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)

For the claim that the satisfaction of certain conditions is sufficient for the application of some concept to serve as part of the (‘reductive’) analysis of that concept, we require the conditions to be specified without employing that very concept. An account of the application conditions of a concept not meeting this requirement, we call analytically circular. For such a claim to be usable in determining the extension of the concept, however, such circularity may not matter, since if the concept (...) figures in a certain kind of intensional context in the specification of the conditions, the satisfaction of those conditions may not itself depend on the extension of the concept. We put this by saying that although analytically circular, the account may yet not be inferentially circular. (shrink)

A definition is given for formulae A 1 ,...,A n in some theory T which is formalized in a propositional calculus S to be (in)dependent with respect to S. It is shown that, for intuitionistic propositional logic IPC, dependency (with respect to IPC itself) is decidable. This is an almost immediate consequence of Pitts' uniform interpolation theorem for IPC. A reasonably simple infinite sequence of IPC-formulae F n (p, q) is given such that IPC-formulae A and B are dependent if (...) and only if at least on the F n (A, B) is provable. (shrink)

A definition is given for formulae $A_1,\ldots,A_n$ in some theory $T$ which is formalized in a propositional calculus $S$ to be (in)dependent with respect to $S$. It is shown that, for intuitionistic propositional logic $\mathbf{IPC}$, dependency (with respect to $\mathbf{IPC}$ itself) is decidable. This is an almost immediate consequence of Pitts' uniform interpolation theorem for $\mathbf{IPC}$. A reasonably simple infinite sequence of $\mathbf{IPC}$-formulae $F_n(p, q)$ is given such that $\mathbf{IPC}$-formulae $A$ and $B$ are dependent if and only if at least (...) on the $F_n(A, B)$ is provable. (shrink)

Informally, we can say that an inference rule is archetypal if any other rule can be transformed to it via some substitution for the propositional variables. It was shown by L. Humberstone that, in the case of classical propositional logic, every non-degenerate binary rule is archetypal and conjectured that this result holds also for all rules in the full language. In this paper we provide a proof of this conjecture and show that it is the unique intermediate logic with this (...) property. (shrink)