Nameability, Identity, Equality and Completeness PhDs in Logic 2017 - - PowerPoint PPT Presentation

Nameability, Identity, Equality and Completeness PhDs in Logic 2017 Bochum

María Manzano Departamento Filosofía y Lógica y Filosofía de la Ciencia Universidad de Salamanca España

USAL

May 2017

• M. Manzano (USAL)

NIEC May 2017 1 / 25

Introduction

Our point of departure

Ramsey (1926): “The preceding and other considerations let Wittgenstein to the view that mathematics does not consist of tautologies, but of what he called ‘equations’, for which I should prefer to substitute ‘identities’ (...) (It) is interesting to see whether a theory of mathematics could not be constructed with identities for its foundations.”

• M. Manzano (USAL)

NIEC May 2017 2 / 25

Introduction

This presentation is divided into two parts

1

The first one is devoted to the concepts of identity and equality in a variety of logical systems that motivates the definition of our system

• f Equational Hybrid Propositional Type Theory (EHPT T ).
• M. Manzano (USAL)

NIEC May 2017 3 / 25

Introduction

This presentation is divided into two parts

1

The first one is devoted to the concepts of identity and equality in a variety of logical systems that motivates the definition of our system

• f Equational Hybrid Propositional Type Theory (EHPT T ).

2

The second part concentrates on the relevant role played by names

• n the three completeness theorems Leon Henkin published last
• century. Our completeness proof for EHPT T owes much to the three

and I will point out the more important debts we have undertaken.

• M. Manzano (USAL)

NIEC May 2017 3 / 25

Identity and equality

Classical logic

‘By the relation of identity we mean that binary relation which holds between any object and itself, and which fails to hold between any two distinct objects.’ (Henkin 1975 Identity as a logical primitive)

• M. Manzano (USAL)

NIEC May 2017 4 / 25

Identity and equality

Classical logic

‘By the relation of identity we mean that binary relation which holds between any object and itself, and which fails to hold between any two distinct objects.’ (Henkin 1975 Identity as a logical primitive) By equality I mean a symbolic relation between terms which is reflexive, transitive and symmetric.

• M. Manzano (USAL)

NIEC May 2017 4 / 25

Identity and equality

Classical logic

‘By the relation of identity we mean that binary relation which holds between any object and itself, and which fails to hold between any two distinct objects.’ (Henkin 1975 Identity as a logical primitive) By equality I mean a symbolic relation between terms which is reflexive, transitive and symmetric. Owing to the central role this notion plays in logic, you can be interested either in:

• M. Manzano (USAL)

NIEC May 2017 4 / 25

Identity and equality

Classical logic

‘By the relation of identity we mean that binary relation which holds between any object and itself, and which fails to hold between any two distinct objects.’ (Henkin 1975 Identity as a logical primitive) By equality I mean a symbolic relation between terms which is reflexive, transitive and symmetric. Owing to the central role this notion plays in logic, you can be interested either in:

how to define it using other logical concepts

• M. Manzano (USAL)

NIEC May 2017 4 / 25

Identity and equality

Classical logic

‘By the relation of identity we mean that binary relation which holds between any object and itself, and which fails to hold between any two distinct objects.’ (Henkin 1975 Identity as a logical primitive) By equality I mean a symbolic relation between terms which is reflexive, transitive and symmetric. Owing to the central role this notion plays in logic, you can be interested either in:

how to define it using other logical concepts

• r, in the opposite scheme.
• M. Manzano (USAL)

NIEC May 2017 4 / 25

Identity and equality

Classical logic

‘By the relation of identity we mean that binary relation which holds between any object and itself, and which fails to hold between any two distinct objects.’ (Henkin 1975 Identity as a logical primitive) By equality I mean a symbolic relation between terms which is reflexive, transitive and symmetric. Owing to the central role this notion plays in logic, you can be interested either in:

how to define it using other logical concepts

• r, in the opposite scheme.

In the first case one investigates what kind of logic is required.

• M. Manzano (USAL)

NIEC May 2017 4 / 25

Identity and equality

Classical logic

‘By the relation of identity we mean that binary relation which holds between any object and itself, and which fails to hold between any two distinct objects.’ (Henkin 1975 Identity as a logical primitive) By equality I mean a symbolic relation between terms which is reflexive, transitive and symmetric. Owing to the central role this notion plays in logic, you can be interested either in:

how to define it using other logical concepts

• r, in the opposite scheme.

In the first case one investigates what kind of logic is required. In the second case, one is interested in the definition of the other logical concepts (connectives and quantifiers) in terms of the identity relation, using also abstraction.

• M. Manzano (USAL)

NIEC May 2017 4 / 25

Lambda and equality in Propositional Type Theory

A Theory of Propositional Types (Henkin 1963)

Can we define with only equality and abstraction the remaining logical symbols?

• M. Manzano (USAL)

NIEC May 2017 5 / 25

Lambda and equality in Propositional Type Theory

A Theory of Propositional Types (Henkin 1963)

Can we define with only equality and abstraction the remaining logical symbols? YES, In PTT

• M. Manzano (USAL)

NIEC May 2017 5 / 25

Lambda and equality in Propositional Type Theory

A Theory of Propositional Types (Henkin 1963)

Can we define with only equality and abstraction the remaining logical symbols? YES, In PTT Truth and falsity, negation, conjunction and quantifiers are defined

• perators.

T n ::= ((λX0X0) ≡ (λX0X0)) F n ::= ((λX0X0) ≡ (λX0T n)) ∀XαA0 := ((λXαA0) ≡ (λXαT n))

• M. Manzano (USAL)

NIEC May 2017 5 / 25

Lambda and equality in Propositional Type Theory

A Theory of Propositional Types (Henkin 1963)

Can we define with only equality and abstraction the remaining logical symbols? YES, In PTT Truth and falsity, negation, conjunction and quantifiers are defined

• perators.

