PHIL222 - General Predicate Logic (GPL)

### Detailed Notes on General Predicate Logic (GPL) – PHIL222 Lecture 14

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#### 1. Limitations of Monadic Predicate Logic (MPL)

- MPL can only handle one-place predicates (properties), e.g., \( Hx \): "x is a husky," \( Dx \): "x is a dog."

- Example:

- Argument:

- P1) All huskies are dogs. \(\forall x (Hx \rightarrow Dx)\)

- P2) Ragnar is a husky. \( Hr \)

- C) Ragnar is a dog. \( Dr \)

- MPL fails for relational statements (e.g., "Brutus betrayed Caesar") because it lacks many-place predicates.

- Requires General Predicate Logic (GPL) to express relations like \( B^2xy \): "x betrays y."

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#### 2. Many-Place Predicates in GPL

- Types of Predicates:

- One-place (properties): \( I^1x \): "x is interesting."

- Two-place (dyadic relations): \( T^2xy \): "x is taller than y," \( N^2xy \): "x is next to y."

- Three-place (triadic relations): \( S^3xyz \): "x is between y and z."

- Examples:

- "Alice is taller than Bob" → \( Tab \) (glossary: \( a \): Alice, \( b \): Bob).

- "Carol is between Alice and Bob" → \( Scab \).

Key Point:

- Order of arguments matters! \( Tab \) ≠ \( Tba \).

- Predicate superscripts (e.g., \(^2\)) can be omitted if clear from context (e.g., \( Hxy \): "x heard y").

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#### 3. Syntax of GPL

- Symbols:

- Names: Constants like \( a, b, c \).

- Variables: \( x, y, z \).

- Predicates: \( A^1, B^2, C^3, \dots \).

- Connectives: \( \neg, \land, \lor, \rightarrow, \leftrightarrow \).

- Quantifiers: \( \forall, \exists \).

- Well-Formed Formulas (wffs):

- Atomic: \( P^n t_1 \dots t_n \) (e.g., \( Tab \)).

- Complex: Built using connectives/quantifiers (e.g., \( \forall x (Hx \rightarrow Dx) \)).

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#### 4. Translating Comparative vs. Absolute Claims

- Absolute properties: Use one-place predicates.

- "Bob is tall" → \( T^1b \).

- Comparative relations: Use two-place predicates.

- "Alice is taller than Bob" → \( T^2ab \).

- Example Argument:

- P1) \( Tab \) (Alice is taller than Bob).

- P2) \( T^1b \) (Bob is tall).

- C) \( T^1a \) (Alice is tall).

- Note: Requires separate predicates for "tall" (\( T^1x \)) and "taller than" (\( T^2xy \)).

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#### 5. Multiple Quantifiers

- Reading Order Matters:

- \( \forall x \exists y Lxy \): "Everyone likes someone (possibly different)."

- \( \exists y \forall x Lxy \): "Someone is liked by everyone."

- Equivalencies:

- \( \forall x \forall y Lxy \equiv \forall y \forall x Lxy \).

- \( \exists x \exists y Lxy \equiv \exists y \exists x Lxy \).

- Non-equivalents:

- \( \forall x \exists y Lxy \) ≠ \( \exists y \forall x Lxy \).

Examples:

- "Something likes everything" → \( \exists x \forall y Lxy \).

- "Everything likes something" → \( \forall x \exists y Lxy \).

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#### 6. Uniform Substitution of Variables

- Replacing bound variables uniformly preserves meaning:

- \( \exists x \forall y Lxy \equiv \exists z \forall y Lzy \equiv \exists x \forall w Lxw \).

- Rule:

- \( \forall x \alpha \equiv \forall y \alpha(y/x) \) (if \( y \) is not free in \( \alpha \)).

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#### Key Takeaways for Study

1. Master Translations:

- Practice converting English to GPL (e.g., "No student hates all teachers" → \( \forall x (Sx \rightarrow \neg \forall y (Ty \rightarrow Hxy)) \)).

2. Understand Many-Place Predicates:

- Distinguish properties (one-place) from relations (two+/three-place).

3. Quantifier Order:

- Recognize how \( \forall \exists \) vs. \( \exists \forall \) changes meaning.

4. Syntax Rules:

- Ensure wffs follow GPL formation rules (e.g., \( \forall x (Px \rightarrow Qx) \) is valid; \( \forall x Px \rightarrow Qx \) is not).

5. Counterexamples:

- Use open truth-tree branches to construct invalid arguments (e.g., \( \exists x Px \not\models \forall x Px \)).

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Practice Exercises:

1. Translate: "Every student admires some professor."

- Answer: \( \forall x (Sx \rightarrow \exists y (Py \land Axy)) \).

2. Evaluate: Is \( \exists x \forall y Lxy \rightarrow \forall y \exists x Lxy \) a tautology?

- Hint: Use truth trees to check validity.

By focusing on these areas, you’ll be prepared for GPL translations, quantifier logic, and syntactic analysis in your test.