PHIL 252 Unit 5 — Categorical Logic and Syllogisms
Core Argument of This Unit
Categorical logic is a formal system for expressing and testing relationships between classes of things. By translating ordinary statements into one of four standard forms (A, E, I, O), building three-term syllogisms, and diagramming with Venn diagrams, we can rigorously test whether conclusions follow necessarily from premises.
Key Ideas
The Four Categorical Statement Types:
| Label | Form | Name | Example |
|---|---|---|---|
| A | All S are P | Universal Affirmative | All dogs are mammals |
| E | No S are P | Universal Negative | No dogs are reptiles |
| I | Some S are P | Particular Affirmative | Some Canadians are teachers |
| O | Some S are not P | Particular Negative | Some snakes are not venomous |
Standard form: Quantifier + Subject + Copula + Predicate
Note: “Some” means at least one, not “most” or “many.” Universal statements (A, E) do not assert existence — only a relationship. Particular statements (I, O) do assert that at least one member exists.
Translation into Categorical Form:
- Rephrase subject and predicate as class terms (nouns, not adjectives)
- Rewrite the verb as “are” or “are not”
- Insert an explicit quantifier (All / No / Some)
- Treat proper names as universal claims (“All people identical to Socrates…“)
Syllogisms: An argument with exactly two premises, one conclusion, and three terms — each appearing in exactly two of the three statements.
- Major term (P): predicate of conclusion, appears in major premise
- Minor term (S): subject of conclusion, appears in minor premise
- Middle term (M): connects both premises, never appears in conclusion
Validity requires the middle term to form a transitive chain. Transitive relations transfer: “is a type of”, “is taller than”, “if..then”. Intransitive relations (loves, is a parent of) block the inference.
Venn Diagram Method (3 circles):
- Draw three overlapping circles: top = M, lower-left = P, lower-right = S
- Shade a region to indicate it is empty (for A and E statements)
- Mark X in a region to indicate at least one member exists (for I and O)
- Graph premises only — never graph the conclusion
- After diagramming both premises: if the conclusion is already represented, the syllogism is valid
Immediate Inference: Drawing a conclusion from a single categorical statement without needing other premises.
| Operation | How | Valid For |
|---|---|---|
| Conversion | Switch S and P | E and I only |
| Contraposition | Switch S and P + replace both with complements | A and O only |
| Obversion | Change quality (aff↔neg) + complement predicate | All four types |
Logical Relations (Square of Opposition):
- Contradiction (A/O, E/I): cannot both be true, cannot both be false
- Contrariety (A/E): cannot both be true, but can both be false
- Subcontrariety (I/O): cannot both be false, but can both be true
- Subalternation: if A is true, I must be true (assuming non-empty class)
Foundational for Unit 7
- Enumerative induction moves from particular statements (I/O) toward universal conclusions (A/E) — exactly the pattern of scientific generalization
- Syllogism structure models how scientific explanations chain premises to conclusions
- Venn diagram thinking (class inclusion/exclusion) supports causal analysis