Predicate Logic

Predicate Logic

Predicate Logic = First Order Logic

Limitation of Propositional Logic

• Cannot represent statements involving objects and their properties
• Cannot express “for all” or “there exists”
• Cannot handle relationships between objects
• Cannot generalize patterns

Example limitations:

• “All students passed” - cannot express relationship between students and passing
• “Some number is prime” - cannot quantify over numbers
• “Every integer has a successor” - cannot express universal property
• Propositional logic treats “John is tall” and “Mary is tall” as unrelated

Predicate Logic

• Extends propositional logic with variables, predicates, and quantifiers
• Also called First Order Logic (FOL)
• Can express properties of objects and relationships
• More expressive power than propositional logic

Components:

1. Variables: x, y, z (represent objects)
2. Predicates: P(x), Q(x,y) (properties/relations)
3. Quantifiers: ∀ (for all), ∃ (there exists)
4. Connectives: ¬, ∧, ∨, →, ↔ (same as propositional logic)
5. Constants: a, b, c (specific objects)
6. Functions: f(x), g(x,y) (map objects to objects)

Predicate

• A predicate is a statement with variables that becomes a proposition when values are assigned
• Notation: P(x), Q(x,y), R(x,y,z)
• P(x) - unary predicate (one variable)
• Q(x,y) - binary predicate (two variables)

Examples:

• P(x): “x is even”
  • P(4) → True (4 is even)
  • P(5) → False (5 is not even)
• Q(x,y): “x > y”
  • Q(5,3) → True
  • Q(2,7) → False
• R(x): “x is a prime number”
• S(x,y): “x is a parent of y”

Truth value:

• Predicate alone has NO truth value
• Becomes proposition when:
  • Variables are assigned values, OR
  • Variables are quantified

Domain (Universe of Discourse)

• Set of all possible values that variables can take
• Must be clearly defined
• Affects truth value of quantified statements

Examples:

• Domain = $\mathbb{Z}$ (Integers): {…, -2, -1, 0, 1, 2, …}
• Domain = $\mathbb{N}$ (Natural numbers): {0, 1, 2, 3, …}
• Domain = Students in a class
• Domain = All humans
• Domain = Real numbers $\mathbb{R}$

GATE Important:

• Same statement can be true in one domain, false in another
• Always check domain carefully

Example:

• $\forall x (x^2 \geq 0)$
  • True if domain = $\mathbb{R}$ (Real numbers)
  • Also true if domain = $\mathbb{Z}$ (Integers)

Quantifiers

Used to specify how many elements in the domain satisfy a predicate

Universal Quantifier (∀) - “For All”

Notation: $\forall x , P(x)$

Meaning: P(x) is true for ALL x in the domain

Truth condition:

• True: P(x) is true for every single element
• False: If even ONE element violates P(x)

Examples:

• $\forall x (x > 0)$ - “All x are positive”

  • False in domain $\mathbb{Z}$ (integers include negatives)
  • True in domain $\mathbb{N}^+$ (positive naturals)

• $\forall x (x^2 \geq 0)$ - “Square of every x is non-negative”

  • True in domain $\mathbb{R}$ (real numbers)

• $\forall x , \text{Student}(x) \to \text{Passed}(x)$ - “All students passed”

Key point: One counterexample makes it false

Existential Quantifier (∃) - “There Exists”

Notation: $\exists x , P(x)$

Meaning: There exists AT LEAST ONE x such that P(x) is true

Truth condition:

• True: If at least one element satisfies P(x)
• False: If NO element satisfies P(x)

Examples:

• $\exists x (x^2 = 4)$ - “There exists x whose square is 4”

  • True in domain $\mathbb{Z}$ (x = 2 or x = -2)

• $\exists x (x > 10)$ - “There exists x greater than 10”

  • True in domain $\mathbb{N}$ (many such numbers exist)
  • False in domain {1, 2, 3, 4, 5}

• $\exists x , \text{Prime}(x) \land \text{Even}(x)$ - “There exists an even prime”

  • True (x = 2)

Key point: One example makes it true

Unique Existential Quantifier (∃!)

