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

Application of Boolean pre-algebras to the foundationsof Computer Science

Marcelo Pereira Novaes

Departamento de Ciencia da ComputacaoUniversidade 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 IdentitySetential 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 SEPI and SE independents

Truth TheoryLogic with Quantifiers and Epistemic Logic


MotivationContext

Regarding Logic, need for a high expressiveness and generalization ina logic system

Regarding Linguistics, attempts to model the Natural Language

Boolean pre-algebras have been used already as semanticrepresentation on Hyperintensional field (not the focus)

Recent equivalence between Boolean pre-algebras and SCI-models(Non-Fregean Logic models)


MotivationCentral 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 algebraIn 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 )


MotivationCentral 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.


MotivationGraphic representation

Figure: Classical Logic



Figure: Modal Logic



Figure: Truth-theory



Figure: Epistemic Logic


MotivationObjective

Show the high expressiveness and some applications of Booleanpre-algebra as a semantic representation.

Specifically,

Detail some results in the literature:

Equivalence between Boolean pre-algebras and SCI-modelsSCI-completeness

Show applications in Modal Logic, Semantically-closed Logic, Logicwith Quantifiers and Epistemic Logic.


Conclusion (Chpt. 7)Summary

The Boolean pre-algebras have an interesting role on ComputerScience foundations as a semantic representation to many logicalsystems.Motivation and a consistent definition were providedThere are some important applications for Boolean pre-algebras suchas on Modal Logic, Truth-theory, Logic with Quantifiers andEpistemic Logic.Contributions were done: they were detailed SCI completeness,ALTs-IDs equivalence, SCI-model - pre-Boolean algebra equivalenceand it was constructed a true truth-teller model.

OutlookSome models were introduced as SCI-model notation, even though theyare equivalent, it should be in Boolean pre-algebras.Did not discussed about drawbacks of structured propositions andnon-Fregean Logic semantic arguments such as the “Slingshotargument”.


ConclusionFuture proposals

Construct some models for the Epistemic Logic ε′KApproach other logical systems such as Intuitionistic Logic, whichsemantic is designed as Heyting Algebras.


Setential CalculusLanguage 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. Beingthe 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 ↔ itcan be any of them.


Setential CalculusDeductive 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 CalculusSemantic

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→, operationswith arity: 1, 2, 2 respectively. We can see f> and f⊥ also as operationswith arity 0.

Definition. Valuation Γ

A valuation Γ : C 7→ M is a function which satisfies: Γ(⊥) = f⊥; Γ(>) = f>

Definition. Assigment γ

An assignment is a function γ : V 7→ M such that for all ϕ,ψ ∈ V , holds:f¬(γ(ϕ)); γ(ϕ→ ψ) = f→(γ(ϕ), γ(ψ));


Setential CalculusSemantic

Definition. Model

Let a valuation v ∈ 2V , a formula ϕ ∈ Fm and a set of formulas Φ ∈ Fmγ is a model of ϕ (γ satisfies ϕ), notation γ |= ϕ, ifγ(ϕ) = 1 [(M, γ) |= ϕ].γ is a model of Φ (γ satisfies Φ), notation γ |= ψ, if γ |= ψ, for everyψ ∈ Φ.

Definition. Logic Consequence

Φ ϕ :⇐⇒ Mod(Φ) ⊆ Mod({ϕ}), where for any set Ψ,Mod(Ψ) := {v ∈ 2V |v |= Ψ}


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, thedefinition is extended to ϕ ≡ ψ ⇐⇒ ϕ =α ψ, where α is the socalled α-congruence.


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 followingabbreviations: > := 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 ↔ itcan be any of them.

Introducing {∧,∨,↔} as new operators, we extend the propositionaldomain. The greater will be the granularity of the propositions.


Setential Calculus with Identity (SCI)Deductive System - Original Approach

TFA’s: Truth Functional Axioms


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


Setential Calculus with Identity (SCI)Deductive System - Alternative Approach

TFA’s: Truth Functional Axioms


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


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≡, operationswith arity: 0, 0, 1, 2, 2 respectively.

Definition. Valuation Γ

A valuation Γ : C 7→ M is a function which satisfies:Γ(c) = fc ; Γ(⊥) = f⊥; Γ(>) = f>; where fc is the proposition referent tothe constant c.

Definition. Assigment γ

An assignment is a function γ : V 7→ 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))



Definition. Valuation γ

A valuation γ : Fm(C )′ 7→ 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.



Definition. SCI-model

Let TRUE and FALSE sets such that M = TRUE ∪ FALSE andTRUE ∩ FALSE = ∅. Then, M = (M, f⊥, f>, f¬, f→, f≡, Γ) is a SCI-modelif 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


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↔ ψOn intensional models, (M, γ) |= ϕ ≡ ψ ⇐⇒ ϕ = ψ.

On Intensional models, extension and intension of a sentence can be putalways in a 1-1 relationship. On Extensional models there is no always 1-1relationship.


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 0 ϕ⇒ Φ 1 ϕ

Completeness: Φ 1 ϕ⇒ Φ 0 ϕ. Factorization bya ≈ 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 forany 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 ≤ apreorder 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 ona 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 elementshave a unique supremum (>) and a unique infimum (⊥).Definition. A Boolean lattice (Boolean algebra) is a distributivecomplemented lattice. (Satisfies a ≤ b ⇐⇒ f→(a, b) = f>)


Boolean pre-algebras (Chpt. 3)Definition

Let the structure Ψ = (M, f>, f⊥, f¬, f∨, f∧, f>,≤Ψ) which1. M is the universe2. f>, f⊥, f¬, f∨, f∧, f→ are functions on M → M (operations on M) witharity 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 thequotient algebra of Ψ/≈Ψ

= (M/≈Ψ, f >, f ⊥, f ¬, f ∨, f ∧, f >,≤M/≈Ψ

) is aBoolean algebra with lattice order ≤Ψ/≈Ψ

such that:

a ≤Ψ/≈Ψb ⇐⇒ a ≤Ψ b and f >, f ⊥, f ¬, f ∨, f ∧, f > are induced operations

for supremum, infimum elements, complement and implication respectivelyon the set of congruence classes ∈ Ψ/M .


