## 1. Avicenna's Logical Works and His Sources



Avicenna's logical works consist of a significant amount of
material unevenly distributed across his various philosophical
*summae*. The most comprehensive treatment of logic is, by far,
the one to be found in the monumental treatment of the
*Sifa`* (*The Cure*), from the 1020s,
where Avicenna extensively engages in systematic analysis as well as
interpretive debate with the Peripatetic tradition. At the opposite
end of the spectrum is the *Isarat
wa-tanbihat* (*Pointers and Reminders*), the
shortest and likely the latest (from the 1030s), where Avicenna
organizes the material in an original way that came to shape the
approach to logic in the Post-Avicennan tradition. In between (not
chronologically, as the work largely reflects Avicenna's early
views from around 1014) lies the account of logic developed in the
*Nagat* (*The Salvation*). While offering a
much abridged treatment (and leaving out dialectic, rhetoric, and
poetics altogether), the latter is methodologically closer to the
*Cure* than it is to *Pointers and Reminders* but far
less ponderous, and offers a good point of access to Avicenna's
logic as a whole.



In light of their relative position in the corpus, the balance of
detail and perspicuity, and--in the case of the *Cure* and
*Pointers*--the relative influence they exerted on the
later tradition, I have chosen in this entry to focus almost
exclusively on these three sources, which also happen to be the ones
that are in better shape in terms of available editions.



Other treatments of logic include the account found in the *Danesname-ye 'Ala`i* (*Philosophy for 'Ala` ad-Dawla*),
in Persian, or the one preserved by the partially extant *al-Masriqiyyun* (*The Easterners*), which reflects an approach no
longer committed to the terminology and exegetical problems of the
Aristotelian tradition, written as it is in opposition to the methods
and doctrines of (Western) Bagdadi Aristotelians (Street 2004).



Finally, interesting insights on certain areas of logic may also be gleaned from additional heterogeneous sources, such as Avicenna's correspondence
(Street 2010a) and texts not directly dealing with
logic, most notably the *Ilahiyyat* (especially books
I and V), which is his major work in metaphysics.



Avicenna's logic is indebted first and foremost to
Aristotle--the First Teacher (*al-mu'allim al-awwal*)--and
the works of the *Organon*: the *Categories*
(*Maqulat*), *On Interpretation*
(*'Ibara*), the *Prior Analytics*
(*Anulutiqa ula* or
*Qiyas*), the *Posterior Analytics*
(*Anulutiqa taniya* or
*Burhan*), the *Topics*
(*Tubiqa* or *Gadal*), and the
*Sophistical Refutations* (*Sufistiqa* or *Mugalata*). As noted above, the Arabic
*Organon* included Porphyry's *Isagoge*
(*Isagugi* or
*Madhal*) as well as Aristotle's *Rhetoric*
(*Hitaba*) and *Poetics*
(*Si'r*).



Besides Aristotle, among the figures we encounter in Avicenna's
logical writings--whether identifiable by name, definite
description, or doctrine--are Plato, Theophrastus, Alexander of
Aphrodisias, Galen, Porphyry, Themistius, and Philoponus from the
Greek tradition; al-Farabi and other
Bagdadi Aristotelians from the Arabic tradition (all
variously involved in exegetical efforts focusing on the books of the
*Organon*, traces of which survive in the glosses to the
famous manuscript Paris BNF, *Arabe* 2346 that preserves
its Arabic translation). Their presence varies from the explicit ascription
of alternative interpretations (e.g., in the case of the
interpretation of the absolute proposition) to the rejection of
competing accounts (the contrast between "the widespread
view" (*al-mashur*) and "the
truth" (*al-haqq*) is a standard device of
Avicenna's style of argument); from the tacit adoption of
systematic distinctions and concepts (especially with regard to
al-Farabi, who is rarely acknowledged) to minute
exegetical points.



In the following, I refer to Avicenna's works by title, book, chapter, page, and line numbers from the Cairo edition for all
works that are part of the *Sifa`*; from Forget's edition of the
*Isarat wa-tanbihat*, and from Danespazuh's
edition of the *Nagat*.


## 2. Definition and Subject Matter of Logic



A standard feature of late ancient Greek commentaries on Aristotle's individual works
is to begin with *prolegomena* dealing with the purpose and
benefit of a certain discipline, determining its subject matter and
rank within the system. Similar issues influence Avicenna's
approach to the question of the nature of logic and what it aims to
accomplish as a discipline.



Logic is a theoretical discipline that provides the tools and methods
required to avoid error in reasoning, where reasoning means acquiring
new knowledge (conceptual or propositional) of what is unknown (*maghul*)
starting from what is known (*ma'lum*), and to attain the optimal epistemic
states legitimately associated with various domains of human
cognition, most notably certainty (*yaqin*) in scientific
discourse (*'ilm*) and persuasion
(*iqna'*) in dialectic and rhetorical contexts.



Accordingly, logic has a twofold nature: it is at the same time (i) a
normative (literally "canonical") instrument, according to
the definition in *Pointers and Reminders*, with respect to any other
discipline, and (ii) a part of philosophy, i.e., an Aristotelian
science with its own internal structure, which presupposes a
well-defined subject.



If the purpose and benefit of logic is to lead from what is known to what is unknown, for example from premises to conclusion
in a valid argument whose premises are asserted, Avicenna's
account of the subject (*mawdu'*) of logic is
inextricably tied with a fundamental distinction in Arabic logic,
which can be traced back at least to al-Farabi, between
the two main ways in which knowledge is available to the human mind:
conception (*tasawwur*) and assertion
(*tasdiq*).



The distinction between conception and assertion marks the boundary
between the acquisition or possession of individual concepts or
notions, and the ascription of truth to statements or judgments. It
provides the foundations of Avicenna's logic (as well as those
of his epistemology) and determines two distinct modes of knowledge
linked on the one hand with (i) concept formation and the domain of
definition (*hadd*) and description (*rasm*), and
on the other with (ii) truth and reasoning (demonstrative and
non-demonstrative syllogism (*qiyas*), depending on the
epistemic strength of the assertions involved).



The ultimate building-blocks of definitions (directly) and of syllogisms
(indirectly) are intelligible notions (*ma'anin
ma'qula*). Logic deals with a special kind of
intelligible notions, which in the *Ilahiyyat* are
characterized as second-order intelligible notions
(*ma'qulat taniya*)
(*Ilahiyyat* I.2, 10-11). Assuming that
first-order notions are things like "human",
"rational", "whiteness", or "stone",
second-order (or more generally, higher-order) notions are either
genuinely distinct concepts such as "subject",
"predicate", "genus",
"proposition", and "syllogism" or, on a weaker
reading, ways to ascribe second-order properties (such as being a
universal or being a genus and so on) to first-order notions. Logic
investigates the salient properties that hold of second-order notions,
such as the properties of being a genus, the characteristics of
propositions and their terms, the types of predication
(essential and accidental), the truth-conditions of propositions, the
conditions under which arguments are valid and those under which
arguments that seem to be valid are not. On the identification of
secondary intelligibles and the subject matter of logic, see (Street 2013) and (Sabra 1980).



Furthermore, logic entertains two levels of analysis, addressing
distinct sets of considerations that are commonly referred to as
formal and material. Formal characteristics are the quantity and
quality of a proposition or the arrangement of terms in an argument,
and in general everything that is necessary and sufficient to proceed
without error in reasoning.



Material aspects of logic involve considerations relative to the types of terms
used in propositions and arguments and the relations holding among the
terms. Thus, for instance, the question whether an argument is
demonstrative or not is typically a question that concerns not only
its form (a demonstrative argument must be valid and consist of well-formed
propositions), but also its matter, namely
whether its terms adequately express the right kinds of necessary and
explanatory relations.



The question what the subject matter of logic is became pressing in later
Arabic logic, as Avicenna's explicit remarks in the
*Ilahiyyat* were taken to identify too narrow a
subject, and conceptions and assertions gained traction as a
plausible, more general alternative to second-order notions
(El-Rouayheb 2012).



On the relation between logic as an instrument and logic as a science,
see *Qiyas* I.2, *Madhal* I.3 and for a
general discussion (Sabra 1980).


## 3. Propositions



Avicenna recognizes two basic types of proposition
(*qadiyya*): (1) categoricals
(*hamliyyat*), atomic subject-predicate
propositions, qualified in various ways; and (2) hypotheticals
(*sartiyyat*), molecular propositions
governed by a main connective, which may express a conditional
(*muttasil*) or a disjunctive (*munfasil*)
statement taking as its parts categoricals, hypotheticals, or a
combination thereof.


### 3.1 Categorical Propositions



Categorical propositions are subject
(*mawdu'*)-predicate
(*mahmul*) propositions expressing a relation
(*nisba*) or judgment (*hukm*) between terms. They
are usually classified according to a number of distinctions and
readings.


1. The first layer of analysis involves their quality
(*kayfiyya*) and quantity (*kammiyya*). An Avicennan
categorical is either (a) determinate (*muhassala*), which
means that it is in turn either a quantified (*mahsura*) or a
singular (*sahsiyya*) proposition (that is to say an a-, e-, i-, or
o-proposition or one which takes a singular term as its subject) or
(b) indeterminate (*muhmala*), when it is neither quantified nor
singular. If a proposition contains at least one negative term it is
called metathetic (*ma'dula*).
2. The second layer of analysis involves a consideration of the
logical form of a categorical proposition as opposed to its surface
expression. Avicenna introduces a distinction between the bipartite
(*tuna`iyya*), the tripartite
(*tulatiyya*), and the quadripartite
(*ruba'iyya*) proposition. In the first case the
copula is not explicit, which is grammatical in Arabic as
in certain cases the juxtaposition of two terms may be sufficient to
yield a predicative statement (A B); in the second case the copula is
explicit but the proposition is not explicitly modalized (A is B); in
the third case the copula is explicit and the proposition is
explicitly modalized (A is necessarily B or A is possibly
 B).[2]
3. The third layer of analysis involves modality, whether implicitly
or explicitly. A categorical proposition is either unqualified
(*mutlaqa*, commonly translated as "absolute") or qualified by a specific modality
(*munawwa'a*). The latter is usually distinguished into
three main types: the necessary, the possible, and the impossible
proposition.



This yields an eight-fold categorization occasionally advocated by
Avicenna himself when he says, for example in the treatment of
hypothetical propositions, that the categoricals embedded in
them may be one of the above eight (*Qiyas* VII.1, 361.1-362.4).


#### 3.1.1 Modality and reading: referential/substantial and descriptional



Avicenna may be credited with one of the most significant reforms in
the analysis of propositional forms in the Peripatetic tradition. His
new catalog of subject-predicate propositions, their relations of
contradiction, subalternation, and conversion, became the center of
gravity of an established tradition of logic.



Avicenna introduces two radical innovations in the Aristotelian
analysis of propositions.


1. *Temporal and alethic modalities:* every categorical
proposition is modalized, either implicitly or explicitly. The
modality may be either temporal (e.g., sometime, always), alethic
(e.g., necessarily, possibly), or a combination of both. This has
significant repercussions on the traditional Aristotelian square of
opposition, which Avicenna rejects as we shall see below, and on the
account of conversion.
2. *Referential/substantial and descriptional reading:* every
categorical proposition is subject to an additional reading, depending
on whether the proposition is taken to express a relation between the
predicate and what is picked out by the subject, (a) as long as what
is picked out by the subject exists (*ma dama
mawgud ad-dat*) or (b) as long as it is
qualified--or "described" (*ma dama
mawsuf*)--by the subject. This move amounts to
adding a temporal parameter that identifies two distinct time frames:
the continued existence of the item picked out by the subject term
(where the subject term is used, as it were, just to fix the
referent), and the time at which such an item is actually qualified by
the subject term. The propositions corresponding to the two readings
are canonically called in the post-Avicennan tradition
"referential/substantial"
 (*dati*)
 and "descriptional" (*wasfi*).[3]



The two parameters jointly yield a variety of types of modalized
propositions, a canonical list of which becomes standard in the
thirteenth century. Below are some notable items that can be found in
various forms in Avicenna's works and which crystallized starting
with Razi (Street 2013; cf. Strobino & Thom
 2016).[4]
 The deeper syntax of these propositions is a subject of dispute
(Hodges 2015). 



Temporal 


* A
referential perpetuity (*da`ima*)
* X1
referential one-sided absoluteness
(*mutlaqa 'amma*)
* X2
referential two-sided absoluteness
(*mutlaqa hassa, wugudiyya la da`ima*)
* Ad1
descriptional unrestricted perpetuity (*'urfiyya
'amma*)
* Xd1
descriptional one-sided absoluteness (*hiniyya mutlaqa*)


Alethic


* Lreferential
necessity
(*daruriyya*)
* M1referential
one-sided possibility (*mumkina 'amma*)
* M2referential
two-sided possibility (*mumkina
hassa*)
* Ld1descriptional
unrestricted necessity
(*masruta 'amma*)
* Md1descriptional
one-sided possibility (*hiniyya
mumkina*)


Mixed Alethic and Temporal


* Treferential
temporal determinate
(*waqtiyya*)
* Ureferential
temporal indeterminate
(*muntasira*)
* X3referential
non-necessary absoluteness (*wugudiyya la
daruriyya*)
* Ld2descriptional
restricted necessity
(*masruta
hassa*)
* Ad2descriptional
restricted perpetuity (*'urfiyya
hassa*)




For a full list of instances of a-, e-, i-, and o-propositions see
 appendix A.


### 3.2 Hypothetical Propositions



Hypotheticals comprise two main sub-types, depending on whether the
component sentences are in connection (*ittisal*) or
in conflict (*'inad*). In the first case, the
connection is analyzed in terms of a relation of following
(*ittiba'*) between an antecedent (*muqaddam*)
and a consequent
 (*talin*).[5]
 In the second case, conflict is expressed in terms of a disjunction
of propositions (or parts, *agza`*).