T n ::= ((λX0X0) ≡ (λX0X0)) F n ::= ((λX0X0) ≡ (λX0T n)) ∀XαA0 := ((λXαA0) ≡ (λXαT n)) Peter Andrews, A Bit of History Related to Logic Based on Equality in [1].

• M. Manzano (USAL)

NIEC May 2017 5 / 25

Lambda and equality in Propositional Type Theory

A Theory of Propositional Types (Henkin 1963)

Can we define with only equality and abstraction the remaining logical symbols? YES, In PTT Truth and falsity, negation, conjunction and quantifiers are defined

• perators.

T n ::= ((λX0X0) ≡ (λX0X0)) F n ::= ((λX0X0) ≡ (λX0T n)) ∀XαA0 := ((λXαA0) ≡ (λXαT n)) Peter Andrews, A Bit of History Related to Logic Based on Equality in [1]. The language also provides a name for each object in the hierarchy.

• M. Manzano (USAL)

NIEC May 2017 5 / 25

Lambda and equality in Propositional Type Theory

A Theory of Propositional Types (Henkin 1963)

Man gave names to all the animals

• M. Manzano (USAL)

NIEC May 2017 6 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

Reflexivity axiom

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

Reflexivity axiom Equals substitution

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

Reflexivity axiom Equals substitution

Second-order logic: Identity can be defined

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

Reflexivity axiom Equals substitution

Second-order logic: Identity can be defined

using Leibniz’s principle x = y ↔Df ∀X(Xx → Xy)

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

Reflexivity axiom Equals substitution

Second-order logic: Identity can be defined

using Leibniz’s principle x = y ↔Df ∀X(Xx → Xy) the property of being the least reflexive relation x = y ↔Df ∀Y (∀zYzz → Yxy)

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

Reflexivity axiom Equals substitution

Second-order logic: Identity can be defined

using Leibniz’s principle x = y ↔Df ∀X(Xx → Xy) the property of being the least reflexive relation x = y ↔Df ∀Y (∀zYzz → Yxy) Both can be used to define identity for individuals as the relation defined by them is ‘genuine’ identity in any standard second order structure,

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

Reflexivity axiom Equals substitution

Second-order logic: Identity can be defined

using Leibniz’s principle x = y ↔Df ∀X(Xx → Xy) the property of being the least reflexive relation x = y ↔Df ∀Y (∀zYzz → Yxy) Both can be used to define identity for individuals as the relation defined by them is ‘genuine’ identity in any standard second order structure,

However, in SOL with general structures the possibility of defining identity is lost and we return to the situation encountered in FOL.

• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality in classical logic

First-order and Higher-order Logics

First-order logic: Identity is a logical primitive concept

We can’t define it, we need axioms and rules to treat equality

Reflexivity axiom Equals substitution

Second-order logic: Identity can be defined

using Leibniz’s principle x = y ↔Df ∀X(Xx → Xy) the property of being the least reflexive relation x = y ↔Df ∀Y (∀zYzz → Yxy) Both can be used to define identity for individuals as the relation defined by them is ‘genuine’ identity in any standard second order structure,

However, in SOL with general structures the possibility of defining identity is lost and we return to the situation encountered in FOL. All identicals are equal, but some equals are more equal than

• thers
• M. Manzano (USAL)

NIEC May 2017 7 / 25

Identity and equality

All identicals are equal, but some equals are more equal than others

• M. Manzano (USAL)

NIEC May 2017 8 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y)

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y) recall that variables are rigid terms

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y) recall that variables are rigid terms

What about equals substitution?

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y) recall that variables are rigid terms

What about equals substitution?

necessity of identity (NI) x = y → (x = y)

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y) recall that variables are rigid terms

What about equals substitution?

necessity of identity (NI) x = y → (x = y) substitutivity of identicals (SI) x = y → (ϕ(x) → ϕ(y))

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y) recall that variables are rigid terms

What about equals substitution?

necessity of identity (NI) x = y → (x = y) substitutivity of identicals (SI) x = y → (ϕ(x) → ϕ(y)) are both sound?

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y) recall that variables are rigid terms

What about equals substitution?

necessity of identity (NI) x = y → (x = y) substitutivity of identicals (SI) x = y → (ϕ(x) → ϕ(y)) are both sound?

These rules turn out to be problematic when terms other than variables are used

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y) recall that variables are rigid terms

What about equals substitution?

necessity of identity (NI) x = y → (x = y) substitutivity of identicals (SI) x = y → (ϕ(x) → ϕ(y)) are both sound?

These rules turn out to be problematic when terms other than variables are used

that is, when we have intensional terms

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

First-order modal logic

In Modal Logic

from an “ontological” point of view equality symbol interpreted as pure global identity M, w g (x = y) iff g(x) = g(y) recall that variables are rigid terms

What about equals substitution?

necessity of identity (NI) x = y → (x = y) substitutivity of identicals (SI) x = y → (ϕ(x) → ϕ(y)) are both sound?

These rules turn out to be problematic when terms other than variables are used

that is, when we have intensional terms these rules only apply for rigids

• M. Manzano (USAL)

NIEC May 2017 9 / 25

Identity and equality in modal logic

Fitting and Meldensohn, (1998)

Term Equality: Melvin Fitting introduces a new relation between intensional terms

• M. Manzano (USAL)

NIEC May 2017 10 / 25

Identity and equality in modal logic

Fitting and Meldensohn, (1998)

Term Equality: Melvin Fitting introduces a new relation between intensional terms

τ1 τ2 ↔Df λx, y.y = x(τ2, τ1)

• M. Manzano (USAL)

NIEC May 2017 10 / 25

Identity and equality in modal logic

Fitting and Meldensohn, (1998)

Term Equality: Melvin Fitting introduces a new relation between intensional terms

τ1 τ2 ↔Df λx, y.y = x(τ2, τ1) M, w g (τ1 τ2) iff (τ1)(w) = (τ2)(w) ‘x = y asserts that the objects that are the values of x and y are the same’

• M. Manzano (USAL)

NIEC May 2017 10 / 25

Identity and equality in modal logic

Fitting and Meldensohn, (1998)

