In his fundamental treatise Organon, Aristotle gives the following definition:
A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that they produce a consequence, and by this, that no further term is required from without in order to make the consequence necessary.
Things that have been stated are known as premises and the one that follows from the premises is known as the conclusion of the syllogism. Considering only two premises Aristotle gave a classification of 19 valid forms of syllogism. The list admits simplification which reduces it to just 8 essentially distinct forms of drawing a conclusion from two premises. Aristotle's withstood more than 2000 years of universal scrutiny until the middle of the 19th century. With the advent of formalism of propositional calculus invented by the English mathematician George Boole(1815-1864), two of his syllogisms were found invalid in that the conclusion might not be always inferred from the premises. (Invalid syllogism are known as fallacies. As one would guess, there are fewer valid syllogism than fallacies. However, most fallacies are obviously such. Aristotle's were tricky.)
Consider two premises: All x are y and All x are z. One of the Aristotle's syllogism supplies the following conclusion: Some y are z. Which may appear supported by the Venn diagram:
Indeed, If all x are y the circle corresponding to x is contained inside the circle corresponding to y, and, for the same reason, is contained in the circle corresponding to z. Therefore, it appears that the y and z circles are bound to intersect which makes the proposition Some y are z true. But it may not. If No x exists then both premises of the syllogism (All x are y and All x are z) are vacuously true. However, the conclusion may be wrong; for then y and z are quite arbitrary.
Lewis Carroll's trilateral diagrams suffer from the same potential inconsistency. In an attempt to avoid it, he insisted that a proposition All m are x makes a statement about existing things: Some m exist, some x exist, and All m are x [Symbolic Logic, Game of Logic, Book II, Chapter III, Para 4]. (Such an assumption of existence is known as existential import.) This is not a commonly (or currently) accepted point of view. According to the modern outlook, the particular propositions ("Some m are x") have existential import, while the universal ones ("All m are x") do not. I speculate that Lewis Carroll's insistence and vigorous critique of the opposite view [Symbolic Logic, Game of Logic, Appendix, Addressed To Teachers] hindered a wider spread of his syllogistic technique.
A valid syllogism is provided by Euler's example:
- Premise 1: All x are y
- Premise 2: No z is y
- Hence, conclusion: No z is x
In general, as in the example above, two premises refer between them to three kinds of objects (like x,y,z in the example) of which one (perhaps in the form of negation) ought to be mentioned twice. These latter are known as the eliminands. Eliminands glue two premises into a meaningful basis for inference. They are not passed on to the conclusion. The other two - the retinends - are so called because they somehow retained and do appear in the conclusion. Following is another example:
- Premise 1: All x are y
- Premise 2: Some z are x
- Hence, conclusion: Some z are y
with x being an eliminand and y and z retinends.
- Aristotle, The Basic Works of Aristotle, Random House, 1941
- K. Devlin, Mathematics: The Science of Patterns, Scientific American Library, 1997, second printing