Zia Movahed
Introduction
After the publication of Ibn-Sina’s al-Shifa: al-Qiyas, edited by S. Zayed, Cairo, 1964, where
Ibn-Sina presented most extensively his theory of the conditional syllogisms and, later on, the
publication of Nabil Shehaby’s translation of it into English in 19731 I think we have all we
should have in hand to evaluate Ibn-Sina’s theory as it is, a theory Ibn-Sina regards as his
important contribution to Aristotalian logic and as a new form of argument “unknown until
now, which I myself discovered”2 (my translation).

My task in this paper is limited. I am going to examine Ibn-Sina’s theory from a purely
logical point of view, and to lay bare the principles he adopted for founding his theory, and
the reason why the newly introduced part of his logic remained undeveloped and eventually
was removed from the text of logic in the later Islamic tradition.
Anyone interested in the long and controversial history of conditional syllogisms or
philosophical, theological or dialectical motivations of the subject should consult the growing
literature of it, now easily available. Here, I only mention Nicholas Rescher’s paper:
Avicenna on the logic of “conditional proposition”, published in 19633. Perhaps that was the
first pioneering paper on the subject in the English language. But at the time of writing his
paper the text of al-Qiyas had not been published. So Rescher wisely remarked: “until it is
available, the present discussion must be viewed as tentative”4. Rescher’s paper is descriptive.
He also recognizes some invalid arguments in the theory without finding out the
methodological reasons for those invalidities. Nabil Shehaby’s introduction to his translation
is also purely descriptive, though informative. In this paper I quote Ibn-Sina’s views in al-
Qiyas from this translation which provides readers with references to the pages and lines of
the Cairo edition mentioned above.
Preliminaries
Aristotelian logic is also called “term logic”. By term, here, is meant concept – term which
stands as the subject or predicate of a categorical proposition. In modern logic concept –terms
are treated as one-place predicates. So in modern terminology one might say that the
Aristotelian logic is a monadic logic. But this is highly misleading. By monadic logic, in the
modern sense, we mean one-place predicate logic based on propositional logic as its
fundamental part. But in the Aristotelian logic this part is missing. So Ibn-Sina’s name for his
theory as “theory of conditional syllogisms” is more appropriate than “theory of propositional
logic”. In fact he never uses letters standing for propositions. He always uses subjectpredicate
forms with letters standing only for concepts, and never writes “If P, then Q” but
always writes “If A is B, then C is D”, with or without a quantifier for the antecedent or
consequent. Of course Ibn-Sina’s theory is meant to have the same status in the Aristotelian
logic that propositional logic has in modern logic. But they are formally worlds apart. They
are founded on a quite different, in fact opposite principle. And this is where the question of
methodology, not properly discussed yet, arises.
Ibn-Sina works within the frame-work of Aristotle’s logic, which is based on the theory of
categorical syllogisms. In a famous passage in the Prior Analytic, Aristotle wrote:
“Many other conclusions also are reached by hypothesis, and these require
further study and clear explanation. What their differences are, and in how
many ways a hypothetical conclusion is effected, will be described later for
the present let us regard this much as evident: that it is impossible to analyse
such syllogisms as these into the figures.”5
But this promise was never carried out, and, more surprisingly, his anticipation that: “it is
impossible to analyse such syllogisms as these into the figures” was never taken seriously by
his followers, notably Ibn-Sina. It is, however, to his credit that he realized the importance of
conditional syllogisms more than many other logicians and in al-Qiyas wrote: “Many theses
in mathematics, physics, and metaphysics are connective (muttasila) or separative (munfasila)
conditionals6, meaning by “connective”, an implication or chance conditional and by
“seperative”, a dis junctive combination.
Ibn-Sina’s Methodology
Ibn-Sina’s overall methodology is to establish a parallelism or correspondence between the
conditional and categorical syllogisms, in fact a reduction of the former to the latter. When
this is done he can claim, in particular in his shorter books and treatises (of which more than
30 authentic ones are recorded), that:
“You must treat the connective conditionals in a quantified form or
indefiniteness, contradiction and conversation as you treat categorical with the
antecedent as a subject and the consequent as a predicate”7 (my translation).
