. Now consider the proposition ‘The vegetable-man breathes’. Is this proposition true or not? How can we assign true predicates to subjects that denote such impossible individuals that do not exist? And if every vegetable-man is a man, does that imply that some men are vegetables? In this section, we will see how Ḫūnaǧī answers these questions.

To give an example of arguing from the impossible and how Ḫūnaǧī applies the externalist/essentialist distinction, let us look at Ḫūnaǧī’s account of the conversion of some modal propositions. Here we focus on the temporal-necessity (waqtīyya) propositions.¹⁹

‘No A is B’ is a true temporal-necessity-proposition only if no A is possibly B at a specific time, t. For example ‘no moon is eclipsed’²⁰ is necessarily true at the time of quadrature (tarbīʿ). Ḫūnaǧī claims that such propositions can be converted, and the conversion of a universal negative proposition with temporal necessity is a particular negative with the modality of perpetuity. Therefore, the conversion of ‘no moon is eclipsed’ with the temporal-necessity mentioned above is ‘some eclipsed is never moon’.

Arabic logicians before Ḫūnaǧī maintained that the universal negative proposition, modified with temporal necessity, is not convertible. Because ‘no moon is possibly eclipsed at the time of quadrature’ is true, but ‘some eclipsed is never moon’ is never true which means that it is necessarily false.²¹ In other words, ‘every eclipsed is a moon’ is necessarily true.

Ḫūnaǧī does not deny that ‘every eclipsed is a moon’ is necessarily true. Moreover, he does not claim that there can be some eclipsed in the external world that is not a moon. Ḫūnaǧī applies the essentialist reading in order to obtain the conversion of ‘no moon is possibly eclipsed at the time of quadrature’. According to the essentialist reading, the subject term of a proposition can refer to actual individuals (al-ʾafrād al-wāqiʿa), possible individuals (mumkināt), and impossible individuals (mumtaniʿāt).²² In the case of the conversion of ‘no moon is possibly eclipsed at the time of quadrature’, it is convertible only in the essentialist reading, and with an impossible subject. The conversion is ‘some eclipsed is never moon’,²³ but the eclipsed that is never moon is an impossible individual, because every eclipsed is a moon. Thus, there cannot be an eclipsed that is not a moon in the external world, and thus the conversion cannot be true in the externalist reading. That is why the temporal necessity proposition, as Ḫūnaǧī maintains, is not convertible in the external reading. The temporal necessity proposition is convertible only if its subject is impossible. There is no impossibility in the external world, but in the essentialist reading, we can have true propositions with impossible subjects.

Ḫūnaǧī thus shows that some implications between propositions-in this case, conversion-are valid only when the subject of the proposition is impossible. And the propositions, namely the original and the converse, are true because conversion is defined as ‘changing the place of the two sides of a proposition while the quality and the truth of the proposition remain the same’.²⁴ Thus, both propositions ‘no moon is possibly eclipsed at the time of quadrature’ and ‘some eclipsed is never moon’ are true, however only in the essentialist reading and when the subjects of the latter are impossible individuals.

We began this section with the example of a vegetable-man. We asked two questions: What is the truth condition for a proposition whose subject is an impossible individual? And what follows from this proposition if it is true?

Now we can answer these questions. According to the essentialist reading of a proposition, only those predicates can be truly affirmed of the subject that are implied by the essence of the subject. Thus, ‘The vegetable-man is a man’ is true, for its being man is implied by its essence.

As for the second question, Ḫūnaǧī discusses the immediate implications between propositions, not only in the externalist reading but also in the essentialist reading which includes impossible individuals – as we saw in the example of the eclipsed that is not a moon.

Let us now turn to the second way in which impossibility shows up in language/logic, namely impossible situations.

3.2. Impossible situations

Consider the following propositional syllogism: ${\displaystyle \frac{\begin{array}{c}\mathrm{Whenever\; two\; is\; odd,\; then\; two\; is\; a\; number}.\\ \mathrm{Whenever\; two\; is\; a\; number,\; then\; two\; is\; even}.\end{array}}{\mathrm{Whenever\; two\; is\; odd,\; then\; two\; is\; even}.}}$In his discussion of propositional syllogisms, Avicenna mentions the above counterexample as a possible objection to the validity of propositional syllogisms.²⁵ It seems that the premises are true, but the conclusion is not acceptable. According to Avicenna, the conclusion is false. However, he does not see the problem in the form of the syllogism. It is a Barbara that is a valid syllogism.²⁶ Rather, he argues that it is not the case in the above example that the premises are true and the conclusion is false. Avicenna maintains that the minor premise is false. He draws a distinction between true through implication (bi-l ilzām) and true in reality (fī nafsihā). As he put it:²⁷

[Text 6] Thus, it is true with regard to implication that whenever two is odd, it is even. But ‘it implies that’ and ‘it is true in reality’ are not the same.

Since the odd two cannot be existent, the conditional, i.e. the minor premise, is not true in reality (fī nafsihā). Thus, the above example is not a counterexample to propositional Barbara. Avicenna claims that a conditional with an impossible antecedent is false (or is false in reality). Ḫāǧi Naṣīr al-Dīn al-Ṭūsī, who follows Avicenna in his answer to this counterexample, makes the distinction between an implication that is true through implication and one that is true in reality more clearly. He calls the former expressional implication (luzūmī-i lafẓī) and the latter real implication (luzūmī-i ḥaqīqī).²⁸ Ṭūsī explains that the real implications (luzūmīy-i ḥaqīqī) are those that do not involve impossible situations (ʾawḍāʿ). As Ṭūsī put it:²⁹

[Text 7] Sometimes the implication in a proposition is not real, rather [it is] with respect to the position of expression, not that it is necessary in reality. Just as it is said: if five is even, then it is a number, because the implication of the consequent is not in reality caused by it [i.e. the content of the antecedent]. This proposition is true in expression and false in meaning because it involves an impossible situation. Thus, the implicative [conditional] is either real or expressional.

About syllogisms from expressional implicative conditionals, explains Ṭūsī:³⁰

[Text 8] As mentioned earlier, connective conditionals are either implicative or coincidental. Implicative conditionals are either real or expressional: a composition of real implications that do not involve impossible situations produces real implicative conclusions without exception. … Simple expressional implicative [conditional] and its combination with real implicative [conditional] produces an expressional implicative conclusion. An example of expressional implicative: if a human neighs she is an animal, and if a human is an animal she is a sensate.

Thus, for Avicenna and Ṭūsī, syllogisms in which the conclusion is obtained through expressional implication do not count as counterexamples, simply because their corresponding propositions are not true in reality.

Ḫūnaǧī disagreed with Avicenna. In order to understand Ḫūnaǧī’s reply to the aforementioned counterexample, we should first see what a conditional’s quantifier quantifies.

