12
Mathematical Aspects of Logic Programming Semantics
Thus, B L,J is the set of all symbols p(d 1 , . . . , d n ), where p is an n-ary predicate
symbol in L and d 1 , . . . , d n ∈ D.
•
1.2.8 Definition Let L be a first-order language, let J be a preinterpretation
for L with domain D, and let T be a logic. A valuation or interpretation for
L (based on J) with values in T is a mapping v : B L,J → T . Let v : B L,J → T
be a valuation and let θ be a J-variable assignment. Then v and θ determine,
inductively, a well-defined truth value in T for any quantifier-free, well-formed
formula F in L by means of the construction of F and the definitions of the
connectives in T . We say that v is a model for F , written v |= F , if v gives
truth value t to F . We sometimes refer to valuations, interpretations, and
models based on J as J-valuations, J-interpretations, and J-models.
In fact, if T is ordered as a complete lattice (and this issue will be considered shortly), then a valuation v gives unique truth value in T , in the
standard way, to any closed well-formed formula F in L: universal quantification corresponds to the infimum of a set of truth values, and existential
quantification corresponds to the supremum of a set of truth values. The term
closed here has, of course, its normal meaning in mathematical logic, namely,
that each variable symbol occurring in F falls within the scope of a quantifier.
(By default, we allow the term closed to apply to formulae with no variable
symbols and no quantifiers.) Once this observation is made, one can go on, in
the standard way, to define at our present level of generality the terms model,
(un)satisfiable, valid, and logical consequence when applied to sets of closed
well-formed formulae.
1.3 Ordered Spaces of Valuations
Following Definition 1.2.8, we will generally denote the set of all valuations for L based on J with values in T by I(B L,J , T ), and we will consider
I(B L,J , T ) as an ordered set. The orderings we have in mind are derived from
orderings on T , and the set B L,J plays no role in this. Therefore, to ease
notation we will work with an arbitrary set X for the rest of this chapter.
Thus, we regard a valuation or interpretation for the time being as simply a
mapping X → T and denote the set of all these by I(X, T ); typical elements
of I(X, T ) will be denoted by u, v, etc. Later on, in applying the results of this
section, we will of course take X to be a set of ground atoms or of J-ground
instances of atoms, and no confusion will be caused. There is, however, a convention we need to establish concerning the terminology “valuation” versus
“interpretation”, as follows.
Mathematical Aspects of Logic Programming Semantics
Thus, B L,J is the set of all symbols p(d 1 , . . . , d n ), where p is an n-ary predicate
symbol in L and d 1 , . . . , d n ∈ D.
•
1.2.8 Definition Let L be a first-order language, let J be a preinterpretation
for L with domain D, and let T be a logic. A valuation or interpretation for
L (based on J) with values in T is a mapping v : B L,J → T . Let v : B L,J → T
be a valuation and let θ be a J-variable assignment. Then v and θ determine,
inductively, a well-defined truth value in T for any quantifier-free, well-formed
formula F in L by means of the construction of F and the definitions of the
connectives in T . We say that v is a model for F , written v |= F , if v gives
truth value t to F . We sometimes refer to valuations, interpretations, and
models based on J as J-valuations, J-interpretations, and J-models.
In fact, if T is ordered as a complete lattice (and this issue will be considered shortly), then a valuation v gives unique truth value in T , in the
standard way, to any closed well-formed formula F in L: universal quantification corresponds to the infimum of a set of truth values, and existential
quantification corresponds to the supremum of a set of truth values. The term
closed here has, of course, its normal meaning in mathematical logic, namely,
that each variable symbol occurring in F falls within the scope of a quantifier.
(By default, we allow the term closed to apply to formulae with no variable
symbols and no quantifiers.) Once this observation is made, one can go on, in
the standard way, to define at our present level of generality the terms model,
(un)satisfiable, valid, and logical consequence when applied to sets of closed
well-formed formulae.
1.3 Ordered Spaces of Valuations
Following Definition 1.2.8, we will generally denote the set of all valuations for L based on J with values in T by I(B L,J , T ), and we will consider
I(B L,J , T ) as an ordered set. The orderings we have in mind are derived from
orderings on T , and the set B L,J plays no role in this. Therefore, to ease
notation we will work with an arbitrary set X for the rest of this chapter.
Thus, we regard a valuation or interpretation for the time being as simply a
mapping X → T and denote the set of all these by I(X, T ); typical elements
of I(X, T ) will be denoted by u, v, etc. Later on, in applying the results of this
section, we will of course take X to be a set of ground atoms or of J-ground
instances of atoms, and no confusion will be caused. There is, however, a convention we need to establish concerning the terminology “valuation” versus
“interpretation”, as follows.