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This is 1st-order logic so it's a bit more complicated, though you're on the right track (what you stated is the completeness theorem for propositional logic, n
by xamuel 7y ago
This is 1st-order logic so it's a bit more complicated, though you're on the right track (what you stated is the completeness theorem for propositional logic, not first-order logic).
Rather than boolean values for the variables, you need to enumerate all possible...
* Ambient universes where the language is interpreted
* Values (taken from the ambient universe) for the constant symbols
* Sets-of-tuples-of-values (from the ambient universe) for the predicate symbols
* Functions-from-tuples-of-values-to-values (from the ambient universe) for the function symbols
- solinent 7y agoYup, my definition of interpretation was limited for pedagogical purposes, but I should have been clearer. The propositional logic version is much simpler to understand, but the implications of the theorem are much more interesting in first-order logic, so introducing all of this nomenclature may be too much for someone who isn't motivated yet.