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Chapter 16
On Descriptional Propositions in
Ibn S¯ın¯a: Elements for a Logical
Analysis
Shahid Rahman and Mohammad Saleh Zarepour
Abstract Employing Constructive Type Theory (CTT), we provide a logical anal-
ysis of Ibn S¯ın¯a’s descriptional propositions. Compared to its rivals, our analysis is
more faithful to the grammatical subject-predicate structure of propositions and can
better reflect the morphological features of the verbs (and descriptions) that extend
time to intervals (or spans of times). We also study briefly the logical structure of
some fallacious inferences that are discussed by Ibn S¯ın¯a. The CTT-framework makes
the fallacious nature of these inferences apparent.
Keywords Ibn s¯ın¯a (Avicenna) ·Modal syllogistic ·Descriptional (was
.f¯ı)
propositions ·Substantial (d
¯¯at¯ı) propositions ·Constructive type theory ·Temporal
logic ·Logical fallacies
16.1 Introduction
In his discussions of the various readings of modal propositions, Ibn S¯ın¯a’s focus is
mostly on a distinction which was later labelled the distinction between descriptional
(was
.f¯ı) and substantial (d
¯¯at¯ı) readings of a modal proposition.1Given that for Ibn S¯ın¯a
all categorical propositions are either implicitly or explicitly modal, the substantial–
1Hasnawi and Hodges (2017, p. 61) have correctly pointed out that ‘substantial’ is not Ibn S¯ın¯a’s
own term. Indeed, as Strobino and Thom (2017, p. 345) have mentioned, it is only in the later stage
of the tradition of Arabic logic that the terminology of ‘substantial’ and ‘descriptional’ became
mainstream. Some of the other names which have been employed to refer to the distinction under
discussion will be mentioned later in the chapter.
S. Rahman
Université de Lille, UMR-CNRS 8163: STL, Lille, France
e-mail: shahid.rahman@univ-lille.fr
M. S. Zarepour (B
)
Munich School of Ancient Philosophy, LMU Munich, Leopoldstr. 13, 80802 Munich, Germany
e-mail: saleh.zarepour@lrz.uni-muenchen.de
© Springer Nature Switzerland AG 2021
M. Mojtahedi et al. (eds.), Mathematics, Logic, and their Philosophies,
Logic, Epistemology, and the Unity of Science 49,
https://doi.org/10.1007/978-3-030- 53654-1_16
411
412 S. Rahman and M. S. Zarepour
descriptional distinction is in some sense applicable to the readings of all categorical
propositions.2This distinction is based on how (i.e., under which conditions) the
predicate of a categorical proposition is true of its subject. According to the substantial
reading, the predicate is true of the subject (perhaps with a certain alethic or temporal
modality) as long as the substance of the subject exists. On the other hand, according
to the descriptional reading, the predicate is true of the subject (again, perhaps with
a certain modality) as long as the substance of the subject is truly described by the
subject. To be clearer, consider the following proposition:
(1) Every Sis P.3
The difference between the substantial and descriptional readings of (1) can be
articulated as follows:
Substantial Reading of (1): Every S, as long as it exists, is P.
Descriptional Reading of (1): Every S, as long as it is S,isP.
It is in principle possible that a proposition is true on one of these readings and
false on the other. It is only the context which determines how a proposition must be
read to be true.4To give an example, consider the following proposition:
(2) Every bachelor is unmarried.
The substantial and descriptional readings of (2) are respectively as follows:
(3) Every bachelor, as long as he exists, is unmarried.
(4) Every bachelor, as long as he is bachelor, is unmarried.
These two propositions have different truth values. Contrary to (3)—which is
false—(4) is true. This is because a bachelor is unmarried only insofar as he is
described as a bachelor. So (4) is true. By contrast, it is in principle possible for
a person who is a bachelor in some period(s) of time to be married in some other
period(s) of time; this is so at least if we assume that ‘as long as’ has a temporal
meaning. In other words, it is not necessary for such a person to be always unmarried.
The mere existence of the substance of this person does not guarantee his being
unmarried. Thus (3) is false. There are, however, other propositions that are true on
the substantial reading. For example, consider the following proposition:
2Street (2002, Sect. 1.1) and Strobino and Thom (2017, Sect. 14.2.1) emphasize that for Ibn S¯ın ¯aall
propositions have either temporal or alethic modality. Absolute propositions are implicitly modal
and all other propositions are explicitly modal. Lagerlund (2009, p. 233) highlights that even the
absolute propositions can be taken to be descriptional.
3Strictly speaking, there is an important difference between a sentence and the proposition expressed
by it. Accordingly, it is a sentence (rather than a proposition) which can be read in different ways.
So what a substantial (respectively, descriptional) reading of a sentence expresses is a substantial
(respectively, descriptional) proposition. Nonetheless, such a clear difference between sentence and
proposition cannot be detected either in Ibn S¯ın ¯a’s own discussion of the substantial–descriptional
distinction or in the secondary literature on this issue. So to remain more focused on the main
points we would like to make—and of course for the sake of simplicity—we do not make the
sentence–proposition distinction bold.
4See Hodges and Johnston (2017, p. 1057).
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 413
(5) Every human is animal.
The predicate Animal is true of every human as long as s/he exists. Put otherwise,
what makes it true to say that every human is animal is the mere existence of human
substances. This means that not only the descriptional but also the substantial reading
of (5) is true.5Indeed, since every human exists if and only if s/he is human, the
substantial and descriptional readings of (5) express one and the same fact.
