On syntactical categories

Abstract

Up to now, most constructed calculi had the following common property: Whenever the rules of formation of a calculus laid down that expressions of the designs ‘Pa’, ‘Qa’, ‘Pb’ were sentences—‘P’ and ‘Q’ being first-level one-place predicates, ‘a’ and ‘b’ individual-symbols—, ‘Qb’ was a sentence too, according to the same rules. This self-imposed restriction of the logicians is historically understandable, since calculi of this common feature have a certain simplicity which differently constructed calculi will not have. So far, calculi of this type have proved to be sufficient for the formalization of mathematics and small parts of other sciences. We may ask ourselves, however, whether such a restriction will still be desirable when attempting to construct calculi covering more ground. It is quite possible that insistence on this kind of simplicity will involve a greater complexity in other respects. Inquiry into types of calculi which do not possess this simplicity should therefore be of some interest. To such an inquiry we are led also from another point of view. More and more stress has been laid in recent researches on the construction of calculi which should show close connection with ordinary languages, and it is obvious that ordinary language does not have the mentioned property. To use an example given by Carnap: Whereas ‘This stone is red,’ ‘Aluminium is red,’ ‘This stone weighs five pounds’ are all meaningful sentences of ordinary English, ‘Aluminium weighs five pounds’ is not and it does not matter in this connection whether we formulate this fact by saying that ‘Aluminium weighs five pounds,’ though grammatically an impeccable sentence, is logically meaningless, or whether we prefer the more modern formulation that this word-sequence does not form a sentence at all.