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

York University- MATH 1090

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11- First Order Semantics

Overview

• Interpretations and domains
• Logically valid formulae

– How does that relate to tautologies?

• Soundness and completeness in first order logic

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Interpretations

• An interpretation, D= (D,M), translates a formula A to AD.
• 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,q,... pD is t or f x,y,... xD is a member of D c ,... cD is a member of D f, ... fD specific function applicable to objects in D ,... D specific predicate applicable to objects in D everything else unchanged

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M

Interpretations (2)

• Given an interpretation D= (D,M), the translation of A, i.e. AD is
• btained 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 AD is xD=yD Therefore, if xD is 2 and yD is 3, then AD is f If A is (x) x=y then AD is (xD) x=3, which is f again If A is (x) (x,y), and D is , then AD is (xD)x3, which is t

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Partial Translation of Formula

• A partial translation of formula A by D with respect to

the variables x1,...,xn, is denoted by and refers to translating A while leaving x1,...,xn untranslated.

• By above definition, ((x)A)D is

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D

n

x x

A ,...,

1

D x

A D x ) (  

Logically Valid Formulae

• Definition. (Model) If AD is t for some A and D, in other

words A is true in the interpretation D, then D is a model

• f 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

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Tautology vs. Valid

• If , then .
• If , it does NOT imply .

– Example: if A is x=x.

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Soundness and Completeness

• Metatheorem. (Soundness in 1st order logic)

If ⊢A then

• Metatheorem. (Gödel’s Completeness Theorem)

If then ⊢A

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