logical arguments

Modus Ponens

If p → q and p is true, then q must also be true.

Modus Tollens

If p → q and q is false (~q), then p must also be false (~p).

Generalization

From a statement p, we can infer p v q.

Specialization

From the conjunction p ^ q, we can infer p and also q.

Conjunction

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

Elimination (Disjunctive Syllogism)

From p v q, if ~q is true, then p must be true; and from p v q, if ~p is true, then q must be true.

Transitivity

If p → q and q → r, then p → r.

Proof by Division into Cases

From p v q, if p → r and q → r, then r must be true.

Contradiction Rule

If ~p implies c (a contradiction), then p must be true.