(Latin verum "true") called a tautology ( altgr. τὸ αὐτό tò autó that same ' and -logy ), also Verum, the logic is a general statement, that is a statement that is always true for logical reasons. Examples of tautologies are statements such as " When it rains, it rains " and " All pigs are pigs ".
Sometimes the term is a tautology for all types of general statements used in part it is limited to those statements that are universal in the bivalent, classical propositional logic. In the latter, the propositional meaning of a compound proposition is a tautology if and only if it is true regardless of whether the sub-statements, of which it is composed, in turn, are true or false.
Formally, the finding that a statement or a tautology is generally valid, as written.
A propositional tautology, for example, the disjunction " It's raining, or it is not raining ": Regardless of whether occurring in her statement " It is raining " is true or not, the whole statement is true: Is " It is raining " is true, then is " it's raining, or it is not raining " true because the first subset of the disjunction is true. Is " It is raining " but wrong, then so is "It is not raining " is true. However, this is again the second subset of the disjunction, so that the whole sentence is true in this case.
If the notion of tautology is used in a broader sense, then statements fall under, are not in the propositional logic, but in other logical systems such as predicate logic or modal logic are universal though. In this sense, for example, the predicate logical general statement " All sheep are sheep," a predicate logic tautology modallogisch general statement "It is possible that it is raining or it is possible that it is not raining " a modallogische tautology.
In many-valued logics, ie in non-classical logics, in which there are more than two truth values , the Tautologiebegriff loses his - alleged or actual - colloquial naturalness and needs to be redefined. One way to take over the Tautologiebegriff in multi-valued logic, is of the truth values of one or more single out and ascribe them special meaning. This singled out pseudo truth values are called pseudo designated truth values . One defines that all those statements are tautologies, which provide a designated truth value for each evaluation of the atoms occur in them. In this solution, the Tautologiebegriff itself remains bivalent, that is, a statement is either a tautology or it is not.
Boundaries and relationships
Examples of tautologies in the two-valued propositional logic
- For each statement A is "If A, then A ' is a tautology - in formal notation:
- For each statement A is " A or not A" is a tautology, since the statement A is always either true or false - in formal notation:
- For each statement A, B "A is a sufficient condition for B, or B is a sufficient condition for A" is a tautology - in formal notation:
- All statements A, B, C " When assuming that A is the case where B is a sufficient condition for C, then the fact that A is a sufficient condition for B, sufficient to ensure that a sufficient O condition for C is " a tautology - in formal notation:
- Often mistaken to be found in the programming: IF ( varText "Hello" ) OR ( varText " good day " ) THEN ...; will return the value TRUE for all truth-possibilities. Such a statement is often spoken in everyday speech with one or, but meant is the logical and (conjunction ). At this point reference is made to the De Morgan's laws.
Central to the logic methods, to examine whether statements ( in any case true) are contingent (ie, in its truth by the truth or falsehood of their basic building blocks dependent) or tautological.
While such an examination of each method is in principle possible using, with the possible for all cases, the truth or falsity of a statement is determined, the so-called tree method is of special importance, since not every case has to be examined here.
In classical propositional logic the task of Tautologieprüfung coincides with the practical importance and intensively investigated satisfiability of propositional logic, because a statement iff is a tautology if its negation is unsatisfiable: To examine whether a statement is a tautology, hence coincides to ascertain whether its negation is satisfiable.