Laws of the Syllogism
The eight general laws of the categorical syllogism (four laws of terms, four laws of propositions) with reasons and examples; followed by the special laws governing conditional, conjunctive, and disjunctive hypothetical syllogisms.
Eight general laws govern all categorical syllogisms. Four laws of terms: (1) exactly three terms; (2) the middle term must not appear in the conclusion; (3) the middle term must be distributed at least once in the premises; (4) no term may be more distributed in the conclusion than it was in the premises (illicit major/minor). Four laws of propositions: (5) no conclusion from two negative premises; (6) if both premises are affirmative the conclusion must be affirmative; (7) if one premise is negative the conclusion must be negative; (8) no conclusion from two particular premises. Violation of any of these laws produces a formal fallacy regardless of the truth of the individual propositions. Special laws govern conditional, conjunctive, and disjunctive hypothetical syllogisms.
a) Laws for the Categorical Syllogism
The eight general laws of correctness for the perfect categorical syllogism are divided into Laws of Terms and Laws of Propositions. Glenn sets them in verse for memorisation:
Laws of Terms
Three terms there must be, neither more nor less; No wider in Conclusion than in Premiss; Conclusion never dares the Middle mention; The Middle, once or twice, has full Extension.
Laws of Propositions
Affirmatives can never breed negation; Two negatives end ever in frustration; Conclusion follows e’er the weaker part; Particulars no argument can start.
I. Laws of Terms
First Law: Three terms there must be, neither more nor less.
The syllogism reasons to the agreement or disagreement of two terms through their relation to a third. It must have — by its very nature — a major, a minor, and a middle term. No more, no fewer.
Note: three terms are required, not merely three words. A word used in two different senses is two terms despite being one word. Such equivocation causes the syllogism to contain four terms and violates this law.
Example of violation (equivocation):
A bank is a place in which money is deposited. This mound of earth is a bank. Therefore this mound of earth is a place in which money is deposited.
Bank is used in two utterly different senses — the argument contains four terms.
Second Law: No wider in Conclusion than in Premiss.
Terms must not have a larger Extension in the conclusion than in the premisses. The conclusion is only the explicit statement of what is implicitly contained in the premisses; it cannot say more than they warrant.
Judge Extension of terms by the two principles already learned:
- The predicate of an affirmative proposition is undistributed (particular).
- The predicate of a negative proposition is distributed (universal).
Example of violation:
Every tiger is a living being. A man is not a tiger. Therefore a man is not a living being.
In the premiss, living being is particular (predicate of affirmative). In the conclusion, living being is universal (predicate of negative). The conclusion expands the term’s Extension — unwarranted.
Third Law: Conclusion never dares the Middle mention.
The middle term must never occur in the conclusion. It is the medium by which the mind reaches the conclusion; having served its purpose, it does not belong in the conclusion. An argument in which the middle term appears in the conclusion is merely a compounding of statements, not genuine reasoning.
Example of violation:
John is lazy. John is a student. Therefore John is a lazy student.
The conclusion merely combines the premisses rather than drawing out something latent in them.
Fourth Law: The Middle, once or twice, has full Extension.
The middle term, in at least one premiss, must be distributed (taken in full Extension). If undistributed in both premisses, the premisses are independent statements with no logical connection — no conclusion can follow.
Example of violation:
Wine is an intoxicant. Whiskey is an intoxicant. Therefore, whiskey is wine.
An intoxicant is undistributed in both premisses (predicate of affirmative propositions). Both statements merely assign wine and whiskey to the class intoxicant, but say nothing about their relation to each other. Draw a circle for intoxicant, with two smaller circles inside for wine and whiskey — the premisses do not specify whether those circles overlap.
II. Laws of Propositions
First Law: Affirmatives can never breed negation.
Two affirmative premisses can never lead to a negative conclusion. Two affirmatives imply no negation; therefore none can appear in the conclusion.
Second Law: Two negatives end ever in frustration.
No conclusion can be drawn from two negative premisses. The syllogism requires the positive assertion of at least one extreme’s relation to the middle term. Two negatives are merely independent denials.
Example:
Man is not a spirit. An angel is not a man. (no conclusion possible)
Nothing in these premisses justifies any inference about the relation of angel to spirit.