Notation: $\exists! x , P(x)$

Meaning: There exists EXACTLY ONE x such that P(x) is true

Expansion: $$\exists! x , P(x) \equiv \exists x , (P(x) \land \forall y , (P(y) \to y = x))$$

Example:

• $\exists! x (x + 5 = 8)$ - “There exists exactly one x such that x + 5 = 8”
  • True (only x = 3)

Nested Quantifiers

• Multiple quantifiers in same statement
• Order matters critically

Examples:

1. $\forall x , \exists y , P(x,y)$

• “For every x, there exists some y such that P(x,y)”
• For each x, we can pick DIFFERENT y
• Example: $\forall x , \exists y , (y > x)$ - True in $\mathbb{Z}$
  • For any integer x, there exists larger integer y

2. $\exists y , \forall x , P(x,y)$

• “There exists one y that works for ALL x”
• Same y must work for every x
• Example: $\exists y , \forall x , (y > x)$ - False in $\mathbb{Z}$
  • No single integer is greater than all integers

Key GATE Point: $$\forall x , \exists y , P(x,y) \not\equiv \exists y , \forall x , P(x,y)$$

More Examples:

3. $\forall x , \forall y , P(x,y)$

• Same as $\forall y , \forall x , P(x,y)$ (order doesn’t matter for same quantifier)
• “For all x and all y, P(x,y) is true”

4. $\exists x , \exists y , P(x,y)$

• Same as $\exists y , \exists x , P(x,y)$ (order doesn’t matter for same quantifier)
• “There exist x and y such that P(x,y) is true”

5. $\forall x , \exists y , \forall z , P(x,y,z)$

• For every x, there exists y (possibly different for each x), such that for all z, P(x,y,z)

Truth Table for Nested Quantifiers:

Statement	Meaning
$\forall x , \forall y , P(x,y)$	P holds for all pairs
$\exists x , \exists y , P(x,y)$	P holds for at least one pair
$\forall x , \exists y , P(x,y)$	For each x, some y works
$\exists x , \forall y , P(x,y)$	Some x works for all y
$\exists y , \forall x , P(x,y)$	Some y works for all x (NOT same as above)

Free and Bound Variables

Bound Variable:

• Variable that is quantified
• Scope limited to quantifier

Free Variable:

• Variable NOT quantified
• Truth value depends on assignment

Examples:

1. $\forall x , P(x, y)$

• x → bound (quantified by ∀)
• y → free (not quantified)
• Truth value depends on value of y

2. $\exists x , (P(x) \land Q(x, y))$

• x → bound
• y → free

3. $\forall x , \exists y , R(x, y, z)$

• x → bound by ∀
• y → bound by ∃
• z → free

4. $\forall x , P(x)$

• x → bound
• No free variables (closed formula)

GATE Important:

• Formula with no free variables = Sentence/Closed Formula
• Formula with free variables = Open Formula
• Only sentences have definite truth values

Scope of Quantifier

• Portion of formula where quantifier applies
• Usually indicated by parentheses

Examples:

1. $\forall x , (P(x) \to Q(x))$

• Scope of ∀x: entire $(P(x) \to Q(x))$

2. $(\forall x , P(x)) \to Q(y)$

• Scope of ∀x: only $P(x)$
• y is free in Q(y)

3. $\forall x , P(x) \land Q(y)$

• Ambiguous! Use parentheses
• Could mean: $\forall x , (P(x) \land Q(y))$ OR $(\forall x , P(x)) \land Q(y)$

Convention:

• Quantifier has lowest precedence
• $\forall x , P(x) \land Q(x)$ means $\forall x , (P(x) \land Q(x))$

Predicate Logic Connectives

Same connectives as propositional logic, but operate on predicates:

• $\neg P(x)$ - NOT
• $P(x) \land Q(x)$ - AND
• $P(x) \lor Q(x)$ - OR
• $P(x) \to Q(x)$ - Implication
• $P(x) \leftrightarrow Q(x)$ - Biconditional

Examples:

• $\forall x , (P(x) \to Q(x))$ - “If x has property P, then x has property Q”
• $\exists x , (P(x) \land Q(x))$ - “There exists x with both properties P and Q”
• $\forall x , (P(x) \lor Q(x))$ - “Every x has property P or property Q”

Negation of Quantifiers (Very Important for GATE)

Rule 1: $$\neg(\forall x , P(x)) \equiv \exists x , \neg P(x)$$

Meaning:

• “NOT all x satisfy P” = “There exists x that does NOT satisfy P”

Example:

• Statement: “All students passed”
• Negation: “At least one student did not pass”

