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### Introduction to Proposition Logic

On this page, I will use capital letters, such as P and Q to refer to statements, which have a "truth value" -- that is, they are true or false.

If I write, simply,

P

this means the same thing as

P is a true statement.

That is, asserting P is the same thing as asserting that P is true.  If P means "it is a sunny day", then asserting P is the same as asserting that it is a sunny day.  It is unnecessary to say "it is true that it is a sunny day".

"not P" means the same thing as "P is false".

### Argument Forms; Rules (or Laws) of inference

 Argument form Rule form Meaning Web Reference Modus Ponens (affirming the antecedent; law of detachment; conditional elimination) P->Q P ———— Q If P then Q. P. therefore Q. Wikipedia Modus Tollens (denying the consequent; law of indirect reasoning) P->Q ¬Q ———— P If P then Q. not Q. therefore not P. Wikipedia Disjunctive Syllogism (Modus Tollendo Ponens) P V Q �P ———— Q P or Q. not P. therefore Q. Wikipedia Disjunction Elimination (proof by cases; separation of cases) P V Q P->R Q->R ———— R P or Q. If P then R. If Q then R. therefore R. Wikipedia Hypothetical Syllogism P->Q Q->R ———— P->R If P then Q. If Q then R. therefore, If P then R. Wikipedia Conjunction Introduction (rule of conjunction; rule of adjunction) P Q ———— P Λ Q P. Q. therefore, P and Q. Wikipedia Rule of Resolution P V Q �P V R ———— Q V R P or Q. (not P) or R. therefore Q or R. Wikipedia Disjunction Introduction (rule of disjunction) P ———— P V Q P. therefore P or Q. Wikipedia Conjunction Elimination (law of simplification) P Λ Q ———— P P and Q. therefore P. Wikipedia Constructive Dilemma P->Q R->S P V R ———— Q V S If P then Q. If R then S. P or R. therefore Q or S. Wikipedia Reductio ad absurdum (negation introduction; proof by contradiction) P->Q P->�Q ———— �P If P then Q. If P then not Q. therefore not P. Wikipedia DeMorgan's Laws (negation of disjunction) �(P V Q) ����� �P Λ �Q Not (P or Q) therefore not P and not Q. Wikipedia DeMorgan's Laws (negation of conjunction) �(P Λ Q) ����� �P V �Q Not (P and Q) therefore not P or not Q. Wikipedia

### Internet references

Wikipedia: List of rules of inference

### Related pages in this website

Truth Value -- a discussion of "truth tables" as a way of illustrating the meaning of logical operations

The webmaster and author of this Math Help site is Graeme McRae.