Arabic logicians, following Avicenna, use quantifiers, not only in categorical propositions, but also over conditionals. In our example ‘whenever’ is a quantifier that signifies the universal quantifier for conditional. The particular quantifier for conditional is ‘sometime’. Although ‘whenever’ (kullamā) and ‘sometime’ (qad yakūnu) are temporal terms, that is not exactly what Arabic logicians mean by them. The universal quantifier (whenever) quantifies over all assumptions (furūḍ), situations (ʾawḍāʿ), and times (ʾazmana). As Ḫūnaǧī put it:³¹

[Text 9] The meaning of these states of connective conditionals and disjunctive conditionals is in accordance with the generality of the assumptions and times for implication and incompatibility. Thus, implication and incompatibility are universal by virtue of their generality with respect to all times and assumptions […] Thus the affirmative implicative conditional is universal if the consequent follows from all situations of the antecedent, not in repetition but in states.

Similarly, a conditional is particular if it is true in some situations,³² and it is individual if it is true in a specific situation.

Ḫūnaǧī’s response to Avicenna’s counterexample is that the major premise, i.e. ‘whenever two is a number then it is even’, is false.³³ There are some impossible assumptions in which two is odd and not even. Thus, it is not true that in all situations if two is a number then it is even, for there are some impossible situations in which two is odd. The major premise is a universal conditional that fails to include the situations in which two is odd.

Ḫūnaǧī considers another possible objection, this time to the third figure, the mood Darapti.³⁴ ${\displaystyle \frac{\begin{array}{c}\mathrm{Whenever\; a\; donkey\; is\; a\; horse,\; then\; it\; is\; an\; animal}.\\ \mathit{Whenever\; a\; donkey\; is\; a\; horse,\; then\; it\; neighs}\mathit{.}\end{array}}{\mathrm{Sometime\; if\; a\; donkey\; is\; an\; animal,\; then\; it\; neighs}.}}$The possible objection that Ḫūnaǧī raises is that if this syllogism is valid, one can conclude the negation of the antecedent of the conclusion. If the syllogism is valid, then the conclusion is true. Since donkeys do not neigh, the consequent of the conclusion is false. Thus, if modus tollens is applied, the antecedent of the conclusion is false, i.e. donkey is not animal, which is obviously false.

Ḫūnaǧī replies:

[Text 10] We do not accept the productivity of this repetitive syllogism [i.e. modus tollens]. The consequent [of the conclusion] is implicate for the antecedent [only] at some times in some assumptions and situations. Thus, the perpetual obtainment [ṯubūt] of the antecedent does not imply the obtainment of the consequent in some [specific] occasion, for the permissibility that its implication is conditioned by some state of affairs [ʾamr] that has not obtained.

For Ḫūnaǧī the above syllogism is valid and thus its conclusion is true, but in a certain impossible situation where donkey is horse, and thus neighs. The truth of the conclusion and its antecedent (which is true in every situation), does not, by applying modus ponens, imply that donkeys actually neigh. Nor does it imply, by applying modus tollens, that donkeys are not actually animal since they do not neigh. As the quantifier of the conclusion shows ‘if donkey is animal, then it neighs’ is true only in some situations, namely those in which the minor antecedent of the minor premise is true, i.e. donkey is horse.

Thus, the implications between conditional propositions and syllogisms consisting of conditionals are discussed with respect to the situations in which they are true or false. Ḫūnaǧī, expanded these situations to include impossible ones. Nevertheless, the conditionals stating impossibilities are considered in the essentialist reading. Their subjects are impossible individuals and do not exist in the external world. Thus, Ḫūnaǧī’s account of impossible situations is based on the essentialist/ externalist distinction.

Of our threefold occurrences of impossibility one remains; contradictory propositions. Let us now look at how Ḫūnaǧī explains the implications from a contradictory pair by rejecting Boethius theses.

3.3. Contradictory propositions³⁵

The connexive principles, also called Boethius theses, are:

• $(A\to B)\to \mathrm{\neg }(A\to \mathrm{\neg B})$

• $(A\to \mathrm{\neg B})\to \mathrm{\neg }(A\to B)$

These theses roughly states that an arbitrary premise does not imply a contradictory pair. Avicenna had advocated the principles of connexivity. These principles became an orthodoxy among Arabic logicians until Ḫūnaǧī rejected them. Before taking a look at Ḫūnaǧī’s rejection of connexive principle, let us look at Avicenna’s account of connexive principles³⁶:

[Text 11] If our saying ‘Never if every A is B, then every J is D’ is true, then our saying ‘Whenever every A is B, then it is not the case that every J is D’ is true. Otherwise, its negation is true which is our saying ‘It is not the case that whenever every A is B, then it is not the case that every J is D’. The meaning of this saying forbids that the negated consequent is implied in every situation where the antecedent [is true]. Thus, there will be inevitably a situation where the antecedent is the case without being followed by the consequent, and it is true in this case with [the truth of] its contradictory. Consequently, it could be [in some situation] that every A is B, and meanwhile every J is D. But we had said ‘Never if every A is B, then every J is D’, and therefore that is unacceptable.

We have already talked about the quantifiers of conditionals, according to which ‘never’ means in no situation, and ‘whenever’ means in every situation. For simplicity, I write ‘$P$’ instead of ‘every A is B’, and ‘$Q$’ instead of ‘every J is D’.³⁷ The two propositions are as follows:

1. In no situation: if P, then $Q$.

2. In every situation: if $P$, then it is not the case that $Q$.

According to Avicenna, if (1) is true, then (2) is true. His proof is a reductio argument:

Suppose (2) is not true. Then its negation:

(3)	There is at least one situation in which $P$ and $Q$ are both true.

Is true.

Now, (3) contradicts (1) and thus (3) cannot be true and its contradictory (2) is true.

Avicenna applies the same argument to prove the converse, i.e. if (2) is true, then (1) is also true.³⁸ Thus, (1) and (2) are equivalent.

Ḫūnaǧī rejected Avicenna’s argument, simply by showing that Avicenna has violated the connexive principle in his proof of the principle. As Ḫūnaǧī put it:³⁹

[Text 12] … [S]ometimes a contradictory pair is inevitably implied by the same antecedent. Is not a reductio proof anything other than a contradictory pair implied by the contradictory of the conclusion? Most assertions in geometry and also in logic [are such that] their contradictories imply a contradictory pair. The books of sciences are full of proofs of such claims [in such a way] that a contradictory pair implies their contradictories. This is so obvious that it needs no further discussion.

Ḫūnaǧī’s main counter-example to connexive principles is reductio proofs one of which Avicenna has used in his argument for connexive principles. Consider Avicenna’s proof above. From (1), (2), and (3) it follows both (1) and the contradictory of (1).⁴⁰ $(1),(2),(3)\Rightarrow (1),\mathrm{\neg }(1)$In this way, Avicenna could reject the truth of (3) and conclude that (2) is true. Unless the premises imply both (1) and the contradictory of (1), which is a contradictory pair, Avicenna could not make his proof.

Moreover, Ḫūnaǧī provides another simple counterexample showing that (1) can be true without (2) being true. Suppose $P$ is ‘Two is even’, and $Q$ is ‘Zayd is at home’. This makes (1) true, but (2) false. Two being even has nothing to do with Zayd being home or Zayd not being home. We have already seen what the conditions are under which an implication is true, i.e. there should be a special connection between the antecedent and the consequent.

