R. Carnap, Fundmentals of Logic and Mathematics
S is called an interpretation of C if the rule of S determine the truth criteria for all sentences of C; if to every formula of C there is a corresponding proposition of S (converse not required).
2. Calculas (formal system)
A system of formal rules which determine certain formal properties and relations of sentences, esp. for the propose of formal deduction.
3. Formal (definition)
A definition of a term in the meta-language is called formal if it refers only to the expressions of the object-language (to the kinds of signs and the order in which they occur in the expressions), not to any extralinguistic objects and expressions, not to the designation of descriptive signs of the object-language.
4. Explicit (definition)
A rule of a calculas such that new sign is to be taken as an abbreviation for a certain expression consisting only of the old signs (i,e, primitive signs or signs defined eariler).
5. Definition in use
A rule of a calculas such that sentences containing the new sign and the old signs are to be taken as an abbreviation for certain sentences containg the old signs only.