Application of Boolean pre-algebras to the foundations
of Computer Science
Marcelo Pereira Novaes
Departamento de Ciˆencia...
Outline
1 Motivation
2 Conclusion (Chpt. 7)
3 Mathematical Logic Background (Chpt. 2)
Classical Propositional Logic and Se...
Motivation
Context
Regarding Logic, need for a high expressiveness and generalization in
a logic system
Regarding Linguist...
Motivation
Central Research Problem
Example
ϕ := ”Salvador is the capital of Bahia”
ψ := ”The age of majority in Brazil is...
Motivation
Central Research Problem
Example
ϕ :=”In 2010, Salvador had a population of about 2,6 million”
ψ := ”In 2010, t...
Motivation
Graphic representation
Figure: Classical Logic
Marcelo Pereira Novaes (Universidade Federal da Bahia)Applicatio...
Motivation
Graphic representation
Figure: Modal Logic
Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of...
Motivation
Graphic representation
Figure: Truth-theory
Marcelo Pereira Novaes (Universidade Federal da Bahia)Application o...
Motivation
Graphic representation
Figure: Epistemic Logic
Marcelo Pereira Novaes (Universidade Federal da Bahia)Applicatio...
Motivation
Objective
Show the high expressiveness and some applications of Boolean
pre-algebra as a semantic representatio...
Conclusion (Chpt. 7)
Summary
The Boolean pre-algebras have an interesting role on Computer
Science foundations as a semant...
Conclusion
Future proposals
Construct some models for the Epistemic Logic K
Approach other logical systems such as Intuiti...
Setential Calculus
Language Syntax
Definition.
Let Fm be the set of formulas, it is the smallest set such as
(i) V ∪ { , ⊥}...
Setential Calculus
Deductive System
Axioms
A1. (ϕ → (ψ → χ)) → ((ϕ → ψ) → (ϕ → χ))
A2. ϕ → (ψ → ϕ)
A3. ¬ϕ → (ϕ → ψ)
A4. (ϕ...
Setential Calculus
Semantic
Definition. Propositional Domain
A propositional domain can be defined as M = (M, f , f⊥, f¬, f→...
Setential Calculus
Semantic
Definition. Model
Let a valuation v ∈ 2V , a formula ϕ ∈ Fm and a set of formulas Φ ∈ Fm
γ is a...
Setential Calculus with Identity (SCI)
Propositional Identity
We can see it as a generalization of the Classical Propositi...
Setential Calculus with Identity (SCI)
Language Syntax
Definition.
Let Fm be the set of formulas, it is the smallest set su...
Setential Calculus with Identity (SCI)
Deductive System - Original Approach
TFA’s: Truth Functional Axioms
A1. (ϕ → (ψ → χ...
Setential Calculus with Identity (SCI)
Deductive System - Alternative Approach
TFA’s: Truth Functional Axioms
A1. (ϕ → (ψ ...
Setential Calculus with Identity (SCI)
Semantic
Definition. Propositional Domain
A propositional domain can be defined as M ...
Setential Calculus with Identity (SCI)
Semantic
Definition. Valuation γ
A valuation γ : Fm(C) → M, such that:
γ( ) = Γ( ) =...
Setential Calculus with Identity (SCI)
Semantic
Definition. SCI-model
Let TRUE and FALSE sets such that M = TRUE ∪ FALSE an...
Setential Calculus with Identity (SCI)
Explicit and Implicit Models
Definition. Extension of a formula
An extension of a fo...
Equivalence between IDA and ALT axioms
Shown equivalence:
IDA
(ID1) ϕ ≡ ϕ
(ID2) ϕ ≡ ψ → ¬ϕ ≡ ¬ψ
(ID3) ϕ1 ≡ ψ1 → (ϕ2 ≡ ψ2 →...
Boolean pre-algebras (Chpt. 3)
Algebraic Semantic - Support definitions
George Boole and the beginning of the Algebraic rep...
Boolean pre-algebras (Chpt. 3)
Definition
Let the structure Ψ = (M, f , f⊥, f¬, f∨, f∧, f , ≤Ψ) which
1. M is the universe
...
Boolean pre-algebras (Chpt. 3)
Ultrafilters
Definition. A filter F with respect to ≤Ψ in a Boolean pre-algebra Ψ is a
non-emp...
