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Lecture8_Logic_and_Algebra

# Lecture8_Logic_and_Algebra - Ling 726 Mathematical...

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Ling 726: Mathematical Linguistics, Lecture 8 Logic and Algebra. V. Borschev and B. Partee, October 14, 2004 p. 1 Lecture 8. Axioms and theories, more examples. Axiomatic description of properties of relations. CONTENTS Axiomatic description of properties and classes of relations ......................................................................................... 1 1. Properties of relations ............................................................................................................................................. 1 1.1. Reflexivity, etc . ............................................................................................................................................... 1 1.2. Symmetry, etc .................................................................................................................................................. 1 1.3. Transitivity, etc ................................................................................................................................................ 1 1.4. Connectedness: ................................................................................................................................................ 2 2. Classes of relations ................................................................................................................................................. 2 2.1. Equivalence relations ...................................................................................................................................... 2 2.2. Tolerance relations .......................................................................................................................................... 2 2.3. Orderings ......................................................................................................................................................... 2 Homework 9 -- mainly algebra review .......................................................................................................................... 3 Reading: Chapter 8: 8.5, Chapter 9: 9.1 – 9.4 of PMW, pp. 206 –211, 247- 253. Axiomatic description of properties and classes of relations We can use PL to define properties and classes of relations which we studied in Lectures 1-3. Below we write axioms for different relations using the same binary predicate symbol R . Models in which these axioms (or sets of axioms) will be true will represent corresponding relations. In all cases below, the domain of quantification is the set A on which the relation is defined. 1. Properties of relations 1.1. Reflexivity, etc. Reflexivity: x R ( x , x ) It is easy to see that in every model in which this axiom holds the relation R will be reflexive. Nonreflexivity: x ¬ R ( x , x ) Irreflexivity: x ¬ R ( x , x ) 1.2. Symmetry, etc. Symmetry: x y ( R ( x , y ) R ( y , x )) Nonsymmetry: x y ( R ( x , y ) & ¬ R ( y , x )) Asymmetry: x y ( R ( x , y ) ¬ R ( y , x )) Antisymmetry: x y (( R ( x , y ) & R ( y , x )) ( x = y )) Note. In Lecture 6 we considered Predicate Logic without equality (we did not introduce the predicate “=”). To deal with it we need additional syntactic and semantic rules. The syntactic rule

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Lecture8_Logic_and_Algebra - Ling 726 Mathematical...

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