Predicate Logic

🎯 Learning Objectives

• Distinguish predicates from propositions
• Use universal ($\forall$) and existential ($\exists$) quantifiers
• Translate between English statements and logical notation
• Negate quantified statements correctly
• Understand nested quantifiers and their order

2.1 Intuition: Beyond Atomic Statements

🔑 Key Insight: Predicates are like functions that return truth values: $P(x)$ is true or false depending on $x$.

2.2 Predicates and Quantifiers

Predicates

• $P(2)$ = True
• $P(4)$ = False
• $P(x)$ is not a proposition until $x$ is specified or quantified.

Universal Quantifier ($\forall$)

Existential Quantifier ($\exists$)

2.3 Quantifier Negation

• Original: $\forall x (\text{Sw}(x) \to \text{Wh}(x))$
• Negation: $\neg \forall x (\text{Sw}(x) \to \text{Wh}(x)) \equiv \exists x \neg (\text{Sw}(x) \to \text{Wh}(x))$
• $\equiv \exists x (\text{Sw}(x) \land \neg \text{Wh}(x))$
• English: "There exists a swan that is not white."

2.4 Nested Quantifiers

Order Matters!

• $\forall x \exists y (y = x^2)$ is true over real numbers (every number has a square)
• $\exists y \forall x (y = x^2)$ is false (no single number equals every square)

2.5 Translating English to Logic

• $\forall x \exists y L(x,y)$ where $L(x,y)$ = "$x$ loves $y$" Example 2: "There is a person who loves everyone."
• $\exists x \forall y L(x,y)$ Example 3: "If all men are mortal and Socrates is a man, then Socrates is mortal."
• $[\forall x (M(x) \to R(x)) \land M(s)] \to R(s)$
• $M(x)$ = "$x$ is a man", $R(x)$ = "$x$ is mortal", $s$ = Socrates

2.6 Multiple Domain Considerations

📊 Formula Summary

✅ Practice Questions

Solution

$\neg \exists x P(x)$ where $P(x)$ = "$x$ is perfect." Equivalent to $\forall x \neg P(x)$. Q2: Negate "Everyone passed the exam." Solution

Original: $\forall x P(x)$. Negation: $\exists x \neg P(x)$ = "Someone did not pass." Q3: Determine truth: $\forall x \in \mathbb{Z} \ \exists y \in \mathbb{Z} \ (x + y = 0)$. Solution

True. For any integer $x$, choose $y = -x$. Then $x + (-x) = 0$. Q4: Determine truth: $\exists y \in \mathbb{Z} \ \forall x \in \mathbb{Z} \ (x + y = 0)$. Solution

False. We'd need a single $y$ such that $x + y = 0$ for ALL $x$. If $y = 0$, then $x = 0$ for all $x$ — false. No such $y$ exists. Q5: Write the logical form of: "Some students in this class have taken every course in the program." Solution

$\exists x (S(x) \land \forall y (C(y) \to T(x,y)))$ where $S(x)$ = "$x$ is a student in this class," $C(y)$ = "$y$ is a course in the program," $T(x,y)$ = "$x$ has taken $y$." Join Discord PreviousPropositional LogicNextQuantifier Applications