The resulting propositional types are conditionals
(*muttasilat*) and disjunctions
(*munfasilat*). Avicenna investigates them both with
respect to basic logical relations--such as contradiction,
subalternation, mutual implication, or conversion--and within the context of
syllogistic inferences.



A distinctive feature of Avicenna's analysis of hypothetical
propositions is quantification. The conditional relation between
antecedent and consequent as well as the relation of conflict between
the parts of a disjunction are embedded within the scope of a
quantifier which may express a universal affirmative, a universal
negative, a particular affirmative, or a particular negative
hypothetical proposition.



The question with respect to *what* conditional and disjunctive
statements are quantified may elicit different answers. In the case of
conditional statements, Avicenna indicates that the quantification
should be understood to apply to the times at which the antecedent is
posited and the states or conditions (in short, additional
assumptions) that can be associated with it. The problem is addressed
for example by (Rescher 1963b), (Movahed 2009), and (Hodges 2013). A
universal affirmative conditional statement is true when the
consequent follows the antecedent at all times the antecedent is
posited and with respect to all states that may legitimately be added
to the
 antecedent:[6]




> 
> 
> If the connection (*ittisal*) is asserted under any
> condition and circumstance (*'ala kull
> istirat wa-wad'*) [at which the
> antecedent] is posited, then the hypothetical conditional proposition
> is universal; (*Qiyas* V.4, 263.1-264.2) 
> 
> 
> 
> In the statement "Always, if C is B, then H is Z" the
> words "always, if" are not only meant to cover what is
> [explicitly] intended (*al-murad*) [by this proposition],
> in such a way that it is as if one were to say "Every time at
> which C is B, H is Z". Rather, [the words "always,
> if"] are also meant to cover every state
> (*hal*) which may be added to the statement
> "Every C is B", in such a way that there is no state
> (*hal*) or condition (*sart*)
> whatsoever that makes C to be B without making H to be Z, when it is
> added to [the antecedent "C is B"]. (*Qiyas*, V.4, 265.1-5)[7]
> 
> 
> 



A classification of hypothetical propositions is given at the
beginning of *Qiyas* V.3, where Avicenna explicitly lists
a first set of permutations and illustrates them with examples. There
seems to be no principled reason to think that Avicenna would not have
entertained the possibility of more complex iterations (where, for
instance, the antecedent of a conditional is itself a conditional
whose antecedent is a disjunction of conditionals whose consequents
are conditionals consisting of categoricals). A hint in this direction
is that he is prepared to entertain the notion of potentially infinite
disjunctions.


#### 3.2.1 Types of hypotheticals



Avicenna considers the following types of hypothetical propositions in
*Qiyas* V.3:





|  |  |  |  |
| --- | --- | --- | --- |
|  |  | Conditional  | Disjunctive  |
| (i) | cat-cat  | [categorical] - [categorical]  | [categorical] [?] [categorical]  |
| (ii) | c-c | (...- ...) - (... - ...) | ( ... - ...) [?] ( ... - ...)
 |
| (iii)  | d-d  | (... [?] ...) - (... [?] ...)  | ( ... [?] ...) [?] ( ... [?] ...)  |
| (iv)  | c-d  | (... - ...) - (... [?] ...)  | (... - ...) [?] (... [?]
...)  |
| (... [?] ...)
- (... - ...)  |
| (v)  | cat-c  | [categorical] - (... - ...)  | (... - ...) [?] [categorical]  |
| (... - ...)
- [categorical]  |
| (vi)  | cat-d  | [categorical] - (... [?] ...)  | (... [?] ...) [?] [categorical]  |
| (... [?] ...)
- [categorical]  |




The first type comprises the most basic kind of hypothetical
propositions, which take categorical propositions as their parts. The
hypotheticals discussed in the development of the hypothetical
syllogistic in *Qiyas* VI are of type (i) only. That is to
say, the premise-pairs of hypothetical syllogisms are either type-(i)
propositions (conditional-conditional; conditional-disjunctive;
disjunctive-disjunctive); or combinations of type (i) propositions
with categoricals (conditional-categorical; disjunctive-categorical).
The former class comprises purely hypothetical syllogisms, the latter
mixed hypothetical syllogisms.



Avicenna illustrates the above types by means of the following
examples (*Qiyas* V.3, 253.1-255.2):





|  |  |  |  |
| --- | --- | --- | --- |
|  |  | Conditional  | Disjunctive  |
| (i)  | cat-cat | **If** [the sun rises], **then** [it
is day]  | **Either** [this number is even]
**or** [<this number> is odd]  |
| (ii)  | c-c  | **If** (always if [it is day], then [the sun has
risen]), **then** (always if [it is night], then [the sun
has set])  | **Either** (always if [the sun rises], then [it is
day]) **or** (sometimes if [the sun rises], then [it is
not day])  |
| (iii)  | d-d  | **If** (either [the body is at rest] or [<the
body is> in motion]), **then** (either [some substances
are at rest] or [<they are> in motion])  | **Either** (either [this fever is yellowish] or
[<this fever is> scarlet]) **or** (either [this
fever is phlegmish] or [<this fever is> melancholic])  |
| (iv)  | c-d  | **If** (always if [the sun rises], then [it is
day]), **then** (either [it is day] or [the sun has not
risen])  | **Either** (if [the sun rises], then
[it is day]) **or** (either [the sun rises] or [it is
day])  |
| **If** (either
[this number is even] or [<this number> is odd]),
**then** (if [<this number> is even] then [<this
number> is not odd])  |
| (v)  | cat-c  | **If** [the sun is the cause of day],
**then** (always if [it is day], then [the sun has
risen])  | **Either** (always if [it is day], then
[the sun has risen]) **or** [the sun is not the cause of
day]  |
| **If** (always
if [it is day], then [the sun has risen]), **then** [the
sun is the cause of day]  |
| (vi)  | cat-d  | **If** [this is a number], **then**
(either [this is even] or [<this is> odd])  | **Either** (either [this is even] or
[<this is> odd]) **or** [<this> is not a
number]  |
| **If** (either
[this is even] or [<this is> odd]), **then** [this
is a number]  |



#### 3.2.2 Types of following



Conditionals express relations of following between an antecedent and
a consequent. Complete conditionals are bi-conditionals.



Avicenna distinguishes between two types of conditionals: implicative
(*luzumi*) and coincidental
(*ittifaqi*). While the former express
necessary relations of following whereby the consequent follows
because or in virtue of the antecedent, the latter have often been
thought to express an accidental connection, though it seems plausible
that Avicenna might simply mean agreement or coincidence
in truth or in being the
 case together (*ma'iyya*).[8]
 When a conditional expresses an implication (*luzum*), it
is called real (*haqiqi* or *'ala
t-tahqiq*); when it expresses coincidence
(*ittifaq*, *muwafaqa* or
*tawafuq*), it is called unqualified (*mutlaq*
or *'ala l-itlaq*). For a characterization
of conditionals see (Karimullah 2014a).


#### 3.2.3 Types of conflict



Avicenna distinguishes among three types of disjunctive statements:
(i) exclusive and exhaustive; (ii) exclusive non-exhaustive; (iii)
inclusive. Put simply, their characteristic properties may be expressed in terms of the
following three relations:


1. \(p \lor^{1} q\) with \(p \leftrightarrow \neg q\)
2. \(p \lor^{2} q\) with \(p \rightarrow \neg q\)
3. \(p \lor^{3} q\) with \(\neg p \rightarrow q\)



Avicenna refers to the first type as the real (or genuine) disjunction
(*munfasila haqiqiyya*) and to the other two
types as unreal (*gayr haqiqi*)
disjunctions. The second and third type were canonically referred to
by Post-Avicennan logicians as the *mani'at
al-gam'* disjunction (preventing the joint affirmation
of the two disjuncts, i.e., compatible with their both being false
together but incompatible with their both being true together) and the
*mani'at al-huluww* disjunction (preventing the
joint negation of the two disjuncts, i.e., compatible with their both
being true together but incompatible with their both being false
together) (El-Rouayheb 2010).



Negative disjunctions express absence of conflict or
incompatibility.


#### 3.2.4 Truth-conditions for conditionals



Implicative conditionals are true (1.1) when the antecedent and the
consequent are both true or (1.2) when the antecedent is false and the
consequent is true or (1.3) when the antecedent and the consequent are
both false, *and* (2) there is a connection of meaning between the two
such that the consequent is inseparable from the antecedent.



Coincidental conditionals are true when the antecedent and the
consequent are both true or when the antecedent is false and the
consequent is true (but false when both the antecedent and the
consequent are false).



Both conditionals are false when the antecedent is true and the
consequent is false. Implicative conditionals with true antecedents
and consequents are also true as coincidental. The question how the
two are related more generally and without such restrictions is in
need of further investigation.



Negative universal (particular) conditionals express the fact the consequent does not necessarily follow from the antecedent (*luzumi*) or is not true together with the antecedent (*ittifaqi*) at any (some) times and under any (some) circumstances. They are equivalent to affirmative conditionals of the
same quantity with a contradictory consequent and express the dual
relation of dependency (or togetherness) of the contradictory of the consequent with respect to the
antecedent at all (some) times and under all (some) circumstances. For
example, a universal negative conditional of the form "Never, if
A is B, then C is D" is logically equivalent to "Always,
if A is B, then C is not D". Negative conditional statements may
express absence of a *luzumi* connection (as in
"Never, if man exists, then void does not exist") or
absence of an *ittifaqi* connection (as in
"Never, if man exists, then void exists")
(*Qiyas* V.5, 279.1-283.9).


#### 3.2.5 Impossible antecedents



Avicenna is keenly interested in conditionals with impossible
antecedents (sophisticated discussions are to be found in his account
of a- and e-conditionals and in connection with *reductio*
arguments) but only in the sense in which an impossibility may genuinely be derived from another impossibility, as it were, out of the meaning of the terms involved,
not in the sense in which an arbitrary consequent may trivially be inferred from
an impossible antecedent. The problem of explosion is recognized and
rejected in post-Avicennan logic, starting with
Hunagi (d. 1248 CE). It seems reasonable to assume, even in
the current state of our fragmentary knowledge of such a vast amount of material,
that both Avicenna's logic and post-Avicennan logic may
generally be committed to an idea of logical consequence that could perhaps
best be captured, in modern terms, by the intuition behind a
relevantist approach to the notion of following. This is
a feature that the Arabic tradition would share with certain strands
of medieval Latin logic. In discussing the problem of conditionals
with impossible antecedents in the post-Avicennan tradition
(El-Rouayheb 2009 and 2010: xxxiii-xl) argues for an even stronger
interpretation in terms of connexive
 logic.[9]


## 4. Contradiction and Conversion



Contradiction (*tanaqud*) and conversion
(*'aks*) are fundamental relations in Avicenna's
logic. The principles that govern them are central in the development
of the syllogistic, both categorical and hypothetical. It is not
surprising, in light of the complexity of Avicenna's analysis of
categorical propositions and of his elaborate account of quantified
hypotheticals, that his answers to questions such as what contradicts
what, what follows from what, and what converts with what all
came under close scrutiny in Post-Avicennan logic, before finding a
stable equilibrium with Katibi (d. 1276 CE), an equilibrium that turns
in several key respects on the interpretation of these relations.


### 4.1 Contradiction of Categorical Propositions



The logical relations usually expressed in the Aristotelian square of
opposition for assertoric propositions and its modal counterparts
become the object of a significant revision in Avicenna's
system, as a result of the richer set of propositions it comprises.
Perhaps the most striking feature is the replacement of the
Aristotelian assertoric with the one-sided absolute (X1),
a proposition containing an
implicit temporal modality ("sometimes" or "at least
once") which, unlike the Aristotelian assertoric, is not
contradicted by a proposition of the same kind, but by a stronger
proposition, namely the "perpetual"
(*da`ima*) A-proposition.



Below is a summary of the relations of entailment (including subalternation) and contradiction for four basic types of propositions, all one-sided, in
the referential/substantial reading; in the case of two-sided propositions,
contradictories are disjunctive statements or involve a disjunctive
predicate (the latter only applies to two-sided possibility propositions):



![a diamond shape with 4 items on each side, for each side arrows point from the first item to the second, second to third, third to fourth. First side (upper middle to middle left): La, Aa, X1a, M1a. Second side (lower middle to middle left): Li, Ai, X1i, M1i. An arrow points from each item in the first side to the corresponding item on the second side (e.g., La to Li). Third side (upper middle to middle right): Le, Ae, X1e, M1e. Fourth side (lower middle to middle right): Lo, Ao, X1o, M1o. An arrow points from each item in the third side to the corresponding item on the fourth side (e.g., Le to Lo). Dotted lines connect the first item of the first side to the fouth item of the fourth side, second item of the first side to the third item of the fourth side (and so on). Similarly for the first item of the third side to the fourth item of the second side and so on. A legend explains that the arrows mean 'entails' and the dotted lines 'contradicts'](fig1.svg)

Figure 1: Referential/substantial necessity,
perpetuity, general absoluteness, and general possibility:
entailment and contradiction




The following are analogous relations among propositions in the
descriptional reading:



![very similar to figure 1, except that the side items are different. A diamond shape with 4 items on each side, for each side arrows point from the first item to the second, second to third, third to fourth. First side (upper middle to middle left): Ld1a, Ad1a, Xd1a, Md1a. Second side (lower middle to middle left): Ld1i, Ad1i, Xd1i, Md1i. An arrow points from each item in the first side to the corresponding item on the second side (e.g., Ld1a to Ld1i). Third side (upper middle to middle right): Ld1e, Ad1e, Xd1e, Md1e. Fourth side (lower middle to middle right): Ld1o, Ad1o, Xd1o, Md1o. An arrow points from each item in the third side to the corresponding item on the fourth side (e.g., Ld1e to Ld1o). Dotted lines connect the first item of the first side to the fourth item of the fourth side, second item of the first side to the third item of the fourth side (and so on). Similarly for the first item of the third side to the fourth item of the second side and so on.](fig2.svg)

Figure 2: Descriptional necessity,
perpetuity, general absoluteness, and general possibility:
entailment and contradiction 




The above relations are central in syllogistic proofs by
*reductio*.


