Propositions

Compound Proposition

The combination of 2 or more simple propositions formed using logical connectors.

Simple Proposition

Propositions that cannot be broken down further.

Negation

The negation of proposition P is denoted by ~P, read as not P.

Conjunction

The conjunction of propositions P and Q is denoted by P^Q, read as P and Q.

If one is F all is F.

Disjunction

The disjunction of propositions P and Q is denoted by PvQ, read as P or Q.

If one is T all is T.

Conditional

The conditional of propositions P and Q is denoted by P → Q, read as if P then Q.

If premise is T then its dependent o the conditional.

If premise is F then the conditional is T.

Biconditional

The biconditional of propositions P and Q is denoted by P <-> Q, read as P if and only if Q or P iff Q.

If its the same it’s T, different, it’s F.

When is a statement considered a proposition?

When it’s a declaration and when you can prove if its true or false.

Converse

If p, then q → If q, then p

Inverse

If p, then q. → If not p, then not q.

Contrapositive

If p, then q. → if not q, then not p.

Tautology

Proposition that’s always true.

Contradiction

Proposition that’s always false.

Contingency

Neither always true or false.

Logically equivalent

If a biconditional is a tautology.

Associative Law

(PvQ)vR = Pv(QvR)

Distributive Law

Pv(Q^R) = (PvQ)^(PvR)

Moragan’s Law

~(P^Q) = ~Pv~Q

Identity Law

P^T = P

PvF = P

Dominion Law

PvT = T

P^F = F

Idempotent Law

PvP = P

P^P = P

Double Negation

~(~P) = P

Commutative Law

PvQ = QvP

Simplification

(P^Q) → P

P^Q/…P

Addition

P → (PvQ)

P/…PvQ

Conjunction

[(P)^(Q)] → (P^Q)

P Q/…P^Q

Modus Ponens

[P^(P→Q)] → Q

P P→Q/…Q

Modus Tollens

[~Q^(P→Q)] → ~P

~Q P→Q/…~P

Hypothetical Syllogism

[(P→Q)^(Q→R)] → (P→R)

P→ Q Q→ R/…P→R

Disjunctive Syllogism

[(PvQ)^(~P)] → Q

PvQ ~P/…Q