Term Equality: Melvin Fitting introduces a new relation between intensional terms

τ1 τ2 ↔Df λx, y.y = x(τ2, τ1) M, w g (τ1 τ2) iff (τ1)(w) = (τ2)(w) ‘x = y asserts that the objects that are the values of x and y are the same’ and the relation defined by it can be taken as the identity relation

• M. Manzano (USAL)

NIEC May 2017 10 / 25

Identity and equality in modal logic

Fitting and Meldensohn, (1998)

Term Equality: Melvin Fitting introduces a new relation between intensional terms

τ1 τ2 ↔Df λx, y.y = x(τ2, τ1) M, w g (τ1 τ2) iff (τ1)(w) = (τ2)(w) ‘x = y asserts that the objects that are the values of x and y are the same’ and the relation defined by it can be taken as the identity relation ‘τ1 τ2 asserts that the terms τ1 and τ2 designate the same object, which is quite a different thing.’

• M. Manzano (USAL)

NIEC May 2017 10 / 25

Identity and equality in modal logic

Fitting and Meldensohn, (1998)

Term Equality: Melvin Fitting introduces a new relation between intensional terms

τ1 τ2 ↔Df λx, y.y = x(τ2, τ1) M, w g (τ1 τ2) iff (τ1)(w) = (τ2)(w) ‘x = y asserts that the objects that are the values of x and y are the same’ and the relation defined by it can be taken as the identity relation ‘τ1 τ2 asserts that the terms τ1 and τ2 designate the same object, which is quite a different thing.’

the formula τ1 τ2 → (τ1 τ2), is not valid.

• M. Manzano (USAL)

NIEC May 2017 10 / 25

Identity and equality in modal logic

Fitting and Meldensohn, (1998)

Term Equality: Melvin Fitting introduces a new relation between intensional terms

τ1 τ2 ↔Df λx, y.y = x(τ2, τ1) M, w g (τ1 τ2) iff (τ1)(w) = (τ2)(w) ‘x = y asserts that the objects that are the values of x and y are the same’ and the relation defined by it can be taken as the identity relation ‘τ1 τ2 asserts that the terms τ1 and τ2 designate the same object, which is quite a different thing.’

the formula τ1 τ2 → (τ1 τ2), is not valid. ‘In fact, (τ1 τ2) express a notion considerable stronger than that

• f simple equality –it has the characteristics of synonymy’
• M. Manzano (USAL)

NIEC May 2017 10 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

NO

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

NO

This lack of expressivity of orthodox modal logic is an obvious weak

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

NO

This lack of expressivity of orthodox modal logic is an obvious weak and ♦ are, in essence, quantifiers over worlds in disguise.

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

NO

This lack of expressivity of orthodox modal logic is an obvious weak and ♦ are, in essence, quantifiers over worlds in disguise.

The basic hybrid language is the modal solution to this deficiency.

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

NO

This lack of expressivity of orthodox modal logic is an obvious weak and ♦ are, in essence, quantifiers over worlds in disguise.

The basic hybrid language is the modal solution to this deficiency. ‘What is a hybrid language? A variant of orthodox modal logic in which the notion of world has been internalized.’

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

NO

This lack of expressivity of orthodox modal logic is an obvious weak and ♦ are, in essence, quantifiers over worlds in disguise.

The basic hybrid language is the modal solution to this deficiency. ‘What is a hybrid language? A variant of orthodox modal logic in which the notion of world has been internalized.’ The two innovations of basic hybrid logic are:

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

NO

This lack of expressivity of orthodox modal logic is an obvious weak and ♦ are, in essence, quantifiers over worlds in disguise.

The basic hybrid language is the modal solution to this deficiency. ‘What is a hybrid language? A variant of orthodox modal logic in which the notion of world has been internalized.’ The two innovations of basic hybrid logic are:

the nominals and

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Pure Extensions, Proof Rules and Hybrid Axiomatics (Blackburn and ten Cate)

In Kripke semantics we have a universe of worlds

1

Can we express in the language identity of worlds?

2

what about the accessibility relation between worlds, can be referred to?

NO

This lack of expressivity of orthodox modal logic is an obvious weak and ♦ are, in essence, quantifiers over worlds in disguise.

The basic hybrid language is the modal solution to this deficiency. ‘What is a hybrid language? A variant of orthodox modal logic in which the notion of world has been internalized.’ The two innovations of basic hybrid logic are:

the nominals and the satisfaction operators.

• M. Manzano (USAL)

NIEC May 2017 11 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

@i ϕ is either true at all worlds, or false at all worlds

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

@i ϕ is either true at all worlds, or false at all worlds

Hybrid logic offers mechanism for naming worlds, asserting equalities between worlds and describing accessibility relation.

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

@i ϕ is either true at all worlds, or false at all worlds

Hybrid logic offers mechanism for naming worlds, asserting equalities between worlds and describing accessibility relation. Formulas of form @ij, together with @i♦j are extremely important

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

@i ϕ is either true at all worlds, or false at all worlds

Hybrid logic offers mechanism for naming worlds, asserting equalities between worlds and describing accessibility relation. Formulas of form @ij, together with @i♦j are extremely important

@ij asserts that i and j name the same point.

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

@i ϕ is either true at all worlds, or false at all worlds

Hybrid logic offers mechanism for naming worlds, asserting equalities between worlds and describing accessibility relation. Formulas of form @ij, together with @i♦j are extremely important

@ij asserts that i and j name the same point. @ij is a modal way of expressing what i = j would express in classical logic.

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

@i ϕ is either true at all worlds, or false at all worlds

Hybrid logic offers mechanism for naming worlds, asserting equalities between worlds and describing accessibility relation. Formulas of form @ij, together with @i♦j are extremely important

@ij asserts that i and j name the same point. @ij is a modal way of expressing what i = j would express in classical logic.

