2. THE ‘CONTEXT THEORY’ OF LOGIC: TRUTH, ASSENT, AND CONDITIONALS

Alfarabi's use of conditional sentences is shaped in crucial ways by the rich legacy of the ‘context theory’ of logic, which he inherited from late antique, Greek commentary tradition on Aristotle's Organon.Footnote ⁵ Alfarabi, like other classical Islamic philosophers, developed many concepts lying at the foundations of his logical doctrines as solutions to, or extensions of, a constellation of problems surrounding the question of how each of the books of the Organon deal with a unitary subject called ‘logic’,Footnote ⁶ as well as the question of how each of the books of the Organon can be distinguished from each other despite this underlying unity. At the risk of schematizing a delicate textual history,Footnote ⁷ we could say that the classical Islamic philosophers accounted for the ultimate unity of the Organon by claiming that the formal theory of the syllogism developed by Aristotle in the Prior Analytics provided the formal principles for demonstrative (Posterior Analytics), dialectical (Topics), rhetorical (Rhetoric), poetical (Poetics), and sophistical (Sophistici Elenchi) modes of argumentation. For Alfarabi, as well as other classical Islamic philosophers, logic was syllogistic.

Yet, by Alfarabi's day, though perhaps as early as Alexander of Aphrodisias (fl. 200),Footnote ⁸ it was recognized that conditional (also called ‘hypothetical [waḍʿī]’)Footnote ⁹ syllogisms were an important subset of the set of the valid deduction schema available in philosophical debate. Thus, in his epitomes of the books of the Organon, we find Alfarabi making extensive use of conditional syllogisms in the rhetoric, dialectic, and demonstration. That being said, the formal properties of conditional premises and syllogisms developed by Alfarabi had to be elastic enough to accommodate their use in a wide variety of argumentative contexts in which the interlocutors engage in debate while harbouring different goals. In a rhetorical exchange between a speaker and an audience, the aim is persuading (qanāʿa, iqnāʿ) an audience;Footnote ¹⁰ in a dialectical exchange, the aim is to discover and then refute (tabkīt, ibṭāl) the position of an opponent if one is the questioner or to defend a position from refutation if one is the respondent;Footnote ¹¹ in poetics it is stimulating the listener's imagination (taḫyīl);Footnote ¹² in demonstration, the aim is eliciting in one's self or in another certainty of the objective truth (burhān) of a proposition. Deborah Black has noted that the classical Islamic philosophers moved away from using premises’ modality, truth-values, or the part of the soul from which premises originate as a way of distinguishing between these syllogistic arts. Black sees Avicenna as the culmination in a trend, in which the distinction between the five syllogistic arts was made to rest on the strength of the audience's assent (taṣdīq) or, more generically, acquiescence (iḏʿān), to the proposition rather than the proposition's content corresponding to states of affairs or not corresponding to them.Footnote ¹³ However, as Black also notes, there is evidence from Kitāb al-Alfāẓ that Alfarabi also formulated a doctrine that distinguishes between the syllogistic arts according to the different grades of the ‘mind's compliance (inqiyād al-ḏihn)’ with the propositions in a syllogistic process of reasoning.Footnote ¹⁴ Thus, it also seems to be Alfarabi's view that syllogism, especially as developed by Aristotle in the Prior Analytics, is a genus for the different species of the syllogism developed in the other five syllogistic books of the Organon. This should be seen as Alfarabi's attempt at providing a partial solution to the problem of the unity of the syllogistic arts, the terms of which Alfarabi largely inherited from the late antique Greek logical commentary tradition.Footnote ¹⁵ The “horizontal” distinction between each of the syllogistic arts (e.g. what distinguishes demonstration from dialectic), as well as the “vertical” distinction between the species of the syllogism and its genus (e.g. in what sense is the syllogism of the Prior Analytics different from the syllogism described in the Topics) rest on Alfarabi's analysis of the mental act of compliance or assent that attaches to propositions in a syllogism-formatted argument. All of the syllogistic arts share in the fact that a mental act of assent, which varies in strength according to the context in which the proposition is expressed, attaches to the propositions from which the premises and conclusion of the syllogism are composed.Footnote ¹⁶ These sentences are, thus, composed of two parts. One part of the sentence is the mental act of assent; the other part is the proposition (e.g. ‘X is Y’) that is the object to which the mind gives its assent. As a consequence of this distinction, it is possible to say that it is ultimately the variation in the ways in which we give assent to propositions, and not necessarily the propositions themselves, that allows us to differentiate among the syllogistic arts. For example, Alfarabi says that there is a generic notion of the syllogism, alluding to the syllogism as outlined in the Prior Analytics, just to the extent that it leads to an unqualified (muṭlaq) act of mental assent. On the other hand, there is a poetic syllogism, which is distinguished from all the other kinds of syllogism, just to the extent that it leads the mind to what Alfarabi calls ‘poetical assent (al-inqiyād al-šiʿrī)’. It is reasonable to take Alfarabi to be claiming that the proposition ‘X is Y’ can be common to both the poetic and demonstrative modes of assent, but what distinguishes the conclusion of a poetic syllogism from a demonstrative syllogism is the modality, or strength, of the mental act itself, not necessarily the propositional content of the judgment.

We can find, therefore, in Alfarabi too a tendency to move away from ‘the objective truth of the proposition which is known, towards the knowing act of evaluating and accepting it as true’.Footnote ¹⁸ In other words, in the context of an argument the modality of the mind's assent to the proposition ‘X is Y’ is just as important to the logical analysis in the thinking of the classical Islamic philosophers as the proposition ‘X is Y’ itself. This is by no means to deny the centrality of objective truth values in Alfarabi's logic, nor should it be taken as denying that all propositions are either true or false (bivalence). Indeed, by insisting, along with the majority of late antique Greek Peripatetics that demonstration represents the telos of logical inquiry despite the existence of other species of syllogism,Footnote ¹⁹ there clearly remains in Alfarabi's mind an ineluctable relation between the act of assent and the assignment of truth-value to a proposition. Indeed, as Black notes in her discussion of Avicenna's ‘imaginative syllogism’, the above passage shows that in Alfarabi's thinking ‘the primary focus of assent remains the determination of what is true. To give one's assent to any proposition necessarily presupposes the prior consideration of whether the proposition is true or false’.Footnote ²⁰ Rather, in Alfarabi as well as Avicenna there is a ‘shift of emphasis from the veracity of the cognitive act as a representation of some object to the way in which the cognition itself is accepted by the knower’.Footnote ²¹ Unlike contemporary accounts of propositions that theorize them as “disembodied” objects that can be assigned objective truth values removed both from the argumentative context and from the speaker who spoke them, to say that a proposition P is true is only to say that a reasoner says that P is true. That being said, a reasoner will say that P is true but entertain different truth-criteria in different argumentative contexts. After considering what relation the proposition P bears to the state of affairs it represents within the particular speech context in which the reasoner places himself, the speaker will then tailor the degree of his assent, or the force with which he says “P is true”, according to the demands of the speech context. Of course, this makes the reasoner's saying ‘P is true’ ambiguous. In Alfarabi's way of thinking, we are able to disambiguate the speaker's context-sensitive assignment of truth to propositions by examining the nature and strength of his assent to the proposition in question.Footnote ²²

In order to illustrate the complex relationship between assent or the mind's compliance to a proposition (taṣdīq, inqiyād al-ḏihn) on the one hand and the proposition's objective truth on the other, it would be helpful to examine what Lear has called the ‘argumentative role’ of each of the propositions in a generic conditional syllogism. Jonathan Lear pointed to the importance of appreciating the argumentative role each premise plays for understanding Aristotle's brief treatment of hypothetical syllogisms in the Prior Analytics. However, since context theory puts perhaps even greater emphasis on the argumentative context, an analysis of the argumentative role of propositions in conditional syllogisms is indispensible.Footnote ²³ Lear says that Aristotle's categorical syllogistic does not recognize the argumentative role of sentences in a deduction. For example, Aristotle's syllogistic does not distinguish between a sentence in a deduction that is merely supposed to be true by two opponents for the sake of argument, or for the sake of probing the logical implications of accepting the sentence as true, and a sentence that is true as such, e.g. a necessary first principle of a science. Aristotle's categorical syllogistic identifies which arguments are valid according to purely formal characteristics of the premises (quantity, quality, etc.). Aristotle does not aim to analyze deductions according to the argumentative role played by each sentence in the deduction. Thus, in Prior Analytics A23, when Aristotle claims that hypothetical syllogisms “are brought about through syllogistic figures”, his claim is that this is so only to the extent that such a hypothetical syllogism contains a categorical syllogism as a proper part of the deduction.Footnote ²⁴ This may be illustrated by considering the following schematized hypothetical argument borrowed from Lear:

In the above hypothetical argument, the part of the argument “but … so P” is a direct syllogism that is formally reducible to one of the valid figures of Aristotle's categorical syllogistic, where “…” represents a series of premises and P represents a categorical proposition that serves as a conclusion of a categorical syllogism of the A-, I-, E- or O-type. Lear believes that Aristotle does not intend to formalize into his syllogistic the act of agreement between the opponents that they accept Q as a necessary result of P's being deduced from a categorical syllogism.

