Completeness in Equational Hybrid Propositional Type Theory

Equational hybrid propositional type theory (EHPTT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EHPTT}$$\end{document}) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra (a nonempty set with functions) and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: (i) Completeness in type theory, (ii) The completeness of the first-order functional calculus and (iii) Completeness in propositional type theory. More precisely, from (i) and (ii) we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, ◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Diamond $$\end{document}-saturated and extensionally algebraic-saturated due to the hybrid and equational nature of EHPTT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EHPTT}$$\end{document}. From (iii), we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system.


Introduction
In [15] and [16] Manzano and Moreno investigate the concepts of identity, equality, nameability and completeness and their mutual relationships on the following areas: first-order logic, second and higher order logic, type theory, first-order modal logic, modal type theory, hybrid type theory and propositional type theory. Concerning identity, we were interested in how to define it using other logical concepts (identity as definiendum) as well as in the opposite scheme (identity as definiens). The conclusion, as explained in the Section "The Kingdom of Identity" of [16], was that identity and lambda are the main logical constants that can be used to define all the other logical constants. As far as equality is concerned, considered as an equivalence relation between terms, the modal environment is crucial to reveal the difference between identity of objects and equality of terms that can receive different denotations at different worlds. Adding to modal logic the hybrid perspective, you are able to express identity of worlds as well as accessibility relation between worlds inside the formal language. Finally, equational logic was chosen because there identity is crucial and there are not interferences with other logical concepts.
Our point of departure is the following quote from Ramsey [18], according to which Wittgenstein maintained that logic is nothing but identities: "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. I have spent a lot of time developing such a theory, and found that it was faced with what seemed to me unsuperable difficulties." The relation of identity is usually understood as the binary relation which holds between any object and itself and which fails to hold between any two distinct objects. By equality we mean a binary relation between terms of the formal language which is reflexive, symmetric and transitive.
What are the logics where equality/identity plays a relevant role? In Henkin [10] we found a logic where equality (≡) and lambda (λ) are the only primitive logical constants, the logic of Propositional Type Theory (PTT). The main results of that paper are the definitions of the other logical constants, as well as the introduction of a calculus based on equality and lambda. The completeness proof is rather curious as it is not adapted from his well known completeness proofs for type theory and first-order logic. In this case, Henkin was able to give a name to each object in the propositional type hierarchy and he obtained the completeness result as an easy corollary.
Another logic where identity is central is Equational Logic (EL) and, therefore, we also incorporate its vision and tools into our paper.
Finally, we wanted to explore intensional contexts, where terms receive different denotations in different worlds. Moreover, when Kripke semantics is used and a domain of worlds appears, we wonder how to treat identity in this domain. That is why we took Hybrid Logic (HL), where the identity relation between worlds can be defined using nominals and the @ operator.
Our proposal is a logic that is able to incorporate all these capabilities. Equational Hybrid Propositional Type Theory (EHPTT) is a combined logic with equational significant ingredients. The language we will use includes as logical symbols ≡ and λ from PTT, but also ♦ and @ from HL. In a previous paper [13] we prove that there is no need to include a special equality symbol for equations as primitive, since it can be defined with lambda and equality. Our language contains variables of all propositional types as well as individual ones; as non-logical constants we have individual and function constants. Finally, the hybrid logic provides nominals to name worlds and formulas of the form @ i ♦j to express that the world named by j is accessible from i.
The structures used to interpret this language include the propositional type hierarchy built upon the set of the two truth values, an algebra, a Kripkean frame with a domain of worlds, an accesibility relation between worlds and the denotation of nominals.
The main result in this paper is the presentation of a set of axioms and the proof that such axiomatization is sound and complete with respect to the intensional Kripke style semantics presented in [13]. Our completeness proof for EHPTT owes much to the three proofs Henkin published last century: (i) Completeness in the Theory of Types [7] (ii), The Completeness of the First-order Functional Calculus [8] and (iii) Completeness of A Theory of Propositional Types [10] (see also [14] for a detailed explanation of these and other Henkin's completeness proofs).
The proof of completeness for our logic EHPTT follows, as usual, from the fact that any consistent set of meaningful sentences has a countable model. To achieve completeness, we use Henkin's method to build the model which is perfectly described by a maximal consistent set of formulas -but not any maximal consistent set will do. To deal with the complexity of our logic, the maximal consistent set needs to be also named and ♦-saturated, as required in hybrid logic. It is also compulsory for the maximal consistent set to be extensionally algebraic-saturated, so that one can distinguish different functions with rigidified constants witnessing such inequalities. The last property is used by Andrews in his book [2] to prove completeness for a full theory of types based on lambda and equality, and we have adapted the property to our hybrid situation.
