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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.