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

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

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

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)

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)

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.

Motivation
Graphic representation
Figure: Classical Logic

Motivation
Figure: Modal Logic

Motivation
Figure: Truth-theory

Motivation
Figure: Epistemic Logic

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.

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”.

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.

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.

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

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));

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}

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.

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.

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

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

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))

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.

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

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.

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

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 )

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
.

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.

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.

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

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

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

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 .

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)).

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)

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.

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)

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

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.

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.

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).

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.

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))

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.

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)

Questions

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