The reflexivity, symmetry and transitivity of equality are validities: @ii @ij → @ji @ij ∧ @jk → @ik

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

@i ϕ is either true at all worlds, or false at all worlds

Hybrid logic offers mechanism for naming worlds, asserting equalities between worlds and describing accessibility relation. Formulas of form @ij, together with @i♦j are extremely important

@ij asserts that i and j name the same point. @ij is a modal way of expressing what i = j would express in classical logic.

The reflexivity, symmetry and transitivity of equality are validities: @ii @ij → @ji @ij ∧ @jk → @ik

@i♦j express the accessibility relation and allows to define relevant properties this relation might have, some are undefinable in orthodox modal logic

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Nominals and the satisfaction operator in hybrid logic

Nominals are special propositional symbols i ∈ NOM

nominals are true at a unique world in any model

Satisfaction operator is a rigidifier operator

@i ϕ is either true at all worlds, or false at all worlds

Hybrid logic offers mechanism for naming worlds, asserting equalities between worlds and describing accessibility relation. Formulas of form @ij, together with @i♦j are extremely important

@ij asserts that i and j name the same point. @ij is a modal way of expressing what i = j would express in classical logic.

The reflexivity, symmetry and transitivity of equality are validities: @ii @ij → @ji @ij ∧ @jk → @ik

@i♦j express the accessibility relation and allows to define relevant properties this relation might have, some are undefinable in orthodox modal logic

like irreflexivity or trichotomy: @i¬♦i @ij ∨ @i♦j ∨ @j♦i

• M. Manzano (USAL)

NIEC May 2017 12 / 25

Zen Philosophy

The book of perfect emptiness

Tang de Ying asked Ge:

• M. Manzano (USAL)

NIEC May 2017 13 / 25

Zen Philosophy

The book of perfect emptiness

Tang de Ying asked Ge: “Did things exist at the dawn of time?”

• M. Manzano (USAL)

NIEC May 2017 13 / 25

Zen Philosophy

The book of perfect emptiness

Tang de Ying asked Ge: “Did things exist at the dawn of time?” Xia Ge answered:

• M. Manzano (USAL)

NIEC May 2017 13 / 25

Zen Philosophy

The book of perfect emptiness

Tang de Ying asked Ge: “Did things exist at the dawn of time?” Xia Ge answered: “If things had not existed at the dawn of time, how could they possibly exist today?

• M. Manzano (USAL)

NIEC May 2017 13 / 25

Zen Philosophy

The book of perfect emptiness

Tang de Ying asked Ge: “Did things exist at the dawn of time?” Xia Ge answered: “If things had not existed at the dawn of time, how could they possibly exist today? By the same token, men in the future could believe that things did not exist today.

• M. Manzano (USAL)

NIEC May 2017 13 / 25

Zen Philosophy

The book of perfect emptiness

The argument can be reformulated in this way

• M. Manzano (USAL)

NIEC May 2017 14 / 25

Zen Philosophy

The book of perfect emptiness

The argument can be reformulated in this way

1

α := If things exist at a given point in time, then at any given previous moment in time things must have existed. α := q → Hq

• M. Manzano (USAL)

NIEC May 2017 14 / 25

Zen Philosophy

The book of perfect emptiness

The argument can be reformulated in this way

1

α := If things exist at a given point in time, then at any given previous moment in time things must have existed. α := q → Hq

2

β := Things exist today. β := @t q

• M. Manzano (USAL)

NIEC May 2017 14 / 25

Zen Philosophy

The book of perfect emptiness

The argument can be reformulated in this way

1

α := If things exist at a given point in time, then at any given previous moment in time things must have existed. α := q → Hq

2

β := Things exist today. β := @t q

3

γ := The dawn of time is previous to all else. γ := @dH ⊥

• M. Manzano (USAL)

NIEC May 2017 14 / 25

Zen Philosophy

The book of perfect emptiness

The argument can be reformulated in this way

1

α := If things exist at a given point in time, then at any given previous moment in time things must have existed. α := q → Hq

2

β := Things exist today. β := @t q

3

γ := The dawn of time is previous to all else. γ := @dH ⊥

4

δ := Things existed at the dawn of time. δ := @d q

• M. Manzano (USAL)

NIEC May 2017 14 / 25

Zen Philosophy

The book of perfect emptiness

The argument can be reformulated in this way

1

α := If things exist at a given point in time, then at any given previous moment in time things must have existed. α := q → Hq

2

β := Things exist today. β := @t q

3

γ := The dawn of time is previous to all else. γ := @dH ⊥

4

δ := Things existed at the dawn of time. δ := @d q

5

To prove {α, β, γ} δ we can use the trichotomy axiom @d t ∨ @d Pt ∨ @t Pd

• M. Manzano (USAL)

NIEC May 2017 14 / 25

Equational Hybrid Propositional Type Theory

A Combined Logic

Equational hybrid propositional type theory EHPT T is a combination of logics where identity plays a relevant role: Propositional type theory, PTT

• M. Manzano (USAL)

NIEC May 2017 15 / 25

Equational Hybrid Propositional Type Theory

A Combined Logic

Equational hybrid propositional type theory EHPT T is a combination of logics where identity plays a relevant role: Propositional type theory, PTT

where all connectives and quantifiers are defined with λ and ≡

• M. Manzano (USAL)

NIEC May 2017 15 / 25

Equational Hybrid Propositional Type Theory

A Combined Logic

Equational hybrid propositional type theory EHPT T is a combination of logics where identity plays a relevant role: Propositional type theory, PTT

where all connectives and quantifiers are defined with λ and ≡

Equational logic, EL

• M. Manzano (USAL)

NIEC May 2017 15 / 25

Equational Hybrid Propositional Type Theory

A Combined Logic

Equational hybrid propositional type theory EHPT T is a combination of logics where identity plays a relevant role: Propositional type theory, PTT

where all connectives and quantifiers are defined with λ and ≡

Equational logic, EL

where all formulas are equations

• M. Manzano (USAL)

