aa r X i v : . [ m a t h . L O ] J un Curry-Howard-Lambek correspondence forintuitionistic belief ∗ Cosimo Perini Brogi
Department of Mathematics · University of Genova [email protected]
Abstract
This paper introduces a natural deduction calculus for intuitionisticlogic of belief
IEL − which is easily turned into a modal λ -calculus giving acomputational semantics for deductions in IEL − . By using that interpre-tation, it is also proved that IEL − has good proof-theoretical properties.The correspondence between deductions and typed terms is then extendedto a categorial semantics for identity of proofs in IEL − showing the generalstructure of such a modality for belief in an intuitionistic framework. Keywords:
Intuitionistic modal logic, epistemic logic, categorial proof theory, modal typetheory, proofs-as-programs.
Introduction
Brouwer-Heyting-Kolgomorov (BHK) interpretation is based on a semanticreading of propositional variables as problems (or tasks), and of logical connec-tives as operations on proofs . In this way, it provides a semantics of mathemat-ical statements in which the computational aspects of proving and refuting arehighlighted. In spite of being named after L.E.J. Brouwer, this approach is rather awayfrom the deeply philosophical attitude at the origin of intuitionism: In BHKinterpretation, reasoning intuitionistically is similar to a safe mode of programexecution which always terminates; on the contrary, according to the foundersof intuitionism, at the basis of the mathematical activity there is a continuousmental process of construction of objects starting with the flow of time underly-ing the chain of natural numbers, and intuitionistic reasoning is what structuresthat process. This reading of the mathematical activity is formally captured by Kripkesemantics for intuitionistic logic [9]: Relational structures based on pre-orders ∗ Main preliminary results were presented in May 2019 during the Logic and Philosophy ofScience Seminar at University of Florence (Italy). A refined presentation was given during theposter session of The Proof Society Summer School at Swansea University (UK) on September9th, 2019. The present version was submitted for publication to Studia Logica in January2020. The reader is referred to [16] for an introduction. For instance, Dummett’s [6] advocates a purely philosophical justification of the wholecurrent of intuitionistic mathematics. natural deduction system
IEL − for the intuitionistic logic of belief is developed and designed with the intent oftranslating it into a functional calculus of IEL − -deductions. In a sense, we definea formal counterpart of Artemov and Protopopescu’s reading of the epistemicoperator for belief by extending the Curry-Howard correspondence between intu-itionistic 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 − , and, in spite of its sim-ple grammar, we show that there is a surprisingly rich categorial structure be-hind the calculus: our λ -system for IEL − -deductions is sound and complete w.r.t.the class of bi-cartesian closed categories equipped with a monoidal pointed end-ofunctor whose point is monoidal.Therefore, by adopting the proofs-as-programs paradigm to give a precisemeaning to the motto “belief-as-verification” we succeed in: ◦ Designing a natural deduction calculus
IEL − for intuitionistic belief whichis well-behaved from a proof-theoretic point of view; ◦ Proving that this calculus corresponds to a modal typed system in whichevery term has a unique normal form, and the epistemic modality acquiresa precise functional interpretation; ◦ Developing a categorial semantics for intuitionistic belief which focuses onidentity 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 thenatural deduction system IEL − and prove – syntactically – that it is logically2quivalent to IEL − ; then we investigate its proof-theoretic properties, provingthat detours can be eliminated from deductions by defining a λ -calculus witha modal operator which captures in a very natural way the behaviour of theepistemic modality on propositions. Finally, in Section 3, we give a categorialsemantics for IEL − -deduction: After recalling the main lines of Curry-Howard-Lambek correspondence, we prove that deductions define – up to normalization– specific categorial structures which subsume Heyting algebras with operatorsand, at the same time, provide a proof-theoretic semantics for intuitionisticbelief.By means of these results we can also see that some claims in [2] concerninga type-theoretic reading of the epistemic operator as the truncation of types arenot correct. The belief modality there defined is ‘weaker’ than inh : Type → Type because of its type-theoretic – hence syntactic – behaviour, validated also froma categorial – hence semantic – point of view: Types truncation equippes bi-cartesian closed categories with an idempotent monad, while we show that thebelief operator we are considering is a more general functor. Axiomatic calculus for intuitionistic belief
Let’s start by recalling the syntax and relational semantics for the logic ofintuitionistic belief as introduced in [2].1.1.