As Ibn S¯ın¯a himself insists, he is the first logician to have focused on the above
distinction and pondered on its fruitfulness for removing some difficulties with Aris-
totle’s syllogistic.6Since the distinction plays a crucial role in Ibn S¯ın ¯a’s syllogistic,
it is discussed in several places in his logical oeuvre.7Moreover, the distinction was
subject to continually heated discussions in Arabic logic after Ibn S¯ın¯a. For instance,
the distinction was accepted by R¯az¯ı and Kh¯una ˇg¯ı, on the one hand, and was seen
as redundant by Ibn Rushd.8The substantial sense of propositions corresponds to
what is called the ‘divided’ sense of propositions in the Latin tradition. However,
although the descriptional sense of propositions plays an important role in Arabic
syllogistic, it has no widely discussed counterpart in the Latin tradition.9These
observations strongly suggest that a comprehensive picture of Arabic syllogistic
from Ibn S¯ın¯a onwards cannot be achieved unless we have a clear logical analysis
of the aforementioned distinction. An effective and popular strategy for providing
such an analysis is to look at the different readings of a proposition through the lens
of modern formal logic. Therefore, it is important to find out which formal language
has the best capacity to capture various aspects of this distinction and the insights
behind it. In the literature, several attempts have been made to formalize the different
readings of propositions in the languages of classical predicate or temporal logics.10
In this chapter, we put forward an alternative based on Martin-Löf’s constructive
type theory (CTT).11 Compared to its rivals, our analysis is more faithful to the
5These examples are adopted from El-Rouayheb (2019, p. 24).
6See al-Qiy¯as (1964, Chapter III.1, p. 126) in which Ibn S¯ın¯a complains that previous philosophers
have not paid enough attention to this distinction.
7A famous passage in which Ibn S¯ın¯a discusses this distinction can be found in the logic part
of al-Iš¯ar¯at (1983, Chap. 4.2, pp. 264–266). For translations of this passage see Street (2005,
pp. 259–260) and Ibn S¯ın ¯a(1984, Chap. 4.2, p. 92). In the logic part of al-Naˇg¯at (1985, pp. 34–
37)—whose translation can be found in Ahmed (2011, Sect. 48)—Ibn S¯ın ¯a proposes six different
readings of necessary propositions. The second and the third readings include respectively substan-
tial and descriptional necessities. This distinction is discussed also in al-Qiy¯as (1964)andMant
.iq
al-Mašriq¯ıy¯ın (1910). Translations of some relevant passages from these two works are provided by
Hodges and Johnston (2017, Appendix A.2). They discuss a distinction between d
.ar¯ur¯ıand l¯azim
propositions in passages from Mant
.iq al-Mašriq¯ıy¯ın that is tantamount to the distinction between
substantial and descriptional readings of propositions.
8See El-Rouayheb (2017, pp. 72 & 81).
9See Street (2002, p. 133).
10See, among others, Rescher and vander Nat (1974), Hodges and Johnston, and Chatti (2019a,
2019b).
11See Martin-Löf (1984). In what follows, a basic familiarity with CTT is assumed. All the
background requirements can be found in Rahman et al. (2018, Chap. 2).
414 S. Rahman and M. S. Zarepour
grammatical subject-predicate structure of propositions and can better reflect the
morphological features of the verbs (and descriptions) that extend time to intervals
(or spans of times). It is worth noting that our focus will mostly be on the analysis of
the descriptional reading of propositions (which can also be called ‘the descriptional
propositions’ for the sake of brevity).
16.2 On What and How
The distinction between substantial and descriptional propositions seems to be related
to Joseph Almog’s famous distinction between what and how a thing is.12 More
precisely, it is relevant to how the subject term of a categorical proposition is true of
its objects in the two different readings we introduced above. The expression ‘ais
S’ can in principle encode two basic forms of predication. The expression encodes
what ais, if Srepresents an essential feature of a. For instance, if Sis a genus of
aor a category to which abelongs, then ‘ais S’ encodes (at least partially) what a
is.13 On the other hand, the expression ‘ais S’ encodes how ais, if Srepresents an
accidental feature of a. For instance, if Sis a description which can be sometimes
but not always true of a, then ‘ais S’ encodes (again, at least partially) how ais.
Returning to the distinction between substantial and descriptional propositions,
it seems that the subject term of a true substantial proposition establishes what its
12See Almog (1991,1996).
13As pointed out by Ranta (1994, p. 55), “the most serious criticism against the type-theoretical anal-
ysis of everyday language comes from intuitionistic thinking” (i.e., from the very same framework
within which CTT is developed). The concern is that although intuitionistic logic is an appro-
priate tool for mathematical reasoning, its application outside mathematics is inappropriate. This
is mainly because, by contrast with mathematical reasoning in which objects are almost always
fully presented, everyday reasoning is usually based on an incomplete presentation of objects. For
example, although a natural number can be fully presented by its canonical expression, giving a full
presentation of a continent seems to be extremely difficult, if not impossible. Stated differently, the
presentations of continents (like many other things) in the natural language is usually incomplete
in the sense that they are usually referred to by expressions which only partially determine what a
continent is. There seems to be no canonical expression of the non-mathematical objects like conti-
nents, humans, trees, etc. One possible way to deal with this concern, as Ranta (1994, pp. 55–56)
suggests, is “to study delimited models of language use, ‘language games’. Such a ‘game’ shows, in
an isolated form, some particular aspect of the use of language, without any pretention to covering
all aspects.” For example, a term like ‘human’, depending on the context, can be partially modelled
by the set of canonical names of the people who are referred to by the term ‘human’ in that specific
context. Accordingly, a set like {John, Mary, Jones, Madeline} can be considered as the interpre-
tation of the term ‘human’ in a certain context. The elements of such a set are fully represented
by the canonical names ‘John’, ‘Mary’, etc. Although we are still far from the full presentation
of humans in flesh and blood, we have a model which enables us to formalize certain fragments
of language in which talking about humans is nothing but talking about those four persons. By
developing such models, we can formalize larger fragments of language. An alternative dialogical
approach for dealing with this concern is put forward by Rahman et al. (2018, Sect. 10.4). This
dialogical alternative is inspired by Martin-Löf (2014).