We have discussed Ḫūnaǧī’s account of inferences from the impossible by dividing the occurrence of impossibilities into three different kinds. However, these three kinds are intertwined. Let us consider the example of the non-animal human. The essentialist reading lets us assign truth to the propositions ‘the non-animal human is an animal’ and ‘the non-animal human is non-animal’, because animal is part of the essence of human, and non-animal is part of the essence of the non-animal human. If connexive principles were valid, these propositions could not be true at the same time. From one antecedent two contradictories have been implied. Let us rewrite it more precisely:

1. x is human and not-animal, thus x is animal.

2. x is human and not-animal, thus x is not animal.

There are two caveats here. First, since the essentialist reading has been applied, there is no difference between x is not animal and x is not-animal. In other words, the negative simple proposition (sālibat al-muḥaṣṣila) and its corresponding affirmative deflected predicate (maʿdūlat al-maḥmūl) have the same truth value.⁴¹ Second, as mentioned, the true predicates of a subject are those that are implied by the essence of the subject. Thus, (i) and (ii) are true. Ḫūnaǧī’s rejection of connexive principles makes it possible for (i) and (ii) to be true together. Moreover, adding conditionals’ quantifiers, i.e. situations, to what the essentialist reading of propositions and the rejection of connexive principles have provided, enables us to make inferences about impossible situations.

Let us now move on to the objections that later Arabic logicians raised against Ḫūnaǧī’s account of the essentialist reading.

4. Universal propositions in the essentialist reading of propositions

Some later Arabic logicians rejected Ḫūnaǧī’s position that impossible individuals are included as the subject of a proposition in the essentialist reading. The first Arabic logician to do so was probably ʾAṯīr al-Dīn al-Abharī (d. 1265). He argued that negative universal propositions are never true in the essentialist reading. He did this by arguing that the contradictory of ‘no J is B’, namely ‘some J is B’, is always true and therefore ‘no J is B’ is always false.⁴²

The argument is as follows:

1. $\mathrm{Every}J\mathrm{and}B\mathrm{is}J.$ (simplification)

2. $\mathrm{Some}J\mathrm{is}B\mathrm{and}J.$ (conversion of 1)

3. $\mathrm{Some}J\mathrm{is}B.$ (2, simplification)

In the essentialist reading the subject of a proposition can be anything, possible or impossible. Suppose some things are J and B. Everything that is J and B is J. This is a logical truth and its conversion is ‘some J and B are J’ which means some Js are B. This is true for any arbitrary B and J. Thus, all particular affirmative propositions are true in the essentialist reading. It follows that all universal negative propositions are false because they are the contradictories of particular affirmative propositions. It is obvious that Abharī’s argument is based on the assumption that an i-proposition is the contradictory of its corresponding e-proposition and that they do not share truth values.

Then, Abharī discusses essentialist universal affirmative propositions. This case is a little more complicated. Abharī discusses two types of such propositions separately. First, the predicate is not part of the subject.⁴³ Second, the predicate is part of the subject.

Let us first consider the first case. As he put it:⁴⁴

[Text 13] And the universal affirmative: if its predicate is not part of its subject, its truth is not certain. For if we say ‘All of those that imply J also imply B’, its negation may be true. For ‘all of those that imply J and imply not-B also imply J’, and ‘all of those that imply J and imply not-B also imply not-B’ thus ‘some of those that imply J imply not-B’, and if [something] implies not-B it may not imply B and it may imply B, because the impossible can imply a contradictory pair.

Let us paraphrase Abharī’s argument. Consider first the universal affirmative proposition: ‘Everything that implies J implies B’, and suppose that B is not part of J. Abharī argues that this proposition cannot be true. He argues from the third figure of propositional syllogism. There are two premises that are true:

1. Everything that implies J and not-B implies J.

2. Everything that implies J and not-B implies not-B.

Therefore, from the third figure:

1. Something that implies J implies not-B.

(3) is the essentialist reading of ‘Some J is not B’ which is the contradictory of ‘Every J is B’. Thus, if the former is true, then the latter is false. However, as Abharī says, the truth of (3) does not imply the falsity of ‘every J is B’. The reason is that Abharī, following Ḫūnaǧī, does not hold the validity of the connexive principles according to one of which if $p$ implies $q$, it does not imply $\mathrm{not}-q$ (suppose $p$ and $q$ are any arbitrary proposition): $(p\to q)\to \sim (p\to \mathrm{\neg q})$Thus, it may be that the Js that are not-B are B too. As Abharī tells us this happens when the subject is impossible.⁴⁵ From impossible a contradictory pair may follow. Thus, this argument does not show that the essential universal affirmative, when the predicate is not part of the subject, is always false across the board. In this case, ‘some J is B’ is both true and false, for some Js are such that they are both B and not B. It seems that by saying that their truths are uncertain, Abharī means the following: depending on which J’s the proposition ‘some J is B’ is about, it can be true or both true and false. The former is when the subject is possible and the latter is when the subject is impossible. I do not think that one needs to specify the referents of ‘some J’. The point is that we only need to consider some J that is not B. It is both B and not B. Thus, ‘some J is B’ is both true and false.

Let us now see what Abharī says about the essential universal affirmative when the predicate is part of the subject. As he put it:⁴⁶

[Text 14] And if its predicate is part of its subject, like in our saying ‘every human is animal’, its truth is certain with the truth of our saying ‘some of those that imply human, they imply not-animal’ because of the proof we have given.

Let us take a closer look at his argument. In the case where the predicate is part of the subject, ‘every J is B’ is true. This is a necessary truth, for the predicate is part of the subject. But it is also true that ‘some J is not-B’. A similar proof to the previous case can apply here. There are two premises: (1) every J and not-B is J, and (2) every J and not-B is not-B. Therefore, from the third figure: ‘Some J is not-B’ which is equivalent to ‘Some J is not B’.⁴⁷

It implies that ‘every J is B’ is both true and false! Thus Abharī suggests that the subject of a proposition in the essential reading be limited only to possibilities. So far, we saw two arguments from Abharī against Ḫūnaǧī’s essential reading of propositions. Abharī argued that:

1. The universal negative is always false in the essentialist reading.

2. The universal affirmative, when the predicate is part of the subject, is contradictory; it is both true and false in the essentialist reading.⁴⁸

Later Arabic logicians such as Naǧm al-Dīn al-Kātibī (d. 1276) and Qutb al-Dīn al-Rāzī al-Taḥtānī (d. 1365) argued that universal propositions, both affirmative and negative, cannot be true in the essentialist reading. Their arguments that all universal negative propositions are false in the essentialist reading are in fact the same as Abharī’s. Let us look at both arguments:⁴⁹

First, consider the universal negative proposition:

1. Every J is B. (assumption)

Now, consider some J that is not B.

(2)	Some J is not B.

Since (2) is the contradictory of (1) and (2) is true, thus (1) is false.

Now consider the universal affirmative proposition:

(3)	No J is B. (Assumption)

Now consider some J that is B. Thus;

(4)	Some J is B.