What strategy shall one use to build the heterogeneous structures needed for EHPTT?
The Kripkean universe of worlds is just the set of equivalence classes of nominals defined by the sentences of the form @ i j in the maximal consistent set; the accesibility relation includes all pairs of equivalent classes of nominals our oracle declares are accesible from one another via sentences of the form @ i ♦j.
How do we construct the rest of the structure? In this part we will follow Henkin's [8] recipe almost to the letter. We will do it, as we did in [3], via a function Φ to be defined using equivalent classes of expressions obtained from the information our oracle provides; namely, the maximal consistent set just built.
To define the domain of individuals and the hierarchy of propositional types, as well as the interpretation of the non-logical constants, we not only define the function Φ as acting on expressions of the form @ i F but, simultaneously, we define the domains D * α for each type, even for propositional types. In this particular case, instead of taking the standard hierarchy as it is usually defined, we do it through the names of the objects in the hierarchy. In essence, we shall take equivalence classes as elements of the domains, but for type αβ we need functions from D * α to D * β . So we define Φ as a map which corresponds, in a proper sense, to the equivalence classes.
Why do we proceed that way in the propositional case? To better integrate our heterogeneous logic ingredients, the propositional type hierarchy is also built with the function Φ, aside from the final result being the standard one, it also entails that each type in the hierarchy is the value of its own name under Φ. In this case, the already mentioned result of PTT concerning the nameability of each type is necessary.
What elements are there in the algebra? From Henkin's first-order completeness proof [7] we take the idea of building the structure out of equivalent classes of individuals using them as objects. Here we do a similar thing, but in our case we need rigidified terms provided by the satisfaction operator @. Nominals and expressions of the form @ i τ play a central role in the completeness result: nominals are the building blocks of the world universe, while @ i τ expressions supply the architectural blueprint in the algebraic construction.
Outline of the paper. The ouline of the paper, organized in five more sections, is the following. Section 2 is an overview of the main results of Henkin's propositional type theory [10]. Also, its language and semantics are formulated in an updated form that makes it easier to use. There, the classical connectives and quantifiers are defined using only lambda and equality. Moreover, the combination of lambda plus equality goes much further, as they provide a name for each object in the propositional type hierarchy. The tool used to name all the objects, the descriptor operator, is also presented and commented in order to understand the beautiful and useful construction of Henkin. Names for propositional types and denotations do match, as we can prove a Nameability Theorem saying that one can associate with each element χ of an arbitrary type a closed formula χ N of the corresponding type such that the interpretation of this sentence turns out to be χ. Finally, Henkin's calculus and the main idea behind its interesting and novel completeness theorem is described.
In Section 3, the syntax and semantics of the EHPTT logic is presented. As it is the result of the combination of PTT, HL and EL, there are characteristics elements of each of them. Furthermore, the integration of the three logics into the new one, is illustrated by the fact that the logical symbols of EHPTT are only four: the lambda operator λ, used for building functions in the hierarchy of types and also the algebraic equations; the equality symbol between expressions of several types ≡; and the modal-hybrid operators of possibility ♦ and satisfaction @. All the remaining logical symbols (propositional connectives, quantifiers, algebraic equations, intensional modal expressions, etc.) can be derived from those. Moreover, the semantics of EHPTT is intensional, since the interpretation of algebraic individual constants and nominals are functions on the set of possible worlds. In addition, it contains the standard hierarchy of types and focuses on equations and the identity relation.
In Section 4, the EHPTT calculus is defined. On the one hand, it extends that of Henkin's in [10]. On the other hand, it contains the most characteristic axioms of the basic hybrid logic that capture the operator @. Likewise, the functional aspect of algebraic logic and propositional type theory is also an essential element.
In Sections 5 and 6, the completeness proof for EHPTT is developed. It is a Henkin-style completeness proof for Type Theory. Thus, in order to build the model of a consistent set of formulas, a maximal consistent set fulfilling the appropriate properties is used. We define a function Φ on rigidified expressions of the form @ i F , and simultaneously the domains of the model that the maximal consistent set describes. Besides, the "names" of the objects in the hierarchy of types play a relevant role, as in Henkin's original proof. This is the "type theoretical" part of the proof. On the other hand, the hybrid element of EHPTT is represented by the central role of the rigidified expressions, and by two important properties of the maximal consistent set used in the proof: those of being named (containing at least a world) and being ♦-saturated. Finally, the equational part of the logic requires another key property of the maximal consistent set of the proof; that of being extensionally algebraic-saturated.

