Model theory
Read as: the class of models in logic L over language L satisfying formula E
Means: This expression denotes the class of models in logic L over language L satisfying formula E. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M sub n satisfies formula E
Means: This expression denotes structure M sub n satisfies formula E. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: q is a subset of p
Means: This expression denotes q is a subset of p. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: Q prime
Means: This expression denotes Q prime. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: sequence c
Means: This expression denotes sequence c. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: formula D sub two
Means: This expression denotes formula D sub two. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: the class of models in logic L satisfying formula E equals the class of models in logic L satisfying formula D
Means: This expression denotes the class of models in logic L satisfying formula E equals the class of models in logic L satisfying formula D. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure N satisfies in logic L formula E
Means: This expression denotes structure N satisfies in logic L formula E. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: language L union P comma Q
Means: This expression denotes language L union P comma Q. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: formula G
Means: This expression denotes formula G. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: abstract logic L together with its satisfaction relation
Means: This expression denotes abstract logic L together with its satisfaction relation. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure N star
Means: This expression denotes structure N star. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: formula F is in L open parenthesis language L union R comma c sub one comma and so on comma c sub n close parenthesis
Means: This expression denotes formula F is in L open parenthesis language L union R comma c sub one comma and so on comma c sub n close parenthesis. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: abstract logic L together with its satisfaction relation
Means: This expression denotes abstract logic L together with its satisfaction relation. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: n
Means: This expression denotes n. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: I open parenthesis sequence a comma sequence b close parenthesis
Means: This expression denotes I open parenthesis sequence a comma sequence b close parenthesis. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: structure N sub zero does not satisfy in logic L formula E
Means: This expression denotes structure N sub zero does not satisfy in logic L formula E. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: the domain of structure N is the set of elements a in the domain of M for which relation R holds of a and parameters b sub one through b sub n
Means: This expression denotes the domain of structure N is the set of elements a in the domain of M for which relation R holds of a and parameters b sub one through b sub n. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the class of models in logic L satisfying formula E
Means: This expression denotes the class of models in logic L satisfying formula E. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the domain of structure M sub n star
Means: This expression denotes the domain of structure M sub n star. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: c sub one
Means: This expression denotes c sub one. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: x
Means: This expression denotes x. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: c
Means: This expression denotes c. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the domain of structure M
Means: This expression denotes the domain of structure M. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: structure M sub n
Means: This expression denotes structure M sub n. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure M satisfies in logic L formula F
Means: This expression denotes structure M satisfies in logic L formula F. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: formula E is in L open parenthesis language L close parenthesis
Means: This expression denotes formula E is in L open parenthesis language L close parenthesis. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M is isomorphic to structure N
Means: This expression denotes structure M is isomorphic to structure N. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the satisfaction relation for logic L
Means: This expression denotes the satisfaction relation for logic L. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: b sub one comma
Means: This expression denotes b sub one comma. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: language L prime
Means: This expression denotes language L prime. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M
Means: This expression denotes structure M. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M sub zero
Means: This expression denotes structure M sub zero. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: sequence a consists of a sub one through a sub n
Means: This expression denotes sequence a consists of a sub one through a sub n. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: p belongs to the family I of partial isomorphisms
Means: This expression denotes p belongs to the family I of partial isomorphisms. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: L open parenthesis language L close parenthesis
Means: This expression denotes L open parenthesis language L close parenthesis. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: Q
Means: This expression denotes Q. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: X
Means: This expression denotes X. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the domain of structure N
Means: This expression denotes the domain of structure N. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: J open parenthesis n comma sequence a comma sequence b close parenthesis
Means: This expression denotes J open parenthesis n comma sequence a comma sequence b close parenthesis. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure M sub n star satisfies formula E in logic L
Means: This expression denotes structure M sub n star satisfies formula E in logic L. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: sequences a and c are finite sequences over the domain of structure M
Means: This expression denotes sequences a and c are finite sequences over the domain of structure M. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: structure N sub n
Means: This expression denotes structure N sub n. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: b sub n
Means: This expression denotes b sub n. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: abstract logic L sub two is at least as expressive as abstract logic L sub one
