We are concerned with showing how ‘labelled’ Natural Deduction presentation systems based on an extension of the so-called Curry-Howard functional interpretation can help us understand and generalise most of the deduction calculi designed to deal with the logical notion of existential quantification. We present the labelling mechanism for ‘’ using what we call ‘ɛ-terms’, which have the form of ‘a’) in a dual form to the ‘Ax.f’ terms of in the sense that the ‘witness’ is chosen at the time of (...) assertion/introduction.With the intention of demonstrating the generality of our framework, we provide a brief comparison with some other calculi of existentials, including the model-theoretic interpretations of Skolem and Herbrand, indicating the way in which some of the classical results on prenex normal forms can be extended to a wide range of logics. The essential ingredient is to look at the conceptual framework of labelled natural deduction as an attempted reconstruction of Frege's functional calculus where the devices of the Begriffsschrift work separately from, yet in harmony with, those of the Grundgesetze 1. (shrink)

Further to the connections between meaning and use, it seems useful to consider the consequences one is allowed to draw from a proposition as something directly related to its meaning/use. And indeed, Wittgenstein's references to the connections between meaning and the consequences, as well as between use and consequences are sometimes as explicit as his celebrated ‘definition’ of meaning as use given in the Investigations. Here we attempt to collect some of these references, discussing how an intuitive basis for the (...) construction of a more convincing proof-theoretic semantics for the mathematical language can arise out of this connection meaning/use/ consequences. (shrink)