Generality

In a formal logic or a calculus , a formula is said to be generally valid or valid if it is fulfilled by any interpretation . The general validity is therefore a special case of the satisfiability of a formula. While the mere satisfiability is already given if there is only a single fulfilling interpretation - a so-called model - all interpretations are models in the case of a general formula .

The central explanation for this concept of interpretation is intuitive as a generalization of the variable assignment in propositional logic understand: the formula is only through the assignment of propositional variables of a propositional formula can be a total write a truth value. In more complex logics, assignments must also be made to the formal components of a formula, which determine the truth value of the overall formula. In predicate logic, for example, a universe is defined and predicate symbols are assigned to predicates (on this universe) and function symbols to functions (on this universe). It is only through this reference to a set of objects in a considered world that it can be determined whether a formula can be fulfilled and whether it is possibly always fulfilled, i.e. generally valid.

The following table lists some closely related terms and synonyms. The columns and are in an equivalence relationship, e.g. B. is generally valid if and only if it cannot be fulfilled. ${\displaystyle F}$ ${\displaystyle \neg F}$ ${\displaystyle F}$ ${\displaystyle \neg F}$

${\displaystyle F}$	Synonyms	condition	${\displaystyle \neg F}$
general	tautological (in propositional logic)	All interpretations meet the formula.	unattainable
achievable	consistent, free of contradictions	There is an interpretation that satisfies the formula.	falsifiable
falsifiable	refutable	There is an interpretation that refutes the formula.	achievable
unattainable	inconsistent, contradicting	No interpretation satisfies the formula.	general

See also

• tautology