Prepositional Logic

Propositional Logic

Proposition

• A declarative statement
• Has a definite truth value: True (T) or False (F)
• Cannot be both true and false simultaneously

Examples:

• “2 + 3 = 5” → Proposition (True)
• “Paris is in France” → Proposition (True)
• “5 > 10” → Proposition (False)
• “Close the door” → Not a proposition (command)
• “What time is it?” → Not a proposition (question)
• “x + 2 = 5” → Not a proposition (depends on x)

Non-propositions:

• Questions
• Commands
• Exclamations
• Statements with variables (until assigned values)

Atomic and Compound Propositions

Atomic Proposition:

• Single, indivisible statement
• Cannot be broken down further
• Denoted by: p, q, r, s, etc.
• Examples:
  • p: “It is raining”
  • q: “The sun is shining”

Compound Proposition:

• Formed by combining atomic propositions using logical connectives
• Examples:
  • $p \land q$: “It is raining AND the sun is shining”
  • $p \lor q$: “It is raining OR the sun is shining”
  • $\neg p$: “It is NOT raining”

Logical Connectives

1. NOT (Negation) $\neg p$

p	$\neg p$
T	F
F	T

• Reverses truth value
• Also written as: ~p, ¬p, !p

2. AND (Conjunction) $p \land q$

p	q	$p \land q$
T	T	T
T	F	F
F	T	F
F	F	F

• True only when BOTH are true
• Also written as: p ∧ q, p · q, p & q

3. OR (Disjunction) $p \lor q$

p	q	$p \lor q$
T	T	T
T	F	T
F	T	T
F	F	F

• True when AT LEAST ONE is true
• Inclusive OR (both can be true)
• Also written as: p ∨ q, p + q, p | q

4. XOR (Exclusive OR) $p \oplus q$

p	q	$p \oplus q$
T	T	F
T	F	T
F	T	T
F	F	F

• True when EXACTLY ONE is true
• $p \oplus q \equiv (p \lor q) \land \neg(p \land q)$
• $p \oplus q \equiv (p \land \neg q) \lor (\neg p \land q)$

5. Implication (Conditional) $p \to q$

p	q	$p \to q$
T	T	T
T	F	F
F	T	T
F	F	T

• False only when p is true and q is false
• Read as: “if p then q”, “p implies q”, “p only if q”
• p = hypothesis/antecedent, q = conclusion/consequent
• Key equivalence: $p \to q \equiv \neg p \lor q$
• Vacuous truth: When p is false, implication is automatically true

6. Biconditional (Double Implication) - $p \leftrightarrow q$

p	q	$p \leftrightarrow q$
T	T	T
T	F	F
F	T	F
F	F	T

• True when both have SAME truth value
• Read as: “p if and only if q”, “p iff q”
• $p \leftrightarrow q \equiv (p \to q) \land (q \to p)$
• $p \leftrightarrow q \equiv (p \land q) \lor (\neg p \land \neg q)$

Operator Precedence (Highest to Lowest) ⭐

1. $\neg$ (NOT)
2. $\land$ (AND)
3. $\lor$ (OR)
4. $\to$ (Implication)
5. $\leftrightarrow$ (Biconditional)

Example: $p \lor q \land r$ means $p \lor (q \land r)$

Truth Table Construction

Steps:

1. Identify number of variables (n)
2. Number of rows = $2^n$
3. List all possible combinations
4. Evaluate expression step by step

Example: $(p \to q) \land (q \to r)$

p	q	r	$p \to q$	$q \to r$	$(p \to q) \land (q \to r)$
T	T	T	T	T	T
T	T	F	T	F	F
T	F	T	F	T	F
T	F	F	F	T	F
F	T	T	T	T	T
F	T	F	T	F	F
F	F	T	T	T	T
F	F	F	T	T	T

Tautology, Contradiction, Contingency

1. Tautology:

• Always true (all rows = T)
• Examples:
  • $p \lor \neg p$ (Law of Excluded Middle)
  • $p \to p$
  • $(p \land q) \to p$
  • $p \to (p \lor q)$
  • $(p \to q) \lor (q \to p)$

2. Contradiction:

• Always false (all rows = F)
• Examples:
  • $p \land \neg p$
  • $(p \to q) \land (p \land \neg q)$

3. Contingency:

• Sometimes true, sometimes false
• Neither tautology nor contradiction
• Examples:
  • $p \land q$
  • $p \to q$
  • $p \lor q$

GATE Important:

• Tautology = Valid formula
• Contradiction = Unsatisfiable formula

Logical Equivalence

Definition: Two propositions are logically equivalent if they have identical truth tables

Notation: $p \equiv q$ or $p \Leftrightarrow q$

Test: Show $(p \leftrightarrow q)$ is a tautology

Important Equivalences:

