Abstract
Equational hybrid propositional type theory ( $$\mathsf {EHPTT}$$ ) 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, $$\Diamond $$ -saturated and extensionally algebraic-saturated due to the hybrid and equational nature of $$\mathsf {EHPTT}$$ . 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.
Highlights
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
The Barcan axioms are 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
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 @ij, which is key in order to have Robinson Diagrams
Summary
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. 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 the algebraic equations; the equality symbol between expressions of several types ≡; and the modal-hybrid operators of possibility ♦ and satisfaction @. The semantics of EHPTT is intensional, since the interpretation of algebraic individual constants and nominals are functions on the set of possible worlds It contains the standard hierarchy of types and focuses on equations and the identity relation. 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. The equational part of the logic requires another key property of the maximal consistent set of the proof; that of being extensionally algebraic-saturated
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