Since (4) is the contradictory of (3) and (4) is true, thus (1) is false.

Neither Kātibī nor Taḥtānī consider the case in which the predicate is not part of the subject. It is not clear to me why Kātibī and Taḥtānī did not mention Abharī’s argument that universal affirmative propositions in the essentialist reading are both true and false when the predicate is part of the subject. In the end of this section, I will suggest a reason for that.

Interestingly, Ṭūsī had argued that if Abharī holds that essentialist universal negative propositions are both true and false, he should also accept that essentialist universal affirmative are both true and false.⁵⁰ Ṭūsī argues against Abharī as follows: Consider an impossible individual, e.g. a non-animal human. For simplicity, let us call this non-animal human h. There are two ways of interpreting h’s essence. One is that h is a human in the sense that a possible human is human. Thus, h is an animal. Since h is non-animal, h is and is not an animal. The second interpretation is that h is not a human in the sense that a possible human is human. A non-animal human is an essence whose being human does not imply that it is an animal; thus no contradiction arises.

According to Ṭūsī, Abharī applied the first interpretation to the essentialist universal affirmative and thus concludes that these propositions are both true and false. Applying the first interpretation to the essentialist universal negative propositions also leads to the same result i.e. these propositions are both true and false.⁵¹

Let us see which interpretation Ḫūnaǧī applied. I will give two arguments that Ḫūnaǧī applied the first interpretation.

The first argument: It is based on Ḫūnaǧī’s definition of the essentialist reading of the subject term of a proposition. As we saw in section 2, for Ḫūnaǧī the subject term of a proposition in the essentialist reading refers to actual, possible, and impossible individuals. In other words, ‘every J’ refers to all actual, possible, and impossible Js. If an impossible J is not a J in the sense that a possible J is J, then the subject has changed, because the impossible does not have the essence that a possible J has, and thus ‘every J’ cannot denote both possible and impossible J. Consider the example of a non-animal human. If the non-animal human is not a human in the sense that possible humans are, its essence differs from possible humans. Thus, we cannot refer to both possible and impossible humans by ‘every human’. Therefore, Ḫūnaǧī must accept the first interpretation.

The second argument: If the non-animal human is not human in the sense that possible humans are, then no impossibility has occurred. What makes a non-animal human an impossible individual is that every human is necessarily an animal. ‘Some human is not animal’ contradicts ‘every human is animal’. The latter is a metaphysical necessity. What contradicts a metaphysical necessity is a metaphysical impossibility. If the human that is not an animal is not a human in the sense of a human that is an animal, no contradiction arises and therefore no metaphysical impossibility arises.

Thus, the first interpretation according to which the non-animal human is an animal must be the correct interpretation.

Let us now revisit the two cases of a universal proposition in the essentialist reading. For now, I will consider only the case in which the predicate is part of the subject. I will argue that both affirmative and negative propositions are both true and false in the essentialist reading.

First, the universal affirmative proposition:

1. Every J is B.

B is part of the essence of J and thus the truth of (i) is necessary (let us say it is true in every situation).

Now consider a J that is not B. We can make this assumption, because the essentialist reading of the subject term of a proposition allow us to include any impossible individual as the subject of a proposition. Thus,

• (ii) Some J is not B.

Kātibī and Taḥtānī argued that since (ii) is true, (i) is false because (i) and (ii) are contradictories. Call the J that is not B ‘d’. B is part of the essence of d, and hence ‘d is B’. This is true of every J that is B. Thus, the Js that are not B are also B, which means (ii) is both true and false. As Abharī argued (i) is also both true and false.

Let us now consider the universal negative proposition.

• (iii) No J is B.

Let me explain this case with an example. Suppose J is square and B is round. No square is possibly round. It is implied by the essence of square that it is not round. Thus it is true of every square that it is not round. That is also true of the round-square, otherwise the round-square would not be an impossible individual. Recall that the round-square is a square in the sense that a possible square is a square. Such an essence is not round. Thus, it is true that some square (namely the round-square) is round. Thus,

• (iv) Some J is B.

But the round-square, which is round, is not round (simply because it is a square and has four corners). Although (iv) is true, it is also false. The same is true for (iii). Thus, Ḫūnaǧī must accept that both universal negative and universal affirmative propositions are both true and false in the essentialist reading.

Nevertheless, Ḫūnaǧī does not mention the falsity of these propositions, and only considers their truth.⁵² More precisely, he discusses what follows from their truth – for example, as we have seen, in his account of conversion. Moreover, Ḫūnaǧī pays attention to propositions that are always false or always true, especially in his discussions of the contrapositions of propositions, but he does not include the essentialist universal negative and affirmative propositions among them. Also, recall his rejection of the principles of connexivity according to which a contradictory pair may be implied by the same antecedent. For Ḫūnaǧī, these implications are true (take $\to$ as implicative relation and $P$ as any arbitrary proposition and $\sim P$ as the contradictory of $P$): $P\mathrm{\&}\sim P\to P$ $P\mathrm{\&}\sim P\to \sim P$

Thus, it seems that according to Ḫūnaǧī’s logic, the truth and falsity of a proposition imply both the truth of that proposition and its falsity. Replace P with ‘every J is B’ or ‘no J is B’. These propositions can be true while their contradictories are true.

Let us look at the consequences of assuming that universal and particular propositions are both true and false in the essentialist reading, and see if this fits into the logic of Ḫūnaǧī. Recall Ḫūnaǧī’s example of the eclipsed that is not a moon.⁵³ Let us consider the following propositions:

1. No moon is eclipsed at t.

2. Some eclipsed is never a moon.

3. Every eclipsed is a moon.

According to Ḫūnaǧī’s predecessors (a) is necessarily true because the moon is not possibly eclipsed at the time of quadrature, and (b) is necessarily false because (c) is necessarily true. Consequently, (a) is not convertible to (b), because of the condition of truth preservation from the convertend to the converse.⁵⁴

Ḫūnaǧī argues that the conversion of (a) is (b). He has defined conversion as ‘changing the place of the two sides of the proposition, with the truth and quality remaining in the converse as they were in convertend’.⁵⁵ Thus, if the converse is true, the conversion is also true.⁵⁶ The main point here is that only truth must be preserved; there are no constraints on the preservation of falsity. This is not different from the way most of his predecessors considered the truth preservation in conversion.⁵⁷

Thus, if Abharī’s argument for the falsity of all negative universal propositions in the essentialist reading works, then the implication from (a) to (b) is trivial, i.e. it is true because (a) is false.

As Street (2014) puts it:

If Taḥtānī is successful in showing that Khūnajī’s essentialist reading makes a- and e-propositions necessarily false, and i- and o-propositions necessarily true, all of Khūnajī’s carefully crafted proofs using the reading are shown to be trivially valid: they all have a universal proposition among the premises, or conclude in a particular proposition. In the eyes of the tradition to which he belonged, Khūnajī’s chapter on conversion is deeply flawed.