Within this theory the validity of a simple sequent like:
P Q, QR  PR
must be given according to the rules of the categorical syllogisms. But before subjecting such
sequent to those rules he has to clothe them in the forms resembling categorical propositions.
Before examining Ibn-Sina’s theory a short reminder of the theory of categorical syllogisms
is in order.
A short summary of the principles of Aristotelian logic
Ibn-Sina’s theory is based on principles among which the following are of immediate interest
for my discussion:
1- A predicative sentence consists of two main parts: subject-term and predicateterm.
The third part is a copula determining the quality of the sentence.
2- There are four types of predicative sentences: universal affirmative (A), universal
negative (E), particular affirmative (I), and particular negative (O). As to the
singular statements it is safe to say that within this theory they have not received
proper treatment (this has its own history in which I am not interested here).
3- Inferences are of two types: immediate and syllogistic.
a. There are different kinds of immediate inferences of which I only mention the
following two principles: the form “All A is B” follows both “some B is A”
(conversion per accidence) and “some A is B” (simple conversions).
b. Categorical syllogism. A categorical syllogism has three sentences, two as
premises and one as conclusion. The two premises must have a term in
common (middle term). This term, which connects the two premises, does not
appear in the conclusion. The terms standing as subject and predicate in the
conclusion are, respectively, called minor and major term. The premise
containing the major term is called the major premise and the one containing
the minor term, the minor premise. The middle term may be the subject in both
premises, or the predicate of both, or the subject only of the minor or the
subject only of the major premise. So we have four figures. Aristotle discusses
only three figures as does Ibn-Sina, who mentions briefly the fourth figure, in
which the middle term is the subject of the minor promise and a predicate of
the major, and discards it.
The Theory of syllogism is a set of rules prescribing which of all possible forms
(moods) of each figure are valid.
Now the fundamental methodology of Ibn-Sina is to embed any inference of
hypothetical syllogisms within the frame-work of the theory of categorical
syllogisms. If we lose sight of this point we will be bound to read many irrelevant
interpretations into it.
In this paper I shall confine my discussion to Ibn-Sina’s analysis of the connective
conditional whose truth-conditions are exactly the same as the material conditional
in the modern sense, i.e. a conditional which is false if and only if the antecedent
is true and the consequent false. But Ibn-Sina’s understanding of this conditional
is a kind of implication where the consequent is related and follows somehow
from the antecedent. He distinguishes this one from the chance conditional with
truth-conditions totally different from the connective conditional. It is in the case
of the latter that Ibn-Sina’s methodology can be seen clearly in application.
Ibn-Sina’s analysis of conditional
In this part I shall try, through systematic stages, to explain Ibn-Sina’s motivation at every
step in reducing conditionals to what I would like to call pseudo-categorical propositions.
1) The first major difficulty is that in “If P then Q” both “P” and “Q” stand for
propositions. How can a conditional consisting of two propositions be reduced to a
single seemingly categorical one?
Ibn-Sina’s way out of this difficulty is to deny that the antecedent and the consequent
of a conditional are sentences (propositions). His interpretation of “If it is so, then it is
so” is as follows:
“When you say ‘If it is so’ it is neither true nor false;
and when you say ‘then it is so’ it is also neither true
nor false provided that ‘then’ fulfils its real function
of indicating that something follows from another.”
8
This argument can lead only to one conclusion: a conditional as a whole is one
proposition. Then after some conflicting remarks, Ibn-Sina concludes that in “if P,
then Q”, “P” and “Q” play the same role respectively that subject- term and predicateterm
play in a categorical proposition.
2) Now the second difficulty arises. In an inference each premise must be one of A, E, I,
and O. Therefore to reduce ‘If P, then Q’, to categorical forms we need to introduce
quantifiers. This is a critical point which may easily give rise to the misinterpretation
of the nature of these quantifiers. Let me explain why.
