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Predicative semantics of modal logics 3
The existential quantifier ∃x· P is simply represented as P : x true , and variable substitution
P [e/x] as P : x (x = e) . An interface x may split into several variables, e.g. (y, z) . For example,
the generic composition P : (y, z) true is the same as the predicate ∃y∃z · P . If the vector is
empty, a generic composition becomes a conjunction: P : R = P ∧ R .
Generic composition has an inverse operator denoted by P / x R, which is the weakest predicate
X such that (X : x R) ⊆ P . It can be defined by a Galois connection:
Def 2 X ⊆ P / x R iff X : x R ⊆ P for any predicate X .
Generic composition and its inverse satisfy a property:
P / x R = ¬ (¬ P : x ˜R) = ∀x0 · (R[x 0 /x, x/x] ⇒ P [x 0 /x])
where ˜R ̂= R[x/x, x/x] is the converse of R for the variable x . Universal quantifier ∀x· P
can then be written as P / x true . Negation ¬ P becomes false / P whose interface is empty.
Implication P ⇒ Q becomes Q / P with an empty interface. Disjunction P ∨ Q is a trivial
combination of negation and implication. Thus all connectives, substitution and quantifiers
become special cases of generic composition and its inverse [2].
Theorem 1 Generic composition and its inverse are complete in the sense that any predicate
that does not contain overlined free variables can be written in terms of generic composition and
its inverse using only the constant predicates and predicate letters.
The theorem shows the expressiveness of generic composition for predicate manipulation. Generic
composition and its inverse form a Galois connection and satisfy the algebraic laws of strictness,
distributivity and associativity.
Law 1
(1) A ⊆ (A : x R) / x R
(3) false : x R = false
(5) A : x (R ∨ S) = (A : x R) ∨ (A : x S)
(7) A / x (R ∨ S) = (A / x R) ∧ (A / x S)
(9) (A : x R) : x S = A : x (R : x S)
(2) (A / x R) : x R ⊆ A
(4) true / x R = true
(6) (A ∨ B) : x R = (A : x R) ∨ (A : x R)
(8) (A ∧ B) / x R = (A / x R) ∧ (A / x R)
(10) (A / x R) / x S = A / x (S : x R) .
The notation is especially useful when the interfaces of the operators in a predicate are not
identical. For example, in the following law we assume that x , y and z are three different
logical variables, A = ∃z · A (independence of the variable z ) and C = ∃y · C (independence of
the variable y ).
Report No. 301,
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