We have seen through Abharī’s arguments and Ṭūsī’s critical examination of them that these apparently problematic particular and universal propositions are both true and false.⁵⁸ Thus, (a) is not only a false proposition; it is also true. In the essentialist reading, then, the truth of particular propositions does not preclude the truth of universal propositions, and vice versa. What remains to be resolved regarding the truth and falsity of essentialist propositions, is whether the falsity of (a) prevents (b) from being implied by (a). According to Ḫūnaǧī’s definition of conversion, truth must be preserved in conversion. If being both true and false for a proposition means that it has a third value other than only true and only false, then (a) is not true in order for the truth to be preserved. This brings us back to what Street mentions: conversion is trivially true in this case.

In contemporary logic, it is usually assumed that a function assigns truth-value to a proposition. However, this is not necessary. There are indeed two ways to interpret a proposition being both true and false:⁵⁹ (1) There are three truth values: only true, only false, and both true and false. In this case, every proposition has only one of these values (2) There are only two truth values: true and false. In this case, some propositions have one and some have both values. The main difference between (1) and (2) is that a function assigns a value to each proposition in (1), while a relation assigns a value or both values to propositions in (2).⁶⁰

According to the second interpretation, a contradictory proposition is the one that has both values of true and false, and not the one that has a third value. I have this interpretation in mind when I say universal propositions are both true and false in the essentialist reading. Thus, in a conversion, if both the convertend and the converse are true and false, they are still both true. According to Ḫūnaǧī’s definition of conversion, truth must be preserved in conversion.⁶¹ (a) is true. Its conversion, namely (b) is also true. Thus, truth has been preserved by changing the sides of the proposition and preserving the quality. There is no further condition for the falsity of (a) and (b). These two propositions are both true and false. Nevertheless, the former implies the latter, i.e. if true, the latter is also true.

There is one more important question we should answer; What about Ḫūnaǧī’s crafted arguments? We have seen that by assuming that particular and universal essentialist propositions are both true and false, Ḫūnaǧī’s account of conversion is not vacuously valid, but do his arguments for the conversion of essentialist propositions support the claim that these propositions are glutty? These questions will indicate whether Ḫūnaǧī himself held essentialist propositions to be both true and false.

So let us consider Ḫūnaǧī’s argument that the converse of ‘No moon is eclipsed at the time of quadrature (t)’ is ‘Some eclipsed is never a moon’. For simplicity let us proceed with an absolute proposition instead of temporal necessity:⁶²

First consider the convertend and the converse:

1. No J is always B.

2. Some B is never J.

Now consider the following syllogism in the third figure (Felapton): ${\displaystyle \frac{\begin{array}{c}(3)\mathrm{Every\; always}\text{-}BisB.\\ (4)\mathrm{No\; always}\text{-}B\mathrm{is}\mathrm{ever}J.\end{array}}{(2)\mathrm{Some}B\mathrm{is}\mathrm{never}J.}}$(3) is self-evident. (4) is the conclusion of the following reductio syllogism consisting of the contradictory of (4) as the minor (5), and (1) as the major premise (Ferio): ${\displaystyle \frac{\begin{array}{c}(5)\mathrm{Some}\mathrm{always}\text{-}B\mathrm{is\; at\; least\; once}J.\\ (1)\mathrm{No}J\mathrm{is}\mathrm{always}B.\end{array}}{(6)\mathrm{Some\; always}\text{-}B\mathrm{is}\mathrm{not}\mathrm{always}B.}}$According to Ḫūnaǧī, since (6) is absurd (ḫulf), (5) is false, and thus (4) is true. Thus, according to the former syllogism, (1) implies (2). This argument is based on two ideas that are in conflict with the idea that universal and particular essentialist propositions are both true and false: First, (5) cannot be true and thus its contradictory, (4) is true. (5) is a particular essentialist proposition and Ḫūnaǧī is taking as an only false proposition.⁶³ Second, (6), which states something is not itself, is absurd. (6) could be a candidate for an example of a proposition which is both true and false. It is a particular essentialist proposition. Moreover, when we can have propositions whose subject is a donkey-horse why not a proposition whose subject is a donkey that is not a donkey?

This raises the following question: How much impossibility is permissible in the essentialist reading? Ḫūnaǧī’s examples of impossibilities are diverse: the odd two, the breathing non-animal, the non-moon eclipsed, the horse donkey, etc. These are of three kinds: (1) The odd two is the number two, but it has the opposite of its implied accident, namely being odd. Ḫūnaǧī’s response to Avicenna’s counterexample about the odd two indicates that considers the odd two which is not even. In this case, the impossible individual lacks its implied accident, rather it has the opposite. (2) The breathing non-animal (and similarly the eclipsed non-moon) is a case in which an individual has the implied accident of an essence without being that essence. (3) The horse-donkey is an individual that has two incompatible essences.

As these three cases show, Ḫūnaǧī’s examples of impossibilities do not include an individual that is J and not J, for example. Apart from this, nothing can prevent us from imagining an impossible individual with any combination of essences or implied accidents. Thus, the non-animal human is also an impossible individual. Although it is not an animal, it is also an animal, because it is a human and animal is part of the essence of human. Accordingly, we can form the following syllogism (Felapton): ${\displaystyle \frac{\begin{array}{c}\mathrm{Every\; non-animal\; human\; is\; an\; animal}.\\ \mathrm{Every\; non-animal\; human\; is\; not\; an\; animal}.\end{array}}{\mathrm{Some\; Animals\; are\; not\; animal}.}}$The conclusion has the form of (6) which Ḫūnaǧī, as we have seen, regards as an absurdity. If (6) is an absurdity for Ḫūnaǧī, then this conclusion must be an absurdity. But this rejects Ḫūnaǧī’s account of impossible individuals. Even if Ḫūnaǧī does not accept that universal and particular essentialist propositions are both true and false, his account of impossible individuals implies a contradictory proposition.

Let me now summarize. As Ṭūsī explained there are two ways of interpreting an impossible J:

First, an impossible J is a J in the sense that a possible J is a J. In other words, the essence of an impossible J consists of, among other things, the essence of the possible J. For example, a non-animal human is a human in the sense that a possible human is a human, i.e. it is an animal, and so on. In this case, as we have seen, particular and universal essentialist are both true and false. Moreover, conversions and contrapositions that Ḫūnaǧī holds to be valid are not trivialy valid, for the truth-preservation from the convertend to the converse. However, Ḫūnaǧī’s arguments no longer work because they are based on the exclusion and exhaustion of truth and falsity.

Second, an impossible J is not a J in the sense that a possible J is a J. In other words, the essence of an impossible J does not consist of the essence of the possible J. For example, a non-animal human is not a human in the sense that a possible human is a human, i.e. it is not the case that the non-animal human is an animal. The arguments of Kātibī and Taḥtānī seem to be based on this interpretation. As we have seen, the main problem with this interpretation is that the subject is changed and ‘every J’ cannot denote both possible and impossible Js. Moreover, as have already pointed out, applying this interpretation does not yield an impossibility. However, even if we accept that the subject has not been changed and the impossible J is an impossibility in this interpretation, Kātibī’s and Taḥtānī’s arguments go through. Thus, all conversions and contrapositions of universal essentialist propositions will be trivially true, because the antecedent of the implication, i.e. the convertend, is false. Moreover, Ḫūnaǧī’s arguments also collapse, because universal essentialist propositions are used in his reductio arguments, such as the one we have met where (3), (4), and (5) are taken to be true by Ḫūnaǧī.