3) In a sentence like:
A triangle is a shape
one can easily introduce a quantifier:
every/some triangle is a shape
But in a conditional like:
If the sun rises, then it is day
it is just meaningless to say:
every/some if the sun rises, then it is day
The reason is obvious. In ‘if P, then Q’, “P” and “Q” are not concepts. There is no extension
here over which quantifiers may range. Here, however, another kind of expression can be
used:
“Always/ under any condition if the sun rises, then it is day.”
It is not the case that Ibn-Sina could have used ordinary quantifiers but he chose not to use
them and used another kind of quantifier. On the other hand, these expressions need not be of
temporal nature. All Muslim logicians are in agreement with Ibn-Sina that:
“In the statement ‘Always: when C is B, then H is Z’ the words
‘Always: when’ are not only meant to generalize the occurrences of
the statement, as if one said: “Every time C is B, then H is Z”, but
then are also meant to generalize the conditions which we may add to
the sentence ‘C is B’ for the antecedent may refer to something which
does not recur and is not repetitive.9”
To emphasize that these expressions are not necessarily of a temporal nature Ibn-Sina
discusses conditionals expressing chance connection. Then in giving the truth-conditions of:
Always: when the man talks, then the donkey brays.
imaging a certain time at which no donkey exists, he writes:
“It might be thought that at this specific time….the proposition
‘always: when man talks, then the donkey brays’ is false. For at this
time there are no donkeys to bray. But this is a false opinion. For the
statement ‘every donkey brays’ is true even if there are no donkeys to
bray.”10
This clearly shows that these expressions are not meant to be only temporal. That is why, I
think, it is a mistake to use temporal operators and translate the conditional mentioned above
into:
t (Rt (P) → Rt (Q)
with interpreting “RtP” as “realization of P at the time t”.
Hereafter I shall call these expressions as pseudo-quantifiers, and show them by s and s
“s” is a variable ranging over any situation temporal or otherwise.
3) The last step is to impose four types on these pseudo-quantified conditionals corresponding
to the four types of categorical propositions. Without going into further details, and based on
my close examinations of Ibn-Sina’s writings as well as the writings of the later Moslem
logicians, the following formalization of the types of conditionals suggests itself:
AC: s (Ps→ Qs)
EC: s (Ps→ ˜Qs)
IC: s (Ps& Qs)
OC: s (Ps& ˜ Qs)
Now by establishing this parallelism between the conditionals and categorical propositions,
Ibn-Sina, as quoted before, claims that all rules of interferences applicable to the categoricals
are equally applicable to conditionals. Now let us examine some cases where Ibn-Sina applies
his theory.
a- Conversion simpliciter
In the conversion simpliciter the antecedent is turned into consequent and the consequent into
an antecedent, while keeping the quality and truth unchanged. This is Ibn-Sina’s first
example:
From stating that ‘Never: when every A is B, then every C is D’ it evidently follows that:
Never: when every C is D, then every A is B11
In symbolism:
From “s (P→ ˜ Q) follows “s (Q→ ˜ P)
This is parallel to the conversion simpliciter of “No A is B” which is “No B is A”
In this theory this inference holds and Ibn-Sina’s proof of it is valid.
Now let us apply the same rule to a universal affirmative. Here from “Always: when every A
is B, then every C is D” we get by the rule corresponding to the universal categorical:
“sometimes: when every C is D, then every A is B” or from “s (Ps→ Qs)”, we get “s (Ps&
Qs)”. Now here parallelism fails. Although from “Every A is B”, given the existential import
of the subject we can get: “some B is A”, but it does not apply to “P” as a sentence. In fact
one consequent of this rule is the following:
From “s (Ps& ˜Ps→ Qs) follows s (Ps & ˜ Ps) & Qs), which is obviously invalid.
