Extensionality in sentence position (Q1820769)

Call an n-ary connective \(\kappa\) extensional if it complies with the principle of replacement of material equivalents: if one substitutes for each sentence \(\phi_ i\) in \(\kappa (\phi_ 1,...,\phi_ n)\) a sentence \(\psi_ i\) with the same truth value as \(\phi_ i\), then the resulting sentence \(\kappa (\psi_ 1,...,\psi_ n)\) will have the same truth value as the original \(\kappa (\phi_ 1,...,\phi_ n)\). Clearly, all truth-functional connectives are extensional in this sense. The converse, however, does not hold. As the author notes, the difference between extensionality and truth-functionality is not always recognized in the literature. The paper offers a characterization of the extensional connectives, both from a semantic and from a more proof-theoretic point of view.