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 415
objects are. By contrast, the subject term of a true descriptional proposition estab-
lishes how its objects are. For example, ‘ais human’ expresses what ais. But ‘a
is bachelor’ expresses how ais.14 So a reasonable expectation of an accurate anal-
ysis of substantial and descriptional propositions is that it must capture the difference
between what and how things are. This shows that classical logic cannot be an eligible
candidate for the frameworks in which such an analysis is supposed to be provided.
This is because those different forms of predication cannot be distinguished in clas-
sical logic, where there is only one way to analyse the expression ‘ais S’, namely as
the propositional function S(a). It is, therefore, an advantage of CTT over classical
logic that the language of the former is sensitive to the difference between these two
kinds of predication.
Suppose that ‘ais A’ expresses what ais. This can be captured in the framework
of CTT as a’s being a member of the category (or domain or type) A. The latter notion
can be represented in the language of CTT as follows:
a:A
In this expression ‘:’ can be read as ‘is’ (in the sense of expressing the what)or,
equivalently, as ‘belongs to’. Moreover, suppose that ‘ais B’ expresses how ais. In
other words, the expression describes aas having the property Band this expression
constitutes a proposition. This can be captured in the language of CTT as follows:
B(a): prop
In this expression ‘prop’ represents the category of propositions. Accordingly,
that ais an object of the type Awhich bears the description Bcan be expressed by
combining the two previous expressions as follows:
B(a): prop,givena:A.
More generally, B(x) constitutes a proposition when an xthat is of the category
A, bear the description B. Formally,
B(x): prop (x:A)
In fact, this expresses the well-formedness of the predicate B(x).15 It determines
the domain upon which the predicate is defined. More explicitly, it clarifies that
Bis predicated upon the objects which belong to the set A. In general, the CTT
formation rules for predicates are in accordance with Plato’s observation that how
something is cannot be asserted without presupposing what that thing is.16 Once
14This picture needs to be refined. As we will shortly see, even in the descriptional reading the
whatness of the objects of the subject term is mentioned, albeit only implicitly.
15In CTT, the well-formation is not only syntactic but also semantic. Consider, for example, the
predicate Hungry. The well-formedness of this predicate can be expressed by ‘Hungry(x): prop (x:
Animal)’, which reveals not only the correct syntactical use of that predicate but also the semantic
domain of the objects of which that predicate can be true.
16In their thorough and meticulous discussion of Plato’s Cratylus, Lorenz and Mittelstrass (1967)
highlight the distinction between naming (Ñνoμ´αζειν)—as establishing what something is—and
stating (λšγειν)—as establishing how something is. They (1967, p. 6) point out that “[t]he subject
416 S. Rahman and M. S. Zarepour
the well-formedness of a predicate has been established, we can produce formal
structures expressing that the predicate is true of some objects. For example, that B
is true of a, which is an arbitrary but fixed element of A, can be expressed by:
B(a)
Similarly, that some or all of the elements of Aare Bcan be expressed, respectively,
by the following expressions:
(∃x:A)B(x)
(∀x:A)B(x)
In all of the latter three expressions the formation rule B(x): prop (x:A)is
presupposed.17
By employing this machinery, the grammatical structure of categorical propo-
sitions can comprehensively be reflected in the language of CTT and this can be
counted as a significant advantage of CTT over classical logic. Consider the proposi-
tion ‘some students are good’. An oversimplified analysis of this proposition within
the classical logic with unrestricted quantification would be as follows:
(∃x)[Student(x)∧Good(x)]
By contrast, in the language of CTT, ‘some students are good’ could be formalized
by this notation:
(∃x:Student)Good(x)
The latter translation restricts Good to the grammatical subject of the proposition
i.e., Student. It singles out the set of those students that are good insofar as they are
students. Quite the contrary, the former translation does not distinguish the subject
and predicate. It coarsely refers to persons who are both good and student, no matter
whether or not that those persons are good at being a student or at something else.
To discuss one of Ibn S¯ın¯a’s own examples, consider the following propositions:
(6) Imraa al-Qays is good.
(7) Imraa al-Qays is a poet.
(8) Imraa al-Qays is a good poet.
Ibn S¯ın¯a argues that (8) cannot be concluded from the conjunction of (6) and (7).18
For him, such an argument is fallacious. But if we translate these propositions in the
language of classical predicate logic, we cannot see why this argument is fallacious.
has to be effectively determined, i.e., it must be a thing correctly named, before one is going to state
something about it”.
17In CTT, the judgment that the proposition B(a) is true is usually represented by ‘B(a)true’. But
as long as we are considering a proposition itself (without making any judgment that it is true) we
do not really need to add ‘true’.
18Ibn S¯ın¯a proposes this example in the logic part of al-Iš¯ar¯at (1983, Chap. 10.1, pp. 501–502). We
are grateful to Alexander Lamprakis for drawing our attention to this example.
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 417
Suppose that ‘a’ refers to Imraa al-Qays, and ‘Poet’ and ‘Good’ represents, respec-
tively, being a poet and being good. An oversimplified translation of (6)–(8) in the
framework of classical logic (with unrestricted domain of quantification) yields:
(6-CL) Good(a)
(7-CL) Poet(a)
(8-CL) Poet(a)∧Good(a)
In such a framework, concluding (8-CL) from (6-CL) and (7-CL) is unproblem-
atic. But this is an undesirable result for Ibn S¯ın¯a. This shows that such a framework
is not suitable for formalizing Ibn S¯ın¯a’s logic. By contrast, in CTT the propositions
(6)–(8) would be translated as follows:
(6-CTT) Good(a): prop,givena:Human.19
(7-CTT) Poet(a): prop,givena:Human.
(8-CTT) Good(a): prop,givena:Poet.
(8-CTT) cannot be concluded from (6-CTT) and (7-CTT); and this is exactly what
Ibn S¯ın¯a expects.20
16.3 Substantial and Descriptional Propositions
Consider the following proposition:
(9) Every moving is changing.21
The substantial and descriptional readings of (9) can be stated respectively as
follows:
(10) Every moving, as long as it exists, is changing.