Third Law: Conclusion follows e’er the weaker part.
The “weaker part” means negation (in quality) and particularity (in quantity). The law means:
- If one premiss is negative and the other affirmative → the conclusion will be negative.
- If one premiss is particular and the other universal → the conclusion will be particular.
Proof that the conclusion must be negative when one premiss is: The affirmative premiss asserts the agreement of one extreme with the middle term. The negative premiss asserts the disagreement of the other extreme with the middle term. Hence the extremes stand in disagreement — the conclusion must be negative.
Proof that the conclusion must be particular when one premiss is: (Summarised) With one particular and one universal premiss, there are insufficient universal terms in the premisses to justify a universal conclusion; full working of the proof is given by Glenn in the text.
Fourth Law: Particulars no argument can start.
If both premisses are particular, no conclusion is possible. The proof proceeds through three cases:
- Both particular affirmatives → no universal term at all in the premisses → middle term cannot be distributed → no syllogism.
- Both particular negatives → violates Second Law of Propositions.
- One particular affirmative, one particular negative → conclusion must be negative (Third Law) → conclusion requires a universal predicate → premisses must supply two universal terms plus a distributed middle → but such premisses contain only one universal term → impossible.
Note for compound categoricals: The general rule is to reduce to simple categorical syllogisms and frame or criticise those by the Eight Laws. Be especially vigilant not to say more in the conclusion than the premisses warrant. Example of a subtle violation:
An infinitely perfect Being is eternal. God alone is an infinitely perfect Being. Therefore, God alone is eternal.
The exclusive particle alone in the minor premiss attaches to infinitely perfect Being. In the conclusion it has been shifted to attach to eternal — saying that only an infinitely perfect being is eternal, which the premisses do not establish. The justified conclusion is only “God is eternal.”
b) Laws for the Hypothetical Syllogism
i. The Conditional Syllogism
Law: From the truth of the antecedent follows the truth of the consequent, but not vice versa; and from the falsity of the consequent follows the falsity of the antecedent, but not vice versa.
| Antecedent | Consequent |
|---|---|
| True | → Consequent true |
| False | → Consequent true or false |
| — | Consequent true → Antecedent true or false |
| — | Consequent false → Antecedent false |
Two valid methods for forming the minor premiss:
-
“Put”-method: affirm the truth of the antecedent.
If God is just, the soul will survive death. / God is just. / Therefore, the soul will survive death.
-
“Take”-method: deny the truth of the consequent.
If the soul does not survive death, God is cruel. / God is not cruel. / Therefore, the soul survives death.
Invalid: concluding to the truth of the antecedent from the truth of the consequent, or to the falsity of the consequent from the falsity of the antecedent.
ii. The Conjunctive Syllogism
Law: From the truth of one component follows the falsity of the other; but from the falsity of one component, it does not follow that the other is true.
| Component | Other |
|---|---|
| True | → Other false |
| False | → Other true or false |
Only one valid method: the “put-take” — affirm one member (put) in the minor premiss, deny the other (take) in the conclusion.
Peter does not sit and stand at the same time. / Peter stands (put). / Therefore Peter does not sit (take).
Invalid: the “take-put” — denying one member and affirming the other — because the members of a conjunctive premiss do not exhaust possibilities.
iii. The Disjunctive Syllogism
Law: From the truth of one member follows the falsity of all the others; from the falsity of one member follows the truth of one of the others.
| Member | Others |
|---|---|
| True | → All others false |
| False | → One of the others true |
Two valid methods:
-
“Take-put”: deny one member, affirm another.
Either it is day or it is night. / It is not day (take). / Therefore, it is night (put).
-
“Put-take”: affirm one member, deny another.
Either it is day or it is night. / It is night (put). / Therefore, it is not day (take).
The student must ensure the major premiss is a complete disjunction, else an unjustified conclusion may follow. “It is spring, summer, or autumn” — this is incomplete; denying autumn does not justify concluding spring or summer, since winter is also possible.
Summary of the Article
We have learned the laws for constructing and criticising syllogisms — the “Laws of Thought.” Each law has been given with its reason; no law has been offered on faith alone. The student will practice reducing arguments to syllogistic form before attempting to judge their validity.