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).