NIEC May 2017 15 / 25

Equational Hybrid Propositional Type Theory

A Combined Logic

Equational hybrid propositional type theory EHPT T is a combination of logics where identity plays a relevant role: Propositional type theory, PTT

where all connectives and quantifiers are defined with λ and ≡

Equational logic, EL

where all formulas are equations

Hybrid logic, HL

• M. Manzano (USAL)

NIEC May 2017 15 / 25

Equational Hybrid Propositional Type Theory

A Combined Logic

Equational hybrid propositional type theory EHPT T is a combination of logics where identity plays a relevant role: Propositional type theory, PTT

where all connectives and quantifiers are defined with λ and ≡

Equational logic, EL

where all formulas are equations

Hybrid logic, HL

where intensional contexts are relevant

• M. Manzano (USAL)

NIEC May 2017 15 / 25

Equational Hybrid Propositional Type Theory

A Combined Logic

Equational hybrid propositional type theory EHPT T is a combination of logics where identity plays a relevant role: Propositional type theory, PTT

where all connectives and quantifiers are defined with λ and ≡

Equational logic, EL

where all formulas are equations

Hybrid logic, HL

where intensional contexts are relevant identity between worlds is expressible

• M. Manzano (USAL)

NIEC May 2017 15 / 25

Equational Hybrid Propositional Type Theory

A Combined Logic

Equational hybrid propositional type theory EHPT T is a combination of logics where identity plays a relevant role: Propositional type theory, PTT

where all connectives and quantifiers are defined with λ and ≡

Equational logic, EL

where all formulas are equations

Hybrid logic, HL

where intensional contexts are relevant identity between worlds is expressible

The challenge is to deal with these heterogeneous components in an integrated system

• M. Manzano (USAL)

NIEC May 2017 15 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms individual variables and constants, function symbols, etc. to form individual terms

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms individual variables and constants, function symbols, etc. to form individual terms

• ther meaningful expressions using:
• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms individual variables and constants, function symbols, etc. to form individual terms

• ther meaningful expressions using:

lambda operator λ,

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms individual variables and constants, function symbols, etc. to form individual terms

• ther meaningful expressions using:

lambda operator λ, equality symbol (with ≡) between expressions of several types PT ∪ AT − {0}

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms individual variables and constants, function symbols, etc. to form individual terms

• ther meaningful expressions using:

lambda operator λ, equality symbol (with ≡) between expressions of several types PT ∪ AT − {0} modal operators ♦ and @i

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms individual variables and constants, function symbols, etc. to form individual terms

• ther meaningful expressions using:

lambda operator λ, equality symbol (with ≡) between expressions of several types PT ∪ AT − {0} modal operators ♦ and @i

As defined operators using λ and ≡ we have:

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms individual variables and constants, function symbols, etc. to form individual terms

• ther meaningful expressions using:

lambda operator λ, equality symbol (with ≡) between expressions of several types PT ∪ AT − {0} modal operators ♦ and @i

As defined operators using λ and ≡ we have:

usual propositional connectives and propositional quantifiers

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Language

Equational hybrid propositional type theory EHPT T algebraic and propositional types TYPES = AT∪ PT propositional variables and nominals are atoms individual variables and constants, function symbols, etc. to form individual terms

• ther meaningful expressions using:

lambda operator λ, equality symbol (with ≡) between expressions of several types PT ∪ AT − {0} modal operators ♦ and @i

As defined operators using λ and ≡ we have:

usual propositional connectives and propositional quantifiers algebraic equations τ ≈ σ :=Df λv1, . . . , vnτ ≡ λv1, . . . , vnσ

• M. Manzano (USAL)

NIEC May 2017 16 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W ,

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W , an accessibility relation between worlds, R

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W , an accessibility relation between worlds, R the standard propositional hierarchy, PT

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W , an accessibility relation between worlds, R the standard propositional hierarchy, PT and a domain of individuals A

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W , an accessibility relation between worlds, R the standard propositional hierarchy, PT and a domain of individuals A the interpretation I of constants and nominals is of an intensional kind

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W , an accessibility relation between worlds, R the standard propositional hierarchy, PT and a domain of individuals A the interpretation I of constants and nominals is of an intensional kind

An interpretation for EHPT T is a pair I = M, g, where M is a structure for EHPT T and g is an assignment of values to variables.

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W , an accessibility relation between worlds, R the standard propositional hierarchy, PT and a domain of individuals A the interpretation I of constants and nominals is of an intensional kind

An interpretation for EHPT T is a pair I = M, g, where M is a structure for EHPT T and g is an assignment of values to variables.

We recursively define for any expression Fa the value of Fa under the interpretation I, denoted by (Fa)I.

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W , an accessibility relation between worlds, R the standard propositional hierarchy, PT and a domain of individuals A the interpretation I of constants and nominals is of an intensional kind

An interpretation for EHPT T is a pair I = M, g, where M is a structure for EHPT T and g is an assignment of values to variables.

We recursively define for any expression Fa the value of Fa under the interpretation I, denoted by (Fa)I. (Fa)I is always defined as a function from W to Da

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Equational Hybrid Propositional Type Theory

The Semantics

The structures M = W , R, A, PT, I used to interpret the language include several domains:

a set of worlds, W , an accessibility relation between worlds, R the standard propositional hierarchy, PT and a domain of individuals A the interpretation I of constants and nominals is of an intensional kind

An interpretation for EHPT T is a pair I = M, g, where M is a structure for EHPT T and g is an assignment of values to variables.

We recursively define for any expression Fa the value of Fa under the interpretation I, denoted by (Fa)I. (Fa)I is always defined as a function from W to Da for rigid expressions the value is a constant function

• M. Manzano (USAL)

NIEC May 2017 17 / 25

Completeness in EHPTT

Henkin’s strategy

Our completeness proof for EHPT T owes much to Henkin’s method and I will point out the more important debts we have undertaken.