Background: Henkin Propositional Types
In this section we will sketch briefly the main results of Henkin's propositional type theory as presented in [10] and also discussed in [15,16].
In the first place, we introduce the formalism and its semantics. We have adapted his language in order to make an easier reading and recognize his theory within ours. Using only lambda and equality, Henkin was able to introduce the classical connectives and quantifiers as defined operators. And not only that, the language also provides a name for each object in the hierarchy. The fundamental tool used to name all the objects is the descriptor operator, which is also defined in the language using, in this case, an election function for each type. Due to the fact that each type is finite, we easily construct such a function without requiring the axiom of choice.
Finally, we recall Henkin's calculus and describe the main idea behind this interesting and novel completeness theorem. Henkin uses the names of the objects of the hierarchy to get that completeness result.

Language and Semantics
According to Henkin's definition, the hierarchy of propositional types, PT, is the least class of sets containing D t as an element, which is closed under passage from D α and D β to D αβ . Here D t is the two truth values set, D t = {T, F }, while D αβ is the set of all functions mapping D α to D β . 1 Let us call PT to the set of type symbols: 2 PT ::= t | αβ for any α, β ∈ PT To build the theory of propositional types, Henkin introduces a formal language with variables for each type, the lambda abstractor, λ, and a collection of equality constants, ≡ α αt , one for each propositional type α ∈ PT. To be more specific, expressions of this theory are either: (1) variables of any type X α , (2) the constants ≡ α αt , (3) A βα B β or (4) λX β B α .
Interpretations of these expressions on the hierarchy PT are recursively defined with the help of assignments, which give values in PT to variables of all types. In particular, for a given assignment g, we recursively define the interpretation V(A α , g) for any A α : (1) V(X α , g) = g(X α ), (2) V(≡ α αt , g) is the identity relation on type α, (3) V(A βα B β , g) is the value of the function V(A βα , g) for the argument V(B β , g) and (4) V(λX α B β , g) is the function of D αβ whose value for any χ ∈ D α is the element V(B β , g χ X α ) of D β . 3 Classical logical constants as defined operators. As we mentioned already, using only the equality ≡ α αt and λ, the remaining connectives as well as the quantifiers ∀X α -for each propositional variable of any propositional type α-are presented as defined operators. 1 Henkin uses the reverse notation; namely, D αβ is the set of all functions from D β to Dα but nowadays it is more common to use the one we have adopted.
2 Even though PT and PT are different sets, it is common to refer to both as types. 3 Here g χ Xα is the Xα− variant of g sending Xα to χ and leaving the rest of values as in g.
The power of the combination of lambda plus equality goes much further, as these symbols provide a name for each object in the propositional type hierarchy. The strategy described in [10] to produce a name for each object in PT rests upon the power of the description operator. In order to define the description operator properly, one fixes one element for each propositional type; this element would serve as the denotation of improper descriptions. The definition is done by induction on types: for type t we just take a t = F ; for type αβ we take the constant function f αβ with value b β for every element of D α , where b β is the element in D β already chosen. Thus, f αβ χ = b β for each χ ∈ D α . Now, using these elements, an election function t (α) can be defined for each type. For any arbitrary type α let t (α) be the function of D α,t ,α such that, for any χ αt ∈ D α,t , we have that (t (α) χ αt ) is the unique element χ α ∈ D α for which (χ αt χ α ) = T , in case there is such a unique element χ α , or else (t (α) χ αt ) = a α if there is no such a particular χ α , or if there are more than one χ α , such that (χ αt χ α ) = T . Having done that Henkin continues: 'We shall show inductively that for each α there is a closed formula ι αt ,α such that (ι αt α ) d = t (α) . Then, for any formula A t and variable X α we shall set (jX α A t ) = (ι αt α (λX α A t )).' 5 Having introduced the description operator, we will produce a name χ N α for each χ α ∈ D α and then prove the renowned "nameability theorem".
Theorem 2 (Nameability Theorem). For each element χ ∈ D α of any arbitrary type α, there exists a sentence χ N of type α such that V(χ N , g) = χ for any assignment g.
Henkin proves this theorem by induction on the hierarchy's construction. Names for the basic object T and F of type t are given by the previous Definition 1. For type αβ , assuming that the theorem is proven for types α and β, we set a name for every function χ αβ which maps every element (χ α ) i of the finite type D α , say D α = {(χ α ) 1 , . . . , (χ α ) q }, to the corresponding value in D β , that is, to (χ αβ (χ α ) i ). To this effect, the names of the objects in D α and D β (whose existence is assumed by induction hypothesis) as well as the descriptor operator are used. To introduce χ N αβ we need to formalize the following: when variable X α is just the name of an object (χ α ) i of type α -that is,

Calculus
For the theory of propositional types Henkin offers a calculus based on λ and equality rules. 6 Let us quote Henkin's seven axioms and Replacement Rule (See Henkin [10], p. 330).
where Y γ is a variable free in A α . 5.2. By the Rule of Replacement we refer to the ternary relation on formulas of type t which holds for A t , C t , D t if and only if A t = (A α ≡ B α ) for some formulas A α and B α and D t is obtained from C t by replacing one occurrence of A α by an occurrence of B α . When this situation holds for A α ≡ B α , C t , D t we shall say that D t is obtained by Rule R from A α ≡ B α and C t .

Completeness.
This calculus is complete. The method of proof is rather different from Henkin's previous completeness theorems for type theory [8] and first-order logic [7]. The next lemma and proposition synthesize the whole idea of the proof which rest upon the Nameability theorem 2 (cf. [10]). The important result from where the completeness theorem easily follows has the amazing form: Lemma 3. For any formula A α and assignment g As you see, the formulation resembles the substitution lemma, but instead of being a semantics metatheorem it is a theorem of the calculus. The lemma is proved in [10], pp. 341-343, by induction on the length of A α .
The obvious question we ask is, how to prove that |= A t implies A t for any formula of type t ? We will see that Lemma 3 implies completeness.
Theorem 4 (Completeness). |= A t implies A t , for any formula of type t.
Proof. If A t is closed then |= A t implies V(A t , g) = T for any assignment g. Thus the Lemma 3 gives A t ≡ (V(A t , g)) N which turns to be A t ≡ T N , where T N is the name of the truth value true.
But using the calculus, in particular, Axiom Schema 2 and the Rule of replacement R, we obtain the desired result, A t .
In case A t were a valid formula, |= A t , but not a sentence, we pass from A t to the sentence ∀X γ 1 Applying the rules of the calculus, we obtain A t .

