=Paper=
{{Paper
|id=Vol-1132/paper3
|storemode=property
|title=Perfect Syllogisms and the Method of Minimal Representation
|pdfUrl=https://ceur-ws.org/Vol-1132/paper3.pdf
|volume=Vol-1132
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==Perfect Syllogisms and the Method of Minimal Representation==
https://ceur-ws.org/Vol-1132/paper3.pdf
Perfect Syllogisms and the Method of Minimal
Representation
Sumanta Sarathi Sharma
School of Philosophy & Culture
Faculty of Humanities and Social Sciences
Shri Mata Vaishno Devi University
Katra, Jammu & Kashmir 182 320, INDIA.
Email: sumantas@gmail.com, ss.sharma@smvdu.ac.in
Abstract—In this paper, the Method of Minimal Representation a perfect syllogism, whereas less-imperfect syllogism takes
(MMR, i.e., an alternative diagrammatic technique to test the fewer steps.
validity of syllogisms) is employed to differentiate perfect (first
figure) from imperfect syllogisms (other figures). It demonstrates Aristotle demonstrated the validity or invalidity of syllo-
that the validity of perfect syllogisms can be exhibited applying gisms using three methods. They are, namely, ecthesis (also
lesser number of rules, which govern the proposed method. This expressed as ekthesis), reductio ad absurdum (also known
is also in accordance with Aristotle’s dictum of ceteris paribus, as reductio per impossible or reductio ad impossible) and
which assumes the superiority of a demonstration that is derived reduction. In the following subsections, I briefly explain these
through fewer postulates or hypotheses. This paper is divided
into four sections. The first section gives a brief exposition on methods taking some examples.
the historical preliminaries of perfect syllogisms. The second
section elaborates the method of minimal representation. In
the penultimate section, we test first figure syllogisms and
contrast them with other figures by the proposed method. In the
A. Ecthesis
concluding section, we demonstrate the primacy of first figure
among all figures. The proof by ecthesis is a method of exposition with which
Aristotle validated a given syllogism. However, he has not
I. I NTRODUCTION used this method extensively. According to Smith, the proof
Syllogism, according to Aristotle, “is [a] discourse in which, by ‘ecthesis or ‘setting out methodology is used several times
certain things being stated, some other thing than that what is by Aristotle to provide an alternative deduction schema for
stated follows [out] of necessity from their being so” [McKeon completing certain syllogistic moods [Smith 1983: 224-232].
1946:66]. The expressions “certain things be stated” forms the He also points, Aristotle never explains why he includes this
premise(s) whereas “some other thing” refer to conclusion. alternative deductions and there is some debate about what
In other words, the implicit assertion(s) in the premise(s) is exactly the procedure is.
made explicit through the conclusion. With reference to this Let us try to understand this procedure taking the example
premise to conclusion transformation, he called the first figure which Aristotle took. Let ‘P belong to all M’ and ‘S belong
syllogisms, perfect. to all M’. Then, ‘P belong to some S’. In order to prove this,
In Organon, he said, “I call that a perfect syllogism1 which Aristotle introduced an element of novelty by assuming, say ‘c
needs nothing other than what has been stated to make it (a member of the class)’ which belong to M. Then ‘c’ belongs
plain what necessarily follows; a syllogism is imperfect, if to P and also ‘c’ belongs to S. It means, that ‘c’ belongs to
it needs either one or more propositions, which are indeed both P and S. If ‘c’ is a common member of P and S, then
necessary consequences of the terms set down, but have not there is something common between P and S. This then entails
been expressly stated as premises”[Ibid:66]. The second and that P belongs to some S.
third figures are imperfect figures2 . Alexander maintains that the constant ‘c’ is a singular
Imperfect syllogisms are not invalid syllogisms, rather ‘po- term given by perception, and the proof by exposition (ec-
tential syllogisms’ where what he means by ‘potentiality’ thesis) consists in a sort of perceptual evidence [Lukasiewicz
is that the imperfect syllogisms are ‘potentially perfect syl- 1955:60]. Lukasiewicz, like any other logician of his time has
logisms’ [Patzig 1968:46].That is to say that an imperfect not accepted it as a proof3 , as perception is not a logical proof
syllogism can be transformed into a perfect syllogism. A [Ibid 60]. In fact, he calls it a method outside the limits of the
more-imperfect syllogism takes more steps to transform into syllogistic [Ibid 45].
