Propositions and Logic

• Logic is the science of evaluating arguments and reasoning.
• Mathematical logic (or symbolic logic) links mathematics and computer science.

Proposition (Statement)

• A proposition is a declarative sentence that is either true or false, but not both.
• "Manila is the capital of the Philippines" is a proposition.
• Questions like "What day is it?" are not propositions.
• Statements like "Mathematics is fun" are propositions, but are ambiguous.

Propositional Variable

• A propositional variable is a variable used to represent a proposition, often using letters like p, q, or r.

Compound Proposition

• A compound proposition combines two or more simple propositions using logical connectives.
• Logical connectives include AND, OR, IF AND ONLY IF, EXCLUSIVE-OR, NOT, and IF THEN.

Atomic Proposition

• An atomic proposition is a simple proposition that is not compound.

Logical Operators

Conjunction (AND, "^")

• p ^ q is true only if both p and q are true; otherwise, it is false.

Disjunction (OR, "V")

• p V q is only false when both p and q are false.

Negation (NOT, "¬")

• If p is true, then ¬p is false, and vice versa.

Conditional (if-then, "→")

• In p → q, p is the antecedent, hypothesis, or premise, and q is the conclusion, consequent, or consequence.
• p → q is only false when p is true and q is false; otherwise, it is true.

Biconditional (if and only if, "↔")

• p ↔ q is true if p and q are both true or both false.

Exclusive-OR (⊕)

• p ⊕ q is false if p and q are both true or both false; otherwise, it is true.

Common Words Associated with Connectives

• Conjunction: and, but, also, moreover
• Disjunction: or
• Negation: not, it is false that, it is not the case that
• Conditional: If p, then q. p implies q. p only if q. p is sufficient condition for q. q is necessary condition for p.
• Biconditional: if and only if, is equivalent to, is necessary and sufficient for
• Exclusive-or: exclusive-or

Tautology

• A tautology is always true for all possible truth values.

Contradiction

• A contradiction is always false for all possible truth values.

Contingency

• A contingency can be either true or false.

Logically Equivalent

• Two compound statements are logically equivalent if their truth values are equal.

Inference

• An inference is a conclusion or opinion formed from known facts and evidence.

Rules of Inference

Modus Ponens (Law of Detachment)

• If p → q is true and p is true, then q is true.

Modus Tollens (Law of Contrapositive)

• If p → q is true and q is false, then p is false.

Hypothetical Syllogism

• If p → q and q → r are true, then p → r is true.

Disjunctive Syllogism

• If p V q is true and p is false, then q is true.

Conjunction

• If p and q are true, then p ^ q is true.

Simplification

• If p ^ q is true, then p is true and q is true.

Addition

• If p is true, then p V q is true.

This article explains the basics of logic, including propositions, propositional variables, and compound propositions. It covers atomic propositions and logical operators such as AND, OR, IF AND ONLY IF, EXCLUSIVE-OR, NOT, and IF THEN. Conjunction (AND) is true only if both propositions are true.