(11) Every moving, as long as it is moving, is changing.
It is not the case that every moving object, as long as its essence exists, is changing.
Rather, it is changing as long as its essence can be described as moving. A moving
object can in principle stop moving at some time without ceasing to exist. So (9) is
true only if it is read in the descriptional sense as (11), rather than in the substantial
sense as (10). To see how the difference between (10) and (11) can be mirrored in
their formal constructions in the language of CTT, a deeper investigation about the
subject and predicate of these propositions needs to be carried out.
At first sight the subject of (9) is Moving and the predicate is Changing. However,
Moving is a description whose bearer is concealed (mud
.mar). Thus it is legitimate
19That Imraa al-Qays belongs to the category Human is not explicitly mentioned in (6) and (7).
But it is necessary be added to the picture. See the next section for more details on this issue.
20Notice that if we simply take Good-Po et (x) as a predicate, then from Good-Poet (a) we cannot
infer either that ais good or that ais a poet.
21The example is borrowed from the logic part of al-Iš¯ar¯at (1983, Chap. 4.2, p. 265).
418 S. Rahman and M. S. Zarepour
to ask what the category of moving objects is. Put otherwise, what is the type of
the things which are supposed in (9) to be Moving?22 (9) itself does not determine
whether moving things are supposed to be, for example, humans, animals, or bodies
(aˇgs¯am) in general. This can be established only by the context in which (9) is stated.
But in any case it is undoubtable that moving things must be considered to be of a
specific category, even if this category is not explicitly mentioned. To preserve the
generality of our analysis we can assume that this category is O. It means that (9)
expresses a fact about those objects of the type Owhich are moving. Depending
on the context, Ocan be replaced with the categories like Human,Animal,or,more
generally, Body.
According to this understanding, every element which lies in the scope of the
universal quantifier of (9) has two different aspects. One aspect reveals what it is
(i.e., it belongs to O) and the other reveals how it is (i.e., it is moving). In other
word, if zis an element in the scope of the universal quantifier of (9)—i.e., if zis
one of those objects that are moving—it can be represented as having the canonical
form < x,b(x) > in which xis of the type Oand b(x) is a method evidencing that
xcan be described as being Moving.b(x) can be seen as a truth-maker or a proof
for the proposition that xis moving.23 The difference between the substantial and
descriptional readings of (6) is rooted in how these two different components are
combined with each other and in the roles each of them plays in the predication. The
descriptional reading of (9)—i.e., (11)—can be formalized in the language of CTT
as follows:
In this translation, left can be interpreted as a projection function which extracts
the left-side element of every z. Similarly, right can be defined as the projection
function which extracts the right-side element of every z. So if z is considered to be
acompositum of the form <x,b(x)> in which xis of the type Oand b(x)isamethod
evidencing that xcan be described as moving, then left(z)=left(<x,b(x)>) =x:O
and right(z)=right(<x,b(x)>) =b(x): Moving(x)(x:O). The above construction
can be seen as involving an anaphora whose head is Moving-O(i.e., it is an object of
the type Othat is Moving). The tail of the anaphora is constituted by the projection
function left(z) which picks out those objects of the type Othat are described as
moving in the grammatical subject.
Philosophically speaking, in the descriptional reading of (9), the two different
aspects of the subject (i.e., the one which reveals what it is and the one which reveals
22On how and why the bearer of the subject of descriptional propositions is concealed see Schöck
(2008, pp. 350–351).
23Atruth-maker is in fact a rudimentary form of what is called proof-object in CTT. See Ranta
(1994, p. 54). However, in the context of this chapter, we assume ‘truth-maker’ and ‘proof-object’
to be synonymous terms. It is also worth mentioning that in the CTT-framework one and the same
true proposition has according to the rule more than one object which makes it true.
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 419
how it is) are merged into a compound unity, so that each constituent carries the infor-
mation about the other. Now it is this compound unity upon which Changing is pred-
icated. More precisely, the subject is assumed to have a whatness (i.e., its belonging
to Owhich can also be seen as the substantial component of the subject) and a
howness (i.e., its being moving which can also be seen as the descriptional compo-
nent of the subject). In the descriptional reading these two aspects are combined with
each other to make a new compound whatness (i.e., Moving-O) upon which another
howness (i.e., Changing) is predicated. Such a compound whatness plays no role in
the substantial reading of (9). In (10), the objects upon which Changing is going to be
predicated are still selected by the description Moving from the domain of the objects
of type O. Nonetheless, the truth of such a substantial predication is not supposed
to be dependent on whether or not those objects preserve the description Moving.
In general, in the substantial predication the objects of predication are selected by a
description but the truth of the predication does not depend on whether or not those
objects preserve the description. By contrast, in the descriptional reading not only
are the objects of predication selected by the description but also the predication is
true only as long as those objects preserve the description. Given these observations,
our proposal for the translation of the substantial reading of (9)—i.e., (10)—goes as
follows:
Here the description Moving is characterizing the domain of the objects of the type
Oupon which Changing is predicated. Nonetheless, the description is not a compo-
nent of the unified whatness upon which Changing is predicated. The subject is not
considered as a compound entity of which the description Moving is an irremov-
able component. Moreover, to highlight the distinction between the substantial and
descriptional reading, instead of left and right, here we use the projection functions
first and second. The main difference between these two couples of the projection
functions is that when one of the functions of the former couple extracts an element
of the pair <x,b(x)> , the selected element carries some piece of information about
the other element. By contrast, what is selected by one of the functions of the latter
couple does not contain any piece of information about the the other element. So
first(z) selects one instantiation of Oand forgets about the second component of z.24
To generalize our formalizations, reconsider the proposition (1)—i.e., ‘every S
is P’—and suppose that Sis a description whose bearers are of the type D.The
descriptional and substantial reading of this proposition can be formalized as follows:
Substantial Reading of (1): {∀z:(x:D|S(x))} P(first(z))
24To put it in more technical language, if in the proposition ‘every B, as long as it exists, is C’, the
bearers of the description Bare of the type A,thenfirst(z): Amust be understood as what Sundholm
(1989, p. 10) calls ‘A-injection’.
