We present a number of, somewhat unusual, ways of describing what Craig’s interpolation theorem achieves, and use them to identify some open problems and further directions.

We formulate a general institution-independent (i.e. independent of the details of the actual logic formalised as institution) version of the Craig Interpolation Theorem and prove it in dependence of Birkhoff-style axiomatizability properties of the actual logic.We formalise Birkhoff-style axiomatizability within the general abstract model theoretic framework of institution theory by the novel concept of Birkhoff institution.

The informal logic movement began as an attempt to develop – and teach – an alternative logic which can account for the real life arguing that surrounds us in our daily lives – in newspapers and the popular media, political and social commentary, advertising, and interpersonal exchange. The movement was rooted in research and discussion in Canada and especially at the University of Windsor, and has become a branch of argumentation theory which intersects with related traditions and approaches (notably formal (...) logic, rhetoric and dialectics in the form of pragma-dialectics). In this volume, some of the best known contributors to the movement discuss their views and the reasoning and argument which is informal logic’s subject matter. Many themes and issues are explored in a way that will fuel the continued evolution of the field. Federico Puppo adds an insightful essay which considers the origins and development of informal logic and whether informal logicians are properly described as a “school” of thought. In considering that proposition, Puppo introduces readers to a diverse range of essays, some of them previously published, others written specifically for this volume. (shrink)

Work on how to axiomatize the subtheories of a first-order theory in which only a proper subset of their extra-logical vocabulary is being used led to a theorem on recursive axiomatizability and to an interpolation theorem for first-order logic. There were some fortuitous events and several logicians played a helpful role.

A form (or pattern) of inference, let us say, explicitlysubsumes just such particular inferences as are instances of the form, and implicitly subsumes thoseinferences with a premiss and conclusion logically equivalent to the premiss and conclusion of an instanceof the form in question. (For simplicity we restrict attention to one-premiss inferences.) A form ofinference is archetypal if it implicitly subsumes every correct inference. A precise definition (Section 1)of these concepts relativizes them to logics, since different logics classify different inferences ascorrect, (...) as well as ruling differently on the matter of logical equivalence which entered into the definitionof implicit subsumption. When relativized to classical propositional logic, we find (Section 2) thatall but a handful of `degenerate' inference forms turn out to be archetypal, whereas matters are verydifferent in this respect for the case of intuitionistic propositional logic (Sections 3 and 4), and an interestingstructure emerges in this case (the poset of equivalence classes of inference forms, with respect tothe equivalence relation of implicitly subsuming the same inferences). Thus a more accurate, if excessivelylong-winded title would be 'Archetypal and Non-Archetypal Forms of Inference in Classical andIntuitionistic Propositional Logic'. Some left-overs are postponed for a final discussion (Section 5).The overall intention is to introduce a new subject matter rather than to have the last word on thequestions it raises; indeed several significant questions are left as open problems. (shrink)