Curry–Howard–Lambek Correspondence for Intuitionistic Belief

This paper introduces a natural deduction calculus for intuitionistic logic of beliefIEL-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {IEL}^{-}$$\end{document} which is easily turned into a modalλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-calculus giving a computational semantics for deductions in IEL-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {IEL}^{-}$$\end{document}. By using that interpretation, it is also proved that IEL-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {IEL}^{-}$$\end{document} has good proof-theoretic properties. The correspondence between deductions and typed terms is then extended to a categorical semantics for identity of proofs in IEL-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {IEL}^{-}$$\end{document} showing the general structure of such a modality for belief in an intuitionistic framework.


Introduction
Brouwer-Heyting-Kolmogorov (BHK) interpretation is based on a semantic reading of propositional variables as problems (or tasks), and of logical connectives as operations on proofs. In this way, it provides a semantics of mathematical statements in which the computational aspects of proving and refuting are highlighted. 1 In spite of being named after L.E.J. Brouwer, this approach is rather away from the deeply philosophical attitude at the origin of intuitionism: In BHK interpretation, reasoning intuitionistically is similar to a safe mode of program execution which always terminates; on the contrary, according to the founders of intuitionism, at the basis of the mathematical activity there is a continuous mental process of construction of objects starting with the flow of time underlying the chain of natural numbers, and intuitionistic reasoning is what structures that process. 2 This reading of the mathematical activity is formally captured by Kripke semantics for intuitionistic logic [9]: Relational structures based on preorders capture the informal idea of a process of growth of knowledge in time which characterises the mental life of the mathematician.
It is worth-noting that the focuses of these semantics are quite different: BHK interpretation stresses the importance of the concept of proof in the semantics for intuitionistic logic; Kripke's approach highlights the epistemic process behind the provability of a statement.
In [2], Artemov and Protopopescu make use of the BHK interpretation to extend-in a sense-the epistemic realm of constructivism: In BHK interpretation we have an implicit notion of proof whose epistemic aspects are modelled by Kripke structures; the construction of a proof for a specific proposition that we carried out as a cumulative mental process gives us sufficient reason for (at least) believing that proposition. It is then possible to cover also traditional epistemic states of belief and knowledge within such a framework, once we recognise the correct clauses for corresponding modal operators: Proving A assures that A is true; proving A is weaker, for we may believe in A even though we do not have a direct proof of A itself.
In [2] the starting point is thus a BHK interpretation of epistemic statements in which knowledge and belief are considered as (different) results of a process of verification. The general idea is that a proof of a (mathematical) statement is a most strict type of verification, and that verifying a statement is a sufficient condition for believing it. At the same time, knowing that a statement is true means, according to this intuitionistic reading, that this very statement cannot be false, since we have a verification of it.
Hence, the proposed intuitionistic account of epistemic states validates a principle of "constructivity of truth" A → A and of "intuitionistic factivity of knowledge" The latter principle appears particularly promising for applying this formal framework in a broader realm than mathematics: In [2] various informal justifications to the proposed account are indeed given, and those authors express the aim of giving new foundations to intuitionistic epistemology.
It appears clear, in any case, that both principles are validated by contemporary mathematical practice: When a statement A is proved, then its very proof can be checked by a human being both by 'pencil and paper' and by computer proof-assistants, so that the whole mathematical community can believe/know that what has been given is a proof of A indeed. Also, intuitionistic factivity of knowledge states that whenever we know that A is true in a (constructive, but maybe) not specified sense-e.g. we do not have direct access to its proof-then A cannot be false, if we agree that knowledge is at least consistent belief. Again, this is common in general mathematical practice, when a theorem is known to hold even by those who cannot develop a proof of it by themselves.
In short, it seems that the BHK computational reading of the epistemic operator provides a finer analysis of truth and of epistemic states from the constructive point of view.
In spite of this, the mentioned paper [2] covers only axiomatic calculi and Kripke semantics for intuitionistic epistemic logics, so that the stimulating question of considering the computational aspects of epistemic states remained informal and at a very beginning stage.
The present paper, on the contrary, is committed to giving a precise, formal analysis of the computational content of intuitionistic belief.
In order to establish some clear facts, a natural deduction system IEL − for the intuitionistic logic of belief is developed and designed with the intent of translating it into a functional calculus of IEL − -deductions. In a sense, we define a formal counterpart of Artemov and Protopopescu's reading of the epistemic operator for belief by extending the Curry-Howard correspondence between intuitionistic natural deduction NJ and simple type theory, to a modal λ-calculus in which the modal connective on propositions behaves according to a single (term-)introduction rule. Furthermore, we establish normalization for IEL − along with other prooftheoretic properties, and, in spite of its simple grammar, we show that there is a surprisingly rich categorical structure behind the calculus: Our λ-system for IEL − -deductions is sound and complete w.r.t. the class of bicartesian closed categories equipped with a monoidal pointed endofunctor whose point is monoidal. Therefore, by adopting the proofs-as-programs paradigm to give a precise meaning to the motto "belief-as-verification" we succeed in: • Designing a natural deduction calculus IEL − for intuitionistic belief which is well-behaved from a proof-theoretic point of view; • Proving that this calculus corresponds to a modal typed system in which every term has a unique normal form, and the epistemic modality acquires a precise functional interpretation; • Developing a categorical semantics for intuitionistic belief which focuses on identity of proofs, and not simply on provability.
The paper is then organised as follows: In Section 1, the axiomatic calculus IEL − and its relational semantics are recalled. In Section 2, we introduce the natural deduction system IEL − and prove-syntactically-that it is logically equivalent to IEL − . Then we investigate some of its proof-theoretic properties: We start by proving that detours can be eliminated from deductions by defining a λ-calculus with a modal operator which captures in a very natural way the behaviour of the epistemic modality on propositions; then we prove a canonicity lemma for this calculus of proof-terms, from which we derive syntactic proofs of the weak and standard disjunction property. Finally, in Section 3, we give a categorical semantics for IEL − -deduction: After recalling the main lines of Curry-Howard-Lambek correspondence, we prove that deductions define-up to normalization-specific categorical structures which subsume Heyting algebras with operators and, at the same time, provide a proof-theoretic semantics for intuitionistic belief.
By means of these results we can also see that some claims in [2] concerning a type-theoretic reading of the epistemic operator as the truncation of types are not correct. The belief modality there defined is 'weaker' than inh : Type → Type because of its type-theoretic-hence syntacticbehaviour, validated also from a categorical-hence semantic-point of view: Types truncation equips bi-cartesian closed categories with an idempotent monad, while we show that the belief operator we are considering is a more general functor. 3 • Axiom scheme of co-reflection A → A;