Other Important Results
Theorem 32 states that all theorems of Henkin's propositional type theory are also provable in our EHPTT calculus. That means that we can just give the reference of Henkin's demonstration, when necessary.
Lemma 5 ([10], p. 340). For each χ αβ ∈ D αβ and χ α ∈ D α we have For each type α and assignment g, we have Using the calculus, in particular Axiom 2 and the rule of replacement Proof. We have already proven its validity as Lemma 6. Then we can use Henkin's completeness Theorem 4 for Propositional Type Theory.

Equational Hybrid Propositional Type Theory (EHPTT)
We are facing the challenge posed by the identity relation and the equality symbol by designing a language with four basic components: propositional type theory, first-order equational logic, modal and hybrid logics.

Syntax
Definition 8 (Type Symbols). Let t and 0 be any fixed objects. The set TYPES = AT ∪ PT of types of EHPTT is defined as follows: • Algebraic Type Symbols, AT ::= 0 | 0 . . . 0 n times (to simplify notation we will write n for 0, . . . , 0 n times ).
• Propositional Type Symbols, PT ::= t | αβ , α, β ∈ PT In the sequel we will use a, b for indiscriminate types.
Definition 9 (Language). The set of meaningful expressions of EHPTT is built on the EHPTT language containing: -a family CON = CON n : n ∈ AT of at most denumerably infinite sets of non-logical constants such that for each n, CON n is a set of functional symbols of type n for n = 0 and CON 0 is a set of individual constants.
-a denumerably infinite set VAR a of variables V a , for each type a ∈ PT ∪ {0}; we will use X α , Y α , Z α , . . . for variables of propositional type α, and v 1 , v 2 , v 3 , . . . for individual variables.
-a denumerably infinite set NOM of nominals.
Definition 10. By recursion we define the set ME a of meaningful expressions of type a.
• Propositional terms (ME α ): As in the previous case, only lambda terms contain bounded variables.
• Formulas (ME t ): The set Free( ) of free variables in expression is defined in the usual way, being lambda functions the only expressions where variables are bounded.

Remark 11.
• As you probably noticed, formulas like F a ≡ G a (with F a , G a ∈ ME a and a ∈ PT ∪ AT − {0}) are rather heterogeneous as they can have both algebraic and propositional components -as in X t ≡ (λv 1 · · · v n c ≡ δ) ∈ ME t (with X t ∈ VAR t , c ∈ ME 0 and δ ∈ ME n for n = 0); they also can be pure algebraic, -like γ ≡ λvv ∈ ME t (with γ ∈ ME 1 )-and pure • All pure algebraic formulas are closed.
• The next definition extends Definition 1 with other logical symbols in terms of λ and ≡, like algebraic equations and propositional quantifiers for all formulas of EHPTT. Having defined ¬ and ∧, we will use the common definitions of ∨ and → using them. We will use as well the operator We extend Definition 10 with the two items below: Definition 12 (Logical operators).
It is a closed formula of type t. These formulas are called "equations". 7 Once we define the semantics for our EHPTT in Section 3.2, it is easy to see that all the introduced connectives behave as expected. That is, with equality, abstraction, nominals, and @ we can define all basic standard operators needed for equational hybrid propositional type theory.
It is important to identify the expressions coming from propositional type theory as a basic ingredient of our combined logic, we will call these expressions "strictly propositional". These expressions are actually Henkin's as defined in [10]. They will play a decisive role in our approach to EHPTT. Next we consider those formulas of our system which correspond to formulas of the ordinary untyped propositional logic.
Definition 13. We say that an expression A β ∈ ME β is strictly propositional if it is a meaninful expression recursively built using only the following rules: Definition 14. We define the class of P -formula to be the least class of formulas containing T N and F N and each variable X t of type t as members, which is such that whenever A t and B t are in the class, then so are ¬A t ,

