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Fuzzy Topological Systems

Apostolos Syropoulos, profile picture
Apostolos Syropoulos

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."

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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
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
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . What is Vagueness?
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.
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?
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.
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.
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
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?
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?
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?
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?
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.
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.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . Defining Vagueness
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.
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.
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.
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.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale Vagueness, Generality, and Ambiguity . . . . . .
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. . . . . . .
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. . . . . . .
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. . . . . . .
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.” . . . . . .
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. . . . . . .
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 . . . . . .
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?). . . . . . .
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?” . . . . . .
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. . . . . . .
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. . . . . . .
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. . . . . . .
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. . . . . . .
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 . . . . . .
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. . . . . . .
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. . . . . . .
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. . . . . . .
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. . . . . . .
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. . . . . . .
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . Lineales
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
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;
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
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.
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.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . The Category 햣헂햺헅﹚(퐒퐞퐭)
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].
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
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.
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￸.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . Logical Connectives
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)
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.
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.
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)
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.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . Topological Systems
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
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.
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.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale Defining Fuzzy Topological Systems . . . . . .
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 . . . . . .
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. . . . . . .
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. . . . . . .
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. . . . . . .
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . Results
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.
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
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).
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . Results cont.
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.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . Let’s be practical!
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.
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 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
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.
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.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . An alternative interpretation
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.”
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.
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.
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.
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.
Fuzzy Topological Systems Syropoulos, de Paiva Vagueness General Ideas Many-valued Logics and Vagueness Dialectica Spaces Fuzzy Topological Spaces Finale . . . . . . Finale!
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
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.
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 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.
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.
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!

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Fuzzy Topological Systems

  • 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!