Boolean pre-algebras (Chpt. 3)
Equivalence between SCI-models and Boolean pre-algebras
Let Ψ = (M, ≤Ψ, f⊥, f , f∧, f∨, f→,...
Boolean pre-algebras (Chpt. 3)
Equivalence between SCI-models and Boolean pre-algebras
Part I: Supporting Lemma
Let Ψ a Bo...
Modal Logic
Introduction
“modality is any word or phrase that can be applied to a given
statement S to create a new statem...
Modal LogicModal LogicPI in terms of SE - Deductive System Syntax:
Similar to Classical Propositional Logic, it introduces...
Modal Logic
PI in terms of SE - Semantic
A S3 model can be defined as the Boolean algebra:
Ψ = (M, f⊥, f , f¬, f∧, f∨, f ) ...
Modal Logic
PI and SE independents - Deductive System
Syntax: Similar to SCI, it introduces
Axioms: (i) TFA’s tautologies
...
Modal Logic
PI and SE independents - Semantic
Model
Ψ = (M, TRUE, NEC, f⊥, f , f , f¬, f∨, f∧, f→, f≡, Γ), where
1. M is a...
Modal Logic
Hyperintensions
Granularity Problem: (1) anti-symetric entailment; (2) handle the
existence of non-principal u...
Truth Theory
Motivation
In a highly expressive language, as the Natural Language, we have the
power to, for example:
1. Ma...
Truth Theory
Motivation
A language strong enough to express the self-reference, a truth-predicate
(true)and use of negatio...
Truth Theory
Study
Origins
Precursor, PhD. Thesis (Strater, 1992) at TU Berlin.
Later developed by (Zeitz, 2000)
Recently ...
Truth-theory
Syntax
Similar to SCI, it introduces T and F
Let V a set of variables, C a set of constants containing the us...
Truth-theory
Syntax
Some translations to the new syntax.
Example
“This statement is false.”⇒ c ≡ Fc
“This statement is tru...
Truth-theory
Deductive System
Axioms:
(i) TFA’s tautologies
(ii) IDA’s tautologies
(iii) Truth-predicate axioms:
(iii.1)ϕ ...
Truth-theory
Semantic
Propositional Domain
A T algebra (propositional domain) M = (M, f¬, f→, f≡, fT, fF ), where M
is the...
Logic with Quantifiers and Epistemic Logic
Epistemic Logic
Propositions are now objects of knowledge, if Ki ϕ is true, than...
Logic with Quantifiers and Epistemic Logic
Epistemic Logic
Logic with quantifiers
Similar to SCI, introduction of quantifier ...
Questions
Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of ...
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Application of Boolean pre-algebras to the foundations of Computer Science

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Senior thesis
Field: Mathematical Logic
Supervisor: Steffen Lewitzka
University: Universidade Federal da Bahia (UFBA)

Abstract:
"Increasing the expressiveness of a logical system is a goal of many fields in Computer Science such as Formal Systems, Knowledge construction, Linguistics, Universal Logic and Model Theory. The increasing of this expressiveness can be reached by the use of non-Fregean Logic, a non-classical logic. In non-Fregean Logic, formulas with the same truth value can have different denotations or meanings (also called situations). This concept breaks the Frege Axiom, the reason for the name non-Fregean Logic. Recently, it was shown that there is an equivalence between Boolean pre-algebras and non-Fregean logic models. This fact linked fields which were already using Boolean pre-algebras to represent their semantic models. In this thesis, an investigation on this equivalence is done and applications are exposed in the fields of Modal Logic, Truth Theory, Logic with Quantifiers and Epistemic Logic."