Axiomatic Calculus for Intuitionistic Belief
as the only inference rule.
We write Γ IEL − A when A is derivable in IEL − assuming the set of hypotheses Γ, and we write IEL − A when Γ = ∅. We immediately have The following properties hold: Proof. See [2]. As stated before, this system axiomatizes the idea of belief as the result of verification within a framework in which truth corresponds to provability, accordingly to the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic. 4 Note also that, in this perspective, the co-reflection scheme is valid, while its converse does not hold: If A is true, then it has a proof, hence it is verified; but A can be verified without disclosing a specific proof, therefore the standard epistemic scheme A → A is not valid under this interpretation. 5

Kripke Semantics for IEL −
Turning to relational semantics, in [2] the following class of Kripke models is given.
• E is a binary 'knowledge' relation on W such that: · if x ≤ y, then xEy; and · if x ≤ y and yEz, then xEz; graphically we have 4 See e.g. [18]. 5 See Williamson's system in [20] for an intuitionistic epistemic logic in which the standard epistemic principle is valid; however it is worth noting that this specific logic is not based on the BHK-semantics.
• v extends to a forcing relation such that A formula A is true in a model iff it is forced by each world of that model; Note that this semantics assumes Kripke's original interpretation of intuitionistic reasoning as a growing knowledge-or discovery process-for an epistemic agent in which the relation E defines an audit of 'cognitively' ≤-accessible states in which the agent can commit a verification.
This semantics is adequate to the calculus: The following hold:

Proof. Soundness is proved by induction on the derivation of A.
Completeness is proved by a standard construction of a canonical model. See [2] for the details.

Natural Deduction for Intuitionistic Belief
We want to develop a semantics of proofs for the logic of intuitionistic belief. In order to do that, we now introduce a natural deduction system which is logically equivalent to IEL − , but which is also capable of a computational reading of the epistemic operator and of proofs involving this kind of modality. Accordingly, the starting point is proving that the calculus IEL − of natural deduction is sound and complete w.r.t. IEL − ; then, we prove that proofs in IEL − can be named by means of λ-terms as stated in the proofsas-programs paradigm for intuitionistic logic also known as Curry-Howard correspondence. By using this formalism, we prove a normalization theorem for IEL − stating that detours can be eliminated from all deductions. Afterwards, we present some results about the proof-theoretic behaviour of IEL − .
In the next section, we extend such a correspondence to category theory for showing the underlying structure of the operator for intuitionistic belief.

