11- First Order Semantics Ref: G. Tourlakis, Mathematical Logic , - PowerPoint PPT Presentation

SC/MATH 1090 11- First Order Semantics Ref: G. Tourlakis, Mathematical Logic , John Wiley & Sons, 2008. York University Department of Computer Science and Engineering 1 York University- MATH 1090 11- First Order Semantics

Overview • Interpretations and domains • Logically valid formulae – How does that relate to tautologies? • Soundness and completeness in first order logic York University- MATH 1090 11- First Order Semantics 2

Interpretations • An interpretation, D = (D,M), translates a formula A to A D . • The two components of a first-order language interpretation are: – Domain D (non empty set, e.g. Natural numbers) – Translator M (a mapping) ┬ ,  ┬ D is t and  D is f p D is t or f p,q,... x D is a member of D x,y,... M c D is a member of D c ,... f D specific function applicable to objects in D f , ...  D specific predicate applicable to objects in D  ,... everything else unchanged York University- MATH 1090 11- First Order Semantics 3

Interpretations (2) • Given an interpretation D = (D,M), the translation of A, i.e. A D is obtained by: – Replacing (  x) with (  x  D) – Keep any bound object variables unchanged – Applying M to every other substrings of A • Examples: If D={1,2,3}, and A is x=y, then A D is x D =y D Therefore, if x D is 2 and y D is 3, then A D is f If A is (  x) x=y then A D is (  x  D) x=3 , which is f again If A is (  x)  (x,y) , and  D is  , then A D is (  x  D)x  3, which is t York University- MATH 1090 11- First Order Semantics 4

Partial Translation of Formula • A partial translation of formula A by D with respect to D the variables x 1 ,...,x n , is denoted by and refers to A ,..., x x 1 n translating A while leaving x 1 ,...,x n untranslated.   D • By above definition, ((  x)A) D is ( x D ) A x York University- MATH 1090 11- First Order Semantics 5

Logically Valid Formulae • Definition. (Model) If A D is t for some A and D , in other words A is true in the interpretation D , then D is a model of A, and is denoted by • Definition. (Universally, Logically, or Absolutely Valid formulae) A formula A in first order logic is valid iff every interpretation D is a model of A. This is denoted by York University- MATH 1090 11- First Order Semantics 6

Tautology vs. Valid • If , then . • If , it does NOT imply . – Example: if A is x=x. York University- MATH 1090 11- First Order Semantics 7

Soundness and Completeness • Metatheorem. (Soundness in 1 st order logic) If ⊢ A then • Metatheorem . (Gödel’s Completeness Theorem) If then ⊢ A York University- MATH 1090 11- First Order Semantics 8