In contrast to Aristotle, there is good reason to believe that Alfarabi developed his theory of conditionals as a way of formalizing the argumentative role that the prior agreement between the opponents plays in a dialectical exchange. Indeed, Alfarabi formalizes both the act of prior agreement between two opponents and the mental act of supposition as a conditional proposition, in the sense that the propositions from which a conditional are composed are given or conceded (waḍʿ) by one or more of the interlocutors. The first sense of waḍʿ relates primarily to a prior act of agreement between two opponents. This sense of waḍʿ, which might be translated as positing or laying down, does not necessarily connote a particular type of mental activity that accompanies the act of positing or laying down of premises. However, Alfarabi assimilates waḍʿ to the notion of farḍ or iftirāḍ, viz. supposition, which does connote a mental process. Thus, the specific meaning of waḍʿ as a prior act of agreement between two opponents to entertain the existence of a connection between an antecedent and consequent can also take on the meaning of mental supposition. In Ğadal Alfarabi says:

According to Alfarabi, the syllogism from a hypothesis described by Aristotle in A23 is a genus for conditional syllogisms generally. The feature that relates them is the fact that in both kinds of syllogism, the antecedent (muqaddam) (or the hypothesis [waḍʿ] in a syllogism from a hypothesis) and the consequent do not have a definite truth values insofar as they are parts of the conditional proposition. Of course, the antecedent and consequent may be objectively true in themselves (fī anfusihimā) when considered on their own. But if the reasoner gives his assent to the conditional (or the hypothesis) as a whole, this does not entail that he gives his assent to the parts of the conditional (or to the hypothesis and what follows from it). In Alfarabi's analysis, the reasoner's assent is given to, or there is compliance of the reasoner's mind with, the propositions that are objectively true (ṣaḥīḥ), which in the case of a conditional syllogism are the conditional itself as a major premise and the minor premise. As a premise, assent can be given to the conditional as a whole, based on the fact that the connection expressed by conditional sentence corresponds to the actual state of affairs. The minor premise as far as its propositional content is identical to antecedent of the conditional or it is the contradictory opposite of consequent. In the case of modus ponens, what distinguishes the antecedent of the conditional and the minor premise is not their propositional content. Rather, the reasoner attaches his assent to the proposition expressed by the minor premise, but does not give assent to the proposition expressed by the antecedent qua member of the conditional (the same proposition is expressed in both instances). In other words, giving assent to a conditional involves the reasoner simultaneously adopting a definite attitude toward the truth of the connection signified by the conditional proposition, and the suspension of his mind's compliance with the antecedent or consequent of the conditional. To justify his view, Alfarabi asks us to consider a situation in which this is not the case, viz. assume for the sake of argument that giving assent to the conditional as a whole is also to give assent to the antecedent and the consequent. Say it is night time and reasoner wants to reason about it being day or night. The reasoner executes a syllogism in modus tollens, with a conditional major premise ‘if the sun is up, then it is day’. Since it is obviously night out, we then want to assert ‘but it is not day’ in order to conclude ‘Therefore, the sun is not up’. However, our assumption has blocked our ability to assert the minor premise, since, having given our assent to the conditional, we have also thereby given our assent to the fact that the sun is up. This leaves us in the undesirable position of having given our assent to a proposition and its contradictory opposite. Though Alfarabi does not mention it, a similar argument can be made for modus ponens. If the conditional ‘if the sun is out, then it is day’ means nothing more than ‘the sun is out, and it is day’, then modus ponens is, technically speaking, nothing more than a petitio principii. Thus, in order not to make nonsense of valid deduction schema, Alfarabi shows that a sharp distinction must be made between the proposition when it is a member of a conditional and when it is not. When it is a member of a conditional that a reasoner has given his assent to, the reasoner has not, in fact, given his assent to the proposition itself, but to the connection between it and the other member of the conditional. However, when the proposition expressed by the antecedent or the consequent appears as a minor premise or a conclusion, then the reasoner's assent does attach to the proposition itself.

On Alfarabi's view assent does not attach to the antecedent and consequent qua constituent parts of a conditional sentence. The reasoner will give his assent to a conditional proposition P saying “P is true” once he has verified that the nature of the connection between the antecedent and consequent is of a strength required by the speech-context. Considerations involving the reasoner's assent are distinct from the question of P's truth-value. The conditional is true just in case there is a certain type of connection (ittiṣāl) between the antecedent and the consequent. As we will see, in Alfarabi's view this connection may be per accidens or per se. If it is per accidens, then the connection between antecedent and consequent is completely coincidental. (P1) “if Dion is walking, then Theon is leaving” is a true conditional per accidens. One type of per se connection is called ‘for-the-most-part’. Read per se with for-the-most-part connection P1 is false, but (P2) “if the sky is clear in winter, then it will be colder” is true. There is also per se necessary connection. P1 and P2 would both be false on this reading of the conditionals, but (P3) “if there is a man, then there is an animal” is true. As I will discuss in greater detail below, it seems that Alfarabi has in mind that the degree or strength of assent given to a conditional proposition “if A, then B” must be in line with the strength of the connection between antecedent and consequent. If the speaker considers P1, he will perhaps notice that the connection between antecedent and consequent is per accidens, and also that P1 is false when read as having per se connection. With these truth-values in hand, depending on the requirements of the speech-context, the reasoner will give assent to P1 “it is true that if Dion is walking, then Theon is leaving”. Yet, his assent will not have the strength of certitude as required by a demonstrative speech-context, but perhaps he would give his assent to P1 with strength appropriate to a rhetorical speech-context. There is, thus, a parity in emphasis in mental representation (truth as correspondence) and mental act (assent-giving) in Alfarabi's account of conditional proposition, which reflects the argumentative backdrop against which the context theory of syllogism was developed by classical Islamic philosophers. Awareness of this backdrop is particularly important when we consider Alfarabi's theory of conditionals and conditional syllogisms. If logic is to give shared, objective criteria for the validity of, for example, demonstrative inferences as well as poetic and rhetorical inferences – viz. in argumentative contexts where the speakers very often lie, tell half-truths, and generally dissimulate, then naturally the aim of our analysis of propositions and syllogisms in such contexts will be aimed less toward the objective truth of the propositions expressed in the argument and more toward what propositions will the audience give its assent to. In non-demonstrative contexts, on the other hand, a wedge is opened up between the proposition's objective truth and the audience's compliance with it. In such contexts, there will be propositions to which assent attaches but whose correspondence to contingent states of affairs is not exact.

In none of these types, however, is the truth of the conditional determined by the straightforward determination of the antecedent and consequent's correspondence to current or possible states of affairs, or the antecedent's and consequent's straightforward assertibility or deniability.Footnote ³¹ There is no question that the objective truth of the antecedent and the consequent is determined by the correlation (or non-correlation) of what it expresses to the state of affairs. However, in the context theory of logic developed by the classical Islamic philosophers, the correlation of the proposition to the state of affairs is not the exclusive unit of analysis. Rather, along side the question of the antecedent's and consequent's being true (or false) sits the question of whether, how, and to what end the listener actively gives his assent to the them. In the non-demonstrative syllogistic arts, what you can get your audience to give its assent to is more important than whether the antecedent and consequent are true or not. The aim of the five syllogistic arts is as much the production of attitudes of different strengths toward a particular proposition as it is the production of a proposition as a conclusion. Nor is there only one attitude that a reasoner attaches to propositions amenable to truth and falsity. Rather, the strength of the reasoner's assent to a proposition varies according to the argumentative context in which the proposition is used. The attaching of different grades of assent to propositions in dialectical and rhetorical contexts is due to the listener's recognition of the contingency of the matter of these propositions. It is often the case that propositions expressed by the conditionals can be said to correspond to the state of affairs, but to a greater or lesser extent, and as a consequence, the mind's compliance with them will be similarly graduated.Footnote ³²

The importance of assent in determining in what senses a conditional is said to be true by an audience has implications in how we think about the syllogistic validity of arguments with conditional premises. For example, an argument constructed according to the schema modus ponens is valid in the truth-valuation sense, if there is no value assignment to the sentences of the premises and the conclusion that makes the former all true but the latter false. If there is such a value assignment, then the argument is invalid. Analogously, an argument in the scheme of modus ponens is valid in the sense of assertibility, if there is no value assignment to the premises and the conclusion that makes all the premises assertible, but the conclusion is either not assertible or is deniable. And, once again, the argument is invalid if there is such a value assignment.Footnote ³³ The difficulty with applying this criterion for validity to Alfarabi's conditional syllogisms is that, depending on the context in which the argument takes place, a listener will give his assent, will be induced to say that a conclusion is true, viz. to say that it corresponds to the state of affairs, given his assent to the premises, but only to a greater or lesser degree. This is especially true in non-demonstrative arts such as dialectic and rhetoric. In dialectical and rhetorical discourses the conclusion may not necessarily follow necessarily from the premises, but it follows in enough cases that a listener will be convinced to give his assent to the conclusion, given that he gives his assent to the premises. In arguments from premises about contingent events or, for example, about ethical analyses of the good and the just that are known to have many exceptions, a listener may be induced to give his assent to propositions talking about what the good and the just are, in spite of his knowledge of exceptions that might otherwise be considered as falsifying the premise.Footnote ³⁴ Analogously, a listener might consider a syllogistic argument good despite his knowledge that in certain instances the conclusion sometimes does not follow from the premises.

3. CONDITIONALS IN ARGUMENTATIVE CONTEXTS

Two of the richest sources for Alfarabi's thoughts about conditionals are two epitomes of Aristotle's Topics, Ğadal and Taḥlīl.Footnote ³⁵ In these two works, Alfarabi gives a precise account of the structure of a dialectical exchange between a questioner (al-sāʾil) and a respondent (al-muğīb). As we will see, these pragmatic conditions determine to a large extent some of the syntactic and semantic properties of Alfarabi's conditional propositions.

Unlike a rhetorical argument, which has the structure of a single individual addressing a passive audience in order to convince them to act in a certain manner,Footnote ³⁶ a dialectical exchange is one involving a single questioner (Q) and a single respondent (R).Footnote ³⁷ The exchange between Q and R is closer to a competition with a winner and a loser than a straightforward deduction. Q's aim in the argument is to discover the view that R has been tasked with defending, and then showing that that view is logically inconsistent.

A dialectical exchange may be characterized as a dialectical competition, the aim of which is primarily refutation. The quaesitum (maṭlūb) – viz. a question in the form of a disjunction of a pair of contradictory or contrary statements whose ‘subject terms are universal’ – organizes the exchange.Footnote ³⁹ For example, the quaesitum may be in the form of ‘Aab or Oab’, e.g. ‘Is all killing injustice or is some killing is not injustice?’, or ‘Aab or Eab’, e.g. ‘Is all killing injustice or is no killing injustice?’.Footnote ⁴⁰ Once R concedes to Q one of the disjuncts of the quaesitum, say Aab,Footnote ⁴¹ by means of question and answer, Q's objective for the rest of the exchange is to elicit further concessions from R. Only these concessions and no others may be used by Q as his set S of premises by means of which he refutes R. Q successfully refutes R's claim ‘Aab’ when a subset of the premises conceded by R S* can be combined into a syllogism, the conclusion of which is identical to the other disjunct of the quaesitum. In this particular case, Q refutes R by constructing a syllogism from S* whose conclusion is ‘Oab’. On the other hand, R successfully defends ‘Aab’ by not conceding premises that, when taken together, Q can use to construct just such a syllogism. Thus, R's objective is to allow Q a set of premises S, no subset of which can be combined to form syllogisms whose conclusion is ‘Oab’. In other words, Q shows R's position is ‘logically inconsistent’ in the sense that he is able to derive from R's set of conceded premises according to the rules of categorical or hypothetical syllogisms the contradictory or the contrary of the thesis R is tasked with defending.

