An intuitionistically complete system of basic intuitionistic conditional logic

We introduce a basic intuitionistic conditional logic $\mathsf{IntCK}$ that we show to be complete both relative to a special type of Kripke models and relative to a standard translation into first-order intuitionistic logic. We show that $\mathsf{IntCK}$ stands in a very natural relation to other similar logics, like the basic classical conditional logic $\mathsf{CK}$ and the basic intuitionistic modal logic $\mathsf{IK}$. As for the basic intuitionistic conditional logic $\mathsf{ICK}$ proposed by Y. Weiss, $\mathsf{IntCK}$ extends its language with a diamond-like conditional modality, but its diamond-conditional-free fragment is also a proper extension of $\mathsf{ICK}$. We briefly discuss the resulting gap between the two candidate systems of basic intuitionistic conditional logic and the possible pros and cons of both candidates.


Introduction
The present paper was written in an attempt to find and vindicate an answer to the question, what is a basic intuitionistic conditional logic.By basic logic we mean a logic that is complete relative to a universal class of suitably defined Kripke models.This basic logic must also be intuitionistic in the sense of being the intuitionistic counterpart of the basic classical conditional logic CK (introduced in [3]; see also [11]).More precisely, its only difference from CK must consist in the fact that the classical reading of the Kripke semantics for CK is replaced by intuitionistic reading, and the classical first-order metalogic of CK is replaced with intuitionistic one.Finally, this logic must be fully conditional in that it must enrich the language of intuitionistic propositional logic with the full set of conditional modalities, both the stronger box-like and the weaker diamond-like .The existing literature on intuitionistic conditional logic is not vast, but it seems to have a clear candidate for the role of a basic system.This candidate is the system ICK, introduced by Y. Weiss in [15] (see also the more detailed exposition in [16]).This research was followed by [4], which was based on a version of Kripke semantics different from that used in [15]; however, this change did not affect the status of ICK which the authors of [4] have shown to be also the system complete relative to the universal class of their preferred variety of Kripke models.ICK also has a claim for the title of the intuitionistic counterpart of CK in that its complete axiomatization results from replacing the classical propositional fragment of CK with the intuitionistic one.
However, ICK only features the stronger conditional modality, , and no attempt is made to show the completeness of ICK relative to an intuitionistic reading of the metatheory of CK.The former shortcoming is openly acknowledged by I. Ciardelli and X. Liu, who write: " Just like ∀ and ∃ are not interdefinable in intuitionistic predicate logic, and and are not interdefinable in intuitionistic modal logic, also and will not be interdefinable in intuitionistic conditional logic.In order to capture might-conditionals in the intuitionistic setting, we need to add to the language as a new primitive...In this extended setting, the properties of might not uniquely determine the properties of .Therefore, it becomes especially interesting to look at the landscape of intuitionistic conditional logics in a setting where the language comprises both operators" [4, p. 830].
In this paper, we are going to propose a different system for the role of basic intuitionistic conditional logic.This system, which we call IntCK, both answers the concern expressed in the above quote by I. Ciardelli and X. Liu and can be shown to enjoy a form of strong completeness relative to an intuitionistic reading of its metatheory.We will also show that the ( )-free fragment of IntCK is a proper extension of ICK, whence it follows that ICK is incomplete relative to intuitionistic reading of its metatheory and hence cannot be viewed as a full basic intuitionistic conditional logic even within the ( )-free fragment of the conditional language.
The rest of this paper is organized as follows.Section 2 introduces the notational preliminaries, followed by Section 3 where we explain the syntax and a form of Kripke semantics of IntCK (in Section 3.1) and then axiomatize the logic (in Section 3.2).Section 4 explores the relation of IntCK to other logics, both propositional (in Section 4.1) and first-order (in Section 4.2).The latter subsection also contains the completeness theorem for IntCK relative to an intuitionistic reading of its Kripke semantics.
