We are now ready to turn to modal logic, which as its name implies, studies logical relations between propositions with attention to the manner or mode (in Arabic jiha) in which the predicate is ascribed to the subject in propositions. Most obviously we have the contrast between somethingâs being asserted as merely possible, and its being asserted as necessaryâa central contrast for Avicennan logic, since what we are working towards is a theory of demonstrative science that admits only necessarily true propositions. Having observed this distinction, Aristotle already began to explore the question of how modalized propositions fit together to form valid arguments. To take just the most obvious example, two merely possible premises yield a merely possible conclusion, whereas two necessary premises yield a necessary conclusion. Avicenna carries on, and greatly expands, this project.1 One fundamental innovation is his claim that all propositions are at least implicitly modalized. If the mode is not explicitly mentioned, a proposition expresses that the predicate belongs to the subject at least once. He calls such propositions âabsolute (muá¹laq)â [T1]. And if we say âtwo is evenâ without adding further qualification, we have expressed a necessarily true proposition, even though the necessity is not made explicit. The upshot is that in a sense there is no such thing for Avicenna as ânon-modalâ logic, because there are no syllogisms whose premises and conclusion lack a modal operator, though it may seem otherwise when this is left tacit.
Furthermore, for Avicenna there are more modes than just necessity, possibility, and impossibility. In the Aristotelian tradition a close connection had been seen between somethingâs being always the case and its being necessary. Aristotle argues that these are mutually entailing in the famous chapter 1.12 of On the Heavens. And obviously something impossible is never the case. That seems to leave merely possible or contingent true propositions to reflect state of affairs that are true sometimes, since if they were never true they would be impossibility propositions, if always true, then necessity propositions. Scholars refer to this as a âstatisticalâ interpretation of modality.2 Avicenna by contrast sharply distinguishes between what we can call âtemporalâ and âalethicâ modes:
|
Temporal |
Alethic |
|---|---|
|
Never |
Impossible |
|
Sometimes |
Contingent (= two-sided or âspecialâ possible [T2])3 |
|
Always |
Necessary |
The real payoff here comes in the second line: it allows Avicenna to consider propositions that might be true but as it happens never are in fact true. For example, âAvicenna is in Floridaâ is not impossible, as some exceedingly unlikely sequence of events might have brought him there. But it never happened. So it would fall into the first line under the heading of âtemporalâ propositions. The upshot is greatly to increase the number of syllogistic forms that need to be considered in modal logic, since now we have to think about what happens when we, say, have a perpetuity premise (one that is always true) in combination with a two-sided possibility premise: what mode or modes would apply to the conclusion?
Nor do the complications end there. Avicenna also distinguishes between substantial (dhÄtÄ«) and descriptional (waá¹£fÄ«) readings of a proposition. For example âbody is extendedâ is true substantially, as âextendedâ is true of body as long as it exists. But âbody is changing place when movingâ is true descriptionally, since it holds true for body only insofar as motion belongs to it. Yet again, therefore, we can expand the total number of possible combinations that yield candidate syllogisms. To take the same example just given, when we ask what happens when one premise is perpetual and the other two-sided possible, in theory at least we need to work out the answer for the case where both premises are true substantively, where both are true descriptionally, and in the two cases where the premises differ in this regard.
In the course of the tradition even compound modal propositions developed. For example, one could say: âAt least once: every human is P, but not necessarily.â (The Arabic terminology for all these different modality propositions is not intuitive; for an explanation see [T21] and T. Street, Najm al-DÄ«n al-KÄtibÄ«'s al-RisÄla al-Shamsiyyah (New York: 2024), Appendix 1 & 2). The total number of modality propositions thus multiplied, on al-ṬūsÄ«âs count, to 217. But al-RÄzÄ«, based on Avicennaâs distinctions and his own, narrowed them down to 13 pertinent types that became standardly considered. This is the figure given in [T21] and [T31], sometimes 15 are distinguished; [T21] is a good place to start since there al-KÄtibÄ« explains each of the types. But thatâs without even getting into the fact that these distinctions are iterated for all the Aristotelian moods (on al-ḤillÄ«âs count [T31], for each of the four syllogistic figures 169 moods can be formed). You will be glad to know that we donât need to go through all possible combinations, though. For one thing, there was a tendency to formulate general rules that cover many cases all at once, something we can already observe with Avicennaâs student BahmanyÄr [T4].
For another thing, the debate over modal logic focused above all on one central problem. Obviously if the premises have the same mode, the conclusion is going to share that mode. So the combinations under dispute were always âmixedâ modal syllogisms, that is, syllogisms with premises of differing modality. What our logicians want to know is, which of these syllogisms are âproductive,â that is, validly yield a conclusion, and what mode will the conclusion have? And here, one combination in particular provoked the most controversy: that of a necessity major premise with a possibility minor premise.
It is with good reason that this case dominates discussions of modal logic in our period. The other mixed combinationâpossibility majors and necessity minorsâwere standardly seen as perfect syllogisms producing a possibility conclusion, without needing a proof. They were seen as instances of dictum de omni, that is, straightforwardly valid because everything that is universally true of a class is true of any subclass (see [T4]). The major premise plays a special role in a modal syllogism, because it contains the major term that reappears as the predicate in the conclusion. The modality expressed by the major premise is what is said to hold between the major and the middle term. It stands to reason that the same modality should hold between the major and the minor term in the conclusion, if the syllogism is valid.
However, the matter is different with necessity majors and possibility minors: According to the above rationale, Aristotle and Avicenna held that they produce necessity conclusions [T3], but for that you need to assume that the possibility in the minor premise is actualized.4 Whether this is a legitimate move was a vital question, because Avicenna derived other valid inference patterns from this one. So if first figure syllogisms with possibility minors are not productive, the entire deductive system is in danger. BahmanyÄr provides an example of the relevant sort of syllogism in [T4]:
Possibility minor premise: every human can possibly moveNecessity major premise: whatever moves is necessarily a bodyTherefore: every human is necessarily a body
Now, this may look strange. How can we infer that something is necessarily the case on the strength of something that is only possibly the case? But as the example shows, at least some inferences of this form are actually rather intuitive. If humans are just a sort of thing that can move, and anything that in fact moves is a body, then surely humans are bodies. The debate in post-Avicennan philosophy becomes quite technical, but fundamentally it boils down to a contest between these two intuitions.
As seen in early engagements with the question, like [T5] and [T8], a rule of thumb is that the modality of the conclusion follows that of the major premise (that is, the premise in which the predicate term of the conclusion appears). In other words, so long as this premise is necessary, so is the conclusion. In [T5] al-SÄwÄ« says that the conclusion will be necessary even if the major premise is absolute, that is, not explicitly modalized. In [T6] he considers an exception, where the major premise âJ is Dâ is applied to subjects that are not always J. For instance if humans who are asleep have closed eyes, and those with closed eyes necessarily cannot see, this doesnât imply that humans cannot see, since humans are sometimes not asleep.
A significant breakthrough then comes with al-RÄzÄ«, who introduces the distinction between essentialist (ḥaqÄ«qÄ«) and externalist (khÄrijÄ«) readings of the subject term [T9], a distinction we have mentioned in the introduction to this volume. As al-RÄzÄ« puts it, the idea is that if the subject term were to be realized externally, then the predicate would apply to it. Now, al-RÄzÄ« still accepts that mixed syllogisms with possibility minors are productive of necessity conclusions [T10, T11]. But his point about the externalist reading opens the door for al-KhÅ«najÄ«âs rejection of this Avicennan doctrine [T13]: if all As are only possibly Bs then perhaps there is not a single A that is ever a B in fact. (A similar worry had already been expressed by al-SuhrawardÄ« [T7].5) In effect al-KhÅ«najÄ« here uses Avicennaâs own insight against him: some genuine possibilities are never realized. And we donât want to infer necessary truths from such unrealized possibilities. Hence, for al-KhÅ«najÄ«, syllogisms with possibility minors are sterile on the externalist reading; on the essentialist reading we simply cannot say whether they are productive or not. A nice example, and explanation, are given by Ibn WÄá¹£il [T26] in his discussion of al-KhÅ«najÄ«âs view.6 Notice how stark is the disagreement with Avicenna: whereas he held that these syllogisms produce a necessity proposition as their conclusion, al-KhÅ«najÄ« and those who followed him held that no conclusion at all validly follows from them, not even a possibility proposition.
Among those who agreed was al-AbharÄ«, who likewise outright denies the productivity of syllogisms with possibility minors [T14, T15, T17]. He extends the point to second and third figure syllogisms [T16, T18]. This solicited a spirited defense of the Avicennan position from al-ṬūsÄ« [T19, T20], as we would expect from him (for a response see al-ḤillÄ« at [T30]). But more generally it seems that most logicians after al-KhÅ«najÄ« at least put constraints on the productivity of these syllogisms. Thus al-KÄtibÄ«, in his very influential logical textbook al-RisÄla al-Shamsiyya, stipulates that the conclusion will follow the modality of the major, but only if the possibility mentioned in the minor premise is at some point actual [T22, T23, T24]. An interesting additional point made in [T24] is his response to an attempt to save the Avicennan view by saying that the subject term in a possibility minor falls within the scope of the predicate term, whether or not the predicate ever actually applies to the subject. To use BahmanyÄrâs example again: humans are just by nature among the things that move, and this is true even if in fact no human ever moved. So the idea here is to push the intuition in favor of Avicennaâs rule by appealing to class inclusion. But al-KÄtibÄ« disagrees. For him, if humans never in fact moved, they would not fall within the relevant class of moving things, so their capacity for motion would not show that they must be bodies.
Alongside al-RÄzÄ«, who is prominent as a defender of the productivity of syllogisms with possibility minors, and al-KhÅ«najÄ«, who is the most influential opponent of this view, another figure who was cited frequently in the later tradition was al-KashshÄ«. In his ḤadÄʾiq he gives a general rule according to which the conclusion in mixed syllogisms carries the same modality as the major premise, except if the minor is necessary and the major a negated perpetual [T12].7 Then, the conclusion follows the minor premise. Among the authors who refer back to him are al-UrmawÄ« [T25] and al-SamarqandÄ« [T27, T28], both of whom provide overviews of the debate up to their time and follow the anti-Avicennan view of al-KhÅ«najÄ«. Finally, al-ḤillÄ« comments on al-KÄtibÄ«âs Shamsiyya and in this context contrasts the new orthodoxy that the minor possibility premise must be actual, to the view of âolder scholarsâ who did not lay down this condition [T29]. Here and at [T31] he mentions a point that makes the actuality requirement seem much less intuitive: how could a conclusion go from being possible to being necessary, just because some other possibility (namely the one articulated in the minor premise) happens to be realized? Using BahmanyÄrâs example one last time, the modal status of the truth that humans are bodies shouldnât be affected by whether or not humans actually move.
As we can see from all this, logicians spent centuries grappling with the implications of Avicennaâs decoupling of possibility from actual occurrence. The course of the debate reveals an underlying, more fundamental disagreement as to the very function of modal syllogisms. Avicennaâs idea was that asserting a possibility about something tells us about the nature of that thing, whether or not the possibility is realized; today we might put the point by saying that if X moves in any possible world, then X is a body in every possible world. Al-KhÅ«najÄ«âs idea was different. He seems to have thought of syllogisms as expressing what we can infer from the observed features of things in our actual world, not from mere possibilities. Al-RÄzÄ«âs contrast between essential and externalist readings seems to offer the crucial insight here. In the end, whether these syllogisms are productive depends on which reading we adopt for the minor premise: on an essentialist reading that accommodates counterfactuals situations, the premise tells us something about what is true in all possible worlds. On an externalist reading, the premise tells us only about the actual world.