420 S. Rahman and M. S. Zarepour
Descriptional Reading of (1): {∀z:(∃x:D)S(x)} P(left(z))
Now we can easily see the advantage of this analysis over one of its rivals which
is proposed in the framework of classical predicate logic. Saula Chatti formalizes
the descriptional reading of (1) as follows:
(∀x)[S(x)⊃(S(x)⊃P(x))] ∧(∃x)S(x)
As she herself pointed out, the above proposition is equivalent to:
(∀x)(S(x)⊃P(x)) ∧(∃x)S(x)
But this is exactly what classical predicate logic proposes for the formalization of
all A-form absolute propositions. So it cannot reflect how the descriptional reading
of a proposition differs from the other possible readings.25
These formalizations enable us to see better how some seemingly contradictory
propositions, like the two below, can be both true at the same time:
(12) Every sitting dog, as long as it exists, can walk.
(13) Every sitting dog, as long as it is sitting, cannot walk.
In (13) the subject must be taken as a compound entity to which the predicate
Cannot-walk applies. Here the projection function left selects only those dogs that
are sitting. In (12), the function first takes the subject in its substantial sense. This
means that although the subject is analysed into Dog and Sitting, when first selects
a dog, its selection does not carry information on whether or not the dog is sitting.
So the formal translations of (12) and (13) go respectively as follows:
(12-CTT) {∀z(x:Dog |Sitting(x))} Can-walk(first(z))
(13-CTT) {∀z:(∃x:Dog)Sitting(x)} Cannot-walk(left(z))
So far so good. But our conception of the roles Ibn S¯ın¯a considers for the substan-
tial–descriptional distinction in different contexts will not be comprehensive until
we understand how the time-parameter and the existence predicate can be added to
the picture.
16.4 Time Parameters
Time parameters add more complexities to the structure of descriptional propositions.
But, fortunately, CTT has the capacity to handle them. Reconsider the following
descriptional proposition propositions:
(11) Every moving, as long as it is moving, is changing.
Which is equivalent to:
25See Chatti (2019b, pp. 113–114). Since Ibn S¯ın¯a considers existential import for A-form
propositions, Chatti emphasizes that the above formulas must include the conjunct ‘(∃x)S(x)’.
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 421
(14) Every moving is changing while it is moving.
Now if we add the time parameter, (14) can be read as follows:
(15) Every moving is changing all the time it is moving.
This proposition is usually rendered as an equivalent of the following proposition:
(16) Every (sometime-)moving is changing all the time it is moving.26
Accordingly, (16) is usually formalized as follows:
(16-CL) (∀x)[(∃t)Moving(t,x)⊃(∀t)(Moving(t,x)⊃Changing(t,x))] ∧
(∃t)(∃x)Moving(t,x)
In this analysis ‘x’ and ‘t’ are variables for respectively the moments of time and
the bearers of the description Moving.‘Moving(t,x)’ can be naturally read as ‘xis
moving at t’.27 Our analysis is, however, quite different. But before presenting our
proposal, we should first discuss some preliminaries on how temporality can be dealt
with in the framework of CTT.
16.4.1 Preliminaries on Temporal Reference in CTT
16.4.1.1 Time Scales
Usually when we are talking about time, we are talking about a specific time scale.
Depending on the length of the temporal units, we can introduce different time
scales. For instance, we can talk about either years, or months, or days, or hours,
etc. These time scales can be represented as, respectively, Year,Month,Day,etc.
28
Each of these time scales is a temporal category. The time scale we are talking about
naturally depends on the context in which the proposition is stated. For example,
when someone is talking about waking up early in the morning, the time scale such a
person considers is probably Day. But when the president of a university is presenting
statistics about their graduates, her/his time scale is likely to be Year. In general, we
can represent the time scale we are talking about as T.
16.4.1.2 Time Spans
An advantage of the CTT-framework as implemented for time reference is that it
provides the opportunity of considering not only moments of time, but also time spans
26Hasnawi and Hodges (2017, p. 61) label such propositions as ‘(a-)’ which can be considered as
an abbreviation for ‘A-form l¯azim’ propositions.
27This formalization is in accordance with what Hodges and Johnston (2017, p. 1061) put forward
following Rescher and vander Nat (1974). The conjunct ‘(∃t)(∃x)Moving(t,x)’ is added to guarantee
the existential import of the proposition.
28For a detailed technical definition of time scales, see Ranta (1994, Sect. 5.1).
422 S. Rahman and M. S. Zarepour
and intervals with a beginning and an end. This is particularly important because
actions like moving, running, etc. do not happen in a moment. Rather they should
be considered as extended events which happen in temporal intervals. Indeed, one
of the main shortcomings of the aforementioned analysis of (16) is that it does not
consider moving and changing as extended events. So it is important to have tools
to express the occurrence of events not only in singular moments of time but also
in temporal intervals. This helps us to formally describe how an object that bears a
specific description in a specific span of time can also bear some other descriptions
in some specific sub-spans of the former span. It is also possible to express how an
object can have the same description with different qualifications in different spans of
time. For example, an object that is moving in a span of time might be slow-moving
in some parts (or sub-spans) of that span and fast-moving in some others. So it seems
to be crucial to see how a span of time can be defined in the framework of CTT.
The category of the spans of a time scale Tcan be defined as the Cartesian product
of Tand the set of natural numbers N.29 More precisely:
span(T)=T×N
To make it clearer, a span of the time scale Tis a pair whose first element refers to
the beginning point of that span in Tand whose second element refers to the number
of temporal units (of the scale T) which must be added to the beginning point to
form the span under discussion. Stated differently, the second element determines
the length of the span. So if d=<t0,n>:span(T), dis a span of the time scale T
which begins at t0and ends at t0+n. The span dcan also be represented as [t0,t0
+n]: span(T). As we will shortly see, the following functions are also useful:
left(d)=begin(d)=t0:T
end(d) =t0+n:T
right(d)=length(d)=n:N
As an example of the spans of time in the time scale Day, consider the following
span:
<14 June 2018, 31> : span(Day)
This span of time begins on 14 July 2018 and extends for 31 days. This is exactly
the interval in which Football World Cup 2018 took place.