System
IEL − Definition 1.1.1.
IEL − is the axiomatic calculus given by: • Axiom schemes for intuitionistic propositional logic; • Axiom scheme K : ✷ ( A → B ) → ✷ A → ✷ B ; • Axiom scheme of co-reflection A → ✷ A ; • Modus Pones A → B A MP B as the only inference rule.We write Γ ⊢ IEL − A when A is derivable in IEL − assuming the set of hypothesesΓ, and we write IEL − ⊢ A when Γ = ∅ .We immediately have Proposition 1.1.2.
The following properties hold:(i) Necessitation rule A ✷ A is derivable in IEL − ;(ii) The deduction theorem holds in IEL − ;(iii) IEL − is a normal intuitionistic modal system.Proof. See [2]. ⊠ As stated before, this system axiomatizes the idea of belief as the resultof verification within a framework in which truth corresponds to provability,accordingly to the Brower-Heyting-Kolgomorov interpretation of intuitionisticlogic. Note also that, in this perspective, the co-reflection scheme is valid, while In fact even when thought of as a modal connective of our base language, the epistemic ✷ is not an idempotent operator. See e.g. [18]. A is true, then it has a proof, hence it is verified;but A can be verified without disclosing a specific proof, therefore the standardepistemic scheme ✷ A → A is not valid under this interpretation. Kripke Semantics for
IEL − Turning to relational semantics, in [2] the following class of Kripke modelsis given.
Definition 1.2.1.
A model for
IEL − is a quadruple h W, ≤ , v, E i where ◦ h W, ≤ , v i is a standard model for intuitionistic propositional logic; ◦ E is a binary ‘knowledge’ relation on W such that: · if x ≤ y , then xEy ; and · if x ≤ y , then if yEz , xEz ; graphically we have y E / / zx ≤ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ E ◦ v extends to a forcing relation (cid:15) such that · x (cid:15) ✷ A iff y (cid:15) A for all y such that xEy .A formula A is true in a model iff it is forced by each world of that model; wewrite IEL − (cid:15) A iff A is true in each model for IEL − .Note that this semantics assumes Kripke’s original interpretation of intu-itionistic reasoning as a growing knowledge – or discovery process – for an epis-temic agent in which the relation E defines an audit of ‘cognitively’ ≤ -accessiblestates in which the agent can commit a verification.This semantics is adequate to the calculus: Theorem 1.2.2.
The following hold: (Soundness) If IEL − ⊢ A , then IEL − (cid:15) A . (Completeness) If IEL − (cid:15) A , then IEL − ⊢ A .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. ⊠ See Williamson’s system in [20] for an intuitionistic epistemic logic in which the standardepistemic principle is valid; however it is worth noting that this specific logic is not based onthe BHK-semantics. Natural Deduction for intuitionistic belief
We want to develop a semantics of proofs for the logic of intuitionistic be-lief. In order to do that, we now introduce a natural deduction system whichis logically equivalent to
IEL − , but which is also capable of a computationalreading of the epistemic operator and of proofs involving this kind of modality.Accordingly, the starting point is proving that the calculus IEL − of naturaldeduction 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 proofs-as-programsparadigm for intuitionistic logic also known as Curry-Howard correspondence.By using this formalism, we prove a normalization theorem for IEL − statingthat detours can be eliminated from all deductions.In the next section, we extend such a correspondence to category theory forshowing the underlying structure of the operator for intuitionistic belief.2.1. System
IEL − Definition 2.1.1.