The full thesis can be found at http://repositorio.ufba.br/ri/handle/ri/1938

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Application of Boolean pre-algebras to the foundations of Computer Science

  1. 1. Application of Boolean pre-algebras to the foundations of Computer Science Marcelo Pereira Novaes Departamento de Ciˆencia da Computa¸c˜ao Universidade Federal da Bahia TCC - Salvador, 2016 Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 1 / 46
  2. 2. Outline 1 Motivation 2 Conclusion (Chpt. 7) 3 Mathematical Logic Background (Chpt. 2) Classical Propositional Logic and Setential Calculus with Identity Setential Calculus with Identity (SCI) 4 Boolean pre-algebras (Chpt. 3) Equivalence between SCI-models and Boolean pre-algebras 5 Applications (Chpt. 4,5 and 6) Modal Logic PI in terms of SE PI and SE independents Truth Theory Logic with Quantifiers and Epistemic Logic Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 2 / 46
  3. 3. Motivation Context Regarding Logic, need for a high expressiveness and generalization in a logic system Regarding Linguistics, attempts to model the Natural Language Boolean pre-algebras have been used already as semantic representation on Hyperintensional field (not the focus) Recent equivalence between Boolean pre-algebras and SCI-models (Non-Fregean Logic models) Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 3 / 46
  4. 4. Motivation Central Research Problem Example ϕ := ”Salvador is the capital of Bahia” ψ := ”The age of majority in Brazil is 18” In Classical Propositional Logic Both formulas represent the same proposition: True. Semantic can be represented as a two-element Boolean algebra In other words, a formula can denote {True,False} or {1,0}, etc... In Non-Fregean Logic ϕ and ψ could denote different propositions. ϕ and ψ would have the same truth-value (e.g. ϕ, ψ ∈ TRUE) Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 4 / 46
  5. 5. Motivation Central Research Problem Example ϕ :=”In 2010, Salvador had a population of about 2,6 million” ψ := ”In 2010, the capital of Bahia had a population of about 2,6 million” Example ϕ := ”I’m going with you and him” ψ := ”I’m going with him and you” Need to express intensions In non-Fregean Logic (ϕ ↔ ψ) → (ϕ ≡ ψ) is not valid (Frege Axiom). (ϕ ≡ ψ) → (ϕ ↔ ψ) is valid. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 5 / 46
  6. 6. Motivation Graphic representation Figure: Classical Logic Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 6 / 46
  7. 7. Motivation Graphic representation Figure: Modal Logic Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 7 / 46
  8. 8. Motivation Graphic representation Figure: Truth-theory Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 8 / 46
  9. 9. Motivation Graphic representation Figure: Epistemic Logic Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 9 / 46
  10. 10. Motivation Objective Show the high expressiveness and some applications of Boolean pre-algebra as a semantic representation. Specifically, Detail some results in the literature: Equivalence between Boolean pre-algebras and SCI-models SCI-completeness Show applications in Modal Logic, Semantically-closed Logic, Logic with Quantifiers and Epistemic Logic. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 10 / 46
  11. 11. Conclusion (Chpt. 7) Summary The Boolean pre-algebras have an interesting role on Computer Science foundations as a semantic representation to many logical systems. Motivation and a consistent definition were provided There are some important applications for Boolean pre-algebras such as on Modal Logic, Truth-theory, Logic with Quantifiers and Epistemic Logic. Contributions were done: they were detailed SCI completeness, ALTs-IDs equivalence, SCI-model - pre-Boolean algebra equivalence and it was constructed a true truth-teller model. Outlook Some models were introduced as SCI-model notation, even though they are equivalent, it should be in Boolean pre-algebras. Did not discussed about drawbacks of structured propositions and non-Fregean Logic semantic arguments such as the “Slingshot argument”. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 11 / 46
