12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music...

12. Predicate Logic Structures

The Lecture

Jouko Väänänen: Predicate logic

What is predicate logic?



Predicate logic deals with properties of elements and relations between elements of a domain.




We can talk about universal properties and existence of solutions of equations.




We can talk about universal properties and existence of solutions of equations.

A basic concept is that of a structure, also called a model.


Unary structure


Unary structureA unary structure M consists of a domain M and a number of subsets of it, called predicates. The predicates are denoted A0,A1,...


A0

M

Unary structureA unary structure M consists of a domain M and a number of subsets of it, called predicates. The predicates are denoted A0,A1,...

One predicate divides the

domain into up to two parts


A0

M

Unary structure

M

A unary structure M consists of a domain M and a number of subsets of it, called predicates. The predicates are denoted A0,A1,...

One predicate divides the

domain into up to two parts

Two predicates divide the

domain into up to four parts

A1A0


Examples


Examples

Women

MenA0M= a set of people

A0= the set of women in M

M


Examples

Women

MenA0M= a set of people

A0= the set of women in M

M

Likes jazz

Likes country music

A1

M= a set of peopleA0= the set of country music lovers in MA1= the set of jazz fans in M

M

A0


Unary structure with three predicates divides the domain into up to 8 parts.

A2A1

A0

M


Example

Likes country music

Likes classical music

Likes jazz

Likes country music and

jazz

Likes classical and country

music

Likes classical music and jazz


Tile models


Tile models

A tile model consists of colored tiles arranged in a row as the five tiles below:


Tile models

A tile model consists of colored tiles arranged in a row as the five tiles below:

The relevant properties of the tiles are: Color. Position: which is left or right of which.


Examples of tile models


A mathematical definition of tile models



A tile model T consists of



A tile model T consists of a finite set T of tiles



A tile model T consists of a finite set T of tiles For each tile x exactly one of the

predicates BT(x) ”x is blue”, RT(x) ”x is red”, YT(x) ”x is yellow” holds.





There is a linear order <T defined on T. If x <T y, we say x is ”left of” y and ”y is right of x”, and write x<Ty.





There is a linear order <T defined on T. If x <T y, we say x is ”left of” y and ”y is right of x”, and write x<Ty.

A linear order on a finite set is a specification of the order of the elements: which is the first, which comes next, etc.


Graphs


Graphs

A graph consists of vertices and edges between the vertices as in:


Graphs


In this picture vertices are blue, edges are red.


Graphs


In this picture vertices are blue, edges are red. Graphs are common in applications.


Graphs


In this picture vertices are blue, edges are red. Graphs are common in applications. Vertices connected by an edge are neighbors.


More graphs


A mathematical definition of graphs



A graph G consists of



A graph G consists of a domain G, called the set of vertices, and



A graph G consists of a domain G, called the set of vertices, and a binary predicate xEy (more exactly xEGy)

for the edge relation. Then x is called a neighbor of y and vice versa.





No vertex is a neighbor of itself. (Antireflexivity)





No vertex is a neighbor of itself. (Antireflexivity)

If xEy then yEx. (Symmetry)


The integers


The integers

The natural numbers are the non-negative integers 0,1,2,...


The integers

The natural numbers are the non-negative integers 0,1,2,...

They have a natural order < in which 0 is the smallest element and for every element x there is a bigger one, namely x+1.

Other structures (some with functions)

Directed graph Equivalence relation Group Field Boolean algebra Lattice Linear order Partial order Tree


12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music...

Documents

Transcript of 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music...