Finally, in Section 5, we briefly discuss the results of the previous sections, and, after drawing some conclusions, describe several avenues for continuing the research lines presented in the paper.The paper also has several appendices where the reader can find the more technical parts of our reasoning which we include for the sake of completeness.

Preliminaries
We use this section to fix some notations to be used throughout this (rather lengthy) paper.We will use IH as the abbreviation for Induction Hypothesis in the inductive proofs, and we will write α := β to mean that we define α as β.We will use the usual notations for sets and functions.As for the sets, we will write X ⋐ Y, iff X ⊆ Y and X is finite.Furthermore, we will understand the natural numbers as the finite von Neumann ordinals.We denote by ω the smallest infinite ordinal; given two ordinals λ, µ, we will use λ ∈ µ and λ < µ interchangeably.Given a (finite) tuple of any sort of objects α = (x 1 , . . ., x n ), we will denote by init(α) and end(α) the initial and final element of α, that is to say, x 1 and x n , respectively.More generally, given any i < ω such that 1 ≤ i ≤ n, we set that π i (α) := x i , i.e. that π i (α) denotes the i-th projection of α.Given another tuple β = (y 1 , . . ., y m ), we will denote by (α) ⌢ (β) the concatenation of the two tuples, i.e. the tuple (x 1 , . . ., x n , y 1 , . . ., y m ).The empty tuple will be denoted by Λ.
We will extensively use ordered couples of sets which we will also call bi-sets.The usual set-theoretic relations and operations on bi-sets will be understood componentwise, so that, e.g.(X, Y) ⊆ (Z, W) means that X ⊆ Z and Y ⊆ Y and similarly in other cases.The relations will be understood as sets of ordered tuples where the length of the tuple defines the arity of the relation.Given binary relations R ⊆ X × Y and S ⊆ Y × Z, we will denote their composition by R • S := {(a, c) | for some b ∈ Y, (a, b) ∈ R & (b, c) ∈ S }.
Given a set X, we will denote by id[X] the identity function on X, i.e. the function f : X → X such that f (x) = x for every x ∈ X.It is clear that a function f : X → Y can be understood as a special type of relation f ⊆ X × Y. Therefore, our notation for the composition of functions is in line with the one used for the composition of relations: namely, given two functions f : X → Y and g : Y → Z, we denote by f • g the function h : X → Z such that h(x) = g( f (x)) for every x ∈ X. 1 In relation to the operation of superposition, functions of the form id[X] have a special importance as a limiting case.More precisely, given a set X and a family F of functions from X to X, we will also assume that the superposition of an empty tuple of functions from F is just id [X].
Furthermore, if f : X → Y is any function, x ∈ X and y ∈ Y, we will denote by f [x/y] the unique function g : X → Y such that, for a given z ∈ X we have: Despite our efforts to accommodate and represent the intuitionistic reading of the classical semantics for conditional logic, the meta-logic of this paper remains classical.Therefore, we presuppose the basic acquaintance with the language of first-order logic and its classical model theory.Finally, in view of the fact that this paper is about intuitionistic conditional logic, we presuppose the basic acquaintance of the reader with both propositional and first-order version of this logic; in particular the reader should know of at least one complete Hilbert-style axiomatization of intuitionistic logic.
3 IntCK, the basic intuitionistic logic of conditionals

Language and semantics
We are going to consider the language L based on a countably infinite set of propositional variables Var and the following set of logical symbols {⊥, ⊤, ∨, ∧, →, , }.We will denote the propositional variables by letters p, q, r, s and the formulas in L by φ, ψ, χ, θ, adding the subscripts and superscripts when it is convenient.