Semantics
In EHPTT the semantic is intensional (i.e., every meaningful expression has an intensional interpretation) while the language has no symbols of intensional types. In practice, a meaningful expression of type a receives as interpretation a function from the set of worlds W to the universe D a , that function is an object of intensional type.
and A is a non empty set -the carrier set of the algebras (in order to unify notation we will use D 0 = A ).
2. PT= D α α∈PT , the hierarchy of standard propositional types, is defined recursively by: 3. I is a function whose domain is the union of the set of nominals with the set of all individual constants and the set of functional symbols such that Definition 16. An assignment g of values to variables is a function having as domain the set VAR = VAR 0 ∪ VAR PT of all variables, and such that for any variable V a ∈ VAR, the value is in the appropriate domain, that is, Observe that the assignment is extensional since the value of any variable of any type is an object of the same type. The interpretation of a variable of type a is redefined below as a constant function of intensional type.
We also define, as usual, that an assignment g is a V a -variant of an assignment g if it coincides with g in all values except perhaps in the value assigned to V a . We will use g θ V a to denote the V a -variant assignment of g whose value for variable V a is θ ∈ D a . Namely, Definition 17. An interpretation for EHPTT is a pair I = M, g , where M is a structure for EHPTT and g is an assignment of values to variables. We denote the interpretation M, g θ V a by I θ V a . Given a structure M and an assignment g we recursively define for any expression F a the interpretation of F a with respect to I, denoted by (F a ) I .
1. Algebraic terms. Let w ∈ W be arbitrary 2. Propositional terms. Let w ∈ W be arbitrary 3. Formulas. Let w ∈ W be arbitrary Definition 18 (Validity, consequence and tautology).
1. We say that a formula A t is true at the world w under the interpretation

A P-formula (Definition 14) which is valid is called a tautology.
Lemma 19. Let A α be any strictly propositional expression (Definition 13) and I = M, g an interpretation, then where w is any element of W and g PT is the restriction of the assignment g to variables of propositional types. V(A α , g PT ) is the classical evaluation as defined 2.1, which corresponds to the definition offered in Henkin [10], p. 326.
Proof. The proof can be done by induction. Proof. Straightforward by induction on the construction of the meaninful expressions.

Variables, Substitution, and Rigidity
Next we define one of the paper's key concept, that of rigid expressions. These expressions are introduced recursively and we will prove that they have the same value at all worlds. Good examples are variables of all types (after all, variable denotations are determined globally and directly by assignment functions), and expressions prefixed by an @ operator (indeed, these operators were designed with rigidification in mind). Rigid expressions play a key role in our axiomatization.
Definition 22 (Rigid meaningful expressions). The set RIGIDS of rigid meaningful expressions is defined inductively as follows: Remark 23. Previous definition introduces the set RIGIDS of rigids expressions by recursion. Expressions of the form @ i F a are called rigidified expressions. Next lemma proves that all the elements in RIGIDS receives a rigid interpretation, namely the interpretation is independent of the evaluation's world. In particular, all strictly propositional expressions (Definition 13) are rigids.
Lemma 24 (Rigids are rigid). Let I be a an interpretation. If A ∈ RIGIDS then (A) Proof. By induction on the construction of rigid expressions. We give the case for: λv 1 · · · v n τ with τ ∈ RIGIDS. Let I = M, g and w, w ∈ W. .
For example: Rigid expressions are well-behaved with respect to substitution, as stated in the following lemma Lemma 26 (Rigid substitution). Let I be an interpretation and w a world on it: Proof. Straightforward by induction on the construction of meaningful expressions, with the help of Coincidence Lemma 21.
Corollary 27. Let B t be a formula.
provided that the variables X 1 α 1 , . . . , X n α n are distinct from one another and that A i α i is rigid and free for X i α i in ϕ for all i = 1, . . . , n.

Nameability in EHPTT
As a consequence of our Lemma 19 we obtain nameability for the strict propositional fragment of our EHPTT logic. In the first place, our Definition 12 extends the Definition 1 of all common logical operators using lambda and propositional equality to cover the needs of of our equational system. The next theorem, proved in [13], states that all the introduced operators behave as usual. That is, with equality, abstraction, nominals, and @ we can define all basic standard operators needed for equational hybrid propositional type theory.
Theorem 28. For every interpretation I the following holds: In the sequel we will use the usual quantifiers as well as the connectives ⊥, ¬, ∨, ∧, → and ↔ as abreviations. Recall that ↔ is a special case of ≡ for type t.
The Nameability Theorem also applies for EHPTT: Theorem 29 (Nameability Theorem). We shall associate, with each element χ of an arbitrary type D α , a closed formula χ N of type α such that (χ N ) I (w) = χ for any interpretation I and world w.
Proof. The proof is obvious using Lemma 19 and Henkin's Nameability Theorem. 8
where γ, δ ∈ ME n and c 1 , . . . , c n ∈ CON 0 do not occur in Δ. From ϕ and (A α ≡ B α ) to infer the result of replacing one occurrence of A α in ϕ by an occurrence of B α , provided that the occurrence of A α in ϕ is not (an occurrence of a variable) immediately preceded by λ.
4. Name: If @ i ϕ and i does not occur in ϕ, then ϕ.

Bounded Generalization:
If @ i ♦j → @ j ϕ and j = i and j does not occur in ϕ, then @ i ϕ.
These are all standard rules drawn from the literature on modal and hybrid logic. For a detailed discussion of the Name and Bounded Generalization rules, see Blackburn and Ten Cate [6]. The restriction in the rigid replacement rule that nominals must replace nominals is standard in hybrid logic; it reflects the fact that nominals embody namelike information, and replacement must respect this. The additional restriction we have imposed (that variables can only be freely replaced by rigid terms and vice-versa) reflects the fact that assignment functions interpret variables rigidly, and replacement must respect this too.