Texts from: Avicenna, BahmanyÄr, al-SÄwÄ«, al-SuhrawardÄ«, AbÅ« al-BarakÄt, al-RÄzÄ«, al-KashshÄ«, al-KhÅ«najÄ«, al-AbharÄ«, al-ṬūsÄ«, al-KÄtibÄ«, al-UrmawÄ«, Ibn WÄá¹£il, al-SamarqandÄ«, al-ḤillÄ«.
[T1] Avicenna, IshÄrÄt, Maná¹iq, 88.4â89.12
[types of necessity: substantial vs descriptional]
Every proposition is either: (a) a general absolute [proposition], that is, a [proposition] in which a judgment is expressed without thereby expressing its necessity or its perpetuity, or anything else concerning [the judgmentâs being applicable] at a certain point in time or in the sense of possibility. Or (b) it is [a proposition] in which some of the above [qualifications] are mentioned, be it necessity, perpetuity without necessity, or existence without perpetuity or necessity.
Necessity may be (a) absolute, as in âGod exists.â Or it may be connected to a condition. The condition may be (b) the duration of a substanceâs existence, as in âHuman is necessarily rational body.â By this we do not mean that human is always and will always be a rational body, because this would be false for each individual human. What we mean by it is that as long as the substance is existent as a human, it is a rational body. The same goes for every negation that resembles this affirmation. [89] Or the condition may be (c) the duration of the subjectâs being described by one of its attributes (mawṣūfan bi-mÄ wuá¸iÊ¿a maÊ¿hu), as in âAll moving things are changing.â The meaning here is not [that this is so] absolutely, or as long as the substance (dhÄt) exists, but [that this is so] as long as the substance that is moving, is moving. There is a difference between this (c) and the first condition (b), because the first condition turns on the substance itself (aá¹£l al-dhÄt), namely âhuman,â whereas here it turns on the substance being described by an attribute attached to it, namely âmoving thing.â A moving thing has a substance and an essence to which both movement and rest may attach; âhumanâ and âblackâ are not like that. Or it may be a condition of [the duration] (d) of the predicate; or (e) of a definite time, as in an eclipse, or (f) of an indefinite time, as in breathing.
Even though necessity on the first condition (a) is not considered to be the absolute necessity that does not turn on any condition, it still shares with it the feature of being either more specific or more general, or of two more specific [necessities] being subsumed under a more general one, if the condition in the conditioned proposition is that the substance is not always existent. What the [types of necessity] share is what is meant by saying ânecessary proposition.â
[T2] Avicenna, IshÄrÄt, Maná¹iq, 90.11â91.13 [trans. Inati, mod.]
[one-sided and two-sided possibility]
By âpossibility (imkÄn)â may be meant (a) that which is implied by negating the necessity of nonexistence, that is, impossibility. This is [the meaning of âpossibleâ] according to the sense first laid down. In this sense, whatever is not possible is impossible. This [type of] possibility may be predicated of the necessary [and not just the contingent].
Or by it may be meant (b) that which is implied by negating both necessity of nonexistence and necessity of existence. This is [the meaning of âpossibleâ] according to the sense that has been laid down according to a specific tradition, such that the primary [i.e., this second] type of possibility may be true of something regardless whether that thing is negated or affirmed, so that it is possible that it is, and also possible that it is not. [91] In other words, it is not impossible that it be, and also not impossible that it not be. As possibility in the primary sense is true for both options [i.e., existing or not], it is properly designated by the name âpossibility.â The necessary does not fall under this kind of possibility, according to which things are either possible, or necessary, or impossible. Whereas what is understood by the first sense (a) is that things are either possible or impossible. âNot possibleâ in the primary sense, i.e., the second and proper sense, means the same as ânot non-necessary.â Hence the necessary is not âpossibleâ in this sense. What is possible in this sense includes things that exist, but whose existence does not necessarily endure, even if it is necessary at a certain point in time, like an eclipse.
One may also say âpossibleâ and understand (c) a third sense by it, which is more specific still than the two senses just mentioned. In this sense the judgment is not necessary at all, not at a specific time as with the eclipse, nor in a certain situation as with change for a moving thing. Rather, it is like âwritingâ [judged to belong to] âhuman.â
There are, then, four notions: (a) necessary (wÄjib), (b) impossible, (c) having a certain necessity (á¸arÅ«ratun mÄ), (d) having no necessity at all.
[T3] Avicenna, ShifÄʾ, Maná¹iq, QiyÄs, 148.9â17 [trans. Street, mod.]
[agreement with Aristotle and al-FÄrÄbÄ« on the mixed modal syllogism]
Know that the Eminent Scholar [al-FÄrÄbÄ«],8 with whom I am most concerned to conduct my discussion, agrees with what I say. Indeed, the First Teacher [Aristotle] believes that if the necessity major in the first figure is joined to a non-necessity minor, the conclusion is necessary. Let us assume all Js are Bs non-necessarily, and all Bs are As necessarily, and it yields what the Eminent Scholar and the First Teacher both agree on, and what you have learned. But why doesnât one of them go on to say that this is not a necessary proposition, but rather must be: all Bs are As necessarily insofar as they are Bs [â¦]? If this is taken into account, then what the detractors say against those who produce a necessity proposition from these premises turns out to be true. That is because what the detractor is saying in this matter is like what the Eminent Scholar had to say about the conversion of the possible proposition.
[T4] BahmanyÄr, Taḥṣīl, 136.5â137.5
[general rules for mixed modal syllogisms in the first figure]
Concerning the mixed possibility and necessity [syllogisms] in the first figure: if the minor premise is a necessity [proposition] and the major a possibility [proposition], then no doubt the conclusion is a possibility [proposition], because it is a dictum de omni (al-maqÅ«l Ê¿alÄ l-kullÄ«). When the major is a necessity [proposition] then the conclusion must be necessary as well. Let us explain this in a way that is easy to understand.
When we say âevery B is necessarily Aââthat is, whatever is said to be B, is perpetually Aâthen if J is said to be B, it is so perpetually, not9 as long as it is described as being B, but as long as the substance J that is described as being B exists. If some J becomes B, then it was already the case beforehand that B is A, and likewise will still be the case afterward that B is A, after [J] ceases to be B. For example:
Every human [J] can possibly move [B]Everything that can move [B] is necessarily a body [A]Therefore, every human [J] is necessarily a body [A]
Given that every moving thing as long as its substance existsâbe it moving or notâis described as being a body, it is true of human insofar as it is a moving thing that it is necessarily a body. That is: as long as its substance is existent, not only insofar as it is [actually] moving, for it is [137] a body before and after it is moving. Not that this [sc. being a body] applies to it [sc. human] only once the latter is moving. For it is not the case that a thing [like a body] acquires a certain property [like moving] only when [the property] exists [i.e. when the body is moving], as this property is applicable even before [the relevant property, i.e. motion] exists; it is thus not the case that if the [property] did not exist, then it would not belong to this [thing].
Likewise, if the major is a negated necessity [proposition], then the conclusion is a negated necessity [proposition as well], as we have explained.
You know now the particular syllogisms among them; the rule for the major premise is that if it is a possibility [proposition] then so is the conclusion, or if it is a necessity proposition, then so is the conclusion.
[T5] Al-SÄwÄ«, Baá¹£Äʾir, 151.10â20
[the modality of the conclusion usually follows the major premise]
As we have now completed [the treatment of] absolute and necessity syllogisms in these three figures, and also the possibility [syllogisms] in the first and second figure, we should now turn to explaining the mixed modal [syllogism] made up from those. If the major is an absolute and the minor a necessity [proposition] in the first figure, then they agreed that the conclusion is an absolute, following the major premise. If the major is a necessity [proposition] then the truth is that the conclusion is so as well. But the received opinion is different from that. To explain: with a necessity conclusion, if we say in the major premise âevery J is Dââthat is, everything that is described as J, whether it is so described perpetually or not, is described as D by necessityâand B is among the things absolutely described by J, then [B] falls under the major premise, and D is said of it necessarily. Thus in this mixed [syllogism], the conclusion follows [the modality] of the major premise, unless the minor is necessary and the major absolute, in the sense that it is conditional on the substanceâs being described in a certain way. Then the conclusion will be necessary, because if B is described by J as long as it exists, and every J is D as long as it is described by J, then B is D, as long as it exists. For the duration of D for it is as long as the duration of J, and J persists for it as long as it exists.
[T6] Al-SÄwÄ«, Baá¹£Äʾir, 151.20â152.11
[âthe most eminent of the later logiciansâ on mixed modal syllogisms]
The most eminent of the later logicians [sc. Avicenna?] said: it is not appropriate for the major premise to be conditioned by [saying that] J is D as long as it described by J, and not perpetually, for then it becomes false. If we say âevery J is D, not perpetually, but only so long as it is J,â then we are judging that every J is not always J. But we had said in the minor premise the same of that which is J, that is: B is what is J perpetually, which is absurd. Let us pursue what he said where he rejects conditioning the major premise by non-perpetuity in a sense, even though the syllogism does not produce a conclusion on this condition.
As for his justification of that by the falsity of the major premise, this is not the case generally and on all considerations and perspectives of predication and subjection, for it is possible that non-perpetuity be made a part of the subject. Then one may say, âeverything that is not always J is Dâ and this is not the respect he mentioned. He made non-perpetuity part of the predicate, saying âevery J is D not always but as long as it is J.â Here, non-perpetuity is part of the predicate and for this reason [152] the major premise is false. If we make the J in the minor premise the predicate of what is described as J always, and here (that is, in the major premise) make the description of every J by J-ness non-perpetual, then since we made the predicate non-perpetual but conditioned on the duration of J-ness, then by necessity J-ness is not perpetual. For if the duration of J-ness were non-existent, then its description as D would be non-perpetual. If one thing is an accompaniment of another that is posited as non-perpetual, then it is itself non-perpetual. As for the perspective from which we made it a part of the subject, in this sense the major premise is not false. For you donât judge of every J that it is described as J, but not always; you make a judgment indicating that what is not always J is among the things described as J. This does not preclude the existence of something that is always described as J. But this J [sc. that which is only temporarily described as J] is not what was predicated in the minor premise. If its predicate is what is always J, then the middle term would not allow for a single syllogism of which it could be part, and therefore no conclusion follows. Hence [comes] the point of view according to which it is said that it is not necessary that in the major premise there be a condition on the duration of J-ness, because either the major is false, or it is true, but then the syllogism has no minor term.
[T7] Al-SuhrawardÄ«, Maná¹iq al-TalwīḥÄt, 53.7â11 [trans. Street, mod.]
