Abstract Sir William Hamilton is remembered for his proposal to extend the four traditional categoricals to eight by quantifying predicate as well as subject terms. He intended the quantifying particles to be understood in a ‘collective’ or ‘cumular’ manner rather than in a ‘distributive’ or ‘exemplar’ one, but commentators from De Morgan onwards have worked primarily from the latter perspective, comforted in the 20th century by the fact that it translates readily into the language of first-order logic with identity. Formal representation of the cumular approach needs more sophisticated resources, and the paper shows how it may be carried out using selection functions in the language of third-order logic. It also reviews a number of variants, some equivalent and others not so, as well as their reductions to second-order logic, and situates historical sources, both before and after Hamilton, with respect to the web of formal constructions.

Abstract The splitting theorem says that any set of formulae has a finest representation as a family of letter-disjoint sets. Parikh formulated this for classical propositional logic, proved it in the finite case, used it to formulate a criterion for relevance in belief change, and showed that AGM partial meet revision can fail the criterion. In this paper we make three further contributions. We begin by establishing a new version of the well-known interpolation theorem, which we call parallel interpolation, use it to prove the splitting theorem in the infinite case, and show how AGM belief change operations may be modified, if desired, so as to ensure satisfaction of Parikh's relevance criterion.

Lehmann, Magidor and others have investigated the effects of adding the non-Horn rule of rational monotony to the rules for preferential inference in nonmonotonic reasoning. In particular, they have shown that every inference relation satisfying those rules is generated by some ranked preferential model. We explore the effects of adding a number of other non-Horn rules that are stronger than or incomparable with rational monotony, but which are still weaker than plain monotony. Distinguished among these is a rule of determinacy preservation, equivalent to one of rational transitivity, for which we establish a representation theorem in terms of quasi-linear preferential models. An important tool in the proof of the representation theorem is the following purely semantic result, implicit in work of Freund, but here established by a more direct argument: every ranked preferential model generates the same inference relation as some ranked preferential model that is collapsed, in the sense of being both injective and such that each of its states is minimal for some formula. We also consider certain other non-Horn rules which are incomparable with monotony but arc implied by conditional excluded middle, and establish a representation result for a central one among them, which we call fragmented disjunction, equivalent to fragmented conjunction, in terms of almost linear preferential models. Finally, we consider briefly some curious Horn rules beyond the preferential ones but weaker than monotony, notably those which we call conjunctive insistence and n-monotony.

Abstract This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gärdenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourrón and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of its axiomatic bases), that fails to imply the proposition being eliminated. In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate “partial meet contraction functions”, which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gärdenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are “relational” and “transitively relational”, are studied in detail, and their connections with certain “supplementary postulates” of Gàrdenfors investigated, with a further representation theorem established.

Given separate though similar proofs of the finite model property for individual modal calculi such as S5, S4, S2, and the Feys-von Wright system T, the problem arises of generalising the arguments and establishing the property for modal calculi en masse. In other words, we would like to be able to show in one fell swoop that any modal calculus satisfying certain general syntactic conditions has the finite model property.

It is well known that the modal calculus S4 has infinitely many non-equivalent formulae in a single proposition letter (in standard terminology, infinitely many modal functions), whilst S5 has only finitely many. However, the situation regarding the intermediate modal calculi S4.2, S4.3, and Prior's Diodorean tense-logic D does not seem to have been settled. In this note we show that each of these systems, together with a certain proper supersystem D* of D, has infinitely many modal functions. This is in contrast with the fact that in the intermediate propositional logics KC and LC, which correspond under the McKinsey-Tarski translations to S4.2 and S4.3, there are only finitely many non-equivalent formulae in a single proposition letter.2