Means: This expression denotes abstract logic L sub two is at least as expressive as abstract logic L sub one. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: language L' equals language L set minus c
Means: This expression denotes language L' equals language L set minus c. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: Gamma
Means: This expression denotes Gamma. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: abstract logic L sub one together with its satisfaction relation
Means: This expression denotes abstract logic L sub one together with its satisfaction relation. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: n is in the natural numbers
Means: This expression denotes n is in the natural numbers. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure N
Means: This expression denotes structure N. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: c'
Means: This expression denotes c'. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: P
Means: This expression denotes P. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: abstract logic L sub one is at least as expressive as abstract logic L sub two
Means: This expression denotes abstract logic L sub one is at least as expressive as abstract logic L sub two. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M is partially isomorphic to structure N
Means: This expression denotes structure M is partially isomorphic to structure N. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: formula E
Means: This expression denotes formula E. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the domain of structure M star
Means: This expression denotes the domain of structure M star. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: relation J holds of n star minus k and sequences a and b
Means: This expression denotes relation J holds of n star minus k and sequences a and b. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: the expansion of structure M by X and b sub one through b sub n satisfies formula F in logic L if and only if structure N satisfies formula E in logic L
Means: This expression denotes the expansion of structure M by X and b sub one through b sub n satisfies formula F in logic L if and only if structure N satisfies formula E in logic L. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M sub n star is partially isomorphic to structure N sub n star
Means: This expression denotes structure M sub n star is partially isomorphic to structure N sub n star. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: c sub n
Means: This expression denotes c sub n. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure N sub n star
Means: This expression denotes structure N sub n star. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure M sub n satisfies in logic L formula E
Means: This expression denotes structure M sub n satisfies in logic L formula E. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: formula D sub structure N
Means: This expression denotes formula D sub structure N. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: the domain of structure N sub n star
Means: This expression denotes the domain of structure N sub n star. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: formula F is in L open parenthesis language L close parenthesis
Means: This expression denotes formula F is in L open parenthesis language L close parenthesis. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: T prime
Means: This expression denotes T prime. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: J
Means: This expression denotes J. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure N sub n star does not satisfy formula E in logic L
Means: This expression denotes structure N sub n star does not satisfy formula E in logic L. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: language L prime
Means: This expression denotes language L prime. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: abstract logic L is at least as expressive as ordinary first order logic
Means: This expression denotes abstract logic L is at least as expressive as ordinary first order logic. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the quantifier rank of formula E is less than or equal to n
Means: This expression denotes the quantifier rank of formula E is less than or equal to n. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: the domain of structure N star
Means: This expression denotes the domain of structure N star. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: L
Means: This expression denotes L. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: sequences a, b, and c are finite sequences over the domain of structure M
Means: This expression denotes sequences a, b, and c are finite sequences over the domain of structure M. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: satisfies
Means: This expression denotes satisfies. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the natural numbers
Means: This expression denotes the natural numbers. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: is at least as expressive as
Means: This expression denotes is at least as expressive as. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: sequence a
Means: This expression denotes sequence a. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: formula A
Means: This expression denotes formula A. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: k
Means: This expression denotes k. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: the class of models in logic L sub one over language L satisfying formula E equals the class of models in logic L sub two over language L satisfying formula F
Means: This expression denotes the class of models in logic L sub one over language L satisfying formula E equals the class of models in logic L sub two over language L satisfying formula F. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: S open parenthesis sequence a comma sequence b comma sequence c close parenthesis
Means: This expression denotes S open parenthesis sequence a comma sequence b comma sequence c close parenthesis. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: structure N sub n does not satisfy formula E
Means: This expression denotes structure N sub n does not satisfy formula E. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure N satisfies in logic L formula D
Means: This expression denotes structure N satisfies in logic L formula D. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: R
Means: This expression denotes R. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: S
Means: This expression denotes S. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: sequence b
Means: This expression denotes sequence b. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: the family I of partial isomorphisms
Means: This expression denotes the family I of partial isomorphisms. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: the universal quantifier
Means: This expression denotes the universal quantifier. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure N satisfies formula D sub structure N
Means: This expression denotes structure N satisfies formula D sub structure N. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: for every x, formula F
