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By Andrew M. Pitts

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4 are derived rules of Fig. 3. e. what can be proved in intuitionistic propositional logic. However, the structure of proofs in the two formulations may be very different. One should note that the presence of rule (Cut ) (which corresponds to the transitivity of in prop-categories) in Fig. 4 is apparently essential for the adjoint formulation to be equivalent to the natural deduction formulation of Fig. 3]). 3. A typical feature of the categorical treatment of logical constructs is the identi cation of which constructs are essentially uniquely determined by their logical properties.

1). Similarly, in the presence of an empty type (cf. 2), z : null] is a stable initial object in C `(Th ). In the presence of product and one-element types we have seen that every object is isomorphic to one of the form x : ]. In this case the presence of disjoint union and empty types, we can conclude that C `(Th ) has all stable nite coproducts. Function types. (Cf. ) In this case, for each pair of types and 0 , the object f : ! 0 ] is the exponential of x0 : 0 ] by x : ] in C `(Th ), with associated evaluation morphism f : !

And a C -property R] 2 PropC ( 1 ] n ] ) for each relation symbol R 1; : : : ; n. For each context = x1 : 1 ; : : : ; xn : n ] and formula over the signature for which the judgement prop ] is derivable, we will de ne a C -property ]]] 2 PropC ( ] ) where ] denotes the product 1 ] n ] . The de nition will be given by induction on the derivation of prop ], since there will be at most one way to derive the well-formedness of a formula in a given context. 1) for forming atomic formulas. 1 specifying the behaviour of the categorical semantics with respect to substitution of terms 44 Andrew M.

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