Boolean pre-algebras (Chpt. 3)Ultrafilters

Definition. A filter F with respect to ≤Ψ in a Boolean pre-algebra Ψ is anon-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⊥ /∈ FAn ultrafilter w.r.t. ≤Ψ is a maximal filter w.r.t. ≤Ψ. In this context, itcan be also called prime filter.


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 apreordered 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.


Boolean pre-algebras (Chpt. 3)Equivalence between SCI-models and Boolean pre-algebras

Part I: Supporting LemmaLet Ψ a Boolean pre-lattice with preorder (so it will be valid forpartial and total orders) ≤Ψ and F a filter w.r.t. ≤Ψ. The followingconditions 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)filterswith regard to ≤Ψ.

Part II: Equivalence


Modal LogicIntroduction

“modality is any word or phrase that can be applied to a givenstatement S to create a new statement that makes an assertion aboutthe mode of the truth of S: about when, where or how S is true, orabout the circumstances under which S may be true.”

Two most common modalities: “possibility” (♦) and “necessity” (�)

Boolean (pre)algebras, (Hyper)intensions and Possible Worldssemantics


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 tautologiesLewis 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 LogicPI 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 anon-empty set of prepositions and ≤ is the lattice order, operationscomplement (f¬), meet (f∧), join (f∨), bound elements infimum (f>) andsupremum (f⊥) and with a ultrafilter TRUE ⊆ M with regard to ≤ (theset of true propositions) and induced unary operation f�, satisfying mainlyfor 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 satisfyconditions 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 LogicPI and SE independents - Deductive System

Syntax: Similar to SCI, it introduces �Axioms: (i) TFA’s tautologiesLewis modal logicS3:(ii) �ϕ→ ϕ)(iii) �(ϕ→ ψ)→ (�ϕ→ �ψ)(iv) �(ϕ→ ψ)→ �(�ϕ→ �ψ)(v) IDA’s tautologiesInference Rules: Modus Ponens and Axiom of Necessitation (ANapplication limited to (i)-(iv)).


Modal LogicPI and SE independents - Semantic

Model

Ψ = (M,TRUE ,NEC , f⊥, f>, f�, f¬, f∨, f∧, f→, f≡, Γ), where1. M is a non-empty set of propositions.2. TRUE ⊆ M is a set of true propositions3. NEC ⊆ set of necessary propositions4. 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 LogicHyperintensions

Granularity Problem: (1) anti-symetric entailment; (2) handle theexistence of non-principal ultrafilters.(1) problem: makes Frege Axiom true, reducing values of denotation of aformula to its truth-value.(2) problem: makes sentences that follow from each other have the samemeaning.(1) solution: consider a pre-algebra. (2) solution: change approach from(Kripke, 1963) back to (Kripke, 1959), called Soft Actualism: primitiveworlds and constructed propositions to primitive propositions andconstructed worlds (ultrafilters).By (1) + (2) solutions it is constructed a high-order logic with subtypesthat solve the granularity problem.


Truth TheoryMotivation

In a highly expressive language, as the Natural Language, we have thepower 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’sstatement 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 TheoryMotivation

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 TheoryStudy

OriginsPrecursor, PhD. Thesis (Strater, 1992) at TU Berlin.Later developed by (Zeitz, 2000)Recently developed by (Lewitzka, 2012)

Relationship between Liar and Theory of Computation resultsGodel First Incompleteness TheoremTarski’s Undefinability Theorem

An elegant way to handle paradoxes. No sentence can denote a theliar 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-theorySyntax

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 closedon: if ϕ and ψ are expressions, then (Tϕ), (Fϕ), (ϕ→ ψ),(¬ϕ), (ϕ ≡ ψ).The usual abbreviations hold.


Truth-theorySyntax

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-theoryDeductive 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)→ ϕInference Rules: Modus Ponens.


Truth-theorySemantic

Propositional Domain

A ε′T algebra (propositional domain) M = (M, f¬, f→, f≡, fT, fF ), where Mis 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 satisfiesthe 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) = fcγ also maps γ(Tϕ) = fT(γ(ϕ)); γ(Fϕ) = fF (γ(ϕ))


Logic with Quantifiers and Epistemic LogicEpistemic Logic

Propositions are now objects of knowledge, if Kiϕ is true, than ϕdenotes 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 factsknown. 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 groupknows the group facts. Furthermore, every agent knows that theothers agents know it, and so on.

It is introduced the notion of Common Knowledge, a commonknowledge among all the agents that satisfies some closure properties.


Logic with Quantifiers and Epistemic LogicEpistemic Logic

Logic with quantifiers

Similar to SCI, introduction of quantifier operators, e.g. ∀Introduction of axioms regarding these new operators in the deductivesystem

Semantically, they are mapped to high order functions, e.g. ∀ ismapped to f∀ : MM 7→ M

Examples

On Epistemic Logic: ∀x .(Kix → x)On Truth-theory: ∀x(Tx ↔ x)


Questions
