The inferential interpretation of a quadruple is as follows. Think of a system of first order inference steps, in which elements of the set A have the role of judgements, which are the premises and conclusions of first-order inference steps. (A first-order inference step is one without local constants, and in which no hypotheses are discharged. In general, the premise of a rule in which subdeductions may use local constants and hypotheses are not basic judgements.)
First-order inference steps are the simplest kind, having as a special case steps with no premises (a particularly severe cardinality restriction). However, to make up for this deficiency, there are no cardinality restrictions on how many premises an inference step requires, so that one may have infinitary inference steps.
The inference steps of such a rule system can be depicted as follows.
A map between rule systems is a way of translating formulas of the domain to formulas of the codomain, in such a way that a proof (partial or total) in the codomain of the translation of a formula can be systematically translated back, in a top-down fashion, to a proof of the formula in the domain. It is a `conservative extension' of the domain.
There is a correspondence between the inference-system and the state-machine interpretations. Given an inference system, there is a `sceptical' state-machine in which the states are judgements, the inputs are (codes for) inference steps, and each output is a premise location within the immediately preceding rule, the point at which further sceptical enquiry is to continue. You tell the machine you have a proof of the initial judgement, and it calls your bluff. Here is a log of the machine's operations, starting in state a1.
state | input | output |
a1 | b1 : B(a1) | c1 : C(a1, b1) |
a2 = d(a1, b1, c1) | b2 : B(a2) | c2 : C(a2, b2) |
a3 = ... | ... | ... |
an = ... | bn : B(an) |
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