Let
IEL − be the calculus extending the propositional frag-ment of NJ – the natural deduction calculus for intuitionistic logic – by thefollowing rule: Γ ... ✷ A · · · Γ n ... ✷ A n [ A , · · · , A n ] , ∆ ...B ✷ − intro ✷ B where Γ and ∆ are sets of occurrences of formulae , and all A , · · · , A n aredischarged. Let’s immediately check that we are dealing with the same logic:
Lemma 2.1.2. Γ ⊢ IEL − A iff Γ ⊢ IEL − A .Proof. Assume Γ ⊢ IEL − A . We proceed by induction on the derivation. • Intuitionistic cases are dealt with NJ propositional rules; • K : ✷ ( A → B ) ✷ A [ A → B, A ] ...B ✷ − intro ✷ B ; • co-reflection: [ A ] ✷ − intro ✷ A →− intro :1 A → ✷ A . See [17] for an introduction. This calculus differs from the system introduced in [5] by allowing the set ∆ of additionalhypotheses is the subdeduction of B . A different calculus, considering the original ✷ -introrule in [5] and a further rule Γ ⊢ A Γ ⊢ ✷ A , even though more symmetric, seems to lack someimportant computational properties, like uniqueness of normal proofs. ⊢ IEL − A . We consider only the ✷ -intro rule:By induction hypothesis, we have Γ ⊢ IEL − ✷ A , · · · , Γ n ⊢ IEL − ✷ A n , and A , · · · , A n , ∆ ⊢ IEL − B . Then we have Γ , · · · , Γ n ⊢ IEL − ✷ A , · · · , ✷ A n , and, by the deduction theorem for IEL − and ordinary logic,∆ ⊢ IEL − A ∧ · · · ∧ A n → B .By co-reflection IEL − ⊢ ( A ∧ · · · ∧ A n → B ) → ✷ ( A ∧ · · · ∧ A n → B ), andby K -scheme IEL − ⊢ ✷ ( A ∧ · · · ∧ A n → B ) → ✷ ( A ∧ · · · ∧ A n ) → ✷ B . Hencewe have ∆ ⊢ IEL − ✷ ( A ∧ · · · ∧ A n ) → ✷ B , whence, by modal logic, we obtain∆ ⊢ IEL − ✷ A ∧ · · · ∧ ✷ A n → ✷ B , which gives Γ , · · · , Γ n , ∆ ⊢ IEL − ✷ B , asdesired. ⊠ Normalization
In order to eliminate potential detours from
IEL − -deduction we introducethe following proof rewritings:( ι ) Γ ... ✷ A [ A ] ✷ − intro ✷ A ❀ Γ ... ✷ A ( δ ) Γ ...~ ✷ A [ ~A ] , ~C...B ✷ − intro :1 ✷ B ∆ ...~ ✷ D [ B, ~D ] , ~E...F ✷ − intro :2 ✷ F ❀ Γ ...~ ✷ A ∆ ...~ ✷ D [ ~A ] ...B ~C, [ ~D ] , ~E...F ✷ − intro :1 ✷ F Note that ( ι ) eliminates a useless application of ✷ -intro, while ( δ ) collapses two ✷ -intros into a single one. 6 emark . Curry-Howard correspondence permits a func-tional reading of proofs in NJ , once one recognises the following mapping: f ≡ A x Ai , where i is the parcel of thehypothesis Af ≡ f A f BA ∧ B t A , s B i , where t A , s B correspondto f and f resp. f ≡ f ′ A ∧ BA π .t A × B , where t A × B correspondsto f ′ f ≡ f ′ A ∧ BB π .t A × B , where t A × B correspondsto f ′ f ≡ f ′ BA → B λx Ai .t B , where t B corresponds to f ′ and i is the parcel ofdischarged hypotheses Af ≡ f A → B f AB t A → B s A , where t A → B , s A corre-spond to f and f resp. f ≡ f ′ AA ∨ B in .t A , where t A corresponds to f ′ f ≡ f ′ BA ∨ B in .s B , where s B corresponds to f ′ f ≡ f ′ A ∨ B [ A ] ...C [ B ] ...CC C ( t, ( x A .t ) , ( y B .t )) where C bounds all oc-currences of x in t andall occurrences of y in t ,and t, t , t correspond to f ′ , the subdeduction of C from A , and the subde-duction of C from B , resp. f ≡ f ′ ⊥ A E A t where t corresponds to f ′ f ≡ f ′ A ⊤ 7−→ ( U t ) where t correspond to f ′ . By imposing specific rewritings we obtain the complete engine of λ -calculusassociated to the propositional fragment of NJ . Since
