Philosophy 244: #4—Adequacy and Extensions

To be a theorem of K is to be derivable from the K-axioms by the K-inference rules. This is a purely syntactical notion, which pays no attention at all what the symbols might mean. To be K-valid, or as we put it, absolutely valid is to be successful in all settings bar none. This is a semantical notion which has a great deal to do with what the symbols mean; soon we will make it more semantical still by restating the definition of validity in the vocabulary of models. The point for now is that the notion of K-theoremhood and the notion and the notion of absolute validity are prima facie as different as they could be. If we want to claim a connection between the two notions the connection will have to be proved. Two things will have to be proved.