1. De Morgan’s Laws (Most Important for GATE): $$\neg(p \land q) \equiv \neg p \lor \neg q$$ $$\neg(p \lor q) \equiv \neg p \land \neg q$$

Extended form: $$\neg(p_1 \land p_2 \land … \land p_n) \equiv \neg p_1 \lor \neg p_2 \lor … \lor \neg p_n$$

2. Double Negation: $$\neg(\neg p) \equiv p$$

3. Implication Equivalences: $$p \to q \equiv \neg p \lor q$$ $$p \to q \equiv \neg q \to \neg p \text{ (Contrapositive)}$$ $$\neg(p \to q) \equiv p \land \neg q$$

4. Biconditional Equivalences: $$p \leftrightarrow q \equiv (p \to q) \land (q \to p)$$ $$p \leftrightarrow q \equiv (p \land q) \lor (\neg p \land \neg q)$$ $$p \leftrightarrow q \equiv (\neg p \lor q) \land (\neg q \lor p)$$

5. Idempotent Laws: $$p \land p \equiv p$$ $$p \lor p \equiv p$$

6. Commutative Laws: $$p \land q \equiv q \land p$$ $$p \lor q \equiv q \lor p$$

7. Associative Laws: $$(p \land q) \land r \equiv p \land (q \land r)$$ $$(p \lor q) \lor r \equiv p \lor (q \lor r)$$

8. Distributive Laws: $$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$ $$p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$$

9. Absorption Laws: $$p \land (p \lor q) \equiv p$$ $$p \lor (p \land q) \equiv p$$

10. Identity Laws: $$p \land T \equiv p$$ $$p \lor F \equiv p$$

11. Domination Laws: $$p \land F \equiv F$$ $$p \lor T \equiv T$$

12. Negation Laws: $$p \land \neg p \equiv F$$ $$p \lor \neg p \equiv T$$

13. Complement Laws: $$\neg T \equiv F$$ $$\neg F \equiv T$$

Implication Variations

Given: $p \to q$ (“if p then q”)

1. Converse: $q \to p$ (“if q then p”)

• NOT logically equivalent to original

2. Inverse: $\neg p \to \neg q$ (“if not p then not q”)

• NOT logically equivalent to original
• Logically equivalent to converse

3. Contrapositive: $\neg q \to \neg p$ (“if not q then not p”)

• Logically equivalent to original (Very important for GATE)

Key Facts:

• $p \to q \equiv \neg q \to \neg p$ (always)
• Converse $\equiv$ Inverse (but both $\not\equiv$ original)
• Original $\equiv$ Contrapositive

Truth Table:

p	q	$p \to q$	Converse $q \to p$	Inverse $\neg p \to \neg q$	Contrapositive $\neg q \to \neg p$
T	T	T	T	T	T
T	F	F	T	T	F
F	T	T	F	F	T
F	F	T	T	T	T

Validity and Satisfiability

Satisfiable:

• At least one truth assignment makes it true
• Has at least one row with T
• Example: $p \land q$ (satisfiable when both true)

Unsatisfiable:

• No truth assignment makes it true
• All rows are F
• Contradiction
• Example: $p \land \neg p$

Valid:

• Every truth assignment makes it true
• All rows are T
• Tautology
• Example: $p \lor \neg p$

Relationships:

• Valid ⇒ Satisfiable (but not vice versa)
• Unsatisfiable ⇒ Not Valid
• $p$ is valid ⟺ $\neg p$ is unsatisfiable

GATE Note:

• Valid = Tautology
• Unsatisfiable = Contradiction
• Satisfiable ≠ Valid (can be contingency)

Arguments and Validity

Argument Structure:

• Premises: $p_1, p_2, …, p_n$
• Conclusion: q
• Form: $p_1, p_2, …, p_n \therefore q$

Valid Argument:

• Conclusion logically follows from premises
• If all premises are true, conclusion MUST be true
• $(p_1 \land p_2 \land … \land p_n) \to q$ is a tautology

Testing Validity: Method 1: Truth table - check if conclusion is true whenever all premises are true Method 2: Prove $(p_1 \land p_2 \land … \land p_n) \to q$ is tautology Method 3: Assume premises true and conclusion false - if contradiction, argument is valid

Example (Valid):

• Premise 1: $p \to q$
• Premise 2: $p$
• Conclusion: $q$
• Test: $(p \to q) \land p \to q$ is tautology ✓

Example (Invalid):

• Premise 1: $p \to q$
• Premise 2: $q$
• Conclusion: $p$
• Test: $(p \to q) \land q \to p$ is NOT tautology ✗ (Fallacy: Affirming the consequent)

