04 — Categorical Logic

All, some, and none — reasoning about categories

Covers: Chapters 4–5 — Categorical Propositions & Categorical Syllogisms (§4.1–5.7) Prerequisite: Lessons 01–03 Unlocks: Lesson 05 (Propositional Logic)

Why Should I Care?

Categorical logic is about reasoning with “all,” “some,” and “none.” In engineering: - “All robots with firmware < v1.20 have the SPI bug.” - “Some sensorbar readings are stale.” - “No robot in the fleet passed the thermal test.”

These statements have precise logical structure, and you can combine them into syllogisms to draw valid conclusions — or detect invalid ones.

PART 1 — CATEGORICAL PROPOSITIONS

1.1 Standard Form

Every categorical proposition relates two categories (called terms). There are exactly four standard forms:

Memory aid: AffIrmo (A and I are affirmative), nEgO (E and O are negative).

• S = subject term (what we’re talking about)
• P = predicate term (what we’re saying about it)
• Quantifier = All / No / Some
• Copula = “are” or “are not”

1.2 Quality and Quantity

• Quality: Affirmative or Negative (does it include or exclude?)
• Quantity: Universal or Particular (all or some?)

1.3 Distribution

A term is distributed when the proposition makes a claim about every member of that term’s category.

Memory trick: Universal → subject distributed. Negative → predicate distributed.

Why distribution matters: It’s the key to detecting invalid syllogisms. If a term is distributed in the conclusion but not in the premises, the syllogism is invalid (the fallacy of illicit distribution).

1.4 Venn Diagrams for Propositions

Represent each proposition visually with two overlapping circles (S and P):

A: All S are P — Shade out the S-only region (no S exists outside P)

    S         P
  ┌───┐   ┌───┐
  │///│   │   │
  │///│───│   │     /// = shaded (empty)
  │///│   │   │
  └───┘   └───┘
  [S outside P is empty → all S must be inside P]

E: No S are P — Shade out the overlap region

    S         P
  ┌───┐   ┌───┐
  │   │   │   │
  │   │///│   │     /// = shaded (empty)
  │   │   │   │
  └───┘   └───┘
  [overlap is empty → no S is P]

I: Some S are P — Place an X in the overlap region

    S         P
  ┌───┐   ┌───┐
  │   │   │   │
  │   │ X │   │     X = at least one member exists here
  │   │   │   │
  └───┘   └───┘

O: Some S are not P — Place an X in the S-only region

    S         P
  ┌───┐   ┌───┐
  │ X │   │   │
  │   │   │   │     X = at least one S exists outside P
  │   │   │   │
  └───┘   └───┘

1.5 The Modern Square of Opposition

The relationships between A, E, I, O:

         A ──── contradictory ──── O
         │                         │
    contrary                  subcontrary
         │                         │
         E ──── contradictory ──── I

• Contradictories (A/O, E/I): One is true, the other is false. Always.
• Contraries (A/E): Can both be false, but cannot both be true.
• Subcontraries (I/O): Can both be true, but cannot both be false.

Practical use: If you know “All robots passed the test” (A) is FALSE, you immediately know “Some robots did not pass the test” (O) is TRUE (contradictory). But you cannot conclude “No robots passed the test” (E) — that might also be false.

1.6 Conversion, Obversion, Contraposition

Operations that transform propositions while preserving truth value (or not):

Conversion (switch S and P): - E: “No S are P” → “No P are S” ✓ (always valid) - I: “Some S are P” → “Some P are S” ✓ (always valid) - A: “All S are P” → “All P are S” ✗ (INVALID! “All dogs are mammals” ≠ “All mammals are dogs”)

Obversion (change quality, replace P with non-P): - A: “All S are P” → “No S are non-P” ✓ - E: “No S are P” → “All S are non-P” ✓ - I: “Some S are P” → “Some S are not non-P” ✓ - O: “Some S are not P” → “Some S are non-P” ✓

Contraposition (switch S and P, replace both with complements): - A: “All S are P” → “All non-P are non-S” ✓ - O: “Some S are not P” → “Some non-P are not non-S” ✓ - E and I: NOT generally valid by contraposition.