We will use ¬φ as an abbreviation for φ → ⊥ and φ ↔ ψ as an abbreviation for (φ → ψ)∧(ψ → φ).The formulas of L are interpreted by the following models: ∅ is a set of worlds, ≤ is a pre-order (i.e., a reflexive and transitive relation) on W, V : Var → P(W) is such that, for every p ∈ Var and all w, v ∈ W it is true that Next, we must have R ⊆ W ×P(W)×W. Thus, for every X ⊆ W, R induces a binary relation R X on W such that, for all w, v ∈ W, R X (w, v) iff R(w, X, v).Finally, the following conditions must be satisfied for every X ⊆ W: Conditions (c1) and (c2) can be naturally reformulated as requirements to complete the dotted parts of each of the following commuting diagrams once the respective straight-line part is given: The models defined are the so-called Chellas models.The alternatives to Chellas models include Segerberg models which use a designated family of subsets of W in place of its full powerset P(W) as in Definition 1.Yet another alternative is to use families of formula-indexed binary relations {R φ | φ ∈ L}.These choices are of no import on the level of the basic conditional logic, in other words, they all induce one and the same logic both in the classical and in the intuitionistic case.We have chosen Chellas models since their definition looks short and simple; yet the Segerberg models can be perhaps ascribed a deeper foundational meaning.For example, the first-order intuitionistic theory T h defined in Section 4.2 gives the intuitionistic encoding of the Segerberg variety of classical conditional semantics rather than the Chellas one.
Our standard notation for models is M = (W, ≤, R, V).Any model decorations are assumed to be inherited by their components, so that, for example M n always stands for (W n , ≤ n , R n , V n ).A pointed model is a structure of the form (M, w), where M is a model and w ∈ W.
The formulas of L are interpreted over pointed models by means of the satisfaction relation | = defined by the following induction on the construction of φ ∈ L: where we assume, for any given φ ∈ L, that φ M stands for the set {w ∈ W | M, w | = φ}.Given a pair (Γ, ∆) ∈ P(L) × P(L), we say that a pointed model (M, w) satisfies (Γ, ∆) and write M, We say that (Γ, ∆) is satisfiable iff some pointed model satisfies it, and that ∆ follows from Γ (and write We say that Γ is satisfiable iff (Γ, ∅) is; and if (M, w) satisfies (Γ, ∅), then we simply write M, w | = Γ.If some of Γ, ∆ are singletons, we may omit the figure brackets; in case some of them are empty, we may omit them altogether.We say that φ ∈ L is Given a model M and an X ⊆ W, we say that X is upward-closed in M iff (∀w ∈ X)(∀v ≥ w)(v ∈ X).Definition 1 clearly implies that, for every Chellas model M and for every p ∈ Var, the set V(p) = p M is upward-closed in M. The latter observation can be lifted to the level of arbitrary formulas: We omit the easy proof by induction on the construction of φ ∈ L. We close this subsection with our first result about IntCK: Proposition 1. IntCK has Disjunction Property.In other words, for all Proof.The right-to-left direction is trivial.As for the other direction, assume, towards contradiction, that φ 1 ∨φ 2 ∈ IntCK, but both φ 1 IntCK and φ 2 IntCK.Then we can choose pointed models (M 1 , w 1 ) and (M 2 , w 2 ) such that M i , w i | = c φ i for all i ∈ {1, 2}; we may assume, wlog, that W 1 ∩ W 2 = ∅.We then choose an element w outside W 1 ∪ W 2 and define the following pointed model (M, w) for which we set: We show that M is indeed a model.The only non-trivial part is the satisfaction of conditions (c1) and (c2) from Definition 1.
As for (c1), assume that some v ′ , v, u ∈ W and X ⊆ W are such that v ′ ≥ v R X u.Then, by defintion of R, we must have either v, u ∈ W 1 or v, u ∈ W 2 .Assume, wlog, that v, u ∈ W 1 .Then we must have, first, that v (R 1 ) X∩W 1 u, and, second, that v ′ ≥ 1 v so that also v ′ ∈ W 1 .But then, since M 1 satisfies (c1), there must be a u ′ ∈ W 1 such that v ′ (R 1 ) X∩W 1 u ′ ≥ 1 u whence clearly also v ′ R X u ′ ≥ u, so that (c1) is shown to hold for M. We argue similarly for (c2).