To illustrate, consider the following highly implausible exchange – in fact, all the examples in this article are highly implausible –between R and Q. R concedes “All killing is injustice” (Aab). It thus falls upon Q to have R concede enough premises that allow him to construct a syllogism that concludes, “So, not all killing is injustice” (Oab). The argument, whatever its plausibility or implausibility, might go as follows:

In the above scenario, R concedes the following set of statements S1 = {Aab, Iaf, Afb, Efb} as well as the concessions that S2 = {for all x, x is by God's decree}, from which immediately follows S3 = {this act of killing is by God's decree} by UI, from which Iaf in S1 follows immediately by EG. Thus, in the above exchange, R concedes all of S = {S1, S2, S3}, but of these Q only needs {Iaf, Efb} to yield the other disjunct of the original quaesitum Oab by Ferio, {Iab, Ebc} entails Oac for any a, b, and c.

Let us now examine how the above considerations affect Alfarabi's thinking about conditional propositions and conditional syllogisms. This will be accomplished primarily by examining Alfarabi's recommendations for how to construct premises and syllogisms, a task which Alfarabi assigns to the topoi (al-mawāḍiʿ).

The topoi (mawāḍiʿ, sing. mawḍiʿ) occupy an important place in Alfarabi's logical theory.Footnote ⁴² Alfarabi does not restrict their use to dialectical investigation. In fact, they are, in his view, central to all of the intellectual disciplines (al-ṣanāʾiʿ al-fikriyya). Alfarabi opens Taḥlīl with the following words:

Thus, despite their application in practically all of the philosophical disciplines, in Alfarabi's view the topoi still provide no more than rules of thumb for constructing premises that are easily adopted into categorical or conditional syllogisms. In this passage, the topoi are called ‘universal premises (muqaddimāt kulliyya)’, but this does not mean that the terms from which they are composed pick out individual objects subsumed under the categories, in the way that ‘animal’ and ‘men’ in the universal premise ‘all men are animals’ pick out individual men and animals. Rather, the topoi are universal in the sense that they are intended as rules for constructing any number of premises, which are likely to gain the assent of the opponent. In making use of the topoi, the speaker is not necessarily interested in constructing true premises, as much as he is interested in constructing premises that are probable, or instill enough conviction in the mind to gain the opponent's assent. In order to gain the opponent's assent, these rules rely on the different ways that a predicate attaches to a subject. Often the topical rule is stated in terms of the five predicables, which for Alfarabi (though not necessarily for Aristotle)Footnote ⁴⁴ are five ways a universal predicate (maḥmūl kullī) Y attaches to a subject (mawḍūʿ) in response to the question “What is X?” (where X is some individual such as Zayd whom we see from afar and ask “What is that?”; Y is a universal predicate if two or more things share in the fact that Y is predicated of them).Footnote ⁴⁵ If, to the question “What is X?”, we respond “X is Y”, then Y is predicated of X in five different ways. The predicate Y is called a “genus [ğins]” when Y constitutes the substance (ğawhar) of two (or more) things X₁ and X₂, in the most generic sense of what it is to be X₁ and what it is to be X₂. The predicate Y is called a “species [nawʿ]” when Y constitutes the substance of two (or more) things X₁ and X₂ in the most specific sense of what it is to be X₁ and what it is to be X₂. The predicate “animal” is, thus, called the “genus” of both Zayd and ʿAmr, because both share in the fact that “animal” is said of both of them in the most generic sense of the questions “What is Zayd?” and “What is ʿAmr?”. The predicate “human” is called the “species” of both Zayd and ʿAmr because both share in the fact that “human” is said of both of them in the most specific sense of the question “What is Zayd?” and “What is ʿAmr?”. On the other hand, if X₁ and X₂ share in the fact that both are called Y but not in way that we are speaking about the substance of X₁ or X₂, then predicate Y is called an “accident [ʿaraḍ]”.Footnote ⁴⁶ If the predicate Y is used to respond to a question about what X₁ is, i.e. a question about X₁'s substance, in such a way that the substance of X₁ is distinguished from the substance of X₂ by the fact that the substance of the former has Y and the latter does not, then Y is called a “differentia [faṣl]”. Thus, “rational” is called a “differentia” because it is a predicate that distinguishes what Plato is from what Bucephalus is, though both are animals. If the predicate Y is used to respond to a question about what distinguishes X₁ from X₂, but not in a way that is connected to the substance of X₁ or X₂, then Y is called a “proprium [ḫāssa]”. The predicate “risible” is a proprium because it is a predicate that is used to distinguish Plato from Bucephalus, though the difference is not at the level of the substance of the individual.

Alfarabi uses topoi to generate categorical as well as conditional premises for a syllogism in a dialectical exchange. First, consider the following prescription for generating universal affirmative or negative categorical premises from “topoi from definition” in Taḥlīl.

We can schematize the scenario described by Alfarabi in the following way. The reasoner is debating a quaesitum such as “either bats are hairy or bats are scaly”. The reasoner is charged with getting his opponent to give his assent to “some bats are hairy”. In order to do so, he looks at the definition of bat, which is “winged mammal”. Finding “mammal” in the definition of bat, the reasoner knows that, in fact, the predicate “mammal” applies not only to some bats but to everything referred to by the term “bat”. This yields the universal affirmative categorical premise “(all) bats are mammals”. This premise is then used as the minor premise in the following first-figure syllogism Barbara (following the premise-order in the Arabic tradition where the minor premise generally appears first) with “all bats are hairy” as the conclusion, the speaker reasons as follows “if (all) bats are mammals and (all) mammals are hairy, then (all) bats are hairy”. If the reasoner is charged with refuting the quaesitum's other disjunct, he needs a syllogism that yields the contradiction or the contrary of the other disjunct in the quaesitum, viz. the conclusion can be “some bats are not scaly” or “no bats are scaly”. Following Alfarabi's prescriptions, the reasoner proceeds as follows. He peruses the definition of “bat” and finds that the term “scaly” is not to be found in “winged mammal”. From this, he is able to construct a universal negative “no mammal is scaly”, which is used as the major premise in the following first-figure syllogism, which yields the contrary of the quaesitum's second disjunct “no bats are scaly” as a conclusion: (Celarent, with minor premise appearing first): “if (all) bats are mammals and (no) mammal is scaly, then no bat is scaly”. The important point to notice in Text 5 is that the topical rules are set out based on relations of inclusion and exclusion that hold between terms falling under the five predicables. The reason why the above “topoi from definition” work is because the definition states the genus (ǧins) “mammal” and differentia (faṣl) “winged”, which allows us to generate a universal affirmative premise based on the fact that genus terms are those that are shared univocally among the species belonging to the genus. Analogously, the fact that “scaly” is not an element in the definition of “bat” entails that “scaly” is neither an element of the substance (ǧawhar) of “bat” that it shares with other objects (viz. it is not the genus of bat), nor is it an element of the substance of “bat” that distinguishes it from other members of the genus (viz. it is not the differentia). From the facts generated by the relations between genus and differentia, we can conclude that no bat is scaly, with the implicit assumption being that no bat is essentially scaly, though it may be scaly accidentally (bi-al-‘araḍ).Footnote ⁴⁸

Consider a topos that generates a conditional premise, an example of which appears in Ğadal: ‘if X is in Y, then the contrary of X is in the contrary of Y’. Possible instantiations of this rule include any number of premises such as the following: ‘if pain is evil, then pleasure is good’, ‘if God is perfect, then creation is deficient’, and ‘if men incline to injustice, then women incline to justice’.Footnote ⁴⁹ In this case, the use of this topos has generated a conditional sentences, which, despite the fact that the antecedent and consequent are indefinite categorical propositions, might reasonably be interpreted as universally quantified; thus, e.g. (P1) ‘if all pain is evil, then all pleasure is good’. Consider a scenario in which Q and R agree to debate the quaesitum ‘is all pain evil or not?’ In this, R decides to defend ‘all pain is evil, in which case Q takes on the task of constructing a syllogism or series of syllogisms that conclude (C) ‘some pain is not evil’ only from premises obtained from R. Using this topos, Q might proceed in the following way. He will try to convince R to concede P1, and then to concede further premises that allow Q to construct a categorical syllogism with (P2) ‘some pleasure is not good’ as the conclusion. With P1, P2 and modus tollens in hand, Q can then directly refute R by forcing him to concede C. It is this use of the topoi that Alfarabi seems to have in mind when he says near the beginning of Taḥlīl:

As in the imaginary exchange between Q and R in this section, Q analyzes the thesis defended by R ‘All pain is evil’ into its subject and predicate parts. The subject and predicate are found to be suited to have the topos ‘if X is in Y, then the contrary of X is in the contrary of Y’ applied to them. Applying this topos to the subject ‘pain’ and predicate ‘evil’ generates a premise ‘if all pain is evil, then all pleasure is good’, which is the key step in constructing the conditional syllogism that refutes R's conceded statement. As alluded to above, the antecedent and the consequent of the conditional are theses obtained from the opponent in a dialectical exchange, and neither R nor Q needs to be committed to their being true or false. Said differently, since they are either hypotheses or derived from hypotheses, Q and R do not take the antecedent and the consequent, as a result of their use in a dialectical context, as being subject to assent or denial, taṣdīq and takḏīb. In particular, Q's interest in the conditional is chiefly as providing a way of constructing a conditional syllogism (in this case, modus tollens) that allows him to falsify R's concession. The conditional does this by providing a connection between the hypothesis (antecedent) and another sentence (consequent). Q rests assured that R will accept the conditional ‘if all pain is evil, then all pleasure is good’ because it is authorized by the topos that ‘if X belongs to Y, then the opposite of X belongs to the opposite of Y’. Nor does it concern Q that this topos generates quite a large number of clearly sophistical conditionals. Yet, in dialectical and rhetorical exchanges it is not of chief importance that the conditional or its parts be true; they only need to be convincing to R so that he will concede them.Footnote ⁵¹

However, it is not the case that the topoi are applied exclusively to the thesis' subject and predicate. In Taḥlīl, which is Alfarabi's most exhaustive treatment of the topoi, the ‘topos from implications’ does not analyze the thesis into its subject and predicate elements. Instead, it operates directly on the thesis, generating a variety of forms of conditional proposition according to what purpose the conditional premise and syllogism will serve in the dialectical exchange.

