Tautology: In logic, a statement is a tautology iff it
is true in each and any case. Typical examples of tautologies in standard
propositional logic are "(pV~p)" i.e. p is true or p is not true (it
rains or it does not rain); "(p>p)" i.e. if p is true then p is true
(if it rains then it rains); and "((p&q) > q)" i.e. if p and q is true
then q is true (if it rains and it is cold then it is cold).
1. One way to show that a statement in standard propositional logic is a
tautology is by making a truthtable for it. The following illustrates
this for the last cited tautology:
p 
q 

(p&q) 
(p>q) 
((p&q) > q) 
T 
T 

T 
T 
T
T T 
T 
F 

F 
F 
F
T F 
F 
T 

F 
T 
F
T T 
F 
F 

F 
T 
F
T F 
As the table shows in the blue column, "((p&q) >
q)"is true in each and any case, whatever the
truthvalues of p and q,
and this depends on the definitons chosen for "&" and ">" that are
also given in the above table.
Note the above holds for standard
propositional logic. There
are other propositional logics, like intuitionist propositional logic
that are less easy to evaluate with the help of truthtables.
2. In some formulations of formal
logic there is an explicit assumption of a logical constant that
represents any tautology. Often, the letter T is used for that constant.
This then also is often combined with a logical constant that represents
any contradiction, for which an
inverse T then often is used.
Two reasons for this inclusion follow. First, this makes it easy to
represent in propositional logic that a statement is true or false,
namely by making it equivalent to a tautology or a
contradiction. Second, because it
allows for a fairly intuitive treatment of fundamental some logical
concepts such as negation, for if one has a constant for contradiction
one can define that a proposition is false iff the proposition implies a
contradiction.
