Logic

Logic is the systematic study of valid rules of inference and reasoning.

Propositional Logic

A proposition is a declarative statement that is either true or false, but not both.

• Conjunction (AND): p ∧ q is true only when both p and q are true
• Disjunction (OR): p ∨ q is true when either p or q (or both) are true
• Negation (NOT): ¬p is true when p is false, and false when p is true
• Implication (IF-THEN): p → q is false only when p is true and q is false
• Biconditional (IFF): p ↔ q is true when p and q have the same truth value

Truth Tables

Truth tables display all possible combinations of truth values for a compound proposition.

Logical Equivalences

Two propositions are logically equivalent if they have the same truth values in all cases.

• De Morgan's Laws: ¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q
• Double Negation: ¬(¬p) ≡ p
• Contrapositive: p → q ≡ ¬q → ¬p

Predicate Logic

Predicate logic extends propositional logic by introducing variables, quantifiers, and predicates.

• Universal Quantifier (∀): “For all”
• Existential Quantifier (∃): “There exists”

Logical Arguments

An argument is valid if the conclusion must be true whenever all premises are true.

• Modus Ponens: If p → q and p, then q
• Modus Tollens: If p → q and ¬q, then ¬p
• Hypothetical Syllogism: If p → q and q → r, then p → r

Common Fallacies

• Affirming the Consequent: If p → q and q, then p (invalid)
• Denying the Antecedent: If p → q and ¬p, then ¬q (invalid)
• Appeal to Authority (without argument)
• Ad Hominem: attacking the person, not the claim