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 Skip Navigation LinksMath Help > First Principles > Logic and Proof > Proposition Logic

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

Basic Principles

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.