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# Lewis Carroll's Diagram and Categorical Syllogisms

Aristotelian logic, or the traditional study of deduction, deals with four so-called categorical or subject-predicate propositions, which can be defined by
S a P ⇔ All S is P (universal affirmative or A proposition),
S i P ⇔ Some S is P (particular affirmative or I proposition),
S e P ⇔ No S is P (universal negative or E proposition),
S o P ⇔ Some S is not P (particular negative or O proposition).
S is called the subject (or minor) term and P is called the predicate (or major) term of the proposition. A categorical syllogism is a deductive argument about categorical propositions in which a conclusion is inferred from two premises. The term M that occurs in both premises is called the middle term. An example of a syllogism is M a P, S i M ⊨ S i P. There are 256 possible triples of categorical propositions, but only 24 valid syllogisms, some of which need existential support (Ex(S), Ex(P), or Ex(M), where Ex means "there is" or "there exists").
The aim of this Demonstration is to falsify (when the checkbox "valid syllogism" is unchecked) the argument, making the premises true and the conclusion false. According to Venn diagrams, a shaded region R means it is empty and + in the region R means existence of at least one element there (proposition Ex(R) is true). (To shade a region, click the appropriate button. A click in a region introduces a new locator (+) and Alt-click removes it.) Such an example is known as a counterexample.
Choosing a valid syllogism means that making the premises true will make the conclusion true as well. So an attempt to make the conclusion false will end in a contradiction (+ in the empty region).

### DETAILS

The so-called figure of a categorical syllogism is determined by the possible position of middle term. There are four figures:
M x P, S x M ⊨ S x P,
P x M, S x M ⊨ S x P,
M x P, M x S ⊨ S x P,
P x M, M x S ⊨ S x P,
where x is a, i, e, or o.
This version of Carroll's diagrams was found in [2], p. 112. See also the Wikipedia entry for Categorical proposition.

[1] R. Audi, ed., The Cambridge Dictionary of Philosophy, Cambridge: Cambridge University Press, 1995 pp. 780–782.
[2] L. Borkowski, Elementy logiki formalnej (Elements of Formal Logic, in Polish), 3rd ed., Warsaw: Wyd, 1976.
[3] L. Carroll, Symbolic Logic and the Game of Logic, New York: Dover, 1958.
[4] I. M. Copi and C. Cohen, Introduction to Logic, 9th ed., New York: Macmillan, 1994 pp. 214–218.

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