Introduction to Metalogic

Definition. A propositional signature Σ is a collection of items, which we call propositional constants. Sometimes these propositional constants are also called elementary sentences, or even atomic sentences. However, we will be using the word “atomic” for a different concept. These propositional constants are assumed to have no independent meaning. Nonetheless, we assume a primitive notion of identity between propositional constants; the fact that two propositional constants are equal or non-equal is not explained by any more fundamental fact. This assumption is tantamount to saying that Σ is a bare set (and it stands in gross violation of Leibniz’s principle of the identity of indiscernibles).