Statement:  a sentence that can be judged to be either TRUE or FALSE

Usually, the letters p, q, r, and s are used to designate a statement (think of them as variables which stand for a statement.)   For all of the following examples, p represents a statement and q represents a statement.

The negation of a statement, ~p  (read as "not p") always has a truth value that is the opposite of the truth value of the original statement.   If p is True, then ~p is False.  If p is False, then ~p is true. 

The conjunction (and) p^q is only true when both p and q are true.

The disjunction (or) pVq is true whenever either or both p or q are true.   It is only false if both are false.

The conditional (if then) p->q is always true except when the first part (antecedent or hypothesis) is true and the second part (conclusion) is false.

The biconditional (If and Only If) p<->q  is true if both sides match and false if they don't match.

Tautology:  A compound statement that is always true, regardless of the truth values of its components.

Logical Equivalence:  Two statements that always have the same truth values are logically equialent.  The biconditional of two logically equivalent statements is a tautology (their truth values always match, so it's true in every case.)

CONVERSE/INVERSE/CONTRAPOSITIVE

These are special types of conditionals (if-then statements)

Converse:  switch the hypothesis & conclusion

Inverse:  negate each side of the conditional

Contrapositive: do both:  negate both sides & switch.

Example:           p->q                                    Example 2:    r->~s

Converse:              q->p                                                            ~s->r

Inverse:                ~p->~q                                                         ~r->s

Contrapositive: ~q->~p                                                           s->~r

The contrapositive is logically equivalent to the original statement.

LAWS OF REASONING:

De Morgan's Laws:

~(p^q) is equivalent to ~pV~q

~(pVq) is equivalent to ~p^~q

(think of it as multiplying both sides of an inequality by a negative number.    Not only do the signs of the numbers change, but you also have to flip the sign.)

Law of the Contrapositive:

If   p->q is true, then ~q->~p is true (If a statement is true, then its contrapositive is true.)

Law of Detachment (Modus Ponens):

Given that a conditional (p->q) is true, and p is true, you can conclude that q is true.  ("If it's raining, then the ground gets wet."  If you know that it's raining, then you can conclude the ground is getting wet.)

Law of Modus Tollens:

Given that a conditional (p->q) is true, and the conclusion q is false, you can conclude that the hypothesis p is false. ("If it's raining, then the ground gets wet."  If you know that the ground isn't getting wet, then you can conclude that it's not raining.)

 Law of Syllogism:

If (p->q) is true, and (q->r) is true, you can conclude that (p->r).  This is sometimes referred to as the chain rule.   ("If it's raining, then the ground gets wet" and "if the ground gets wet, then it gets slippery."   If you know that it's raining, then you can directly conclude that the ground is getting slippery.)

Law of Disjunctive Inference:

If you know that (pVq) is true, and you know that either p is false or q is false, then you can conclude that the other is true.  (If you know that one of them is true, and it isn't this one, then it must be the other one!)

Negation of a Conditional

~(p -> q) is equivalent to  p^~q