Parts of a syllogism: A: a universal affirmative proposition--All S is P [(x)(Sx -> Px)]. E: a universal negative proposition--No S is P [(x)(Sx -> -Px)]. I: a particular affirmative proposition--Some S is P [(x)(Sx & Px)]. O: a particular negative proposition--Some S is not P [(x)(Sx & -Px)]. The predicate of an affirmative proposition is regarded as having particular quantification, the predicate of a negative proposition, universal. S: subject of the conclusion. P: predicate of the conclusion. M: the middle term. The Major Premise of a syllogism contains the predicate of the conclusion and the middle term. The Minor Premise contains the subject of the conclusion and the middle term. The fourfigures, possible combinations of middle terms as subjects or predicates of major or minor premises, are: 1st 2nd 3rd 4th M P P M M P P M S M S M M S M S ---- ---- ---- ---- S P S P S P S P All the possiblemoods, or kinds of propositions in the two premises (the moods that turn out to be valid in some figure are in bold face): Major Premise:AAAAIIIIEEEEOOOO Minor Premise:AEIOAEIOAEIOAEIO 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.10) In extensional logic, if both premises are universal, the conclusionis universal. (See DARAPTI, etc., and "In Defense of Bramantip")These moods have premises that are both particular or both negative and so do not produce valid syllogisms: Major Premise: II EE OOO Minor Premise: IO EO EIO One more mood is always invalid: Major Premise: I In this mood the major term would have particular Minor Premise: E quantification in the major premise; but, since the - conclusion would have to be negative, it would have Conclusion: O universal quantification there, violating rule 3. Other moods are eliminated in each figure. The vowels in the names for the moods give the types of propositions in the major premise, the minor premise, and then the conclusion, respectively. First Figure: BARBARA, CELARENT, DARII, FERIO M P S M 1) The minor premise must be affirmative. 2) The major premise must be universal. Second Figure: CESARE, CAMESTRES, FESTINO, BAROCO P M S M 1) One premise must be negative. 2) The major premise must be universal. Third Figure:DARAPTI, DISAMIS, DATISI,FELAPTON, BOCARDO, FERISON 1) The minor premise must be affirmative. M P 2) The conclusion must be particular. M S Fourth Figure:BRAMANTIP, CAMENES, DIMARIS,FESAPO, FRESISON 1) If the major premise is affirmative, the minor premise must be universal. 2) If the minor premise is affirmative, the conclusion must be particular 3) If either premise is negative, the major must be universal The names of the moods are in a code that tells how to convert the syllogism in question to a syllogism of the first figure, which was regarded as more perfect: s: means that the subject and predicate of the preceding proposition should be exchanged, without changing the quantity. p: means that the subject and predicate of the preceding proposition should be exchanged, while changing the quantity of the proposition. m: exchange the major and minor premises. c: anreduction to BARBARA by contradicting the conclusion, using it as a premise and deriving the contradiction of the premise followed by "c", which becomes aindirectreductio ad absurdumof the denial of the mood, e.g. BOCARDO, extended to the derivation of the conclusion from the original premises: {1} 1. (x)(Mx & -Px) P, Premise {2}2. (x)(Mx -> Sx)P / (x)(Sx & -Px) {3} 3. -(x)(Sx & -Px) P {3} 4. (x)-(Sx & -Px) 3 Q, Quantifier Exchange {3} 5. (x)(-Sx v --Px) 4 R 42, De Morgan {3} 6. (x)(-Sx v Px) 5 R 29,Double Negation {3}7. (x)(Sx -> Px)6 R 53, Material Implication {2} 8. Ma -> Sa 2 US, Universal Specification {3} 9. Sa -> Pa 7 US {2,3} 10. Ma -> Pa 8,9 Th 1, Hypothetical Syllogism {2,3}11. (x)(Mx -> Px)10 UG, U Generalization,Barbara{3} 12. -(x)-(Mx -> Px) 11 Q {2,3} 13. -(x)-(Mx -> --Px) 12 R 29, Double Negation {2,3} 14. -(x)(Mx & -Px) 13 D, Definitional Interchange {1,2,3} 15. (x)(Sx & -Px) 1,14 Th 8, Duns Scotus {1,2} 16. -(x)(Sx & -Px) -> (x)(Sx & -Px) 3,15 C, Conditionalization {1,2} 17. (x)(Sx & -Px) 16 Th 16, Clavius, QED This reductio ad absurdum proof also shows how proofs with an existential premise and conclusion can be constructed without using the Existential Specification rule.

The Arch of Aristotelian Logic