In this topos, the subject and predicate play no role at all. Instead of the predicate ‘belonging to’ a subject as in a categorical thesis, this topos describes the thesis as ‘being by virtue’ of something else, or being absent or eliminated by virtue of the presence or absence of something else. Whereas the former approach relates terms from the categories to terms from the categories, the latter is a way of relating facts, conditions or events to each other, in the way that (the fact, condition or event of) its being day is by virtue of the (fact, condition or event of the) sun's being out. ‘Implications’, between antecedent and consequent for Alfarabi are best expressed using conditionals because they are a linguistic form that speakers conventionally use to express the way in which the presence or absence of a fact, condition, or event is somehow by virtue of (i.e. connected to, responsible for, related to, causally associated with, prior or posterior to) the presence or absence of another fact, condition, or event. Indeed, Alfarabi mentions that this topos is used most often to argue – often speciously in Alfarabi's view – about causal relations between events (e.g. the cutting of this or that nerve being the cause of paralysis) or between substances (e.g. the sense in which the being of animal causes the being of human). Thus, it seems that the chief virtue of the ‘topoi from implications’ in relation to arguments like this is that they allow us to formalize arguments about causes as conditionals that are then used to construct conditional syllogisms. Thus, Alfarabi's presentation of the ‘topoi from implications’ might be used in a dialectical context in the following way.

As usual, in the dialectical context described above, this topos from implication is being used negatively, i.e. in order to show that the thesis defended by R cannot be correct. Clearly, R and Q are hardly any closer to discovering what the object is, though they are that much closer to knowing what the object is not. The ‘implications’ that Q peruses in this example are data about phenomena that bear some sort of connection to the phenomenon expressed in the thesis. Though Alfarabi says that this topos is usually used to talk about causes,Footnote ⁵⁴ nightfall causes bats to appear in the sky only in a secondary or derivative sense. If the fact that bats come out at night is due to features of their physiology or their nocturnal feeding patters, then perhaps these are more accurately called ‘causes’ of bats taking wing at nightfall. If this is so, then nightfall might be called an inseparable yet accidental concomitant of bats flying at night rather than a cause as such. The point here, however, is that in both cases R intuitively accepts the conditional ‘if bats are out, then night has fallen’ because he recognizes (without necessarily knowing why) that the phenomenon expressed in the antecedent somehow or other is by virtue of the existence of the phenomenon expressed in the consequent. This topos is thus not concerned with investigating the nature of connection between the phenomena (causal, purely accidental, inseparable accidental, relational, mathematical, etc.), but simply takes advantage of conventional intuitions to generate conditional propositions. Throughout, the aim of the opponents in the schematic argument described above is investigating the phenomena expressed in the antecedent and consequent, not with the connection expressed by the conditional sentence as a whole. R gives his assent to the conditional because it expresses a real connection that exists between the phenomena expressed in the antecedent and consequent. This connection, however, is taken at face value, and does not itself become the object of investigation in the debate.

With respect to the foregoing discussion of dialectic and the topoi with respect to conditional sentences, the argumentative context in which conditional sentences are used endows Alfarabian conditionals with peculiar, but deeply-ramified formal properties. As we observed in §2, the reasoner does not give his assent to the antecedent and consequent qua parts of a conditional when he gives assent to the conditional as whole. However, after examining the structure of dialectical and rhetorical argument, we can see the reason for this seeming peculiarity. As we have observed in this section, conditionals arise in argumentative contexts in which the interlocutors debate a particular thesis that is subject to dispute. The interlocutors may harbour motives for engaging in argument that are geared more toward getting an audience, or a single listener to give its assent to a desired opinion rather than showing that the opinion is true. In the arguments with conditional syllogisms described above, it is usually the case that the disputed thesis or the thesis we want the listener to give his assent to is the antecedent, the consequent, or their contradictory opposites. Thus, as we saw in the case of conditional topoi, assent to the antecedent, the consequent, or their contradictory opposites is usually the very point under dispute. To that extent, it would be circular to expect the parties to the debate to give their assent to the conditional based on their prior knowledge of the truth of either the antecedent or the consequent. Rather, the parties to the debate will be induced to give their assent to the conditional based on their recognition that the connection between the antecedent and the consequent, the nature of which is usually dictated by the topoi, corresponds to the states of affairs. Consider the two conditionals generated from the two topoi discussed above: (a) ‘if all pain is evil, then all pleasure is good’, and (b) ‘if bats are out, then night has fallen’. The advertised purpose of the topoi is to generate premises that are, if not true, then at least engender some sort of compliance in the listener's mind; that is, they do not at first glance strike the listener as so implausible as to be rejected outright. As for (b), the reason why R might be tempted to accept it is due to his observations of bat behaviour in the past. In particular, he has no doubt observed that, without exception, bats never fly while the sun is up. In Alfarabi's technical language, this could be expressed by saying that the bats' being (wuğūd) implies the sun's elimination (irtifāʿ). In the examples explicitly discussed by Alfarabi, the reasons that justify R's belief about why, say, bats only come out at night, are not important in the course of the argument. The point is that it is R's recognition of a connection between the phenomena that leads R to concede the conditional sentence expressing this connection. In fact, since Q and R are trying to settle what exactly the black shape flying in front of them is, their assent to the conditional cannot be based on their assent to the antecedent or consequent, or their knowledge that they are true or false. In (a), the topoi ‘if Y belongs in X, then the opposite of Y belongs in the opposite of X’ describes a certain connection that a reasoner might reasonably accept as existing between two things, events, states, or conditions which are expressible in a subject-predicate format. The point of the argument between Q and R with (a) as a major premise is to settle, even if negatively, whether all pain is evil. R is thus not likely to give assent to the conditional ‘if all pain is evil, then all pleasure is good’ based on his conviction that the consequent is true without any consideration given to the truth of the antecedent. In this case, R might feel justified in conceding ‘if all pain is evil, then all pleasure is good’ is due to a certain, obviously specious but no less convincing, connection we recognize to exist between opposites such as pain/pleasure and evil/good. Thus, in Alfarabi's way of thinking, the use of conditionals in a dialectic, and presumably a fortiori, in a rhetorical, context is not interpretable as the speaker asserting, or adopting a definite cognitive attitude toward, either the antecedent or the consequent. Rather, he gives his assent to the conditional to the extent that he recognizes that the connection between the antecedent and consequent that the conditional signifies really does exist, viz. that the antecedent and consequent are connected in such a way that the listener will give his assent to the consequent given his assent to the antecedent. Clearly, then, Alfarabi's view of conditionals is not amenable to an interpretation in which the truth or falsity of the conditional is in any way dependent on the truth or falsity of its constitutive parts. This is not merely to say, however, that Alfarabi's conditionals do not appear to be amenable to a truth-functional interpretation of ‘if…then…’ sentences as material implication. Material conditionals are true in all cases in which the antecedent is false. By contrast, in the context of a dialectical or rhetorical exchange, it does not make sense for two interlocutors to make a hypothesis (waḍʿ) about something they already know to be false or impossible. It seems that in Alfarabi's way of thinking about the use of conditionals in dialectical exchanges, a conditional with a false or impossible antecedent violates a pragmatic assumption that conditionals be useful in debate. Nor are Stalnaker-Lewis interpretations appropriate either. In order to evaluate whether or not an Alfarabi conditional is true, we are not required to look to see whether the consequent is true according to a ‘stock of beliefs’ that are hypothetically altered to make the antecedent true.Footnote ⁵⁵ Rather, the focus is on what the listener will give his assent to given that he gives his assent to the antecedent.Footnote ⁵⁶ In non-demonstrative contexts, for example, what is objectively true and what the audience gives its assent to are not coextensive.

Finally, it is also important to realize that the most common form of atomic sentences that serve in the antecedent and consequent position in a conditional are categorical propositions of the form Aab, Eab, Iab or Oab for any distinct a and b that can be plausibly located in one of the ten categories. Nor is there any reason to believe that Alfarabi would disallow singular terms as subjects: ‘s is b’ or ‘s is not b’, where s is a this and b is plausibly located in one of the ten categories. What is more, we find in APCA that Alfarabi allows fact-like or event-like propositional contents to be expressed by the antecedent and consequent of a conditional. For example, the following ‘event-like’ propositions are used as antecedents and consequents: ‘Zayd comes’, ‘ʿAmr departs’, ‘Sirius rises in the morning’, ‘the day will be hot and the rain will cease’, ‘it is day’, and ‘the sun is out’.Footnote ⁵⁷ While none of these propositions is obviously expressible in the form of a universal or particular categorical proposition, it is easy to see how they could serve as one part of a dialectical quaesitum, e.g. ‘Either Zayd comes or he doesn't’, or ‘the rain will cease or it won't’. Thus, despite Alfarabi's claim in Ğadal that ‘the only thing sought from a quaesitum is whether the predicate belongs to the subject or not’, which, if taken strictly, would restrict the types of quaesitum to disjunctions of subject-predicate categorical propositions, practically speaking, Alfarabi does not seem strictly committed to such a view.Footnote ⁵⁸ What such a view does exclude, however, is the very possibility that we can have a debate or argue about the truth and falsity of a conditional, or that we can try to elicit assent to a conditional as the main objective of an argument. This is true to the extent that Alfarabi does not seem to consider the case in which the quaesitum itself is composed of conditionals, i.e. that the quaesitum can be of the form ‘either if P, then Q, or not-(if P, then Q)’. In order to develop such an argumentative scheme, Alfarabi would have to develop, one, a conditional syllogistic that allows nested conditional; two, a conditional syllogistic that can yield conditional conclusions; and three, a doctrine of conditional contradiction. In the logical structure of the dialectical exchange, the disjuncts of the quaesitum circumscribe the choice of opinions R can defend and Q can refute. Say, for instance, that R decides to defend ‘if P, then Q’. Then Q's task is to get R to concede premises that allow Q to construct a conditional syllogism that yields the contradictory of ‘if P, then Q’ as a conclusion. This single requirement requires that Q know what the contradictory of a conditional is, that he be able to construct a conditional syllogism in the moods of modus tollens or modus ponens whose antecedent and consequent are the conditional ‘if P, then Q’ or its contradiction, and that he be able to generate the contradictory opposite of ‘if P, then Q’ as a conclusion. But Alfarabi does not theorize at any length about any of these three items in his extant logical works.