IEL − consists also of rules for ✷ , we need to extend the grammar ofsuch a typed λ -calculus as follows: T ::= | | p | A → B | A × B | A + B | ✷ A See [7] for a clear introduction to the topic. ::= x | E ( t ) | U ( t ) | λx : A.t : B | t t | ( t , t ) | π ( t ) | π ( t ) | in ( a : A ) | in ( b : B ) | C ( t, x.t , y.t ) | box [ ~x : ~A | ~z ] .~t : ~ ✷ A in ( s : B ) : ✷ B .As for NJ , a modal λ -calculus is obtained by decorating IEL − -deductions withproof names. Proof rewritings can be then expressed by imposing appropriatereductions of λ -terms: box [ x ] .t in x > ι t box [ x , · · · , x i − , x i , x i +1 , · · · , x n | ~z ] . ( t , · · · , t i − , ( box [ ~y | ~w ] .~s in t i ) in r > δ > δ box [ x , · · · , x i − , ~y, x i +1 , · · · , x n | ~z, ~w ] . ( t , · · · , t i − , ~s, t i +1 , · · · , t n ) in r [ t i /x i ] Assuming this reading of deductions as programs, normalization now be-comes just the execution of a program written in our modal λ -calculus; nor-malization then assures consistency of IEL − , its analyticity, and hence its de-cidability. However, the quest for normalizing natural deduction systems is notlimited to the proofs-as-programs paradigm, and its origins are actually at thevery core of proof theory: We refer the reader to [12] and [19] for the technicaland historical aspects of the research field, respectively.We write ✄ for the transitive closure of the relation obtained by combining > ι and > δ . 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. We can now prove that every deduction in
IEL − can be uniquely reduced toa proof containing no detours. Theorem 2.2.1.
Strong normalization holds for
IEL − .Proof. We define a translation | − | from the λ -calculus of IEL − -deductions totyped λ -calculus with products, sums, empty and unit types: | | := | | := | p | := p | A → B | := | A | → | B || A × B | := | A | × | B || A + B | := | A | + | B || ✷ A | := ( | A | → q ) → q Note that ι = is just a special case of δ =. We decide to adopt this redundant system ofrewriting since ι = has a straightforward interpretation in category theory: see Section 3.2. This function is introduced in [8] to prove detour-elimination for the implicational frag-ment of basic intuitionistic modal logic IK by reducing the problem to normalization of simpletype theory. Here we adopt the mapping to consider also product, co-product, empty, and unittypes, keeping the original strategy due to [4]. A different proof based on Tait’s computabilitymethod [15] should also be possible and is under development by the author. x | := x | c | := c | E ( t ) | := E ( | t | ) | U ( t ) | := U ( | t | ) | λx t | := λx. | t || ts | := | t || s || ( t, s ) | := ( | t | , | s | ) | π i ( t ) | := π i ( | t | ) | C ( t, x.t , y.t ) | := C ( | t | , x. | t | , y. | t | ) | in i ( t ) | := in i ( | t | ) | box [ x , · · · , x n | ~z ] . ( t , · · · , t n ) in s | := λk. | t | ( λx . · · · | t n | ( λx n . k | s | ) · · · )where q is specific atom type. Then it is easy to see that ✷ = is preserved bythis mapping. Therefore, since typed λ -calculus with products, sums, emptyand unit types is strongly normalizing, so is our modal λ -calculus, and IEL − also . ⊠ Lemma 2.2.2.
The modal λ -calculus of IEL − -deductions has the Church-Rosserproperty.Proof. It is straightforward to prove weak Church-Rosser property for our cal-culus. By Theorem 2.2.1, the modal λ -calculus of IEL − -deductions has theChurch-Rosser property. ⊠ Corollary 2.2.3.
