To determine what deductions are it does not seem sufficient to know that the premises and conclusions are propositions, or something in the field of propositions, like commands and questions. It seems equally, if not more, important to know that deductions make structures, which in mathematics we find in categories, multicategories and polycategories. It seems also important to know that deductions should be members of particular kinds of families, which is what is meant by their being in accordance with rules.
Functions of LanguageIn a terminology like that of the old logic, the notion of deduction will be for us primarily a hypothetical and not a categorical notion. (This use of categorical should not be confused with categorial, which is found later in this paper, and which, according to the Oxford English Dictionary [23], means "relating to, or involving, categories"; unfortunately, in mathematical category theory categorical dominates in the sense of categorial.) The distinction between categorical and hypothetical is found when we speak about categorical and hypothetical proofs. The latter is a proof under hypotheses, while the former depends on no hypothesis. Both may involve deduction, but we will be concerned here with deduction as found in hypothetical proofs.Schroeder-Heister (together with P. Contu in [22], Sect. 4,in [20], Sect. 3,and in [21]; see also [8]) states that the reigning semantics-both classical semantics based on model theory and constructivist proof-theoretic semantics-is based on dogmas, the main one of which may be formulated succinctly by saying that categorical