Context Mereology

We show that under a small number of assumptions, it is possible to interpret truth in a context as a quantification over truth in 'atomic' or pointlike contexts, which are transparent to all the connectives. We discuss the necessary assumptions and suggest conditions under which they are intuitively reasonable. Introduction This note is inspired by the context logic originally introduced by J. McCarthy (McCarthy 1993) and R. V. Guha, (Guha 1991) and subsequently developed by others. It is intended to address the issue discussed in (Makarios, Heuer & Fikes 2006), viz. the transparency of contextual assertions to the propositional connectives. The central construction of context logic is ist(c P). Here c denotes a context, which is supposed to be a 'bearer of truth' in some broad sense, and P denotes a proposition, which is some entity that can be said to be true or false in a context. A proposition may fail to have a truth-value in a context, so that ist(c P) and ist(c ~P) might both be false. Examples of contexts include time-intervals, where ist means that the proposition holds during the time-interval; believers, in which ist means that the proposition is believed; information sources such as databases or documents, in which ist means that the information source is a provenance for the proposition; and modal-alternative, imaginary or counterfactual worlds or situations, in which ist asserts that a proposition is true in the world or situation. Most of the intuitions underlying the development here arise from the first application, where contexts are thought of as time-intervals, but the formal results apply to any kind of context which satisfies the axioms. Ist and its dual It seems to be widely assumed that ist distributes over conjunctions in its second argument: that is, that ist(c P&Q;) implies ist(c P) & ist(c Q). It is easy to see intuitively that this corresponds to the idea that the proposition P is true throughout the context c. However, not all attributions of truth to a situation distribute over conjunction. For example, a recent day of hard traveling might be summarized by saying "On April 30 I was in seven states" meaning, of course, that I was at various times of the day in one of seven states, not that I was in all seven states all of the day. In this case, the relevant notion of 'truth in' seems to not distribute over conjunction: for (true-in 300405 (I am in Texas)) & (true-in 300405 (I am in Mississippi)) can be true, when clearly (true-in 300405 ((I am in Texas) & (I am in Mississippi)) ) must be false. It is intuitively clear what is meant, however: this is a different notion of 'true in' from the ist sense, and in fact it is precisely the classical dual, definable as wist(c P) =df ~ist(c ~P) Clearly, wist distributes over disjunction but not (in general) over conjunction; and true-in, as used above, is wist rather than ist: it means intuitively "at some time during" rather than "throughout". The relationship between ist and wist is exactly analogous to the usual duality between the universal and existential quantifiers, and between the strong and weak modal operators. In fact, the two operators can be viewed as indexed modalities, with the particular context providing the index, and the standard transliteration of modal propositional logic into first-order logic then maps them into patterns of quantification. If we think of a context as a time-interval, and a time-interval as a set of points, then ist(c P) translates into for all p in c, P(p), and wist(c P) translates into there exists a p in c with P(p). This kind of translation provides for a useful and natural reduction of 'ist' language to a simpler subcase where the basic relationship between a proposition and a 'context-point' is transparent to all the connectives and hence merges ist and wist into a single relationship, which we could paraphrase as 'P is true at c'. The various cases of truth-in-a-context being opaque to the connectives, such as the example given above of wist(c P&Q;) not being identical in meaning to wist(c P) & wist(c Q) , can then all be explained by the patterns of quantification; in this case, the fact that exists(x)(P(x) & Q(x) is not logically equivalent to (exists(x) P(x)) & (exists(x) Q(x)). Propositional context logic then reduces to classical quantifier logic plus a very simple, completely transparent, notion of true-at-a-contextpoint. The purpose of this note is to identify some general conditions under which this reduction of contexts to sets of context-points can be done. We will show that a small number of axioms, only one of which is controversial, suffice. Axioms Following this analogy with modal logic, and to reduce notational clutter in what follows, we will adopt modality-style notation and write [c]P for ist(c P) and P for wist(c P); the brackets are intended to suggest the box-diamond notation commonly used to indicate modalities. Definition 1. P =df ~ [c](~P) The first axiom is the basic assumption about ist, that it distributes over conjunction. Axiom 1. [c](P & Q) iff ( [c]P and [c]Q ) The second is a kind of internal coherence principle for contexts, that an overt contradiction cannot be true throughout a context: Axiom 2. ~ [c](P & ~P) Lemma 1. [c]P implies P Proof. Assume [c]P, and suppose ~P. By def1, [c]~P. By axiom 1, [c](P&~P) contradicting axiom 2. QED Parts of Contexts Now we introduce a relationship of parthood on contexts. Intuitively, a part of a context is some piece or aspect of it which is also considered to be a context, and can be distinguished by there being a proposition which has a different truth-value in that part than in some other part. In the case of a time-interval, parthood seems to correspond naturally to being a subinterval. We will write cP iff there is a d<c with [d]P This could be stated as an implication, since "if" is a consequence of earlier axioms. Finally, we assume that parthood and truth have an extensional relationship: Axiom 6. (Separation) c<d or there is a proposition P with