System IEL −
Definition 2.1.1. Let IEL − be the calculus extending the propositional fragment of NJ-the natural deduction calculus for intuitionistic logic as presented in [17]-by the following rule: where Γ and Δ are sets of occurrences of formulae, and all A 1 , · · · , A n are discharged.
Note that this calculus differs from the system introduced in [5] by allowing the set Δ of additional hypotheses is the subdeduction of B.
Let's immediately check that we are dealing with the same logic presented in [2] We proceed by induction on the derivation.
• Intuitionistic cases are dealt with NJ propositional rules; • co-reflection: Conversely, assume Γ IEL − A. We consider only the -intro rule: By induction hypothesis, we have and, by the deduction theorem for IEL − and ordinary logic,

Normalization
In order to eliminate potential detours from IEL − -deduction we introduce the following proof rewritings: Note that (ι) eliminates a useless application of -intro, while (δ) collapses two -intros into a single one.
Remark 1. (Proofs-as-programs). Curry-Howard correspondence permits a functional reading of proofs in NJ, once one recognises the following mapping: where t A , s B correspond to f1 and f2 resp.
where t A→B , s A correspond to f1 and f2 resp. . .
where C bounds all occurrences of x in t1 and all occurrences of y in t2, and t, t1, t2 correspond to f , the subdeduction of C from A, and the subdeduc- where t correspond to f .
By imposing specific rewritings-which are clearly defined in [7]-we obtain the complete engine of λ-calculus associated to the propositional fragment of NJ.
Since IEL − consists also of a rule for , we need to extend the grammar of such a typed λ-calculus as follows: As for NJ, a modal λ-calculus is obtained by decorating IEL − -deductions with proof names.
Concerning the modality, we see that in the proof-term box (s : B) : B for the -intro rule, a term s of type B-with x of type A 1 , · · · , A n among its free variables-is used along with terms t of type A 1 , · · · , A n in order to obtain a term of type B not depending on the variables x.
Proof rewritings can be then expressed by imposing appropriate reductions of λ-terms: . s in ti), ti+1, · · · , tn) in r > δ box[x1, · · · , xi−1, y, xi+1, · · · , xn].(t1, · · · , ti−1, s, ti+1, · · · , tn) in r [ti/xi] Assuming this reading of deductions as programs, normalization now becomes just the execution of a program written in our modal λ-calculus; it assures then consistency of IEL − , being also useful to prove its analyticity, and hence its decidability. Accordingly, the quest for normalizing natural deduction systems is not limited to the proofs-as-programs paradigm, and its origins are actually at the very core of proof theory: We refer the reader to [12] and [19] for the technical and historical aspects of the research field, respectively.
We write for the transitive closure of the relation obtained by combining > ι and > δ , together with standard rewritings of λ-calculus as presented in Remark 1. An algebra of λ-terms is then obtained by considering the reflexive, symmetric, transitive closure = of , i.e. by combining the reflexive, symmetric, transitive closure ι = and δ = of > ι and > δ , respectively, along with standard rewriting rules for typed λ-calculus. 6 We can now prove that every deduction in IEL − can be uniquely reduced to a proof containing no detours. Proof. We define a translation 7 |−| from the λ-calculus of IEL − -deductions to typed λ-calculus with products, sums, empty and unit types: 6 Note that ι = is just a special case of δ =. We decide to adopt this redundant system of rewriting since ι = has a straightforward interpretation in category theory: See Section 3.2. 7 This function is introduced in [8] to prove detour-elimination for the implicational fragment of basic intuitionistic modal logic IK by reducing the problem to normalization of simple type theory. Here we adopt the mapping to consider also product, co-product, empty, and unit types, keeping the original strategy due to [4]. A different proof based on Tait's computability method [15] should also be possible and is under development by the author.
where q is specific atom type. Then it is easy to see that = is preserved by this mapping: If t t in one step, then |t| |t |, where indicates the usual relation made of βη-reductions with permutations. Therefore, since typed λ-calculus with products, sums, empty and unit types is strongly normalizing, 8 so is our modal λ-calculus, and IEL − also. In other terms: If IEL − was not normalizing, then we would have an infinite -reduction starting from, say, t : A. By the previous note, this would lead to an infinite -reduction starting from |t| : |A|, contradicting strong normalization of typed λ-calculus with ×, +, 1, 0.