### 4.2 Contradiction of Hypothetical Propositions



In the case of hypothetical propositions, contradiction is determined
only with respect to the main quantifier of the conditional or the
disjunction and does not affect the component statements. In other
words, it should not be confused with another relation involving
negation, namely mutual implication (*talazum*). The
contradictory of a conditional a-proposition is a conditional
o-proposition with the same antecedent and consequent and the
contradictory of a conditional e-proposition is a conditional
i-proposition with the same antecedent and consequent. The same
applies to the case of disjunctions.



Avicenna is keen to point out on various occasions throughout the
treatment of conditionals in the *Qiyas* that what matters
are the quantificational aspects of the *relation* between
antecedent and consequent (the *nisba* or the
*hukm*), not the quantity and quality of the antecedent
and consequent themselves. In particular, he vigorously rejects the
view that the contradictory of a conditional statement is a
conditional statement of the same quality and quantity whose
consequent is denied. In his view, the contradictory of (a-\(\mathbb{C}\))aa is
 (o-\(\mathbb{C}\))aa, not (a-\(\mathbb{C}\))ao. In other words, "Always, if every A is B,
then every C is D" is contradicted by "Not always, if every A is
B, then every C is D" and not by "Always, if every A is B, then not every C is D" which is in fact a stronger statement equivalent to its contrary, namely  (e-\(\mathbb{C}\))aa ("Never, if every A is B, then every C is D").[10]
On the other hand, (o-\(\mathbb{C}\))aa is logically equivalent to (i-\(\mathbb{C}\))ao
("Sometimes, if every A is B, then not every C is D"),
which is therefore another contradictory of (a-\(\mathbb{C}\))aa. A set of basic relations is represented in the
following diagram



![An 8 sided diagram. On the left side the vertices are (a-\(\mathbb{C}\))aa, (e0c)ao, (o-\(\mathbb{C}\))ao, (i-\(\mathbb{C}\))aa with double headed arrows connecting the first two items to each other and also the last two items to each other, a single arrow points from the second item to the third to make up three of the sides of the diagram. In addition the first item has an arrow pointing to the fourth item. Similary on the right side except the vertices are (e-\(\mathbb{C}\))aa, (a-\(\mathbb{C}\)ao), (i-\(\mathbb{C}\))ao and (o-\(\mathbb{C}\))aa. Between the first item of the left side and the first item of the right side is the word 'contraries' and between the last item of the left side and the last item of the right side is the word 'subcontraries'. Dotted lines connect the first item of one side to the third and fourth items of the second side and also from the second item of each side to the third and fourth of the other.](fig3.svg)

Figure 3: Relations among conditionals: mutual implication, subalternation, contrariety, and contradiction) 



### 4.3 Conversion of Categorical Propositions



(Some of) the rules of conversion for categorical propositions
constitute another major departure from Aristotle. On Avicenna's
model, the absolute universal negative proposition, X1-e,
fails to convert due to its being
a one-sided temporal proposition (in general it is not true that if no
A is always B, then no B is always A, as in "No animal always
sleeps" and "Nothing that sleeps is always an
animal"). L-a and L-i
propositions do not convert as L-i
propositions but rather as M1-i
propositions (for a general
presentation and an account of the proofs see Street 2002, Thom 2003).
L-e and Ld1-e
propositions convert as such. In
the first case, the proof is one of the most controversial points in
Avicenna's system, directly connected with the validity of
first-figure syllogisms with a one-sided possibility, M1,
minor premise. The conversion of
Ld1-propositions as Ld1-propositions is in fact the minimal
essential requirement for the productivity of second-figure moods
involving absolutes in XX and XM mixes. For Avicenna's account of why an Ld1-proposition
may legitimately be regarded as an
absolute proposition, see *Qiyas* I.4 and *Burhan* II.1, discussed in (Strobino
forthcoming).


### 4.4 Conversion of Hypothetical Propositions



The conversion of hypothetical propositions is explicitly discussed
only in a short chapter of the *Qiyas* (VII.3), where
Avicenna briefly talks about the status of (e-\(\mathbb{C}\)) propositions
(conversion does not apply in a genuine sense to disjunctions, whose parts are equivalent: "Always, either every A is B or C is
D" is the same statement as "Always, either every C is D
or every A is B"). This, however, should not suggest that
conversion is less important in the case of hypotheticals than it is
in the case of categoricals: it is in fact an essential method of proof for
certain hypothetical syllogisms. Additional rules may be extrapolated from the
extensive discussions of syllogisms involving conditionals in
*Qiyas* VI.1, VI.2, and VI.4.



In the conversion of conditional propositions, the parameters that matter are the
quantity of the proposition to be converted and that of its
converse, and the type of conditional (*luzumi* or
*ittifaqi*), while the antecedent and the consequent are
treated in a way analogous to the way in which terms are treated in
the conversion of categoricals. The following are notable as well as
frequently used conversion principles:

 

\[
\begin{align}
(\eC^{i})pq & [?] (\eC^{i})qp & (\textit{Qiyas}, \textrm{ VII.3}) \\
(\aC)pq & [?] (\iC)qp\\
(\iC)pq & [?] (\iC)qp 
\end{align}
\]

### 4.5 Inferential Relations among Hypotheticals



The analysis of inferential relations among hypotheticals includes the
mutual implication (*talazum*) of (i) conditional
statements and (ii) disjunctive statements, as well as (iii) the interaction
between the two classes of hypotheticals. Besides its intrinsic interest, such an analysis is
often required for the proof validity of hypothetical syllogisms, in addition
to standard techniques such as conversion, ecthesis
(*iftirad*), and *reductio*. The treatment of
conditionals is found in *Qiyas* VII.1, while
*Qiyas* VII.2 is devoted to the analysis of disjunctions
and the connection between the two.


#### 4.5.1 Conditionals



In *Qiyas* VII.1, Avicenna considers a basic set of
quantified conditional statements with quantified antecedents and
consequents. Assuming the four basic forms of quantified conditional
statements (a-\(\mathbb{C}\)), (e-\(\mathbb{C}\)), (i-\(\mathbb{C}\)), and (o-\(\mathbb{C}\)) and all permutations of a-,
e-, i-, o-propositions as antecedents and consequents, Avicenna
generates four groups of sixteen conditional propositions (see
 Appendix B)
 and argues that any of those forms is logically equivalent to the
contradictory conditional with a contradictory consequent, based on a
*reductio* proof that may easily be generalized. As
illustrated above (contradiction of conditional propositions), the
(a-\(\mathbb{C}\))aa proposition is for instance equivalent to the (e-\(\mathbb{C}\))ao
proposition ("Always, if every A is B, then every C is D"
if and only if "Never, if every A is B, then not every C is
D"). If that were not the case, then (i-\(\mathbb{C}\))ao would be true. But
then (o-\(\mathbb{C}\))aa would be true, which contradicts the initial assumption.
For an account of Avicenna's *reductio* proof of the fact
that "Always, if every A is B, then every C is D" implies
"Never, if every A is B, then not every C is D"--at
*Qiyas* 367.11-368.1--see (El-Rouayheb 2009:
209-210 and 2010:
 xxxiii-xxxiv).[11]


#### 4.5.2 Disjunctions



The scheme of classification of disjunctive propositions is analogous
to that of conditionals. Disjunctive statements consisting of two
parts are quantified as a-, e-, i-, and o-propositions (where negative
propositions express absence of conflict or incompatibility between
the disjuncts at all (some) times and under all (some) circumstances)
with sixteen permutations for each case depending on the quality and
quantity of the parts (see
 Appendix B).
In this case, however, the situation is complicated by the fact that
different types of disjunctions have different logical properties,
with specific considerations applying to each type. Besides, in the
schematic representations offered below, I prefer to retain the
quantificational features of the disjuncts, as Avicenna seems to
differentiate explicitly between real and unreal disjunctions in
virtue, among other things, of their having affirmative or negative
parts, another point that deserves more attention (for a preliminary
analysis see Street 1995). In general, for Avicenna disjunctions and
conditionals are not inter-definable without qualification.


1. Avicenna addresses the relation between disjunctions and conditionals
first. The main distinction is between the case in which the
disjunction is (i) real (exclusive and exhaustive) and the case in
which it is (ii) unreal (to be taken here in the sense of inclusive).[12] 
In case (i), the negation of a disjunct implies the affirmation of the other disjunct and the affirmation of a disjunct implies the negation of the other disjunct. In case (ii), the negation of a disjunct implies the affirmation of the other disjunct, but the affirmation of a disjunct does not imply the negation of the other disjunct. Thus, for instance the following
relations hold, as instances of the schemata \((\aD^1)pq \rightarrow
(\aC)\neg pq\) and \((\aD^1)pq \rightarrow (\aC)p\neg q\):



\[\begin{align}
(\aD^{1})\ra^{1}\ra^{2} & \rightarrow (\aC)\ro^{1}\ra^{2 }\\
(\aD^{1})\ra^{1}\ra^{2} & \rightarrow^{ } (\aC)\ro^{2}\ra^{1}\\
(\aD^{1})\ra^{1}\ra^{2} & \rightarrow (\aC)\ra^{1}\ro^{2 }\\
(\aD^{1})\ra^{1}\ra^{2} & \rightarrow (\aC)\ra^{2}\ro^{1}
 \end{align}
 \]


By contrast, if the disjunction is unreal (inclusive), only the
following relations hold:

 

\[\begin{align}
(\aD^{3})\ra^{1}\ra^{2} & \rightarrow (\aC)\ro^{1}\ra^{2 }\\
(\aD^{3})\ra^{1}\ra^{2} & \rightarrow^{ } (\aC)\ro^{2}\ra^{1}
 \end{align}
 \]
2. In analogy with the case of conditionals, Avicenna also discusses
inferential relations among disjunctions (*Qiyas* VII.2,
379.17-381.10), where notable cases are, for real affirmative
disjunctions with affirmative parts:

 

\[\begin{align}
(\aD^{1})\ra^{1}\ra^{2} & \rightarrow (\eD)\ro^{1}\ra^{2}\\
(\aD^{1})\ra^{1}\ra^{2} & \rightarrow (\eD)\ro^{2}\ra^{1}
 \end{align}
 \]


Thus, an exclusive and exhaustive universal affirmative disjunction
entails a proposition that expresses universal absence of conflict
between one of the disjuncts and the contradictory of the other
("from 'Always either every A is B or every C is D'
one can infer 'Never either not every A is B or every C is
D' and 'Never either not every C is D or every A is
B'", *Qiyas* VII.2, 380.1-4). The converse is
not generally true, as the negative disjunction may be true in virtue of
an impossible disjunct that is not in conflict with the other
(according to the semantics of negative disjunctions discussed in
*Qiyas* V.5). The same holds for (i-\(\mathbb{D}\)1) and
(o-\(\mathbb{D}\)1) propositions.
3. The relations among disjunctions and conditionals, a theme of obvious
interest from the standpoint of modern logic, reflect the complexity
of the above distinctions. Avicenna investigates relations such as,
for instance, the triangulation between (a-\(\mathbb{D}\)),
(a-\(\mathbb{C}\)), and (e-\(\mathbb{D}\)) propositions
(*Qiyas* VII.2, 381.11-17):



![(a-\(\mathbb{D}\)^1)pq with arrows pointing to (e-\(\mathbb{D}\)^1)!pq and to (a-\(\mathbb{C}\)^l)!pq and also an arrow pointing from the last to the second to the last item](diagram1.svg)



This is only an example of a more complex network of inferential
relations that should be complemented with an exhaustive list and an
accurate analysis of the inferences used in the treatment of
syllogisms, especially in *Qiyas* VI.


## 5. Syllogistic



The culmination of Avicenna's formal logic is the development of
an elaborate theory of the syllogism that brings together various
strands from ancient and late ancient logic, filtered through
Peripatetic lenses, and produces a new and original system.



The theory of the syllogism (*qiyas*), in its main
divisions, occupies a prominent part in all of Avicenna's
logical works. The doctrinal skeleton remains the same in spite of a
significant re-arrangement of the material, especially in the
*Isarat*. In the works of the middle period,
especially the *Nagat* and the extensive treatment of the
*Qiyas*, Avicenna follows the path charted by the
*Prior
 Analytics*.[13]


##### Types of syllogisms: connective and repetitive



Among the several innovations Avicenna may be credited with is the
codification of a canonical division, echoing distinctions already
brewing in ancient and late ancient logic, of syllogisms in a number
of categories that became standard conceptual vocabulary in later
Arabic logic.



Syllogisms divide into two main types: (1) connective syllogisms
(*iqtirani*) and (2) repetitive syllogisms
(*istitna`i*).



Connective syllogisms are defined as those in which neither the
conclusion nor its contradictory is explicitly part of the premises,
and hence as those syllogisms in which the conclusion is only
potentially contained in the premises. Connective syllogisms are
characterized by a partial overlap or sharing between the premises (of
a middle term or of a proposition discharged in the conclusion).



Repetitive syllogisms are defined as those in which either the
conclusion or its contradictory is explicitly part of the
premises.



Further discussion of syllogistic arguments include the (i) compound
syllogism (*qiyas murakkab*), which involves multiple
syllogisms such that the conclusion of a syllogism serves as a premise for
the next syllogism, whether by an explicit concatenation
(*mawsul*) or an implicit one
(*mufassal*), as well as other forms that came to
be discussed, in later Arabic logic, under the rubric of concomitants
of the syllogism (*lawahiq al-qiyas*). These
variously connect to chapters of the *Prior Analytics* and include:
(ii) the proof through the impossible or *reductio ad absurdum*
(*qiyas al-half* or *kalam ila
l-muhal*), which in Avicenna is interestingly
analyzed in terms of a combination of a connective and a repetitive
syllogism, (iii) the conversion of the syllogism (*'aks
al-qiyas*), (iv) the syllogism from opposites (*qiyasat mu`allafa min muqaddamat mutaqabila*), and (v)
circular proof (*bayan ad-dawr*).