• M. Manzano (USAL)

NIEC May 2017 18 / 25

Completeness in EHPTT

Henkin’s strategy

Our completeness proof for EHPT T owes much to Henkin’s method and I will point out the more important debts we have undertaken. We will follow Henkin’s strategy to prove completeness:

• M. Manzano (USAL)

NIEC May 2017 18 / 25

Completeness in EHPTT

Henkin’s strategy

Our completeness proof for EHPT T owes much to Henkin’s method and I will point out the more important debts we have undertaken. We will follow Henkin’s strategy to prove completeness:

That is, as a corollary of the theorem saying that any consistent set have a model

• M. Manzano (USAL)

NIEC May 2017 18 / 25

Completeness in EHPTT

Henkin’s strategy

Our completeness proof for EHPT T owes much to Henkin’s method and I will point out the more important debts we have undertaken. We will follow Henkin’s strategy to prove completeness:

That is, as a corollary of the theorem saying that any consistent set have a model

We have an arbitrary consistent set of meaningful expressions of type t and we want to prove that it has a countable model

• M. Manzano (USAL)

NIEC May 2017 18 / 25

Completeness in EHPTT

Henkin’s strategy

Our completeness proof for EHPT T owes much to Henkin’s method and I will point out the more important debts we have undertaken. We will follow Henkin’s strategy to prove completeness:

That is, as a corollary of the theorem saying that any consistent set have a model

We have an arbitrary consistent set of meaningful expressions of type t and we want to prove that it has a countable model

as in any Henkin completeness proof, we shall build the model which is perfectly described by a maximal consistent set of sentences.

• M. Manzano (USAL)

NIEC May 2017 18 / 25

Completeness in EHPTT

Henkin’s strategy

Our completeness proof for EHPT T owes much to Henkin’s method and I will point out the more important debts we have undertaken. We will follow Henkin’s strategy to prove completeness:

That is, as a corollary of the theorem saying that any consistent set have a model

We have an arbitrary consistent set of meaningful expressions of type t and we want to prove that it has a countable model

as in any Henkin completeness proof, we shall build the model which is perfectly described by a maximal consistent set of sentences. And not only that, we build the model out of the expressions contained in the maximal consistent set

• M. Manzano (USAL)

NIEC May 2017 18 / 25

Completeness in EHPTT

Henkin’s strategy

Our completeness proof for EHPT T owes much to Henkin’s method and I will point out the more important debts we have undertaken. We will follow Henkin’s strategy to prove completeness:

That is, as a corollary of the theorem saying that any consistent set have a model

We have an arbitrary consistent set of meaningful expressions of type t and we want to prove that it has a countable model

as in any Henkin completeness proof, we shall build the model which is perfectly described by a maximal consistent set of sentences. And not only that, we build the model out of the expressions contained in the maximal consistent set As you will see, the nameability of the objects involved is the clue.

• M. Manzano (USAL)

NIEC May 2017 18 / 25

Completeness in EHPTT

Henkin’s strategy

In the first step we extent the consistent set to the required maximal consistent one.

• M. Manzano (USAL)

NIEC May 2017 19 / 25

Completeness in EHPTT

Henkin’s strategy

In the first step we extent the consistent set to the required maximal consistent one.

From First-order completeness proof we learnt that we need witnesses for all existential formulas

• M. Manzano (USAL)

NIEC May 2017 19 / 25

Completeness in EHPTT

Henkin’s strategy

In the first step we extent the consistent set to the required maximal consistent one.

From First-order completeness proof we learnt that we need witnesses for all existential formulas each ♦-formula will be witnessed by a nominal (as ♦ counts as a quantifier)

• M. Manzano (USAL)

NIEC May 2017 19 / 25

Completeness in EHPTT

Henkin’s strategy

In the first step we extent the consistent set to the required maximal consistent one.

From First-order completeness proof we learnt that we need witnesses for all existential formulas each ♦-formula will be witnessed by a nominal (as ♦ counts as a quantifier) also, we use nominals to build the worlds (we ask the maximal consistent set to be named)

• M. Manzano (USAL)

NIEC May 2017 19 / 25

Completeness in EHPTT

Henkin’s strategy

In the first step we extent the consistent set to the required maximal consistent one.

From First-order completeness proof we learnt that we need witnesses for all existential formulas each ♦-formula will be witnessed by a nominal (as ♦ counts as a quantifier) also, we use nominals to build the worlds (we ask the maximal consistent set to be named)

in the algebraic part we will ask our maximal consistent set to be extensionally algebraic-saturated so that we can distinguish different functions with appropriate constants witnessing such inequalities.

• M. Manzano (USAL)

NIEC May 2017 19 / 25

Completeness in EHPTT

Henkin’s strategy

In the first step we extent the consistent set to the required maximal consistent one.

From First-order completeness proof we learnt that we need witnesses for all existential formulas each ♦-formula will be witnessed by a nominal (as ♦ counts as a quantifier) also, we use nominals to build the worlds (we ask the maximal consistent set to be named)

in the algebraic part we will ask our maximal consistent set to be extensionally algebraic-saturated so that we can distinguish different functions with appropriate constants witnessing such inequalities.

we do not have proper first-order quantifiers in the algebraic part, the closer we get to it is with the algebraic equations (with ≈)

• M. Manzano (USAL)

NIEC May 2017 19 / 25

Completeness in EHPTT

The model

This part of our proof follows Henkin’s recipe almost to the letter. We will do it, as we did in Completeness in Hybrid Type Theory [2], via a function Φ to be defined on equivalent classes of expressions.

• M. Manzano (USAL)

NIEC May 2017 20 / 25

Completeness in EHPTT

The model

This part of our proof follows Henkin’s recipe almost to the letter. We will do it, as we did in Completeness in Hybrid Type Theory [2], via a function Φ to be defined on equivalent classes of expressions.

so, we define by induction on types the domains {D∗

a} and

simultaneously

• M. Manzano (USAL)

NIEC May 2017 20 / 25

Completeness in EHPTT

The model

This part of our proof follows Henkin’s recipe almost to the letter. We will do it, as we did in Completeness in Hybrid Type Theory [2], via a function Φ to be defined on equivalent classes of expressions.

so, we define by induction on types the domains {D∗

a} and

simultaneously the mapping Φ

• M. Manzano (USAL)

NIEC May 2017 20 / 25

Completeness in EHPTT

The model

This part of our proof follows Henkin’s recipe almost to the letter. We will do it, as we did in Completeness in Hybrid Type Theory [2], via a function Φ to be defined on equivalent classes of expressions.

so, we define by induction on types the domains {D∗

a} and

simultaneously the mapping Φ

Prior to the model construction we define equivalent relations between nominals and other types of expressions, the maximal consistent set just built provides the measuring rule.

