A propositional calculus is a calculus for propositional logic . He derives new statements from a given set of statements, which follow from the given statements in a logical manner. In general, the statements from which one derives are called premises ; the derived statements are called conclusions . The derivation of a conclusion from a set of premises is called an argument .
A distinction is made in principle:
- Semantic validity (conclusion): For classical propositional logic, semantic validity is defined as follows: An argument is semantically valid if and only if, provided that all premises are true, the conclusion is also true.
- Syntactic validity (derivation): An argument is syntactically valid if and only if the conclusion can be derived from the premises with the help of the axioms and inference rules of the selected propositional calculus.
A calculus is correct if only conclusions can be derived from it. A calculus is complete when all its consequences can be derived from it. For classical propositional logic, calculi can be given that are correct and complete.
Various propositional calculi are also decision procedures for the validity of arguments, that is, they allow you to determine for any argument within a finite time whether the argument is valid or not. Propositional calculi , which are decision-making procedures, are, for example, the propositional tree calculus or the propositional resolution calculus .
Concrete propositional calculi are given in the following articles:
- In the article propositional logic an axiomatic propositional calculus is given.
- In the article Baumkalkül a proposition calculus according to Beth , a refutation calculus, is given.
- In the article Existential Graphs the graphical propositional calculus of the alphagraphs is given.
- In the article resolution (logic) a refutation calculus is given, which is especially important for automatic proofing.
- In the articles Systems of Natural Inference and Sequential Calculus, rule calculi are given.