### 5.1 Connective Syllogistic



Connective syllogisms are divided into two main types: (1) categorical
(*hamli*) and (2) hypothetical
(*sarti*) syllogisms.



(1) Categorical syllogisms are those whose premises and conclusions
are all and only categorical propositions. Their treatment constitutes
one of Avicenna's most original contributions. It follows paths
and adopts techniques that are familiar in the Aristotelian tradition
but the resulting system is original and based on distinctive
intuitions, largely due to the sophisticated understanding of modality
that emerges in the analysis of propositions.



(2) The hypothetical syllogistic is a new development, at least with
respect to Aristotle. Although the model is likely indebted to late
ancient Greek sources, the genuinely innovative character of its
constituent elements (quantified hypothetical propositions) turns this
area into something rather different from its most natural Greek
(Galen) or Latin (most notably Boethius) terms of comparison, even
leaving aside other differences.



The hypothetical syllogistic investigates arguments in which at least one
of the premises is a hypothetical proposition (of type (i), namely one
whose parts are themselves categoricals, though Avicenna does not seem
to put any restriction on more complex argument forms). Purely hypothetical
syllogisms are those in which the combination of the premises involve
*only* hypotheticals (conditional-conditional;
conditional-disjunction; disjunction-disjunction). Mixed hypothetical
syllogisms are those in which the combination of the premises involves
a hypothetical (conditional or disjunction) and a categorical.



The extent to which hypothetical syllogisms are treated in
Avicenna's various logical works varies significantly. A great
deal of material is left out of the abridged treatments (for instance
in the *Nagat* and in the *Isarat*)
compared to the extensive analysis provided in the *Qiyas*
(and to that promised but never delivered--as far as we
know--in the frequently announced *Kitab
al-Lawahiq*, the *Book of the Appendices*). The
criteria of selection are not just pragmatic (hypothetical syllogisms
are significantly more complex, if only in virtue of the compounding
effect generated by an additional layer of quantification) but also
based on the idea that some combinations are more natural (perhaps
more intuitive) than others.



The entire edifice of Avicenna's connective syllogistic rests on the assumption that certain argument forms are self-evidently productive
(*bayyina, bayyina bi-anfusiha, bayyinat
al-intag*) or perfect (*kamila*) and that all
other forms can be reduced (*rugu'*) to them by
standard methods of proof. The latter include (i) proofs based on
conversion (*'aks*) (alongside other kinds of
transformation of premises or conclusions into logically equivalent premises or
conclusions, in the case of hypothetical syllogisms), (ii) proofs
based on setting out a term (*iftirad*)--the
counterpart of the Aristotelian notion of *ecthesis*, and (iii) proofs
through the impossible (or *reductio ad absurdum*,
*qiyas al-half*). Depending on the circumstances,
one, two or all three methods may be used to prove productivity.


#### 5.1.1 Categorical syllogistic



Categorical syllogisms are argument forms consisting of triplets of
terms (*hudud*) arranged in two premises
(*muqaddamat*) and a conclusion (*natiga*
or, insofar as it is implicitly contained in or aimed at by the premises,
*matlub*). The two premises share a term with one
another and one each with the conclusion, for example, "Every B
is C; every A is B; therefore every A is C" (BaC,
AaB [?] AaC). The term shared by the premises is
called middle term (*hadd awsat*). The subject of
the conclusion is called minor term (*hadd
asgar*), the predicate of the conclusion is called
major term (*hadd akbar*). The premise that shares with
the conclusion the minor term is called minor premise (*muqaddama
sugra*). The premise that shares with the
conclusion the major term is called major premise (*muqaddama
kubra*).


##### Figures and moods



The middle term may be (i) subject of the major premise and predicate
of the minor premise; (ii) predicate of both; or (iii) subject of
both. The resulting configurations are called (i) first figure
(*sakl awwal*), (ii) second figure (*sakl
tanin*), and (iii) third figure (*sakl
talit*). Like Aristotle, Avicenna does not
contemplate a fourth
 figure.[14]
 Each figure contains a number of valid moods
(*durub*), identified by the quantity, quality, and
modality of the premises and conclusion. For the sake of simplicity, I
use standard vocabulary for the treatment of the Aristotelian syllogistic
(in particular the list of mnemonic tags developed in the Latin middle
ages for the identification of
 moods).[15]
 The moods of the first figure are Barbara, Celarent, Darii, Ferio.
The moods of the second figure are Cesare, Camestres, Festino, Baroco.
The moods of the third figure are Darapti, Felapton, Datisi, Disamis,
Bocardo, and Ferison. Each of the above will be productive with
certain combinations of modalized premises and not with others.



The discussion of productive moods in the *Qiyas* and in the
*Nagat* follows the order and method of presentation
of the *Prior Analytics* and focuses on different modal mixes
(*ihtilatat*) that are then distinguished
by figure (for example, the LX or the LM combinations in the first,
second, and third figure). In the *Isarat*
Avicenna chooses a different approach and discusses productivity by
figure, not by mix. This enables him to present more effectively a
number of general principles governing inferences in each figure, and
most importantly modifications to the so-called rule of the major for
the first figure (Street forthcoming). The rule of the major tells us
that the modality of the conclusion in first-figure syllogisms follows
the modality of the major premise, except in two cases that will be
discussed below (Ld1LL and XMM). Given the logical
priority ascribed by Avicenna to the first figure, this move
represents a significant gain in terms of simplicity.



Below is a schematic summary of the productive combinations of
modalized premises in each figure (the combinations in square brackets
are productive but not always explicitly identified by Avicenna).





|  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
|  | XX | LL  | LX  | MM  | MX  | ML  |
| First

figure  | XXX

[XXM]  | LLL

[LLM]  | LXL

XLX

Ld1LL  | MMM  | MXM

XMM

[XMX]

[MXX]  | LML

MLM

[LMM]  |
| Second

figure  | only with

X converting as itself  | LLL  | LXL

XLL  | no moods  | only with

X converting as itself  | LML

MLL

[LMM]

[MLM]  |
| Third

figure  | XXX

[XXM]  | LLL  | LXL

XLX  | MMM  | MXM

XMM

[XMX]

[XXM]

[MXX]  | LML

MLM

[LMM]  |




In the first figure, according to the terse account offered in the
*Isarat*, if the minor premise is an actuality
proposition (X
or L), the conclusion follows the modality
of the major (X, L
or M). There are, however, two exceptions in
which the conclusion follows the modality of the minor premise. The
first exception is Ld1LL, where the major is a
descriptional necessity proposition and the conclusion, like the
minor, is an L-proposition (stronger than
the major). The second exception is XMM, where the major is an
absolute proposition and the conclusion, like the minor, is
an M-proposition proposition (weaker than
the major). In the remaining two cases in which the minor is
an M-proposition, namely MMM and LML, the
rule of the major applies.



MMM and MLM moods in the first figure are perfect (*Qiyas*
IV.1, 181.2-4, 184.3-6, and IV.3, 199.5-6, where the
latter are explicitly characterized as instances of the *maqul
'ala l-kull* or *dictum de omni*). LML moods, by
contrast, require proof.



Some of the most interesting (and problematic) aspects of
Avicenna's modal logic emerge in connection with the proofs used
to establish the productivity of certain notable moods. For instance,
the controversial idea that the possible may be assumed to be actual
in the context of a *reductio* proof (as well as in the proof
of e-conversion of necessity propositions) is justified by intuitions
that have sparked heated controversies in the post-Avicennan tradition
(similar issues arise in the Greek tradition; for a philosophical
interpretation of this technique, see Malink & Rosen
2013). Another problem is the potential circularity of the proof of
Barbara LML, which, even if we leave aside the issue of assuming the
possible to be actual, seems to rest on Baroco LXL, which in turn
depends on Barbara LML (Street 2002).



A crucial intuition that lies at the core of Avicenna's
understanding of modality is connected with the different readings
introduced above (referential/substantial and descriptional). If a
property holds necessarily of something (picked out by the middle
term) under the referential/substantial reading, then it also
necessarily holds of whatever the middle term may be true of: 



> 
> 
> In sum, be aware of the fact that that for which it is possible to be
> necessary is always necessary (*ma yumkinu an
> yasira daruriyyan fa-huwa daruriyyun
> da`iman*) and that it may [also] be possible in the more
> general sense (*wa-amkanahu l-imkan
> al-a'amm*). (*Qiyas* IV.3, 203.4-5)
> 
> 
> 
> 



A similar line of reasoning applies to the case of LXL moods
(*Qiyas* IV.3, 202, *Nagat* 71) and,
*mutatis mutandis*, to Ld1LL
moods with regard to the minor
premise.



Avicenna explicitly formulates a generalized modal version of the
*dictum de omni* at *Qiyas* IV.1, 184.3-5:




> 
> 
> What possibly holds of what possibly holds [of something] possibly
> holds of [that something] in an evident way (*mumkin al-mumkin
> mumkin zahir al-imkan*), just as what necessarily
> holds of what necessarily holds [of something] necessarily holds [of
> that something] (*daruri
> ad-daruri daruri*) and what
> holds of what holds [of something] holds of [that something]
> (*mawgud [for wugud] al-mawgud
> mawgud*).
> 
> 
> 



Perhaps the most striking feature of Avicenna's system is the
productivity (and foundational role) of first-figure syllogisms with a
possibility minor premise, which has traditionally been regarded as
problematic by many of Avicenna's commentators. The case in
support of this view hinges primarily on a certain reading of the
terms and the idea that possibility propositions express relations of
compatibility (or separability) among conceptions or essences, which
Avicenna often spells out in terms of natures (of the subject and
predicate) (Thom 2008, 2012).



A noteworthy property of the second figure is that as a result of
Avicenna's replacement of the Aristotelian assertoric with his
own general absolute, there are, *strictly speaking*, no
productive moods out of two absolute premises, due to their failure to
convert as themselves (Street 2002). Avicenna, however, is committed
to the productivity of the familiar four moods in the second figure
when the absolute is taken in a qualified sense, according to which it
converts as itself, namely when it is at least as strong as
an Ld1-proposition (the same
applies to second-figure XM moods). Avicenna consistently discusses
this essential requirement in his logical works, often in conjunction
with the claim that this qualified reading is the default sense of the
e-proposition as well as being the one adopted in the sciences
(Strobino forthcoming).



An account of Avicenna's modal syllogistic influenced by
considerations of this sort and aiming to look at its logical
properties in tandem with Avicenna's metaphysics has been
recently offered by (Thom 2012). The project is inspired by principles
that are similar to those underpinning the most significant recent
developments in the interpretation of Aristotle's modal
syllogistic (Malink 2013). A model that stresses the two-dimensional
character of Avicenna's logic and reads Avicenna as reducing
alethic modalities to temporal modalities only has been proposed by
(Hodges and Johnston 2017).


#### 5.1.2 Hypothetical syllogistic



The hypothetical propositions discussed above may be used as premises
and conclusions of syllogistic arguments.


##### Types of hypothetical syllogism



Avicenna investigates five main types of hypothetical syllogisms (with
further internal divisions). The first three are combinations of two
conditional premises (type 1), a conditional premise and a disjunctive
premise (type 2), or two disjunctive premises (type 3), and are called
pure (*sirfa*) hypothetical syllogisms. The last two are
combinations of a conditional and a categorical premise (type 4,
subdivided into two classes), or of a disjunctive and a categorical
premise (type 5), and are called mixed
(*muhtalata*) hypothetical
 syllogisms.[16]



In general, there are some structural features that need to be kept in
mind, while looking at this part of Avicenna's logic.
Hypothetical syllogisms are connective (*iqtirani*) syllogisms
and, as we have seen above, this means by definition that the premises
are connected by a shared element. The latter may be either (i) a term
(types 4 and 5, where the sharing occurs between a term in the
categorical premise and a term in one of the constituent propositions of the
hypothetical (conditional or disjunctive) premise) or (ii) a proposition (types 1, 2,
and 3). In certain cases, the arrangement is such that it is possible to identify
three figures in a way analogous to categorical
syllogisms, whether the shared part is a term (types 4 and 5 again) or
a proposition (type 1). In other cases, no such partition emerges,
especially in the absence of an intrinsic order of the parts (type 3).



Another standard feature of Avicenna's discussion, especially
where premises are of different kinds (type 2, 4, and 5), is that
arguments are classified in function of (i) which premise is major and
which is minor; (ii) what part is shared (for conditionals, whether it
is part of the antecedent or of the consequent; for disjunctions, whether
the disjunction is real or unreal, and whether the shared part is
affirmative or negative).



On several occasions--most notably but not exclusively in the
context of the abridged treatments--Avicenna simplifies the
analysis and avoids a detailed discussion of individual productive
moods by providing general criteria for productivity, determined
according to these different parameters. Again, this approach is
structurally similar to the one adopted in the treatment of the categorical syllogistic but the complexity
is higher due to additional variables at play with hypotheticals. It
is worth noting that in Post-Avicennan logic, starting in particular
with Hunagi, considerable attention is paid to
this part of Avicenna's system, and the discovery of additional
moods as well as the rejection of moods accepted by Avicenna were
often associated with a critique and revision of these criteria
(El-Rouayheb 2010).


##### Type 1: hypothetical syllogisms with two conditional premises (*Qiyas* VI.1)



In hypothetical syllogisms with two conditional premises, the shared
part is a proposition which plays the same role as the middle term in
categorical syllogisms. Depending on whether it is (i) the consequent
of the major premise and the antecedent of the minor, (ii) the
consequent of both or (iii) the antecedent of both, arguments of this
kind will fall into one of the three familiar figures. The moods in
each figure are determined here in function of the quality and
quantity of the conditional premises and conclusion, unlike in the
case of combinations involving a categorical and a conditional, where figures
and moods, as we shall see below, are determined by the categorical
premise and another categorical proposition, which is either the consequent or the
antecedent of the other (conditional) premise.



