# Aristotelian Syllogisms

### after Raymond McCall, Basic Logic (Barnes & Noble, 1967); symbolic apparatus from Elementary Logic, by Benson Mates (Oxford, 1972)

```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 four figures, 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 possible moods, 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:          AAAA     IIII       EEEE        OOOO
Minor Premise:          AEIO     AEIO       AEIO        AEIO

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 conclusion
is 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:  an indirect reduction to BARBARA by contradicting the conclusion,
using it as a premise and deriving the contradiction of the premise
followed by "c", which becomes a reductio ad absurdum of 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.
```