[an exception to the rule that the modality of the conclusion follows that of the major premise]
You should know that the conclusion in first-figure syllogisms follows the major premise in the mixed-premise syllogisms, because the major term has greater scope than the minor term with respect to what is judged to hold of the middle term; except when the minor premise is possible and the major is existential (wujÅ«diyya). If we say âpossibly every J is Bâ and âin actual existence (bi-l-wujÅ«d) every B is A,â it is known from the nature of possibility that it may never actually occur; so if the J is never described as B, it does not follow that A comes to it actually, but only potentially. So itâs [merely] possible.
[T8] AbÅ« al-BarakÄt, MuÊ¿tabar, vol. 1, 148.9â149.5
[the modality of the conclusion follows the major premise]
If the propositions are necessary, then their conclusions are likewise necessary in the first and the second figure, as well as in the productive moods [constructed] in them. The sterile moods, as well as the examples and proofs, are the same as those we adduced to specify the generality and specificity of the terms in each of the moods.
With regard to the first figure, this is because the minor term is among [the instances] of the middle term, so the former is [whatever] the latter is. So if the major term is judged to hold necessarily of the middle term, then the same goes for [the middle termâs] being predicated of the minor term. The predication in the conclusion does not have greater scope than that of the major premise. [â¦]
[mixed modality extended to the case of existential and mental premises]
[148.23] In the first figure, [when the premises are merely possible the conclusion is] likewise possible, because when it is possible that the minor term applies to the middle term, and possible that the middle term applies to the major term, [149] then it is [merely] possible that the minor term applies to the major term, regardless whether the possibility is an existential or a mental one. So the conclusion is likewise [possible]. If [the syllogism] is a mix of both [mental and existential possibility premises], then the conclusion is a mental [possibility proposition], not an existential one. If it is of the kind of mental possibility that is necessary in the existential sense (fÄ« l-wujÅ«d), then the judgement in the conclusion mixes possibility and necessity. Sometimes it is necessary and sometimes possibleâyou donât know. But the judgment in it is by way of mental possibility.
[T9] Al-RÄzÄ«, Maná¹iq al-Mulakhkhaá¹£, 140.11â143.3
[essentialist vs externalist readings of the subject term]
When we say âEvery J,â this is sometimes used with reference to a essence (ḥaqÄ«qa) and sometimes with reference to external existence (al-wujÅ«d al-khÄrijÄ«). As for the first [sc. the essentialist reading], when we say âEvery Jâ there are five conditions to be considered.
(a) We donât mean by it J as a universal, nor the whole [of the Js] insofar as they are a whole, but we mean each instance of J. [141] The difference between the three concepts just mentioned will be fully explained later on.
(b) We donât mean by it that whose essence is J only, or that which is described as J, but something broader than that, namely that of which it is true to say that it is J. If we meant by âEvery Jâ that whose essence is J, the judgmentâs scope could not extend from the middle term to the minor term. And if we meant by it that which is described [as J], then no subject would need anything else.
(c) We donât mean by it that which is described with J-ness in the extra-mental world, but something broader than this, namely that of which, if it were to exist in the extramental world, it would be true to say that it is J, regardless of whether it actually exists in the extra-mental world or not. We can say âEvery triangle is a shape,â even if no triangle exists in the extra-mental world, as meaning that everything that, were it to exist [extra-mentally] and is a triangle, would inevitably be a shape when it existed.
(d) We donât mean by it that which is J always, or not always, but something broader than these two, from which these two [senses] should be distinguished. Namely that, when we say âEvery Jâ and then stop; or when we say âEvery J, [142] whether necessary or not, whether always or not,â this is [the meaning] it has. These modalities are part of the subject, but they are not absolute or modalized on that account, but on account of the quality by which the predicates are true of [the subjects].
(e) Al-FÄrÄbÄ« claimed that by saying âEvery Jâ we donât express that J-ness obtains in actual fact (ḥuṣūl bi-l-fiÊ¿l), but rather [we express] everything that can possibly be described by J-ness. But this is a linguistic issue. For whoever says âEvery Jâ may mean by it whatever he wants, but language precludes [what al-FÄrÄbÄ« proposed], because âblackâ does not include any object that is devoid of blackness at all times, even though it would be possible that it have âblackâ as an attribute.
Having considered these matters, [we may conclude that] âEvery Jâ allows for two readings. First [it may mean]: every J, according to the points mentioned above, in the state of being J. Second [it may mean]: everything of which it is true to say that it is J in actual fact, regardless whether the judgment about it is [considered to apply] now, before, or later. There is a difference between these two notions, because on the first account it is not true to say âEvery moving thing is at rest,â whereas on the second account it could be true [sc. if every moving thing is at rest at some other time].
As for the second, [the externalist reading]: here what we mean by âEvery Jâ is each of what exists in the extra-mental world in terms of instances of J, or everything that is present among the instances of J. On this account, if no heptagons exist in the extra-mental world, it would not be true to say âEvery heptagon is a shape,â whereas if no shapes except triangles existed in the extra-mental world, it would be true to say âEvery shape is a triangle.â Whereas on [143] the first [essentialist] reading, both would be false.
Thus far the discussion of the conditions expressed in the subject of an affirmative proposition; they are expressed in negative propositions in exactly the same way, with no difference.
[T10] Al-RÄzÄ«, Maná¹iq al-Mulakhkhaá¹£, 273.5â274.7
[the productivity of syllogisms with possibility minors and necessity majors]
If the minor premise is a possibility [proposition], then the major premise is either necessary, perpetual, non-necessary and non-perpetual, or any [of the above].
In the first case [sc. a possibility minor and necessity major], the conclusion is necessary, because the major premise signifies that whatever the middle term holds true of, in whatever way, also has the major term hold true of it, necessarily and at all times of its existence, both before the middle term occurs, and still thereafter. The minor premise, meanwhile, signifies that the middle term may possibly hold true of the minor term. On the assumption that this possible thing actually occurs, the major term holds true of the minor term necessarily. Everything that is necessarily true when something possible is stipulated as actually occurring is [itself] necessary, regardless whether this possible thing is actually occurring or not. For it is inconceivable that it becomes necessary [only] at the point when the possible thing actually occurs, and that it wasnât necessary already before.
Proof by reductio. Given the syllogism
Possibly: every J is BNecessarily: every B is A
If we say:
Not necessarily: Every J is A
Then the contraposition [of this] is either âNecessarily: not some J is Aâ or what amounts to the same thing, by two-sided possibility (bi-l-imkÄn al-khÄṣṣ)10 [i.e. âPossibly but not necessarily: every J is Aâ].
So adopting the first [version of the contraposition], we then take it with the major premise like this [274]:
|
Necessarily: not some J is A |
|
|
Necessarily: every B is A |
|
|
[to infer] |
Necessarily: some J is not B |
But [by the minor premise of the original syllogism] all [J] is possibly [B], so this is a contradiction.
If we adopt the second [version of the contraposition], then we take it together with the minor premise like this:
|
Possibly: every J is B |
|
|
Possibly: not some J is A |
|
|
[to infer] |
Possibly: not some B is A |
But [by the major premise of the original syllogism] all [B] is necessarily [A], so this too is a contradiction.
Let it not be said that this can be disproven with the example
Possibly: every sperm is a humanNecessarily: every human is an animalNecessarily: every sperm is an animal
For we say: we have clarified that the major premise here is not necessary absolutely, but only conditionally.
[T11] Al-RÄzÄ«, Maná¹iq al-Mulakhkhaá¹£, 274.8â276.12
[possibility minors with all other types of majors]
As for the second case, which was that when the major premise is perpetual, the conclusion is inevitably perpetual [too], the same argument (as in T10) can be applied.
As for the third case, which was that when the [minor premise] is neither necessary nor perpetual, then the conclusion is a two-sided possibility, regardless whether the major premise is a two-sided possibility, or a non-necessary existential, or a non-perpetual existential, whether a determinate temporal necessity or an indeterminate temporal necessity.11 For the major premise signifies that whatever the middle term holds true of, in whatever way, the major term also holds true of it with the same modality expressed in the major premise. The minor premise signifies that the middle term is possibly true of the minor term. If this possible thing occurs, then the major term holds true of the minor term with this same modality. But if it does not hold true, then it may be that the major term is not true of the minor, because its holding true of it depends on the middle term holding true of it. When the middle term does not hold true of it, then neither does the major term hold true of it, [275] though it may be that it holds true even in the absence of this dependence [i.e. the major term just happens to hold even though the middle does not], and the shared extension is that of two-sided possibility. [â¦]
[276.8] As for the fourth case, which was that if the major premise accepts both necessity and non-necessityâthere are four types of propositions like this: one-sided possibility,12 general absolute, general conventional, and general conditionedâthen the conclusion is a universal one-sided possibility proposition. Because [suppose it] accepts necessity, if it is true by itself and thus necessary, then the conclusion is necessary, too. Otherwise, it is a two-sided possibility. But the shared extension is one-sided possibility.
[T12] Al-KashshÄ«, ḤadÄʾiq, fols. 89v2â12
[productivity rules for mixed modal conditionals]
When the minor term belongs to the middle term in actuality in the case of two possibility premises, then this amounts to eleven kinds of propositions. And when the major premise is a proposition in which the duration of the predicate, as the description of the subject, is not expressed, and this can involve both conditioned and conventional premises, this amounts to nine kinds of propositions. In all of these, the conclusion follows the [modality of] the major premise, because the meaning of the major premise in all of them is that whatever holds true of the middle term, however it may be, also holds of the major term (or its negation), either without explicit modality if it is an absolute, or with explicit mention of a specific modality if it is modalized. The minor term, as that of which the middle term holds true, has the major term predicated of it without that modality, except in cases where the minor premise is necessary and the major premise a perpetual negation. In this case the conclusion follows [the modality of] the minor premise. Otherwise, the falsity of the minor premise follows.
[T13] Al-KhÅ«najÄ«, Kashf al-AsrÄr, 272.4â274.17
[Avicennaâs arguments that syllogisms with possibility minors are productive]
The Master [Avicenna] claimed that all of these [modal syllogisms] are productive. He made the conclusion in [syllogisms with] necessity majors necessary, while the conclusion from non-necessity [propositions] is a two-sided possibility, and the one from premises including both is a one-sided possibility. He said that if the syllogism is in the first figure and stands in need of a proof, it is not possible to provide the proof by means of the converse of the minor or major premise, insofar as one would be relying on something less clear [to prove what is more clear], but in another way.
He demonstrates the first [sc. that syllogisms with necessity majors produce necessity conclusions] in several ways.
First, suppose the contradictory of the conclusion is trueânamely âpossibly, some Js are not Aââand we stipulate this possibility to exist, and make it the minor premise, so that it produces a conclusion with the major premise in the second figure, contradicting the [original] minor premise. It does not follow from our stipulation that âsome Js are not A in realityâ becomes impossible, for that was also possible, and nothing impossible follows from the stipulation that [something possible] occurs. Rather, because it is impossible, it does not follow from a true premise. Hence, âsome Js are not Asâ as a one-sided possibility is false.
[273] Second, we add the contradictory of the conclusion after stipulating its occurrence to the minor premise so that this produces the contradictory of the major premise in the third figure, in the same way as just mentioned.
Third, we stipulate the minor premise to be existent, so that there follows that which contradicts the contradiction of the conclusion, and this implies an absurdity. But this is implied not by stipulating the minor premise to occur in actual reality due to its possibility, nor by the true major premise, but by that which is doubted in it: the contradictory of the conclusion.