  12. 12. Conclusion Future proposals Construct some models for the Epistemic Logic K Approach other logical systems such as Intuitionistic Logic, which semantic is designed as Heyting Algebras. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 12 / 46
  13. 13. Setential Calculus Language Syntax Definition. Let Fm be the set of formulas, it is the smallest set such as (i) V ∪ { , ⊥} ⊆ Fm (ii) If ϕ, ψ ∈ Fm, so ¬ϕ, (ϕ → ψ), (ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ ↔ ψ) ∈ Fm. Being the connective basis {¬, →} : and the following abbreviations: := x0 → x0; ⊥ := ¬ ; ϕ ∧ ψ := ¬(ϕ → ¬ψ); ϕ ∨ ψ := ¬ϕ → ψ; ϕ ↔ ψ := (ψ → ψ) ∧ (ψ → ϕ) (iii) The precedence of operators follow the order: ¬, ∨ and ∧, →, ↔, where → is right-associated while the others are left-associated, for ↔ it can be any of them. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 13 / 46
  14. 14. Setential Calculus Deductive System Axioms A1. (ϕ → (ψ → χ)) → ((ϕ → ψ) → (ϕ → χ)) A2. ϕ → (ψ → ϕ) A3. ¬ϕ → (ϕ → ψ) A4. (ϕ → ψ) → (¬ϕ → ψ) → ψ Inference Rules (MP) Modus Ponens Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 14 / 46
  15. 15. Setential Calculus Semantic Definition. Propositional Domain A propositional domain can be defined as M = (M, f , f⊥, f¬, f→, Γ), where M = {f , f⊥} is the universe of propositions and f¬, f→, operations with arity: 1, 2, 2 respectively. We can see f and f⊥ also as operations with arity 0. Definition. Valuation Γ A valuation Γ : C → M is a function which satisfies: Γ(⊥) = f⊥; Γ( ) = f Definition. Assigment γ An assignment is a function γ : V → M such that for all ϕ, ψ ∈ V , holds: f¬(γ(ϕ)); γ(ϕ → ψ) = f→(γ(ϕ), γ(ψ)); Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 15 / 46
  16. 16. Setential Calculus Semantic Definition. Model Let a valuation v ∈ 2V , a formula ϕ ∈ Fm and a set of formulas Φ ∈ Fm γ is a model of ϕ (γ satisfies ϕ), notation γ |= ϕ, if γ(ϕ) = 1 [(M, γ) |= ϕ]. γ is a model of Φ (γ satisfies Φ), notation γ |= ψ, if γ |= ψ, for every ψ ∈ Φ. Definition. Logic Consequence Φ ϕ : ⇐⇒ Mod(Φ) ⊆ Mod({ϕ}), where for any set Ψ, Mod(Ψ) := {v ∈ 2V |v |= Ψ} Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 16 / 46
  17. 17. Setential Calculus with Identity (SCI) Propositional Identity We can see it as a generalization of the Classical Propositional Logic New operator ≡, called Propositional Identity (PI). ϕ ≡ ψ means ϕ and ψ have the same denotation (also called “situation”). It is defined as ϕ ≡ ψ : ⇐⇒ ϕ = ψ. For logic with quantifiers, the definition is extended to ϕ ≡ ψ ⇐⇒ ϕ =α ψ, where α is the so called α-congruence. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 17 / 46
  18. 18. Setential Calculus with Identity (SCI) Language Syntax Definition. Let Fm be the set of formulas, it is the smallest set such as (i) V ∪ { , ⊥} ⊆ Fm (ii) If ϕ, ψ ∈ Fm, so ¬ϕ, (ϕ → ψ), (ϕ ≡ ψ), (ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ ↔ ψ) ∈ Fm. Being the connective basis {¬, →, ≡} : and the following abbreviations: := x0 → x0; ⊥ := ¬ ; ϕ ∧ ψ := ¬(ϕ → ¬ψ); ϕ ∨ ψ := ¬ϕ → ψ; ϕ ↔ ψ := (ψ → ψ) ∧ (ψ → ϕ) (iii) The precedence of operators follow the order: ¬, ∨ and ∧, →, ↔, where → is right-associated while the others are left-associated, for ↔ it can be any of them. Introducing {∧, ∨, ↔} as new operators, we extend the propositional domain. The greater will be the granularity of the propositions. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 18 / 46
