A talk given at the 8th Panhellenic Logic Symposium, July 7, 2011, Ioannina, Greece.

1. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
.
. Fuzzy Topological Systems
Apostolos Syropoulos﷪ Valeria de Paiva﷫
﷪Greek Molecular Computing Group
. . . . . .
Xanthi, Greece
asyropoulos@gmail.com
﷫School of Computer Science
University of Birmingham, UK
valeria.depaiva@gmail.com
8th Panhellenic Logic Symposium
July 7, 2011
Ioannina, Greece

2. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Outline
1. Vagueness
General Ideas
Many-valued Logics and Vagueness
Fuzzy Sets and Relations
2. Dialectica Spaces
3. Fuzzy Topological Spaces
4. Finale

3. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?

4. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.

5. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.

6. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.

7. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.

8. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples

9. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?

10. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?

11. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?
Is Betelgeuse a huge star?

12. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?
Is Betelgeuse a huge star?
If you cut one head off of a two headed man, have you
decapitated him?

13. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?
Is Betelgeuse a huge star?
If you cut one head off of a two headed man, have you
decapitated him?
Eubulides of Miletus: The Sorites Paradox.

14. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems possible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?
Is Betelgeuse a huge star?
If you cut one head off of a two headed man, have you
decapitated him?
Eubulides of Miletus: The Sorites Paradox.
The old puzzle about bald people.

15. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Defining Vagueness

16. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Defining Vagueness
Bertrand Russell: Per contra, a representation is vague when
the relation of the representing system to the represented
system is not one-one, but one-many.

17. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Defining Vagueness
Bertrand Russell: Per contra, a representation is vague when
the relation of the representing system to the represented
system is not one-one, but one-many.
Example: A small-scale map is usually vaguer than a large-scale
map.

18. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Defining Vagueness
Bertrand Russell: Per contra, a representation is vague when
the relation of the representing system to the represented
system is not one-one, but one-many.
Example: A small-scale map is usually vaguer than a large-scale
map.
Charles Sanders Peirce: A proposition is vague when there are
possible states of things concerning which it is intrinsically
uncertain whether, had they been contemplated by the
speaker, he would have regarded them as excluded or
allowed by the proposition. By intrinsically uncertain we
mean not uncertain in consequence of any ignorance of the
interpreter, but because the speaker’s habits of language were
indeterminate.

19. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Defining Vagueness
Bertrand Russell: Per contra, a representation is vague when
the relation of the representing system to the represented
system is not one-one, but one-many.
Example: A small-scale map is usually vaguer than a large-scale
map.
Charles Sanders Peirce: A proposition is vague when there are
possible states of things concerning which it is intrinsically
uncertain whether, had they been contemplated by the
speaker, he would have regarded them as excluded or
allowed by the proposition. By intrinsically uncertain we
mean not uncertain in consequence of any ignorance of the
interpreter, but because the speaker’s habits of language were
indeterminate.
Max Black demonstrated this definition using the word chair.

20. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
. . . . . .

21. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
. . . . . .

22. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
A term or phrase is ambiguous if it has at least two specific
meanings that make sense in context.
. . . . . .

23. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
A term or phrase is ambiguous if it has at least two specific
meanings that make sense in context.
Example: He ate the cookies on the couch.
. . . . . .

24. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
A term or phrase is ambiguous if it has at least two specific
meanings that make sense in context.
Example: He ate the cookies on the couch.
Bertrand Russell: “A proposition involving a general
concept—e.g., ‘This is a man’—will be verified by a number of
facts, such as ‘This’ being Brown or Jones or Robinson.”
. . . . . .

25. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
A term or phrase is ambiguous if it has at least two specific
meanings that make sense in context.
Example: He ate the cookies on the couch.
Bertrand Russell: “A proposition involving a general
concept—e.g., ‘This is a man’—will be verified by a number of
facts, such as ‘This’ being Brown or Jones or Robinson.”
Example: Again consider the word chair.
. . . . . .

26. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Many-valued Logics and Fuzzy Set Theory
. . . . . .

27. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
. . . . . .

28. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
. . . . . .

29. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
Jan Łukasiewicz defined a three-valued logic to deal with future
contingencies.
. . . . . .

30. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
Jan Łukasiewicz defined a three-valued logic to deal with future
contingencies.
Emil Leon Post introduced the formulation of 푛 truth values,
where 푛 ≥ ﷡.
. . . . . .

31. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
Jan Łukasiewicz defined a three-valued logic to deal with future
contingencies.
Emil Leon Post introduced the formulation of 푛 truth values,
where 푛 ≥ ﷡.
C.C. Chang proposed MV-algebras, that is, an algebraic model
of many-valued logics.
. . . . . .

32. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
Jan Łukasiewicz defined a three-valued logic to deal with future
contingencies.
Emil Leon Post introduced the formulation of 푛 truth values,
where 푛 ≥ ﷡.
C.C. Chang proposed MV-algebras, that is, an algebraic model
of many-valued logics.
In 1965, Lotfi A. Zadeh introduced his fuzzy set theory and
its accompanying fuzzy logic.
. . . . . .

33. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Fuzzy (Sub)sets

34. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to a set to a certain degree.

35. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to a set to a certain degree.
Given a universe 푋, a fuzzy subset 퐴 of 푋 is characterized by
a function 퐴 ∶ 푋 → ﹚, ﹚ = [﷟, ﷠], where 퐴(푥) = 푖 means that 푥
belongs to 퐴 with degree equal to 푖.

36. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to a set to a certain degree.
Given a universe 푋, a fuzzy subset 퐴 of 푋 is characterized by
a function 퐴 ∶ 푋 → ﹚, ﹚ = [﷟, ﷠], where 퐴(푥) = 푖 means that 푥
belongs to 퐴 with degree equal to 푖.
Example: One can easily build a fuzzy subset consisting of all
the tall students of a class.

37. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to a set to a certain degree.
Given a universe 푋, a fuzzy subset 퐴 of 푋 is characterized by
a function 퐴 ∶ 푋 → ﹚, ﹚ = [﷟, ﷠], where 퐴(푥) = 푖 means that 푥
belongs to 퐴 with degree equal to 푖.
Example: One can easily build a fuzzy subset consisting of all
the tall students of a class.
Assume that 푈 and 푋 are nonempty sets. A binary fuzzy
relation 푅 in 푈 and 푋 is a fuzzy subset of 푈 × 푋. The value
of 푅(푢, 푥) is interpreted as the degree of membership of the
ordered pair (푢, 푥) in 푅.

38. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Categorical Model of Linear Logic
. . . . . .

39. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Categorical Model of Linear Logic
Categories form a very high-level abstract mathematical theory
that unifies all branches of mathematics.
. . . . . .

40. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Categorical Model of Linear Logic
Categories form a very high-level abstract mathematical theory
that unifies all branches of mathematics.
Categories have been used to model and study logical systems.
. . . . . .

41. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Categorical Model of Linear Logic
Categories form a very high-level abstract mathematical theory
that unifies all branches of mathematics.
Categories have been used to model and study logical systems.
Dialectica spaces, which were introduced by de Paiva, form a
category which is a model of linear logic.
. . . . . .

42. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Categorical Model of Linear Logic
Categories form a very high-level abstract mathematical theory
that unifies all branches of mathematics.
Categories have been used to model and study logical systems.
Dialectica spaces, which were introduced by de Paiva, form a
category which is a model of linear logic.
Dialectica categories have been used to model Petri nets, the
Lambek Calculus, state in programming, and to define fuzzy
petri nets.
. . . . . .

43. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Categorical Model of Linear Logic
Categories form a very high-level abstract mathematical theory
that unifies all branches of mathematics.
Categories have been used to model and study logical systems.
Dialectica spaces, which were introduced by de Paiva, form a
category which is a model of linear logic.
Dialectica categories have been used to model Petri nets, the
Lambek Calculus, state in programming, and to define fuzzy
petri nets.
Recent development: fuzzy topological systems, that is, the
fuzzy counterpart of Vickers’s topological systems.
. . . . . .

44. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales

45. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, ﷠, ⊸) is a lineale if

46. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, ﷠, ⊸) is a lineale if
(퐿, ≤) is a poset;

47. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, ﷠, ⊸) is a lineale if
(퐿, ≤) is a poset;
∘ ∶ 퐿 × 퐿 → 퐿 is an order-preserving multiplication, such that
(퐿, ∘, ﷠) is a symmetric monoidal structure; and

48. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, ﷠, ⊸) is a lineale if
(퐿, ≤) is a poset;
∘ ∶ 퐿 × 퐿 → 퐿 is an order-preserving multiplication, such that
(퐿, ∘, ﷠) is a symmetric monoidal structure; and
for any 푎, 푏 ∈ 퐿 exists the largest 푥 ∈ 퐿 such that 푎 ∘ 푥 ≤ 푏, then
this element is denoted 푎 ⊸ 푏 and is called the
pseudo-complement of 푎 with respect to 푏.

49. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, ﷠, ⊸) is a lineale if
(퐿, ≤) is a poset;
∘ ∶ 퐿 × 퐿 → 퐿 is an order-preserving multiplication, such that
(퐿, ∘, ﷠) is a symmetric monoidal structure; and
for any 푎, 푏 ∈ 퐿 exists the largest 푥 ∈ 퐿 such that 푎 ∘ 푥 ≤ 푏, then
this element is denoted 푎 ⊸ 푏 and is called the
pseudo-complement of 푎 with respect to 푏.
The quintuple (﹚, ≤, ∧, ﷠, ⇒), where ﹚ is the unit interval,
푎 ∧ 푏 = ︌︈︍{푎, 푏}, and 푎 ⇒ 푏 = ⋁{푐 ∶ 푐 ∧ 푎 ≤ 푏}
(푎 ∨ 푏 = ︌︀︗{푎, 푏}), is a lineale.

50. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
The Category 햣헂햺헅﹚(퐒퐞퐭)

51. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
The Category 햣헂햺헅﹚(퐒퐞퐭)
Objects Triples 퐴 = (푈, 푋, 훼), where 푈 and 푋 are sets and 훼
is a map 푈 × 푋 → ﹚.

52. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
훼
?
. . . . . .
The Category 햣헂햺헅﹚(퐒퐞퐭)
Objects Triples 퐴 = (푈, 푋, 훼), where 푈 and 푋 are sets and 훼
is a map 푈 × 푋 → ﹚.
Arrows An arrow from 퐴 = (푈, 푋, 훼) to 퐵 = (푉, 푌, 훽) is a pair
of 퐒퐞퐭 maps (푓, 푔), 푓 ∶ 푈 → 푉, 푔 ∶ 푌 → 푋 such that
훼(푢, 푔(푦)) ≤ 훽(푓(푢), 푦),
or in pictorial form:
푈 × 푌
︈︃푈 ×푔-
푈 × 푋
≥
푓 × ︈︃푌
?
푉 × 푌
훽
- ﹚

53. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
The Category 햣헂햺헅﹚(퐒퐞퐭) cont.

54. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
The Category 햣헂햺헅﹚(퐒퐞퐭) cont.
Arrow Composition Let (푓, 푔) and (푓′, 푔′) be the following arrows:
(푈, 푋, 훼)
(푓,푔)
⟶ (푉, 푌, 훽)
(푓′,푔′)
⟶ (푊, 푍, 훾).
Then (푓, 푔) ∘ (푓′, 푔′) = (푓 ∘ 푓′, 푔′ ∘ 푔) such that
훼￵푢, ￴푔′ ∘ 푔￷(푧)￸ ≤ 훾￵￴푓 ∘ 푓′￷(푢), 푧￸.

55. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Logical Connectives

56. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Logical Connectives
Given objects 퐴 = (푈, 푋, 훼) and 퐵 = (푉, 푌, 훽)

57. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Logical Connectives
Given objects 퐴 = (푈, 푋, 훼) and 퐵 = (푉, 푌, 훽)
Tensor Product 퐴 ⊗ 퐵 = (푈 × 푉, 푋푉 × 푌푈 , 훼 × 훽), where 훼 × 훽 is the
relation that, using the lineale structure of 퐼, takes the
minimum of the membership degrees.

58. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Logical Connectives
Given objects 퐴 = (푈, 푋, 훼) and 퐵 = (푉, 푌, 훽)
Tensor Product 퐴 ⊗ 퐵 = (푈 × 푉, 푋푉 × 푌푈 , 훼 × 훽), where 훼 × 훽 is the
relation that, using the lineale structure of 퐼, takes the
minimum of the membership degrees.
Linear Function-Space 퐴 → 퐵 = (푉푈 × 푌푋 , 푈 × 푋, 훼 → 훽), where
again the relation 훼 → 훽 is given by the implication in
the lineale.

59. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Logical Connectives
Given objects 퐴 = (푈, 푋, 훼) and 퐵 = (푉, 푌, 훽)
Tensor Product 퐴 ⊗ 퐵 = (푈 × 푉, 푋푉 × 푌푈 , 훼 × 훽), where 훼 × 훽 is the
relation that, using the lineale structure of 퐼, takes the
minimum of the membership degrees.
Linear Function-Space 퐴 → 퐵 = (푉푈 × 푌푋 , 푈 × 푋, 훼 → 훽), where
again the relation 훼 → 훽 is given by the implication in
the lineale.
Product 퐴 × 퐵 = (푈 × 푉, 푋 + 푌, 훾), where
훾 ∶ 푈 × 푉 × (푋 + 푌) → ﹚ is the fuzzy relation that is
defined as follows
훾￴(푢, 푣), 푧￷ = 
훼(푢, 푥), if 푧 = (푥, ﷟)
훽(푣, 푦), if 푧 = (푦, ﷠)

60. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Logical Connectives
Given objects 퐴 = (푈, 푋, 훼) and 퐵 = (푉, 푌, 훽)
Tensor Product 퐴 ⊗ 퐵 = (푈 × 푉, 푋푉 × 푌푈 , 훼 × 훽), where 훼 × 훽 is the
relation that, using the lineale structure of 퐼, takes the
minimum of the membership degrees.
Linear Function-Space 퐴 → 퐵 = (푉푈 × 푌푋 , 푈 × 푋, 훼 → 훽), where
again the relation 훼 → 훽 is given by the implication in
the lineale.
Product 퐴 × 퐵 = (푈 × 푉, 푋 + 푌, 훾), where
훾 ∶ 푈 × 푉 × (푋 + 푌) → ﹚ is the fuzzy relation that is
defined as follows
훾￴(푢, 푣), 푧￷ = 
훼(푢, 푥), if 푧 = (푥, ﷟)
훽(푣, 푦), if 푧 = (푦, ﷠)
Coproduct 퐴 ⊕ 퐵 = (푈 + 푉, 푋 × 푌, 훿), where 훿 is defined similar
to 훾.

61. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Topological Systems

62. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Topological Systems
A triple (푈, ⊧, 푋), where 푋 is a frame and 푈 is a set, is a
topological system if

63. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Topological Systems
A triple (푈, ⊧, 푋), where 푋 is a frame and 푈 is a set, is a
topological system if
when 푆 is a finite subset of 푋, then
푢 ⊧ ﾓ푆 ⟺ 푢 ⊧ 푥 for all 푥 ∈ 푆.

64. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Topological Systems
A triple (푈, ⊧, 푋), where 푋 is a frame and 푈 is a set, is a
topological system if
when 푆 is a finite subset of 푋, then
푢 ⊧ ﾓ푆 ⟺ 푢 ⊧ 푥 for all 푥 ∈ 푆.
when 푆 is any subset of 푋, then
푢 ⊧ ﾕ푆 ⟺ 푢 ⊧ 푥 for some 푥 ∈ 푆.

65. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Defining Fuzzy Topological Systems
. . . . . .

66. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Defining Fuzzy Topological Systems
A triple (푈, 훼, 푋), where 푋 is a frame, 푈 is a set, and 훼 ∶ 푈 × 푋 → ﹚
a binary fuzzy relation, is a fuzzy topological system if
. . . . . .

67. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Defining Fuzzy Topological Systems
A triple (푈, 훼, 푋), where 푋 is a frame, 푈 is a set, and 훼 ∶ 푈 × 푋 → ﹚
a binary fuzzy relation, is a fuzzy topological system if
when 푆 is a finite subset of 푋, then
훼(푢,ﾓ푆) ≤ 훼(푢, 푥) for all 푥 ∈ 푆.
. . . . . .

68. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Defining Fuzzy Topological Systems
A triple (푈, 훼, 푋), where 푋 is a frame, 푈 is a set, and 훼 ∶ 푈 × 푋 → ﹚
a binary fuzzy relation, is a fuzzy topological system if
when 푆 is a finite subset of 푋, then
훼(푢,ﾓ푆) ≤ 훼(푢, 푥) for all 푥 ∈ 푆.
. . . . . .
when 푆 is any subset of 푋, then
훼(푢,ﾕ푆) ≤ 훼(푢, 푥) for some 푥 ∈ 푆.

69. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
Defining Fuzzy Topological Systems
A triple (푈, 훼, 푋), where 푋 is a frame, 푈 is a set, and 훼 ∶ 푈 × 푋 → ﹚
a binary fuzzy relation, is a fuzzy topological system if
when 푆 is a finite subset of 푋, then
훼(푢,ﾓ푆) ≤ 훼(푢, 푥) for all 푥 ∈ 푆.
. . . . . .
when 푆 is any subset of 푋, then
훼(푢,ﾕ푆) ≤ 훼(푢, 푥) for some 푥 ∈ 푆.
훼(푢, ⊤) = ﷠ and 훼(푢, ⊥) = ﷟ for all 푢 ∈ 푈.

70. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Results

71. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Results
The collection of objects of 햣헂햺헅ﹴ(퐒퐞퐭) that are fuzzy topological
systems and the arrows between them form the category
퐅퐓퐨퐩퐒퐲퐬퐭퐞퐦퐬.

72. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Results
The collection of objects of 햣헂햺헅ﹴ(퐒퐞퐭) that are fuzzy topological
systems and the arrows between them form the category
퐅퐓퐨퐩퐒퐲퐬퐭퐞퐦퐬.
Any topological system (푈, 푋) is a fuzzy topological system
(푈, 휄, 푋), where
휄(푢, 푥) = 
﷠, when 푢 ⊧ 푥
﷟, otherwise

73. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Results
The collection of objects of 햣헂햺헅ﹴ(퐒퐞퐭) that are fuzzy topological
systems and the arrows between them form the category
퐅퐓퐨퐩퐒퐲퐬퐭퐞퐦퐬.
Any topological system (푈, 푋) is a fuzzy topological system
(푈, 휄, 푋), where
휄(푢, 푥) = 
﷠, when 푢 ⊧ 푥
﷟, otherwise
The category of topological systems is a full subcategory of
햣헂햺헅ﹴ(퐒퐞퐭).

74. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Results cont.

75. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Results cont.
Assume that 푎 ∈ 퐴, where (푈, 훼, 푋) is a fuzzy topological
space. Then the extent of an open 푥 is a function whose
graph is given below:
￴푢, 훼(푢, 푥)￷ ∶ 푢 ∈ 푈.
Now, the collection of all fuzzy sets created by the extents of
the members of 퐴 correspond to a fuzzy topology on 푋.

76. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Let’s be practical!

77. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Let’s be practical!
Physical Interpratation 푈 set of programs that generate bit stream
and elements of 푋 are assertions about bit streams.

78. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Let’s be practical!
Physical Interpratation 푈 set of programs that generate bit stream
and elements of 푋 are assertions about bit streams.
Example If 푢 generates the bit stream 010101010101…, and
“starts 01010” ∈ 푋, then 푥 ⊧ starts 01010.

79. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Let’s be practical!
Physical Interpratation 푈 set of programs that generate bit stream
and elements of 푋 are assertions about bit streams.
Example If 푢 generates the bit stream 010101010101…, and
“starts 01010” ∈ 푋, then 푥 ⊧ starts 01010.
Fuzzy Bit Streams Assume that 푥′ is a program that produces bit
streams like the following
0 1 0 1 0 1 0 1 0

80. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Let’s be practical!
Physical Interpratation 푈 set of programs that generate bit stream
and elements of 푋 are assertions about bit streams.
Example If 푢 generates the bit stream 010101010101…, and
“starts 01010” ∈ 푋, then 푥 ⊧ starts 01010.
Fuzzy Bit Streams Assume that 푥′ is a program that produces bit
streams like the following
0 1 0 1 0 1 0 1 0
Explanation The bits above are distorted because of some
interaction with the environment.

81. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Let’s be practical!
Physical Interpratation 푈 set of programs that generate bit stream
and elements of 푋 are assertions about bit streams.
Example If 푢 generates the bit stream 010101010101…, and
“starts 01010” ∈ 푋, then 푥 ⊧ starts 01010.
Fuzzy Bit Streams Assume that 푥′ is a program that produces bit
streams like the following
0 1 0 1 0 1 0 1 0
Explanation The bits above are distorted because of some
interaction with the environment.
Result 푥′ satisfies the assertion “starts 01010” to some
degree.

82. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Finale!

83. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Finale!
We have briefly presented

84. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Finale!
We have briefly presented
The notion of vagueness.

85. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Finale!
We have briefly presented
The notion of vagueness.
Dialectica categories.

86. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Finale!
We have briefly presented
The notion of vagueness.
Dialectica categories.
Fuzzy topological systems.

87. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Finale!
We have briefly presented
The notion of vagueness.
Dialectica categories.
Fuzzy topological systems.
A “real life” application of fuzzy topological systems.

88. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics and
Vagueness
Fuzzy Sets and Relations
Dialectica Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Finale!
We have briefly presented
The notion of vagueness.
Dialectica categories.
Fuzzy topological systems.
A “real life” application of fuzzy topological systems.
Thank you so much for your attention!