1 In an interesting study, perfect syllogisms have been further divided into
two sets [Lehman 1972]. However, in this paper, I have not considered that 3 Until late twentieth century, diagrammatic representations or reasoning
approach. based on diagrams were not accepted as proofs. They were at most understood
2 The fourth figure that was later added by Galen is also imperfect. as schemes based on heuristics.
17
B. Reductio ad impossible Logicians theorized various overtures to understand the na-
4
It is one of the oldest method used in any system of ture of perfect syllogisms though there exists no exact logical
reasoning. This is performed in two steps. In the first step, analysis of the proofs Aristotle gives to reduce the imperfect
the negation of the conclusion is assumed. In the second step, syllogisms to the perfect [Lukasiewicz 1955:47]. Roy opines,
the falsity (as a result of contradiction) of this negation of the the dictum de omni et nullo is directly applicable to first figure
conclusion is proved. If the negation of the original conclusion [Roy 1958:216]. Whereas, Patterson claims “this is misleading
is proved to be false, then the original conclusion is proved not because Aristotle nowhere explicitly formulates the dictum
true. This is called reductio ad absurdum or what Aristotle and casts it in the role of logical touchstone, but because the
termed it as reductio ad impossible. Roy called it reductio per dictum itself should be seen as a reflection or encapsulation of
impossible [Roy 1958:]. Aristotelian convictions about the (small number of) ways in
Let us take an example from third figure, ‘if R belongs to which one item can relate predicatively to another [Patterson
some S, and P to all S, P must belong to some R’. Aristotle 1993:375].
says, that this can be shown valid using reduction, reductio ad In a[nother] word[s], perfect syllogisms are ‘self-evident’
impossible as well as ecthesis5 . Let us examine reductio here. syllogism [Patzig 1968:45]. He further opines, that the defined
Suppose, P does not belong to any R. It means, P belongs ‘necessity’ (as per the definition of syllogism) not only occurs
to no R. Then, R belongs to no P. But P belongs to all S but also ‘appears’ or is transparent. In an imperfect syllogism,
and R belongs to some S, thus, P must belong to some R. this defined ‘necessity’ undergoes certain operations before it
This makes our initial supposition false. Therefore, the given ‘appears’ or becomes transparent. This observation of Patzig
syllogism is valid. also supports the claim of Kneales that Aristotle’s thought
The method of reductio was one of the prevalent methods of was guided by diagrams which makes this necessity ‘appear’
reasoning during those times. Aristotle has used it in order to evident [Kneale & Kneale 1962:72]. Flannery has successfully
show his systems coherence to the existing rationality as this provided a rationale for the notion of perfect syllogisms using
expression many a times occurs in Organon that a syllogism is diagrams [Flannery 1987:455-471].
“possible to demonstrate it [or show its validity] also [using] Another interesting remark made by Roy is that ‘reduction’
reductio per impossible”. reveals the essential unity of all forms of syllogistic inference
(Roy 1958:218) as all the figures get transformed into the
C. Reduction first figure. Flannery’s rationale, Roy’s remark along with
This refers broadly to a method of transforming the moods Kneales’ claim, points to an important fact that in order to
of one figure to moods of another figure [Roy 1958:216]. More cognize the idea of perfect syllogisms, one requires visual
precisely, the transformation of second and third figure moods aid and clarification. Similarly, when Patzig says, that defined
into first figure is reduction. A first figure valid syllogism is necessity ‘appears’ transparent, what he means is that the line
accepted as an axiom by Aristotle. It is a necessary truth of reasoning (in case of perfect syllogisms) creates a picture in
and thus requires no further explanation. He shows that a our mind, which our rationality grasps. Thus, in what follows, I
form is valid by showing how to deduce its conclusion from intend to understand this notion of perfect-ness with the help of
its premises. A deduction is a series of steps leading from a new diagrammatic technique to test the validity of syllogism,
the premises to the conclusion, each of which is either an called the ‘method of minimal representation’.