  19. 19. Setential Calculus with Identity (SCI) Deductive System - Original Approach TFA’s: Truth Functional Axioms A1. (ϕ → (ψ → χ)) → ((ϕ → ψ) → (ϕ → χ)) A2. ϕ → (ψ → ϕ) A3. ¬ϕ → (ϕ → ψ) A4. (ϕ → ψ) → (¬ϕ → ψ) → ψ IDA’s: Identity Axioms (ID1) ϕ ≡ ϕ (ID2) ϕ ≡ ψ → ¬ϕ ≡ ¬ψ (ID3) ϕ1 ≡ ψ1 → (ϕ2 ≡ ψ2 → (ϕ1 → ϕ2) ≡ (ψ1 → ψ2)) (ID4) ϕ1 ≡ ψ1 → (ϕ2 ≡ ψ2 → (ϕ1 ≡ ϕ2) ≡ (ψ1 ≡ ψ2)) Inference Rules (MP) Modus Ponens Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 19 / 46
  20. 20. Setential Calculus with Identity (SCI) Deductive System - Alternative Approach TFA’s: Truth Functional Axioms A1. (ϕ → (ψ → χ)) → ((ϕ → ψ) → (ϕ → χ)) A2. ϕ → (ψ → ϕ) A3. ¬ϕ → (ϕ → ψ) A4. (ϕ → ψ) → (¬ϕ → ψ) → ψ ALT’s: Alternative Axioms for IDA’s (ALT1) ϕ ≡ ϕ, (ALT2) (ϕ ≡ ψ) → (ϕ → ψ), leads to (ϕ ≡ ψ) → (ϕ ↔ ψ) (ALT3) (ψ ≡ ψ ) → (ϕ[x := ψ] ≡ ϕ[x := ψ ]) Inference Rules (MP) Modus Ponens Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 20 / 46
  21. 21. Setential Calculus with Identity (SCI) Semantic Definition. Propositional Domain A propositional domain can be defined as M = (M, f⊥, f , f¬, f→, f≡, Γ), where M is the universe of propositions and f⊥, f , f¬, f→, f≡, operations with arity: 0, 0, 1, 2, 2 respectively. Definition. Valuation Γ A valuation Γ : C → M is a function which satisfies: Γ(c) = fc; Γ(⊥) = f⊥; Γ( ) = f ; where fc is the proposition referent to the constant c. Definition. Assigment γ An assignment is a function γ : V → M such that for all ϕ, ψ ∈ V , holds: γ(x) = fx , being fx a proposition on M correspondent to the variable x; f¬(γ(ϕ)); γ(ϕ → ψ) = f→(γ(ϕ), γ(ψ)); γ(ϕ ≡ ψ) = f≡(γ(ϕ), γ(ψ)) Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 21 / 46
  22. 22. Setential Calculus with Identity (SCI) Semantic Definition. Valuation γ A valuation γ : Fm(C) → M, such that: γ( ) = Γ( ) = f ; γ(⊥) = Γ(⊥) = f⊥; γ(c) = Γ(c) and for the variables, it follows the previous definition. Definition. Proposition A proposition is a denotation γ(ϕ) ∈ M of a formula ϕ under valuation γ. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 22 / 46
  23. 23. Setential Calculus with Identity (SCI) Semantic Definition. SCI-model Let TRUE and FALSE sets such that M = TRUE ∪ FALSE and TRUE ∩ FALSE = ∅. Then, M = (M, f⊥, f , f¬, f→, f≡, Γ) is a SCI-model if it satisfies: (i): f ∈ TRUE, f⊥ ∈ FALSE (ii): f¬(a) ∈ TRUE ⇐⇒ a ∈ FALSE (iii): f→(a, b) ∈ TRUE ⇐⇒ a ∈ FALSE or b ∈ TRUE (iv): f≡(a, b) ∈ TRUE ⇐⇒ a = b Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 23 / 46
  24. 24. Setential Calculus with Identity (SCI) Explicit and Implicit Models Definition. Extension of a formula An extension of a formula is the equivalence class ϕ = {ϕ ≡ ψ}. Definition. Intension of a formula An intension of a formula is its syntax representation. Result. SCI-model classification SCI-models can be one of two types: Intensional or Extensional. On extensional models, (M, γ) |= ϕ ≡ ψ ⇐⇒ ϕ ↔ ψ On intensional models, (M, γ) |= ϕ ≡ ψ ⇐⇒ ϕ = ψ. On Intensional models, extension and intension of a sentence can be put always in a 1-1 relationship. On Extensional models there is no always 1-1 relationship. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 24 / 46
  25. 25. Equivalence between IDA and ALT axioms Shown equivalence: IDA (ID1) ϕ ≡ ϕ (ID2) ϕ ≡ ψ → ¬ϕ ≡ ¬ψ (ID3) ϕ1 ≡ ψ1 → (ϕ2 ≡ ψ2 → (ϕ1 → ϕ2) ≡ (ψ1 → ψ2)) (ID4) ϕ1 ≡ ψ1 → (ϕ2 ≡ ψ2 → (ϕ1 ≡ ϕ2) ≡ (ψ1 ≡ ψ2)) ALT (ALT1) ϕ ≡ ϕ, (ALT2) (ϕ ≡ ψ) → (ϕ → ψ) (ALT3) (ψ ≡ ψ ) → (ϕ[x := ψ] ≡ ϕ[x := ψ ]) Shown Strong Completeness: Correctness: Φ ϕ ⇒ Φ ϕ Completeness: Φ ϕ ⇒ Φ ϕ. Factorization by a ≈ b ⇐⇒ f≡(a, b) ∈ TRUE Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 25 / 46