Every
IEL − -deduction reduces uniquely to a deduction withoutdetours.Proof. By Theorem 2.2.1 and Lemma 2.2.2, any term of the modal λ -calculusof IEL − -deductions has a unique normal form. ⊠ Categorial Semantics for intuitionistic belief If λ -calculus gives a computational semantics of proofs in NJ – and, as weshowed in the previous section, in IEL − also – category theory furnishes the toolsfor an ‘algebraic’ semantics which is proof relevant – i.e. contrary to traditionalalgebraic semantics based on Heyting algebras and to relational semantics basedon Kripke models, it focuses on the very notion of proof, distinguishing betweendifferent deductions of the same formula.In this perspective, the correspondence between proofs and programs is ex-tended to consider arrows in categories which have enough structure to capturethe behaviour of logical operators. The so-called Curry-Howard-Lambek corre-spondence can be then summarized by the following table This means that if t ✄ t ′ in one step, then | t | >> | t ′ | , where >> indicates the usual βη -reductions with permutations. See [7] and [1]. In other terms, if
IEL − was not normalizing, then we would have an infinite ✄ -reductionstarting from, say, t : A . By the previous note, this would lead to an infinite >> -reductionstarting from | t | : | A | , contradicting strong normalization of typed λ -calculus with × , + , , . See [14] for the general result relating strong normalization and Church-Rosser theorem. ogic Type Theory Category Theory proposition type objectproof term arrowtheorem inhabitant element-arrowconjunction product type producttrue unit type terminal objectimplication function type exponentialdisjunction sum type coproductfalse empty type initial objectHere we see that cartesian product models conjunction, and exponential modelsimplication. Any category having products and exponentials for any of its ob-jects is called cartesian closed (CCCat) ; moreover, if it has also coproducts– modelling disjunction – it is called bi-cartesian closed (bi-CCCat) . Thereader 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 defi-nitions are recalled and then used to provide
IEL − with an adequate categorialsemantics.3.1. Monoidal Functors, Pointed Functors, and Monoidal NaturalTransformations
Definition 3.1.1.
Given a CCCat C , an endofunctor F : C → C is monoidal when • there exists a natural transformation m A,B : F A × F B → F ( A × B ); • there exists a morphism m : → F , preserving the monoidal structure of C . These are called structure morphisms of F .It is quite easy to see that a monoidal endofunctor on the category of logicalformulas induces a modal operator satisfying K -scheme, as proved in [5]. Definition 3.1.2.
Given any category C , an endofunctor F : C → C is pointed iff there exists a natural transformation π : Id C ⇒ F π A : A → F AA π A / / f (cid:15) (cid:15) F A F f (cid:15) (cid:15) B π B / / F Bπ is called the point of F . ⊤ and ⊥ correspond to empty product – the terminal object – and empty coproduct– the initial object . See [11] for the corresponding commuting diagrams and the definition of monoidal cate-gory.