• M. Manzano (USAL)

NIEC May 2017 20 / 25

Completeness in EHPTT

The model

This part of our proof follows Henkin’s recipe almost to the letter. We will do it, as we did in Completeness in Hybrid Type Theory [2], via a function Φ to be defined on equivalent classes of expressions.

so, we define by induction on types the domains {D∗

a} and

simultaneously the mapping Φ

Prior to the model construction we define equivalent relations between nominals and other types of expressions, the maximal consistent set just built provides the measuring rule.

The equivalence relation is ‘being equals at the eyes of the our oracle ∆’

• M. Manzano (USAL)

NIEC May 2017 20 / 25

Completeness in EHPTT

The model

This part of our proof follows Henkin’s recipe almost to the letter. We will do it, as we did in Completeness in Hybrid Type Theory [2], via a function Φ to be defined on equivalent classes of expressions.

so, we define by induction on types the domains {D∗

a} and

simultaneously the mapping Φ

Prior to the model construction we define equivalent relations between nominals and other types of expressions, the maximal consistent set just built provides the measuring rule.

The equivalence relation is ‘being equals at the eyes of the our oracle ∆’

nominals i, j are equivalent when @ij ∈ ∆

• M. Manzano (USAL)

NIEC May 2017 20 / 25

Completeness in EHPTT

The model

This part of our proof follows Henkin’s recipe almost to the letter. We will do it, as we did in Completeness in Hybrid Type Theory [2], via a function Φ to be defined on equivalent classes of expressions.

so, we define by induction on types the domains {D∗

a} and

simultaneously the mapping Φ

Prior to the model construction we define equivalent relations between nominals and other types of expressions, the maximal consistent set just built provides the measuring rule.

The equivalence relation is ‘being equals at the eyes of the our oracle ∆’

nominals i, j are equivalent when @ij ∈ ∆ individual terms τ0, τ

0 are equivalent when τ0 ≈ τ 0 ∈ ∆

• M. Manzano (USAL)

NIEC May 2017 20 / 25

Completeness in EHPTT

The model

This part of our proof follows Henkin’s recipe almost to the letter. We will do it, as we did in Completeness in Hybrid Type Theory [2], via a function Φ to be defined on equivalent classes of expressions.

so, we define by induction on types the domains {D∗

a} and

simultaneously the mapping Φ

Prior to the model construction we define equivalent relations between nominals and other types of expressions, the maximal consistent set just built provides the measuring rule.

The equivalence relation is ‘being equals at the eyes of the our oracle ∆’

nominals i, j are equivalent when @ij ∈ ∆ individual terms τ0, τ

0 are equivalent when τ0 ≈ τ 0 ∈ ∆

expressions Aa, Ba of any type a ∈ PT ∪ AT − {0} are equivalent when Aα ≡ Bα ∈ ∆

• M. Manzano (USAL)

NIEC May 2017 20 / 25

Completeness in EHPTT

The model

Nominals and expressions of the form @iAa play a central role in the completeness result:

• M. Manzano (USAL)

NIEC May 2017 21 / 25

Completeness in EHPTT

The model

Nominals and expressions of the form @iAa play a central role in the completeness result:

nominals are the building blocks of the world universe

• M. Manzano (USAL)

NIEC May 2017 21 / 25

Completeness in EHPTT

The model

Nominals and expressions of the form @iAa play a central role in the completeness result:

nominals are the building blocks of the world universe accessibility relation is given by @i♦j ∈ ∆

• M. Manzano (USAL)

NIEC May 2017 21 / 25

Completeness in EHPTT

The model

Nominals and expressions of the form @iAa play a central role in the completeness result:

nominals are the building blocks of the world universe accessibility relation is given by @i♦j ∈ ∆ @iFa expressions supply the architectural blueprint in the algebraic construction.

• M. Manzano (USAL)

NIEC May 2017 21 / 25

Completeness in EHPTT

The model

Nominals and expressions of the form @iAa play a central role in the completeness result:

nominals are the building blocks of the world universe accessibility relation is given by @i♦j ∈ ∆ @iFa expressions supply the architectural blueprint in the algebraic construction.

What about the hierarchy of propositional types?

• M. Manzano (USAL)

NIEC May 2017 21 / 25

Completeness in EHPTT

The model

Nominals and expressions of the form @iAa play a central role in the completeness result:

nominals are the building blocks of the world universe accessibility relation is given by @i♦j ∈ ∆ @iFa expressions supply the architectural blueprint in the algebraic construction.

What about the hierarchy of propositional types?

From the propositional type development we learned that each type has a name and we will use it in the construction.

• M. Manzano (USAL)

NIEC May 2017 21 / 25

Completeness in EHPTT

The model

Nominals and expressions of the form @iAa play a central role in the completeness result:

nominals are the building blocks of the world universe accessibility relation is given by @i♦j ∈ ∆ @iFa expressions supply the architectural blueprint in the algebraic construction.

What about the hierarchy of propositional types?

From the propositional type development we learned that each type has a name and we will use it in the construction. That is why, instead of taking the standard propositional types as it, PT, we will take the values of function Φ on closed rigid expressions of any type α ∈ PT.

• M. Manzano (USAL)

NIEC May 2017 21 / 25

Completeness in EHPTT

The model

Nominals and expressions of the form @iAa play a central role in the completeness result:

nominals are the building blocks of the world universe accessibility relation is given by @i♦j ∈ ∆ @iFa expressions supply the architectural blueprint in the algebraic construction.

