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Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek

T - Tautology

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 truth-table 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 truth-values 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 truth-tables.

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.

Literature:

Carnap, Cartwright, Quine, Stegmüller, Wittgenstein

Original: Aug 26, 2004                                                Last edited: 12 December 2011.