A Class Room Logic / Deductive and Inductive, with Special Application to the Science and Art of Teaching

by George Hastings McNair

4. SPECIAL CANONS OF THE FOUR FIGURES.

As a deductive exercise in clear, logical thought, the indirect proof involved in establishing certain principles underlying the four figures, is of immense value. On no account should this section be omitted. The mere fact that it appears to be a difficult section is proof positive that the student is in need of just such exercises.

Canons of the first figure.

(1) The minor premise must be affirmative.

(2) The major premise must be universal.

Problem: The minor premise must be affirmative.

Data: Given the form of the first figure, which is,

M — G

S — M

S — G

Proof: (1) If the minor premise is not affirmative then it must be negative; because affirmative and negative propositions, being contradictory in nature, admit of no middle ground.

(2) If the minor premise is negative, the conclusion must be negative; for the reason that a negative premise necessitates a negative conclusion.

(3) If the conclusion is negative then its predicate, G, must be distributed; since all negatives distribute their predicates.

(4) If the predicate of the conclusion, which is the major term, is distributed, then it must be distributed in the premise where it occurs, which is the major premise; for any term which is distributed in the conclusion must be distributed in the premise where it occurs.

(5) If the major term, which is the predicate of the major premise, is distributed, then the major premise must be negative; because only negatives distribute their predicates.

(6) The result of this argument, then, gives two negative premises, and we know from rule 3 that a conclusion from two negatives is untenable.

(7) Since the minor premise cannot be negative, it must be affirmative.

Problem: To prove that the major premise must be universal.

Data: Given the form of the first figure:

M — G

S — M

S — G

Proof: (1) The predicate of the minor premise, M, which is the middle term, is undistributed; because no affirmative proposition distributes its predicate.

(2) The middle term must be distributed in the major premise; since in any syllogism the middle term must be distributed at least once.

(3) As the middle term, M, used as the subject of the major premise, must be distributed, then the major premise must be universal; because only universals distribute their subjects.

Epitome.

In the first figure, the minor premise must be affirmative, since making it negative necessitates making the major premise negative also; the major premise must be universal in order to distribute the middle term at least once.

Special canons of the second figure.

(1) One premise must be negative.

(2) The major premise must be universal.

Problem: To prove that one premise must be negative.

Data: Given the form of the second figure:

G — M

S — M

S — G

Proof: (1) The middle term, M, is the predicate of both premises.

(2) The middle term must be distributed at least once, according to rule 3.

(3) Hence one premise must be negative; since only negatives distribute their predicates.

Problem: To prove that the major premise must be universal.

Data: Given the form of the second figure:

G — M

S — M

S — G

Proof: (1) As one premise must be negative, it follows that the conclusion must be negative according to rule 6.

(2) If the conclusion is negative, then its predicate, G, the major term, must be distributed; since all negatives distribute their predicates.

(3) When distributed in the conclusion, the major term, G, must also be distributed in the major premise, where it is used as the subject. See rule 4.

(4) Hence the major premise must be universal; for only universals distribute their subjects.

Epitome.

In the second figure one premise must be negative in order to distribute the middle term at least once; and the major premise must be universal that the major term, which is distributed in the conclusion, may be distributed in the premise where it occurs.

Canons of the third figure.

(1) The minor premise must be affirmative.

(2) The conclusion must be particular.

Problem: To prove that the minor premise must be affirmative.

Data: Given the form of the third figure, which is,

M — G

M — S

S — G

Proof: (1) Suppose the minor premise were negative, then the conclusion would have to be negative, and this would distribute the predicate G.

(2) A distributed predicate would necessitate its being distributed in the major premise.

(3) But G, being the conclusion of the major premise, could be distributed only by a negative proposition.

(4) This would result in two negatives; therefore no conclusion could be drawn, if the minor premise were negative.

Problem: To prove that the conclusion must be particular.

Data: Given the form of the third figure:

M — G

M — S

S — G

Proof: (1) The minor term, which is the predicate of the affirmative minor premise, is undistributed; because no affirmative distributes its predicate.

(2) If undistributed in the premise, then the minorterm must remain undistributed in the conclusion, where it is used as the subject.

(3) The conclusion must, then, be particular; since all universals distribute their subjects.

Epitome.

In the third figure, unless the minor premise be affirmative, there can be no conclusion; since a negative minor would necessitate a negative major. An affirmative minor compels a particular conclusion, in order that the minor term, in the conclusion, may remain undistributed.

Canons of the fourth figure.

(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.

Problem: To prove that if the major is affirmative, the minor must be universal.

Data: Given the form of the fourth figure:

G — M

M — S

S — G

Proof: (1) If the major premise is affirmative, then its predicate which is the middle term, M, is undistributed; for no affirmative distributes its predicate.

(2) The middle term must then be distributed in the “minor” according to rule 3.

(3) Then the “minor” must be universal; since only universals distribute their subjects.

Problem: To prove that if the minor is affirmative, the conclusion must be particular.

Data: Given the form of the fourth figure:

G — M

M — S

S — G

Proof: (1) If the minor premise be affirmative, then S, its predicate, must be undistributed; because no affirmative distributes its predicate.

(2) Since S is undistributed in the minor premise, it must remain undistributed in the conclusion where it is used as the subject.

Problem: To prove that if either premise is negative, the major must be universal.

Data: Given the form of the fourth figure:

G — M

M — S

S — G

Proof: (1) If one of the premises is negative, then the conclusion must be negative according to rule 6.

(2) If the conclusion is negative, then the predicate, G, must be distributed.

(3) If G is distributed in the conclusion, it must be distributed in the major premise.

(4) The major premise must be universal; as G is used as its subject, and only universals distribute their subjects.

Epitome.

In the fourth figure, if the “major” is affirmative, the “minor” must be universal in order to distribute the middle term. If the minor is affirmative, the conclusion must be particular; otherwise the fallacy of illicit minor would result. If either premise is negative, the major must be universal to avoid the fallacy of illicit major.