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Nameability, Identity, Equality and Completeness PhDs in Logic 2017 Bochum Mara Manzano Departamento Filosofa y Lgica y Filosofa de la Ciencia Universidad de Salamanca Espaa USAL May 2017 M. Manzano (USAL) NIEC May 2017 1 / 25

  1. 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

  2. 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

  3. 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

  4. 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 others M. Manzano (USAL) NIEC May 2017 7 / 25

  5. Identity and equality All identicals are equal, but some equals are more equal than others M. Manzano (USAL) NIEC May 2017 8 / 25

  6. Identity and equality in modal logic First-order modal logic In Modal Logic M. Manzano (USAL) NIEC May 2017 9 / 25

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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? x = y → � ( x = y ) necessity of identity ( NI ) M. Manzano (USAL) NIEC May 2017 9 / 25

  13. 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? x = y → � ( x = y ) necessity of identity ( NI ) substitutivity of identicals ( SI ) x = y → ( ϕ ( x ) → ϕ ( y )) M. Manzano (USAL) NIEC May 2017 9 / 25

  14. 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? x = y → � ( x = y ) necessity of identity ( NI ) substitutivity of identicals ( SI ) x = y → ( ϕ ( x ) → ϕ ( y )) are both sound? M. Manzano (USAL) NIEC May 2017 9 / 25

  15. 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? x = y → � ( x = y ) necessity of identity ( NI ) 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

  16. 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? x = y → � ( x = y ) necessity of identity ( NI ) 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

  17. 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? x = y → � ( x = y ) necessity of identity ( NI ) 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

  18. 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

  19. Identity and equality in modal logic Fitting and Meldensohn, (1998) Term Equality: Melvin Fitting introduces a new relation between intensional terms τ 1 � τ 2 ↔ Df � λ x , y . y = x � ( τ 2 , τ 1 ) M. Manzano (USAL) NIEC May 2017 10 / 25

  20. Identity and equality in modal logic Fitting and Meldensohn, (1998) Term Equality: Melvin Fitting introduces a new relation between intensional terms τ 1 � τ 2 ↔ Df � λ x , y . y = x � ( τ 2 , τ 1 ) M , w � g ( τ 1 � τ 2 ) iff ( τ 1 ) � ( w ) = ( τ 2 ) � ( 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

  21. Identity and equality in modal logic Fitting and Meldensohn, (1998) Term Equality: Melvin Fitting introduces a new relation between intensional terms τ 1 � τ 2 ↔ Df � λ x , y . y = x � ( τ 2 , τ 1 ) M , w � g ( τ 1 � τ 2 ) iff ( τ 1 ) � ( w ) = ( τ 2 ) � ( 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

  22. Identity and equality in modal logic Fitting and Meldensohn, (1998) Term Equality: Melvin Fitting introduces a new relation between intensional terms τ 1 � τ 2 ↔ Df � λ x , y . y = x � ( τ 2 , τ 1 ) M , w � g ( τ 1 � τ 2 ) iff ( τ 1 ) � ( w ) = ( τ 2 ) � ( 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 ‘ τ 1 � τ 2 asserts that the terms τ 1 and τ 2 designate the same object, which is quite a different thing.’ M. Manzano (USAL) NIEC May 2017 10 / 25

  23. Identity and equality in modal logic Fitting and Meldensohn, (1998) Term Equality: Melvin Fitting introduces a new relation between intensional terms τ 1 � τ 2 ↔ Df � λ x , y . y = x � ( τ 2 , τ 1 ) M , w � g ( τ 1 � τ 2 ) iff ( τ 1 ) � ( w ) = ( τ 2 ) � ( 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 ‘ τ 1 � τ 2 asserts that the terms τ 1 and τ 2 designate the same object, which is quite a different thing.’ the formula τ 1 � τ 2 → � ( τ 1 � τ 2 ) , is not valid. M. Manzano (USAL) NIEC May 2017 10 / 25

  24. Identity and equality in modal logic Fitting and Meldensohn, (1998) Term Equality: Melvin Fitting introduces a new relation between intensional terms τ 1 � τ 2 ↔ Df � λ x , y . y = x � ( τ 2 , τ 1 ) M , w � g ( τ 1 � τ 2 ) iff ( τ 1 ) � ( w ) = ( τ 2 ) � ( 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 ‘ τ 1 � τ 2 asserts that the terms τ 1 and τ 2 designate the same object, which is quite a different thing.’ the formula τ 1 � τ 2 → � ( τ 1 � τ 2 ) , is not valid. ‘In fact, � ( τ 1 � τ 2 ) express a notion considerable stronger than that of simple equality –it has the characteristics of synonymy ’ M. Manzano (USAL) NIEC May 2017 10 / 25

  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

  26. 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 Can we express in the language identity of worlds? 1 M. Manzano (USAL) NIEC May 2017 11 / 25

  27. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 to? M. Manzano (USAL) NIEC May 2017 11 / 25

  28. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 to? NO M. Manzano (USAL) NIEC May 2017 11 / 25

  29. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 to? NO This lack of expressivity of orthodox modal logic is an obvious weak M. Manzano (USAL) NIEC May 2017 11 / 25

