What is a syllogism, or a categorical syllogism, or a syllogistic argument?
Encyclopedias have long sections on the term. Gordon Clark’s definition from the Glossary of his Logic is a good one.
Syllogism – An argument composed of two premises and a conclusion, with the predicate of the conclusion in one, the subject of the conclusion in the other, and a third term in the two premises. (Logic, Glossary, PB, p.123)
The syllogism’s elements consist of three and only three propositions. Two of the propositions are reasons (premises) in support of the third proposition, the conclusion. These propositions share three terms in a definite arrangement.
The subject of the conclusion of the argument is the minor term, the predicate of the same proposition is the major term, and the middle term is in both premises but never in the conclusion.
The premise with the predicate term of the conclusion is the major premise.
The premise with the minor term, either as subject or predicate, is the minor premise.
A syllogism has 3 propositions: the major premise, placed 1st, the minor premise, placed 2nd, and the conclusion (placed 3rd).
- The major term located in the major premise and in the conclusion (the conclusion’s predicate term).
- The minor term is located in the minor premise and in the conclusion (the conclusion’s subject term).
- The middle term is located in the major premise and the minor premise, never in the conclusion.
M -- P |
P -- M |
M -- P |
P -- M |
S -- M |
S -- M |
M -- S |
M --S |
.:. S -- P |
.:. S -- P |
.:. S -- P |
.:. S -- P |
Each may have 1 of 4 Forms (A, E, I, O) as conclusion: 16 x 4 = 64 total permutations.
With 4 figures, the total number is 64 x 4 = 256 syllogistic argument forms.
[(4 x 4 x 4) = 64; (64 x 4) = 256]
The three propositions share 3 terms. Here is the structure of the form of a syllogism:
Major Premise: [MiddleTerm M]- is- [Major Term P]
Minor Premise: [Minor Term S]- is - [MiddleTerm M]
Conclusion: [Minor Term S]- is - [Major Term P]
As you can see the elements of a Syllogism have a definite arrangement. Variation in the structure of a syllogism can be obtained by the different locations of the Middle Term in the Premises. There are four positions known as figures 1, 2, 3, and 4.
- 1st Figure: The middle term is the subject of the major premise & the predicate of the minor premise.
- 2nd Figure: The middle term is in the predicates of both premises.
- 3rd Figure: The middle term is in the subjects of both premises.
- 4th Figure: The middle term in the predicate of the major premise & the subject of the minor premise.
PREMISES |
1st FIGURE |
2ndFIGURE |
3rd FIGURE |
4th FIGURE |
Major |
M - P |
P - M |
M -P |
P - M |
Minor |
S - M |
S - M |
M - P |
M - P |
Conclusion |
.:. S - P |
.:. S - P |
.:. S - P |
.:. S - P |
The propositions of a syllogism have 4 Forms (A, E, I, O) expressing 2 properties: Quality (Affirmative or Negative) and Quantity (Universal or Particular).
FORMS |
CATEGORICAL |
QUALITY |
QUNANTITY |
A Form |
All S is p. |
Affirmative |
Universal |
E Form |
No S is p. |
Negative |
Universal |
I Form |
Some S is p. |
Affirmative |
Particular |
O Form |
Some S is not p. |
Negative |
Particular |
FORMS |
A |
E |
I |
O |
A |
A - A |
A - E |
A - I |
A - O |
E |
E - A |
E - E |
E -I |
E - O |
I |
I - A |
I - E |
I - I |
I - O |
O |
O - A |
O - E |
O - I |
O - O |
TOTALS |
4 |
4 |
4 |
4 |
Each term of a syllogism is either distributed or undistributed. If the term is modified by all, no, or not, it is distributed. If it is modified by some, it is undistributed.
- A Form subject term is distributed; predicate term undistributed.
- E Form subject term is distributed; predicate term is distributed.
- I Form subject term is undistributed; predicate term is undistributed.
- O Form subject term is undistributed; predicate term is distributed.
FORM |
PROPOSITION |
SUBJECT TERM |
PREDICATE TERM |
A |
All S is not P. |
Distributed |
Undistributed |
E |
No S is not P. |
Distributed |
Distributed |
I |
Some S is P. |
Undistributed |
Undistributed |
O |
Some S is not P. |
Undistributed |
Distributed |
Clark formulates Five (5) Rules for determining the validity of a syllogism based on the distribution of its terms. (Logic, PB. pp. 78-79)
- Rule 1: Two negative premises do not imply a conclusion.
- Rule 2: Two affirmative premises do not imply a negative conclusion.
- Rule 3: An affirmative premise and a negative premise do not imply an affirmative conclusion.
- Rule 4: Two premises, in both of which the middle term is undistributed, do not imply a conclusion.
- Rule 5: Two premises in which a given term is undistributed do not imply a conclusion in which that term is distributed.
A syllogism is an argument with three categorical propositions having three terms arranged in a precise configuration, which maintain the integrity of a standard categorical syllogism. If the configuration conforms to the rules for obtaining necessary consequences from premises, the syllogism is valid; if not, then invalid. If the valid syllogism consists of true propositions, then the syllogism is sound; otherwise, unsound.
2. Socrates is a man.
3. Therefore, Socrates is mortal.
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