Fun With Syllogisms Types of Syllogisms Categorical Syllogism – deals with categories and whether or not things belong in certain categories Disjunctive Syllogism – involves a choice and contains an either/or statement Either Tom is a cat, or he is on a mat. Tom is not a cat. Therefore, Tom is on a mat. Hypothetical Syllogism – created through a chain. When the chain is created properly, the syllogism is said to be valid. The chain is made of one or various if/then statements. ex. If you make lots of money, you will be happy. You make lots of money. Therefore, you are happy. If you sit down to rest, you will have energy. If you have energy, you will go for a run. Therefore, if you sit down you will go for a run Recall: The Major Premise of a syllogism contains the Major premise predicate of the conclusion and the middle Minor premise term. The Minor Premise contains the subject Conclusion of the conclusion and the middle term. Figure of the Syllogism predicate term middle term middle term subject term All cats are on mats. Garfield is a cat. Therefore, Garfield is on a mat. subject term predicate term Figure 1 Figure 2 Figure 3 Figure 4 Major premise M-P P-M M-P P-M Minor premise S-M S-M M-S M-S Conclusion S-P S-P P-S P-S All dogs are cats All dogs are cats Some mice are dogs Some dogs are mice Therefore, some mice are cats Therefore, some mice are cats All mammals are animals All mammals are animals Some ‘things with four legs’ are mammals Some mammals are ‘things with four legs’ Therefore, some ‘things with four legs’ are animals Therefore, some ‘things with four legs’ are animals Types of Syllogisms The types of syllogisms (we have already seen these) are referred to with the vowels A, E, I, and O. A– Universal affirmative All A’s are B’s E– Universal negative No A’s are B’s I– Particular affirmative Some A’s are B’s O– Particular negative Some A’s are not B’s So we can refer to a specific type with a specific figure in the following manner: AAA-1 We have already seen many examples of this syllogism. All philosophers are Greek. All people with beards are philosophers. Therefore, all people with beards are Greek. Rules of the syllogism 1) There are only three terms in a syllogism (by definition). 2) The middle term is not in the conclusion (by definition). 3) The quantity of a term cannot become greater in the conclusion. 4) The middle term must be universally quantified in at least one premise. 5) At least one premise must be affirmative. 6) If one premise is negative, the conclusion is negative. 7) If both premises are affirmative, the conclusion is affirmative. 8) At least one premise must be universal. 9) If one premise is particular, the conclusion is particular. Venn Diagrams Venn diagrams can be used to test or show the validity of a syllogism. These can be used in two different ways. All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Examples: Identify the type, the figure, and use a Venn diagram to show the argument. All cats are on mats. All mice are on mats. Therefore, all cats are mice. No cats are on mats. Jill is not a cat. Therefore, Jill is on a mat. All dogs have fur. Some dogs have fleas. Therefore, some things with fur have fleas.