  30. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 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

  31. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 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

  32. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 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

  33. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 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

  34. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 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

  35. 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 Can we express in the language identity of worlds? 1 what about the accessibility relation between worlds, can be referred 2 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

  36. Nominals and the satisfaction operator in hybrid logic Nominals are special propositional symbols i ∈ NOM M. Manzano (USAL) NIEC May 2017 12 / 25

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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 @ i j , together with @ i ♦ j are extremely important M. Manzano (USAL) NIEC May 2017 12 / 25

  42. 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 @ i j , together with @ i ♦ j are extremely important @ i j asserts that i and j name the same point. M. Manzano (USAL) NIEC May 2017 12 / 25

  43. 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 @ i j , together with @ i ♦ j are extremely important @ i j asserts that i and j name the same point. @ i j is a modal way of expressing what i = j would express in classical logic. M. Manzano (USAL) NIEC May 2017 12 / 25

  44. 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 @ i j , together with @ i ♦ j are extremely important @ i j asserts that i and j name the same point. @ i j is a modal way of expressing what i = j would express in classical logic. The reflexivity, symmetry and transitivity of equality are validities: @ i i @ i j → @ j i @ i j ∧ @ j k → @ i k M. Manzano (USAL) NIEC May 2017 12 / 25

  45. 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 @ i j , together with @ i ♦ j are extremely important @ i j asserts that i and j name the same point. @ i j is a modal way of expressing what i = j would express in classical logic. The reflexivity, symmetry and transitivity of equality are validities: @ i i @ i j → @ j i @ i j ∧ @ j k → @ i k @ 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

  46. 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 @ i j , together with @ i ♦ j are extremely important @ i j asserts that i and j name the same point. @ i j is a modal way of expressing what i = j would express in classical logic. The reflexivity, symmetry and transitivity of equality are validities: @ i i @ i j → @ j i @ i j ∧ @ j k → @ i k @ 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 @ i j ∨ @ i ♦ j ∨ @ j ♦ i M. Manzano (USAL) NIEC May 2017 12 / 25

  47. Zen Philosophy The book of perfect emptiness Tang de Ying asked Ge: M. Manzano (USAL) NIEC May 2017 13 / 25

  48. 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

  49. 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

  50. 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

  51. 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

  52. Zen Philosophy The book of perfect emptiness The argument can be reformulated in this way M. Manzano (USAL) NIEC May 2017 14 / 25

  53. Zen Philosophy The book of perfect emptiness The argument can be reformulated in this way α : = If things exist at a given point in time, then at any given 1 previous moment in time things must have existed. α : = q → Hq M. Manzano (USAL) NIEC May 2017 14 / 25

  54. Zen Philosophy The book of perfect emptiness The argument can be reformulated in this way α : = If things exist at a given point in time, then at any given 1 previous moment in time things must have existed. α : = q → Hq β : = Things exist today. 2 β : = @ t q M. Manzano (USAL) NIEC May 2017 14 / 25

  55. Zen Philosophy The book of perfect emptiness The argument can be reformulated in this way α : = If things exist at a given point in time, then at any given 1 previous moment in time things must have existed. α : = q → Hq β : = Things exist today. 2 β : = @ t q γ : = The dawn of time is previous to all else. 3 γ : = @ d H ⊥ M. Manzano (USAL) NIEC May 2017 14 / 25

  56. Zen Philosophy The book of perfect emptiness The argument can be reformulated in this way α : = If things exist at a given point in time, then at any given 1 previous moment in time things must have existed. α : = q → Hq β : = Things exist today. 2 β : = @ t q γ : = The dawn of time is previous to all else. 3 γ : = @ d H ⊥ δ : = Things existed at the dawn of time. 4 δ : = @ d q M. Manzano (USAL) NIEC May 2017 14 / 25

  57. Zen Philosophy The book of perfect emptiness The argument can be reformulated in this way α : = If things exist at a given point in time, then at any given 1 previous moment in time things must have existed. α : = q → Hq β : = Things exist today. 2 β : = @ t q γ : = The dawn of time is previous to all else. 3 γ : = @ d H ⊥ δ : = Things existed at the dawn of time. 4 δ : = @ d q To prove { α , β , γ } � δ we can use the trichotomy axiom 5 @ d t ∨ @ d Pt ∨ @ t Pd M. Manzano (USAL) NIEC May 2017 14 / 25

  58. 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

  59. 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

  60. 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

  61. 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

  62. 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

  63. 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

  64. 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

  65. 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

  66. 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

  67. 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

  68. 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

  69. 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 other meaningful expressions using: M. Manzano (USAL) NIEC May 2017 16 / 25

  70. 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 other meaningful expressions using: lambda operator λ , M. Manzano (USAL) NIEC May 2017 16 / 25

  71. 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 other 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

  72. 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 other 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

  73. 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 other 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

  74. 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 other 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

  75. 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 other 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 λ v 1 , . . . , v n τ ≡ λ v 1 , . . . , v n σ M. Manzano (USAL) NIEC May 2017 16 / 25

  76. 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

  77. 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

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