  26. 26. Boolean pre-algebras (Chpt. 3) Algebraic Semantic - Support definitions George Boole and the beginning of the Algebraic representation Structured propositions vs Relational semantic Definition. A pre-order ≤ is a binary relation defined over a set which for any a,b,c elements, it is valid: a ≤ a (Reflexivity) If a ≤ b and b ≤ c, then a ≤ c (Transitivity) Definition. A pre-ordered set is the pair (S, ≤), where S is a set and ≤ a preorder over it. Definition. A partially ordered set is the pair (S, ≤), where S is a set and ≤ a partial order (antisymmetric pre-order). An antisymmetric relation on a set can be defined as for all a,b elements: a ≤ b and b ≤ a ⇒ a = b. Definition. A lattice is a partially-ordered set which each two elements have a unique supremum ( ) and a unique infimum (⊥). Definition. A Boolean lattice (Boolean algebra) is a distributive complemented lattice. (Satisfies a ≤ b ⇐⇒ f→(a, b) = f ) Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 26 / 46
  27. 27. Boolean pre-algebras (Chpt. 3) Definition Let the structure Ψ = (M, f , f⊥, f¬, f∨, f∧, f , ≤Ψ) which 1. M is the universe 2. f , f⊥, f¬, f∨, f∧, f→ are functions on M → M (operations on M) with arity 0,0,1,2,2,2 respectively. 3. ≤Ψ on M is a pre-order. Definition. It is a Boolean pre-algebra if: a ≈Ψ b : ⇐⇒ a ≤Ψ b and b ≤Ψ a is a congruence relation on M and the quotient algebra of Ψ/≈Ψ = (M/≈Ψ , f , f ⊥, f ¬, f ∨, f ∧, f , ≤M/≈Ψ ) is a Boolean algebra with lattice order ≤Ψ/≈Ψ such that: a ≤Ψ/≈Ψ b ⇐⇒ a ≤Ψ b and f , f ⊥, f ¬, f ∨, f ∧, f are induced operations for supremum, infimum elements, complement and implication respectively on the set of congruence classes ∈ Ψ/M . Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 27 / 46
  28. 28. Boolean pre-algebras (Chpt. 3) Ultrafilters Definition. A filter F with respect to ≤Ψ in a Boolean pre-algebra Ψ is a non-empty subset F ⊆ M, s.t. for all a,b ∈ M the filter axioms holds: (i) if a ∈ F and a ≤Ψ b, then b ∈ F (ii) if a,b ∈ F, then f∧(a, b) ∈ F (iii) f ⊥ /∈ F An ultrafilter w.r.t. ≤Ψ is a maximal filter w.r.t. ≤Ψ. In this context, it can be also called prime filter. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 28 / 46
  29. 29. Boolean pre-algebras (Chpt. 3) Equivalence between SCI-models and Boolean pre-algebras Let Ψ = (M, ≤Ψ, f⊥, f , f∧, f∨, f→, f→) a Boolean pre-lattice with a preordered set (≤Ψ, M) and induced operations infimum (f⊥), supremum (f ), complement (f¬), join (f∨), meet (f∧) and implication (f→). M = (M, TRUE, f⊥, f , f∧, f∨, f→, f≡) a SCI-model. We want to show that Ψ and M are equivalent. It is done in two parts. First, proving a supporting lemma, then proving the desired statement. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 29 / 46
  30. 30. Boolean pre-algebras (Chpt. 3) Equivalence between SCI-models and Boolean pre-algebras Part I: Supporting Lemma Let Ψ a Boolean pre-lattice with preorder (so it will be valid for partial and total orders) ≤Ψ and F a filter w.r.t. ≤Ψ. The following conditions are equivalents: (i) a ≤Ψ b ⇐⇒ f→(a, b) ∈ F, for all a, b ∈ M (ii) F = {a ∈ M| a ≈Ψ f } (iii) F is the smallest filter. It is the intersection of all (ultra)filters with regard to ≤Ψ. Part II: Equivalence Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 30 / 46
  31. 31. Modal Logic Introduction “modality is any word or phrase that can be applied to a given statement S to create a new statement that makes an assertion about the mode of the truth of S: about when, where or how S is true, or about the circumstances under which S may be true.” Two most common modalities: “possibility” (♦) and “necessity” ( ) Boolean (pre)algebras, (Hyper)intensions and Possible Worlds semantics Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 31 / 46
  32. 32. Modal LogicModal LogicPI in terms of SE - Deductive System Syntax: Similar to Classical Propositional Logic, it introduces Definition: ϕ ≡ ψ := (ψ ↔ ψ) Axioms: (i) TFA’s tautologies Lewis modal logic S3: (ii) ϕ → ϕ) (iii) (ϕ → ψ) → ( ϕ → ψ) (iv) (ϕ → ψ) → ( ϕ → ψ) Inference Rules: Modus Ponens and Axiom of Necessitation Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 32 / 46
