alwaystrueFormel

AlwaystrueFormel is a term used in logic and formal methods to denote a formula that is true under every interpretation and assignment. In standard terminology, such formulas are called tautologies or valid formulas, meaning they are true in every possible model and valuation.

Formally, a propositional formula F is an alwaystrueFormel if for every valuation v, the interpretation of F under v yields true. In predicate logic, this corresponds to formulas that are true in every structure and for every variable assignment. The concept is often denoted by the semantic consequence symbol or by the notation ⊨ F, indicating that F is logically valid.

Common examples in propositional logic include p ∨ ¬p, the law of excluded middle, and P ∨ ¬P for any proposition P. In first-order logic, examples include tautological implications such as ∀x (P(x) → P(x)) and A → A, which are true under all interpretations. Notably, the status of certain formulas can differ between classical and non-classical logics; for instance, p ∨ ¬p is a tautology in classical logic but not generally provable in intuitionistic logic.

AlwaystrueFormeln have practical uses in formal verification, automated theorem proving, and logical simplification. They serve as fundamental truths against which more complex formulas can be tested or simplified. While the term itself is descriptive and commonly understood in educational contexts, it is often treated as synonymous with the notion of a tautology or valid formula in most formal systems.