17
Order and Logic
1.3.4 Theorem Let X be an arbitrary set. Then the following statements
hold.
(a) In case T is the truth set T W O, the set I(X, T ) is a complete lattice in
the ordering [ t .
(b) In case T is the truth set T HREE, the set I(X, T ) is a complete upper semi-lattice in the ordering [ k , but not a complete lattice, and is a
complete lattice in the ordering [ t .
(c) In case T is the truth set FOU R, the set I(X, T ) is a complete lattice in
each of the orderings [ k and [ t .
Furthermore, in each case and in each ordering, the set I(X, T ) is a Scott
domain whose compact elements are precisely those valuations v for which
the set {x ∈ X | v(x) = ⊥} is finite, where ⊥ denotes the appropriate bottom
element.
•
Notice that the order structure here is independent of the actual logic involved, as distinct from the underlying truth set. Thus, for example, Kleene’s
strong and weak three-valued logics give rise to precisely the same order structure on I(X, T HREE); the difference between them is in the definitions of
the connectives, rather than in their order structure.
We next take up the point made in Remark 1.3.1, concerning the representation of a valuation in terms of the sets on which it takes various truth
values in T W O, T HREE, or FOU R.
Let v be a valuation, and let v = v
−1
1
u
(u), let v f = v
− (f ), let v
1
t = v
− (t),
and let v b = v
−1 (b); these sets are pairwise disjoint subsets of X, and some
may be empty. A valuation v taking values in T W O is clearly completely
determined by the subset I = v t of X and therefore can be identified with
I. A valuation taking values in T HREE can be identified either with the
pair I = (v t , v f ) of subsets of X or with the pair I = (v t , v u ). The former
choice will be made when we are concerned with the ordering [ k , so that
the bottom element is u and this is also the “default” value in the sense that
v u = X \ (v t ∪ v f ). The latter choice will be made when we are concerned with
the ordering [ t , so that the bottom element is f and this is also the default
value in that v f = X \ (v t ∪ v u ). Finally, a valuation v with values in FOU R
can be identified either with the triple I = (v t , v f , v b ) of subsets of X when
u is the bottom and default value or with the triple I = (v t , v u , v b ) when f is
the bottom and default value.
Conversely, a subset I of X determines a valuation v : X → T W O with
the property that v(x) = t if and only if x ∈ I. Given the ordering [ k , a pair
I = (I t , I f ) of disjoint subsets of X determines a valuation v : X → T HREE
which takes value t on I t , takes value f on I f , and, by default, takes value u
on X \ (I t ∪ I f ). Similarly, given the ordering [ t , a pair I = (I t , I u ) of disjoint
subsets of X determines a valuation v : X → T HREE which takes value t on
Order and Logic
1.3.4 Theorem Let X be an arbitrary set. Then the following statements
hold.
(a) In case T is the truth set T W O, the set I(X, T ) is a complete lattice in
the ordering [ t .
(b) In case T is the truth set T HREE, the set I(X, T ) is a complete upper semi-lattice in the ordering [ k , but not a complete lattice, and is a
complete lattice in the ordering [ t .
(c) In case T is the truth set FOU R, the set I(X, T ) is a complete lattice in
each of the orderings [ k and [ t .
Furthermore, in each case and in each ordering, the set I(X, T ) is a Scott
domain whose compact elements are precisely those valuations v for which
the set {x ∈ X | v(x) = ⊥} is finite, where ⊥ denotes the appropriate bottom
element.
•
Notice that the order structure here is independent of the actual logic involved, as distinct from the underlying truth set. Thus, for example, Kleene’s
strong and weak three-valued logics give rise to precisely the same order structure on I(X, T HREE); the difference between them is in the definitions of
the connectives, rather than in their order structure.
We next take up the point made in Remark 1.3.1, concerning the representation of a valuation in terms of the sets on which it takes various truth
values in T W O, T HREE, or FOU R.
Let v be a valuation, and let v = v
−1
1
u
(u), let v f = v
− (f ), let v
1
t = v
− (t),
and let v b = v
−1 (b); these sets are pairwise disjoint subsets of X, and some
may be empty. A valuation v taking values in T W O is clearly completely
determined by the subset I = v t of X and therefore can be identified with
I. A valuation taking values in T HREE can be identified either with the
pair I = (v t , v f ) of subsets of X or with the pair I = (v t , v u ). The former
choice will be made when we are concerned with the ordering [ k , so that
the bottom element is u and this is also the “default” value in the sense that
v u = X \ (v t ∪ v f ). The latter choice will be made when we are concerned with
the ordering [ t , so that the bottom element is f and this is also the default
value in that v f = X \ (v t ∪ v u ). Finally, a valuation v with values in FOU R
can be identified either with the triple I = (v t , v f , v b ) of subsets of X when
u is the bottom and default value or with the triple I = (v t , v u , v b ) when f is
the bottom and default value.
Conversely, a subset I of X determines a valuation v : X → T W O with
the property that v(x) = t if and only if x ∈ I. Given the ordering [ k , a pair
I = (I t , I f ) of disjoint subsets of X determines a valuation v : X → T HREE
which takes value t on I t , takes value f on I f , and, by default, takes value u
on X \ (I t ∪ I f ). Similarly, given the ordering [ t , a pair I = (I t , I u ) of disjoint
subsets of X determines a valuation v : X → T HREE which takes value t on