Axioms
We will give the logical axioms as general schemas.

Barcan Axioms:
(a) Prop: Remark 30. This set of axioms exhibits that the combination of the three logics (propositional type theory, equational logic and hybrid modal logic) is not obtained just by putting together the axioms of each one. There are axioms that come from a specific logic (e.g., Henkin PTT axiom 4, distributive axioms, algebraic functionality, etc), as well as axioms that integrate attributes from more than one logic (λ-conversion axioms, equality axioms, etc.). There are also others that are just reformulations for the combined language (e.g., tautologies). One interesting axiom is the quantifier axiom that substantiates the importance of the Henkin's nameability and the finiteness of D α in the hierarchy of propositional types. The Barcan axioms are also well known in first order modal logic, they are connected with the fact that in our semantics the algebras at each world are over the same set A, that is, we are dealing with constants domains.
Since our semantics is intensional, some axioms have to be restricted to rigid (or rigidified) terms and/or formulas.
Definition 31. A deduction of ϕ is a finite sequence ξ 1 . . . ξ n of expressions in ME t such that ξ n := ϕ and for every 1 ≤ i ≤ n, either: -ξ i is an axiom or -ξ i is obtained from previous expressions in the sequence using the rules.
We will write ϕ whenever we have such a sequence and we will say that ϕ is an EHPTT-theorem of the calculus.

EHPTT Contains PTT
It is easy to see that the calculus of our EHPTT logic contains that of Henkin's PTT.
Theorem 32. If PTT A t then EHPTT A t for any A t formula of PTT.
The proofs of Henkin's Axioms 2, 3 and 5 in our EHPTT calculus are in the appendix (Theorems 68, 70 and 69).

Maximal Consistent Sets
In this section we define and explore maximal consistent sets of EHPTT sentences with various useful properties, prove the variant of Lindenbaum's Lemma we shall require and then introduce several equivalence relations using a saturated maximal consistent set.
Definition 33. A set Δ ⊆ ME t is inconsistent (or contradictory) iff for every ϕ ∈ ME t , Δ ϕ. Δ is consistent iff it is not inconsistent. Δ is a maximally consistent set iff Δ is consistent and whenever ϕ ∈ ME t and ϕ / ∈ Δ, then Δ ∪ {ϕ} is inconsistent.
The following lemmas state some well known consequences of the definitions and rules of the calculus.
5. If Δ is consistent, then for all ϕ ∈ ME t such that Δ ϕ we have Δ ∪ {ϕ} is consistent. 6. Δ is consistent iff every finite subset of Δ is consistent.

Maximal Consistent, Named, ♦−saturated and Extensionally Algebraic-Saturated Sets
Definition 36. Let Γ be a set of sentences.
1. Γ is named iff one of its elements is a nominal.
3. Γ is extensionally algebraic-saturated if for all expressions @ i γ ≡ @ j δ, with γ, δ ∈ ME n , there are rigid terms @ k 1 τ 1 , . . . , @ k n τ n ∈ ME 0 such that We must now prove that any consistent set of formulas can be extended to a maximal consistent set with all three desirable properties. We need, in short, the following version of Lindenbaum's Lemma: Lemma 37 (Extended Lindenbaum). Let Γ be a consistent set of sentences. Then Γ can be extended to a maximal consistent set Γ ω which is named, ♦-saturated and extensionally algebraic-saturated.
Proof. Let {i n } n∈ω be an enumeration of a countably infinite set of new nominals, {c n } n∈ω an enumeration of a countably infinite set of new constants of type 0, and {ϕ n } n∈ω an enumeration of the sentences of the extended language. We will build {Γ n } n∈ω , a family of subsets of ME t , by induction: • Assume that Γ n has already been built. To define Γ n+1 we distinguish four cases: c 1 , . . . , c m are the first constants of type 0 not in Γ n or ϕ n and k 1 , . . . , k n are the first nominals not in Γ n or ϕ n .
First we show by induction that each Γ n is consistent. For the base case, suppose Γ 0 is inconsistent. Hence Γ ∪ {i 0 } ⊥, then, by Theorem 54, Γ i 0 → ⊥ and by Name * (Theorem 59), Γ ⊥, which is impossible. Now assume as inductive hypothesis that Γ n is consistent. Γ n+1 has only four possible forms: 1. Γ n+1 = Γ n is consistent by the induction hypothesis.