The moods presented below are not unrestrictedly productive. In several
cases their productivity depends on whether the conditional is
implicative (*luzumi*) or coincidental
(*ittifaqi*). I list moods that Avicenna regards to
be valid once the required qualifications are added, but for the sake
of simplicity the latter are not expressed in the notation.



The first figure comprises the four characteristic moods, which are
described as perfect (*kamil*). Avicenna discusses at some
length the status of this figure with respect to whether the
conditionals it expresses are *luzumi* or
*ittifaqi*.




First figure




|  |  |
| --- | --- |
| Hyp. Barbara \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) | \((\textrm{a-}\mathbb{C})rq\); \((\textrm{a-}\mathbb{C})pr [?] (\textrm{a-}\mathbb{C})pq\)[17] |
| Hyp. Celarent \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) | \((\textrm{e-}\mathbb{C})rq\); \((\textrm{a-}\mathbb{C})pr [?] (\textrm{e-}\mathbb{C})pq\) |
| Hyp. Darii \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) |  \((\textrm{a-}\mathbb{C})rq\); \((\textrm{i-}\mathbb{C})pr [?] (\textrm{i-}\mathbb{C})pq\) |
| Hyp. Ferio \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) |  \((\textrm{e-}\mathbb{C})rq\); \((\textrm{i-}\mathbb{C})pr [?] (\textrm{o-}\mathbb{C})pq\) |



Second figure




|  |  |
| --- | --- |
| Hyp. Cesare \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) |  \((\textrm{e-}\mathbb{C})qr\); \((\textrm{a-}\mathbb{C})pr [?] (\textrm{e-}\mathbb{C})pq\) |
| Hyp. Camestres \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) |  \((\textrm{a-}\mathbb{C})qr\); \((\textrm{e-}\mathbb{C})pr [?] (\textrm{e-}\mathbb{C})pq\) |
| Hyp. Baroco \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) |  \((\textrm{a-}\mathbb{C})qr\); \((\textrm{o-}\mathbb{C})pr [?] (\textrm{o-}\mathbb{C})pq\) |
| Hyp. Festino \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) |  \((\textrm{e-}\mathbb{C})qr\); \((\textrm{i-}\mathbb{C})pr [?] (\textrm{o-}\mathbb{C})pq\) |




Again, the conditions of productivity depend on the kind of
conditional expressed in each of the premises. For example, 



> 
> 
> If the negative [premise] is [the negation] of an implication, which is compatible with
> [mere] coincidence, and the affirmative [premise] affirms coincidence, then the syllogism will not be productive at
> all. (*Qiyas* VI.1, 300.2-3) 
> 
> 
> 



On the other hand "If the affirmative premise is an implication,
the combination will be productive" (*Qiyas* VI.1, 300.9-10) regardless of the status of the negative premise.



All of the above moods may be proved, according to Avicenna, either by
converting one of the premises or by *reductio* (and in the
case of Baroco, by the analogue of an ecthetic proof, where the
instantiated element is a specified "case and time").




Third figure




|  |  |
| --- | --- |
| Hyp. Darapti \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) | \((\textrm{a-}\mathbb{C})rq\); \((\textrm{a-}\mathbb{C})rp [?] (\textrm{i-}\mathbb{C})pq\) |
| Hyp. Felapton \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) | \((\textrm{e-}\mathbb{C})rq\); \((\textrm{a-}\mathbb{C})rp [?] (\textrm{o-}\mathbb{C})pq\) |
| Hyp. Datisi \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) | \((\textrm{a-}\mathbb{C})rq\); \((\textrm{i-}\mathbb{C})rp [?] (\textrm{i-}\mathbb{C})pq\) |
| Hyp. Disamis \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) | \((\textrm{i-}\mathbb{C})rq\); \((\textrm{a-}\mathbb{C})rp [?] (\textrm{i-}\mathbb{C})pq\) |
| Hyp. Bocardo \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) | \((\textrm{o-}\mathbb{C})rq\); \((\textrm{a-}\mathbb{C})rp [?] (\textrm{o-}\mathbb{C})pq\) |
| Hyp. Ferison \(\mathbb{C}\textrm{-}\mathbb{C}\textrm{-}\mathbb{C}\) | \((\textrm{e-}\mathbb{C})rq\); \((\textrm{i-}\mathbb{C})rp [?] (\textrm{o-}\mathbb{C})pq\) |




Just as in the first and second figure, these moods are typically proved
by conversion or by *reductio*.


##### Type 2: hypothetical syllogisms with a conditional and a disjunctive premise (*Qiyas* VI.2)



Hypothetical syllogisms consisting of a conditional and a disjunctive
premise are second, in terms of complexity, only to the combination of
conditionals and categoricals (type 4 below). The proof of the
productive moods, in addition to the standard methods seen above,
often involves turning a conditional premise into a disjunctive
premise or vice versa (the analysis of these inferential relations occupies
Avicenna in *Qiyas* VII.1-2, but he uses them freely
throughout *Qiyas* VI, before explicitly discussing them
in the subsequent treatise). Similarly, after showing that a given
premise pair yields a hypothetical conclusion of a certain kind,
Avicenna often shows that it also yields a hypothetical conclusion of
the other kind, whether logically equivalent or simply implied by the
former.



In general, for this type of premises, the classification of arguments
rests on the following criteria: which premise--major or
minor--is conditional and which is disjunctive; whether the
shared part (here always a
 proposition)
 is the antecedent or the consequent of the conditional;[18] whether the
disjunction is real or not; whether the parts of the disjunction are
affirmative or negative; and whether the conditional is an implication
or
 not.[19]



Avicenna discusses various combinations
(*ta`lifat*) of syllogisms, identified first on the basis of their major and minor premises, and then differentiated according to the
other parameters. Individual moods, in analogy with categoricals, are
identified in function of the quality and quantity of their
(hypothetical) premises. The combinations are accordingly distributed
into two main groups. In the first group, syllogisms have a
disjunctive major and a conditional minor. In the second group,
syllogisms have a conditional major and a disjunctive minor.




Group 1 (disjunctive major, conditional
minor)




Case 1: the disjunction is real and the
shared part is the consequent of the conditional



Case 2: the disjunction is unreal and the
shared part is the consequent of the conditional



Case 3: the disjunction is real and the
shared part is the antecedent of the conditional



Case 4.1: the disjunction is unreal, the
shared part is the antecedent of the conditional and is
affirmative



Case 4.2: the disjunction is unreal, the
shared part is the antecedent of the conditional and is negative




Group 2 (conditional major, disjunctive
minor)




Case 5: the disjunction is real and the
shared part is the antecedent of the conditional



Case 6.1: the disjunction is unreal, the
shared part is the antecedent of the conditional and is
affirmative



Case 6.2: the disjunction is unreal, the
shared part is the antecedent of the conditional and is negative



Case 7: the disjunction is real, the shared
part is the consequent of the conditional



Case 8: the disjunction is unreal, the
shared part is the consequent of the conditional





An example of case 1, from the first group, is the following (*Qiyas* VI.2,
305.8-10):




\[
(\aD^{1})rq; (\aC)pr \vdash (\aC)p\neg q
\]


> 
> 
> Always, either C is D or A is B; Always, if H is Z, then C is D
> [?] Always, if H is Z, then A is not B
> 
> 
> 




but also (*Nagat* 82, xix)




\[
(\aD^{1})rq; (\aC)pr \vdash \aD^{1})pq
\]


> 
> 
> Always, either C is D or A is B; Always, if H is Z, then C is D
> [?] Always, either H is Z or A is B
> 
> 
> 




Avicenna's discussion, besides its intrinsic logical merits, is
enriched by several examples (especially counterexamples put forward
to show that certain moods are not productive) taken from the domains
of physics and metaphysics.


##### Type 3: hypothetical syllogisms with two disjunctive premises (*Qiyas* VI.3)



The third type of hypothetical syllogisms is, by contrast, the simplest.
It involves premise pairs with two disjunctions. Avicenna notes that
as a result of the status of real disjunctions, which are both exclusive
and exhaustive, there can be no combination involving two such
premises. The proof of this claim involves an interesting point about
the meaning of quantification in the case of disjunctions already
raised at *Qiyas* V.5 (particular disjunctions express
non-exhaustive alternatives).



Furthermore, this type is not arranged in figures because disjuncts
are symmetrical (*mutakafi`*), unlike the antecedent
and the consequent of a conditional, which also implies that there is
no distinction between major and minor premise (all permutations are
equivalent, as they are not sensitive to order).



The combinations considered in this case all involve at least an
unreal disjunction, which may be paired with a real disjunction, the
negation of a real disjunction or with another unreal disjunction,
where the shared part may be affirmative or negative.



The first mood, where the two premises are both affirmative and one has a negative part, is as follows (*Qiyas* VI.3, 321.4-5)



\[
(\aD^{3})r\neg q; (\aD^{1})pr \vdash (\aD^{1/3})\neg p\neg q
\]


The proof is based on turning the two disjunctive premises into the
conditional statements that logically follow from them, namely
\((\aD^{3})r\neg q \vdash (\aC)\neg r\neg q\), and \((\aD^{1})pr
\vdash (\aC)p\neg r\). From such conditional premises, one may easily derive the conditional conclusion \((\aC)p\neg q\), which in turn entails the disjunctive proposition \((\aD^{1/3})\neg p\neg q\).



An example with affirmative universal unreal disjunctive premises sharing a negative part 
is as follows (*Qiyas* VI.3, 323.14-324.5)

 

\[
(\aD^{3})\neg rq; (\aD^{3})p\neg r \vdash (\oD^{3})pq
\]


which is equivalent to the following syllogism

 

\[
(\aC)rq; (\aC)rp \vdash (\iC)pq
\]


in virtue of inferential relations holding among disjunctions and
 conditionals.[20]


##### Type 4: hypothetical syllogisms with a conditional and a categorical premise (*Qiyas* VI.4-5)



The first three types are pure hypothetical syllogisms. The fourth and
fifth are mixed hypothetical syllogisms consisting of a hypothetical
premise combined with a categorical.



The hypothetical syllogism consisting of a conditional and a
categorical premise is the most prominent case in the extensive
treatment of the *Qiyas*. The analysis occupies two chapters whose sequential order seems to be all but accidental. The classification of arguments
follows familiar criteria but in this case the order of priority is
different. The first parameter depends on what the shared part is. There
are two main types according to whether the connection between major
and minor premise obtains in the consequent (*Qiyas* VI.4)
or in the antecedent of the conditional premise (*Qiyas*
VI.5). The second parameter depends on whether the major is
categorical or conditional.



What is shared in this case is a term that connects the categorical
premise with another categorical, namely the antecedent or the
consequent of the conditional premise. This entails that syllogisms of
this type are in fact arguments "containing" a categorical
syllogism whose conclusion is either implied by or implies another
proposition. This is perhaps the reason why the main division is given
in terms of whether the sharing occurs (i) in the consequent or (ii)
in the antecedent, and also the reason why the former case is taken to
be more relevant (or at least more common) in the sciences. For in
case (i), all syllogisms will yield the conclusion of the
corresponding categorical syllogism subject to a condition, namely as
the consequent of a conditional statement; whereas in case (ii) the
interaction between the quantification of the conditional and the
quantification of the antecedent yields moods that only partially overlap with
the standard set of categorical moods and that are subject to various
qualifications.



Both kinds of considerations--on the more natural character of
certain inference patterns and on their applicability to the sciences
(*Qiyas* V.1, *Nagat*
82)--indirectly confirm that the hypothetical syllogistic is an
integral part of Avicenna's theory and practice of philosophical and
scientific discourse.



Last, the two types of mixed hypothetical syllogisms are the only cases in which
Avicenna systematically investigates quantified atomic propositions
(an obvious consequence of the fact that he is looking at inferences
which, in conditional form, very much resemble those analyzed in the
categorical syllogistic).



In *Qiyas* VI.4, which generally deals with the group of
syllogisms where the sharing occurs in the consequent of the
conditional premise, Avicenna investigates two cases: the first with a
categorical major and a conditional minor, the second with a conditional
major and a categorical minor. It is not hard to see how this approach naturally yields conditional versions of the familiar figures and moods, in function of the position of the shared term.



In each case and figure, Avicenna distinguishes the moods also
according to the kind of quantified conditionals involved in the conditional
premise and in the conditional conclusion.