What about the hierarchy of propositional types?

From the propositional type development we learned that each type has a name and we will use it in the construction. That is why, instead of taking the standard propositional types as it, PT, we will take the values of function Φ on closed rigid expressions of any type α ∈ PT. and we prove: χα = Φ(χNAME

α

) for all type α ∈ PT and any χα ∈ Dα (where Dα are the standard echelons of the hierarchy built on Dt = {T, F})

• M. Manzano (USAL)

NIEC May 2017 21 / 25

Completeness

Nominalism

As we have already mentioned, the models Henkin builds for type theory are in accordance with a Nominalistic position, as Henkin himself affirms in his paper On Nominalism, published in 1953.

• M. Manzano (USAL)

NIEC May 2017 22 / 25

Completeness

Nominalism

As we have already mentioned, the models Henkin builds for type theory are in accordance with a Nominalistic position, as Henkin himself affirms in his paper On Nominalism, published in 1953. As you remember, the general models contain the nameable types, those sets and functions that are definable

• M. Manzano (USAL)

NIEC May 2017 22 / 25

Completeness

Nominalism

As we have already mentioned, the models Henkin builds for type theory are in accordance with a Nominalistic position, as Henkin himself affirms in his paper On Nominalism, published in 1953. As you remember, the general models contain the nameable types, those sets and functions that are definable

We did something similar and even the method is the same

• M. Manzano (USAL)

NIEC May 2017 22 / 25

Completeness

Nominalism

As we have already mentioned, the models Henkin builds for type theory are in accordance with a Nominalistic position, as Henkin himself affirms in his paper On Nominalism, published in 1953. As you remember, the general models contain the nameable types, those sets and functions that are definable

We did something similar and even the method is the same

In first-order logic, terms are used to build the ground level

• M. Manzano (USAL)

NIEC May 2017 22 / 25

Completeness

Nominalism

As we have already mentioned, the models Henkin builds for type theory are in accordance with a Nominalistic position, as Henkin himself affirms in his paper On Nominalism, published in 1953. As you remember, the general models contain the nameable types, those sets and functions that are definable

We did something similar and even the method is the same

In first-order logic, terms are used to build the ground level

We use nominals and rigid terms

• M. Manzano (USAL)

NIEC May 2017 22 / 25

Completeness

Nominalism

As we have already mentioned, the models Henkin builds for type theory are in accordance with a Nominalistic position, as Henkin himself affirms in his paper On Nominalism, published in 1953. As you remember, the general models contain the nameable types, those sets and functions that are definable

We did something similar and even the method is the same

In first-order logic, terms are used to build the ground level

We use nominals and rigid terms

Nameability theorem of Propositional type theory

• M. Manzano (USAL)

NIEC May 2017 22 / 25

Completeness

Nominalism

As we have already mentioned, the models Henkin builds for type theory are in accordance with a Nominalistic position, as Henkin himself affirms in his paper On Nominalism, published in 1953. As you remember, the general models contain the nameable types, those sets and functions that are definable

We did something similar and even the method is the same

In first-order logic, terms are used to build the ground level

We use nominals and rigid terms

Nameability theorem of Propositional type theory We have just shown that in this proof the Nominalistic position is more revealing than ever.

• M. Manzano (USAL)

NIEC May 2017 22 / 25

Two papers

To appear

A technical paper:

• M. Manzano (USAL)

NIEC May 2017 23 / 25

Two papers

To appear

A technical paper: Completeness in Equational Hybrid Propositional Type Theory by Manzano, Martins, and Huertas.

• M. Manzano (USAL)

NIEC May 2017 23 / 25

Two papers

To appear

A technical paper: Completeness in Equational Hybrid Propositional Type Theory by Manzano, Martins, and Huertas. A philosophical one:

• M. Manzano (USAL)

NIEC May 2017 23 / 25

Two papers

To appear

A technical paper: Completeness in Equational Hybrid Propositional Type Theory by Manzano, Martins, and Huertas. A philosophical one: Identity, Equality, Nameability and Completeness by Manzano and Moreno.

• M. Manzano (USAL)

NIEC May 2017 23 / 25

References

Andrews, P., A Bit of History Related to Logic Based on Equality in [6] Areces, C. Blackburn, P. Huertas, A. Manzano, M. [2014] Completeness in Hybrid Type Theory, DOI 10.1007/s10992-012-9260-4. J Philos Logic (Journal of Philosophical Logic) 43 (2-3) pp 209-238. Springer Blackburn, P. Cate, T. [2006]. Pure Extensions, Proof Rules and Hybrid Axiomatics. Studia Logica. vol. 84, pp. 277-322. Fitting, M, Meldensohn [1998]. First-order modal logic. Kluwer Academic Publishers. Henkin, L. [1949]. The completeness of the first order functional

• calculus. The Journal of Symbolic Logic. vol. 14, pp. 159-166.

Henkin, L. [1950]. Completeness in the theory of types.The Journal of Symbolic Logic. vol. 15. pp. 81-91.

• M. Manzano (USAL)

NIEC May 2017 24 / 25

References

Henkin, L. [1953]. Some Notes on Nominalism. The Journal of Symbolic Logic. vol. 18. pp. 19-29. Henkin, L., A theory of propositional types, Fundam. Math. 52 (1963), pp. 323—344. Henkin, L., Identity as a logical primitive, Philosophia 5 (1975),

• pp. 31—45.

Manzano, M., Extensions of first order logic. Cambridge Univ. Press., Cambridge, 1996. Manzano, M. Martins, M.A. and Huertas, A. [2014]. A Semantics for Equational Hybrid Propositional Type Theory. Bulletin of the Section

• f Logic. 43(3/4): 121-138. Łód´

z University Press Manzano, M. Sain, I. and Alonso, E. (eds) [2014]. The Life and Work

• f Leon Henkin. Essays on His Contributions. Springer Basil.
• M. Manzano (USAL)

NIEC May 2017 25 / 25