Equivalence Relations Using a Saturated Maximal Consistent Set
Let Δ be a maximal consistent named, ♦-saturated and extensionally algebraic-saturated. We define an equivalence relation on the set NOM of nominals.
Definition 38. Let Δ be a maximal consistent named, ♦-saturated and extensionally algebraic-saturated. Define for all i, j ∈ NOM the relation ∼: Proposition 39. The relation i ∼ j is an equivalence on the set NOM of nominals.
Proof. It is trivial to prove that ∼ is an equivalence relation by using the theorems of maximal consistency and applying reflexivity (Axiom 4) and theorems of symmetry (Theorem 63) and transitivity (Theorem 64).
And then, we define the equivalence class [i] = {j ∈ NOM : i ∼ j}.
Definition 40. Let Δ be a maximal consistent which is named, ♦ saturated and extensionally algebraic-saturated. The accessibility relation is a binary relation on W defined by Proposition 41. R is well defined.
Proof. To prove that the definition is independent of the representatives, we use Theorem 60 (Nom) and Theorem 66, the definition of the equivalence relation and that Δ is maximal consistent.
Next, we define another equivalence relation ∼ for expressions of algebraic types.
Definition 42. Let Δ be a maximal consistent which named, ♦ saturated and extensionally algebraic-saturated. We define a relation ∼ on rigidified algebraic expressions of any type n: • Equivalence relation on type 0: @ i τ ∼ @ j τ iff @ i τ ≈ @ j τ ∈ Δ, for all rigid terms of the form @ i τ or @ j τ .
Proposition 43. The relation ∼ is an equivalence on the set of rigid algebraic expressions of any type n.
Proof. In both cases, it is trivial to prove that ∼ is indeed an equivalence relation. For terms of type n = 0, it follows from Axioms 4, 4 and 4. For type 0 we do not need to prove that the relation ∼ is an equivalence relation, as equations of this form @ i τ ≈ @ j τ are defined as λv@ i τ ≡ λv@ j τ . Using Barcan Axiom 4 we see that this very relation could also be defined with @ i λvτ ≡ @ j λvτ and we have already proved that the relation @ i λvτ ∼ @ j λvτ is an equivalence.
And then, we define the equivalence classes for all algebraic types. In the next section we will build a model using the set of all classes of rigidified terms of type 0 as the domain of individuals.

Completeness
In this section we will build a structure using the information in the set Δ which is maximal consistent, named, ♦-saturated and extensionally algebraic-saturated. The domain W and the relation R are defined (Definition 40) in the usual way based on the equivalence relation ∼ on nominals (Definition 38).
To define the domain of individuals and the hierarchy of propositional types, as well as the interpretation of the non-logical constants, we define a function Φ acting on expressions of the form @ i F and, simultaneously, we define the domains D * a for each type, even for propositional types. In this particular case, instead of taking the standard hierarchy as it, we build it through the names of the objects in the hierarchy. These names play a very relevant role, similar to the one played in Henkin's original proof. The decision was made in order to easy our final objective, namely, the Completeness Theorem.
In essence, we shall take equivalence classes as elements of the domains, but for type αβ we need functions from D * α to D * β . So we define Φ as a map which corresponds, in a proper sense, to the equivalence classes.

Definition of the Function Φ and Associated Domains
Function Φ for algebraic types. On the first place we define a function Φ on rigidified algebraic expressions, as well as the domain of individuals of the Δ-model to be defined afterwards.
The equivalence relation used in this definition is the one introduced at the end of the previous section, namely, Definition 42.
Lemma 45 (Algebraic types). Φ is well defined for all algebraic types.
Type n. In order to see that it is well defined we have to show that 1. Φ is independent of the representatives.
2. Φ respects the equivalence relation on type n, i.e., We prove both items for n = 1, for the remaining algebraic types it is similar.
from Axiom 4 and properties of maximal consistent sets.
Remark 46. Obviously, The function Φ on type 0 characterized the universe of individuals. As we do not have variables of type n = 0, we are not defining universes D * n . On type n the function will be used to define the interpretation of functional symbols as the value under Φ, guaranteeing that such an interpretation is a n-ary function on the domain of individuals.

Definition of Φ for propositional types.
Lemma 47 (Propositional types). Given a maximal consistent set Δ, which is named, ♦ saturated and extensionally algebraical-saturated, there exists a family of domains D * α α∈P T and a function Φ satisfying: 1. Φ is well defined over the set CME α ∩ RIGIDS and so we define: 3. χ α = Φ(χ N α ) for each type α ∈ PT and any χ α ∈ D α (where D α are the standard echelons of the hierarchy built on D t = {T, F }) Proof. The proof is by induction on propositional types by simultaneously defining the function Φ and the hierarchy D * Let us see that the 4 conditions above are satisfied.
1. Since Δ is maximal consistent, exactly one of this conditions A t ∈ Δ or ¬A t ∈ Δ holds, for all A t ∈ CME t ∩ RIGIDS. That proves the first requirement. 3. D t = {T, F }, according to the standard definition. We need to prove that Φ(T N ) = T and Φ(F N ) = F . As already defined, T N := ((λX t X t ) ≡ (λX t X t )) and F N := ((λX t X t ) ≡ (λX t T n )). It is obvious that both are in the set CME t ∩RIGIDS as both are strictly propositional (Definition 13) sentences.T N ∈ Δ, since T N , using our Axiom 4. Hence, Φ(T N ) = T . F N ∈ Δ, since F N ≡ ¬T N (Theorem 67) and Lemma 35 on maximal consistent sets. Therefore , Φ(F N ) = F .
Type αβ , α, β ∈ PT. Assume the two domains D * α and D * β are defined as well as the function Φ acting on CME α ∩ RIGIDS and CME β ∩ RIGIDS. Assume as well the induction hypothesis. In particular: for all elements A α , A α ∈ CME α ∩ RIGIDS and B β , B β ∈ CME β ∩ RIGIDS.
A αβ ∈ CME αβ ∩ RIGIDS} and we need to prove that Φ is independent of the particular representatives chosen.