PART 2 — CATEGORICAL SYLLOGISMS

2.1 What Is a Syllogism?

A categorical syllogism is an argument with: - Exactly two premises and one conclusion - Exactly three terms, each appearing exactly twice - Each proposition in standard form (A, E, I, or O)

Terms: - Major term (P): The predicate of the conclusion - Minor term (S): The subject of the conclusion
- Middle term (M): Appears in both premises but NOT in the conclusion

Standard form layout:

Major premise:  [contains M and P]
Minor premise:  [contains M and S]
Conclusion:     [S ... P]

Example:

All SPI-connected devices (M) are susceptible to clock jitter (P).     [A]
All sensorbars (S) are SPI-connected devices (M).                       [A]
∴ All sensorbars (S) are susceptible to clock jitter (P).               [A]

This is Figure AAA-1 — valid. (We’ll explain figures and moods shortly.)

2.2 Mood and Figure

Mood = the letter combination of the three propositions (e.g., AAA, EIO, AOO).

Figure = the position of the middle term:

There are 4 × 4 × 4 = 64 possible moods, and 4 figures, giving 256 possible syllogistic forms. Only 15 are valid (unconditionally, in the Boolean interpretation).

2.3 Venn Diagram Test for Validity

To test a syllogism with Venn diagrams: 1. Draw three overlapping circles (S, M, P). 2. Diagram the universal premise first (shading). 3. Then diagram the particular premise (X mark). 4. Check: does the diagram already show the conclusion? - If yes → Valid - If no → Invalid

Rule: Only diagram the premises. Never diagram the conclusion. Then check if the conclusion appears automatically.

2.4 The Five Rules for Valid Syllogisms

Instead of drawing Venn diagrams every time, you can apply these rules:

1. The middle term must be distributed at least once. - Violation → Undistributed Middle - “All P are M. All S are M. ∴ All S are P.” — M is never distributed (predicate of A propositions).

2. If a term is distributed in the conclusion, it must be distributed in a premise. - Violation → Illicit Major or Illicit Minor - If P is distributed in conclusion but not in the premise → Illicit Major.

3. Two negative premises are not allowed. - Violation → Exclusive Premises

4. A negative premise requires a negative conclusion (and vice versa). - Violation → Drawing Affirmative from Negative (or Negative from Affirmative)

5. If both premises are universal, the conclusion cannot be particular. - (Boolean interpretation — no existential import for universals)

Practical workflow: Check rules 3 → 4 → 1 → 2 → 5 in order. First violation found = invalid.

2.5 Common Valid Forms (Named Syllogisms)

Engineering example (Barbara):

All devices with firmware < v1.20 (M) have the SPI timing bug (P).
All robots in building 3 (S) have firmware < v1.20 (M).
∴ All robots in building 3 (S) have the SPI timing bug (P).

Engineering example (Camestres):

All robots that passed testing (P) have firmware ≥ v1.24 (M).
No robots in fleet X (S) have firmware ≥ v1.24 (M).
∴ No robots in fleet X (S) passed testing (P).

2.6 Common Invalid Forms (Fallacies)

Undistributed Middle:

All robots with the bug (P) run Linux (M).
All OKS robots (S) run Linux (M).
∴ All OKS robots (S) have the bug (P).    ← INVALID!

M (run Linux) is predicate of two A-propositions → never distributed. Lots of things run Linux without having the bug.

Illicit Major:

All SPI devices (M) are affected by the bug (P).
No I2C devices (S) are SPI devices (M).
∴ No I2C devices (S) are affected by the bug (P).    ← INVALID!

P is distributed in the E conclusion but not in the A premise. The bug might affect I2C devices for a different reason.

Key Takeaways

1. Four standard forms: A (All S are P), E (No S are P), I (Some S are P), O (Some S are not P). Every categorical claim can be put in one of these forms.
2. Distribution is the key concept for checking validity — track which terms talk about “all” of their category.
3. The five rules let you check any syllogism quickly without Venn diagrams.
4. Undistributed middle is the most common syllogistic fallacy in everyday reasoning.
5. Translate engineering claims into standard form to check their logic: “All X are Y” / “No X are Y” / “Some X are Y” / “Some X are not Y.”