Fourth, if the major term is not necessarily [true of] the minor term, once it is described by the middle term, inevitably it becomes necessarily [true of] it, and what was not necessary then becomes necessary. This is absurd, because what we mean by ânecessaryâ is that which is necessary absolutely and whose essence remains existent. The transformation from necessity to non-necessity and vice-versa only because of a change in attributes and states is impossible.
[replies]
Reply to the first: you unduly assumed that an absurdity follows if the contradictory of the conclusion occurs in actual fact, or if it is jointly present with the major premise itself. But the first is impossible, because it may be that the major premise is not true even if the contradictory of the conclusion is taken to occur in actual fact. Hence, nothing absurd follows that would be the conclusion of the syllogism, and there is nothing impossible about the occurrence of something in actual fact and the non-existence of its possible concomitants. If what is assumed is the second [i.e., if the contradictory of the conclusion is jointly present with the major premise itself], then we concede the hypothetical [premise], and the impossibility of the antecedent, but [that does not mean] that the contradictory of the conclusion follows from itâeven if the other premise were trueâbecause it may be that something is possible in itself, but it is absurd that it occur together with something else that is possible or necessary. Absurdity does not arise from just one part of the combination, but only from both [parts] together.
Let it not be said: this infringes on all syllogisms of the reductio ad absurdum form. For the conclusion sought here is the falsity of the conclusionâs contradictory, and if its occurrence together with something else implies an absurdity or a falsity, then there follows [274] from its falsity the falsity of both and the truth of the remaining parts [of the syllogism], because whereas the falsity of the combination does necessitate the falsity of some of its parts, its impossibility does not necessitate the impossibility of its parts. You have now taken note of this matter regarding the necessity of the converse negation, and there will be more clarification about this shortly.
And let it not be said here: the contradictory of the conclusion together with the major premise implies the falsity of the minor premise, because a possibility [premise] together with a necessity [premise] implies a necessity in the second figure, so that they are false in combination, and hence the contradictory of the conclusion is false. Or we claim that the conclusion is perpetual and its contradictory âSome J is not B by general absoluteness (bi-l-iá¹lÄq al-Ê¿Ämma),â together with the necessary major premise, is in the second figure productive of a necessity proposition contradicting the minor possibility premise. For we explained that a possibility [premise] together with a necessity [premise] in the second figureâeven though the syllogism about this consists of two necessity [premises]âis not productive of a necessity proposition. Rather, [it produces] a perpetuity proposition which does not contradict the minor premise, and no absurdity follows from this. This is also the reply to the second and third [of Avicennaâs points].
Reply to the fourth: by no means do we concede that the minor termâif it is not necessarily [true of] the major termâis described by the middle term in actual fact, so that what you said about its becoming necessary once it is described by the middle term would come to pass. This is because possibility is never just one of two disjuncts (lÄ yunÄfÄ« al-khuluww dÄʾiman) and its description in itself (fÄ« nafs al-amr) does not require it to be so in actual fact at all, such that the transformation [from possibility to necessity] you mentioned would follow. This would only be so if you claimed that its being described by the middle term held in itself (fÄ« nafs al-amr) and the transformation would be likewise so. Or else, if you claimed that the transformation was only so on account of its being described by the middle termâeven if it didnât occur this wayâthen what we mean by âtransformationâ would be that in this case the major premise is true. But this leads to the same issues as with the first point, and what you have learned there applies here too.
[T14] Al-AbharÄ«, Kashf al-ḥaqÄʾiq, 138.5â139.7
[disagreement with al-RÄzÄ« on mixed modal syllogisms]
Regarding the first figure: the minor premise in it is either actual or possible. If it is possible, then the major premise is either a necessity premise or a perpetual or a two-sided possibility premise that applies either to one of its parts or to the whole.
Concerning the first option, [al-RÄzÄ«] claimed that the conclusion is a necessity proposition, because the major premise signifies that everything of which the middle term holds true in actual fact has the major term hold true of it by necessity; and the minor premise signifies that the middle term is possibly true of the minor term. On the assumption that this possibility occurs, the major term will hold true of the minor term by necessity, and everything whose being is necessarily true on the assumption that a possible event actually occurs is necessary, regardless of whether this possible event occurs or not. This is because it is impossible that something become necessary on account of its occurrence despite being non-necessary in itself (fÄ« nafs al-amr).
But this calls for further inquiry. For we do not concede that, on the assumption that this possible actually occurs, [139] the major term will necessarily hold true of the minor term. It is only necessarily [true] if the major premise remains true on that assumption. Assuming that it is accepted [as true, the syllogism] cannot be productive. It is only productive if the following proposition is true: âFor everything of which the middle term holds true on this assumption, the major term necessarily holds true of it.â But this is rejected. What is true on this assumption is rather: âEverything that is a B in itself (fÄ« nafs al-amr) is necessarily A.â From this it does not follow that âEverything that is B, on the assumption that B obtains when [this thing] is J, it is necessarily A.â Hence, it does not follow that the major term is necessarily true of the minor term on the assumption that the middle term obtains for [the minor term].
[T15] Al-AbharÄ«, KhulÄá¹£at al-AfkÄr, 279.7â280.20
[disagreement with Avicenna on first figure mixed modal syllogisms]
This is the gist of [Avicennaâs] account after cutting down its longwindedness. But most of it is unconvincing.
As for the first figure, we do not concede that if the minor premise is an actuality premise, then the conclusion in which it is repeated follows [the modality] of the major premise. This is because a minor premise that is a general absolute or an existential [premise], together with a general conventional [premise], produces a general absolute [proposition] as a conclusion, whereas with a restricted perpetuity premise it produces a non-perpetual existential conclusion. Nor do we concede that in a repetitive [syllogism of this form, the conclusion] is a necessity premise, because the major term is perpetually true of what the middle term holds true of in actual fact, and the middle term is necessarily true of the minor term, so that the major term is perpetually true of the minor term and the conclusion is a perpetuity proposition. Moreover, we say: we do not concede that if the minor premise is possible, then together with a major premise that is also possible, it produces a possibility conclusion.
As to his words: âJ falls under B potentially, so that what is true of B potentially, is true of J,â we say: didnât you say that it follows from this that the major term is true of the minor term potentially? This is because the major term is substantially (bi-l-dhÄt) possible of that for which the middle term holds true in actuality. But it does not follow from this that it is possibly true of that for which the middle term is true potentially. Hence it does not follow that it obtains for the minor term potentially.
[280] As to his words: âthe major term is possibly true of the middle term, which is possibly true of the minor term,â we say: we do not concede that [the major term] is possibly true of the middle term. Rather, it is possibly true of the substance of which the middle term holds true in actuality, and the minor premise signifies that the middle term is possibly true of the minor term. It doesnât follow from this that the substance of which the middle term holds true in actuality is possibly true of the minor term, so as to imply that the major term is possibly true of the minor term.
His saying âtogether with an existential or a special conventional premise it produces a two-sided possibility [conclusion]â is obviously unconvincing.
As to his words, âtogether with a general absolute and a general conventional premise it produces a one-sided possibility conclusion,â we say: we do not concede this.
As to his words, âif the minor term obtains potentially, then the conclusion is a two-sided possibility [proposition],â we say: we do not concede this either. The argument he produced earlier to show this is unconvincing.
As to his words, âtogether with a necessity [premise] it produces a necessity conclusion,â we say: we do not concede that if the middle term is stipulated to be existent for the minor term in actual fact, then the major term is necessarily true of the minor term. As for the major premise, we do not concede that its truth is preserved on that assumption. Nor do we concede that it signifies that the major term is necessarily true of the minor term on that assumption, because what it signifies is that for everything of which the middle term holds true in actual fact, in itself (fÄ« nafs al-amr), the major term is necessarily true of it. But here it was assumed that the minor term is described by the middle term not âin itselfâ (fÄ« nafs al-amr), and hence it does not follow that the major term is necessarily true of it.
Concerning the second point, we do not concede that two necessity premises in a second figure [syllogism] jointly produce a necessity conclusion. Nor do we concede that two possibility premises in the third figure produce a two-sided possibility conclusion. All of this is impossible on our view, and we reject it.
[T16] Al-AbharÄ«, KhulÄá¹£at al-AfkÄr, 280.21â281.9
[disagreement with Avicenna on mixed modal syllogisms: second and third figures]
As for the second figure, we do not concede that if one of its two premises is a necessity proposition, then the conclusion is a necessity proposition [as well]. If this is shown by a reductio proof, then it turns on mixing possibility [281] and necessity premises in the first figure, which then produce a necessity conclusion. But it has already been said that this is unconvincing. If it is shown by conversion, then it turns on a negated necessity premise that is opposed to itself, and the deficiency of this has likewise already been discussed.
Moreover, we say: we do not concede that a conventional premise together with a possibility premise produces a one-sided possibility conclusion, for this cannot be shown by either reductio or conversion.
As for the third figure, we do not concede that if the minor is a possibility premise, then [the syllogism] is productive. This claim depends on the idea that the possibility premise converts to a one-sided possibility premise, and that the possibility premise can be the minor premise in the first figure. All this was already said to be unconvincing. The deficiency of his words, âif the minor premise is an actuality premise, then the conclusion follows the [modality of] the major premise,â is the same as what you have learned about the first figure.
[T17] Al-AbharÄ«, KhulÄá¹£at al-AfkÄr, 286.11â288.21
[disagreement with al-RÄzÄ« on first figure mixed modal syllogisms, cf. T11]
This is the summary of [al-RÄzÄ«âs] account after cutting down on its verbosity, and most of it is wrong. As for the first figure, we do not concede that if the minor premise is a possibility premise and the major a necessity premise, then it produces a conclusion [with the same modality] as the major premise. Concerning the argument that he adduced for this, its weakness has already been discussed in connection with the Master [Avicennaâs] position.
The ImÄm [al-RÄzÄ«] advanced a critique regarding mixed necessity and possibility [syllogisms], saying: âwhen we truly say âpossibly: every sperm is a humanâ and ânecessarily: every human is an animal,â [287] it does not follow that ânecessarily: every sperm is an animal.ââ¯â He replied to this by saying: âwe do not concede that the major premise is an absolute necessity proposition; rather, it is general conditioned proposition.â But this is unconvincing, because the absolute necessity proposition is such that its predicate cannot be possibly separated from the subject as long as its substance persists. Animality cannot possibly be separated from humanity as long as its substance exists, so the major premise is necessary. But we reject the minor premise, for the substance which is truly said to be a sperm cannot be described as âhuman.â Rather, a sperm can possibly turn into a human, but then the form of the sperm ceases to belong to the matter and the form of humanity arises in it instead.
We have taken the trouble [to present] for them an argument to show that a possibility minor together with a necessity major produces a necessity conclusion. Namely, that if it is true to say âevery J is possibly B, and every B is necessarily A, therefore every J is necessarily Aââbecause if we stipulate the truth of âevery J is B in actual factâ this entails the truth of âevery B is necessarily Aââthen there follows from this and from âevery J is necessarily Aâ either the truth of ânot: every J is perpetually Bâ or âevery J is necessarily A.â From this it follows what was claimed.