Rules of Inference (Very Important for GATE)

1. Modus Ponens (MP): $$\frac{p \to q, \quad p}{\therefore q}$$

2. Modus Tollens (MT): $$\frac{p \to q, \quad \neg q}{\therefore \neg p}$$

3. Hypothetical Syllogism (HS): $$\frac{p \to q, \quad q \to r}{\therefore p \to r}$$

4. Disjunctive Syllogism (DS): $$\frac{p \lor q, \quad \neg p}{\therefore q}$$

5. Addition (Add): $$\frac{p}{\therefore p \lor q}$$

6. Simplification (Simp): $$\frac{p \land q}{\therefore p}$$

7. Conjunction (Conj): $$\frac{p, \quad q}{\therefore p \land q}$$

8. Resolution: $$\frac{p \lor q, \quad \neg p \lor r}{\therefore q \lor r}$$

9. Constructive Dilemma: $$\frac{(p \to q) \land (r \to s), \quad p \lor r}{\therefore q \lor s}$$

10. Destructive Dilemma: $$\frac{(p \to q) \land (r \to s), \quad \neg q \lor \neg s}{\therefore \neg p \lor \neg r}$$

Common Fallacies (Invalid Arguments)

1. Affirming the Consequent: $$\frac{p \to q, \quad q}{\therefore p} \quad \text{(INVALID)}$$

2. Denying the Antecedent: $$\frac{p \to q, \quad \neg p}{\therefore \neg q} \quad \text{(INVALID)}$$

Conjecture (Logic Context)

• A proposition assumed or proposed to be true
• Requires proof or counterexample
• Common in mathematical reasoning
• Until proven, remains a conjecture
• Once proven, becomes a theorem

Example:

• Goldbach’s Conjecture: Every even integer > 2 is sum of two primes (unproven)

Normal Forms

Literal: Variable or its negation (p, ¬p)

Conjunctive Normal Form (CNF):

• AND of ORs
• $(p \lor q) \land (\neg p \lor r) \land (q \lor \neg r)$
• Product of Sums (POS)

Disjunctive Normal Form (DNF):

• OR of ANDs
• $(p \land q) \lor (\neg p \land r) \lor (q \land \neg r)$
• Sum of Products (SOP)

Every formula can be converted to CNF or DNF

GATE Focus Points

1. Implication truth table - Know when $p \to q$ is false
2. De Morgan’s laws - Distribute negation
3. Valid vs Satisfiable vs Tautology - Understand differences
4. Argument validity - Test using truth tables or rules
5. Conversion: $p \to q \equiv \neg p \lor q$
6. Contrapositive equivalence: $p \to q \equiv \neg q \to \neg p$
7. Difference between implication and converse - Not equivalent
8. Rules of inference - Modus Ponens, Modus Tollens, etc.
9. Logical equivalences - All laws listed above
10. CNF/DNF conversion

GATE Common Traps

1. Converse ≠ Implication

   • $p \to q \not\equiv q \to p$

2. Inverse ≠ Implication

   • $p \to q \not\equiv \neg p \to \neg q$

3. Only Contrapositive = Implication

   • $p \to q \equiv \neg q \to \neg p$

4. Implication is false only in one case

   • True antecedent, False consequent

5. Valid ≠ Satisfiable

   • Valid means always true (tautology)
   • Satisfiable means at least once true

6. De Morgan’s - watch the flip

   • AND becomes OR, OR becomes AND
   • Don’t forget to negate each part

7. Affirming consequent is invalid

   • $p \to q, q \not\Rightarrow p$

8. Denying antecedent is invalid

   • $p \to q, \neg p \not\Rightarrow \neg q$

Quick Formula Reference

$$p \to q \equiv \neg p \lor q$$ $$\neg(p \land q) \equiv \neg p \lor \neg q$$ $$\neg(p \lor q) \equiv \neg p \land \neg q$$ $$p \leftrightarrow q \equiv (p \to q) \land (q \to p)$$ $$p \to q \equiv \neg q \to \neg p$$ $$\neg(p \to q) \equiv p \land \neg q$$ $$p \oplus q \equiv (p \land \neg q) \lor (\neg p \land q)$$

Final Insights

1. Master implication - Most GATE questions involve $\to$
2. Know De Morgan’s laws cold - Used everywhere
3. Contrapositive is your friend - Equivalent to original
4. Truth tables never lie - When in doubt, construct one
5. Valid arguments preserve truth - Tautology test
6. Logical equivalence is transitive - Chain them
7. Precedence matters - NOT > AND > OR > IMPLIES > IFF
8. Practice conversion - Between different forms
9. Rules of inference - Memorize common patterns
10. Fallacies are tricky - Know what’s invalid