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# Venn Diagrams and Syllogisms

This Demonstration lets you verify 24 valid syllogisms using Venn diagrams with only one element in the domain. The domain only needs two elements, denoted by "+" and "×", to show that a syllogistic form is not valid.
The universal set is divided into eight subsets by , , and . If a subset is shaded, it is empty. A white subset does not guarantee that it contains an element, but if the sign "+" or "×" is in a subset, then it does have an element. If "+" or "×" is in a shaded subset, there is a contradiction. So the statement that a subset is empty is true if it is shaded, false if either "+" or "×" is in it, and otherwise the statement is undecided.

### DETAILS

A monadic formula of first-order logic is one for which all nonlogical symbols are one-place predicates.
Theorem. If is a monadic sentence that is satisfiable, then is true in some interpretation whose domain contains at most members, where is the number of one-place predicate letters and is the number of variables in .
Therefore there is an effective procedure for deciding whether or not a monadic sentence is valid [1, p. 250].
Syllogistic forms are monadic sentences if considered as sentences of the form with predicate letters , , and .
References
[1] G. S. Boolos and R. C. Jeffrey, Computability and Logic, Cambridge, UK: Cambridge University Press, 1974.
[2] L. Carroll, Symbolic Logic and The Game of Logic, New York: Dover Publications, 1958.
[3] I. M. Copi and C. Cohen, Introduction to Logic, 9th ed., New York: Macmillan Publishers, 1994 pp. 214–218.
[4] J. M. Bocheński, A History of Formal Logic, 2nd ed. (I. Thomas, trans. and ed.), New York: Chelsea Publishing Company, 1970 p. 235.
[5] Wikipedia. "Syllogism." (Aug 3, 2016) en.wikipedia.org/wiki/Syllogism.
[6] Wikipedia. "Venn Diagram." (Aug 3, 2016) en.wikipedia.org/wiki/Venn_diagram.
[7] J. Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings," Philosophical Magazine Series 5, 10(59), 1880 pp. 1–18. doi:10.1080/14786448008626877.

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