  33. 33. Modal Logic PI in terms of SE - Semantic A S3 model can be defined as the Boolean algebra: Ψ = (M, f⊥, f , f¬, f∧, f∨, f ) where (M,≤) is a preordered set, M is a non-empty set of prepositions and ≤ is the lattice order, operations complement (f¬), meet (f∧), join (f∨), bound elements infimum (f ) and supremum (f⊥) and with a ultrafilter TRUE ⊆ M with regard to ≤ (the set of true propositions) and induced unary operation f , satisfying mainly for all a,b,c ∈ M: (i) f ∈ TRUE, f⊥ /∈ TRUE (ii)f¬(a) ∈ TRUE ⇐⇒ a /∈ TRUE (iii)f→(a, b) ∈ TRUE ⇐⇒ a /∈ TRUE or b ∈ TRUE. It also satisfy conditions for the other operators (iv) f (a) ∈ TRUE ⇐⇒ a = f (v) f (a) ≤ a (iv) f (f→(a, b)) ≤ f→(f (a), f (b)) (vi) f (f→(a, b)) ≤ f (f→(f (a), f (b))) It also satisfy Boolean algebra properties such as f→(a, b) = f . Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 33 / 46
  34. 34. Modal Logic PI and SE independents - Deductive System Syntax: Similar to SCI, it introduces Axioms: (i) TFA’s tautologies Lewis modal logicS3: (ii) ϕ → ϕ) (iii) (ϕ → ψ) → ( ϕ → ψ) (iv) (ϕ → ψ) → ( ϕ → ψ) (v) IDA’s tautologies Inference Rules: Modus Ponens and Axiom of Necessitation (AN application limited to (i)-(iv)). Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 34 / 46
  35. 35. Modal Logic PI and SE independents - Semantic Model Ψ = (M, TRUE, NEC, f⊥, f , f , f¬, f∨, f∧, f→, f≡, Γ), where 1. M is a non-empty set of propositions. 2. TRUE ⊆ M is a set of true propositions 3. NEC ⊆ set of necessary propositions 4. f⊥, f , f , f¬, f∨, f∧, f→, f≡, functions with arity (operations of type) 0,0,1,1,2,2,2,2 respectively. 5. Γ : C → M is a function Gamma, satisfying Γ( ) = f , Γ(⊥) = f⊥. It will satisfy also semantic conditions on SCI A model can be seen as a Boolean pre-algebra. (PI refines SE) Lemma. (ϕ ≡ ψ) → (ϕ ↔ ψ) Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 35 / 46
  36. 36. Modal Logic Hyperintensions Granularity Problem: (1) anti-symetric entailment; (2) handle the existence of non-principal ultrafilters. (1) problem: makes Frege Axiom true, reducing values of denotation of a formula to its truth-value. (2) problem: makes sentences that follow from each other have the same meaning. (1) solution: consider a pre-algebra. (2) solution: change approach from (Kripke, 1963) back to (Kripke, 1959), called Soft Actualism: primitive worlds and constructed propositions to primitive propositions and constructed worlds (ultrafilters). By (1) + (2) solutions it is constructed a high-order logic with subtypes that solve the granularity problem. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 36 / 46
  37. 37. Truth Theory Motivation In a highly expressive language, as the Natural Language, we have the power to, for example: 1. Make references about the truth of propositions (or statements) Example What Foo meant to say is not true. (That Foo’s statement is not true) 2. Make inferences about the own statement Example What Foo stated is exactly the opposite of what Bar stated. (Foo’s statement is the opposite of the Bar’s statement) 3. Make statements as a composition of them Example I’m saying the truth right now. (informal Truth-teller) Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 37 / 46
  38. 38. Truth Theory Motivation A language strong enough to express the self-reference, a truth-predicate (true)and use of negation, can lead us to contradictions. Example I’m lying right now. (Liar Paradox) Two main ways to approach them: Do not permit self-reference and truth-predicates Handle paradoxes Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 38 / 46