We say, however, that from this only one of the two things follows, because from the truth of a [conditional] implication follows either the contradictory of the antecedent or the consequent itself.13 If what occurs is the consequent itself, then it is one of the two things [that follows]. If what occurs is the contradictory of the antecedent, and âevery B is necessarily Aâ is true, then there follows the falsity of âevery J is B in actual fact, therefore, not: every J is perpetually B.â We say that from this it follows that what was claimed holds true. For, since one of the two things is true, if âevery J is B in actual factâ is true, then âevery J is necessarily Aâ is [as well], based on the fact that an exclusive disjunction implies a conditional composed of the contradictory of one of the parts and the other part itself [i.e.: p or q = if not-p then q]. If it is necessary on this assumption, it must also be necessary in itself (fÄ« nafs al-amr), because it is impossible for something that is not necessary to turn into something necessary on the assumption that [288] the predicate exists and possibly obtains for the subject. The answer to this is to reject that the disjunction implies the truth of a conditional composed of the contradictory of one of the parts and the other part itself, based on what has become clear to us regarding the falsity of this principle.
As for the second case [sc. the major premise is perpetual], we do not concede that it is productive of a perpetuity conclusion. We have seen that the argument mentioned earlier was unconvincing. But, even assuming that it is accepted here, it does not lead to a perpetuity [conclusion], because we say: we do not concede that what is not perpetual cannot possibly be perpetual, just on the assumption of something that is [merely] possible; the fact that no such impossibility arises is obvious, and needs no proof.
As for the third case [sc. the minor premise is neither necessary nor perpetual], we do not concede that if the middle term obtains in actual fact, then the conclusion is a two-sided possibility [proposition]. This is connected to what was said earlier concerning the rejection of a possibility minor with a necessity major.
As for the fourth case [sc. major premise is neutral with respect to necessity], we do not concede that if the universal major premise is not necessarily true, then the two-sided possibility proposition is true, because it may be that the nexus of the predicate to some instances of the subject term is a two-sided possibility [relation], while for others it is a necessity [relation]. From the falsity of the universal major necessity premise, it does not follow that it is true as a universal two-sided possibility premise.
Concerning [al-RÄzÄ«âs] statement, âif the minor premise allows for perpetuity and non-perpetuity and the major premise is a special conventional [premise], by joining them no apodictic judgment occurs,â this too is unconvincing. Because it is only the case that no apodictic judgment occurs by joining them, if the apodictic judgment does not occur on account of the concept susceptible to perpetuity in the minor premise, whose major premise is a special conventional [proposition], and it is true of a non-perpetual proposition. The apodictic judgment occurs because its being true of the perpetuity is impossible; otherwise, the major term would be both perpetually true of the minor term and non-perpetually true of it. Since this is so, by joining them, an apodictic judgment occurs.
[T18] Al-AbharÄ«, KhulÄá¹£at al-AfkÄr, 288.22â290.6
[disagreement with al-RÄzÄ« on mixed modal syllogisms in the remaining figures]
Concerning the second figure, we say: we do not concede that if one of the premises is a necessity proposition, the conclusion is so as well. As to his saying, âif the other premise is a necessity proposition, then [289] the predicate necessarily holds true of one of the terms, and is necessarily negated of the other term, so that between the two terms there is an obvious distinction,â we say: we do not concede this. For if this is shown by reductio, then it turns on mixing possibility and necessity premises in the first figure, which then produce a necessity conclusion. Whereas if it is shown by conversion, then it turns on a negated necessity premise that is opposed to itself. We have gone over the fact that this is unconvincing.
As to his saying, âif one of two premises is a perpetuity proposition, then the conclusion is a perpetuity proposition as well,â we say: we do not concede this, because if one of them is a perpetuity proposition and the other a possibility proposition, then [the syllogism] is not productive. For one may deny that it is possible that one of two co-extensive terms holds perpetually of the other, and vice-versa, even though neither can be denied of the other [for a limited time].
As to his saying, âif the major premise is one of two conventional (Ê¿urfiyyatayn) or conditioned (mashrÅ«á¹atayn) propositions taken with a possibility [minor], then they produce a one-sided possibility conclusion,â this is rejected. For [such a syllogism] is not productive, whether by conversion or by reductio.
When he says, âtwo conditioned premises produce a general conditioned conclusion,â this too is rejected. This would only follow if the negative general conditioned premise converted to a general conditioned proposition, but we have gone over the fact that this is unconvincing.
Concerning the third figure, we do not concede that the modality of the conclusion is the same as in the first figure. For, if the minor premise is a necessity, perpetuity, conventional or conditioned premise and the major premise one of two conventional or conditioned premises, then the modality of the respective conclusions are different from what follows in the first figure.
Concerning the fourth figure, if in the first two moods the premises are possibility propositionsâsince we can turn the major premise into a minor and the minor into a major premiseâthis comes down to the mixed possibility [syllogisms] in the first figure, and we have gone over the fact that this is unconvincing. If only the major premise is a possibility proposition, this comes down to the mixed case [290] with a possibility and actuality minor in the first figure. If only the minor is a possibility proposition, then upon conversion of the arrangement it produces a possibility proposition, but it does not convert to the intended conclusion, since we have shown that the possibility proposition is not acknowledged as converting. If both are actuality premises, then so is the conclusion, which is more specific than a one-sided possibility proposition. As for the three remaining moods, in the cases where they come down to the conversion of the minor premise to a false premise in the second figure, then in these cases they are false; but if to a true one, then true. But we wonât extend the discussion to the details here.
[T19] Al-ṬūsÄ«, TaÊ¿dÄ«l, 206.10â207.7
[against al-Abharī on the productivity of first figure syllogisms with possibility minors]
[Al-Abharī] said: concerning the first figure [syllogism], if its minor premise is a possibility premise, then the productivity condition is not satisfied, because the minor term might not fall under the middle term.
I say: if what is meant by âfalling under (indirÄj)â is that there is an affirmative judgment of the minor term in actuality, then we do not concede that this is a condition for productivity. But if [he means] that the affirmative judgment could be made, then we do not concede that the condition is not satisfied in a possibility minor.
He said: if the middle term is stipulated in actuality together with a necessity majorâso that the major term becomes necessary for the minor term, and thus necessary in itself (fÄ« nafs al-amr) because it is impossible to transform something that is not necessary into something necessaryâthen we reject the major on this assumption.
I say: it is impossible for this [kind of] minor premise that the stipulation of its actual occurrence be opposed to the truth of the major premise, simply because it is a possibility proposition, and it must not be inconceivable (muḥÄl) of itself. [Note that] I do not say, âinconceivable in respect of something else,â but only that we can rule out the impossibility of the necessity major in itself. Therefore, the rejection of the major premise on the assumption of the actual occurrence of the minor premise in this combination allows for the possible to imply something that is in itself inconceivable.
Having gone through this, we say: stipulating the minor premise to be an actuality premise is not a [207] condition for productivity in this combination. So the production [of the conclusion] is not undermined by any faultiness that is down to [the minor premiseâs being merely possible]. Rather [adding that the possibility is actually realized] is a way to show that [the syllogism] is in fact productive; but eliminating this way of showing [its productivity, sc. by saying the minor is only possible and not actual] does not eliminate the judgment [sc. that the conclusion does follow from a syllogism with a mere possibility minor].
Something that in itself indicates the productivity of this combination is that the substance (dhÄt) of the minor term in this combination cannot possibly differ from the substance of the middle term. Otherwise, it would not be possible to predicate the middle term of it. Hence, the necessity judgment that the major term is [part of] the substance of the middle term also applies to the very substance of the minor term by itself. The same applies to the judgment of possibility. The existence judgment differs [from the necessity judgment] because it may apply only to the description (waá¹£f) of the middle term and not its substance; [in that case] it is unknown whether it actually applies to the substance of the minor term.
[T20] Al-ṬūsÄ«, TaÊ¿dÄ«l, 207.20â208.13
[on composite mixed modal syllogisms and the proof for their validity]
[Al-AbharÄ«] said: perhaps there is no syllogism with true premises to be included among the composite syllogisms. If we say, for example: âEvery J is B, alwaysâ and âEvery B is A as long as it is B, not-always,â and consider [these statements] to be jointly true, then an absurdity follows.
[208] I say: syllogisms like this one are not productive, and a demonstration [for this] is straightforward. For the two premises are not jointly true in themselves (fÄ« nafs al-amr). They are productive as an implication only if we accept that B is negated of some, because in potentiality [the two premises as stated] include the following two premises: âsome things that are B are always Bâ and âeverything that is B is B, but not always,â therefore: âsomething that is B always is not B.â Because the conclusion is not composed of the minor and the major term, [such cases] do not count as a syllogism.
[He said]: you should know that in the first figure [the syllogism] implies the conclusion in such a way that its productivity is clear, no matter whether its two premises are true or stipulated to be true. As for the other three figures, the demonstration of [a syllogismâs] productivity is only achieved when its two premises are true, not if they are only stipulated to be true. This is because conversion is not permissible on this assumption. Likewise with reductio ad absurdum. For a falsity may imply an absurdity, either by being impossible by itself or in respect of something else.
I say: we do not concede that if the proof by conversion or reductio cannot be achieved [in these cases], then it cannot be achieved at all. For demonstration is not confined to these two methods. The true demonstration, which is called âdemonstration of the reason why,â would work for it. This is so even though the demonstrations by conversion and by reductio are not declared to be false by what we have mentioned.
[T21] Al-KÄtibÄ«, Shamsiyya, 213.22â216.5
[simple and compound modal propositions]
Predicates must relate to subjects, whether affirmatively or negatively, in a certain way (min kayfiyya), such as ânecessarily,â âperpetually,â ânot-necessarily,â or ânot-perpetually.â Such a qualification is called the âmatter (mÄdda)â of the proposition, and the word expressing it is called the âmode (jiha)â of the proposition.
There are thirteen modal propositions into which it is usual to inquire. Some of them are simple, that is to say, their true nature is simply an affirmation or negation; and some are [214] compound, that is to say, their true nature is composed at the same time of an affirmation and a negation.
There are six simple modal propositions.
(a) Necessary absolute. It judges that the predicate is necessarily affirmed or denied of the subject for so long as the essence of the subject exists, as when we say, âNecessarily, every human is animalâ and âNecessarily, no human is stone.â
(b) Perpetual absolute. It judges that the predicate is perpetually affirmed or denied of the subject as long as the essence of the subject exists. [The same examples are then given as under (a).]
(c) General conditioned. It judges that the predicate is necessarily affirmed or denied under the condition that a certain attribute (waá¹£f) [applies to] the subject, as when we say âNecessarily, every writer is moving the fingers as long as he writesâ or âNecessarily, a writer does not keep his fingers at rest as long as he writes.â
(d) General conventional. It judges that the predicate is perpetually affirmed or denied of the subject under the condition that a certain attribute [applies to] the subject. Same examples [as for the general conditioned].
(e) General absolute. It judges that the predicate is actually affirmed or denied of the subject, as when we say âEvery human without exception [lit. âwith general absolutenessâ: bi-l-iá¹lÄq al-Ê¿Ämm] is breathing,â or âNo human is breathing, without exception.â
(f) General possible. It judges that there is no absolute necessity on the side of what is contrary to the judgment, as when we say âBy a general possibility all fire is hot,â or âBy a general possibility what is warm is not cold.â
[compound modal propositions]
The compound modal propositions are seven in number.
