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Relations and Expressions |
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We proceed semantically; that is, from a semantic
motivation and intent. In all questions the test is whether a solution abides
by the semantics of the situation. Consider the relation types rel(L) and expressions expr(L) in a typed logical language L. Semantically, both evaluate to relations; that is, subsets of Cartesian products. To be more precise, assume in a logical model A that all entity types α have received an interpretation as a domain (set) A(α). On the one hand, a relation type ρ with valence n and signature (α1, … , αn) has an assumed interpretation A(α) Í A(α1)´ … ´A(αn). On the other hand an expression φ with its signature of a set of n free typed variables sign(φ) = {x1:α1, … , xn:αn} has an inductively defined interpretation A(φ) Í ÕA(α1)´ … ´A(αn). So relation types and expressions represent the same idea, one with a primitive specification, the other with an inductive definition. Normally, one considers any relation type ρ to be an expression through the use of n canonical variables (–1:α1, … , –n:αn); or for that matter, any chosen set of appropriately typed variables (x1:α1, … , xn:αn). However, from a formal linguistic point-of-view, we want to view expressions as relations – any expression φ with n free typed variables (x1:α1, … , xn:αn) is a relation type (in an expression ontological language) with valence n and signature (α1, … , αn). So the passage from expressions to relation types has discarded the set of free variables arity(φ) = {x1, … , xn} and the bindings (x14 α1, … , xn4 αn), which abstractly is a map ¶ : arity(φ) ® ent(L). On the one hand, we view entity types and relation types as linguistic symbols of a logical language L. On the other hand, we view entity types and expressions as a logical language expr(L). But expressions have the added components: variables, connectives and quantifiers. These notions seem quite linguistic in nature. However, in model theory they are assumed and considered to be extra-linguistic logical symbols. We do the same here and assume the following. ○ There is a fixed and adequately large set of variables var. ○ There is a set of logical connectives {Ù, Ú, Ø, →, ↔} representing conjunction, disjunction, negation, implication and logical equivalence. ○ There is a set of quantifiers {", $} representing universal and existential quantification. All of these are extra-linguistic, meaning not contained in logical languages. But there is a question about signature. (to be continued) |
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Please send questions,
comments and suggestions about this page to: Robert E. Kent rekent@ontologos.org |
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