Therefore, the second interpretation implies more damage to Ḫūnaǧī’s logic. In the first interpretation, only Ḫūnaǧī’s argument for the validity of conversions and contrapositions are affected⁶⁴; they are not valid, whereas in the second interpretation, not only his arguments but also his account of essentialist propositions are rejected. In other words, the subject of an essentialist proposition cannot denote both possible and impossible individuals. More importantly, the conversions and contrapositions in the second interpretation are trivially true – even if we accept that the subject of an essentialist proposition denotes both possible and impossible individuals.

One may rightly ask whether we need an argument for the conversion of these propositions whose subjects are impossible such as ‘No moon is possibly eclipsed at the time of quadrature’. Why cannot we just imagine a lunar-eclipsed which is not a moon? Ḫūnaǧī did not provide any criteria for which impossible individuals qualify as the subject of an essentialist proposition, nor did he explicate the background metaphysics of the essentialist reading.⁶⁵ The question of what makes an impossible individual the subject of a proposition is not answered in Kašf al-asrār. If any arbitrary combination of essences can be the subject of an essentialist proposition, there will be no need for argumentation in order to obtain, for example, the conversion of ‘No moon is possibly eclipsed at the time of quadrature’. We can just imagine the lunar eclipsed at the time of quadrature. We only need a proposition that satisfies the condition of the converse of the convertend. Thus, the main conflict here is between the inclusion of impossible individuals as the subjects of essentialist propositions on the one hand, and Ḫūnaǧī’s reductio arguments which are based on the absurdity of a self-contradictory proposition like ‘some B is not B’ and the exclusion and exhaustion of truth and falsity one the other hand.

5. Concluding remarks

One of Ḫūnaǧī’s most important contributions to Avicennian logic was formulating inferences about the impossible. His account is based on the essentialist/externalist distinction that he borrowed from Rāzī. We have examined three aspects of Ḫūnaǧī’s account of inferences about the impossible: immediate implications between propositions whose subjects are impossible, syllogisms about impossible situations, and implications from a contradictory pair. In all these three cases, Ḫūnaǧī is an innovative logician who departs from and argues against Avicenna. More importantly, we discussed the objections that later logicians raised against his account of essentialist propositions. These objections convinced many logicians to restrict the domain of subjects of essentialist propositions to possible individuals. Abharī had argued that all universal negative propositions in Ḫūnaǧī’s essentialist reading are false (without being true), and all universal affirmative propositions (at least when the predicate is part of the subject) are both true and false. It seems, however, that Kātibī’s and Taḥtānī’s objections had a greater impact on later Arabic logicians. They argued that all universal propositions in the essentialist reading are false and all particular propositions in the essentialist reading are true.

In his Taʿdīl al-miʿyār, Ṭūsī argues that if Abharī claims that Ḫūnaǧī’s universal affirmative propositions are both true and false in the essentialist reading, he must accept that universal negative propositions are also both true and false in the essentialist reading.⁶⁶ Ṭūsī drew a distinction between two ways of construing the essence of an impossible J: first, the impossible J is a J in the sense that a possible J is, and second, otherwise. Ṭūsī’s argument, however, only goes through if one accepts the first interpretation.

Based on Ṭūsī’s remarks, I evaluated both interpretations and their consequences for Ḫūnaǧī’s logic. I argued that the first interpretation, which implies that essentialist universal propositions are both true and false, has less harmful consequences than the second interpretation, which does not imply that both essentialist universal propositions are both true and false. In the first interpretation, only Ḫūnaǧī’s argument for the validity of conversions and contrapositions in the essentialist reading are affected while his conversions and contrapositions are preserved from being trivially valid, whereas in the second interpretation not only his arguments but also his account of essentialist propositions are rejected, since the subject of an essentialist proposition cannot denote both possible and impossible individuals. Even if we accept that the subject of an essentialist proposition denotes both possible and impossible individuals, the conversions and contrapositions in the second interpretation are trivially valid because all convertends are false. Therefore, the objections that later Arabic logicians raised against Ḫūnaǧī’s account of essentialist reading shows that Ḫūnaǧī’s logic cannot be sustained as it is.

Arabic and Persian Texts:

[Text 1] والمطلق هو ما كان من طبيعة الممكن، وحصل الآن موجوداً، بعد أن كان ممكنا أن يوجد، وألا يوجد، وممكن أيضاً ألا يوجد في المستقبل .

[Text 2] وهذه الصفة ليست صفة الإمكان والصحة. فإن قولنا: كل أبيض، لا يفهم منه البتة أنه كل ما يصح أن يكون أبيض، بل كل ما هو موصوف بأنه أبيض … وهذا الفعل ليس في الأعيان فقط، فربما لم يكن الموضوع ملتفتا إليه من حيث هو موجود في الأعيان … بل من حيث هو معقول بالفعل موصوف بالصفة على أن العقل يصفه بأن وجوده بالفعل يكون كذا، سواء وجد أو لم يوجد.

[Text 3] إذا قلنا “كل ج” فهذا يستعمل تارةً بحسب الحقيقة وتارةً بحسب الوجود الخارجي .

[Text 4] ولانعني به ما يكون موصوفاً بالجيمية في الخارج، بل مايكون أعمّ منه وهو الذي لو وُجد في الخارج لصدق عليه أنّه ج، سواء كان في الخارج أو لم يكن. فإنّه يمكننا أن نقول “كل مثلث شكل” ولو لم يكن شيء من المثلثات موجوداً في الخارج، بل على معني أنّ كل ما إذا وجد وكان مثلثا فإنّه لابدّ وأن يكون بحيث متى وجد كان شكلاً.

[Text 5] وكان معناه: كلّ ما هو ملزوم للجيم ملزوم للباء.

[Text 6] فصادق على سبيل الإلزام أن الاثنين كلما كان فردا يكون زوجا و ليس “أن يلزمه” و “أن يكون حقا” شيء واحد.

[Text 7] و گاه بود که لزوم در قضیه حقیقی نبود، بل بحسب وضع لفظ باشد، نه آنک فی نفس الامر واجب بود، چنانک گویند: اگر پنج زوج است پس عدد است، چه لزوم تالی نه باین علتست فی نفس الامر. و این قضیه در لفظ صادق بود و بمعنی کاذب، چه مشتمل بر وضع محالیست. پس لزومی یا حقیقی بود یا لفظی.

[Text 8] و متصلات چنانک گفته آمده است لزومی باشد یا اتفاقی، و لزومی حقیقی بود یا لفظی: اما تألیف از مقدمات لزومی حقیقی که بر اوضاع محال مشتمل نباشد نتایج لزومی حقیقی دهد بی اشتباه، … و لزومی لفظی بسیط و مختلط با لزومی حقیقی نتیجه لزومی لفظی دهد. لزومی لفظی مثالش: اگر انسان صهال بود حیوان بود و اگر انسان حیوان بود حساس بود .