**Case 1:** categorical major, conditional minor, sharing
in the consequent of the conditional; in three figures
(*Qiyas* VI.4, 325.2-331.9)



**First figure:** predicate of the consequent of the
conditional = subject of the categorical major


1. universal affirmative conditional premise



|  |  |
| --- | --- |
| Hyp. Barbara \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | BaC; \((\textrm{a-}\mathbb{C})p\textrm{-AaB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AaC}\)  |
| Every B is C; Always, if H is Z, then every A is B
[?] Always, if H is Z, then every A is C (where *p*
stands for "H is Z") |
| Hyp. Celarent \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\) |  BeC; \((\textrm{a-}\mathbb{C})p\textrm{-AaB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AeC}\) |
| Hyp. Darii \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\) |  BaC; \((\textrm{a-}\mathbb{C})p\textrm{-AiB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AiC}\) |
| Hyp. Ferio \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | BeC; \((\textrm{a-}\mathbb{C})p\textrm{-AiB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AoC}\) |
2. particular affirmative conditional premise: Avicenna simply states that there
are four productive moods, presumably the analogues of the above where the minor
premise and the conclusion are i-conditionals.
3. universal negative conditional premise: in this case, the fact that
the conditional premise is negative affects the categorical
consequent, which is denied at all times and under all circumstances.
Avicenna presents three moods that are in fact equivalent, in virtue
of the relations of mutual implication discussed in *Qiyas*
VII.1, to three moods of case (a) (he does not explicitly discuss the
hypothetical equivalent of Celarent, which is also productive via the
corresponding Ferio):



|  |  |
| --- | --- |
| Hyp. Barbara\* \(\textrm{Cat-}(\textrm{e-}\mathbb{C})\textrm{-}(\textrm{e-}\mathbb{C})\)  | BaC; \((\textrm{e-}\mathbb{C})p\textrm{-AoB} [?] (\textrm{e-}\mathbb{C})p\textrm{-AoC}\)[21] |
| [Hyp. Celarent\* \(\textrm{Cat-}(\textrm{e-}\mathbb{C})\textrm{-}(\textrm{e-}\mathbb{C})\)  | BeC; \((\textrm{e-}\mathbb{C})p\textrm{-AoB} [?] (\textrm{e-}\mathbb{C})p\textrm{-AiC}\)]  |
| Hyp. Darii\* \(\textrm{Cat-}(\textrm{e-}\mathbb{C})\textrm{-}(\textrm{e-}\mathbb{C})\)  | BaC; \((\textrm{e-}\mathbb{C})p\textrm{-AeB} [?] (\textrm{e-}\mathbb{C})p\textrm{-AeC}\) |
| Hyp. Ferio\* \(\textrm{Cat-}(\textrm{e-}\mathbb{C})\textrm{-}(\textrm{e-}\mathbb{C})\)  | BeC; \((\textrm{e-}\mathbb{C})p\textrm{-AeB} [?] (\textrm{e-}\mathbb{C})p\textrm{-AaC}\) |
4. particular negative conditional premise: Avicenna states that
there are four productive moods, which must presumably be particular
versions of the previous case, if we supplement the latter with a
counterpart of BeC; \((\textrm{e-}\mathbb{C})p\textrm{-AoB} [?]
(\textrm{e-}\mathbb{C})p\textrm{-AiC}\) (Avicenna does not discuss
this mood explicitly, but it is compatible with the general criteria
of productivity he adopts, namely that the categorical major should be
universal and the consequent of the conditional negative).



**Second figure:** predicate of the consequent of the
conditional = predicate of the categorical major


1. universal affirmative conditional premise



|  |  |
| --- | --- |
| Hyp. Cesare \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | CeB; \((\textrm{a-}\mathbb{C})p\textrm{-AaB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AeC}\) |
| Hyp. Camestres \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | CaB; \((\textrm{a-}\mathbb{C})p\textrm{-AeB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AeC}\) |
| Hyp. Festino \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | CeB; \((\textrm{a-}\mathbb{C})p\textrm{-AiB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AoC}\) |
| Hyp. Baroco \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | CaB; \((\textrm{a-}\mathbb{C})p\textrm{-AoB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AoC}\) |

When the conditional is particular, Avicenna contends that there
are four additional moods. Negative (universals) are treated in
general terms: their criteria of productivity require the categorical
major and the consequent of the conditional minor to be both
affirmative or both negative, and the categorical to be universal. For
example,



|  |  |
| --- | --- |
| Hyp. Cesare\* \(\textrm{Cat-}(\textrm{e-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | CeB; \((\textrm{e-}\mathbb{C})p\textrm{-AoB} [?] (\textrm{e-}\mathbb{C})p\textrm{-AiC}\) |

is valid in virtue of the familiar rules of
 transformation.[22]



**Third figure:** subject of the consequent of the
conditional = subject of the categorical major


1. universal affirmative conditional premise



|  |  |
| --- | --- |
| Hyp. Darapti \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | BaC; \((\textrm{a-}\mathbb{C})p\textrm{-BaA} [?] (\textrm{a-}\mathbb{C})p\textrm{-AiC}\) |
| Hyp. Felapton \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | BeC; \((\textrm{a-}\mathbb{C})p\textrm{-BaA} [?] (\textrm{a-}\mathbb{C})p\textrm{-AoC}\) |
| Hyp. Datisi \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | BaC; \((\textrm{a-}\mathbb{C})p\textrm{-BiA} [?] (\textrm{a-}\mathbb{C})p\textrm{-AiC}\) |
| Hyp. Disamis \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | BiC; \((\textrm{a-}\mathbb{C})p\textrm{-BaA} [?] (\textrm{a-}\mathbb{C})p\textrm{-AiC}\) |
| Hyp. Bocardo \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | BoC; \((\textrm{a-}\mathbb{C})p\textrm{-BaA} [?] (\textrm{a-}\mathbb{C})p\textrm{-AoC}\) |
| Hyp. Ferison \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\)  | BeC; \((\textrm{a-}\mathbb{C})p\textrm{-BiA} [?] (\textrm{a-}\mathbb{C})p\textrm{-AoC}\) |

When the conditional premises is quantified differently, the moods
will be revised forms of the above, just as in the case of the first
and second figure.



**Case 2:** conditional major, categorical minor, sharing
in the consequent of the conditional; in three figures
(*Qiyas* VI.4, 331.10-336.8).



**First figure:** subject of the consequent of the
conditional = predicate of the categorical minor


1. universal affirmative conditional premise



|  |  |
| --- | --- |
| Hyp. Barbara \((\textrm{a-}\mathbb{C})\textrm{-Cat-}(\textrm{a-}\mathbb{C})\)  | \((\textrm{a-}\mathbb{C})p\textrm{-BaC; AaB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AaC}\)[23] |
| 
Always, if H is Z, then every B is C; Every A is B
[?] Always, if H is Z, then every A is C |
| Hyp. Celarent \((\textrm{a-}\mathbb{C})\textrm{-Cat-}(\textrm{a-}\mathbb{C})\)  | \((\textrm{a-}\mathbb{C})p\textrm{-BeC; AaB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AeC}\) |
| Hyp. Darii \((\textrm{a-}\mathbb{C})\textrm{-Cat-}(\textrm{a-}\mathbb{C})\)  |  \((\textrm{a-}\mathbb{C})p\textrm{-BaC; AiB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AiC}\) |
| Hyp. Ferio \((\textrm{a-}\mathbb{C})\textrm{-Cat-}(\textrm{a-}\mathbb{C})\)  |  \((\textrm{a-}\mathbb{C})p\textrm{-BeC; AiB} [?] (\textrm{a-}\mathbb{C})p\textrm{-AoC}\) |



When the conditional premise is (e-\(\mathbb{C}\)), (i-\(\mathbb{C}\)) or (o-\(\mathbb{C}\)), the moods are
four in each case, in virtue of the canonical equivalences.



**Second and third figure:**



The four conditional moods of the second figure (where the predicate
of the consequent of the conditional is the predicate of the
categorical minor) and the six conditional moods of the third figure
(where the subject of the consequent of the conditional is the subject of
the categorical minor) are treated in a similar way. In both cases the
moods correspond to those of the relevant categorical figures, in each
case with four variations depending on the quality and quantity of the
conditional major. The transformations are the same as above. Avicenna
occasionally gives a summary of the conditions of productivity (for
example, in second-figure moods where the conditional premise is
a universal negative, the consequent must be particular and have the
same quality as the categorical premise) without ever analyzing the
case of particular conditional premises explicitly.




Last, it is worth noting that Avicenna explicitly characterizes the
hypothetical syllogisms with an affirmative conditional major, a
categorical minor and a shared term in the consequent of the
categorical, as those in which the condition of productivity is that
the categorical and the consequent of the conditional should be related in the same way as in the connections
(*iqtiranat*) of first-figure categorical
syllogisms. In this case,



> 
> 
> the conclusion is a conditional proposition whose consequent would be
> the conclusion [produced] by the two categorical [propositions], were they to be taken in isolation [i.e., not as part of
> a hypothetical syllogism]. (*Qiyas* VI.4, 331.13) 
> 
> 
> 



For similar reasons, the same applies to the previous case (the
combination of a categorical major and a conditional minor). In this connection,
Avicenna notes that the only difference between categorical syllogisms
and hypothetical syllogisms with such characteristics is that in the
former the conclusion "follows without qualification, while here [it
follows] if something is posited" (*Qiyas* VI.4,
325.14-5).



The second major division of hypothetical syllogisms consisting of a
conditional and a categorical premise obtains when the sharing occurs in
the antecedent. The analysis of these syllogisms is the subject of
*Qiyas* VI.5, which is the only place in the corpus where
they are treated explicitly. Elsewhere (e.g., *Nagat*
82), hypothetical syllogisms of this kind are said to be less used in
the sciences. In spite of the structural analogy to the cases
discussed above (the conclusion is still a conditional proposition,
even though here the categorical "fragment" occupies the
place of the antecedent), the way the quantification of the
conditional and the quantification of the antecedent interact generate
results that do not generally mirror categorical syllogisms. The
classification is parallel to the previous case and consists of two
main groups: the first with a conditional major and a categorical
minor, the second with a categorical major and a conditional minor.
Each group is divided into three figures, as above, and the moods of
each figure are analyzed with respect the four combinations of quality
and quantity of the conditional premise.


##### Type 5: hypothetical syllogisms with a disjunctive and a categorical premise (*Qiyas* VI.6)



The second kind of mixed hypothetical syllogisms consists of a
disjunctive and a categorical premise (or several categorical
premises). It is at least briefly mentioned in all of Avicenna's
logical works and receives a more extensive treatment in
*Qiyas* VI.6, 349.1-354.12. Like some of the types discussed above, this
kind of hypothetical syllogism is indirectly acknowledged to have an
application outside the domain of formal logic, as Avicenna contends
that, with certain qualifications, it expresses a mode of reasoning
similar to that involved in the method of induction
(*Qiyas* VI.6, *Burhan* I.7). This obtains when a hypothetical syllogism of this type expresses a
complete division, the predication is real
(*haqiqi*), and the subject is the subject of
the disjunctive proposition, in which case it is called "divided
syllogism" (*qiyas muqassam*).



Avicenna explores different possibilities in turn. With the help
of the shorter treatment in the *Nagat*, an
underlying structure may be identified, where the order is inverted
with respect to the way in which materials are arranged in the *Qiyas*. He distinguishes
between two main groups of syllogisms, one (i) in which the
categorical premise is only one; the other (ii) in which multiple
categoricals are involved.



In the first case, the most natural arrangement is when the minor is
an affirmative categorical and its predicate is the subject of the
disjunctive major. The example at *Nagat* 82, xvi "Everything countable is either even or
odd; every large quantity is countable; therefore, every large
quantity is either even or odd" is in line with the general
scheme "Always, every B is either H or Z; every C is B;
therefore, always, every C is either H or Z" given at
*Qiyas* VI.6, 353.8-9. In the *Nagat* Avicenna
asserts that this arrangement comes in four combinations, and both in the
*Qiyas* and in the *Nagat* he refers to the
above as (a mood of) the first figure. Admissible variants seem to be
those involving a particular major and minor, or a universal negative
major (or affirmative with negative parts). The second figure is said
to be non-productive (and is consistently not mentioned in the
*Nagat*), while the third figure is said to conclude
by conversion but is not discussed any further.



In the second case, where the major is categorical and the minor
disjunctive, the major consists in fact of multiple categoricals, which must
be as many in number as the parts of the disjunction (to ensure a
proper connection). Every categorical and every disjunct are connected
by a term; and all the parts of the disjunction share a
 term.[24]
 This yields two basic sub-cases, each arranged in figures. A schema
from the first figure associated with the notion of a complete
induction is "Every C and H and Z is A; every B is either C or H
or Z; therefore every B is A". It is illustrated by the example 



> 
> 
> Every animal is a body and every plant is a body and every mineral is
> a body; everything that moves is either an animal or a plant or a
> mineral; therefore everything that moves is a body.
> (*Nagat* 82, xvii)
> 
> 
> 


##### Syllogisms of hypotheticals sharing an incomplete part (two disjunctions) (*Qiyas* VI.6, 354.13-356.6)



Hypothetical syllogisms whose premises share an incomplete part
are perhaps best understood in opposition to two types of purely
hypothetical syllogisms discussed above, namely those consisting of a
pair of disjunctive premises and those consisting of a disjunctive and
a conditional premise. In both cases, the shared part was a whole
proposition. Syllogisms involving hypotheticals that share an
incomplete part are a variation on the latter, determined by the fact
that the shared element is a term instead of a whole proposition. They
may involve either (i) two disjunctive premises or (ii) a conditional
and a disjunctive premise with one term in common (in the latter case,
with a conditional minor). An example is given at *Nagat*
82, xx.



Avicenna's cursory treatment of this case may be at the origin
of a much more elaborate development in the analysis of this class of hypothetical syllogisms in Post-Avicennan logic, starting with
Kassi (El-Rouayheb 2010).


### 5.2 Repetitive Syllogistic



The repetitive (*istitna`i*) syllogistic
covers inference patterns such as *modus ponens* and *modus
tollens* (in their conditional and disjunctive variants). It
constitutes the other major division of the syllogistic along with the
connective (*iqtirani*) syllogistic and seems to be regarded by Avicenna as somehow
dependent on it (a systematic
discussion of the reducibility of the former to the latter occupies
part of *Qiyas* IX.1, where Avicenna
discusses among other things the idea that the major premise of *modus
ponens* may be in a sense redundant).



Repetitive syllogisms consist of (i) a hypothetical premise
(conditional or disjunctive) containing the conclusion or its negation as one of its
parts, and (ii) another premise which asserts or denies
(and thereby "repeats") part of the hypothetical premise. Example of
notable argument forms of this kind are the following

 

\[\begin{align}
p \rightarrow q, p & [?] q && (\textit{modus ponens})\\
p \rightarrow q, !q & [?] !p && (\textit{modus tollens})
\end{align}\]


where the hypothetical premise is a conditional (Avicenna also
considers the case of bi-conditionals), and

 

\[\begin{align}
p \lor^{1} q, \neg p & [?] q\\
p \lor^{1} q, p & [?] \neg q\\
p \lor^{3} q, \neg p & [?] q && (\textrm{disjunctive syllogism})\\
\end{align}\]


where the hypothetical premise is a disjunction (either real or
unreal).


