Z Logic and its Consequences

This paper provides an introduction to the specification language Z from a logical perspective. The possibility of presenting Z in this way is a consequence of a number of joint publications on Z logic that Henson and Reeves have co-written since 1997. We provide an informal as well as formal introduction to Z logic and show how it may be used, and extended, to investigate issues such as equational logic, the logic of preconditions, the issue of monotonicity and both operation and data refinement

The Standard Logic of Z is Inconsistent

. We demonstrate the logic contained in the draft Z standard is inconsistent. 1. Introduction The specification language Z has been very widely used and commended, but it has not yet received the kind of mathematical attention one would hope for a formal method. Despite more than a decade of effort, the standard source for the formalisation of Z [Nic95] still contains many unclarities and incompletenesses. Worse, as this short note demonstrates, the proposed logic it contains is inconsistent. 2. The inconsistency proof Firstly, the rule (BindSel) enables us to obtain the usual closure axiom for bindings. Suppose the alphabet set of b is f\Delta \Delta \Delta l i \Delta \Delta \Deltag. Then we have: \Delta \Delta \Delta b:l i = b:l i \Delta \Delta \Delta \Delta \Delta \Delta b:l i = b:l i \Delta \Delta \Delta b = hj \Delta \Delta \Delta l i V b:l i \Delta \Delta \Delta ji Secondly, we may use this closure axiom to obtain more natural equality congruence rules for binding projecti..