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Logic for Two: Part A

(extended abstract)

John Slaney Robert Meyer

This abstract covers the first part of [4]. Our aim is to introduce a wide class of nonclassical logics to a general audience, showing that far from being esoteric and contrived they sustain an extremely natural, readily comprehensible reading. The systems considered here were given a syntactic treatment by us in [3] where they are presented as labelled natural deductive calculi, provided with a canonical reading in terms of sequents and motivated by ruminations on bodies of information", but not characterised semantically.

It must be emphasised that little of the formal material to be found here is new. The main source on which it draws is the work of Meyer and Routley reported in the `Semantics of Entailment' papers and to be found in [2]. This in turn was part of a body of research involving Dunn, Fine, Urquhart, Belnap and further back Curry and others. A presentation of the formal logic, along with commentary, may be found in [1] which dates from over twenty years ago but which, we suggest without much immodesty, is still in advance of many of the re-inventions of this particular wheel.

1 Multi-agent Inference

Agents have perspectives not only on what is the case but also on what inferences are warranted. That being so, the implicit beliefs, or theory, of agent a according to agent b may be different from those according to agent c or from those of b according to a. It is therefore important to study not only a's theory, but more generally a's theory according to b, which we might take to be the set of conclusions obtainable by taking a's theory as premises and b's as the determinant of available inference. It is evident that in general the logic of multi-agent systems needs to allow for a more elaborate notion of closure than the usual monadic one. We have always seen a binary notion of inference as natural: in order to draw conclusions you need some propositions to serve as premises and you need some inferential laws. That is, you need two sources, X and Y .

Note first what the closure of X under `Y is not. It is not the closure of X [ Y under ` . That is, we do not expect

X`Y A () X [ Y ` A

Plausibly, we do not even expect the weaker commutativity condition

X `Y A () Y `XA

Therefore we do not expect to be able to represent the notion of (binary) closure by means of a unary modal connective and we do not expect the usual apparatus of possible worlds semantics to be adequate to model it.