Logic Study Guide: Validity, Categorical, and Propositional Logic

Validity

An argument is valid if the conclusion logically follows from the premises (no possible situation where premises are true and conclusion is false).

Soundness

A valid argument with all true premises.

Standard Form A

All S are P.

Standard Form E

No S are P.

Standard Form I

Some S are P.

Standard Form O

Some S are not P.

Square of Opposition

Shows logical relationships (contraries, subcontraries, contradictories).

Conversion

Switch S and P (valid for E and I).

Obversion

Change quality (A ↔ E, I ↔ O) and replace predicate with its complement (valid for all).

Contraposition

Switch S and P and replace both with complements (valid for A and O).

Negation

~ : Negation

Conjunction

· : Conjunction (and)

Disjunction

v : Disjunction (or)

Conditional

⊃ : Conditional (if...then)

Biconditional

≡ : Biconditional (if and only if)

Logical Truth

Always true.

Contradiction

Always false.

Contingency

Sometimes true.

Modus Ponens

MP: Modus Ponens (P ⊃ Q, P ⊢ Q)

Modus Tollens

MT: Modus Tollens (P ⊃ Q, ~Q ⊢ ~P)

Disjunctive Syllogism

DS: Disjunctive Syllogism (P v Q, ~P ⊢ Q)

Hypothetical Syllogism

HS: Hypothetical Syllogism (P ⊃ Q, Q ⊃ R ⊢ P ⊃ R)

Simplification

SIMP: Simplification (P · Q ⊢ P)

Conjunction

CONJ: Conjunction (P, Q ⊢ P · Q)

Addition

ADD: Addition (P ⊢ P v Q)

Double Negation

DN: Double Negation (~ ~P ⊢ P)

DeMorgan's

DeM: DeMorgan's (~(P · Q) ≡ ~P v ~Q; ~(P v Q) ≡ ~P · ~Q)

Conditional Proof

Assume antecedent, derive consequent, discharge to form implication.

Indirect Proof

Assume negation of conclusion, derive a contradiction, discharge assumption as false.