(a) Special conditioned. It is the same as the general conditioned, with the restriction that [the relation of the subject to the predicate] is not perpetual in regard to the substance [of the subject]. If it is affirmative, as âNecessarily, every writer moves his fingers as long as he writes, but not perpetually,â it is composed of the affirmative general conditioned and negative general absolute propositions. Whereas, if it is negative, as âNecessarily, the fingers of a writer are not at rest as long as he writes, but not perpetually,â it is composed of the negative general conditioned and of the general affirmative absolute.
[215] (b) Special conventional. It is the same as the general conventional, with the restriction that [the relation does] not [hold] perpetually in reference to the substance [of the subject]. If it is affirmative, then it is composed of the affirmative general conventional and of the negative general absolute, whereas if it is negative it is composed of the negative general conventional and of the affirmative general absolute. Same examples [as for the special conditions].
(c) Non-necessary existential. It is the same as the general absolute, with the restriction that [the relation does] not [hold] of necessity in reference to the substance [of the subject]. If it is affirmative, as âHuman is actually risible, but not necessarily,â it is composed of the affirmative general absolute and the negative general possible. Whereas if it is negative, as âHuman is not actually risible, but not necessarily,â it is composed of the negative general absolute and the affirmative general possible.
(d) Non-perpetual existential. It is the same as the general absolute, with the restriction of non-perpetuity in reference to the substance [of the subject]. Whether it be affirmative or negative it is composed of two general absolute [propositions], one affirmative and the other negative. Same examples [as for the non-necessary existential].
(e) Temporal (waqtiyya). It judges that necessarily, the predicate is affirmed or denied of the subject during a definite period of the subjectâs existence, with the restriction that [the relation] does not [hold] perpetually in regard to the substance [of the subject]. If it is affirmative, as âNecessarily, an eclipse of the moon takes place during the time the earth is placed between the sun and the moon, but this is not the case perpetually,â it is composed of the affirmative absolute temporal and the negative general absolute. Whereas if it is negative, as âNecessarily, no eclipse of the moon takes place when the earth, moon and sun are at right angles, but this is not the case perpetually,â it is composed of the negative absolute temporal and the affirmative general absolute.
(f) Spread (muntashira). It judges that necessarily, the predicate is affirmed or denied of the subject during an indefinite period of the existence of the subject, with the restriction that [the relation] does not [hold] perpetually in regard to the substance [of the subject]. If it is affirmative, as âNecessarily, every human is breathing at times, but this is not the case perpetually,â it is composed of the affirmative absolute spread [proposition] and the negative general absolute. Whereas if it is negative, as âNecessarily, no human is breathing at some time, but this is not the case perpetually,â it is composed of the negative absolute spread [proposition, which is not composite] and the affirmative general absolute.
(g) Special possible. It judges that there is no absolute necessity, either for existence or nonexistence. It makes no difference whether it is affirmative, as âBy [216] special possibility, every human is a writer,â or negative, as âBy special possibility, every human is not a writer.â It is composed of two general possible propositions, one affirmative, the other negative. The general rule here is that, if a proposition is restricted by non-perpetuity, it indicates that it is a general absolute proposition, while if it is restricted by non-necessity, then it is a general possible proposition disagreeing in mode but agreeing in quantity.
[T22] Al-KÄtibÄ«, Shamsiyya, 226.22â227.3 (trans. Rescher, mod.)
[productivity of mixed modal syllogisms in the first figure]
As to the first figure, its condition for productivity is that the minor is an actuality proposition. The conclusion here is the same as the major [in modality], if [the minor] is other than one of the two conditioned and the two conventional propositions. In these cases [i.e. if the minor is one of these four], then [the modality of the conclusion is] the same as the minor, removing from [the conclusion] the restriction of non-necessity or non-perpetuity found in the minor, if the major is one of the two generals, and after adding the non-perpetual to [the conclusion] if [the major] is one of the two specials.
[T23] Al-KÄtibÄ«, Munaṣṣaá¹£, fols. 120v13â121r7
[commentary on T10: criticism of al-RÄzÄ« on syllogisms with possibility minors]
Know that the ImÄm [al-RÄzÄ«] mentions three preliminary points concerning these mixed modal syllogisms and the explication of their conclusions. The first is that if the minor premise is one of the possibility propositions, then the major premise must be one of the following: (a) necessary absolute, (b) perpetual absolute, (c) non-necessary and non-perpetual, or (d) any of them, that is: any of the first three types. If it is a necessary absolute, then the conclusion is a necessary absolute as well. He justifies this by saying, âthe major premise signifies that whatever the middle termâ actually âholds true of, in whatever wayââthat is, regardless whether it is necessarily true of it actually, whether substantially or descriptively, or for a certain time; or always substantially or descriptively; or neither necessarily at all nor always in [combination] with one of these notionsââalso has the major term hold true of it,â necessarily and âat all timesâ and respects before, together with, or after the obtaining of the middle term in actuality. âThe minor premise signifies that the middle term may possibly hold true of the minor termâ and nothing that is possible implies an absurdity when it is stipulated to occur. Then it is stipulated that the middle term is true in actuality of the minor term. On this assumption the major term is true of the minor term in the sense of the necessary absolute, that is: at all times at which the essence of the minor term exists. There is then a perfect syllogism. This is because in this case it is true to say âWhatever the minor term holds true of in actuality, of that the middle term [also] holds true in actuality,â and âWhatever the middle term holds true of in actuality, of that the major term is necessarily true, at all times at which its substance exists,â which then produces the conclusion âthe major term applies to each instance of the minor term at all times at which its essence exists.â If it is necessary on this assumption, then it must be necessary in itself (fÄ« nafs al-amr) too, regardless of whether the possible actually occurs in actuality or not. For if it werenât necessary in itself, then it would be impossible that it become necessary on this assumption, because what is not necessary in itself cannot be necessary when something possible is assumed. Otherwise, the possible would imply that something that was not necessary before can be transformed [into something necessary], which is absurd. Thus, he shows that this mixed modal syllogism is productive of a conclusion in the necessary absolute.
But this calls for further inquiry, for we do not concede that the arrangement of the abovementioned syllogism [is valid], on the assumption that the middle term is true of the minor term in actuality. The arrangement would be valid only if the universal necessity major premise were true on this assumption. But this is rejected, because on this assumption the numerous instances of its subject would severally have to be necessary.14 It cannot be said in this case, concerning the necessity of the instances of the minor term taken together, that if the necessity conclusion is not true, then the possible implies no absurdity. For if the major premise that is necessary in itself (fī nafs al-amr) is not true on the assumption of the truth of the minor premise in actual fact, then an absurdity does follow from the possible, because the contradictory of the necessary is impossible. If [the major premise] is true, it implies a necessity conclusion because of some part and we stipulate it to be non-necessary in itself, and the possible again implies an absurdity. For the necessary cannot be true of something non-necessary, because we say: by no means do we concede that if the necessity major premise is true, that on this assumption a syllogism comes together at all. It only comes together if the minor term falls under the subject term of the major premise. And it only falls under it if what is meant by the major premise is that everything that is the middle term in actuality, absolutely and regardless of whether it is so in itself or on this assumption, also is the major term. But this is not so. Rather what is meant by it is that everything that is actually the middle term in itself is the major term. If that is indeed the meaning, then it could be that the minor term is not in actuality in itself [the major term], even if it is described as being so on this assumption. Since the syllogism does not come together, no conclusion results, let alone a necessary one.
[T24] Al-KÄtibÄ«, JÄmiÊ¿ al-daqÄʾiq, fols. 71v9â72r15
[mixed modal syllogisms in the first figure are only productive with actuality minors]
Let us investigate the mixed modal syllogisms in which the thirteen [types of modal propositions are combined] with one another in the four figures. Concerning the first figure, the condition of its productivity with regard to modality is that its minor premise be an actuality premise, because if it were a possibility premise, then it would not follow that the minor term falls under the middle term on which judgment is passed in the major premise. For the major premise signifies that whatever the middle term applies to in actuality, the major term applies to it also (or is denied of it, with the modality expressed in it). And the minor premise indicates the possibility that the middle term is true of the minor term. But the possibility that one thing hold true of another does not require that it holds true of it in general, because it may be that something is possibly true of something else, without in fact applying to it at all. Since this is so, there is no apodictic judgment that the minor term falls under the major term, nor is there any apodictic judgment of the conclusion. So clearly there is no apodictic judgment of the productivity of the syllogism in this figure, unless the minor premise is an actuality premise.
[objection: productivity with possibility minors can be saved by an argument from extension]
Let it not be said: if we say âEvery J is possibly Bâ and âEvery B is necessarily Aâ this produces the conclusion âEvery J is necessarily Aâ because B, which is predicated possibly of all instances of J, must have a more general extension than it when taken absolutely, or [at least] be co-extensive with it. For what is more specific than somethingâwhether absolutely or in some respectâand is distinct from it, cannot be predicated of each of its instances. This being so, the major term predicated necessarily of each of its instances is necessarily predicated of each of the instances [72r] of the minor term. If the predicate of all instances has a more general extension in some modality, then it must also be predicated of a more specific extension of the instances in that modality, because the more specific extension among the instances is just part of the more general extension over those instances. Likewise, the predicate of all instances could be co-extensive with another [term], and in a given modality the predicate of all the instances will be co-extensive with the other [term] in this modality, because of the overlapping of the extensions of the two [terms] in the instances.
[reply to the objection]
Because we say: we do not concede that what is predicated of each of the instances of the subject term in a possibility proposition must have a more general extension than the subject term taken absolutely, or be co-extensive with it. There is no proof for this, unless each single one of the Js [has the predicate] in actuality. âCapable of laughingâ in actuality has a more specific extension than âhumanâ taken absolutely, even though both are predicated by possibility of all instances of âhuman.â Likewise, âwhitenessâ in relation to âanimalâ has a more general extension in some respect, and a more specific extension in another, even though it is predicated by possibility of all its instances. It is true that possibly every animal is white, but one of the two distinct things is possibly true of each of the instances that has the other [that is, what is not-white is possibly white, and vice-versa]. So it is true when we say âPossibly: every mobile thing is at restâ and also true when we say âPossibly: every resting thing is mobile.â In this case one can go so far as [to say that] some instances of mobile things cannot be at rest, while some instances of resting things cannot be moving. But this is due to external factors. The instances of each of the two in respect of its essence accepts the other [e.g. the moving, as such, is always possibly at rest]. This being so, from the fact that something is necessarily predicated of everything that is truly said to be actually B, and that B is possibly predicated of all instances of J, it does not follow that this thing is necessarily predicated of all instances of J. For it may be that the B predicated of all instances of J by possibility has a more specific extension than J taken absolutely, or in some respect, or that it is distinct from it. There is no requirement that predication of each of the instances of these things by necessity [guarantees] predication of everything of which these things are possibly true, because it may be that their predication of it is only potential. Hence, it does not fall under the subject term of the second premise, and no productivity follows.
[T25] Al-UrmawÄ«, Maá¹ÄliÊ¿, 74.6â76.9
[the productivity of syllogisms with possibility minors]
The Master [Avicenna] and the Imam [al-RÄzÄ«], and those who follow them, claimed that (a) possibility minors together with necessity [majors] are productive of a necessity [conclusion]; (b) that together with a non-necessity major they are productive of a two-sided possibility [conclusion]; (c) and that together with all others, they are productive of a one-sided possibility [conclusion].
