Definition: A formula of propositional logic is a tautology if the formula itself is always true regardless of which valuation is used for the propositional variables.
There are infinitely many tautologies.
Example
This formula has only 1 variable, A. To valuate this formula A, we must assign one of the truth values true or false, and assign ~A to the other truth value.
tautology 
meaning  
A 
˅ 
~ 
A 
Law of excluded middle: for any proposition, either that proposition is true, or its negation is. 
T 
T 
F 
T 
A or not A 
F 
T 
T 
F 
not A or A 
Verifying with tautologies
Why should we want to have tautologies? A tautology is a semantic way to verify that premises are true as well as their conclusion.
How do we determine whether a formula is a tautology? We 1^{st} evaluate every variable and then the operand(s) for its possible valuations, till every part of the formula is mapped. For every variable in a formula, we have 2 values, T and F. If a formula has n variables, then there are 2^{n} possible valuations of the formula.
For example (A˅B) has 2 variables, then this statement has 2^{2} (=2×2=4) possible valuations. When a statement has 3 variables, then it has 2^{3} (=2x2x2=8) possible valuations.
Tautological implication
Suppose formula S is a label for formula (A˅(B˅~B)). If R implies S, when every valuation that causes R to be true also causes S to be true, then we have a tautology.
However, suppose formula S is a label for (A˄(B˅~B)). Any valuation that makes A false, will make the formula S false, so that S is not a tautology.
Contradiction
Definition: A contradiction is a statement that is false under all circumstances.
If R is false for all values of its variables, so that there is no truth valuation that causes R to be true, then R is a contradiction.
What happens when we have a contradiction?
For example, in R implies S, the outcome of R→S for R=false, is always true, regardless of the truth or falsity of S. Say R is a label for a formula (p˄~p), which is a contradiction; then R is always false.
contradiction 
meaning  
p 
˄ 
~ 
p 

T 
F 
F 
T 
p and not p 
F 
F 
T 
F 
not p and p 
A contradiction does not eliminate anything, and therefore it implies everything. So a contradiction is useless for purposes of logic or evidence because it is always false, and anything can follow from it.
Thank you for your thourough contribution to my day! remarks: THe term U is not explained – there seems to be a mistake in the second proof “However, suppose formula S is a label for (A˄(B˅~B)). Any valuation that makes A false, will make the formula S false, so that S is not a tautology”Noticing that if S is false R> S cannot be a tautology. It doesn’t matter to the total argument, that S is not a tautology. Actually it is a contradiction is R and S are held in R>S
R = (A v (B v ~B)) > S = (A ^ (B v ~B)) 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1
Date: Wed, 20 Mar 2013 04:59:09 +0000
To: timecoach@hotmail.com
U=Universe, which is ~A. U and A together form the context. U makes A distinguishable from ~A.
However, suppose formula S (but I mistakenly wrote R) is a label for (A˄(B˅~B)). Any valuation that makes A false (when evaluating per row), will make the formula S false, so that S is not a tautology (as a formula).
R→S. S is a label for (A˄(B˅~B)), which is not a tautology 1100. Then, if R is a tautology, then R→S is not a tautology 1100 (as a formula).
(A ˄ (B ˅ ~ B))
1 1 1 1 0 1 A=T, T˄T=T
1 1 0 1 1 0 A=T, T˄T=T
0 0 1 1 0 1 A=F, F˄T=F
0 0 0 1 1 0 A=F, F˄T=F
But if S is a label for (A˅ (B˅~B)), which is a tautology 1111, then, if R is a tautology, then R→S is a tautology 1111 (as a formula).
(A ˅ (B ˅ ~ B))
1 1 1 1 0 1 A=T, T˅T=T
1 1 0 1 1 0 A=T, T˅T=T
0 1 1 1 0 1 A=F, F˅T=T
0 1 0 1 1 0 A=F, F˅T=T