This document discusses fuzzy topological systems and vagueness. It begins by defining vagueness and giving examples of vague terms and borderline cases. It describes views on vagueness from Bertrand Russell, Charles Sanders Peirce, and others. It discusses the differences between vagueness, ambiguity, and generality. It then covers many-valued logics and fuzzy set theory, and how Aristotle discussed future contingencies and truth values. It poses the question of determining the truth value of the proposition "At Christmas there will be snow in Athens, Greece."

1. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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
Segundo Congresso Brasileiro de Sistemas Fuzzy
November 6–9, 2012
Natal, Rio Grande do Norte, Brazil

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

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

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

16. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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
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
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
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.

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

22. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambiguity, 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
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambiguity, 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
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambiguity, 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
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
Vagueness, Generality, and Ambiguity
Vagueness, ambiguity, 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
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
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
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 “At Christmas there will be snow in Athens,
Greece?”
. . . . . .

29. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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 “At Christmas there will be snow in Athens,
Greece?”
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
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 “At Christmas there will be snow in Athens,
Greece?”
Jan Łukasiewicz defined a three-valued logic to deal
with future contingencies.
Emil Leon Post introduced the formulation of 푛 truth
values, where 푛 ≥ 2.
. . . . . .

31. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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 “At Christmas there will be snow in Athens,
Greece?”
Jan Łukasiewicz defined a three-valued logic to deal
with future contingencies.
Emil Leon Post introduced the formulation of 푛 truth
values, where 푛 ≥ 2.
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
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 “At Christmas there will be snow in Athens,
Greece?”
Jan Łukasiewicz defined a three-valued logic to deal
with future contingencies.
Emil Leon Post introduced the formulation of 푛 truth
values, where 푛 ≥ 2.
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
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
Categorical Model of Linear Logic
. . . . . .

34. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.
. . . . . .

35. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.
. . . . . .

36. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.
. . . . . .

37. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.
. . . . . .

38. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.
. . . . . .

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

40. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, 1, ⊸) is a lineale if

41. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, 1, ⊸) is a lineale if
(퐿, ≤) is a poset;

42. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, 1, ⊸) 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

43. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, 1, ⊸) 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.

44. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Lineales
A quintuple (퐿, ≤, ∘, 1, ⊸) 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 (I, ≤, ∧, 1, ⇒), where I is the unit interval,
푎 ∧ 푏 = {푎, 푏}, and 푎 ⇒ 푏 = ⋁{푐 ∶ 푐 ∧ 푎 ≤ 푏}
(푎 ∨ 푏 = {푎, 푏}), is a lineale.

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

46. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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 → I, where I = [0, 1].

47. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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 → I, where I = [0, 1].
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
id푈 ×푔-
푈 × 푋
≥
푓 × id푌
?
푉 × 푌
훽
- I

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

49. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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￸.

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

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

52. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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
I, takes the minimum of the membership
degrees.

53. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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
I, 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.

54. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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
I, 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) → I is the fuzzy relation that
is defined as follows
훾￴(푢, 푣), 푧￷ = 
훼(푢, 푥), if 푧 = (푥, 0)
훽(푣, 푦), if 푧 = (푦, 1)

55. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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
I, 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) → I is the fuzzy relation that
is defined as follows
훾￴(푢, 푣), 푧￷ = 
훼(푢, 푥), if 푧 = (푥, 0)
훽(푣, 푦), if 푧 = (푦, 1)
Coproduct 퐴 ⊕ 퐵 = (푈 + 푉, 푋 × 푌, 훿), where 훿 is defined
similar to 훾.

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

57. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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

58. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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 ⊧ ﾒ 푆 ⟺ 푢 ⊧ 푥 for all 푥 ∈ 푆.

59. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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 ⊧ ﾒ 푆 ⟺ 푢 ⊧ 푥 for all 푥 ∈ 푆.
when 푆 is any subset of 푋, then
푢 ⊧ ﾔ 푆 ⟺ 푢 ⊧ 푥 for some 푥 ∈ 푆.

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

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

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

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

64. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
Defining Fuzzy Topological Systems
A triple (푈, 푋, 훼), where 푋 is a frame, 푈 is a set, and
훼 ∶ 푈 × 푋 → I a binary fuzzy relation, is a fuzzy topological
system if
when 푆 is a finite subset of 푋, then
훼(푢, ﾒ 푆) ≤ 훼(푢, 푥) for all 푥 ∈ 푆.
when 푆 is any subset of 푋, then
훼(푢, ﾔ 푆) ≤ 훼(푢, 푥) for some 푥 ∈ 푆.
훼(푢, ⊤) = 1 and 훼(푢, ⊥) = 0 for all 푢 ∈ 푈.
. . . . . .

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

66. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.

67. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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, 푋, 휄), where
휄(푢, 푥) = 
1, when 푢 ⊧ 푥
0, otherwise

68. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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, 푋, 휄), where
휄(푢, 푥) = 
1, when 푢 ⊧ 푥
0, otherwise
The category of topological systems is a full
subcategory of 햣헂햺헅ﹴ(퐒퐞퐭).

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

70. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
Results cont.
Assume that 푎 ∈ 퐴, where (푈, 푋, 훼) 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.

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

72. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.

73. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.

74. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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

75. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.

76. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.

77. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
An alternative interpretation

78. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
An alternative interpretation
Bart Kosko has argued that fuzziness “measures the
degree to which an event occurs, not whether it occurs.
Randomness describes the uncertainty of event
occurrence.”

79. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
An alternative interpretation
Bart Kosko has argued that fuzziness “measures the
degree to which an event occurs, not whether it occurs.
Randomness describes the uncertainty of event
occurrence.”
According to this thesis, expression 훼(푢, 푥) would denote
the degree to which program 푢 will output a bitstream
that will satisfy assertion 푥.

80. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
An alternative interpretation
Bart Kosko has argued that fuzziness “measures the
degree to which an event occurs, not whether it occurs.
Randomness describes the uncertainty of event
occurrence.”
According to this thesis, expression 훼(푢, 푥) would denote
the degree to which program 푢 will output a bitstream
that will satisfy assertion 푥.
When 훼(푢, 푥) = 0, 푢 will not generate a bitstream that
satisfies 푥.

81. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
An alternative interpretation
Bart Kosko has argued that fuzziness “measures the
degree to which an event occurs, not whether it occurs.
Randomness describes the uncertainty of event
occurrence.”
According to this thesis, expression 훼(푢, 푥) would denote
the degree to which program 푢 will output a bitstream
that will satisfy assertion 푥.
When 훼(푢, 푥) = 0, 푢 will not generate a bitstream that
satisfies 푥.
When 훼(푢, 푥) = 1, 푢 will definitely generate a bistream
that satisfies 푥.

82. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
Dialectica
Spaces
Fuzzy
Topological
Spaces
Finale
. . . . . .
An alternative interpretation
Bart Kosko has argued that fuzziness “measures the
degree to which an event occurs, not whether it occurs.
Randomness describes the uncertainty of event
occurrence.”
According to this thesis, expression 훼(푢, 푥) would denote
the degree to which program 푢 will output a bitstream
that will satisfy assertion 푥.
When 훼(푢, 푥) = 0, 푢 will not generate a bitstream that
satisfies 푥.
When 훼(푢, 푥) = 1, 푢 will definitely generate a bistream
that satisfies 푥.
This new physical interpretation is based on the
assumption that the programs behave dynamically and
nondeterministic.

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

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

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

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

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

88. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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.

89. Fuzzy
Topological
Systems
Syropoulos,
de Paiva
Vagueness
General Ideas
Many-valued Logics
and Vagueness
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!