### 5.3 *Reductio ad absurdum* (*qiyas al-half* )



Avicenna's account of *reductio* is an interesting case
in which several of his technical distinctions come into play. A
*reductio* is a compound syllogism (*qiyas
murakkab*)--i.e., a concatenation of
syllogisms--consisting of a connective hypothetical syllogism
and of a repetitive syllogism. Both categorical and hypothetical
propositions may be proved by *reductio*.



At *Nagat* 91, i, Avicenna offers a straightforward example of a
r*eductio* proof of AaB in the categorical syllogism CaB; AaC
[?] AaB, where (1) is the connective hypothetical syllogism (with a conditional minor) and (2) the
repetitive syllogism (an instance of *modus tollens*):


1. CaB; If !AaB, then AoB [?] if ! AaB, then AoC
2. If !AaB, then AoC; AaC [?] AaB



First syllogism (connective hypothetical of type 4, with a conditional minor)





|  |  |  |
| --- | --- | --- |
| i. | CaB |  (major premise of the initial syllogism = major premise of 1) |
| ii. |  If ! AaB, then AoB  | (minor premise of 1, based on a/o contradiction) |
| iii. | If ! AaB, then AoC  | (i, ii, Hyp. Baroco \(\textrm{Cat-}(\textrm{a-}\mathbb{C})\textrm{-}(\textrm{a-}\mathbb{C})\), *Qiyas* VI.4, 328). |



Second syllogism (repetitive)





|  |  |  |
| --- | --- | --- |
| iv. | If ! AaB, then AoC  | (conclusion of 1 = major premise of 2) |
| v. | AaC  | (minor premise of the initial syllogism = minor premise of 2) |
| vi. |  AaB  | (iv, v, *modus tollens*) QED  |




In *Qiyas* VIII.3 Avicenna discusses a version of
*reductio* in which the conclusion aimed at
(*matlub*) is a hypothetical (conditional)
proposition:



Hypothetical case: (a-\(\mathbb{C}\))*qr*; (o-\(\mathbb{C}\))*pr* [?]
(o-\(\mathbb{C}\))*pq*


1. (a-\(\mathbb{C}\))*qr*; If !(o-\(\mathbb{C}\))*pq*, then (a-\(\mathbb{C}\))*pq* 
[?] if !(o-\(\mathbb{C}\))*pq*, then (a-\(\mathbb{C}\))*pr*
2. If !(o-\(\mathbb{C}\))*pq*, then (a-\(\mathbb{C}\))*pr*; (o-\(\mathbb{C}\))*pr*
[?] (o-\(\mathbb{C}\))*pq*



For a recent analysis of the logical properties of *reductio*
in Avicenna and its general significance see (Hodges 2017).


## 6. Demonstration



The theory of demonstration represents for Avicenna not only an
essential part of logic but its true culmination. Its central problems
are the identification of (i) the conditions under which
*certainty* (*yaqin*) may be attained in making
assertions within the context of a scientific discipline and the
identification of (ii) the conditions under which completeness,
relevance, and accuracy may be achieved in the process of concept
formation that leads to definitions. This determines in turn the
nature of the principles assumed in each science for the derivation of
its own theorems, the way in which boundaries between disciplines
are drawn, and more generally the overall architecture of scientific
knowledge.



Avicenna's original contributions in this area include: (1)
re-casting the model of the *Posterior Analytics* in terms of
the two fundamental notions of conception and assertion (Strobino 2010
and 2015b), associating them with (2) a more systematic, and
ultimately simplified, treatment of the four Aristotelian questions
(if, that, what, why) (Strobino 2015b); (3) a peculiar understanding
of the notion of necessity in light of Avicenna's own distinction between the
referential/substantial and the descriptional reading (Strobino
2015a); (4) a sophisticated account of per se, linked with the
theory of the predicables (Strobino 2016b); (5) an elaborate model for the classification of the
sciences (Strobino 2017); (6) an innovative classification of that-
and why-demonstrations (Strobino forthcoming); and (7) a commitment to
the applicability of formal logic as a whole to scientific discourse,
for instance through the explicit acknowledgment of the fact that the principles
and conclusions of demonstrative arguments may be both categorical and
hypothetical propositions, and that categorical as well as
hypothetical (and repetitive) syllogisms play a role in the
sciences.



In this area, Avicenna's work reflects in complex ways the
influence not only of al-Farabi but also of the Greek
commentary tradition, especially Themistius and Philoponus, whose
exegetical work on the *Posterior Analytics*, now lost in
Arabic, was available to Avicenna. This becomes especially evident
through implicit or explicit references to the commentators
(*mufassirun*) and their commentaries (*tafasir,
suruh*) disseminated in his own *Kitab
al-Burhan* (Eichner 2010, Strobino 2012, forthcoming).


### 6.1. Conception and Assertion



The distinction between conception
(*tasawwur*) and assertion (*tasdiq*)
comes into play in several crucial ways in Avicenna's logic of
scientific discourse.



The distinctive mark of demonstrative
arguments--certainty--is characterized in terms of assertion. An assertion *p* is certain if and only if one holds (i) the
justified belief that *p* *and* (ii) the justified belief that
*p* cannot be otherwise; a demonstrative argument is one in
which certainty is transferred from premises to conclusion.
Demonstrative assertions are explicitly identified by Avicenna in the
framework of a broader classification of various types of assertions
of decreasing epistemic strength (certain, endoxic, suppositional,
fallacious), which are in turn associated with a canonical taxonomy of
logical disciplines (demonstration, dialectic, rhetoric, sophistic)
(Gutas 2012; Hasnawi 2013; Black 1990).



Furthermore, Avicenna's flavor of Aristotelian foundationalism,
i.e., the view that in every science there are first principles on which everything else depends and which do not depend in
turn on anything else, is expressed in terms of the need for (i)
immediate assertions (*tasdiqat*) that are not
grounded on further assertions and (ii) for the conceptions
(*tasawwurat*) of primitive notions.



The notion of conception is also crucial for Avicenna's theory
of per se (*dati*), where a basic distinction between types of scientific
predicates depends on whether they are conceptually inseparable from
their subjects or not (Strobino 2016b).


### 6.2 Questions



The four Aristotelian questions--if, that, what, and
why--discussed in *Posterior Analytics* B1-2 are
subject to a more systematic analysis in Avicenna's
*Burhan* I.5 and IV.1. In *Burhan* I.5, in particular, they are rearranged in three pairs with
several internal subdivisions: (i) what (*ma*), with
respect to the meaning of the name or with respect to the essence;
(ii) why (*lima*), with respect to unqualified existence or
with respect to existence in a given state; and (iii) if (*hal*)/that
(*anna*), again with respect to unqualified existence or with respect to existence in
a given state. The notion of existence in a state in (ii) and (iii) encapsulates the case of attribute ascription in its causal and factual version, respectively. The three pairs of questions may apply to both subject
and predicate of a scientific statement.



Avicenna explores various relations and concludes that what- and
if-questions, in some of their declensions, are more fundamental in
character, as why-questions ultimately reduce to questions about the
essence of something, and if-questions express crucial necessary
conditions for any science, namely the existence of basic entities in
its domain and the assumption of certain fundamental predicative
claims about them.



The classification of questions is inspired by the idea of reducing them
to the two basic paths of scientific knowledge: conception and
assertion and their discursive counterparts, definition and syllogism
(Strobino 2015b).


### 6.3 Necessity



One of the single most important theoretical revisions of the
Aristotelian model concerns the notion of necessity (*darura*), which in fact
takes on a new role in Avicenna's system in a number of
different ways. At a general level, there is an epistemic notion
associated with the idea of certainty mentioned above; then there is a
technical sense involved in Avicenna's reading of necessity
statements (which are unsurprisingly a major ingredient of scientific
theories); and finally there is a notion of necessity associated with
scientific predication, which in a way is the most fundamental of
all.



Take "triangle" as a subject of predication and consider
the following two sets of attributes: (i) "being a
plane figure enclosed by three straight lines" or simply
"being a plane figure", and (ii) "having three
angles" or "having the sum of the internal angles equal to
two right angles". All of these attributes are necessarily true of
"triangle" but, according to Avicenna, they are necessarily true of
their subject in two quite different ways. The first two attributes,
"being a plane figure" and "being a plane figure
enclosed by three straight lines" are *essential*
attributes of "triangle" (the former is its genus, the latter is its
definition); they are *constitutive* of it in a way that
the other attributes are not. Furthermore, "being a figure" is
*conceptually inseparable* from "triangle" and *contained*
in its definition, in the sense that we cannot properly define
"triangle", in its true nature, without appealing to that notion. By contrast
"having the sum of the internal angles equal to two right angles"
is not at all required to define "triangle", even if it is necessarily
implied by it, and is in fact an attribute that we typically *prove* to
hold of its subject. Essential attributes of the first kind are captured
by definitions, and in general have an explanatory role with respect
to attributes of the other kind. They explain the fact that attributes
of the other kind hold of their subjects. In Aristotle's theory
of science, these two types of attributes are usually designated by
the technical term "per se".



Avicenna also originally puts his own distinction between the
referential/substantial and descriptional reading to use in his theory
of science. Each of the above two types of necessity (essential and
non-essential) can hold on either reading. In particular, the
descriptional reading, which Avicenna takes to be the default sense of
necessity in scientific discourse, allows us to study the properties
of a given object *qua* object of a certain kind (a bronze
triangle *qua* triangle) and to make true, necessary
predications about it as *long as it is a triangle* (but not as
long as it exists, as a bronze triangle could be re-cast into a
different shape and lose some of the properties that necessarily hold
of it *qua* triangle) (Strobino 2015a).


### 6.4 Per se, Inseparability, Containment, and Implication



Avicenna's elaborate account of per se predication elevates the
role of a technical distinction internal to the Aristotelian model and
assigns it a larger and more systematic theoretical relevance, where
three levels of discourse are linked in an original way: (i) the
theory of per se itself, (ii) a distinction between types of metaphysical
necessity of different strength, captured by the notion of
containment (*tadammun*) and necessary implication (*iltizam*), and (iii) a distinction between
types of inseparability, which grounds the other two. All three levels
express in different ways the dichotomy between essential and
non-essential necessity (Strobino 2016b, forthcoming).



(i) The notion of a per se attribute is ultimately linked with the
idea that in a scientific theory the admissible terms must be salient
properties of the objects under investigation. Avicenna argues that it
is only through this regimented conceptual vocabulary that the two
defining conditions of scientific knowledge (necessity and explanatory
character) can be met by a theory. In doing so, he provides a
systematic account of per se which elaborates on Aristotle's
cursory remarks in *Posterior Analytics* A4 and A22. The Aristotelian
analysis of per se is based on the relation of "being part of the definition, i.e. of the essence, of something": a
term A is per se 1 with respect to another term B if and only if A is part of the
definition of B while a term A is per se 2 with respect to another term B if and only if B
is part of the definition of A.



Avicenna connects the definition and classification of the two primary
types of per se relation to the concepts of constituent
(*muqawwim*)--the counterpart of per se 1--and
implicate (*lazim*)--the counterpart of per se
2--two characteristic notions of his system associated with
different kinds of metaphysical necessity.



(ii) These different kinds of metaphysical necessity are the
counterparts of two types of entailment which Avicenna calls
containment and necessary implication and which express necessary
connections of different strength between terms. Containment is the
relation between a notion and its intensional parts (i.e., the
constituents of its definition, which are ontologically part of its
essence). Necessary implication is a weaker form of necessity holding
between two notions that are merely inseparable.



(iii) The above pairs (per se 1 and 2, and containment and necessary
implication) are associated with a further distinction between two
types of inseparability. The theme is developed by Avicenna especially in connection with the *Isagoge*, most notably in *Madhal* I.5, I.6, and I.8. The first type is conceptual inseparability (the
counterpart of per se 1 and containment); the second type is
inseparability in imagination (the counterpart of per se 2 and
necessary implication) (Strobino 2016b).



Another interesting feature of Avicenna's account of per se is
the attempt to develop systematically the Aristotelian criteria and to
identify the sets of per se predicates relevant for a given kind or
domain of scientific discourse. This has a significant impact on his
taxonomy of the interrelations among the sciences.


### 6.5 Division of the Sciences



Avicenna's classification (or division) of the sciences is a
natural complement to his treatment of per se as well as being one of the
most innovative attempts to develop systematically the material on the subordination of scientific disciplines in *Posterior Analytics* A7, A9,
and A13. It is also important for an historical reason.
Avicenna's account of the division of the sciences is in fact
the only part of his *Burhan* that actually had a direct
impact on the Western Latin tradition.



The source of Avicenna's classification is a chapter from his
*Burhan* famously translated and encapsulated by the
twelfth-century Toledan philosopher and translator Dominicus
Gundissalinus (ca. 1110-1190) in his *De divisione philosophiae*. In the first
part, Avicenna presents a basic classification of the types of
hierarchical relations holding between various pairs of sciences, based on the corresponding
relations holding between their underlying subjects. At the root of the division
is a distinction between sciences that differ because (a) they have
two distinct subjects or (b) because they treat one and the same
subject in different ways. He articulates in full the
primary divisions of scientific domains, offering an analysis and
categorization of basic types of relations and dependencies (most notably, parthood
and subordination). In the second part, Avicenna develops a complementary classification
of the criteria of identity and distinctness for the sciences focusing
on the ways in which sciences may have elements in common. In this context, he addresses
the question from a different perspective--that of the canonical
three elements of an Aristotelian science--including not only the
subjects, but also scientific principles and questions (*masa`il*), i.e., the
conclusions of scientific syllogisms (the theorems of a science).



The use of three fundamental criteria--subject/genus
(*mawdu'/gins*), principles
(*mabadi`*), and per se attributes
(*'awarid datiyya*)--to
determine identity, overlap, or distinctness between domains of
scientific inquiry is in line with Aristotelian orthodoxy. But
Avicenna takes the criteria to a whole different level and articulates
a complex network of relations of superordination and subordination
whereby scientific knowledge is rigidly organized in hierarchical
structures that reflect underlying ontological relations between their
objects and attributes. For a detailed account see (Hugonnard-Roche 1988) and (Strobino 2017).


