In this paper we define a logical framework, called &;lambda;TT, that is well suited for semantic analysis. We introduce the notion of a fibration L1 : F1 x C1 being internally definable in a fibration L2 : F2 x C2. This notion amounts to distinguishing an internal category L in L2 and relating L1 to the externalization of L through a pullback. When both L1 and L2 are term models of typed calculi L1 and L2, respectively, we say that L1 is an internal typed calculus definable in the frame language L2. We will show by examples that if an object language is adequately represented in λTT, then it is an internal typed calculus definable in the frame language λTT. These examples also show a general phenomenon: if the term model of an object language has categorical structure S, then an adequate encoding of the language in λTT imposes an explicit internal categorical structure S in the term model of λTT and the two structures are related via internal definability. Our categorical investigation of logical frameworks indicates a sensible model theory of encodings.
['Yuxi Fu']
Categorical properties of logical frameworks