10n the present setting, a pointed endofunctor on the category of logical for-mulas ‘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 thefollowing commute: F A × F B κ A × κ B (cid:15) (cid:15) m F A,B / / F ( A × B ) κ A × B (cid:15) (cid:15) G A × G B m G A,B / / G ( A × B )and m F / / F ( ) κ (cid:15) (cid:15) m G / / G ( )3.2. Categorial 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 amonoidal pointed endofunctor K whose point κ is monoidal.Now we can check adequacy of these models. Theorem 3.2.2 (Soundness) . Let C be an IEL − -category. Then there is acanonical interpretation J − K of IEL − in C such that ◦ a formula A is mapped to a C -object J A K ; ◦ a deduction t of A , · · · , A n ⊢ IEL − B is mapped to an arrow J t K : J A K × · · · × J A n K → J B K ; ◦ for any two deductions t and s which are equal modulo ✷ = , we have J t K = J s K .Proof. By structural induction on f : ~A ⊢ IEL − B . The intuitionistic cases areinterpreted according to the remarks about CCCats at the beginning of thissection. We overload the notation using h ✷ , m, κ i for the monoidal pointedendofunctor of C , its structure morphisms, and its point.The deduction f : Γ ⊢ ✷ A · · · f n : Γ n ⊢ ✷ A n g : [ A , · · · , A n ] , C , · · · , C m ⊢ B ✷ B is mapped to( ✷ J g K ) ◦ m J A K , ··· , J A n K , J C K , ··· , J C m K ◦ J f K × · · · × J f n K × κ J C K × · · · × κ J C m K , m X , ··· ,X n is defined inductively as m X , ··· ,X n − ,X n := m X ×···× X n − ,X n ◦ ( m X , ··· ,X n − ) × id ✷ X n . It is straightforward to check that the categorification of ι = holds by functo-riality of ✷ .The relation δ = is also valid by naturality of m and κ : The reader is invited tocheck that κ must be monoidal in order to model correctly the following specialcase Γ ...A ✷ − intro ✷ A · · · Γ n ...A n ✷ − intro ✷ A n [ A , · · · , A n ] , C , · · · , C m ...B ✷ − intro :1 ✷ B ❀❀ Γ ...A · · ·· · · Γ n ...A n C , · · · , C m ...B ✷ − intro ✷ B . ⊠ It remains to show that this interpretation is also complete.
Theorem 3.2.3 (Completeness) . If the interpretation of two
IEL − -deductionsis equal in all IEL − -categories, then the two deductions are equal modulo ✷ = .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 ; • identities are given by assuming a hypothesis; • composition is given by transitivity of deductions.Then M has a bi-cartesian closed structure given by the properties of con-junction, implication, and disjunction in NJ .Moreover, the modal operator ✷ induces a functor K by mapping A to ✷ A ,and A , · · · , A n ...B ✷ ( A ∧ · · · ∧ A n ) [ A ∧ · · · ∧ A n ] ...B ✷ − intro ✷ B Everything reduces to long categorial calculations. ι = , and preserves composition as a special caseof δ =.The structure morphism is given by ✷ A ∧ ✷ B ✷ A ✷ A ∧ ✷ B ✷ B [ A ] [ B ] A ∧ B ✷ ( A ∧ B )whose properties follow as a special case of δ =.The point is given by A ✷ A and its characteristic property is given as aspecial case of δ =. Finally such a point is monoidal by δ = up to ∧ -detours.Then if an equation between interpreted IEL − -deductions holds in all IEL − -categories, then it holds also in M , so that those deductions are equal w.r.t. ✷ =. ⊠ Remark . In [3], truncation – there called “brackettypes” – is defined in a first order calculus with types, and showed to behave likea monad. Similarly, in [13], ( n -)truncation is defined as a monadic idempotentmodality within the framework of homotopy type theory.We have just seen that despite the truncation does eliminate all compu-tational significance to an inhabitant of a type – turning then a proof of aproposition into a simple verification of that statement – the belief modalitydefined in [2] does not correspond to that operator on types.Actually, after considering the potential applications of IEL ( − ) prospectedby Artemov and Protopopescu outside the realm of mathematical statements,that should be not surprising at all: The categorial semantics of IEL − -deductionssubsumes the interpretation of truncation as an idempotent monad, since sucha functor is just a special case of monoidal pointed endofunctor with monoidalpoint.It might be interesting thus to consider the relationship between truncationand the belief modality from a purely syntactic perspective, by comparing thestructural properties of a potential simple type theory with bracket types andour modal λ -calculus for intuitionistic belief. Conclusion