This is not to say that Alfarabi did not formulate or discuss such doctrines in other works that we no longer possess. Indeed, it is highly unlikely that this could be the case. Alfarabi must have been aware that there were late antique logicians, such as Galen and Boethius,Footnote ⁵⁹ who espoused the view that the conditionals ‘if A, then B’ and ‘if A, then not-B’ are logically inconsistent, in the sense that the speaker will never assert both simultaneously.Footnote ⁶⁰ Indeed, Avicenna explicitly mentions the existence of such a view (and goes to great lengths to refute it) in book 5 of Qiyās of Šifāʾ. Similarly, it is well-documented that late antique Peripatetics like Boethius developed hypothetical syllogistics that allowed conditionals and nested conditionals as premises and conclusions. The truly perplexing question in my view is this: why was not Alfarabi sufficiently impressed by the importance of these issues to include them in his epitomes, even if only to mention them in passing? The above discussions in this section and the last perhaps provide at least a partial explanation. In Alfarabi's use of conditionals, it seems to be the case that the nature and purpose of the quaesitum determines what formal features of conditionals are developed and what formal features are not. Not only in Ğadal, but also in Taḥlīl, which presents practical guidelines for constructing arguments in all of the sciences, arguments tend to be limited to disputes about predicable-type relationships (species/genus, definition, differentia, property accidental or otherwise, individual substance (i.e. this-es), and existence/non-existence). As a consequence, Alfarabi consistently presents the quaesitum, a disjunction of two opposite sentences with interrogative force, in the form of a categorical proposition such as ‘is Aab or Oab?’ Of course, the purpose of the quaesitum is to provide the basic standard by which a thesis is overthrown in an argument, and to organize the initial conditions of the inquiry (what Q aims to derive, what R thesis defends). Yet, the formal property of the disjuncts as exclusively categorical propositions circumscribes the scope of the types of question that can be asked in such exchanges. In other words, it is difficult to ask questions about causes, for example, which are, as we saw in the discussion of the ‘topoi from implications’ best formalized in conditional sentences. Yet, the quaesitum-based dialectical exchange which focuses on arguments organized around predicable-type questions makes it so that conditionals do not appear as elements of the quaesitum. But the placement of conditionals in the quaesitum demands the explicit formulation of both a rule about the formal contradiction of a conditional, and a doctrine of conditional syllogism that, at the very least, could countenance conditionals as conclusions of conditional syllogisms. It is one thing for R to concede ‘if A, then B’ as a thesis, but Q must know, at least with regard to form, what is required to contradict such a conditional in order to refute R's concession. Also, the quaesitum-based argument format requires that Q refute R by producing the contrary or contradictory of R's thesis as a conclusion of a conditional syllogism. Thus, Alfarabi would also be required to develop a conditional syllogistic that can not only take conditionals (and nested conditionals) as premises, but also produce conditionals (and nested conditionals) as conclusions. Alfarabi's silence on these important points is, at first glance, surprising. It seems, however, that when we consider the dialectical backdrop in which conditionals are used, we see that the quaesitum-based argumentative structure that focuses on predicable-type topoi and arguments makes the development or discussion of such doctrines unimportant if not unnecessary.

4. CLASSIFICATION OF IMPLICATION (LUZŪM) IN APCA: CONDITIONS OF ASSENT, CONTRADICTION

In APCA §58 Alfarabi directly links the notion of necessary implication (luzūm bi-al-ḍarūra) to the formation of connective conditional propositions.

It is difficult to overemphasize the importance of this move since it allows (1) all implication relations to be formalized in the language of conditional propositions; and (2) it allows us to construct valid syllogisms based on the standard schema, primarily modus tollens (MT) and modus ponens (MP), with conditional major premises; finally, (3) it allows us to reason demonstratively about sentences between which relations of necessary implication exist. Nevertheless, this should not make us lose sight of the fact that Alfarabi also recognizes types of implication that are not necessary. In other words, Alfarabi says that an antecedent implies a consequent even if the connection between them is not necessary. Thus, in this section, out task will be to explore how Alfarabi speaks about the different grades of implication that exist according to the different strengths of connection between antecedent and consequent. In Alfarabi's treatment of conditionals, the conditional sentences (1) ‘if Paul comes, Peter goes’, (2) ‘if it is windy, then it is cool’, and (3) ‘if this shape is square, then it has four sides’ are formalized indifferently as ‘if P, then Q’. However, Alfarabi would also say that (1) signifies accidental implication (luzūm bi-al-ʿaraḍ), (2) implication for the most part (ʿalā al-akṯar), and (3) necessary implication. Thus, the notion of connection underlies each of these types of conditional, but the nature of the connection in each is different.

In fact, we are already quite familiar with Alfarabi's discussion of implication from his development of the ‘topoi from implication’ from Taḥlīl. The ‘topoi from implications (mawāḍiʿ min al-lawāzim)’ are rules for forming conditional sentences that rely on common intuitions about by-virtue-of relations – these are, as Alfarabi notes, often causal relationsFootnote ⁶² – that hold between pairs of events, states, substances, accidental or essential properties, or phenomena, taken broadly. These topoi rely, in particular, on the reasoner's intuitions about how some Y (be it a state, event, substance, property or phenomenon) is somehow or other ‘by virtue of (bi…)’ X (be it some other state, event, substance, property or phenomenon). With this intuition in hand, the reasoner is then justified in constructing a conditional sentence where the sentence expressing Y is the consequent and the sentence expressing X is the antecedent. These topoi allow us to say ‘X implies Y’, to the extent that it is possible to say that Y is (or is not) by virtue of the being (or non-being) of X.

Alfarabi does not take the notion of implication as the object of investigation in Taḥlīl, but he does, and in quite some detail, in APCA.Footnote ⁶³ As we have seen in previous sections, Alfarabi's classification of the types of implication in APCA is sensitive to the argumentative contexts in which conditionals expressing implicative relations appear as premises. Alfarabi's tendency to give pride of place to what he calls ‘necessary implication (luzūm bi-al-ḍarūra)’ is characteristic of the well-known, if not somewhat problematic, Peripatetic lionization of demonstration and demonstrative premises, while still attempting to maintain a place for dialectical, rhetorical and even poetical reasoning.Footnote ⁶⁴ As such, Alfarabi's classification of implication also includes subdivisions such as ‘per accidens implication (luzūm bi-al-ʿaraḍ)’ and implication for the most part (ʿalā al-akṯar), which, based on the examples Alfarabi provides, seem to correspond to types of implication in rhetorical and dialectical conditional syllogisms respectively. Thus, despite Alfarabi's giving pride of place to necessary implication, it is important to keep in mind that, for Alfarabi, there is no single “correct” reading of conditionals of the form ‘if P, then Q’. Rather, the sort of implication expressed by conditional sentences, divided according to the weakness or strength of the connection between the antecedent and consequent, depends crucially on the argumentative context in which the conditional is deployed.

As in Taḥlīl, in APCA two things are called implicates (mutalāzimān), or to stand in a relation of implication, when ‘if one of them is, then the other is by virtue of the being [of the first] (iḏā wuǧida aḥaduhumā wuǧida al-āḫaru bi-wuǧūdihi)’.Footnote ⁶⁵ As we saw in the previous two sections, Alfarabi does not strictly delimit the types of sentential content that can be expressed by implicates, in the sense that he allows content that is amenable to expression in subject-predicate sentences, and content that is not, e.g. sentences talking about facts and states of affairs. Nevertheless, as we discussed in the last section, Alfarabi does not deem it important to consider in what sense an implication might be by virtue of the being of another implication, i.e. a relation of implication between pairs of implications. In general then, the types of thing that are said to imply something else are, broadly speaking, states of affairs on the one hand and predicable-type objects on the other.

Alfarabi's analysis of implication begins with a division of the types of implication according to whether the antecedent implies the consequent per se (bi-al-ḏāt) or per accidens (bi-al-ʿaraḍ). This analysis of implication, in the final analysis, reduces to an examination of the strength of the connection between the antecedent and the consequent. The antecedent's implying the consequent accidentally signifies the weakest form connection between antecedent and consequent (if there is any connection at all), whereas the antecedent's implying the consequent necessarily signifies the strongest connection between antecedent and consequent. As an example of per accidens implication, Alfarabi gives us the conditional proposition ‘if Zayd comes, then ʿAmr departs’. In this case, there is not any clear principle that gives insight into why ʿAmr's departure, as expressed in the consequent, should be by virtue of Zayd's arrival, as expressed in the antecedent. Alfarabi's words suggest that it simply happens to be the case that ʿAmr's departure and Zayd's arrival coincide. Thus, it seems that Zayd's arrival implies ʿAmr's departure in a purely accidental way, in the sense that there is no underlying principle requiring the inseparability or perpetual concomitance of ʿAmr's departure and Zayd's arrival. In this weakest sense of implication, it would be unintuitive, but no less correct, to say that, for example, ‘Socrates is white’ implies ‘the sun is round’.

On the other hand, an antecedent implies the consequent per se when the following of the latter from the former is not accidental, in the sense that there is presumably some underlying principle, or set of them, that determines that when the antecedent is, the consequent is by virtue of the antecedent.

It is clear that the two types of per se implication presented here, namely, implication for-the-most-part and implication of necessity, do not hold between antecedent and consequent out of chance. The consequent's (the intensification of heat and cessation of rain) being implied by the antecedent (Sirius' passing the zenith in the morning) for-the-most-part seems to be due to the fact that astronomical phenomena have some sort of regular, law-like, though not entirely determinative effect on the weather and other natural processes in the world. In other words, implication per se, but for the most part, seems to be due to physical laws that are, nevertheless, not entirely deterministic. For example, as a matter of fact, cloudless winter days in Montréal are generally colder than days where there are clouds. Of course, it happens that sometimes a winter day is clear and unusually warm. Yet, in spite of knowledge of these exceptions, Alfarabi would not say that someone who says ‘it is a clear, winter day in Montréal’ implies ‘it is a cold’ has spoken wrongly. Rather, Alfarabi wants to include conditional premises, likely in dialectical arguments, that while not true eternally and unchangingly, are true often enough to be acceptable in non-demonstrative contexts. The connection between the antecedent and consequent in this type of implication is strong enough to be ‘widely-accepted’, but not so strong that it is falsified by an instance or instances of the antecedent not coinciding with the consequent.

