Alternative forms of propositional calculus for a given deduction theorem

In a propositional calculus based on combinatory logic it is necessary to have a restriction on the deduction theorem for implication as otherwise Curry's paradox results (see [5]). In [1] and [2] we restricted the deduction theorem for implication as follows: where Δ is any sequence of obs and HX stands for "X is a proposition". Motivation for this deduction theorem was given in [2] using the following three valued tables (that for implication also appears in Kleene [6]). where N can stand for "neither J nor F" and Γ (negation) can be defined by CP(ΞHI), 1 A question that arises is: to what extent are the entries in the third column and the third row of the table for implication uniquely determined by DTP, modus ponens and the (fairly obvious) rule: H IhHI? 1. Here P stands for implication. EHI, which can be interpreted as stating that all propositions are provable, is taken as the "standard false" proposition. Given that SHI is assigned F the table for Γ follows from that for D. After part (iii) on page xx, below we assume for SHI: ZH\,HXhX.