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Metaphysical adequacy obtains if there are no contradictions between the facts we wish to represent and our representation of them. Thus, for example, a representation of the world as a collection of non-interacting particles would be metaphysically inadequate, since this denies an obvious and central fact about the world, namely that particles do interact. Conversely the representation of the world as a collection of particles which interact through forces between each pair of particles is metaphysically adequate, as is a representation of the world as a giant quantum mechanical wave function.

Instead of interpreting predicates as sets we can interpret them as characteristic functions. We can then think of minimising a predicate at a particular 'point' (argument of the characteristic function) by changing its value at that point from true to false. The pointwise circumscription of a predicate within the context of a given set of formulae then states that the predicate cannot be further minimised at any point. The pointwise circumscription of P in A is the FO schema: A & Vx~(P(x) & A(A,y(P(y) & x / y ) ) For example, again taking A to be Block(a) & Block(b), the pointwise circumscription of Block in A can be simplified to Block(a) & Block(b) & Vx~(Block(x) & x / a & x # b ) which again states that the only objects which are blocks are a and b.

M,w| = A) and (M,w| = B). (Vw,)(wRw,)(M,wl| = A). Clause 4 requires some comment. It states that A is known to be true at a world w in M if, and only if, A is true in all worlds which are alternative (via R) to w. Depending on the properties of R the above notion of knowledge (belief) can be completely characterised by the following sound and complete axiom systems: A, A2 A3 A4 all instances of propositional tautologies (KA & K(A 3 B)) ZD KB KADA KA => KKA ~KA^K~KA R,: A,A ZD B/B R2: A / K A {modus ponens) Each of the axioms A3-A5 corresponds to some property of R.

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