b- Syllogism
More revealing is Ibn-Sina’s proof of the third mood of the third figure of conditional
syllogisms. Here I quote him in detail:
“This mood is compounded of two universal affirmative propositions
always: when C is D, then H is Z;
and
always: when C is D, then A is B
therefore
sometimes: when H is Z, then A is B”
Then, by reductio, he gives the following proof:
“Let (the conclusion) be
‘Never: if H is Z, then A is B’
If we add to it:
‘always: when C is D, then A is B
both will yield the following conclusion:
‘Never: if C is D, then A is B’
This is contradiction”12
This proof is carried out in the same way that the proof of its corresponding mood of the
categorical syllogism:
Every A is B
Every A is C
and given the existential presupposition that “some A exists”:
we have:
Some B is C
Now for comparison, and to see it clearly, Ibn-Sina’s proof in symbolism is as following:
s (Ps→ Qs)
s (Ps→ Rs)
therefore,
s (Qs& Rs).
Now by reductio,
˜s (Qs& Rs)
or
s (Qs→˜ Rs)
from this and the first premise we get
s (Ps→˜ Rs)
Now Ibn-Sina claims this conditional is contradictory to the second premise, I .e.
s (Ps→Rs)
This is how he understands the negation of his quantified conditional. Rescher believes that
by so doing: “He has, in effect, broadened the categories of “conjunctive” and “disjunctive”
propositions beyond their original characterization”13. Rescher, I believe, fails to note the
reductive nature of Ibn-Sina’s quantifiers used for conditionals, a reduction which is supposed
to reduce every quantified conditional to the corresponding categorical proposition. The proof
under discussion is defective for three reasons:
1. Propositions are not concepts with extensions and so are not obtainable by existential
import;
2. s (Ps→Rs) and s (Ps→˜Rs) are not contradictory;
3. s (Qs& Rs) is not a consequence of s (Ps→ Qs) and s (Ps→Rs). So here reductio ad
absurdum has no useful application.
All this shows the limitations inherent in the Aristotelian syllogism as the building blocks of
propositional logic. In fact I think that Ibn-Sina’s theory suffers from violating a principle so
fundamental to all sciences and in particular to logic and mathematics: the principle of
structuring the complex out of the simples. As Lukasrewiez rightly observes even in the
limited theory of Aristotle’s syllogism, Aristotle had to use theses of propositional logic “to
reduce syllogisms of the second and third figures to the syllogisms of the first figure”.14
Propositional logic, as Frege shows us, is the most simple and fundamental part of logic upon
which more complex and complicated logics should be founded. But Ibn-Sina’s theory is
exactly the other way round. I examined only some simple cases of the application of his
theory. When we come to his more complicated conditional syllogisms many inferences
become so involved and lead to invalid syllogisms. No wonder that logicians following Ibn-
Sina found the theory so difficult and confusing that eventually regarded it as dispensable in
practice and not worthy of serious consideration.
Whether Ibn-Sina’s theory can be saved by introducing ontology of situations or a kind of the
Davidsonian ontology of events for the quantified conditionals corresponding to existential
import for the categorical propositions would be a matter of further research, which I am not
pursuing here. But if that could be done, many of the invalid inferences, including some
mentioned so far, would be turned into valid ones.
Putting, however, your finger on the short comings of a work of a past master who lived more
than one thousand years ago and judging his theory from modern point of view without
mentioning his great innovations and ingenious insights into the subject is certainly unfair. I
would like to end by mentioning briefly only a few of Ibn-Sina’s many remarkable insights
on the conditionals:
1. Ibn-Sina is quite aware of the differences between conditionals and categorical
propositions and the impossibility of reducing the former to the latter generally. So he
writes:
“The person who thought that the proposition: ’Always: when
A is B, then H is Z’ is predicative because ‘Always: when this
is a man, then he is an animal’ is equal to ‘Every man is an
animal’ is mistaken for the following reasons”15
Ibn-Sina’s reasons are best summarized in one of his shorter books as follows:
A difference between the antecedent and the consequent, on
the one hand, and the subject and the predicate, on the other
hand, is that the subject and the predicate can be single terms,
but the antecedent and the consequent can never be… Another
difference between the antecedent and the consequent of the
conditional, and the subject and the predicate of the categorical
is that it is possible to ask about a subject predicate proposition
whether or not the predicate belongs to the subject. For
example when someone says “Zia is alive” you may ask
whether he is or he is not. But when someone utters a
conditional you cannot ask whether or not the consequent
belongs to the antecedent16.
