Categorical Syllogisms and their Validity with the help of Diagrams
Sumanta S. Sharma Ph.D. Student Humanities & Social Sciences IIT Kanpur
Some valid Categorical Syllogisms in traditional interpretation befall invalid in the modern point of view. Euler Circles and its variants evaluate the validity as per the traditional interpretation whereas Venn Diagrams and its modifications examine the modern point of view. Hence, we fail to find any standard diagrammatic technique, which incorporates both the points of view together. The present article explores the possibility of developing an alternative diagrammatic technique to test the validity of Categorical Syllogisms. The proposed technique also attempts to test the validity in both the formats. Keywords: Euler Circles, Venn Diagrams, Method of Minimal Representation
ote Editor's N The present paper is an adaptation for NERD, IIT Kanpur that was previously published with Springer. See,
Particular Negative Proposition (O) (Some S is not P). Suppose we have an argument, which has exactly two premises and one conclusion. For example: All snakes are reptiles. All cobras are snakes. Therefore, all cobras are reptiles.
Sharma, S.S.: Method of Minimal Representation: An Alternative Diagrammatic Technique to Test the Validity of Categorical Syllogisms. LNAI 5223, 412-414. Springer, Heidelberg (2008)
Introduction This article is divided into two parts. In the first part, we will attempt to understand what is meant by Categorical Syllogisms and how to assess its validity with the help of existing diagrammatic techniques. In the concluding section, we will attempt to develop an alternative diagrammatic technique to test the validity in both the traditional and modern frameworks. Categorical Syllogism
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Here, we will deal with four kinds of propositions. They are: Universal Affirmative Proposition (A) - (All S is P) Universal Negative Proposition (E) - (No S is P) Particular Affirmative Proposition (I) - (Some S is P)
It contains exactly three terms (snakes, reptiles and cobras), each of which occurs twice. This type of argument in deductive logic is called a categorical proposition. There are several methods to test the validity of categorical syllogisms. It can be divided under the following heads: a) Formal Rules. b) Diagrammatic Rules. a) Formal Rules
The first rule explicate that a syllogism must consist of exactly three terms each of which is used in the same sense throughout the argument. The second rule expounds that the middle term must be distributed at least once in the premises. The third rule states that no term can be distributed in the conclusion unless it is distributed in the premise. The fourth rule
illustrates that from two negative premises no conclusion can be drawn. The fifth rule demonstrates that if one premise is negative, the conclusion must be negative and vice versa. The last and final rule, which is a later addition, concerns existential import. It states that no valid syllogism with particular conclusion can have two universal premises. It can be seen that the above stated rules are technical, and thus we refrain from discussing them here. b) Diagrammatic Rules
Diagrammatic methods are also used to test the validity of Categorical Syllogisms. They have been employed in pursuit of reasoning, as a heuristic tool to explore the proof of any given problem. Now, it has proved that the status of diagrams is not that of a second grade citizen but rather an effective tool for proof systems. A complete discussion on the status of diagrams is found in Shin, S.J.: The Logical Status of Diagrams. Cambridge University Press, New York (1994). In this article, we will concentrate chiefly on Euler Circles and Venn Diagrams.
Euler Circles Euler proposed circles or closed curves to illustrate relation between classes. The representation for A, E, I, and O type of propositions are given below:
A: All S is P
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E: No S is P
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I: Some S is P
O: Some S is not P