Syllogism
A syllogism is a deductive argument composed of exactly two premises, one conclusion, and three terms — each term appearing in exactly two of the three statements. Syllogisms are the formal engine of categorical logic: they demonstrate how class relationships chain together to produce necessary conclusions.
How It Appears Per Course
PHIL 252
The culmination of Unit 5’s categorical logic system. Syllogisms bring together the four categorical statement types, transitivity, and Venn diagram testing into a single unified framework.
Anatomy of a Syllogism
| Component | Definition | Location |
|---|---|---|
| Major term (P) | Predicate of the conclusion | Major premise |
| Minor term (S) | Subject of the conclusion | Minor premise |
| Middle term (M) | Links the two premises — never in conclusion | Both premises |
| Major premise | Contains major term + middle term | Written first |
| Minor premise | Contains minor term + middle term | Written second |
Major Premise: All birds are egg-layers. (M → P)
Minor Premise: All ducks are birds. (S → M)
──────────────────────────────────────────────
Conclusion: All ducks are egg-layers. (S → P)
Transitivity — the Key to Validity
A syllogism is valid only if the relationship it encodes is transitive: if A is R to B, and B is R to C, then A must be R to C.
Transitive relations (support valid syllogisms): “is a type of”, “is older than”, “is inside of”, “if…then”
Intransitive relations (block valid syllogisms): “is a parent of”, “loves”, “is a friend of”
Example of intransitivity: “Alice loves Bob. Bob loves Carol. ∴ Alice loves Carol.” — does not follow.
Mood and Figure
- Mood: The pattern of A/E/I/O statements used (e.g., AAA, EIO)
- Figure: The four possible arrangements of the middle term across the two premises Together, mood and figure determine whether a syllogism form is valid. Only a handful of the 256 possible combinations are valid.
Venn Diagram Test (3 Circles)
Setup: Three interlocking circles
- Top = Middle term (M)
- Lower-left = Major term (P)
- Lower-right = Minor term (S)
Procedure:
- Translate all three statements into categorical form
- Graph the major premise first (shading for universal, X for particular)
- Graph the minor premise (same rules)
- Do NOT graph the conclusion
- Inspect: If the diagram of the two premises already shows the conclusion is true, the syllogism is valid
If an X falls on a boundary between shaded and unshaded regions, move it to the unshaded area.
Syllogism Enthymemes
A syllogism with one premise left unstated — the audience supplies it. In a chained enthymeme, two syllogisms are linked such that the conclusion of the first is the unstated premise of the second. See Enthymeme.
Cross-Course Connections
CategoricalStatements — the A/E/I/O forms are the building blocks
Validity — syllogism validity is tested structurally (and with Venn diagrams)
Enthymeme — syllogisms often appear with missing premises in real arguments
ImmediateInference — single-statement inferences feed into multi-premise syllogisms
Key Points for Exam/Study
- Three terms, two premises, one conclusion — each term in exactly two statements
- Never diagram the conclusion — this is a common error
- Major premise is always written first; minor premise second
- The middle term must be the “bridge” — if its relation isn’t transitive, the syllogism fails
- Shading = region is empty; X = at least one member exists
- If both premises are particular (I/O), validity is rarely achieved — Venn diagram usually shows no overlap is guaranteed