A Note on Spector’s Quantifier-Free Rule of Extensionality

In this note we show that the so-called weakly extensional arithmeticin all finite types, which is based on a quantifier-free rule ofextensionality due to C. Spector and which is of significance in thecontext of G¨odel's functional interpretation, does not satisfy the deductiontheorem for additional axioms. This holds already for PI^0_1-axioms. Previously, only the failure of the stronger deduction theoremfor deductions from (possibly open) assumptions (with parameterskept fixed) was known