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List of logic systems


This article contains a list of sample Hilbert-style deductive systems for propositional logic.

Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent and its main property is that it is syntactically complete, otherwise said that no new axiom not already consequence of the existing axioms can be added without making the logic inconsistent. Many different equivalent complete axiom systems have been formulated. They differ in the choice of basic connectives used, which in all cases have to be functionally complete (i.e. able to express by composition all n-ary truth tables), and in the exact complete choice of axioms over the chosen basis of connectives.

The formulations here use implication and negation as functionally complete set of basic connectives. Every logic system requires at least one non-nullary rule of inference. Classical propositional calculus typically uses the rule of modus ponens:

We assume this rule is included in all systems below unless stated otherwise.

Frege's axiom system:


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