It is noteworthy that since 0 is a member of N, every singular moment of the
time scale Tcan be considered as a span of the length 0. In other word, every tof
the time scale Tcorresponds to <t, 0> which is a member of span(T). This shows
that everything expressible by the terminology of singular moments of time is also
expressible by the terminology of time spans, though the other way around does not
hold.
29For a detailed technical definition of time spans, see Ranta (1994, p. 115).
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 423
16.4.1.3 Saturation Versus Enrichment
There are at least two different approaches for dealing with temporal reference in
the CTT-framework.30 More clearly, a proposition which expresses the occurrence
of an event (or fact) in a span of time can be seen in at least two different ways. Such
a proposition can be seen either as an incomplete propositional function that can be
saturated by that specific span of time or as an event (or a fact) that can be timed by a
timing function. These two formal terminologies are translatable into each other. This
means that everything expressible by one of these two approaches is also expressible
by the other. Nonetheless, there is a significant philosophical difference between
these two approaches. In the first approach time is primitive. Temporal entities (i.e.,
singular moments of times or time spans) are independent entities which can be put
as the arguments of propositional functions. So, ontologically speaking, complete
propositions in some sense depend on these temporal entities. By contrast, in the
second approach, events (or facts, or truth-makers of the propositions which express
those events) are primitive individuals which can be put as the arguments of the
timing functions. Thus, in a sense, time is dependent on events. Inspired by François
Recanati’s terminology, we call these two approaches, respectively, ‘saturation’ and
‘enrichment’.31
According to the saturation approach, ‘Aoccurs at the span dof the time scale T’
can be formalized as a propositional function Athat is saturated by d. So:
A(d): prop (d:span(T))
By contrast, according to the enrichment approach Aitself is a fully saturated
proposition which is made true by different events (or facts) at different time spans.
Equivalently, it has different truth-makers or proofs at different time spans. These
truth-makers can be timed by a timing function. Informally speaking, the timing
function operates upon the set of truth-makers (or justifications, or proofs) of Aand
determines the time span in which such a truth-maker is obtained.32 So if xis a truth-
maker of the proposition A(i.e., if xis an event or fact whose occurrence makes
Atrue), then the timing function τwould determine the span of time in which xis
obtained. So the role of τcan be defined as follows33:
30See Ranta (1994, Sect. 5.4).
31This terminology is borrowed from Recanati (2007a,2007b).
32Recall that as pointed out before, it is assumed that a proposition has different truth-makers (or
proofs or justifications). In the present context this amounts to the assumption that a proposition
has different truth-makers during different time spans. That a proposition is true in a specific time
span is equivalent to that one of its truth-makers is obtained in that time span.
33See Ranta (1994, p. 108).
424 S. Rahman and M. S. Zarepour
A:prop
τ(x): span(T)(x:A)
For example, that a human xis running in the time span dcan be expressed by
the saturation approach as follows:
Running(x,d): prop (x:Human,d:span(T))
Quite differently, the same proposition can be formalized by the enrichment
approach as follows:
Running(x): prop (x:Human)
τ(b(x)) =d:span(T)(x:Human, b(x): Running(x)).34
In this formalization b(x) is a truth-maker or evidence for the proposition
Running(x); and τis a timing function which determines the time span in which
b(x) is obtained. So in a sense the time span dis eventually defined by that specific
truth-maker of Running(x) that is obtained in that span. In other words, the time span
dis given by the operation of the timing function τupon the event which makes
Running(x) true. Borrowing Aristotelian terminology, we can say that in the enrich-
ment approach time elements are measurements—i.e., timing operations—of (and,
consequently, dependent on) events.
After explaining these preliminary points, we are now well equipped to analyse
the temporal interpretation of descriptional propositions through both the saturation
and the enrichment approaches.
16.4.2 Descriptional Propositions Relativized by Saturation
Reconsider the proposition (15):
(15) Every moving is changing all the time it is moving.
As we mentioned, moving and changing are extended events which happen in
time spans, rather than in singular moments of time. So it is plausible to restate (15)
in the language of time spans. If we do so, the result would be something like the
following:
(17) Every moving is changing in all the spans in which it is moving.
34In order to avoid notational complexity we omitted one variable within the timing function.
Indeed, strictly speaking, the correct formalization must be τ(x,b(x)) =d:span(T)(x:Human,
b(x): Running(x)).
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 425
If we suppose again that the bearers of the description Moving are of the type O
and that our time scale is T, then our proposal for the logical analysis of (17), in the
saturation approach, goes as follows:
(17-CTT-S) {∀z:(∃d:span(T)) ((∃x:O)Moving(d,x))}
Changing(left(z),left(right(z))
This can be glossed as:
(17-CTT-S*) Every zthat is an element of the set of those objects that are moving
at some time span dis subject to change at the time span in which it is moving.
More precisely, here zis a variable for those time spans dat which some xof the
type Ois moving. So zcan be considered as a pair of the canonical form <d,<x,b(x)
. Thus, left(z) gives the first constituent of zwhich is some time span dat which the
moving thing is moving. The right constituent of zis the pair of the moving object x
and the evidence b(x) which shows that xbears the description Moving at d. In other
words, in the time span d,b(x) is the truth-maker of the proposition that ‘xis moving’.
Hence, while left(z) yields some time span d,left(right(z)) provides the object that
is moving at that time span. This is the object of which the grammatical predicate
Changing is true. To generalize this approach, consider the following proposition:
(18) Every Sis Pin all the spans in which it is S.