immediate inference6 from the previous step or an inference II. M ETHOD OF M INIMAL R EPRESENTATION
from two previous steps. Method of Minimal Representation7 is an alternative dia-
A simple question which arises at this juncture: How grammatic scheme for testing syllogisms [Sharma 2008:412-
Aristotle shows a given syllogism valid by transforming the 415]. The purpose of developing this method is to differentiate
terms of the proposition? He shows them valid by changing traditional and modern valid syllogisms. Let us in brief, revisit
a syllogism of second or third figure to first figure. The next the history of valid syllogism (in numbers) before explaining
question then will be: Why Aristotle transforms second or third the proposed method.
figure syllogism into first figure? It is because, the first figure
is perfect figure and the valid syllogisms of this figure are A. Valid Syllogisms
perfect syllogisms. Hence to establish that the conclusion of a Aristotle discusses fourteen syllogisms belonging to the
perfect syllogism follows from the premises, one should need first, second and third of the traditional figures [Patzig
to do no more than state the syllogism itself [Lear 1980:2]. 1968:132] as valid. With the development of fourth figure and
considering the weakened moods8 , twenty-four syllogisms are
4 It is one of the oldest, if not the oldest method used as a proof. Pre-
socratics have used it along with counter-example technique. However, it is 7 This method was originally presented as Method of Least Representation at
difficult to comment on their precedence. the First World Congress on the Square of Opposition, Montreux, Switzerland
5 Ecthesis is seldom used by Aristotle for proving validity of syllogisms. (June 2007). Later, it was revised (based on the comments and suggestions),
6 Conversion, Obversion and Contraposition (sometimes, Inversion as well) and was named as Method of Minimal Representation. See Sharma (2008,
are the immediate inferences. They play a vital role in determining the validity 2012).
of syllogisms. It is also contended that rules of immediate inferences were 8 Weakened moods refer to those syllogisms, where we replace a universal
principally developed to serve the purpose of validation of syllogism using conclusion with its sub-altern. For e.g., if AAA-1 is valid then AAI-1 must
the method of reduction. also be valid since I proposition is the sub-altern of A proposition.
18
traditionally valid. If we leave the weakened moods, then there
are nineteen valid syllogisms. However, if we consider modern
interpretation, only fifteen of these are valid.
The Aristotlelian methods (as discussed in the previous
section) along with mnemonics (developed by Petrus Hispanus
[Keynes 1906:329]) were used to test syllogisms. Formal
syllogistic rules, which follows the late ninteenth and early
twentieth century development in logic is widely used these Fig. 2. E proposition
days. Nonetheless, Euler circles (for traditional valid syllo-
gisms) and Venn-Peirce system (for modern valid syllogisms)
also gained currency for testing syllogisms from the standpoint 3) Particular Affirmative Proposition (I) - Some S is P:
of diagrams. Here, a right-angled triangle (S) is drawn from the right bottom
However, none of the above diagrammatic technique is able edge of the rectangle (P) containing it. The arrows suggest that
to demonstrate whether a syllogism is valid/invalid in both the possibility of finding S is both inside as well as outside P.
traditional and modern readings9 . The motivation to develop
a unified diagrammatic scheme for interpreting syllogisms in
both traditional and modern readings came from here10 . In the
next subsection, I explain the method of drawing propositions
and consequently syllogisms.
B. Drawing Propositions and Syllogisms
There are two basic shapes11 that are employed in this
method. They are, namely, rectangles and right angled trian- Fig. 3. I proposition
gles. These shapes act as initial tools for drawing a proposition.