Means: This expression denotes for every x, formula F. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the expansion of structure M by relation X comma b sub one comma and so on comma b sub n
Means: This expression denotes the expansion of structure M by relation X comma b sub one comma and so on comma b sub n. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: formula F
Means: This expression denotes formula F. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: first row: the domain of M sub n is the set of a in the domain of M for which R holds of a and n; the domain of N sub n is the set of a in the domain of N for which S holds of a and n. second row: the finite sequences in M sub n and N sub n are selected by the same predicates R and S
Means: This expression denotes first row: the domain of M sub n is the set of a in the domain of M for which R holds of a and n; the domain of N sub n is the set of a in the domain of N for which S holds of a and n. second row: the finite sequences in M sub n and N sub n are selected by the same predicates R and S. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: the pair of sequences a and b belongs to the family I
Means: This expression denotes the pair of sequences a and b belongs to the family I. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure M sub zero satisfies in logic L formula E
Means: This expression denotes structure M sub zero satisfies in logic L formula E. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: Gamma sub zero is a subset of Gamma
Means: This expression denotes Gamma sub zero is a subset of Gamma. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: X is a subset of M superscript n plus one
Means: This expression denotes X is a subset of M superscript n plus one. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the set of finite sequences over the domain of structure M
Means: This expression denotes the set of finite sequences over the domain of structure M. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: S prime
Means: This expression denotes S prime. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: structure N satisfies formula D sub structure M
Means: This expression denotes structure N satisfies formula D sub structure M. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: I
Means: This expression denotes I. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: formula E is in L sub one open parenthesis language L close parenthesis
Means: This expression denotes formula E is in L sub one open parenthesis language L close parenthesis. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the nonstandard number n star
Means: This expression denotes the nonstandard number n star. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: I sub n open parenthesis sequence a comma sequence b close parenthesis
Means: This expression denotes I sub n open parenthesis sequence a comma sequence b close parenthesis. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: language L is a subset of language L'
Means: This expression denotes language L is a subset of language L'. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M star
Means: This expression denotes structure M star. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: structure N does not satisfy in logic L formula E
Means: This expression denotes structure N does not satisfy in logic L formula E. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: structure M does not satisfy in logic L formula E
Means: This expression denotes structure M does not satisfy in logic L formula E. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the models of formula F in logic L over language L prime are exactly the reducts whose expansion by some domain element is a model of formula E over language L
Means: This expression denotes the models of formula F in logic L over language L prime are exactly the reducts whose expansion by some domain element is a model of formula E over language L. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M satisfies in logic L formula E
Means: This expression denotes structure M satisfies in logic L formula E. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: formula E prime belongs to logic L over language L prime
Means: This expression denotes formula E prime belongs to logic L over language L prime. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: language L
Means: This expression denotes language L. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: ordinary first order logic F together with its satisfaction relation
Means: This expression denotes ordinary first order logic F together with its satisfaction relation. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: formula D sub structure M
Means: This expression denotes formula D sub structure M. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: the atomic formula R of x and constants c sub one through c sub n
Means: This expression denotes the atomic formula R of x and constants c sub one through c sub n. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M and structure N agree on all formulas of quantifier rank at most n
Means: This expression denotes structure M and structure N agree on all formulas of quantifier rank at most n. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure N sub zero
Means: This expression denotes structure N sub zero. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: I sub n open parenthesis the empty set comma the empty set close parenthesis
Means: This expression denotes I sub n open parenthesis the empty set comma the empty set close parenthesis. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure M star
Means: This expression denotes structure M star. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: the atomic formula R of x and constants c sub one through c sub n
Means: This expression denotes the atomic formula R of x and constants c sub one through c sub n. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the class of models in logic L satisfying formula G equals the class of models in logic L satisfying formula E intersect the class of models in logic L satisfying formula F
Means: This expression denotes the class of models in logic L satisfying formula G equals the class of models in logic L satisfying formula E intersect the class of models in logic L satisfying formula F. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure N and structure M agree on all formulas of quantifier rank at most n
Means: This expression denotes structure N and structure M agree on all formulas of quantifier rank at most n. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure M prime
Means: This expression denotes structure M prime. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: a
Means: This expression denotes a. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M sub n star
Means: This expression denotes structure M sub n star. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: P prime
Means: This expression denotes P prime. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: formula B
Means: This expression denotes formula B. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the set of finite sequences over the domain of structure N