On the other hand, necessary implication, which is also classified under the per se division, represents the strongest type of implication between the antecedent and consequent envisioned by Alfarabi. Furthermore, just as demonstrative categorical syllogisms must have premises in which the predicate's being in the subject is necessary, Alfarabi likely envisioned conditionals with a necessary connection between antecedent and consequent as being primarily suited to demonstrative conditional syllogisms. In a general sense, necessary implication represents a type for which it is impossible that the antecedent ever be without being accompanied (or not accompanied) by the consequent. However, Alfarabi subdivides per se necessary implication into complete implication (luzūm tāmm) and incomplete implication (luzūm ġayr tāmm). Alfarabi describes the former in the following words:

It is important to note that the order of the antecedent and consequent in this type of implication relation is unimportant. Whichever of X or Y happens to be (ittafaqa), then the other is by virtue of it. This is not the case with incomplete following, in the sense that which proposition is treated as the given is important. Alfarabi says:

With respect to incomplete following, the order in which the antecedent and consequent are spoken is important, since if X is given, then it is necessary that Y is, though the converse does not follow. Consider, for example, two of Alfarabi's favourite examples: (a) ‘if the sun is up, then it is day’, and (b) ‘if human is, then animal is’. In (a), the antecedent and the consequent stand in a complete relation of implication. This means that the connection between the antecedent and consequent is such that it is impossible for the consequent to not be when it is given that the antecedent is. But since the implication is complete, it does not matter which of ‘it is day’ or ‘the sun is up’ is given to be. This means that in complete implication the antecedent and consequent are convertible, meaning that switching the order of the antecedent and consequent – whichever of the pair of propositions is hypothesized as the given – will not affect the truth of the conditional. In (b), on the other hand, the ordering of the antecedent and consequent is important insofar as it is important which of ‘human is’ and ‘animal is’ is assumed to exist. If human is taken to be, then it is impossible that human is but there is no animal. However, on the assumption that animal is, it is possible for an animal to be and it not be a human. Thus, for (b) to be true, the antecedent must be ‘man is’. For (b) to be false is just for the assumption (in this case, that ‘animal is’) to leave it open to a situation in which, given that the antecedent is, the consequent still is not.

Although conversion does not hold for incomplete implication, contraposition does. As in Taḥlīl, in APCA Alfarabi also enlists the aid of the notion of the removal (irtifāʿ) as an alternative way of talking about necessary types of implication.

By an ad absurdum argument about man and animal, Alfarabi suggests, without formally proving it, that contraposition holds for incomplete implication, viz. X's incompletely implying Y entails that Y's not being incompletely implies that X is not (‘if X is, then Y is’ entails ‘if Y is not, then X is not’). It is perhaps worth dwelling on Alfarabi's proof of contraposition of incomplete implication at length. The proof runs as follows. Assume X incompletely implies Y, i.e. assume ‘if X is, then necessarily Y is’. Then, in order to generate a contradiction later, assume further that it is not the case that Y's not being incompletely implies X's not being, i.e. assume ‘not-(if Y is not, then necessarily X is not’). This step forces Alfarabi to give voice to his intuitions about what the contradictory of conditional is. According to Alfarabi, the conditional sentence ‘if animal is not, then human is not’ signifies a connection of incomplete necessity between the antecedent and the consequent, or, said differently, signifies that the antecedent incompletely implies the consequent. Thus, the meaning of this conditional is that it is impossible that human be and animal not be, or, informally, it is impossible for there to be something outside the soul called ‘human’ but not ‘animal’.Footnote ⁷¹ To contradict this statement is to say that it is possible at one and the same moment that animal is not but, somehow, human is. Informally, this would be to say that it is possible for there to be an object outside the soul that is simultaneously a human but not an animal. Thus, it seems that Alfarabi's intuition about the contradictory of a conditional is that it is in the form of a conjunction in which the leading conjunct is identical to the original antecedent and the final conjunct is the contradictory of the original consequent. And the conjunction itself signifies, as Alfarabi says, temporal coincidence. This conclusion, viz. that the contradiction of a conditional sentence in Alfarabi is formally a conjunction rather than another conditional as in the contradiction of indicative conditionals, is borne out by the rest of Alfarabi's “proof” of the validity of the rule of contraposition. The ad absurdum assumption allows Alfarabi to construct a conditional syllogism with (P1) the original assumption ‘if human is, then animal is’, and (P2) ‘human is’ which he obtains from the ad absurdum assumption that there is in fact something that is human.Footnote ⁷² P1 and P2 with modus ponens yields ‘animal is’, whereas the other conjunct ‘animal is not’ was assumed ad absurdum, yielding the desired contradiction. Alfarabi could not have reasoned in the way he does here in order to generate this ad absurdum argument if the contradiction of a conditional were another conditional. On the other hand, it is perfectly consistent with Alfarabi's way of talking here to say that, in general, the contradiction of a conditional ‘if P, then Q’ is the conjunction ‘P and not-Q’. Yet, the formal similarity of Alfarabi's rule of conditional contradiction with the negation of the material conditional should not make us lose sight of the fact that they could not be more different with respect to their semantics. The material conditional ‘if P, then Q’ is false just when its contradictory is true, viz. when P is true and Q is false. Alfarabi's conditional ‘if P, then Q’ is also false just in case its contradiction is true. But the contradiction of ‘if P, then Q’ will be true when it is possible that the state of affairs expressed by P coincide with absence of the state of affairs expressed by Q. In this case, the modality makes the current truth-value of P and Q irrelevant. It is the consistency of the state of affairs expressed by P with the absence of the state of affairs expressed by Q that is signified by the contradiction of Alfarabi's conditionals.

Having considered Alfarabi's comments in APCA, Ğadal, and Taḥlīl, we are now in a position to provide a reasonable conjecture about the truth-conditions for conditionals. As I discussed in Section 2, to say that a conditional is true is to say only that a reasoner says that the conditional is true. As we have noted throughout this paper, however, a reasoner harbours different criteria for calling a conditional true in different contexts. Thus, the onus for making suitable distinctions between the notions of truth in context falls on the strength with which the mind's assent attaches to the conditional in question. Broadly speaking, the context-sensitivity of the modality of the reasoner's mental assent to a conditional makes room for a variety of different attitudes that the reasoner can adopt towards it. For a theory of conditionals, this means that the ambiguity of the reasoner's assessment that a conditional is true is resolved by looking to the different strengths of the implication relation between antecedent and consequent. The weaker the connection between the antecedent and consequent, the weaker the mental assent to the conditional composed of them. In determining whether or not to give assent to a proposition, the reasoner looks to see if, and to what extent, the connection between antecedent and consequent correspond to what is the case. However, the current truth-values of the antecedent and consequent are not necessarily relevant to the reasoner's decision to call a conditional true or false. Rather, it is his observation of the variation in the frequency with which the connection between the antecedent and the consequent mirrors what is the case that accounts both for the variation of the strengths of implication in Alfarabi's conditionals, and for the variation in the grades of assent to those conditionals. As we saw at the beginning of this section Alfarabi does not feel the need to consistently use an overt linguistic marker that indicates the modality of implication between antecedent and consequent signified by the conditional expression. On the face of it, there is no overt linguistic indicator that when the reasoner says that ‘if human is, then animal is’ and ‘if it is December in Montreal, then it is cold out’ are true, he holds each conditional to different criteria.Footnote ⁷³ Yet, for the reasoner to say that a conditional is true means different things in different argumentative contexts. When the reasoner says that a conditional is true in a demonstrative argument, this means that he has observed that the consequent is always true given the truth of the antecedent. This means that there is necessary implication between the antecedent and consequent, and further, that the reasoner gives the strongest form of assent to the conditional in question. On the other hand, the reasoner also says that a conditional is true in a dialectical argument. But in this context, this should be taken to mean that he has observed that the consequent is almost always true given the truth of the antecedent. There is, thus, for-the-most-part implication between the antecedent and the consequent, and further, that the reasoner gives fairly strong assent to the conditional. And when he says a conditional is true in a rhetorical argument, this means only that he has observed that the consequent was true at least once when the antecedent was true. This also means there is a per accidens type of implication between the antecedent and consequent, and further, the reasoner gives the weakest form of assent to the conditional.

Truth-Conditions for Alfarabi's necessary, for-the-most-part, and per accidens conditionals Footnote ⁷⁴

It might seem amiss to claim that the sentence ‘if it is December in Montreal, then it is cold out’ is true with the explicit admission that the antecedent may very well be true and the consequent come out false. Yet, it is not amiss when we realize that the conditional is not true according to the conditions required for necessary conditionals, but is according to the conditions for ‘for-the-most-part’ conditionals. This fact is corroborated by the observation that ‘if it is December in Montreal, then it is cold out’ will likely draw the interlocutor's assent in a dialectical or rhetorical debate because it is rarely the case that the antecedent is true and the consequent false. On the other hand, since the conditional signifies a statistical rather than a sempiternal connection between two states of affairs, the conditional is false when read as a necessary conditional, and the connection indicated by the conditional is not likely to elicit assent from the respondent in the case that the interlocutors are engaged in a demonstrative argument. In short, a true conditional is a conditional that a reasoner calls true in a context. The reasoner calls a conditional true according to the observed frequency with which the consequent is true given the antecedent. Without inconsistency, he may call one and the same conditional true in one context and false in another. This ostensive inconsistency is resolved by looking to the degree of assent the reasoner gives to the consequent given the truth of the antecedent. He may call a conditional such as ‘if human is, then animal is’ and ‘if it is December in Montreal, then it is cold out’ true, but his assent to the former is strong, whereas his assent to the latter is significantly weaker. Alfarabi prefers to speak about the strength and weakness that inheres in propositions generally and conditionals in particular in terms of the notion of the ‘site of opposition (mawḍiʿ al-ʿinād)’ to the proposition, which may be characterized as the doubt or reservations that the reasoner has about the proposition in question. In the case of conditionals, the basis of this site of opposition will be the frequency with which the reasoner observes the consequent coming out false while the antecedent is true. In the case of necessary conditionals, this never happens so there is no site of opposition to such a conditional. In the case of rhetorical conditionals, the site of opposition is potentially much greater, since in a rhetorical argument the audience may be induced to say that a conditional is true based on a single instance in which the antecedent and consequent coincide.