[Text 9] والإعتبار في هذه الأحوال في الشرطيّات إنّما هو بعموم الفروض والأزمنة للّزوم والعناد، فكليّة اللزوم والعناد بعمومها بحسب جميع الأزمنة والفروض، … فالمتّصلة اللزومية الموجبة إنّما تكون كليّة إذا كان التالي يتبع كلّ وضع للمقدّم لا في المرار بل في الأحوال.

[Text 10] لا نسلّم إنتاج هذا القياس الاستثناييّ، فإنّ التالي لازم للمقدّم في بعض الأزمنة على بعض الفروض والأوضاع، فلم يلزم من ثبوت المقدّم في نفسه دائماً ثبوت التالي في وقت، لجواز أن يكون لزومه بشرط أمرٍ لم يثبت .

[Text 11] إذا صدق قولنا: ليس البتة إذا كان كل آ ب فكل جـ د، صدق قولنا: كلما كان كل آ ب فليس كل جـ د، وإلا صدق نقيضه وهو قولنا: ليس كلما كان كل آ ب فليس كل جـ د. ومعني هذا الكلام هو منع أن يكون هذا التالي السالب لازما لكل وضع للمقدم، فيكون هناك لا محالة وضع مرة من المرات يوضع فيها هذا المقدم خاليا عن متابعة هذا التالي إياه، فيكون الصادق حينئذ معه نقيضه. فيكون حينئذ قد كان كل آ ب ومعه كل جـ د، وقد قلنا: ليس البتة إذا كان كل آ ب فكل جـ د، هذا الخلف .

[Text 12]  … فإنّ النقيضين ربّما لزما مقدّماً واحداً محالاً، وهل قياس الخلف إلّا لزوم النقيضين معاً لنقيض المطلوب؟ وأكثر الدعاوى في الهندسيّات بل في المنطق نفسه يستلزم نقائضها الشيء ونقيضه، بل الكتب العلميّة مشحونة بإثبات الدعاوى بملازمة النقيضين نقائضها، وذلك لا يُخفى حتّى يُحتاج إلى الإطناب فيه .

[Text 13] وأمّا الموجبةُ الكلّيّةُ فإن لم يكنْ محمولُها جزءاً من الموضوعِ، لم يَحصُلِ الجزمُ بصدقها؛ لأنّا إذا قلنا: “كلُّ ما هو ملزومُ ج فهو ملزومُ ب"، كان نقيضُه محتملَ الصدقِ؛ لأنَّ “كلَّ ما هو ملزومُ ج وملزومُ لاب فهو ملزومُ ج” و “كلُّ ما هو ملزومُ ج وملزومُ لاب فهو ملزومُ لاب"، فـ"بعضُ ما هو ملزومُ ج فهو ملزومُ لاب"؛ وإذا كان ملزومُ لاب احتملَ أن لا يكون ملزومَ ب واحتملَ أن يكونَ؛ إذ الممتنعُ يحتملُ أن يكونَ ملزوماً للنقيضَين .

[Text 14] فإن كان محمولُها جزءاً من الموضوعِ كقولنا: “كلُّ إنسانٍ حيوانٌ” حصلَ الجزمُ بصدقِها مع صدقِ قولِنا: “بعضُ ما هو ملزومُ الإنسانِ فهو ملزومُ اللاحيوان” لما بَيّنّا من البرهانِ .

Acknowledgments

I would like to thank the two anonymous reviewers, Peter Adamson, Andreas Kapsner, and Tony Street for their comments on this paper. This paper is part of a project supported by Deutsche Forschungsgemeinschaft (DFG). I am very grateful for their generous support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Deutsche Forschungsgemeinschaft.

Notes

1 See Street (2014, 461).

2 Street (2014) admits that he could not find an answer to these objections in the works of Arabic logicians.

3 Absolute propositions are those propositions that are not modified by modalities such as necessary, possibility, or temporality. Nevertheless, according to Arabic logicians, all propositions are modal in one way or another. Roughly speaking, an absolute proposition is one in which the predicate is true of the subject at least once. For more details, see Street (2000).

4 Al-Fārābī (2012 108).

5 Many Arabic logicians found al-Fārābī’s idea in contrast with what is common (ḫalāf al-ʿurf). Ṭūsī, for example, objected that Fārābī’s thesis implies that a sperm is also rational, for an sperm is a potential human, and it is possible for it to become human (Ṭūsī, 1947, 162). One may correctly object that Ṭūsī confuses possibility with potentiality.

6 Qiyās (1964, 21).

7 Išārāt (2008, 90).

8 Al-ʿibāra (1970, 112).

9 One may ask if impossibles can not be the subjects of propositions how can we talk about them? Avicenna’s answer is that the impossibles do not exist in the mind, but their concepts exist in the mind and that enables us to talk about and refer to them. For more, see Bäck (1992).

10 Mulakhkhaṣ (2002, 140–141). Translation from Street (2014). Street (2014) also discusses the essentialist/externalist distinction but confines his discussion to Rāzī’s version of the definition of the essentialist proposition and does not go through the details of Ḫūnaǧī’s definition which is based on an implication relation.

11 I have translated ‘ḥaqīqat’ as essentiality instead of ‘real’ which could have been a more literal. This is to emphasize that in the essentialist reading only those predicates are true of a subject that are implied by the essence of the subject. For more on the translation of ‘ḥaqīqīyya' as ‘essentialist' (see Street 2014, 462). For an essentialist reading of subject term of a proposition I simply use ‘essentialist proposition'. In some places, especially in his discussion of contraposition, Ḫūnaǧī applies the essentialist reading to the predicates of propositions as well. But we will not discuss these cases in this paper. So when I use ‘essentialist proposition', I mean a proposition under an essentialist reading of its subject term.

12 Rāzī applies the distinction to his account of the conversion of propositions, but the distinction is present in most parts of Ḫūnaǧī’s logic. Moreover, many of his objections, and his replies to objections raised by other logicians are based on this distinction.

13 Ḫūnaǧī (2010, 84).

14 Ḫūnaǧī (2010, 148).

15 That may explain the coinage of ‘essential’ (‘ḥaqīqīya’). See also Street (2014, 462).

16 Ḫūnaǧī (2010, 130).

17 For a brief review of Ḫūnaǧī’s innovations (see El-Rouayheb 2010).

18 For one thing, he applies the essentialist/ externalist distinction, in his discussion of contraposition, not only to subjects of propositions but also to predicates of proposition. Consequently, there will be four different readings of a proposition.

19 In fact what follows is true of the negative universal of seven modal propositions: the temporal necessity (waqtīyya), the spread necessity (muntašira), general possibility (mumkina ʿāmma), special possibility (mumkina ḫāṣṣa), non-perpetual existential (wujūdīyya lā-dāʾima), non-necessary existential (wujūdīyya lā-ḍarūrīyya), and the general absolute (muṭlaqa ʿāmma). The temporal necessity is the strongest one among these seven modal propositions. As Ḫūnaǧī discusses this modal proposition and draws some conclusions about other six propositions, we limit our discussion, here, to this modal proposition, namely the temporal necessity proposition. For more details, see Street (2014). Street provides a clear translation and a comprehensive commentary on the section on conversion from Ḫūnaǧī’s Kašf al-Asrār.

