A tautology ( ancient Greek ταυτολογία from ταὐτό t'autó [from τὸ αὐτό] "the same" and logic ), also called verum ( Latin verum "true"), is a generally valid statement in logic , that is, a statement that consists of logical reasons is always true. Examples of tautologies are statements such as "If it rains, then it rains" or "The weather changes or it stays as it is."
Sometimes the term tautology is used for all types of generally valid statements, sometimes it is restricted to statements that are generally valid in two-valued, classical propositional logic. In the latter, propositional logic , a compound statement is a tautology if and only if it is true, regardless of whether the partial statements from which it is composed are themselves true or false.
Formally, the statement that a statement is generally valid or a tautology is written as.
A propositional tautology is, for example, the disjunction "Either it's raining or it's not raining": Regardless of whether the statement "It's raining" is true or not, the whole statement is true: Is "It's raining" true, then "It is raining or it is not raining" is true because the first clause of the disjunction is true. But if “it's raining” is false, then “it's not raining” is true. But this in turn is the second sub-clause of the disjunction, so that the whole clause is also true in this case.
If the term tautology is used in a broader sense, then it also includes statements that, although not in propositional logic, are generally valid in other logical systems such as predicate logic or modal logic . In this sense, for example, the predicate logic generally valid statement “All sheep are sheep” is a predicate logic tautology, the modal logic generally valid statement “It is possible that it will rain or it is possible that it will not rain” is a modal logic tautology.
In multi-valued logics , i.e. in non-classical logics in which there are more than two truth values, the term tautology loses its - supposed or actual - colloquial naturalness and must be redefined. One possibility of adopting the concept of tautology in multi-valued logic is to pick one or more of the truth values and assign them special importance. These singled out pseudo-truth values are called designated pseudo-truth values. One defines that all those statements are tautologies that deliver a designated truth value for each evaluation of the atoms occurring in them. With this solution, the tautology concept itself remains bivalent, that is, a statement is either a tautology or it is not a tautology.
Delimitations and connections
- Tautology and theorem
- The concept of tautology is a semantic concept, i.e. defined from the meaning of a statement. It must be clearly distinguished from the syntactic concept theorem: A statement is called a theorem if it can be derived from a logical calculus using the axioms and inference rules of this calculus. In general, however, when setting up a calculus for logical purposes, one tries to formulate it in such a way that the theorems that can be derived from it are actually tautologies. In this case one speaks of a correct calculus. If a calculus is constructed in such a way that all tautologies can be derived from it, then it is called complete. For classical propositional logic and for first-order predicate logic , it is possible to give calculi that are both correct and complete. For the second level predicate logic, the Trachtenbrot theorem says that the general statements cannot be enumerated .
- Tautology and contradiction
- A contradiction is always a false statement. Thus in classical logic a statement is a tautology if and only if its negation is a contradiction, and a statement is a contradiction if and only if its negation is a tautology.
- Tautology and satisfiability
- A statement is called satisfiable if it can become true, i.e. it is not a contradiction. A proposition is a tautology if and only if its negation cannot be fulfilled.
- Tautology and analytically true sentences
- In traditional philosophical terminology, tautologies are, in the logical sense, a subclass of analytically true propositions. They are in contrast to synthetic formulas.
Examples of tautologies in bivalent propositional logic
- For every statement A , "If A , then A " is a tautology - in formal notation:
- For every statement A , "A or not A " is a tautology, since the statement A is always either true or false - in formal notation:
- For all statements A, B, C , “If, assuming that A is the case, B is a sufficient condition for C , then the fact that A is a sufficient condition for B is sufficient for A to be a sufficient one Condition for C and vice versa "is a tautology - in formal notation:
- The following error is conceivable in the programming: IF (varText "Hello") OR (varText "Hello") THEN ...; will deliver the value TRUE for all truth possibilities . Such a statement is often spoken in everyday language with a or , but what is meant is the logical and ( conjunction ). At this point we refer to De Morgan's laws .
Of central importance for logic are methods to check whether statements are contingent (that is, their truth depends on the truths or falsities of their basic building blocks) or tautological (true in any case).
While such a test is in principle possible with the help of any method with which the truth or falsity of a statement can be determined for all possible cases, the so-called tree method is of particular importance, since not every individual case has to be checked.
In classical propositional logic, the task of the tautology test coincides with the practically significant and intensively investigated satisfiability problem of propositional logic , because a statement is a tautology precisely when its negation is unsatisfiable: To check whether a statement is a tautology coincides with it, to check whether their negation can be fulfilled.