Descriptions in Mathematical Logic

Studia Logica 43 (3):281 - 294 (1984)
After a discussion of the different treatments in the literature of vacuous descriptions, the notion of descriptor is slightly generalized to function descriptor Ⅎ $\overset \rightarrow \to{y}(x)$ , so as to form partial functions φ = Ⅎ $y(\overset \rightarrow \to{x}).A(\overset \rightarrow \to{x},y)$ which satisfy $\forall \overset \rightarrow \to{x}z(z=\phi \overset \rightarrow \to{x}\leftrightarrow \forall y(A(\overset \rightarrow \to{x},y)\leftrightarrow y=z))$ . We use (intuitionistic, classical or intermediate) logic with existence predicate, as introduced by D. S. Scott, to handle partial functions, and prove that adding function descriptors to a theory based on such a logic is conservative. For theories with quantification over functions, the situation is different: there the addition of Ⅎ yields new theorems in the Ⅎ-free fragment, but an axiomatisation is easily given. The proofs are syntactical.
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