(a) But several objections have been raised against the first [claim].
First, one may take the contradictory of the conclusion as an absolute [proposition] or stipulate it to [be true] in actual fact, and then add it to the major premise so that in the second figure it produces the contradictory of the minor premise. Reply: we do not concede that the major necessity premise in the second figure produces a necessity proposition.
Second, one may add [the contradictory of the conclusion] to the minor possibility premise, so that in the third figure it produces the contradictory of the major premise. Reply: we do not concede that the minor possibility premise in the third figure is productive.
Third, if the minor premise obtains in actual fact, it implies a necessity conclusion. But [the conclusion] is also necessary if it is assumed to not occur, because if something is necessary on any possible assumption, then it is necessary on [a given] possible assumption. Reply: we do not concede that the major premise is true on the assumption that the minor premise obtains in actual fact, because there might be more individuals falling under the subject term of the major premise [to which the predicate in the major premise does not apply].
(b) The same objections have been raised against the second [claim], and also [the second claim] would change the structure of reductio syllogisms.
(c) Against the third [it has been argued] (c1) that if the major premise is necessarily true, then the conclusion is a necessity proposition as well; [75] if not, then [the conclusion] is a two-sided possibility proposition. Either way [the conclusion] is a general possibility proposition.15 Also, (c2) the conclusion in [the first] figure follows the major premise in being indeterminate with regard to the necessity and the duration of the two attributes. If [just] one of these is determined, then [the conclusion] follows the minor premise in being indeterminate with regard to existence and necessity as well, if there is no necessity in the major premise.
(c1) As for the first objection, it is obvious that [the syllogism] is not included [among the productive ones].16 Al-Kashshī claimed that the minor necessity premise together with the major premise as a perpetual negation produces a necessity proposition by conversion and reductio (khulf). Reply: as to conversion, it is denied that the mentioned syllogism in the second figure produces a necessity proposition. As to reductio, it is denied that a possibility proposition together with a perpetuity proposition in the second figure is productive.
(c2) Concerning the second objection: the middle term describes the major term perpetually, then the major term thereby holds true of the minor term on account of that fact; but if it is perpetually true by necessity, the major term thereby holds true of the minor term. This does not require the determination of existence, I mean the non-perpetuity and the non-necessity of the minor premise, because the major termâeven if it lasts as long as the middle term lastsâmay not be restricted only to the time it [is true of] the middle term, so that it is true of it even if the middle term no longer subsists. Nor does the major premise on its own require necessity in this case, because it may be that the necessity of the major term is determined by the middle term, so that it will not hold true once the middle termâs possibility is not realized. Nor does the minor premise on its own require it, because if the perpetuity of the middle term for the major term is not necessary, then it may be that the major term is denied, even if the middle term holds true by necessity.
[76] Al-KashshÄ« claimed that the minor necessity premise together with a major premise that is a negated general conventional (Ê¿urfiyya Ê¿Ämma) is productive of a necessity proposition by conversion and reductio. But you have learned the answers to both already.
If it is said: a major premise that is a general conditioned (mashrÅ«á¹a Ê¿Ämma), together with a minor perpetuity premise, produces a necessity [conclusion]. For the necessity of the major term, being perpetual thanks to the middle termâs perpetuity, and lasting as long as the substance of the minor term, yields [a conclusion that] is true as long they are true. Then we say: this necessity is only on condition of the description of the middle term, which is not what was being sought in the conclusion.
[T26] Ibn WÄá¹£il, Sharḥ al-Jumal, 130.12â131.9 [trans. El-Rouayheb, mod.]
[Ibn WÄá¹£il reports al-KhÅ«najÄ«âs argument to deny the productivity of syllogisms with possibility minors]
You should know that this proof [provided by Avicenna] is most unconvincing. For we do not concede that [even] on the assumption of the truth of the minor as actual [and not merely possible], the universal major remains true. Rather, it is possible that on this supposition the middle term will include what the major term is not true of necessarily, in which case the major is necessarily true only of a particular, and the conclusion will not be fully achieved. We give an example that clarifies the objection: let us suppose that Zayd has never ridden, except on a donkey. Then it would be true that every horse is possibly ridden by Zayd, and on our assumption clearly everything ridden by Zayd in actual fact is necessarily a donkey. Yet it is not true that every horse is necessarily a donkey. If we suppose the possibility minor to be true actually, then the horse would be one of the individuals that are ridden by Zayd and would fall under [the description] âWhat is ridden by Zayd,â and the major would cease to be true as a universal. [â¦]
[131.3] The example just mentioned on this matter is a counterexample which makes known the sterility of the mentioned syllogism [with a possibility minor], if the proposition is understood in accordance with external existence [i.e. on an externalist reading, cf. T9]. If it is understood in accordance with the true nature [i.e. on an essentialist reading, cf. T9]âmeaning that when we say every J is B, then all that were it to exist would be J, would when it existed, be Bâthen the counter-example does not apply. Yet the objection we have offered remains. For it is enough for our purposes that there may be more individuals falling under the subject-term of the major premise, on the assumption that the minor premise is actually [and not merely possibly] true, and it is not incumbent on us to adduce a concrete (bi-mÄdda) illustration.
We say then: if the two premises of the syllogism are externalist propositions and the minor is a possibility proposition, then it is known to be sterile by the counterexample we have adduced. If they are essentialist propositions, then judgment is suspended, and it is unknown whether it is productive or sterile.
[T27] Al-SamarqandÄ«, Qisá¹Äs, 351.15â355.5
[on the replies to Avicennaâs position that possibility minors in the first figure are productive of necessity conclusions]
They [sc. the later logicians] replied to [Avicennaâs] three arguments as follows.
Firstly, the conjunction of possibility premises may imply a contradiction, just as two sides of a possibility proposition with both sides being possible, when conjoined, imply a contradiction [in other words, P is possible and not-P is possible, but âP and not-Pâ is impossible]. In this case the impossibility of the conclusionâs contradictory does not follow.
But this reply is unconvincing. For if their conjunction implies a contradiction, then it is [already] contradictory. In which case it must surely be that some of its parts are contradictory, either in and of themselves (fī nafs al-amr), or on the assumption that some of the other [parts] hold true while the major premise is true in the first figure, so that there follows the impossibility of occurring in actual fact. In this case there follows the impossibility of the possibility premise whose contradictory would have been the conclusion, because the possibility of something contradictory is contradictory.
[352] The same goes for the second and the third [arguments]. But the answer the Master [Avicenna] gave was to say: we do not concede that the actuality minor together with a necessity major produces a necessity conclusion, but rather that two necessity premises in the second figure produce such a conclusion. The inquiry into this will come below.
As for the second argument, we do not concede that the possibility minor is productive in the third figure. The inquiry into this follows as well.
As for the third argument, we do not concede that if we stipulate the minor to be an actuality proposition, then the major remains true, and they are productive. Rather, it is possible that [the major] is false because further instances [falling under the subject term are counter-examples], in which case the situation is like in the example mentioned. For if we stipulate that the donkey is ridden by Zayd, then the major does not remain true [cf. T26]. For in this case, it is not true to say: âeverything that is ridden by Zayd is a horse.â Given that this is not productive, no co-presence of contradictories follows. [â¦]
[355.1] As for the general rule for the mode of the conclusion in this figure, it is that the conclusion follows the [modality] of the major premise if the major is not among the descriptional premises (waá¹£fiyyÄt), by which I mean the two conditioned and the two conventional premises. If it is one of these four, then it follows the minor [in its modality], be it non-necessary or necessary, even if the major is necessary. It follows [the minor] as well if it is non-existential, even if the major is existential.
[T28] Al-SamarqandÄ«, Qisá¹Äs, 367.1â17
[on al-RÄzÄ«âs position that possibility minors produce one-sided possibility conclusions]
The ImÄm [al-RÄzÄ«] claimed that if the possibility minorâbe it one-sided or two-sidedâtogether with the six negated converse propositions, produces a one-sided possibility conclusion. He argued for this on the basis that if major premises are negated, then they additionally signify that the middle and the major term are not both true. If the minor possibility premise signifies that the minor term might be described by the middle term, then the judgment that this âmightâ be the case must allow for a disjunction with the major term, inferring from the possibility of the negated element the possibility of the negation. If [the minor possibility premise] is affirmative, then it additionally signifies that the major term cannot be separated from the middle term. When we judge the minor premise to allow for a disjunction of the minor from the middle term, then in this case it must also allow for its disjunction from the major term, inferring from the disjunction of the antecedent that it allows for the disjunction from the consequent.
Al-KÄshshÄ« reckoned that the possibility minor is not productive, except with negations of the converse of the major, so that it goes back to the original [problem]. But this is absurd. Neither point is true, except for possibility minors with necessity majors or with the two conditioned premises, as has been discussed before.
Concerning what the ImÄm [al-RÄzÄ«] said about the three remaining premises, namely the perpetual and the two conventional ones as possibility premises, this is not right either. For inferring from the possibility of the negated element the possibility of the negation is only correct if the negated premise is necessary. Otherwise this is not correct. It may be that the major term is necessarily true of the minor, as you learned from the fact that something that is perpetually negated of a thing may be possible for it, and vice-versa. But the major term is not implied [to hold true of] the middle term in these three premises, because the middle term may be separated from it.
[T29] Al-ḤillÄ«, QawÄÊ¿id, 355.14â357.4
[mixed modal syllogisms]
He [al-KÄtibÄ«] said: Second Part on Mixed Modal Syllogisms. Concerning the first figure, its condition for productivity is that the minor is an actuality proposition.
I say: you should know that the four figures, just as they are sterile when not conditioned by quantity and quality, so they are sterile when not conditioned by modality in the way we have mentioned.
[356] Concerning the first figure, it is conditioned by the modality of actuality in the minor premise, which means that the minor premise is one of the eleven propositions that are not possibility propositions. For the predicate of the major premise is true only of what [falls under] the middle term in actual fact, as we have established in the verification of quantified propositions. And the minor term does not [fall under] the middle term in actual fact, so that the middle term need not extend to it.
The older scholars considered the possibility minor to produce, together with a necessity major, a necessity conclusion, on the assumption that the occurrence of the possible proposition would make the conclusion necessary, which then would be necessary in itself (fÄ« nafs al-amr). Otherwise the possible proposition would imply an absurdity, namely that something that is not necessary is necessary when it is assumed to be possible. From this and from a possibility minor results a one-sided possibility, because assuming the possible proposition to occur, the possibility proposition is true, so it is also true in itself. Otherwise, that which is not possible in itself would be possible on the assumption that the possible proposition occurs, which is absurd. Likewise, if the major premise is an existential proposition and the minor a possibility proposition, then [the conclusion] is only a two-sided possibility, because of the arrangement of the minor premise with one of its parts as a syllogism productive of one sided-possibility (li-l-Ê¿Ämma), and with the other as a syllogism productive of a [degree of] generality (li-l-Ê¿Ämma) different from the first, i.e. specific two-sided possibility.
