Week 7 - Syllogism

Part 1 — Taxonomy and Form (syllogisms1.pdf)

What is a categorical syllogism?

• An argument with exactly:
  • 2 premises + 1 conclusion
  • Only categorical propositions (A/E/I/O types)
  • Exactly 3 terms:
    • Minor term (S): subject of the conclusion; appears only in the 2nd premise
    • Major term (P): predicate of the conclusion; appears only in the 1st premise
    • Middle term (M): appears in both premises, not in the conclusion

Figures (position of M relative to S and P)

• Figure 1:
  • Major: M–P
  • Minor: S–M
  • Conclusion: S–P
• Figure 2:
  • Major: P–M
  • Minor: S–M
  • Conclusion: S–P
• Figure 3:
  • Major: M–P
  • Minor: M–S
  • Conclusion: S–P
• Figure 4:
  • Major: P–M
  • Minor: M–S
  • Conclusion: S–P

Mood (A/E/I/O types of each line)

• Mood is a 3-letter sequence: (Major)(Minor)(Conclusion)
  • A: All S are P (universal affirmative)
  • E: No S are P (universal negative)
  • I: Some S are P (particular affirmative)
  • O: Some S are not P (particular negative)
• Form = Mood + Figure (e.g., AAA-1)

Categorical proposition structure

• Quantifier (all/some) + Subject + Copula (is/are) + Qualifier (affirm/negate) + Predicate
• Quantifiers: universal (“all”), particular (“some”)
• Qualifiers: affirmative (no “not”), negative (“not”)

Venn diagram basics (Boolean stance)

• A (All S are P): shade S-outside-P
• E (No S are P): shade S∩P
• I (Some S are P): put x in S∩P
• O (Some S are not P): put x in S-outside-P

Square of Opposition (overview)

• Aristotelian stance (assumes S non-empty):
  • A and E are contraries (can’t both be true)
  • I and O are subcontraries (can’t both be false)
  • A ⇒ I, E ⇒ O (subalternation)
  • A vs O, E vs I are contradictories
• Boolean stance (no existence presupposition):
  • Only contradictories remain (A↔O, E↔I)

---

Part 2 — Evaluation Methods in Boolean Logic (syllogisms2.pdf)

Method 1: Venn Diagram Technique (3-circle)

Steps:

1. Draw 3 overlapping circles (top-left = minor S, top-right = major P, bottom = middle M; or as directed).
2. Diagram the premises only.
3. The syllogism is valid iff the diagram already depicts the conclusion.

Key insight:

• If x’s location is ambiguous and does not force the conclusion’s region, the argument is invalid.

Method 2: Rule Technique (Boolean stance)

Distribution concept:

• A: All S are P → S distributed, P undistributed
• E: No S are P → S distributed, P distributed
• I: Some S are P → S undistributed, P undistributed
• O: Some S are not P → S undistributed, P distributed (stipulation aligns rules with Venn results)

Rules (R1–R5):

1. The middle term (M) must be distributed in at least one premise.
2. Any term distributed in the conclusion must be distributed in its corresponding premise.
3. No valid syllogism has two negative premises.
4. A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.
5. Two universal premises require a universal conclusion (existential fallacy if violated).

Agreement:

• Venn diagram method and rule technique are equivalent in Boolean logic.

Tips:

• “All EIOs are valid” (MacDonald’s Rule): every EIO mood is valid in its figure—verify by diagrams/rules.
• Some forms can be rejected quickly by rules (e.g., OOO-3 violates R3; EOE-4 violates R3/R4, etc.).

---

Part 3 — Aristotelian Stance (syllogisms3.pdf)

Stance difference

• Assumes the S-term refers (non-empty class).
• Strengthens relations among A/E/I/O (traditional square holds, not just contradictories).

Square (Aristotelian):

• Contraries: A vs E (can’t both be true)
• Subcontraries: I vs O (can’t both be false)
• Subalternation: A ⇒ I, E ⇒ O
• Contradictories: A vs O, E vs I (can’t both be true or both be false)

Venn diagram adjustments (Aristotelian)

• A: “All S are P, and there are S”
  • Shade S-outside-P
  • Place x in S∩P (to mark existence)
• E: “No S are P, and there are S and there are P”
  • Shade S∩P
  • Place x in S-outside-P and x in P-outside-S
• I, O: same as Boolean (already include existence via x in S).
• Consequence: Some syllogisms invalid in Boolean become valid in Aristotelian if they only failed by existential fallacy (R5).

Rule Technique (Aristotelian)

• Drop R5. Keep R1–R4.
  • If a syllogism violated only R5 in Boolean logic, it becomes valid here.
  • Violations of R1–R4 remain invalid.

Validity relationships across stances

• Every Boolean-valid syllogism is Aristotelian-valid.
• Some Aristotelian-valid syllogisms are Boolean-invalid (those that only failed R5).

Example

• EAO-3:
  • Boolean: invalid (fails R5; needs existence to infer particular conclusion from universal premises).
  • Aristotelian: valid (existence presupposition supplies the needed x).

---

Practice and Identification

Identifying form (mood + figure)

• Example:
  • Major: All mammals are animals (A)
  • Minor: All humans are mammals (A)
  • Conclusion: All humans are animals (A)
  • Figure 1 (M–P; S–M; S–P) → AAA-1

Quick diagnostics

• Distributed terms:
  • If conclusion distributes S or P, the matching term must be distributed in its premise (R2).
• Negativity:
  • Two negative premises → invalid (R3).
  • Negative conclusion iff exactly one negative premise (R4).
• Existential fallacy:
  • Universal premises with a particular conclusion → invalid in Boolean (R5), but fine in Aristotelian.

Single-proposition inferences

• Given A true:
  • Boolean: O false; no info on E, I.
  • Aristotelian: O false, E false (contraries), I true (subalternation/subcontraries).
• Given A false:
  • Both stances: O true (contradictory).
  • Aristotelian: E could be T or F; I could be T or F (only constraints: contraries/subcontraries).

---

Common Pitfalls

• Confusing figures by misplacing the middle term.
• Forgetting distribution in O-propositions applies to P.
• Assuming existence in Boolean diagrams (don’t place x unless required by I/O or forced by Aristotelian A/E).
• Drawing the conclusion into the diagram before testing validity (only diagram premises).

Workflow Summary

1. Classify mood and figure → Form.
2. Choose method:
   • Venn (diagram premises; check if conclusion is already represented).
   • Rules (R1–R5 for Boolean; R1–R4 for Aristotelian).
3. If Boolean-invalid solely by R5, then Aristotelian-valid.
4. Use square of opposition appropriately:
   • Boolean: only contradictories.
   • Aristotelian: full traditional square.

Valid Categorical Syllogisms

Below are the valid moods by figure for each stance. Forms are given as MOOD–FIGURE.

Valid Boolean Syllogisms

Figure	Valid Forms
First	AAA, EAE, AII, EIO
Second	EAE, AEE, EIO, AOO
Third	IAI, AII, OAO, EIO
Fourth	AEE, IAI, EIO

Valid Aristotelian Syllogisms

Figure	Valid Forms
First	AAI, EAO
Second	AEO, EAO
Third	AAI, EAO
Fourth	AEO, EAO, AAI