Rule 2: $$\neg(\exists x , P(x)) \equiv \forall x , \neg P(x)$$

Meaning:

• “There does NOT exist x satisfying P” = “All x do NOT satisfy P”

Example:

• Statement: “There exists a perfect square that is negative”
• Negation: “All perfect squares are non-negative”

Nested Quantifiers Negation:

1. $\neg(\forall x , \exists y , P(x,y))$ $$\equiv \exists x , \neg(\exists y , P(x,y))$$ $$\equiv \exists x , \forall y , \neg P(x,y)$$

2. $\neg(\exists x , \forall y , P(x,y))$ $$\equiv \forall x , \neg(\forall y , P(x,y))$$ $$\equiv \forall x , \exists y , \neg P(x,y)$$

General Rule:

• Push negation inward
• Flip each quantifier (∀ becomes ∃, ∃ becomes ∀)
• Negate the predicate at the end

GATE Major Trap:

• $\neg \forall x , P(x) \not\equiv \forall x , \neg P(x)$
• Correct: $\neg \forall x , P(x) \equiv \exists x , \neg P(x)$

Logical Equivalence in Predicate Logic

Definition: Two formulas are equivalent if they have same truth value for all interpretations

Important Equivalences:

1. Distributive Laws: $$\forall x , (P(x) \land Q(x)) \equiv (\forall x , P(x)) \land (\forall x , Q(x))$$ $$\exists x , (P(x) \lor Q(x)) \equiv (\exists x , P(x)) \lor (\exists x , Q(x))$$

WARNING - NOT Equivalent: $$\forall x , (P(x) \lor Q(x)) \not\equiv (\forall x , P(x)) \lor (\forall x , Q(x))$$ $$\exists x , (P(x) \land Q(x)) \not\equiv (\exists x , P(x)) \land (\exists x , Q(x))$$

2. De Morgan’s for Quantifiers: $$\neg(\forall x , P(x)) \equiv \exists x , \neg P(x)$$ $$\neg(\exists x , P(x)) \equiv \forall x , \neg P(x)$$

3. Implication: $$\forall x , (P(x) \to Q(x)) \equiv \forall x , (\neg P(x) \lor Q(x))$$

4. Contrapositive: $$\forall x , (P(x) \to Q(x)) \equiv \forall x , (\neg Q(x) \to \neg P(x))$$

5. Quantifier Commutation (Same Type): $$\forall x , \forall y , P(x,y) \equiv \forall y , \forall x , P(x,y)$$ $$\exists x , \exists y , P(x,y) \equiv \exists y , \exists x , P(x,y)$$

6. Different Quantifiers - NOT Commutative: $$\forall x , \exists y , P(x,y) \not\equiv \exists y , \forall x , P(x,y)$$

7. Prenex Normal Form:

• All quantifiers moved to front
• Example: $\forall x , \exists y , (P(x) \to Q(y))$ is in prenex form

Validity and Satisfiability

Valid (Logically True):

• True for ALL interpretations (all domains, all predicates)
• Example: $\forall x , P(x) \to \exists x , P(x)$
  • “If P holds for all, then P holds for at least one”

Satisfiable:

• True for AT LEAST ONE interpretation
• Example: $\exists x , P(x)$
  • Can be true in some domain

Unsatisfiable (Contradiction):

• False for ALL interpretations
• Example: $\forall x , P(x) \land \exists x , \neg P(x)$
  • “All x have P AND some x doesn’t have P”

Relationships:

• Valid ⇒ Satisfiable
• Unsatisfiable ⇒ Not Valid
• $F$ is valid ⟺ $\neg F$ is unsatisfiable

Conjecture (Predicate Logic)

• A statement with quantifiers assumed to be true
• Requires formal proof or counterexample
• Common in mathematical reasoning

Examples:

• Goldbach’s Conjecture: $\forall n , (n > 2 \land \text{Even}(n) \to \exists p , \exists q , (\text{Prime}(p) \land \text{Prime}(q) \land n = p + q))$
• Collatz Conjecture: Properties of iterative sequences

