Even though it is not very often admitted, partial functions do play a signiicant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a three-valued logic decades ago. This approach allows rejecting certain unwanted formulae as faulty, which the simpler two-valued ones accept. We have developed resolution and tableau c...

You can trust your friends. You should trust your friends. Not all of your friends all of the time: you can reasonably trust different friends to different degrees, and in different domains. Still, we often trust our friends, and it is often reasonable to do so.Why is this? In this paper I explore how and whether friendship gives us reasons to trust our friends, reasons which may outstrip or co...

The interface specification of a procedure describes the procedure’s behaviour using preand postconditions. These preand postconditions are written using various functions. If some of these functions are partial, or underspecified, then the procedure specification may not be well-defined. We show how to write preand postcondition specifications that avoid such problems, by having the preconditi...

We present an illative system Is of classical higher-order logic with subtyping and basic inductive types. The system Is allows for direct definitions of partial and general recursive functions, and provides means for handling functions whose termination has not been proven. We give examples of how properties of some recursive functions may be established in our system. In a technical appendix ...

In Isabelle, there are several possibilities when one wants to support partial functions. One could put Isabelle to its intended use by directly embedding a logic of partial functions in Isabelle's meta-logic. Alternatively, one could investigate partial functions in one of Isabelle's already well-developed object logics, e.g., ZF or HOL. In this paper we pursue the second approach and describe...

We introduce the notion of effectoid as a way of axiomatising the notion of “computational effect”. Guided by classical algebra, we define several effectoids equationally and explore their relationship with each other. We demonstrate their computational relevance by applying them to global exceptions, partiality, continuations, and global state.