If the bearers of Sare of the type Dand our time scale is T, then the logical
analysis of (18) in the language of CTT and based on the saturation approach goes
as follows:
(18-CTT-S) {∀z:(∃d:span(T)) ((∃x:D)S(d,x))} P(left(z),left(right(z)))
As we previously mentioned, objects that have a description in a specific span
can be described as having other properties in some specific sub-spans of the former
span. Now we are well equipped to formalize some of the propositions which express
such situations. Consider the following example:
(19) Everyone who studies mathematics as an undergraduate spends the first year studying
calculus.
To formalize this proposition we can take our time scale to be Year. We can also
take ‘Math(d,x)’ to represent that xstudies mathematics as an undergraduate in the
span d(where d:span(Year) and x:Human). More precisely, we assume that xstarts
studying mathematics at begin(d) and graduates at end(d). If so, the first year of d
can be referred to by the following function:
426 S. Rahman and M. S. Zarepour
first-year(d)=<begin(d), 1 > : span(Year)
So first-year(d) refers to the span of time which begins at begin(d) and extends
for 1 year. Now if ‘Calculus(d,x)’ expresses that xstudies calculus in d, then the
formal interpretation of (19) would be as follows:
(19-CTT-S) {∀z:(∃d:span(Year)((∃x:Human)Math(d,x))} Calculus(first-
year(left(z)),left(right(z))
Here again zis the pair < d,<x,b(x) > > in which dis a span of time scale Year,x
is a human, and b(x) is the evidence that xcan be described as studying mathematics
at the undergraduate level. To generalize this example, suppose that for every time
span d, the function s-period(d) determines a specific period of d.Sofirst-year is
an instance of this kind of functions. But s-period(d) can be defined to determine,
for example, the first quarter, the second third, or any other specific part of d.Now
consider a proposition of the following general form:
(20) Every Sis Pin a s-period of the time span in which it is S.
This can be formalized as follows:
(20-CTT-S) {∀z:(∃d:span(T)((∃x:D)S(d,x))} P(s-period(left(z)),left(right(z))
Developing this approach would help us to formalize some other types of temporal
propositions which play a crucial role in the temporal logic of Ibn S¯ın¯a. (21) is one
such proposition:
(21) Every (sometime-)Sis Psometime while it is S.35
The proposition has been usually formalized as follows:
(21-CL) (∀x)[(∃t)S(t,x)⊃(∃t)(S(t,x)∧P(t,x))] ∧(∃t)(∃x)S(t,x)36
To analyse (21) using the saturation approach of CTT, we need to add a time span
quantifier on the predicate side. Accordingly, (21) can be formalized as:
(21-CTT-S) {∀z:(∃d1:span(T)((∃x:D)S(d1,x))} (∃d2:span(T)) [S(d2,
left(right(z)) ∧P(d2,left(right(z))]
Here zmust still be considered as a pair of the canonical form <d1,<x,b(x).
Informally, (21-CTT-S) says that for every object xof the type Dthat is Sin a time
span d1, there is a time span d2in which that object is Pwhile it is S. Now we can
turn to the enrichment approach.
35Hasnawi and Hodges (2017, p. 61) label such propositions as ‘(a-m)’, which can be considered
as an abbreviation for ‘A-form muw¯afiq’ propositions.
36This formalization is suggested by Hodges and Johnston (2017, p. 1061), following Rescher and
vander Nat(1974). Again, the conjunct ‘(∃t)(∃x)S(t,x)’ is added to preserve the existential import.
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 427
16.4.3 Descriptional Propositions Relativized by Enrichment
As we previously mentioned, in the enrichment approach time elements are not
primitive and have no independent existence. They are dependent on events which
make propositions true. In other words, they are dependent objects—i.e., functions.
Since this philosophical conception of time is closer to how Ibn S¯ın ¯a understands this
notion, it is more plausible to analyse his temporal propositions based on the enrich-
ment approach (rather than based on the saturation approach in which time elements
are primitive and have independent existent).37 To see how temporal propositions
can be formalized by the enrichment approach, reconsider the proposition (17):
(17) Every moving is changing in all the spans in which it is moving.
In the enrichment approach, this proposition can be formalized as:
(17-CTT-E) {∀z:(∃x:O)(Moving(x)} Changing(left(z)) AT(τ(right(z)))
Here, zis a pair of the canonical form < x,b(x)>.Soleft(z)=xand right(z)
=b(x). As a result, τ(right(z)) amounts to τ(b(x)). Since b(x) is a truth-maker of
‘xis moving’, τ(b(x)) yields a time span within which xis moving. Finally AT is
an operator that operates upon propositions. Informally speaking, for every span d
and every proposition A,AAT ( d) indicates that Ais the case within the time span
d.SoChanging(left(z)) AT(d) means xis changing within the time span d. Putting
together all of these observations, what (17-CTT-E) says is that every xof the type
Ois changing in all the time spans in which it is moving. In other words, if xis
moving in d, it would also be changing in this span. To generalize this formalization,
reconsider (18):
(18) Every Sis Pin all the spans in which it is S.
If we suppose that the bearers of the description Sare of the type D, then (18) can
be analysed as:
(18-CTT-E) {∀z:(∃x:D)(S(x)} P(left(z)) AT(τ(right(z)))
Finally, reconsider (21):
(21) Every (sometime-)Sis Psometime while it is S.
To formalize (21) in the enrichment approach we need to add a temporal quantifier
on the side of predicate. In this respect, there is no difference between this approach
and the saturation approach. (21) can be formally analysed as:
(21-CTT-E) {∀z:(∃x:D)(S(x)} (∃d:span(T))[τ(right(z)) =d∧P(left(z)) AT(d))]
Here again zis a pair of the canonical form < x,b(x) > in which xis an object of
the type Dand b(x) is a witness (or proof) for that xis S.Soτ(right(z)) gives a time
37For Ibn S¯ın¯a time is the number or magnitude of motion. Although he does not explicitly talk
about events, his definition of time shows that he does not consider an independent existence for it.