5. CONDITIONAL PROPOSITIONS AND INFERENTIAL VALIDITY

Consistent with Alfarabi's lionization of demonstrative methods, Alfarabi is less concerned with per accidens implication and per se implication for-the-most-part than with necessary implication, whether complete or incomplete. The reason for this is that Alfarabi seems to believe that these types of following are not suited to carrying out deductions in which a true conclusion follows of necessity from a set of true premises. Yet, it is only natural for Alfarabi to entertain such a view of syllogisms given Aristotle's generic definition of the syllogism at the opening of the Prior Analytics, which states that a discourse is a syllogism if, inter alia, it is impossible that the conclusion be false given that the admission that the premises are true. But if this is so, then there is a problem. Consider an example of per se implication of the for-the-most-part variety in the following conditional: (P1) ‘if Sirius passes the zenith in the morning, then it will be hot’. For Alfarabi, this means that there is a ‘for-the-most-part’ implication relation between the sentences, ‘Sirius reached its zenith in the morning’ and ‘it will be hot’. An antecedent implies a consequent for-the-most-part if the connection between the state of affairs expressed by the antecedent and the consequent holds with law-like regularity. Observing this fact, a reasoner gives assent to the conditional when he sees that the consequent is true in most cases in which the antecedent is true. Say that in the course of an argument a respondent gives his assent to P1. This means that the respondent, seeing that it is hardly ever the case that Sirius reaches its zenith in the morning but the temperature remains mild, gives his assent to the conditional. Then, as it turns out the questioner and respondent observe that today is a day that Sirius reached its zenith in the morning, and thus the respondent also feels obliged to assert (P2) ‘but Sirius passes the zenith in the morning’. Now, Alfarabi classes modus ponens, along with modus tollens, as conditional syllogistic figures in Qiyās and Madḫal,Footnote ⁷⁵ and, thus, given that the respondent has conceded that both P1 and P2 are true, the canonical notion of syllogistic validity tells us that it should be impossible that the conclusion be false. If we take impossibility statistically, as Alfarabi normally does, the inferential validity of modus ponens requires that there never be a case in which P1 and P2 are true and the conclusion is false, i.e. there is never a situation in which Sirius passes the zenith in the morning and yet the day is rainy and mild. Yet, consider the truth conditions for for-the-most-part conditionals like P1 above. P1 is true if and only if it is statistically more frequent that Sirius reaches its zenith in the morning and it is a hot than it is that Sirius reaches its zenith and it mild or cold. So P1 will still be true and an interlocutor will still give his assent to P1 knowing that there are cases in which Sirius reaches its zenith in the morning and the weather remains cold, rainy or mild. Such a notion of implication makes room for the following scenario: Sirius reaches its zenith in the morning and the weather remains cold. Thus, P1 is true and the respondent will give his assent to it because its truth is based on statistical frequency and he will also be willing to give his assent to P2 since he observed Sirius reach its zenith in the morning. Yet the respondent will still not give his assent to the conclusion that it is a hot day, for the simple reason that it is not. So modus ponens with conditional premises read ‘for-the-most part’ (and, a fortiori, per accidens) is classically invalid.

What are we to make of this result? Deborah Black has observed that Alfarabi, like other classical Islamic philosophers, countenanced the use of fallacious inferences schemes such as ‘denying the antecedent’ (DA) and ‘affirming the consequent’ (AC) in rhetorical, and especially, enthymemic forms of reasoning.Footnote ⁷⁶ Thus, one approach might be to bite the bullet, so to speak, and admit that in arguments in rhetorical or dialectical contexts in which for-the-most-part conditionals are admissible, arguments in modus ponens are just as invalid as arguments in which we deny the antecedent or affirm the consequent. This approach would be consistent with the general tendency in Alfarabi to hold up demonstration as the genuine form of syllogistic, and to hold up the other species of syllogism as inferior. On this reading, rhetoric and dialectic are inferior to demonstration because they produce ‘variable opinions’ rather than certainty,Footnote ⁷⁷ admit premises of contingent rather than necessary matter,Footnote ⁷⁸ false premises rather than eternally true premises,Footnote ⁷⁹ premises derived authority and sense perception rather than from first principles,Footnote ⁸⁰ and now also because they admit the use of fallacies in syllogistic argumentation.Footnote ⁸¹ Yet, this would ignore important distinctions that Alfarabi explicitly makes in Kitāb al-Ḫaṭāba between syllogisms that are genuinely productive such as modus ponens and modus tollens, and those that are only ostensibly (fī al-ẓāhir) productive such as DA and AC. In reference to conditional enthymemic reasoning, Alfarabi notes that valid inferences and ostensibly valid inferences are rhetorically persuasive only if we suppress the asserted minor premise (mustaṯnā, lit. the “repeated” premise) in each.Footnote ⁸² Yet, the reason for withholding the asserted minor in each case is different. In the case of modus ponens and modus tollens, syllogisms which will be recognized as genuinely valid (ṣaḥīḥ) by the audience, the asserted minor is withheld. For example, the minor is suppressed in the case of modus ponens so the speaker avoids further questioning about the propriety of asserting the antecedent in the minor premise.Footnote ⁸³ The minor is withheld in the case of modus tollens, the validity of which will also be recognized by the audience, so that the locus of opposition (mawḍiʿ al-ʿinād) to the view being argued over, the persistence of which is necessary to the process of persuading one's audience, is not revealed.Footnote ⁸⁴ For if the locus of opposition to the viewpoint stands exposed, the audience will turn to view the viewpoint with a more critical eye, changing the modality of their assent to the opinion from persuasion to refutation.Footnote ⁸⁵ By contrast, Alfarabi says that the assertion of the minor premise must be suppressed in the case of fallacious arguments such as DA and AC because the audience will realize that the arguments are, in fact, formal fallacies – in Alfarabi's words, the particular combination of premises is ‘unsound’ or ‘corrupt’, sc. fāsid – and, as a consequence, the arguments will lose their persuasive power.

Thus, even in argumentative contexts in which the interlocutors are willing to admit weaker forms of conditionals such as per accidens and for-the-most-part, Alfarabi still holds that the audience distinguishes between formally valid syllogisms such as MP and MT, and invalid premise combinations such as AC and DA. And the audience makes this distinction despite the fact that the conclusions do not follow necessarily from the premises in any of these schema, whether in classically valid schema such as MP and MT, or classically invalid schema such as AC and DA, when weaker forms of conditional are used as major premises. If the primary aim of employing syllogisms in demonstrative contexts is the generation of a conclusion from the premise set necessarily, the same cannot be said for syllogisms in non-demonstrative contexts. In employing syllogisms in these latter types of context, the primary aim is eliciting assent or compliance in the audience's mind with the conclusion rather than generating a true conclusion of necessity. The notion of validity aptly characterizes demonstrative arguments to the extent that the concern in a valid argument is with the truth of the conclusion given the truth of the premises. In dialectical and rhetorical arguments on the other hand the speaker is concerned with eliciting assent to the conclusion given that the audience gives its assent to the premises. It seems more appropriate, then, to speak about the ‘success’ and ‘failure’ or the ‘felicity’ or ‘infelicity’ of a rhetorical or dialectical argument, to the extent that calling an argument ‘successful’ or a ‘failure’, ‘felicitous’ or ‘infelicitous’ makes it clear that the aim of these types of argument is inculcating mental compliance in the audience rather than generating a true conclusion of necessity from a pair of premises.

Consider the inference with the for-the-most-part conditional major premise (P1) ‘if it is December in Montreal, then it is cold out’. This conditional is true because it is more often the case that it is December in Montreal and it is cold than it is December in Montreal and it is warm or mild. This is so, despite the explicit admission that there exists at least one case in which it is December in Montreal and it is not cold. As we observed above, however, if we demand that an argument in MP, which uses P1 as a major premise, satisfy the requirement of classical validity, then this argument will be invalid because given a warm December day in Montreal, the speaker may get his audience to concede P1 and P2 ‘but it is now December in Montreal’, and the conclusion ‘it is cold’ may still be false. Yet, consider: there is a sense in which a mother who argues with her son about dressing for the cold during his visit to Montreal in December is making a good argument when she gets him to concede that, on the one hand, it is hardly ever the case that December in Montreal is warm, and, on the other hand, that he will soon be in Montreal in December. The fact that we feel, even if just intuitively, that this is a good argument is registered by our genuine surprise upon hearing that her son returned to inform her that he did not, in fact, need his jacket after all because the weather during his visit was unusually warm. Our reaction is not so much to say that her inference was invalid, but to remark at the exceptionality of Montreal's weather. Moreover, if her son were to be advised in the same fashion on another occasion, he would still likely be convinced to give his assent to the premises once more, since the connection between cold weather and Montreal in December still holds for the most part.

Alfarabi's discussion from Kitāb al-Ḫaṭāba suggests that we hold MP to be a good (or successful, or felicitous) argument, even if the argument is classically invalid because a for-the-most-part conditional is used as a major premise. It seems that at the heart of this issue is the requirement, as stipulated in Aristotle's generic definition of the syllogism in the Prior Analytics, that the connection between the premises and the conclusion is too stringent in all but the demonstrative syllogistic arts. This requirement brings schema like MP and MT to the same level as DA and AC, when it is intuitive in Alfarabi's eyes that arguments in the former are good and those in the latter are not. Speaking in reference to using formal fallacies such as AC in rhetorical debate, Deborah Black notes that despite the fact that rhetorical discourse often makes use of argument schema that ‘do not formally entail their conclusions,

It seems that an analogous situation should be entertained with respect to MP and MT. That is to say, despite the fact that the truth of premises in a syllogism with a for-the-most-part conditional as a major premise do not necessitate the truth of the conclusion, there is still a sense in which the assent to the for-the-most-part conditional followed by the assertion of the antecedent as a minor premise will elicit, but in a weaker way, assent to the consequent as a conclusion of the syllogism, even if the entailment is not of the strength required by Aristotle in his generic definition of the syllogism.