The later scholars, however, rejected the truth of the major premise on the assumption that the minor actually occurs. But this is a mistake, because otherwise what they were trying to avoid will be forced on them after all. For if a necessity or possibility major were false on the assumption that the possible [proposition] occurs, then the possible [proposition] would imply an absurdity. The falsity of the necessary or possible [proposition] is absurd with regard to its essence. [This is so] on an existential [reading], in which case it is possible because it accepts a judgment that follows the description of the middle term without [reference] to its essence, and then it is something whose occurrence is unknown with respect to the essence of the minor term, so that it is possible for them to reject the truth of the actuality [proposition]. But [it is not so] on a possibility [reading], for the possible proposition will follow in any case.
According to the doctrine of the older scholars, the combinations in this figure are 179 productive moods that obtain from the 13 moods taken by themselves. According to the doctrine of the author [al-KÄtibÄ«], 26 of them are dropped, that obtain from the mood of two possibility [propositions] among the 13 moods.
[T30] Al-ḤillÄ«, Jawhar, 107.18â109.11
[on al-ṬūsÄ«âs position regarding the productivity of modal syllogisms]
[Al-Ṭūsī] said: considering the modalities [of syllogisms], we say: if the minor premise is negative, it implies an affirmative, and in connection with the major premise it produces a conclusion that is potentially that which the affirmative implies.
[108] I say: the composite negative premise which implies an affirmative, like the special conventional, the special conditioned, and the existential propositions, may be employed as a minor in the first figure, because they accommodate a tie to both affirmation and negation. But a conclusion is produced only when the affirmative tie is considered. Then it produces in potentiality the conclusion that the affirmative would produce. For example, if the following are true:
No J is B as long as it is J but not alwaysEvery B is necessarily A
it follows that:
Every J is necessarily A
because the minor is composed of a negative general conventional, and the affirmative general absolute is âevery J is B.â If we turn the minor into a major, the conclusion follows as we said. [â¦]
[108.10] [Al-Ṭūsī] said: actuality minors that require that the minor term falls under the middle term in actuality, together with essentiality majors, produce a conclusion that is like the major premise [in modality], because the minor term contains some parts of the middle term, so whatever goes for the latter term also goes for the former.
I say: the minor premise in this figure is either an actuality or a possibility proposition. If it is an actuality proposition, then the major is either essential or descriptional. If it is essential, then the conclusion [follows the modality] of the major premise. So if we say, for example,
Every J is B absolutelyEvery B is necessarily A
then we judge in the minor premise that J is some parts of B, so that what is true of [B] is also true of J, because we said in the major premise that every B is necessarily A, comprising all parts that taken together are J. Thus J is necessarily A, which was the intended conclusion.
[Al-Ṭūsī] said: possibility minors together with major premises that are not qualified as either necessary or perpetual produce a possibility conclusion, because the minor term only falls under the middle term potentially.
I say: concerning possibility minors in this figure, some hold that they are not productive at all. The author [al-ṬūsÄ«], [109] may God have mercy on him, adopted the doctrine of AbÅ« Ê¿AlÄ« [Avicenna] concerning its productivity. Here is the detailed account (tafṣīl). The major is either an absolute necessary, or an absolute perpetual, or non-necessary and non-perpetual. If it is non-necessary and non-perpetual, then the conclusion is a possibility propositionâeither one-sided, if the major is not composite, or two-sided, if the major is composite.
As for the one-sided possibility conclusion, this is because if the minor is stipulated to occur, then the syllogism produces a conclusion like the major premise [in modality], as was said before. And when it is true in the same way as the major premise, then on the assumption that [the major premise] is possible, it [i.e. the conclusion] is possible. Otherwise what is not possible would be possible, given that the possible actually occurs, and this is absurd. The reason it is not an actuality [conclusion] is that the minor term [may] not actually fall under the middle term.
As for the two-sided possibility conclusion, [the conclusion follows the major premise in modality] because if the possibility [minor] is composed of one of the parts of the major premise, it produces a one-sided possibility conclusion. If, however, it is composed of another part different from this part in quality, then it will produce a conclusion different from the first conclusion; and from them the two-sided possibility is composed.
[T31] Al-ḤillÄ«, AsrÄr, 126.15â129.1
[mixed modal syllogisms in the first figure]
Each figure has [productivity] conditions additional to those we already mentioned, in consideration of the modalities without which no productivity could occur. There are 13 modal propositions if they are counted by themselves. From them 169 moods can be formed, but in each figure all those that do not satisfy the conditions mentioned are subtracted.
The first figure: here the later scholars stipulated that the minor must be an actuality premise. They said that otherwise, the minor term may not fall under the middle term, and so the judgment could not extend to the major term. Hence, 26 moods are subtracted and two possibility premises from the thirteen [productive modal propositions]. But some of the verifiers responded to this by saying: we do not concede that actually falling [under the middle term] is a condition for productivity. [127] As for the Master [Avicenna], he considered this figure to be productive on the assumption of the [mere] possibility of its minor premise.
For the major is either (a) necessary or (b) not. (a) In the first case, the conclusion is necessary too, because if the middle term does actually come to [fall under] the minor term, then the conclusion is necessary. So the necessity holds just on the assumption of possibility [for the minor term]: it holds in itself (fī nafs al-amr), because what is not necessary cannot turn into something necessary [just] by assuming something possible. (b) In the second case it is (b1) a one-sided possibility [conclusion] if the major premise is incomposite, because assuming the minor term to occur, the conclusion is either a one-sided possibility [proposition], if the major premise is a one-sided possibility, or (b2) an actuality [proposition], if the major premise is an actuality premise. On either assumption it will be a one-sided possibility. Otherwise, what is not a one-sided possibility in itself, would become a possibility assuming that something possible occurs, which is absurd. (b3) Finally, it is a two-sided possibility if the major premise is composite, because of the ordering of the minor premise together with its two parts, so that both extremes are productive, in which case the two-sided possibility is true.
They asserted that there is no proof needed for [syllogisms] with a possibility minor. But those with an actuality minor do need a proof, as we mentioned. And it is allowed that the minor premise be negative, because it can be transformed into an affirmative: the affirmative is productive, and then it is turned into a negative. [Also itâs allowed] to have both [premises] be negations.
The later scholars said: on the assumption that the possible actually occurs, there are further instances falling under the subject-term of the major premise, but the remaining ones are not implied in this case. Yet they fall into that which they sought to avoid, for they only resorted to this cunning strategy to guard against the implication that the actual occurrence of the possible yields something inconceivable, and they allowed the necessity major to be false, on the assumption that the possible actually occurs. But this is just to assert the implication of the possible, which is absurd in itself.
Some of the verifiers explained the necessity conclusion by the converse of [the proposition] âeverything that is not necessary [128] on account of its essence is impossibly necessary on account of itself.â They claim that the necessity premise in this argument is the contraposition of âeverything that is not impossibly necessary is necessary.â They also say that the essence of the minor term is the essence of the middle term, otherwise it would be impossible to ascribe possibility to it. So what is judged regarding the essence of the middle term is the same what is judged regarding the essence of the minor term. Now, they deny the latter and leave it at that. But the essence of the minor term is not the same as the essence of the middle term in actuality, but only possibly. Hence, the judgment regarding the middle term does not imply the unity of the two essences in actuality, except if the judgment is one of actuality.
You should know that, according to the rules set out by AbÅ« Naá¹£r [al-FÄrÄbÄ«], the productivity of such cases was deemed to be evident. Yet suspicions persisted. Some of the first [logicians] sought to prove the productivity of two possibility premises by saying that the major term is possible for the middle term, which in turn is possible for the minor term; and what is possible for something possible, is itself possible. But the Master [Avicenna] said that this is not a proof, for what is to be proven is assumed alongside it. Rather, [according to him] the truth is that the fact that its conclusion necessarily follows is clear, and there is no need for such a proof.
The later scholars said: the major term is possible for what falls under the middle term, and the description of the middle term is possible for the minor term, so there is no identity [between the two terms, i.e. the middle term is not used in the same sense]. But they shy away from verification, and simply give up the discussion.
Some of the later scholars made [syllogisms] with perpetuity majors and possibility minors productive of a perpetuity conclusion, based on what we have mentioned regarding necessity. But this is of no help, for what is not necessary cannot become necessary just on the assumption of something possible. As for the perpetual, it could then wind up being no longer perpetual on the assumption of the possible. [129] Or at best, the perpetuity would be made a concomitant of necessity, as one group indeed has held.
For an overview see R. Strobino and P. Thom, âThe Logic of Modality,â in C. Dutilh-Novaes and S. Read (eds), The Cambridge Companion to Medieval Logic (Cambridge: 2016), 342â369; a detailed presentation of Avicennaâs modal syllogistic is R. Strobino, âIbn Sinaâs Logic,â in E.N. Zalta (ed), The Stanford Encyclopedia of Philosophy, 2018. See also T. Street, âMedieval and Modern Interpretations of Avicennaâs Modal Syllogistic,â in F. Opwis & D.C Reisman (eds), Islamic Philosophy, Science, Culture, and Religion: Essays in Honor of Dimitri Gutas (Leiden: 2012), 232â256.
On this see e.g. J. Hintikka, Time and Necessity (Oxford: 1973).
A quick explanation is in order here: in both Arabic and English the word âpossible (mumkin)â is ambiguous. It can mean either ânot impossible,â in which case it includes both the contingent and necessary, or âcontingent,â in which case it excludes the necessary as well as the impossible. In English we capture this by calling the first sense âone-sided possibility,â the second sense âtwo sided-possibilityâ; in Arabic the distinction was called âgeneral (Ê¿Ämm)â vs. âspecial (khÄṣṣ)â possible.
For a philosophical discussion of this strategy, see M. Malink and J. Rosen, âProof by Assumption of the Possible in âPrior Analyticsâ 1.15,â Mind 122 (2013), 953â986.
On SuhrawÄrdÄ«âs modal syllogistic, see T. Street, âSuhrawardÄ« on Modal Syllogisms,â in A. Akasoy & W. Raven (eds): Islamic Thought in the Middle Ages (Leiden: 2008), 163â178.
See also the introduction to al-KhÅ«najÄ«, A. Kashf al-asrÄr Ê¿an ghawÄmiḠal-afkÄr. Ed. by K. El-Rouayheb (Tehran; Berlin: 2010).
For example:
Always, winged is not said of human
Necessarily, winged is said of flighted animal
Therefore: Necessarily, flighted animal is not said of human
In other words, if a predicate never applies to something and is such that it is necessarily true of something else, the latter is necessarily false of the former.
For the identification of this figure as al-FÄrÄbÄ« see T. Street, ââ¯âThe Eminent Later Scholarâ in Avicennaâs Book of the Syllogism,â Arabic Sciences and Philosophy 11 (2001), 205â218.
Reading lÄ for lahu after dÄʾiman.
As a reminder: âproper (khÄṣṣ)â or two-sided possibility means neither necessary nor impossible, cf. [T2] above.
That is, whether at a given time or dispersed across several times.
And, in contrast to the kind of possibility mentioned in the previous note, âgeneral (Ê¿Ämm)â or one-sided possibility means simply not impossible, i.e. either contingent or necessary.
That is, from âif p then qâ one can infer âeither not-p or q.â
That is, âAll As are Bsâ is necessarily true only if there are no actual instances of A that are not B.
That is, because both necessary and two-sided possible fall under one-sided possibility.
Reading li-lÄ-indirÄj.