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12. Predicate Logic Structures
The Lecture
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Jouko Väänänen: Predicate logic
What is predicate logic?
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Jouko Väänänen: Predicate logic
What is predicate logic?
Predicate logic deals with properties of elements and relations between elements of a domain.
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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.
![Page 5: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/5.jpg)
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.
![Page 6: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/6.jpg)
Jouko Väänänen: Predicate logic
Unary structure
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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,...
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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
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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
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Jouko Väänänen: Predicate logic
Examples
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Jouko Väänänen: Predicate logic
Examples
Women
MenA0M= a set of people
A0= the set of women in M
M
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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
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Jouko Väänänen: Predicate logic
Unary structure with three predicates divides the domain into up to 8 parts.
A2A1
A0
M
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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
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Jouko Väänänen: Predicate logic
Tile models
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Jouko Väänänen: Predicate logic
Tile models
A tile model consists of colored tiles arranged in a row as the five tiles below:
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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.
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Jouko Väänänen: Predicate logic
Examples of tile models
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Jouko Väänänen: Predicate logic
Examples of tile models
![Page 20: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/20.jpg)
Jouko Väänänen: Predicate logic
Examples of tile models
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Jouko Väänänen: Predicate logic
Examples of tile models
![Page 22: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/22.jpg)
Jouko Väänänen: Predicate logic
Examples of tile models
![Page 23: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/23.jpg)
Jouko Väänänen: Predicate logic
Examples of tile models
![Page 24: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/24.jpg)
Jouko Väänänen: Predicate logic
A mathematical definition of tile models
![Page 25: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/25.jpg)
Jouko Väänänen: Predicate logic
A mathematical definition of tile models
A tile model T consists of
![Page 26: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/26.jpg)
Jouko Väänänen: Predicate logic
A mathematical definition of tile models
A tile model T consists of a finite set T of tiles
![Page 27: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/27.jpg)
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.
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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.
![Page 29: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/29.jpg)
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.
![Page 30: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/30.jpg)
Jouko Väänänen: Predicate logic
Graphs
![Page 31: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/31.jpg)
Jouko Väänänen: Predicate logic
Graphs
A graph consists of vertices and edges between the vertices as in:
![Page 32: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/32.jpg)
Jouko Väänänen: Predicate logic
Graphs
A graph consists of vertices and edges between the vertices as in:
![Page 33: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/33.jpg)
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.
![Page 34: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/34.jpg)
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.
![Page 35: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/35.jpg)
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.
![Page 36: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/36.jpg)
Jouko Väänänen: Predicate logic
More graphs
![Page 37: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/37.jpg)
Jouko Väänänen: Predicate logic
More graphs
![Page 38: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/38.jpg)
Jouko Väänänen: Predicate logic
More graphs
![Page 39: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/39.jpg)
Jouko Väänänen: Predicate logic
More graphs
![Page 40: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/40.jpg)
Jouko Väänänen: Predicate logic
More graphs
![Page 41: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/41.jpg)
Jouko Väänänen: Predicate logic
More graphs
![Page 42: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/42.jpg)
Jouko Väänänen: Predicate logic
More graphs
![Page 43: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/43.jpg)
Jouko Väänänen: Predicate logic
A mathematical definition of graphs
![Page 44: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/44.jpg)
Jouko Väänänen: Predicate logic
A mathematical definition of graphs
A graph G consists of
![Page 45: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/45.jpg)
Jouko Väänänen: Predicate logic
A mathematical definition of graphs
A graph G consists of a domain G, called the set of vertices, and
![Page 46: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/46.jpg)
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.
![Page 47: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/47.jpg)
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)
![Page 48: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/48.jpg)
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)
![Page 49: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/49.jpg)
Jouko Väänänen: Predicate logic
The integers
![Page 50: 12. Predicate Logic Structures - math.helsinki.fi · Jouko Väänänen: Predicate logic ... music lovers in M A 1= the set of jazz fans in M M A 0. Jouko Väänänen: Predicate logic](https://reader034.vdocuments.mx/reader034/viewer/2022051508/5ac5d5e87f8b9aae1b8e4990/html5/thumbnails/50.jpg)
Jouko Väänänen: Predicate logic
The integers
The natural numbers are the non-negative integers 0,1,2,...
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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.
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Other structures (some with functions)
Directed graph Equivalence relation Group Field Boolean algebra Lattice Linear order Partial order Tree
Jouko Väänänen: Predicate logic