Valid
Valid
In the terminology of Carnap, a sentence (or class of sentences) is valid if it is a consequence of the null class of sentences, contra-valid if every sentence is a consequence of it. The notion of consequence here refers to a full set of primitive formulas and rules of inference for the language or logistic system (q.v.) in question, known as c-rules, and including (in general) non-effective rules. If the notion of consequence is restricted to depend only on the d-rules — i.e., the subclass of the c-rules which are effective — it is then called d-consequence or derivability, and the terms corresponding to valid and contravalid are demonstrable and refutable respectively.
The formulas and the c-rules of the language in question may include some which are extralogical in character — corresponding, e.g., to physical laws or to matters of empirical fact. Carnap makes an attempt (which, however, has been questioned) to define in purely syntactical terms when a relation of consequence is one of logical consequence. If the notion of consequence is restricted to that of logical consequence, the terms corresponding to valid and contra-valid are analytic and contradictory respectively. If the c-rules are purely logical in character, the class of analytic sentences coincides with that of valid sentences, and the class of contradictory sentences with that of contravalid sentences.
The explicit definition of analyticity (etc.) for a particular language of course requires statement of the c-rules. Actually, in the case of his “Language II,” Carnap prefers to define analytic and contradictory first, and consequence in terms of these.
Part of the purpose of the definition of analyticity is to secure that every logical sentence is either analytic or contradictory. (The corresponding situation with demonstrability and refutability is impossible in many significant cases in consequence of Gdel’s theorem — see logic, formal, 6.)
Refer further to the article syntax, logical, where references to the literature are given. A.C.