Lemma 2.2.2. The modal λ-calculus of IEL − -deductions has the Church-Rosser property.
Proof. It is straightforward to prove weak Church-Rosser property for our calculus. By Theorem 2.2.1, the modal λ-calculus of IEL − -deductions has the Church-Rosser property. 9

Corollary 2.2.3. Every IEL − -deduction reduces uniquely to a deduction without detours.
Proof. By Theorem 2.2.1 and Lemma 2.2.2, any term of the modal λ-calculus of IEL − -deductions has a unique normal form.

Further Proof-Theoretic Properties
Knowing that IEL − -deductions normalise, it is possible to investigate further on the relevant aspects of our system from the perspective of proof theory.
Following [7], we will say that a deduction (or, equivalently, a proof-term) is neutral, iff it is a hypothesis, or its last rule is an elimination rule (E-rule, for short).
We immediately have the following Proof. Straightforward induction on the height of the deduction.
By the previous lemma we can easily prove canonicity of proof-terms:

. (Canonicity). If IEL − A is normal, then the last rule of the deduction is the appropriate introduction rule.
Proof. By Lemma 2.3.1, since Γ = ∅ here, and the deduction is normal, its last rule cannot be an E-rule.
It is now straightforward to apply the canonicity lemma in order to obtain the following results:

Corollary 2.3.4. Reflection rule is admissible in IEL
Another interesting question is whether IEL − satisfies the subformula property. 10 In the present paper, we are mainly focused on the computational aspects of the intuitionistic modality for belief, and that property would be relevant to these aims only for it implies the decidability of the calculus. We have already developed the main lines of a general proof for the property, and obtained some preliminary results for answering positively the question (modulo minor tweaks); however, proving that the principle holds indeed for IEL − involve checking many minor details, which make the intermediate proofs quite long. We prefer then to discuss the status of this principle in another work (in preparation) focused on the structural properties of our system.

Categorical Semantics for Intuitionistic Belief
If λ-calculus gives a computational semantics of proofs in NJ-and, as we showed in the previous section, in IEL − also-category theory furnishes the tools for an 'algebraic' semantics which is proof relevant-i.e. contrary to traditional algebraic semantics based on Heyting algebras and to relational semantics based on Kripke models, it focuses on the very notion of proof, distinguishing between different deductions of the same formula.
In this perspective, the correspondence between proofs and programs is extended to consider arrows in categories which have enough structure to capture the behaviour of logical operators. The so-called Curry-Howard-Lambek correspondence can be then summarized by the following Here we see that cartesian product models conjunction, and exponential models implication. Any category having products and exponentials for any of its objects is called cartesian closed (CCCat); moreover, if it has also coproducts-modelling disjunction-it is called bi-cartesian closed (bi-CCCat): and ⊥ correspond to empty product -the terminal object 1 -and empty coproduct -the initial object 0-respectively. The reader is referred to the classic [10] for the details of such completeness result.
For our calculus, in order to capture the behaviour of the epistemic modality, some more structure is required: In the following subsections some basic definitions are recalled and then used to provide IEL − with an adequate categorical semantics. • a morphism

Monoidal Functors, Pointed Functors, and Monoidal Natural Transformations
preserving the monoidal structure of C. 11 These are called structure morphisms of F.
It is quite easy to see that a monoidal endofunctor on the category of logical formulas induces a modal operator satisfying K-scheme, as proved in [5].
In the present setting, a pointed endofunctor on the category of logical formulas 'represents' the co-reflection scheme.
Since we want to give a semantics of proofs-and not simply of derivability-in IEL − , we need a further notion from category theory. Definition 3.1.3. Given a monoidal category C, and monoidal endofunctors F, G : C → C, a natural transformation κ : F ⇒ G is monoidal when the following commute: and