2. Ibn-Sina realizes that some conditionals are in fact equivalent to some
categoricals. So he distinguishes what are now called general conditionals from
material conditionals:
[T]he connective in which the antecedent and consequent share
one part can be reduced to predicative propositions – as when
you say, for example, “If a straight line falling on two straight
lines make the angle on the same side such and such, the two
straight lines are parallel”. This is equivalent in force (fi
quwwati) to the predicative proposition: “Every two straight
lines on which another straight line fells in a certain way are
parallel” 17
3. Ibn-Sina’s classifications of connective and separative (disjunctive) conditionals,
their various combinations and his truth-functional treatment of them, within the
limitations of Aristotelian logic is, perhaps, unprecedented. Thus Rescher writes:
[A] fully articulated theory of logic of hypothetical and
disjunctive proposition is apparently first to be found in the
logic treatises of Avicenna18.
Conclusion:
Ibn-Sina’s theory of hypothetical syllogisms is supposed to be the
missing part of Aristotle’s logic. Ibn-Sina, by introducing quality and
quantity to the conditional, tries to reduce each conditional to a form
corresponding to its corresponding categorical in order to make the
rule of inferences which are applicable to the categoricals equally
applicable to the conditionals. But the parallels between the two logics
break down. Whether by introducing a kind of Davidsonian ontology
for situations or events and providing it with an existential import we
could save Ibn-Sina’s theory of invalid consequences remains to be
seen.
Acknowledgement
I would like to express my gratitude to Professor Wilfrid Hodges for
commenting on this paper and encouraging me to publish it soon. I
also would like to acknowledge the support I received from The Iranian
Research Institute of Philosophy during my sabbatical leave in Oxford.
Zia Movahed
References:
1. Nabil Shehaby. The Propositional Logic of Avicenna, A
Translation of al-Shifa: al-Qiyas, D.Reidle, Dordrecht-Holland,
1973.
2. Ibn-Sina, Daneshname Alaei, Anjoman-e-Asar-e melli, Tehran,
1952, P.89.
3. Nicholas Rescher. “Avicenna on the Logic of Conditional
Propositions”, Notre Dame Journal of Formal Logic, No. 1, January
1963, pp.48-58.
4. Ibid. p. 55.
5. Aristotle. Prior Analytics,50a39-50b5, trans. By H.P. Cook and H.
Tredennik, Loeb Classical Library,Vol.1, P. 389.
6. Nabil Shehaby, p. 35.
7. Al-Isharat wa al-Tanbihat, ed. Mojtaba Zareei, Boostan Kitab Qum,
2008, p. 82.
8. Nabil Shehaby, 39.
9. Ibid. 63.
10. Ibid. 63-64.
11. Ibid. 180.
12. Ibid. pp. 97-98 (an error in Shehaby’s translation has been
corrected).
13. Rescher, p. 50.
14. “On History of Logic of Proposition”, in Polish Logic, ed.
StorrmcCall, OUP, 1967, P. 79.
15. Nabil Shehaby, p. 62.
16. Ibn-Sina, Danesh-Nameh Alai, ed. and tr. Farhang Zabeeh
Martinus Nijhoff, The Hague, 1971, p. 25.
17. N. Shehaby, p. 55.
18. N. Rescher. “Avicenna on Logic of Conditional Proposition”, Notre
Dame Journal of Formal Logic, Vol. IV, No. 1, January 1963.

 

Source : irip.ir