Apart from these, arrows are used to show the possible location 4) Particular Negative Proposition (O) - Some S is not P:
of the shape of subject term of the proposition. We also label Here, a right-angled triangle (S) is drawn from the right bottom
the rectangles or triangles for pinpointing them. The scheme edge of the rectangle (P) outside it. The arrows suggest that
for propositions is as under: the possibility of finding S is both inside as well as outside P.
1) Universal Affirmative Proposition (A) - All S is P: Here,
a small rectangle (S) is drawn from the right bottom edge of
the larger rectangle (P) containing it. The arrow shows that
the possibility of finding S is inside P only.
Fig. 4. O proposition
Summarizing the above:
Fig. 1. A proposition
• The Universal Propositions (A and E) are represented
2) Universal Negative Proposition (E) - No S is P: Here,
by a smaller rectangle inside a greater rectangle.
two disjoint rectangles (S and P), of equal areas, are drawn.
The arrow shows that the possibility of finding S is outside P
• The Particular Propositions (I and O) are represented by a
only.
right-angled triangle inside or outside a greater rectangle.
9 It may also be noted here that no standard digrammatic technique attempts
to address the notion of perfect syllogism with its diagram.
• In a Universal Affirmative Proposition (A), a rectangle
10 After explaining this diagrammatic technique, I with the help of an is drawn from the right bottom edge approximately
example clarify, how to use the same diagram for traditional and modern less than one-fourth of the area of greater rectangle
interpretation. containing it.
11 I use ‘shape’ intentionally so as to distinguish it from ‘figure’. ‘Shape’
means a geometrical structure used in the MMR whereas ‘figure’ in syllogistic
is that which determines the position of middle term in a syllogism. • In a Universal Negative Proposition (E), two disjoint
19
rectangles, of around equal areas, are drawn.
• In a Particular Affirmative Proposition (I), a right-
angled triangle is drawn from the right bottom edge
approximately less than one-fourth of the area of greater
rectangle containing it.
• In a Particular Negative Proposition (O), a right-angled
triangle is drawn from the right bottom edge approxi-
mately less than one-fourth of the area of greater rectan- Fig. 6. No P is M and No S is M
gle outside it.
C. Additional Parameters D. Inference Rules
Apart from the above rules, certain additional conventions Let us take an example to show how this method can test
are also followed while drawing the propositions and even- the validity of a syllogism in both traditional and modern
tually syllogisms. In these propositions, the subject class understanding. Darapti i.e., AAI-3 has the following structure:
(smaller rectangle or right-angled triangle) is represented in All P is M
the predicate class (greater rectangle) with less than one- All S is M
fourth occupation. The prescription for drawing rectangles Therefore, Some S is P.
or right-angled triangles approximately less than one-fourth We first draw a small rectangle M inside the greater rectangle
of the area of greater rectangle needs to be elucidated here. P along with arrows. Similarly, keeping the small rectangle M
This convention has been included to nullify the possibility there, we draw another greater rectangle S as shown below:
of any unnecessary overlapping of basic figures. This is the
case because while representing syllogisms we may require
drawing basic shapes (rectangles/right-angled triangles) inside
a rectangle. Furthermore, to avoid any needless overlapping,
we always draw the second premise in the syllogism from
another edge of the greater rectangle. Let us take the following
example:
Suppose, we have to draw All P is M and All S is M in a
syllogism, then this is drawn as:
Fig. 7. All P is M and All S is M
In the traditional interpretation, we need to find that there
shall be a part common part between S and P. In the diagram,
it is portrayed by M. Therefore, AAA-3 is valid in the
traditional interpretation. However, in the modern viewpoint,
Fig. 5. All P is M and All S is M we need to have a right-angled triangle S inside rectangle P. If
we examine the diagram In the next section, we test syllogisms
Similarly, arrows become important for validating syllo- with MMR in each first, second and third figure.
