Intuitionistic Logic - Syntax - Hilbert-style Calculus

Hilbert-style Calculus

Intuitionistic logic can be defined using the following Hilbert-style calculus. Compare with the deduction system at Propositional calculus#Alternative calculus.

In propositional logic, the inference rule is modus ponens

  • MP: from and infer

and the axioms are

  • THEN-1:
  • THEN-2:
  • AND-1:
  • AND-2:
  • AND-3:
  • OR-1:
  • OR-2:
  • OR-3:
  • FALSE:

To make this a system of first-order predicate logic, the generalization rules

  • -GEN: from infer, if is not free in
  • -GEN: from infer, if is not free in

are added, along with the axioms

  • PRED-1:, if the term t is free for substitution for the variable x in (i.e., if no occurrence of any variable in t becomes bound in )
  • PRED-2:, with the same restriction as for PRED-1

Read more about this topic:  Intuitionistic Logic, Syntax

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