Defining the Δ-structure
With the domains already defined it is straightforward to complete the definition of the required structure by defining W, R and I. The domain W and the relation R were introduced in Section 5.2.

Propositional expressions.
• For variables of any propositional type in PT − {t}, say α. We want to , by Axiom 4, max. consist. and props. of Φ • For lambda expressions λX α A β of propositional type α, β we want to prove that We will prove that both functions (λX α A β ) I ([i]) and give the same value for each argument. Axiom 4, the properties of Δ as a maximal consistent set and the properties of Φ respecting equivalent relation based on Δ), we can use the induction hypothesis to get , β conversion (4) , by ind. hypothesis , by Lemma 26 By using the induction hypothesis and the properties of Φ we get ((g(X 1 Finally using n-times Lemma 26, Therefore, both functions ) are the same.

• For any expression
Step (1) holds by definition, (2) by induction hypothesis and (3) is by definition of Φ. Using Axiom 4 (Rigid function application) and the properties of maximal consistent sets we get that justifies the last step in our proof, as Φ respects the equivalence based on Δ.
• For any expression @ j A α of propositional type α = t we want to prove that , by def.

Formulas.
• For variables of type t we prove that ) using the same argument used in the previuos case for propositional variables of type α = t.
• For expression A αβ B α of propositional type β = t, the proof is similar to the corresponding case above for type αβ with β = t.
• (@ j ϕ) I ([i]) can be proved as we did for similar expressions above.
Theorem 51 (Henkin's Theorem). Every consistent set of sentences has a model.

Conclusions and Future Work
Identity and equality can be considered in different contexts with a diversity of meanings; for example: (1) in an algebraic context, equational identity is used to build equational theories and equational classes which are the basis of Universal Algebra; (2) in the context of propositional type theory, identity takes the form of the biconditional connective at the first level and it plays an important role in order to define other connectives and quantifiers with the help of lambda operator; (3) in Hybrid logic, identity between worlds can be defined by the formula @ i j, which is key in order to have Robinson Diagrams.
The logic we discuss in this paper, EHPTT, combines these three different logics, namely, propositional type theory, equational logic and hybrid modal logic. These logics have the three identities we describe above, and more. This combined nature is rather obvious in the syntax, as well as in the list of axioms and rules we include in the proof system we propose. In most sentences the three components are present and we have to create an harmonious unity. In the new logic the equational terms and formulas receive intensional interpretations, and that change is the origin of a completely novel environment. In the new landscape, most of the axioms and rules dealing with equality are restricted to rigid expressions. Intentionally, we did not make any effort to get a smaller set of axioms; on the contrary, we group the axioms in order to explicitly show the heterogeneous nature of our logic and to help the reader identify the origin of each axiom (however, it will be an interesting exercise to find an independent set of axioms for our logic).
The formal achievement of this paper is the axiomatic calculus and the proof that such axiomatization is sound and complete with respect to the intensional Kripke style semantics presented in [13]. Our completeness proof for EHPTT is inspired in the work of Henkin on completeness, namely in the three proofs Henkin published last century (see [7], [8] and [10]).
There are still open questions that we would like to address in the near future. First, we would like to study the application of Henkin's method employed in the proof of completeness for Propositional Type Theory to a Many-valued Propositional Type Theory. We think that one can build the type hierarchy from the enlarged finite set of truth values as Henkin did from {T, F }, and use a similar nameability theorem to achieve completeness. In our logic EHPTT we assume that the universe of individuals is the same in each world; however, there are circumstances where some elements do not exist in some worlds. We think all this could be accommodated in a specific logic, by developing a propositional type theory allowing more truth values, and so partial functions appearing in the algebraic part could receive an adequate treatment.
Finally, the proliferation of logical systems used in mathematics, computer science, philosophy and linguistics (to what we also have contributed with our system EHPTT) makes their relationships between and their possible translations into one another a pressing issue. Translation between logics has been scarcely formulated as an ambitious paradigm whose tools would serve for handling the multiplicity of logics. From a strictly pragmatic perspective, the translation can allow, for example, borrowing proof systems or completeness theorems from one to another. We would like to study translation from the logics we have been studying into well behaved and studied logics, like many-sorted logic, general logics or labelled deduction systems.
Theorem 56. The two theorems below are easily obtained in our calculus: Theorem 57.
Theorem 58. The following are derivable Name If i → ϕ then ϕ where i ∈ NOM does not occur in ϕ Paste ♦ If (@ i ♦j ∧ @ j ϕ) → ψ and j = i and j does not occur in ϕ and ψ, then @ i ♦ϕ → ψ Corollary 59. The following is also provable Name * If Δ i → ϕ then Δ ϕ where i ∈ NOM does not occur in Δ∪{ϕ}.
Theorem 67. F N ≡ ¬T N