  39. 39. Truth Theory Study Origins Precursor, PhD. Thesis (Strater, 1992) at TU Berlin. Later developed by (Zeitz, 2000) Recently developed by (Lewitzka, 2012) Relationship between Liar and Theory of Computation results Godel First Incompleteness Theorem Tarski’s Undefinability Theorem An elegant way to handle paradoxes. No sentence can denote a the liar statement. The liar does not exist. We can express equations that cannot be satisfied by any model i2 = −1, in a more formal way: (i ∈ R s.t. f∗(i, i) = f−(0, 1)) The Liar: c ≡ Fc. There is no model which satisfies this equation. Suppose there exist a model which satisfies it. By Fc constructive definition for a model (we will see it next), c ∈ TRUE =⇒ Fc ∈ FALSE. In the same way c ∈ FALSE, Fc ∈ TRUE. By definition TRUE ∩ FALSE = ∅. Contradiction. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 39 / 46
  40. 40. Truth-theory Syntax Similar to SCI, it introduces T and F Let V a set of variables, C a set of constants containing the usual and ⊥. Then, let Fm(C)’ be the smallest set of formulas with V,C and closed on: if ϕ and ψ are expressions, then (Tϕ), (Fϕ), (ϕ → ψ),(¬ϕ), (ϕ ≡ ψ). The usual abbreviations hold. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 40 / 46
  41. 41. Truth-theory Syntax Some translations to the new syntax. Example “This statement is false.”⇒ c ≡ Fc “This statement is true.” ⇒ c ≡ Tc “The next statement is true”. ⇒ (1) c ≡ Td (2) d ≡ ϕ. “This previous statement is not true.” ⇒ c ≡ ϕ d ≡ Tc. “This statement is false or ϕ is true.” ⇒ c ≡ (Fc ∨ Tϕ). Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 41 / 46
  42. 42. Truth-theory Deductive System Axioms: (i) TFA’s tautologies (ii) IDA’s tautologies (iii) Truth-predicate axioms: (iii.1)ϕ → (Tϕ) and (Tϕ) → ϕ (iii.2)¬ϕ → (Fϕ) and (Fϕ) → ϕ Inference Rules: Modus Ponens. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 42 / 46
  43. 43. Truth-theory Semantic Propositional Domain A T algebra (propositional domain) M = (M, f¬, f→, f≡, fT, fF ), where M is the universe of propositions and f¬, f→, f≡, fT, fF , operations with arity: 1, 2, 2, 1, 1 respectively. Model Definition. Let TRUE and FALSE sets such that M = TRUE ∪ FALSE. Then, M = (M, TRUE, f¬, f→, f≡, fT, fF , FALSE, Γ) is a model if it satisfies the SCI-model truth-conditions plus: (i): fT(a) ∈ TRUE ⇐⇒ a ∈ TRUE (ii): fF (a) ∈ TRUE ⇐⇒ a ∈ FALSE A model for c ≡ Tc, where c ∈ TRUE was constructed. Γ now also maps Γ(c) = fc γ also maps γ(Tϕ) = fT(γ(ϕ)); γ(Fϕ) = fF (γ(ϕ)) Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 43 / 46
  44. 44. Logic with Quantifiers and Epistemic Logic Epistemic Logic Propositions are now objects of knowledge, if Ki ϕ is true, than ϕ denotes a true proposition (statement) that is known by the agent i. TRUE is the set of facts TRUEi is the set of facts known by the agent i. Let KNOWN = ∪TRUEi , for all i agents. It is the set of all the facts known. KNOWN ⊆ TRUE and usually expected: KNOWN ⊂ TRUE. It is introduced the concept of a group of agents. There are facts known by a group of agents. Every agent in the group knows the group facts. Furthermore, every agent knows that the others agents know it, and so on. It is introduced the notion of Common Knowledge, a common knowledge among all the agents that satisfies some closure properties. Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 44 / 46
  45. 45. Logic with Quantifiers and Epistemic Logic Epistemic Logic Logic with quantifiers Similar to SCI, introduction of quantifier operators, e.g. ∀ Introduction of axioms regarding these new operators in the deductive system Semantically, they are mapped to high order functions, e.g. ∀ is mapped to f∀ : MM → M Examples On Epistemic Logic: ∀x.(Ki x → x) On Truth-theory: ∀x(Tx ↔ x) Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 45 / 46
  46. 46. Questions Marcelo Pereira Novaes (Universidade Federal da Bahia)Application of Boolean pre-algebras to the foundations of Computer ScienceTCC - Salvador, 2016 46 / 46

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