gisms as given under with the following example. Suppose,
III. MMR AND P ERFECT S YLLOGISMS
we have to draw No P is M and No S is M in a syllogism,
then this is drawn as three disjoint rectangles as given below: In this section, we take three examples from each figure
to demonstrate the simplicity of first figure and the relative
If the conclusion says that No S is P, then it seems from the complexities associated with second and third figures.
above that it holds. However, the arrows (s and p) depicts that
S or P can be anywhere except inside M. Thus, S and P can A. MMR and the First Figure
be overlapping or can fully be contained inside. Therefore, we Let us take AAA-1:
cannot conclude No S is P, and the given syllogism is invalid. All M is P
In the next subsection, I discuss the inference rules briefly. All S is M
20
Therefore, all S is P. order to tacle this difficulty, we draw the minor premise first13 .
Nonetheless, if we draw the minor premise first, then also
We first draw the rectangle M inside the largest rectangle P it requires suitable accommodation (of inference rules and
as given in the first premise. We then draw rectangle S inside additional parameters) to integrate the major premise after that
the larger rectangle M. We label the diagram appropriately in this diagram. After taking all these into consideration, the
and draw the arrows as per rules. given syllogism can be drawn in the following fashion:
Fig. 8. All M is P and All S is M
Fig. 10. Some M is not P and All M is S
B. MMR and the Second Figure
Let us take AOO-2: Though, we draw the above diagram, putting the arrows in
All P is M the right positions is also difficult. However, the above diagram
Some S is not M can be drawn for the given syllogism using MMR considering
Therefore, some S is not P. the inference rules and additional parameters.
The above examples, show that the second and third figure14
In this, we first draw the smaller rectangle P inside greater poses problems (as they are tangled) while drawing with the
rectangle M. The minor premise i.e., some S is not M poses help of MMR.
a problem (to be integrated) in this diagram. In order to avoid
IV. C ONCLUSION
any intersection, so as to preserve the conventions of minimal
representation, we draw it in the following way: This article shows that the syllogisms of first figure are
simple15 (or easier) to draw using the proposed method. In
the first figure, we use only the rules prescribed in Draw-
ing Propositions and Syllogisms. However, the second and
third figures are relatively complex and thus require (apart
from the above rules) the usage of Additional parameters.
Moreover, the syllogisms of second and third figures need
careful considerations (to suitably incorporate both the major
and minor premises along with the positioning of arrows) for
its final diagram to be drawn. MMR depicts the clarity and
simplicity of first figure and its line of reasoning. The clarity
and simplicity of first figure makes it perfect.
ACKNOWLEDGMENT
Fig. 9. All P is M and Some S is not M
I thank Dr. James Burton (University of Brighton) and Dr.
Lopamudra Choudhury (Jadavpur University) for organizing
C. MMR and the Third Figure this workshop on Diagrammatic Logic and Cognition and
Considering OAO-2: providing me with an oppurtunity to present this work. I am
Some M is not P grateful to the anonymous reviewers, for their valuable com-
All M is S ments and suggestions, which helped immensely to improve
Therefore, some S is not P
13 It may further be noted that drawing the minor premise first and then the
The problem it (this syllogism) poses first is that which
major premise, is to use additional parameters as explained in Section II-C.
premise may be drawn first. If we draw the major premise 14 I have not considered fourth figure syllogisms in this paper, in order
first, we find it difficult to integrate the minor premise12 . In to keep Aristotelian notion of figures intact. Nonetheless, we face similar
difficulties (as in second and third figure) while drawing the fourth figure
12 In fact, it is not only difficult but impossible to draw the minor premise syllogisms as well.
later, if we draw the major premise before. 15
21
this paper. I also thank Dr. Ranjan Mukhopadhay (Vishwab-
harati University) for Patzig and Mr. Varun K. Paliwal (SMVD
University) for helping me with LATEX preparation.
R EFERENCES
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