Binding Logic: Proofs and Models

We define an extension of predicate logic, called Binding Logic, where variables can be bound in terms and in propositions. We introduce a notion of model for this logic and prove a soundness and completeness theorem for it. This theorem is obtained by encoding this logic back into predicate logic and using the classical soundness and completeness theorem there. In predicate logic, only a quantifier can bind a variable. Function and predicate symbols cannot. In contrast, in the informal language of mathematics and in languages used in computer science, binders are pervasive, as they are used to express functions, relations, definite descriptions, fix-points, sums, products, integrals, derivatives, etc. There are many ways to express such constructions without explicit binding symbols using, for instance, combinators [7], de Bruijn indices [4], higher-order abstract syntax [18, 21], . . . However, as binders are everywhere in mathematics and in computer science, we believe that they should be studied as a basic notion, and not just encoded. The first contribution of this paper is to define a logic with binders, called Binding Logic, that is a simple extension of predicate logic: proof rules are the same and no extra axiom is added. A first question, about binding logic, is that of the provability of the extensionality scheme (∀x (t = u)) ⇒ Λx t = Λx u where Λ is a unary function symbol binding one variable in its argument. This scheme cannot be derived in a simple way from the axiom of equality [15]: ∀y ∀z (y = z ⇒ Λx y = Λx z) because substituting t for y and u for z renames the variable x to avoid captures. We shall prove that this scheme is indeed independent. Thus, if we want binders to be extensional, we must add this scheme, but Binding logic can also be used, if we do not, for instance, when we define weak reduction in λ-calculus or when we want to distinguish various algorithms that implement the same function. To prove such independence results, we want to construct counter-models. Thus the notion of model of predicate logics needs to be extended to Binding logic. The last and main contribution of this paper is to define such a notion of model, and prove that is sound and complete with respect to deduction. In predicate logic, a term t that contains a free variable x does not have a unique denotation in a given modelM. Instead, for each element a of the domain M of this model, we can define a denotation JtKa/x. It is tempting to define the absolute denotation of the term t as the function from M to M mapping a to JtKa/x. Then, we would obtain the denotation of Λx t by applying the denotation of the symbol Λ to the function denoted by the term t. The denotation of the symbol Λ would then be a functional Λ̂. However, with such a notion of model, some non provable propositions are valid in all models, e.g. the extensionality scheme. Indeed, if t and u are two terms containing the free variable x such that ∀x (t = u) is valid, then JtKa/x and JuKa/x are always equal. Hence the terms t and u denote the same function, and thus the terms Λx t and Λx u have the same denotation. As we want to have the choice to take this extensionality scheme or not, it should be valid in some models, but not all. In other words, extensionality should not be wired in the notion of model. Thus we need to consider a more general notion of model, leaving room to interpret a term containing a free variable as an “intensional function”. When, in the term Λx t, the body t contains other free variables than x, we interpret it as an intensional function of several arguments. Thus, to be able to define the denotation of the term Λx t, we must be able to apply the functional Λ̂ to intensional functions of various arities. Hence, we shall define the denotation of Λ as a family of functionals Λ̂0, Λ̂1, Λ̂2, . . . each of these taking as arguments functions of some arity. Once the notion of model is defined, we enter the proof of the completeness theorem. The proof proceeds by giving a provability preserving translation from binding logic to a formulation of predicate logic, called Deduction modulo. This translation is based on de Bruijn indices and explicit substitutions. Then, we relate the models of a theory in binding logic with those of its translation in Deduction modulo. It should be noticed that de Bruijn indices and explicit substitutions are used in the proof of the completeness theorem, but not in the definition of the notion of model itself. Hence we can imagine other completeness proofs using other methods such as a translation to higher-order logic, based on higher-order abstract syntax or a direct construction of a model à la Henkin. A potential difficulty in such a direct proof is to construct the elements of the model, that have to be intensional, and thus must differ from the functions used in Henkin’s construction [14, 2].