### 6.6 Why- and That-Demonstration



Avicenna's re-interpretation of the distinction between a
demonstration that something is the case and a demonstration why
something is the case is related (i) to the classification of per se,
i.e., to the types of predicates used in demonstrative arguments,
and (ii) to the division of the sciences (Strobino forthcoming).



Avicenna distinguishes between two main types of demonstration in function of the types of predicates involved (in first-figure
categorical syllogisms): (a) those in which the middle term expresses
a per se 2 attribute of the minor, in which case the major may be
either (aa) a per se 2 attribute of the middle or (ab) a per se 1
attribute of the middle; and (b) those in which the middle term
expresses a per se 1 attribute of the minor, in which case the major is
in most cases (ba) a per se 2 attribute of the middle, and only in two exceptional cases (bb) a per se 1 attribute of the middle (the reason is that
usually demonstrations do not aim to prove that per se 1 attributes
hold of their subjects, these relations being self-evident and captured by
definitions).



While proving an assertion, a demonstration may merely provide
an inferential justification of it or, in addition to that, also the terms that causally explain why things are the way they are. All demonstrations of the
first kind are that-demonstrations; all demonstrations of the second
kind are why-demonstrations.



Why-demonstrations may express why (i) a certain attribute holds of a
subject or (ii) both why it holds of that subject and why it exists.
In this connection, Avicenna aims to draw a distinction between two different kinds of
explanation. The fact that the attribute of having the sum of the internal angles
equal to two right angles holds of isosceles triangles *because* they
are triangles is an example of the first kind: the attribute is true of
something in virtue of its being true of a per se 1 attribute (in this case its genus) of that thing. The fact that vines are deciduous because they are wide-leaved
is an example of the second kind. The distinction is parallel to a
distinction proposed for Aristotle between application arguments and
subject-attribute demonstrations (McKirahan 1992). Avicenna's
account of per se fits neatly into this scheme, where case (aa)
corresponds to why-demonstration of type (ii), and case (ba)
corresponds to why-demonstrations of type (i) (Strobino
forthcoming).



That-demonstrations are also divided into two types, depending on
whether the context is one in which there is (iii) a causal relation
between subject and predicate or (iv) not. Case (iii) is the most
common type of that-demonstration, which typically obtains when the
middle term (in the first figure) is the effect of the existence of
the major instead of being its cause (the opposite of type-(ii)
why-demonstration). Case (iv) is a peculiar kind of that-demonstration which Avicenna
includes to account for immediate predications where an attribute
directly follows, without mediation, from something's essence,
which is the only case in which a that-demonstration may be associated
with certainty; or for cases where two attributes that are both
effects of a cause are used to prove one another of a subject
(Strobino 2016a, forthcoming).


### 6.7 Logical Forms of Premises and Conclusions



A point that Avicenna borrows from al-Farabi (Strobino
2017) is the acknowledgment of the fact that the logical forms of scientific
statements (whether premises or conclusions) include both categorical
and hypothetical propositions and that demonstrative arguments may be
both categorical and hypothetical (or even repetitive) syllogisms.
What may look like a minor technical point has in fact significant
implications for a better understanding of some of Avicenna's
motives in developing his formal logic.


### 6.8 Formal Logic and the Logic of Scientific Discourse



Avicenna is open about the fact that the project undertaken in
the investigation of formal aspects of logic goes beyond its application in
the sciences (*Qiyas*
 IV.1).[25]



At the same time he is also open about the fact that the formal apparatus
developed in the syllogistic is not exclusively analyzed for the sake
of studying its properties as a logical system. In fact, on multiple
occasions Avicenna tells us that the various tools developed for
instance in the *Qiyas* are used in the sciences, or offers
remarks on theoretical questions by specifying the default
interpretation in the context of scientific reasoning. This is the case (just to mention a few significant examples) in
the discussion of *reductio ad absurdum* and conditionals with
impossible antecedents and consequents (*Qiyas*
V.4-5); in his account of descriptional necessity as the
canonical reading of necessity propositions (*Burhan*
II.1, *Qiyas* II.2, *Nagat* 123); in
the acknowledgment that premises and conclusions of scientific syllogisms may
be categorical as well as hypothetical propositions
(*Qiyas* V.1, *Burhan* II.6); when he contends
that certain hypothetical syllogisms are less used in the sciences
(*Nagat* 82); when he draws connections between the
two areas (*Qiyas* VI.5, *Burhan* I.7:
induction and *qiyas muqassam*; *Burhan* IV.1:
if-question and conditional reasoning); or in maintaining the conditional character
of certain principles in the particular sciences (*Burhan* II.7)
(Strobino 2017).



While the formal development of logic undeniably acquires for Avicenna
an autonomous status that invites its own independent theoretical
questions, to see it as an entirely separate enterprise, detached from
application in the context of scientific discourse, would betray a
misconception of his intentions as well as of his practice.


## 7. Fallacies



The treatment of fallacies addresses one of the complementary
conditions expressed by the definition of logic, which is supposed to
deal not only with what is required in order to infer correctly what
is unknown from what is known, but also with what is required to avoid
error in the case of inferences that appear legitimate without being
so. There is a parallel in the case of conception and definition:
logic provides rules to identify and construct definitions and at the
same time instructions on how to avoid common errors in the process.
Avicenna's analysis of fallacious reasoning attempts to combine
(or perhaps force) Aristotle's treatment from the
*Sophistical Refutations* into a broader scheme driven by more
systematic concerns, which even in the extensive treatment of the
*Cure* is rather compact. The analysis of error in defining is
usually covered in connection with the *Posterior Analytics* and
the *Topics* (for instance at *Burhan* IV,
*Gadal*, IV, *Nagat* 149). I will focus
here only on the former.



The *Nagat* treatment perfectly captures this complex
strategy. It starts with a canonical distinction between fallacies as
(i) dependent on expression (*lafz*) or (ii) dependent on
meaning (*ma'na*)--the familiar distinction
between *de dictione* and *extra dictionem*
(language-dependent and language-independent) fallacies--and
pairs it with the claim that fallacies are also either dependent on
(iii) the form or on (iv) the matter of an argument, an aspect that is
present also in the cursory classification given in the
*Isarat* (Street forthcoming).



The analysis continues with an attempt to "deduct" the
fallacies based on the idea that when a false conclusion is reached,
then at least one of the characteristic conditions of syllogistic
reasoning must not have been met. It then proceeds to articulate a
detailed account of the different ways in which failure to meet those
conditions corresponds to different items in Aristotle's famous
list of thirteen fallacies, which Avicenna occasionally groups or divides in
slightly different ways, especially in the case of those that are dependent
on expression.



The discussion ends with a recapitulation that yields something very
close to Aristotle's list, even though in a few cases it is hard
to pinpoint where exactly certain items belong in the systematic
analysis.



The primary reasons of error may be abstracted from a strong
characterization of a syllogism. If a syllogistic argument is (a)
arranged according to one of the figures and its mood is productive,
(b) it has primary distinct parts (terms) and secondary distinct parts
(premises), (c) its premises are true, (d) distinct from and (e)
better known than the conclusion, then a true conclusion must follow.
Otherwise, at least one of conditions (a)-(e) has not been
met.



Avicenna unequivocally identifies two of the fallacies that are not
dependent on expression with (d) and (e), namely *petitio
principii* (*musadara 'ala
l-matlub al-awwal*) and positing what is not a cause as
a cause (*ahd* or *wad' ma laysa
bi-'illa 'illatan*). The latter is identified with
circular proof (*bayan ad-dawr*), which is consistent with
the doctrine of *Qiyas* IX.4 and *Burhan*
II.1.



The fallacy of consequent (*iham 'aks
al-lawazim* or *iham al-'aks*) can be
traced back to a subdivision of (c).



The fallacy of combining many questions (*gam'
al-masa`il*) is listed in the *Nagat* without being discussed and it is altogether absent from the
*Isarat*. In *Safsata* I.3,
however, Avicenna seems to treat this case, too, as a subdivision of (c).



Two other canonical fallacies of type (ii), namely the fallacy of
accident (*ma bi-l-'arad*) and *secundum
quid* (*su` i'tibar al-haml* or
*igfal tawabi' al-haml*) are listed
at the end of the *Nagat* section without being discussed, although
they, too, may presumably be classified as subdivisions of (c), namely
errors based on taking a premise to be true which merely appears to be
true.



The case of *ignoratio elenchi* is more problematic. The last
available slot in the *Nagat* list is called
"sterility of the connection" (*'aqm
al-qarina*) and is seemingly not discussed in the preceding
analysis. A possibility is to associate it with case (a), namely with
the error resulting from failing to recognize an argument as invalid,
which would however distort the original Aristotelian meaning. It may
be that Avicenna fails (or refuses) to recognize *ignoratio
elenchi* in Aristotle's sense and perhaps turns into
something like the error resulting from a lack of awareness of the form of
the syllogism, which leads to accept an invalid argument as valid.
This would be consistent with the *Isarat*
vocabulary, which introduces the "ignorance of the
composition" (*gahl at-ta'lif*) as a
fallacy while at the same time failing to mention *ignoratio
elenchi*.



In the *Nagat*, fallacies dependent on expressions
seem to be variously associated with cases (a)-(c). Avicenna
divides this category into two main sub-cases involving (i) a
simple expression (*lafz mufrad*) or (ii) a compound
expression (*lafz murakkab*).



In case (i), error may result from the association of multiple
meanings with an expression either in its core form
(*gawhar*) or in a derivative form (*hay`a*).
The first case is proper equivocation (*istirak
al-ism*) (even though Avicenna uses the term also to refer
generically to other types of language-dependent fallacies) and is in
fact divided into a number of different sub-types (*lafz
mustarak, musakkak, mutasabih, lafz
manqul, musta'ar, magaz*). The second
case is where Avicenna puts the counterpart of the fallacies of accent
and form of expression (or inflection).



Case (ii) includes the canonical fallacies of composition
(*tarkib al-lafz*) and division
(*tafsil al-lafz*).



The case of amphiboly (judging from the examples) seems to be treated
as an independent type, associated with (b), but its omission in the
final *Nagat* list may suggest that Avicenna, in line with what he does in the
*Isarat*, possibly regards it as a sub-type of
equivocation.



The treatment of fallacies is complemented by a parallel
classification of fallacious assertions (propositions that are deemed
to be true because of a systematic error in their evaluation)
consistently discussed throughout Avicenna's logical works
(e.g., *Burhan* I.4, *Isarat*
I.6).


## Abbreviations



I use standard abbreviations for quantified categorical propositions:
a-, e-, i-, and o-propositions (universal affirmative, universal
negative, particular affirmative, and particular negative).




|  |  |  |
| --- | --- | --- |
| AaB |  "Every A is B" |  (*kull* A B) |
| AeB |  "No A is B" |  (*la say` min* A B) |
| AiB |  "Some A is B" |  (*ba'd* A B) |
| AoB  | "Not every A is B"  | (*laysa kull* A B) |



In the case of modalized categoricals, the modality is prefixed (for
the one-sided absolute proposition, I omit the identifier X1
unless required to avoid ambiguity).
A full list of abbreviations is given below. For example, Ld1-AaB
stands for "Every A is
necessarily B as long as it is A", M1-AiB
for "Some A is possibly
B", and A-AeB for "No A is ever
B".



In the case of hypotheticals, I indicate first between parentheses
quantification and type (conditional or disjunctive), followed by an
indication of the quality and quantity of the parts (or, when the latter is not required, by simple propositional variables); in some cases the syntactic structure of an embedded categorical is made explicit in the above canonical form. Below are a few examples of expressions frequently used in this entry:




|  |  |
| --- | --- |
| (a-\(\mathbb{C}\))aa |  "Always, if every A is B, then every C is
D" |
| (e-\(\mathbb{C}\))ia |  "Never, if some A is B, then every C is
D" |
| (i-\(\mathbb{C}\))oe |  "Sometimes, if not every A is B, then no C is
D" |
| (o-\(\mathbb{C}\))ee |  "Not always, if no A is B, then no C is
D" |
|  |
| (a-\(\mathbb{D}\))aa |  "Always, either every A is B or every C is
D" |
| (e-\(\mathbb{D}\))ia  | "Never, either some A is B or every C is
D" |
| (i-\(\mathbb{D}\))oe |  "Sometimes, either not every A is B or no C is
D" |
| (o-\(\mathbb{D}\))ee  | "Not always, either no A is B or no C is
D" |
| (a-\(\mathbb{C}\))*pq* |  "Always, if *p*, then *q*" |
| (a-\(\mathbb{C}\))*p*-AaB |  "Always, if *p*, then every A is B" |



Occasionally, a more fine-grained distinction between types of
conditionals (*luzumi* or *ittifaqi*) or types of disjunctions (exclusive and
exhaustive; exclusive but non-exhaustive; inclusive) is required.
Similarly, it is sometimes necessary to identify antecedents and
consequents of conditionals (for conversion and inferential relations
with disjunctions). In such cases the above abbreviations are expanded
to include the required additional information. For instance to
express the claim that a universal affirmative exclusive and
exhaustive disjunction implies a universal affirmative
*luzumi* conditional in which antecedent and
consequent are contradictories and may be any of the parts of the
disjunctive, I will write that
(a-\(\mathbb{D}\)1)a1a2 implies
(a-\(\mathbb{C}\)*l*)a1o2,
(a-\(\mathbb{C}\)*l*)a2o1,
(a-\(\mathbb{C}\)*l*)o1a2, and
(a-\(\mathbb{C}\)*l*)o2a1, where superscript
"*l*" stands for *luzumi*, and the
superscript numbers are used to track the disjuncts where they appear
as antecedent or consequent of the conditional statements.