20 In Arabic there are different words for the eclipse of the moon and the eclipse of the sun. I have translated ‘ḫusūf’ as ‘eclipse’ instead of ‘lunar eclipse’. That is for the sake of simplicity. Wherever I am using ‘eclipse’ in this paper, I mean lunar eclipse.

21 Ḫūnaǧī (2010, 129).

22 (Ḫūnaǧī 2010, 130).

23 See footnote 21.

24 Ḫūnaǧī (2010, 129).

25 Qiyās (1964, 296–297).

26 It is worth noting that Avicenna and post-Avicennan logicians quantified conditionals and therefore recognized A-, I-, E- and O-conditionals, and also applied the figures and moods of the categorical syllogisms to wholly hypothetical syllogisms.

27 Qiyās (1964, 297). Also, see Qiyās (1964, 239), where he says: ‘and know that if a speaker says “if five were even it would be a number”, what he says is in some respect right (ḥaqq) and in some respect not right. This saying is right so much that it is an implication for the speaker (yalzimu al-qāʾil bi-hī), and it is not true in reality (or in fact)’.

28 Ṭūsī (1947, 259–260).

29 Ṭūsī (1947, 81).

30 Ṭūsī (1947, 259).

31 Ḫūnaǧī (2010, 204). This is the way Avicenna had defined quantifiers for conditionals (Qiyās 1964, 262–263).

32 I use ‘situation’ as including both assumptions and times. This is in fact the way Arabic logicians often use the term.

33 Ḫūnaǧī (2010, 319).

34 Ḫūnaǧī (2010, 320).

35 This section is mainly concerned with Ḫūnaǧī’s rejection of connexive principles. El-Rouayheb (2009) goes into more detail about the history of the rejection of connexive principles in Arabic logic and provides a comprehensive account of their relevance to inferences from inconsistent premises. For this reason, I will be brief in this section. The keen reader may wish to consult El-Rouayheb’s work. Yet, I have tried to add that Ḫūnaǧī’s argument for rejecting the principles of connexivity targets Avicenna’s argument for these principles which is a reductio argument.

36 Qiyās (1964, 366–367).

37 Nothing will change. Arabic logicians did not use letters for propositions, though they used latter for terms in categorical propositions.

38 To be more precise he proves that if ‘in every situation if $P$, then $Q$’ is true, then ‘in no situation if $P$, then it is not the case that $Q$’ is true.

39 Ḫūnaǧī (2010, 208).

40 For more on this, i.e. Ḫūnaǧī’s argument against connexive principles from reductio proof, see (El-Rouayheb 2009).

41 For one thing, the predicate has been predicated truly of the subject by supposing the existence of the subject. Recall that in the essentialist reading one reads ‘if it existed, it would be … '.

42 Abharī (2018, 177).

43 Which means that B is an implied accident (ʿaraḍ al-lāzim) for J, and not its essence.

44 Abharī (2018, 177–178).

45 Abharī (2018, 195).

46 Abharī (2018, 178).

47 One may argue that ‘some J is not B’ is different from ‘some J is not-B’. It is true, but the difference, which results in a difference between the truth values, is dependent on whether the subject exists or not. Here such a question does not arise, because according to the essential reading everything as the subject of the proposition is treated as if it exists. So there is no difference between subjects of the propositions with regard to their existence and nonexistence.

48 I dismiss the case in which the predicate is not part of the subject. This case raised no problem for Ḫūnaǧī’s essentialist reading. The truth of such propositions are not certain, though some are both true and false. To be fair, for any J that implies B, we can consider a J that does not imply B, and vice versa. If that is true, these propositions are also both true and false.

49 Kātibī (2019, 440) and Taḥtānī (2014, 193).

50 Ṭūsī (1974, 163).

51 For more discussions on Ṭūsī’s objections to Abharī on implications between a whole and one of its part (such as between J &B and B) see Thom (2010). Thom fails to see that some part of Ṭūsī’s objection to Abharī are about Abharī’s objections to Ḫūnaǧī’s account of eesentialist propositions. The main concern of Thom (2010) is implication of the part from the whole.

52 This does not deny that these propositions are both true and false. After all, according to Ḫūnaǧī, there are only two truth-values, truth and falsity. Accordingly, a proposition that is contradictory has both values. As Priest (2014) shows, it is not necessary to invoke a third value, namely both true and false, to include contradiction in the semantics.

53 As we saw in footnote. 19, Ḫūnaǧī considered seven modal propositions among which temporal necessity proposition is the strongest.

54 See section 3.1, and Ḫūnaǧī (2010, 129–130).

55 Ḫūnaǧī (2010, 129).

56 See Street (2014) for a comprehensive commentary and an English translation of Ḫūnaǧī’s arguments.

57 The most notable exception is Rāzī. As Kātibī explains in his commentary on Kašf al-asrār, Rāzī maintained that both truth and falsity must be preserved in conversion (Kātibī 2019, 385). For more see Rāzī (2002, 184–185).

58 This discussion was not mentioned in Street (2014). Thom (2010) discusses Ṭūsī’s objection to Abharī’s account of implication of the part from the whole, but does not do so thoroughly, nor does he recognize its relation to Ḫūnaǧī’s account of the essentialist reading.

59 Obviously when we are not assuming truth value gaps.

60 It is possible to give both interpretation for the same logic. Yet, what else must be changed beside truth values is the designated value in that logic, i.e. the value that is preserved in a valid entailment. See Priest (2014). He argues that for some logics such as LP, both interpretations are equivalent, though the designated values in (1) are true and both true and false, and the designated value in (2) is true. If the second interpretation is applied, the designated value is true, for there is no other value other than false (For obvious reasons it does not seem reasonable to take false be the designated value).

61 The same is true of contraposition, syllogism, and the notion of implication.

62 Ḫūnaǧī (2010, 129). For a better explanation of Ḫūnaǧī’s arguments see Street (2014).

63 Remember that (1) is convertible only in the essentialist reading and thus the propositions in the reductio argument are essentialist.

64 These are the two main topics that Ḫūnaǧī’s essentialist reading play an important role. His account of modal syllogistic is completely based on the externalist reading.

65 Rāzī had applied the essentialist reading as a result of his rejection of Avicenna’s mental existence. Accordingly, non-actual possible and impossible individuals that are the subjects of essentialist propositions do not have any existence.

66 Nevertheless, there is an important point to note. Ṭūsī’s objection to Abharī’s argument should not be taken as an endorsement of Ḫūnaǧī’s position. Ṭūsī was probably familiar with Ḫūnaǧī’s logic at the time that he wrote Tanzīl al-afkār i.e. 1258. As a purist Avicennian logician, Ṭūsī could not agree with Ḫūnaǧī on his account of essentialist propositions, and he never endorsed the essentialist reading according to which the subjects of propositions can be impossible individuals. Thanks to the referee who mentioned this point.