Syllogisms and Venn-Diagram Validity Testing

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A syllogism is valid when its conclusion must be true if its premises are true. Validity concerns the form of the inference, not whether the premises happen to be true.

Notebook source

Notebook pp. 82–83 — syllogism types, rules, and Venn-diagram tests (file not included in this release).

Three families that should not be conflated

Family	Canonical form	Example
Categorical	relations among classes using all, no, or some	All philosophers are thinkers; some philosophers are teachers; therefore some thinkers are teachers.
Conditional	a conditional plus a premise about one side	If the alarm is set, the light is on; the alarm is set; therefore the light is on.
Disjunctive	an “or” statement plus elimination of an option	Either the key is in the desk or the bag; it is not in the desk; therefore it is in the bag.

The notebook source compresses these families together. Venn diagrams are specifically a decision procedure for categorical syllogisms; conditional and disjunctive arguments are better tested with inference rules or truth tables.

Categorical propositions

Form	Name	Diagram instruction
All S are P	universal affirmative (A)	shade the part of S outside P
No S are P	universal negative (E)	shade the S–P overlap
Some S are P	particular affirmative (I)	place an `X` in the S–P overlap
Some S are not P	particular negative (O)	place an `X` in the part of S outside P

Shading means a region is empty. An X means at least one thing exists in that region. If the premises do not tell you which side of a boundary contains that thing, place the X on the boundary.

Venn-diagram procedure

Draw three overlapping circles for the minor term, major term, and middle term.
Diagram universal premises first by shading empty regions.
Diagram particular premises second with X marks.
Ignore the conclusion while diagramming the premises.
Ask whether the completed premise diagram already contains everything asserted by the conclusion.
If yes, the form is valid; if not, it is invalid.

Valid notebook example

All philosophers are thinkers.
Some philosophers are teachers.
Therefore, some thinkers are teachers.

The existential mark for the second premise lies inside philosopher and teacher. Because the first premise places every philosopher inside thinker, that same mark must also be in thinker. The conclusion follows.

Invalid notebook example

No fish are birds.
Some eagles are not fish.
Therefore, some eagles are not birds.

The second premise permits the eagle to fall either inside or outside the bird circle, so the conclusion is not forced. A single countermodel—an eagle that is a bird but not a fish—is enough to show invalidity.

Core categorical checks

Exactly three terms should appear, each used consistently.
The middle term must be distributed at least once.
A term distributed in the conclusion must be distributed in its premise.
Two negative premises yield no valid conclusion.
A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.
Two particular premises do not force a conclusion.
Do not infer existence from universal premises alone in modern Boolean logic.

These verbal rules are useful diagnostics; the Venn test is usually safer because it exposes precisely where a purported conclusion exceeds the premises.

Conditional and disjunctive checks

Valid forms:

Modus ponens                 Modus tollens
If P, then Q.                If P, then Q.
P.                           Not Q.
Therefore Q.                 Therefore not P.
 
Disjunctive syllogism
P or Q.
Not P.
Therefore Q.

Common invalid forms:

Affirming the consequent     Denying the antecedent
If P, then Q.                If P, then Q.
Q.                           Not P.
Therefore P.                 Therefore not Q.

Also distinguish inclusive or—at least one disjunct is true—from exclusive or—exactly one is true. Ordinary language does not settle that automatically.

Retrieval questions

Why does truth of the premises not by itself establish validity?
What does shading mean, and what does an X mean?
Why is the eagle–bird–fish argument invalid?
Which inference rule is confused by affirming the consequent?
When does a universal statement carry existential import?