Our original intent has been to make precise the computational significanceof the motto “belief-as-verification” which leads in [2] to the introduction ofepistemic modalities in the framework of BHK interpretation. In particular,despite some claims contained in that paper, we were not sure how to relate thebelief 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 easilyturned into a modal λ -calculus, showing that the epistemic operator behavesdifferently from truncation. As stated before, truncation has been considered only in a first-order context – i.e. work-ing within dependent types. It should be possible, however, to define truncation for simpletypes by imposing further reductions to terms of the system mimicking the rules involving(intensional) equality in bracketed types. IK in [5] and [8], we developed a proof-theoreticsemantics for intuitionistic belief based on monoidal pointed endofunctors withmonoidal points on bi-CCCats. Even from this ‘structural’ perspective, themodal operator differs from type-theoretic truncation, so that the reading ofbelief as the result of verification seems to be just a heuristic interpretation ofthat specific modality.Having established so, some general questions naturally arise: • how could be the original motivation of co-reflection scheme A → ✷ A –i.e. the interpretation of ✷ as a verification operator on propositions –correctly captured, from a computational point of view, by intuitionisticlogic of belief? • does the possible extension IEL of IEL − obtained by adding the eliminationrule Γ ⊢ ✷ A Γ ⊢ ¬¬ A recover the intuitionistic reading of epistemic states asresults of verification in a formal way – i.e. as type-theoretic truncation?In our opinion, these problems are strongly related: In fact, it seems plausiblethat the additional elimination rule provides IEL with an adjunction between ✷ and ¬¬ which has still to be checked and deserves a fine grained analysis andcomparison with type truncation.Moreover, it might be interesting to consider similar modalities in differentsettings, including first order logic and linear logics. References [1]
Y. Akama , On Mints’ reduction for ccc-calculus , in International Confer-ence on Typed Lambda Calculi and Applications, Springer, 1993, pp. 1–12.[2]
S. Artemov and T. Protopopescu , Intuitionistic epistemic logic , TheReview of Symbolic Logic, 9.2 (2016), pp. 266–298.[3]
S. Awodey and A. Bauer , Propositions as [types] , Journal of Logic andComputation, 14(4) (2004), pp. 447–471.[4]
P. de Groote , On the strong normalisation of intuitionistic naturaldeduction with permutation-conversions , Information and Computation,178.2 (2002), pp. 441–464.[5]
V. de Paiva and E. Ritter , Basic constructive modality , Logic withoutFrontiers: Festschrift for Walter Alexandre Carnielli on the occasion of his60th Birthday, (2011), pp. 411–428.[6]
M. Dummett , Elements of intuitionism , Oxford University Press, 2000.[7]
J.-Y. Girard, P. Taylor, and Y. Lafont , Proofs and types , CambridgeUniversity Press, 1989.[8]
Y. Kakutani , Calculi for intuitionistic normal modal logic , arXiv preprintarXiv:1606.03180, (2016). 149]
S. A. Kripke , Semantical analysis of intuitionistic logic i , in Studies inLogic and the Foundations of Mathematics, vol. 40, Elsevier, 1965, pp. 92–130.[10]
J. Lambek and P. J. Scott , Introduction to higher-order categoricallogic , vol. 7, Cambridge University Press, 1988.[11]
S. Mac Lane , Categories for the working mathematician , vol. 5, SpringerScience & Business Media, 2013.[12]
S. Negri, J. Von Plato, and A. Ranta , Structural proof theory , Cam-bridge University Press, 2008.[13]
E. Rijke, M. Shulman, and B. Spitters , Modalities in homotopy typetheory , arXiv preprint arXiv:1706.07526, (2017).[14]
M. H. Sørensen and P. Urzyczyn , Lectures on the Curry-Howard iso-morphism , vol. 149 of Studies in Logic and the Foundations of Mathematics,Elsevier, 2006.[15]
W. W. Tait , Intensional interpretations of functionals of finite type i , TheJournal of Symbolic Logic, 32.2 (1967), pp. 198–212.[16]
A. Troelstra , Principles of Intuitionism , Springer, 1969.[17]
D. van Dalen , Logic and Structure , Springer, 4th ed., 2008.[18]
D. van Dalen and A. Troelstra , Constructivism in Mathematics. AnIntroduction I , vol. 121 of Studies in Logic and the Foundations of Mathe-matics, Elsevier, 1988.[19]
J. von Plato , The development of proof theory , in The Stanford Encyclo-pedia of Philosophy, E. N. Zalta, ed., Metaphysics Research Lab, StanfordUniversity, winter 2018 ed., 2018.[20]