A Theory of Truthmaker Content II: Subject-matter, Common Content, Remainder and Ground

We continue with the development of the theory of truthmaker content begun in part I, dealing with such ‘non-standard’ topics as subject matter, common content, logical remainder and ground. This is by no means an exhaustive list of topics that might have been considered but it does provide an indication of the nature and scope of the theory. As before, the paper is divided into an informal exposition and a technical addendum. Both can be read independently of the other but it would be helpful, in either case, to have the first part of the paper at hand.

One feature of great interest in the present account is that it deals with a number of the topics that lack an adequate treatment either within the possible worlds account or under a structural conception of propositions. The notion of common content, for example, can be readily handled within the present framework but cannot be properly handled in the other two frameworks without either introducing or deriving something like the present conception of verification. Thus we should not simply regard the present ‘extensional’ conception of content as a mere approximation to a structural conception but as an important conception in its own right.

Another feature of interest is that many of the notions we shall consider are obtained by ‘lifting’ corresponding notions at the level of verifying states to the level of propositions. Take again the notion of common content. There is a notion of common part for states, what it is they have in common; and this may then be extended from states to propositions. Likewise for the notion of remainder. We may subtract one state from another; and, similarly, we may consider the result of subtracting one proposition from another. This process of lifting gives the theory a familiar mathematical feel while also providing a firm intuitive foundation for the derived notions.

A final feature of general interest concerns duality. We are all familiar with the notion of duality in classical logic; disjunction, for example, is dual to conjunction and will therefore behave in a related way. We may think of the duality in classical logic as arising from the possibility of ‘reversing’ the relation of classical consequence. Thus if a conjunction classically entails its conjuncts then a disjunction will be classically entailed by its disjuncts. Conjunction and disjunction are not dual in this way within the present framework. But they are dual in a different way. Instead of replacing the relation of conjunctive part (or containment) by its converse, we replace it by the notion of disjunctive part (or entailment). Thus whereas a conjunction will ‘contain’ its conjuncts, a disjunction will be ‘entailed’ by its disjuncts. We might call this ‘horizontal’ duality. But there will also be a ‘vertical’ duality of the familiar sort. Thus in the case of both containment and entailment we may consider the notions obtained by reversing these relations. We thereby obtain a much richer theory, in which the two aspects of duality operate independently of one another.

Let us now give an informal exposition of the topics of interest to us before proceeding to the more formal exposition.