Propositional Calculus - Example 1. Simple Axiom System

Example 1. Simple Axiom System

Let, where, are defined as follows:

• The alpha set, is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

• Of the three connectives for conjunction, disjunction, and implication (, and ), one can be taken as primitive and the other two can be defined in terms of it and negation . Indeed, all of the logical connectives can be defined in terms of a sole sufficient operator. The biconditional can of course be defined in terms of conjunction and implication, with defined as .
  Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set partition as follows:

• An axiom system discovered by Jan Łukasiewicz formulates a propositional calculus in this language as follows. The axioms are all substitution instances of:

• The rule of inference is modus ponens (i.e., from and, infer ). Then is defined as, and is defined as .