Categorical Completeness
Finally, we introduce the models by which we want to capture IEL − . Definition 3.2.1. An IEL − -category is given by a bi-CCCat C together with a monoidal pointed endofunctor K whose point κ is monoidal. Now we can check adequacy of these models.
• for any two deductions t and s which are equal modulo =, we have t = s .
Proof. By structural induction on f : A IEL − B. The intuitionistic cases are interpreted according to the remarks about bi-CCCats at the beginning of this section. We overload the notation using , m, κ for the monoidal pointed endofunctor of C, its structure morphisms, and its point.
The relation δ = is also valid by naturality of m and κ: The reader is invited to check that κ must be monoidal in order to model correctly the following special case 12 It remains to show that this interpretation is also complete. Proof. We proceed by constructing a term model for the modal λ-calculus for IEL − -deductions. Consider the following category M: • its objects are formulae; • an arrow f : A → B is an IEL − -deduction of B from A modulo =; • identities are given by assuming a hypothesis; 12 Everything reduces to long categorical calculations.
• composition is given by transitivity of deductions.
Then M has a bi-cartesian closed structure given by the properties of conjunction, implication, and disjunction in NJ.
Moreover, the modal operator induces a functor K by mapping A to A, and . . .
which preserves identities by ι = , and preserves composition as a special case of δ =.
The structure morphism is given by whose properties follow as a special case of Remark 2. (Belief and truncation). In [3], truncation-there called "bracket types"-is defined in a first order calculus with types, and showed to behave like a monad. Similarly, in [13], (n-)truncation is defined as a monadic idempotent modality within the framework of homotopy type theory.
We have just seen that despite the truncation does eliminate all computational significance to an inhabitant of a type-turning then a proof of a proposition into a simple verification of that statement-the belief modality defined in [2] does not correspond to that operator on types.
Actually, after considering the potential applications of IEL (−) prospected by Artemov and Protopopescu outside the realm of mathematical statements, that should be not surprising at all: The categorical semantics of IEL − -deductions subsumes the interpretation of truncation as an idempotent monad, since such a functor is just a special case of monoidal pointed endofunctor with monoidal point.
It might be interesting thus to consider the relationship between truncation and the belief modality from a purely syntactic perspective, by comparing the structural properties of a potential simple type theory with bracket types and our modal λ-calculus for intuitionistic belief. 13

Conclusion
Our original intent has been to make precise the computational significance of the motto "belief-as-verification" which leads in [2] to the introduction of epistemic modalities in the framework of BHK interpretation. In particular, despite some claims contained in that paper, we were not sure how to relate the belief operator with type truncation.
In the present paper, we have addressed these questions and have developed a 'proof-theoretically tractable' system for intuitionistic belief that can be easily turned into a modal λ-calculus, showing that the epistemic operator behaves differently from truncation.
Moreover, by extending some results concerning categorical semantics for the basic intuitionistic modal logic IK in [5] and [8], we developed a proof-theoretic semantics for intuitionistic belief based on monoidal pointed endofunctors with monoidal points on bi-CCCats. Even from this 'categorical' perspective, the modal operator differs from type-theoretic truncation, so that the reading of belief as the result of verification seems to be just a heuristic interpretation of that specific modality.
Having established so, some general questions naturally arise: • How could be the original motivation of co-reflection scheme A → Ai.e. the interpretation of as a verification operator on propositionscorrectly captured, from a computational point of view, by intuitionistic logic of belief?
• Does the possible extension IEL of IEL − obtained by adding the elimination rule Γ A Γ ¬¬A recover the intuitionistic reading of epistemic states as results of verification in a formal way-i.e. as type-theoretic truncation?
In our opinion, these problems are strongly related: In fact, it seems plausible that the additional elimination rule provides IEL with an adjunction between and ¬¬ which has still to be checked and deserves a fine grained analysis and comparison with type truncation.
Moreover, it might be interesting to consider similar modalities in different settings, including first order logic and linear logics.
Acknowledgements. These results were obtained during my first year of PhD studies at DiMa of University of Genoa. After discussing a preliminary version of categorical semantics for intuitionistic belief during the Logic and Philosophy of Science Seminar of University of Florence in May 2019, I had the great opportunity to present more refined results during the poster session of The Proof Society Summer School in September 2019: My gratitude goes to all logicians of the Computational Foundry of Swansea University who hosted the event, along with the invited lecturers and the participants. In particular, I wish to thank Reuben Rowe for an enlightening discussion on the way to Rhossili about the behaviour of the in my formal system. A sincere acknowledgement goes to an anonymous reviewer for suggesting further investigations on the proof theory of the calculus I originally presented to this journal: Her/His comments have been very important for structuring the final version of this work.
Funding Open access funding provided by Universitá degli Studi di Genova within the CRUI-CARE Agreement.
Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.