This suffices to convince us that the enrichment approach is preferable to the saturation approach.
For a detailed discussion on Ibn S¯ın¯a’s view regarding time, see Lammer (2018, Chap. 6).
428 S. Rahman and M. S. Zarepour
span within which xis S. Accordingly, what (21-CTT-E) says is that for every xof
the type Dthat is S, there are some spans din which x is both Sand and P.
16.5 Existence With and Without Existence Predicate
According to the ontological system underlying CTT, the fact that a type has been
instantiated amounts to showing that the type is not empty. In other words, we should
understand the instances of types as witnessing the existence. Therefore, we do not
really need to capture the existential import of propositions by adding conjuncts
which guarantee the existence of the subject. This can be counted as another advan-
tage of our analysis over some of the earlier studies.38 If the import is automatically
guaranteed, then a fortiori there is no need to the existence predicate. It is so, at least,
unless the existence predicate is defined in some way that allows us to distinguish
those instantiations that witness existence from those that do not.
It is worth mentioning that associating the instantiation of a type to the existence
of its elements does not prevent us from considering different sorts of existence.
Indeed, each instantiation can be understood as representing the kind of existence
associated to the type they instantiate. For example, the members of the type N(i.e.,
natural numbers) do not have the same form of existence as the members of the type
Man. Each type has its own form of existence depending upon how the process of its
instantiation is defined. Technically speaking, such a process is defined by the rules
that introduce the canonical elements of the type under discussion. For example,
the notion of existence associated with the set of natural numbers is construction by
mathematical induction. By contrast, members of the type Human have a completely
different mode of existence.39
Having made these points, if someone still insists on adding the existence predicate
to the picture, there seems to be no technical difficulty in the way of fulfilling this
desire. To give examples we can reconsider the logical analysis provided in (18-CTT-
S):
(18-CTT-S) {∀z:(∃d:span(T)) ((∃x:D)S(d,x))} P(left(z),left(right(z)))
The existence predicate can be added to the side of subject as follows:
(18-CTT-S*){∀z:(∃d:span(T)) ((∃x:D)S(d,x)∧Exists(d,x))}
P(left(z),left(right(z)))
If desired, the existence can also be added to the side of the predicate as follows:
(18-CTT-S**) {∀z:(∃d:span(T)) ((∃x:D)S(d,x)∧Exists(d,x))}
P(left(z),left(right(z))) ∧Exists(left(z),left(right(z)))
38See notes 25, 27, and 36.
39The existence of the subject matter of non-mathematical propositions can be presented either by
‘logical games’ or by dialogical verification procedures. See note 13.
16 On Descriptional Propositions in Ibn S¯ın¯a: Elements … 429
By following the same approach, the existence predicate can be added to the
propositions relativized by enrichment. However, it should be borne in mind that the
existence predicate cannot be introduced rigorously unless we clarify exactly how it
is formed. For example, we can say that the objects of the category Oexist if and only
if they are temporally and spatially indexed.40 So if the time scale and the category
of location are represented by, respectively ‘T’ and ‘L’, the existence predicate can
be defined as follows:
Exists(x,y,z): prop (x: O,y: L, z:T).
Obviously, such a definition of this predicate needs to be embedded into the logical
structure of the preceding sentences. We will leave the task of modifying the formal
structure to the reader.
16.6 Conclusion
In this chapter we provided a new logical analysis of Ibn S¯ın¯a’s descriptional propo-
sitions in the framework of CTT. Assuming an anaphoric structure for propositions,
we showed that the grammatical predicate of a descriptional proposition is true of an
anaphoric tail that encodes not only what the object of predication is, but also how
it is. By contrast, in the case of a non-descriptional propositions the anaphoric tail
only encodes what the object of predication is.
Our analysis has at least three advantages over its rivals. First, it better reflects
the grammatical structure of propositions. Second, it is quite flexible for capturing
different temporal features of propositions. In particular, it can enable us to formalize
not only sentences which talk about singular moments of time, but also sentences
which include actions extended in time spans. Third, the existential import of the
universal propositions is automatically guaranteed by the instantiation of the types
about which those propositions talk. So we do not need to consider an additional
conjunct to our translations just to make sure that the existential import is preserved.
Nonetheless, as we saw, there is no obstacle in the way of adding the existence
predicate either to the side of subject or to that of the predicate.
Our main focus in this chapter was on how some propositions can be formalized
in the framework of CTT. We did not touch on how these formal constructions can
be put into the syllogism. This should be postponed to a future project. Nonetheless,
there is an important insight about the theory of the syllogism which can be seen
from here. We showed that the anaphoric structure is quite general and is applicable
even when the temporal dimensions of propositions are completely put aside. But
if we analyse the propositions of a syllogism as such anaphoric structures, then it
would be obvious that the subject-term must always contain a descriptional element
in relation to the individual of a domain shared by the premises. So, in a sense, it is
40An alternative approach is based on the introduction of the notion of ontological dependence. See
Rahman and Redmond (2015).
430 S. Rahman and M. S. Zarepour
assumed by the premises that the kind of the involved object in the inference to be
drawn is known. This is one of the general insights on the theory of the syllogism
which can be acquired from Ibn S¯ın¯a.
Acknowledgements We are thankful to Leone Gazziero (STL), Laurent Cesalli (Genève), and
Tony Street (Cambridge), leaders of the ERC-Generator project “Logic in Reverse: Fallacies in
the Latin and the Islamic traditions,” and to Claudio Majolino (STL), associated researcher to
that project, for fostering the research leading to the present study. We should also thank Vincent
Wistrand (UMR: 8163, STL) and Alexis Lamprakis (München) for many fruitful discussions from
which we have benefited a lot. The present paper was written while Mohammad Saleh Zarepour
was a Humboldt Research Fellow at LMU Munich. We are thankful to Alexander von Humboldt
Foundation for their support.
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