English to Predicate Logic Translation

Domain: All people

English	Predicate Logic
”All students are intelligent”	$\forall x , (\text{Student}(x) \to \text{Intelligent}(x))$
“Some student is intelligent”	$\exists x , (\text{Student}(x) \land \text{Intelligent}(x))$
“No student is lazy”	$\forall x , (\text{Student}(x) \to \neg \text{Lazy}(x))$
“Not all students passed”	$\neg \forall x , (\text{Student}(x) \to \text{Passed}(x))$
“Every student loves some professor”	$\forall x , (\text{Student}(x) \to \exists y , (\text{Professor}(y) \land \text{Loves}(x,y)))$
“Some professor is loved by all students”	$\exists y , (\text{Professor}(y) \land \forall x , (\text{Student}(x) \to \text{Loves}(x,y)))$

Key Pattern:

• “All A are B” → $\forall x , (A(x) \to B(x))$ (use →)
• “Some A is B” → $\exists x , (A(x) \land B(x))$ (use ∧)
• “No A is B” → $\forall x , (A(x) \to \neg B(x))$

GATE Trap:

• $\forall x , (\text{Student}(x) \land \text{Passed}(x))$ is WRONG for “All students passed”
• Correct: $\forall x , (\text{Student}(x) \to \text{Passed}(x))$

GATE Focus Points

1. Difference between ∀ and ∃
   • ∀ = all, ∃ = at least one
2. Negation of quantified statements
   • Flip quantifier, negate predicate
3. Order of quantifiers
   • $\forall x , \exists y \not\equiv \exists y , \forall x$
4. Bound vs free variables
   • Quantified vs not quantified
5. Valid vs satisfiable statements
   • Valid = always true, Satisfiable = sometimes true
6. Translation from English to logic
   • “All” uses →, “Some” uses ∧
7. Scope of quantifiers
   • Use parentheses clearly
8. Distributive property limitations
   • Works for ∀ with ∧, ∃ with ∨ only
9. Contrapositive in predicate logic
   • $\forall x (P(x) \to Q(x)) \equiv \forall x (\neg Q(x) \to \neg P(x))$
10. Nested quantifier negation
    • Push negation through, flip each quantifier

Common GATE Traps

1. Universal with AND (Wrong):

   • “All students passed” ≠ $\forall x , (\text{Student}(x) \land \text{Passed}(x))$
   • Correct: $\forall x , (\text{Student}(x) \to \text{Passed}(x))$

2. Existential with Implication (Wrong):

   • “Some student passed” ≠ $\exists x , (\text{Student}(x) \to \text{Passed}(x))$
   • Correct: $\exists x , (\text{Student}(x) \land \text{Passed}(x))$

3. Negation Flip:

   • $\neg \forall x , P(x) \not\equiv \forall x , \neg P(x)$
   • Correct: $\neg \forall x , P(x) \equiv \exists x , \neg P(x)$

4. Quantifier Order:

   • $\forall x , \exists y , P(x,y)$ very different from $\exists y , \forall x , P(x,y)$

5. Distribution Error:

   • $\exists x , (P(x) \land Q(x)) \not\equiv (\exists x , P(x)) \land (\exists x , Q(x))$

6. Free Variables:

   • Formula with free variables has no definite truth value

Quick Formula Reference

$$\neg(\forall x , P(x)) \equiv \exists x , \neg P(x)$$ $$\neg(\exists x , P(x)) \equiv \forall x , \neg P(x)$$ $$\forall x , (P(x) \land Q(x)) \equiv (\forall x , P(x)) \land (\forall x , Q(x))$$ $$\exists x , (P(x) \lor Q(x)) \equiv (\exists x , P(x)) \lor (\exists x , Q(x))$$ $$\forall x , (P(x) \to Q(x)) \equiv \forall x , (\neg Q(x) \to \neg P(x))$$ $$\forall x , \forall y , P(x,y) \equiv \forall y , \forall x , P(x,y)$$ $$\forall x , \exists y , P(x,y) \not\equiv \exists y , \forall x , P(x,y)$$

Final Insights

1. Quantifier negation is critical - flip and negate
2. Order matters for different quantifiers - $\forall \exists \not\equiv \exists \forall$
3. Use → for “all”, ∧ for “some” - standard pattern
4. Bound variables are dummy variables - can be renamed
5. Free variables prevent truth assignment - only sentences have truth values
6. Distributive laws are limited - only certain combinations work
7. Domain affects truth - always check universe of discourse
8. Nested quantifiers need care - work from outside in
9. Prenex form simplifies - move all quantifiers to front
10. Practice translation - English ↔ Predicate Logic is key skill