Soundness and Completeness of The Rules
The crucial properties of this set of rules are that they are sound and complete. Informally this means that the rules are correct and that no other rules are required. These claims can be made more formal as follows.
We define a truth assignment as a function that maps propositional variables to true or false. Informally such a truth assignment can be understood as the description of a possible state of affairs (or possible world) where certain statements are true and others are not. The semantics of formulae can then be formalized by defining for which "state of affairs" they are considered to be true, which is what is done by the following definition.
We define when such a truth assignment satisfies a certain wff with the following rules:
- satisfies the propositional variable if and only if
- satisfies if and only if does not satisfy
- satisfies if and only if satisfies both and
- satisfies if and only if satisfies at least one of either or
- satisfies if and only if it is not the case that satisfies but not
- satisfies if and only if satisfies both and or satisfies neither one of them
With this definition we can now formalize what it means for a formula to be implied by a certain set of formulae. Informally this is true if in all worlds that are possible given the set of formulae the formula also holds. This leads to the following formal definition: We say that a set of wffs semantically entails (or implies) a certain wff if all truth assignments that satisfy all the formulae in also satisfy
Finally we define syntactical entailment such that is syntactically entailed by if and only if we can derive it with the inference rules that were presented above in a finite number of steps. This allows us to formulate exactly what it means for the set of inference rules to be sound and complete:
- Soundness
- If the set of wffs syntactically entails wff then semantically entails
- Completeness
- If the set of wffs semantically entails wff then syntactically entails
For the above set of rules this is indeed the case.
Read more about this topic: Propositional Calculus
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