
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 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 
Wikipedia: List of rules of inference
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.