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 .
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