Means: This expression denotes the set of finite sequences over the domain of structure N. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: n star minus k is greater than zero
Means: This expression denotes n star minus k is greater than zero. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: formula D is the finite disjunction of the formulas D sub M for structures M that satisfy formula E in logic L
Means: This expression denotes formula D is the finite disjunction of the formulas D sub M for structures M that satisfy formula E in logic L. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: structure M prime satisfies formula E prime in logic L
Means: This expression denotes structure M prime satisfies formula E prime in logic L. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure M is elementarily equivalent to structure N in logic L
Means: This expression denotes structure M is elementarily equivalent to structure N in logic L. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: for every x, if R holds of x and constants c sub one through c sub n, then formula F
Means: This expression denotes for every x, if R holds of x and constants c sub one through c sub n, then formula F. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: formula F is in L sub two open parenthesis language L close parenthesis
Means: This expression denotes formula F is in L sub two open parenthesis language L close parenthesis. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: structure N sub n does not satisfy in logic L formula E
Means: This expression denotes structure N sub n does not satisfy in logic L formula E. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: T
Means: This expression denotes T. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: q belongs to the family I of partial isomorphisms
Means: This expression denotes q belongs to the family I of partial isomorphisms. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: I sub n open parenthesis the empty set comma the empty set close parenthesis
Means: This expression denotes I sub n open parenthesis the empty set comma the empty set close parenthesis. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: sequence c consists of a sub one through a sub n followed by b
Means: This expression denotes sequence c consists of a sub one through a sub n followed by b. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: b is in M
Means: This expression denotes b is in M. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: the domain of structure M together with its finite sequences
Means: This expression denotes the domain of structure M together with its finite sequences. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: the existential quantifier
Means: This expression denotes the existential quantifier. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: the set of structure M such that structure M satisfies in logic L formula E
Means: This expression denotes the set of structure M such that structure M satisfies in logic L formula E. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: F
Means: This expression denotes F. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: T open parenthesis sequence a comma b comma sequence c close parenthesis
Means: This expression denotes T open parenthesis sequence a comma b comma sequence c close parenthesis. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: L open parenthesis language L close parenthesis is a subset of L open parenthesis language L' close parenthesis
Means: This expression denotes L open parenthesis language L close parenthesis is a subset of L open parenthesis language L' close parenthesis. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: formula D sub one
Means: This expression denotes formula D sub one. In compactness and Lowenheim Skolem properties for abstract logics, it proves that partial isomorphism implies equivalence for a normal logic with the Lowenheim Skolem property.
Read as: formula D
Means: This expression denotes formula D. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: abstract logic L sub two together with its satisfaction relation
Means: This expression denotes abstract logic L sub two together with its satisfaction relation. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Read as: ordinary first order logic is at least as expressive as abstract logic L
Means: This expression denotes ordinary first order logic is at least as expressive as abstract logic L. In Lindstrom's maximality theorem, it reduces abstract sentences to first-order sentences and derives a contradiction from compactness.
Read as: the expansion of structure M by relation a
Means: This expression denotes the expansion of structure M by relation a. In abstract logics and their normality conditions, it defines model classes, normal abstract logics, relativization, and expressive strength.
Defines an abstract logic as a language-indexed collection of sentences together with a satisfaction relation between structures and those sentences. It identifies ordinary first-order logic as the principal example.
Defines the class of models of an abstract sentence and says that two structures are elementarily equivalent in an abstract logic when they satisfy exactly the same sentences of that logic.
Lists the seven closure and invariance requirements for a normal abstract logic: language monotonicity, finite expansion dependence, isomorphism invariance, renaming, Boolean closure, quantification, and relativization.
Defines one abstract logic to be at least as expressive as another when every sentence of the first has a sentence of the second with the same class of models; mutual expressibility defines equivalence.
An abstract logic has compactness when a set of its sentences is satisfiable whenever every finite subset is satisfiable.
An abstract logic has the downward Lowenheim Skolem property when every satisfiable set of its sentences has an enumerable model.
States that for a normal abstract logic with the downward Lowenheim Skolem property, partially isomorphic structures satisfy the same sentences of the abstract logic.
Shows one ambient structure containing expanded copies of structures M and N. Each copy contains its original substructure, and a two-headed arrow labeled I connects the sequence domains used for the internal partial isomorphism.
A rounded rectangle represents the ambient structure. Two nested circles on the left represent M inside M star, two nested circles on the right represent N inside N star, and a bidirectional curved arrow labeled I links their sequence regions.
If membership in the model class of an abstract sentence is invariant under equivalence through some finite quantifier rank n, the sentence has the same models as a first-order sentence.
States that a normal logic with compactness and the downward Lowenheim Skolem property is no more expressive than first-order logic and hence is equivalent to first-order logic.
Defines the domains of M sub n and N sub n, and their finite-sequence domains, as the elements selected by the coding predicates R and S at index n.