Next, the following claim can be shown by a straightforward induction on the construction of φ ∈ L: Claim.For every i ∈ {1, 2}, every v ∈ W i , and every φ ∈ L, we have It follows now that M, w i | = φ i for all i ∈ {1, 2}, and, since we have w ≤ w 1 , w 2 , Lemma 1 implies that M, w | = φ i for all i ∈ {1, 2}, or, equivalently, that M, w | = c φ 1 ∨ φ 2 , contrary to our assumption.The obtained contradiction shows that IntCK must have Disjunction Property.

Axiomatization
In this subsection, we obtain a sound and (strongly) complete axiomatization of IntCK.We consider the Hilbert-style axiomatic system ICK, given by the following list of axiomatic schemes: A complete list of axioms of intuitionistic propositional logic Int (A0) Besides the axioms, ICK includes the following rules of inference: We assume the standard notion of a proof in a Hilbert-style axiomatic system for ICK, namely as a finite sequence of formulas, where every formula is either an axiom or is derived from earlier formulas by an application of a rule.A proof is a proof of its last formula.A derivation from premises is a finite sequence of formulas, where every formula is either a premise, or a provable formula, or is derived from earlier formulas by an application of (MP).Given a Γ ∪ {ψ} ⊆ L, we write Γ ⊢ ψ iff ψ there exists a derivation of ψ from some (possibly zero) elements of Γ as premises; we omit the figure brackets in case Γ is either a singleton or empty.
On the other hand, we write Γ ⊢ ψ iff there is a derivable rule allowing to infer φ from Γ, i.e. iff there is a finite sequence of formulas in which the last formula is ψ, and every formula in the sequence is either in Γ, or a provable formula, or is obtained from earlier formulas by an application of one of the inference rules.Thus, we have In particular, it follows from these conventions that φ is provable iff ⊢ φ iff ⊢ φ.
Before we go on to prove the soundness and completeness of ICK relative to IntCK, we would like to quickly address the relations between ICK and the intuitionistic propositional logic Int.
The language L i of Int is the { , }-free fragment of L; a complete axiomatization of Int is provided by (A0) together with (MP).Int is one of the best researched non-classical propositional logics with numerous detailed expositions to be found in the existing literature (see, e.g., [5,Ch. 5] for a quick textbook level introduction).The following lemma sums up the relations between Int and ICK: Lemma 2. The following statements hold: 1.If Γ, ∆ ⊆ L i are such that Γ | = Int ∆, and Γ ′ , ∆ ′ ⊆ L are obtained from Γ, ∆ by a simultaneous substitution of L-formulas for variables, then Γ ′ ⊢ ∆ ′ .Moreover, Deduction Theorem holds for ICK in that for all Γ ∪ {φ, ψ} ⊆ L we have Proof (a sketch).Part 1 is trivial.As for Part 2, its (⇐)-part is also trivial, and its (⇒)-part follows from an observation that, given a proof of φ ∈ L i in ICK we can turn it into a proof in Int by replacing all its subformulas of the form χ θ with ⊤ and all its subformulas of the form χ θ with ⊥.
Turning now to the question of soundness and completeness of ICK relative to IntCK, we observe, first, that ICK only allows us to deduce theorems of IntCK: The proof proceeds by the usual method, i.e. we show that all the axioms are valid and that the rules of ICK preserve the validity.We are now going to show the converse of Lemma 3, and we start our work by proving some theorems and derived rules in ICK, which we collect in the following lemma: Lemma 4. Let φ, ψ, χ ∈ L. The following theorems and derived rules can be deduced in ICK: The sketch of its proof is relegated to Appendix A. A bi-set (Γ, ∆) ∈ P(L) × P(L) is called consistent iff for no ∆ ′ ⋐ ∆ do we have that Γ ⊢ ∆ ′ . 2 Note that, since ICK extends Int, and also in view of Lemma 2.1, this definition allows for the following equivalent form: Lemma 5. A bi-set (Γ, ∆) ∈ P(L)×P(L) is inconsistent iff, for some m, n ∈ ω some φ 1 , . . ., φ n ∈ Γ and some ψ 1 , . . ., ψ m ∈ ∆ we have: n i=1 φ i ⊢ m j=1 ψ j , or, equivalently, ⊢ n i=1 φ i → m j=1 ψ j .
