12. Predicate Logic Structures

The Lecture

Jouko Väänänen: Predicate logic

What is predicate logic?

Jouko Väänänen: Predicate logic

What is predicate logic?

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

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.

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.

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

Jouko Väänänen: Predicate logic

Unary structure

Jouko Väänänen: Predicate logic

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,...

Jouko Väänänen: Predicate logic

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

Jouko Väänänen: Predicate logic

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

Jouko Väänänen: Predicate logic

Examples

Jouko Väänänen: Predicate logic

Examples

Women

MenA0M= a set of people

A0= the set of women in M

M

Jouko Väänänen: Predicate logic

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

Jouko Väänänen: Predicate logic

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

A2A1

A0

M

Jouko Väänänen: Predicate logic

Example

Likes country music

Likes classical music

Likes jazz

Likes country music and

jazz

Likes classical and country

music

Likes classical music and jazz

Jouko Väänänen: Predicate logic

Tile models

Jouko Väänänen: Predicate logic

Tile models

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

Jouko Väänänen: Predicate logic

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.

Jouko Väänänen: Predicate logic

Examples of tile models

Jouko Väänänen: Predicate logic

Examples of tile models

Jouko Väänänen: Predicate logic

Examples of tile models

Jouko Väänänen: Predicate logic

Examples of tile models

Jouko Väänänen: Predicate logic

Examples of tile models

Jouko Väänänen: Predicate logic

Examples of tile models

Jouko Väänänen: Predicate logic

A mathematical definition of tile models

Jouko Väänänen: Predicate logic

A mathematical definition of tile models

A tile model T consists of

Jouko Väänänen: Predicate logic

A mathematical definition of tile models

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

Jouko Väänänen: Predicate logic

A mathematical definition of tile models

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.

Jouko Väänänen: Predicate logic

A mathematical definition of tile models

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.

Jouko Väänänen: Predicate logic

A mathematical definition of tile models

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.

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

Jouko Väänänen: Predicate logic

Graphs

Jouko Väänänen: Predicate logic

Graphs

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

Jouko Väänänen: Predicate logic

Graphs

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

Jouko Väänänen: Predicate logic

Graphs

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

In this picture vertices are blue, edges are red.

Jouko Väänänen: Predicate logic

Graphs

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

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

Jouko Väänänen: Predicate logic

Graphs

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

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

Jouko Väänänen: Predicate logic

More graphs

Jouko Väänänen: Predicate logic

More graphs

Jouko Väänänen: Predicate logic

More graphs

Jouko Väänänen: Predicate logic

More graphs

Jouko Väänänen: Predicate logic

More graphs

Jouko Väänänen: Predicate logic

More graphs

Jouko Väänänen: Predicate logic

More graphs

Jouko Väänänen: Predicate logic

A mathematical definition of graphs

Jouko Väänänen: Predicate logic

A mathematical definition of graphs

A graph G consists of

Jouko Väänänen: Predicate logic

A mathematical definition of graphs

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

Jouko Väänänen: Predicate logic

A mathematical definition of graphs

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.

Jouko Väänänen: Predicate logic

A mathematical definition of graphs

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)

Jouko Väänänen: Predicate logic

A mathematical definition of graphs

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)

If xEy then yEx. (Symmetry)

Jouko Väänänen: Predicate logic

The integers

Jouko Väänänen: Predicate logic

The integers

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

Jouko Väänänen: Predicate logic

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

Jouko Väänänen: Predicate logic