1 / 7

6.3 Interpretations

6.3 Interpretations. In Sentential logic, truth semantics comes first, truth trees later. Good order! In this chapter, quantified trees come first, the value-assignment semantics comes later. Bad order! Disadvantage: Undecidability of first-order logic. Definitions.

alaura
Download Presentation

6.3 Interpretations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 6.3 Interpretations • In Sentential logic, truth semantics comes first, truth trees later. Good order! • In this chapter, quantified trees come first, the value-assignment semantics comes later. Bad order! • Disadvantage: Undecidability of first-order logic.

  2. Definitions • An interpretationmaking a set of formulas true is a model of the set. • An interpretation M of a set S of formulas of QL is an ordered set < D, [] >, where D is M’s nonempty domain , and [] is a function assigning: • truth values to sentential letters in S • elements of D to constants in S • sets of n-tuples of elements of D to n-ary predicates in S

  3. Two Semantic Components • D: General individual domain for x (not quantifier) to range over. • []: Value-assignment function: • Sentential letter truth value • constant object in D • unitary predicate subset of D • binary predicate subset of pairs in DxD • n-ary predicate subset of n-tuples Dn

  4. Truth Conditions by M • [p]m = T iff p] = T • [Ra1 …an]m=T iff <[a1], …, [an]> is in [R] • [¬A]m = T iff [A]m = F • [(A&B)]m = T iff [A]m=[B]m=T • [(AVB)]m = T iff [A]m=T or [B]m=T • [(A→B)]m = T iff [A]m=F or [B]m=T • [(A↔B)]m = T iff [A]m=[B]m

  5. Truth Conditions for Quantifiers • c-variant M’ of M, just like M but [ ]’may assign different object to c • [uA]m = T iff [Ac/u]m’ = T for some c-variant M’ of M (there is some [a] in D s.t. A(a) = T) • [uA]m = T iff [Ac/u]m’ = T for every c-variant M’ of M (every [a] in D s.t. A(a) = T)

  6. 6.4 Constructing Interpretations from Trees • Consider only a finished, open tree • Consider an open branch including nodes: Fab, Fbb, Ga, ¬Gb, ¬Faa The model M = < D, [ ] > is [F]={<a, b>, <b, b>}, [G]={<a>}, D = {a, b}

  7. Rules • D = {a1, …, an, …}, where a1, an, … are the constants appearing in atomics on the open branch. • [p] = T if p is on the branch; F, otherwise. • [R] = {<a1, …, an>, <b1, …, bn>, …}, where R(a1, …an), R(b1, …,bn), and so on, appear on the branch.

More Related