In short, the context theory forces on the logician an analysis of conditionals in which the connection between the antecedent and consequent varies according to the dialectical context in which the conditional proposition is used. In a demonstrative context, the connection between antecedent and consequent is necessary; in a dialectical it is for-the-most part; in a rhetorical it is per accidens. Each of these conditionals is true in suitable contexts in the sense that the one's interlocutor is willing to admit that the connection between the antecedent and consequent expressed by the conditional proposition corresponds to the actual state of affairs in the way required by dialectical context. Thus, an interlocutor will not give his assent to a for-the-most-part conditional in a demonstrative context because the connection between the antecedent and consequent in such a context must be sempiternal. On the other hand, he will give his assent to a for-the-most-part conditional in a dialectical or rhetorical exchange where, for example, premises that the reasoner knows may be false elicit assent from the listener nevertheless.Footnote ⁸⁸ Yet, our discussion above shows that not only does the context demand different grades of assent in relation to the actual state of affairs, but also a sensitivity to the dialectical context demands variable degrees in the strength of the connection between the premises and conclusion of the conditional syllogism. In other words, it requires the admission that dialectical contexts require different grades of syllogistic validity. MP is syllogistically (and classically) invalid with a for-the-most-part conditional because it may happen that it is admitted that the premises are all true and the conclusion still happens to come out false. However, it seems inappropriate to require a single, monolithic criterion of syllogistic validity in all dialectical contexts, as required by many contemporary formal accounts of logical validity. Ancient philosophers – and the classical Islamic philosophers are not exceptional in this regard – proved to be extremely adept at adapting the notion of the syllogism as developed in the Prior Analytics to the demands of the different species of syllogism that they took Aristotle to be developing in the five syllogistic arts of the Organon. I suggest, rather, that a conditional syllogism in MP executed in a dialectical or rhetorical context with a for-the-most-part conditional major entails its conclusion, but in a weaker sense than in a demonstrative argumentative context, for example. Alfarabi considers just such a possibility in Kitāb al-Ḫaṭāba. Once again returning to the question of the problem of how to distinguish between the five syllogistic arts in the face of their underlying syllogistic unity, Alfarabi moves to distinguish between unqualified (or ‘simpliciter’, muṭlaq) and qualified senses of the term ‘syllogism’ based this time on how strongly (or weakly) the premises entail the conclusion. In the face of claims that a wide variety of clearly distinct argument forms, such as enthymemes, induction, and analogy, can somehow all still meaningfully be grouped under the name ‘syllogism’,

Alfarabi observes that ‘syllogism’, taken in its most generic sense, means a set of premises that necessarily entail the conclusion. In terms of conditional syllogisms, this means that ‘conditional syllogism’ in an unqualified sense denotes a premise set composed of a conditional major premise and another, non-conditional minor premise, which, if asserted, necessitate the assertion of the conclusion. This generic sense is made to align exactly with the sense in which the premises in a demonstrative syllogism must entail their conclusion, namely, of necessity. In the case of a demonstrative syllogism, in order to ensure that the premises necessitate the conclusion, the connection between the antecedent and the consequent of the conditional major premise must be necessary too, whether of the complete or the incomplete variety. Thus, in the case of demonstrative syllogisms, the strength of the connection between the antecedent and consequent indexes the strength of the premises' entailment of the conclusion. Said differently, the strength of the following of the conclusion from the premises is limited by the strength of the connection between the antecedent and consequent of the conditional major premise. This observation may be generalized to the two other types of conditional discussed by Alfarabi in APCA. In a conditional syllogism with a for-the-most-part conditional major premise, the frequency with which the connection between the antecedent and consequent expressed by the conditional proposition corresponds to the actual state of affairs is the frequency with which the premises and the conclusion will be true together. In a conditional syllogism with a per accidens conditional major premise, the frequency with which the conclusion will be true given that the premises are true is just the frequency with which the consequent of the per accidens conditional is true given that the antecedent of the per accidens conditional is true.

In sum, for each argumentative context, viz. in each argument in which the interlocutors agree to use syllogisms that will elicit only a certain degree of assent, the context theory of logic assigns an appropriate syllogistic art. Alfarabi distinguishes between these syllogistic arts in various ways. At times he recognizes that the distinction between them is at the level of the premises' truth values; at other times he distinguishes between them at the level of the modality of their premises; at other times, he seems inclined to distinguish them, like Avicenna would after him, with the rank of assent that the premises and conclusion elicit from the listener. The result of the above discussion is to realize that he also distinguishes between the different syllogistic arts – regardless of whether the syllogism deployed in the argument is categorical, conditional, or per impossibile – according to the way in which the premises entail the conclusion. Depending on the pragmatic conditions under which the argument takes place, not only will the interlocutors come to an implicit or explicit agreement about the level of assent they require from their opponent's premises, but they will also expect or even explicitly stipulate the suitable manner in which the conclusion follows from the premises vis-à-vis the context; in other words, they will implicitly expect or explicitly stipulate a suitable notion of validity vis-à-vis the context. Indeed, this result should strike us as intuitive: of course, we do not expect the premises of arguments in historical disciplines to entail their conclusions in the same way that premises in, say, mathematics.

Conditional syllogisms represent a special case of this general rule. In the case of conditional syllogisms such as MP and MT, the variability in the strength of the premises' entailment of the conclusion is entirely determined by the strength of the implication relation between the antecedent and the consequent. The stronger the connection between antecedent and consequent, the more circumscribed the locus of opposition to assent to the conclusion given truth of the premises. In a demonstrative syllogism, there is no locus of opposition to the connection between the antecedent and consequent since the connection is sempiternal, and as a consequence, there is strong assent to the conclusion given the premises. In rhetorical arguments, the locus of opposition to the connection between the antecedent and consequent is greater due to the weakness of the per accidens type of implication. And, as a consequence, the strength of assent to the conclusion given the truth of the antecedent is greatly diminished as well. Our degree of assent to a conclusion ‘so it is not a triangle’ is very strong given that we give our assent to ‘if this figure is a triangle, then it has only three sides’ and ‘but it does not have only three sides’. But this is not chiefly due to the strength of our assent to the minor premise. Rather, we are willing to give a high rank of assent to the conclusion because we recognize that it is quite impossible for the consequent of the conditional to be false and the antecedent of the conditional to be true. On the other hand, if we are arguing about weather patterns, then we are strongly inclined to give our assent to the conclusion ‘so it is not December in Montreal’ given that we give our assent to the conditional major premise ‘if it is December in Montreal, then it is cold’ and the minor premise ‘but it is not cold’. The locus of opposition to the conclusion, viz. the knowledge that the conclusion may, in fact be false, is magnified in this argument by our knowledge that there are instances when Montreal is warm in December, though rarely. Yet, in spite of this locus of opposition, the conclusion elicits our assent. A context theory-based concept of validity must take into account the fact that, unconsciously or consciously, two interlocutors will adopt different validity criteria based on the pragmatic conditions under which their argument takes place.

With these considerations in mind, call a syllogistic argument in a context in which the interlocutors agree, explicitly or implicitly, to require that the conclusion follows necessarily from the premises, an argument in a demonstrative context. Call a syllogistic argument in a context in which the interlocutors agree, explicitly or implicitly, to require that the conclusion follow from the premises for the most part, an argument in a dialectical context. Finally, call a syllogistic argument in a context in which the interlocutors agree, explicitly or implicitly, to require that the conclusion follow from the premises per accidens, an argument in a rhetorical context. Since the aim of arguments executed in demonstrative contexts is generating a true conclusion from true premises, say that a conditional syllogism in a demonstrative context is valid when it is never the case that the conclusion is false and the premises are true; it is invalid if there is such a case. On the other hand, call a conditional syllogism in a dialectical context valid (in the sense of ‘successful’ or ‘felicitous’ discussed above) when it is hardly ever the case that the audience fails to give assent to the conclusion though it does give its assent to the premises; it is invalid (in the sense of ‘infelicitous’, ‘fails’) if it is often the case that the audience fails to give its assent to the conclusion but does give its assent to the premises. For similar reasons, call a conditional syllogism in a rhetorical context valid when there is at least one instance in which the audience gives its assent to the conclusion and the premises together; it is invalid in the case that there is never an instance in which it gives its assent to both together.Footnote ⁹¹

The crucial test for this view of syllogistic validity for conditional syllogisms is whether it can make meaningful distinctions between valid inference schema such as MT and MP and fallacious ones such as DA and AC. Maintaining this distinction becomes particularly important in the case of arguments with for-the-most-part and per accidens conditional major premises, since in Text 14 Alfarabi is so emphatic in holding that, for example, MT is, as he says, ‘genuinely productive (fī al-ḥaqīqa)’, whereas DA is only ostensibly productive (fī al-ẓāhir).

In order for a conditional syllogism with for-the-most-part conditionals to be valid in a dialectic context, it must be hardly ever the case that the conclusion is false while the premises are true. Take the usual example of a true for-the-most-part conditional as the major premise ‘if it is December in Montreal, then it is cold out’ and let the minor premise be ‘but, it is not cold out’. This premise pair yields the conclusion by MT ‘then it is not December in Montreal’. This inference is valid in a dialectic context because it is hardly ever the case that we will deny the conclusion, viz. we affirm that it is December in Montreal, while also giving assent to the premises, viz. we give assent to the fact that it is warm or mild out (minor premise) and that it is hardly ever the case that it is warm or mild out and December in Montreal (conditional major premise). Now consider the fallacy DA with the conditional major ‘if it is December in Montreal, then it is cold out’ and the minor premise ‘but it is not December in Montreal’. The purported conclusion is ‘then it is not cold out’. In order for this inference to be valid it would have to be hardly ever the case that we do not give assent to the conclusion in spite of our giving assent to the premises. Assent to the conditional premise is assent to the fact that it rarely happens that it is warm out and it is December, as well as assent to the fact that there are, nevertheless, instances in which it is warm out and it is December in Montreal. What sort of assent can the assertion of the minor premise ‘but it is not December in Montreal’ in combination with the conditional major elicit? In fact, the minor premise is irrelevant to the information provided by the conditional: the conditional tells us only about the connection between the weather in Montreal in December, and nothing about the weather in any other time of the year. Thus, it may very often happen that the conclusion ‘then it is not cold out’ is false while the premises are true. DA is invalid. Consider AC, the other common fallacy, with the same conditional major ‘if it is December in Montreal, then it is cold out’ and the minor premise ‘but it is cold out’. The purported conclusion is ‘then it is December in Montreal’. Once again, in order to be valid, it must be hardly ever the case that we deny the conclusion, viz. we deny that it is December in Montreal, but we nevertheless give assent to the pair of premises, viz. we give assent to the fact that it is, in fact, cold out, and that it sometimes happens, though rarely, that it is warm or mild and it is December in Montreal. What sort of assent does the minor premise in combination with the conditional major elcit vis-à-vis the conclusion ‘it is December in Montreal’? The conditional tells us that it is cold given that it is December in Montreal. In other words, it points out only a single condition, from all of the countless conditions, under which the statement ‘it is cold out’ is true. Thus, even if the conclusion ‘then it is December in Montreal’ is not ruled out by the combination of premises, more often than not the conclusion will be false due to the large number of other conditions under which ‘it is cold out’ is true. AC is invalid.