As for Part 5, assume its hypothesis and suppose, towards contradiction that (Γ It follows now from (18), that (Γ, ∆) is not consistent and thus also not maximal, contrary to our initial assumption.The obtained contradiction shows that the bi-set must have been consistent.For Part 6, assume its hypothesis and suppose that (Γ 0 ∪{φ ).Again, we set ψ := n i=1 ψ i , χ := m j=1 χ j , and θ := k r=1 θ r , and reason as follows: By Γ 1 ⊆ Γ, we know then that also (χ → θ) ∈ Γ.Now, since clearly Γ ⊢ χ, it follows that Γ ⊢ θ, whence, by Parts 1 and 3, we know that Γ ⊢ θ r for some 1 ≤ r ≤ k, which clearly contradicts the consistency of (Γ, ∆).The obtained contradiction shows that the bi-set We observe, next, that we can use the usual Lindenbaum construction to extend every consistent bi-set to a maximal one: Lemma 8. Let (Γ, ∆) ∈ P(L)×P(L) be consistent.Then there exists a maximal (Ξ, Θ) ∈ P(L)×P(L) such that Γ ⊆ Ξ and ∆ ⊆ Θ.
Next, we define the canonical model M c for ICK: • For all (Γ 0 , ∆ 0 ), (Γ 1 , ∆ 1 ) ∈ W c and X ⊆ W c , we have ((Γ 0 , ∆ 0 ), X, (Γ 1 , ∆ 1 )) ∈ R c iff there exists a φ ∈ L, such that all of the following holds: First of all, we observe that the definition of R c does not depend on the choice of the representative formula φ ∈ L. The following lemma provides the necessary stepping stone: Proof.Assume the hypothesis of the Lemma.We will show that in this case we must have ⊢ φ ↔ ψ.Suppose not, and assume, wlog, that φ → ψ.Then ({φ}, {ψ}) must be consistent and thus extendable to some maximal (Γ 0 , ∆ 0 ) ⊇ ({φ}, {ψ}).But then clearly in contradiction with our initial assumptions.The obtained contradiction shows that we must have ⊢ φ ↔ ψ.The application of (RA ) and (RA ) then yields that also ⊢ (φ χ) ↔ ψ χ and ⊢ (φ χ) ↔ ψ χ for every χ ∈ L, whence our Lemma clearly follows.
We have to make sure that we have indeed just defined a model: Lemma 10.The structure M c , as given in Definition 2, is a model.
It is also clear from Definition 2 and Lemma 9 that ≤ c is a pre-order, and that R c ⊆ W c ×P(W c )×W c is well-defined.So it only remains to check the satisfaction of conditions (c1) and (c2) from Definition 1.
Basis.If φ = p ∈ Var, then the lemma holds by the definition of M c .If φ ∈ {⊤, ⊥}, then we reason as in the case of Int.

Relations with other logics
In the existing literature, one can find several systems which can be viewed as natural companions to IntCK.On the one hand, there are different intensional propositional logics, which either treat conditionals from a viewpoint similar to that of IntCK, or extend Int with similar additional connectives, or both.On the other hand there is FOIL, the first-order version of Int, which is naturally viewed as a super-system for IntCK, in that IntCK can be seen as isolating a special subclass of intuitionistic first-order reasoning which is relevant to handling conditionals.In both cases, one can expect that IntCK will display some sort of natural relation to each of these logics.This section is devoted to looking into some examples of such relations.

Intensional propositional logics
Due to the great number of systems in this class that can be related to IntCK, we only confine ourselves to mentioning a few prominent examples; and, considering the length of this paper, most of our claims will only be supplied with a rather sketchy proof.We will mostly consider logical systems given by complete Hilbert-style axiomatizations.If S is such a system and A 1 , . . ., A n is a finite sequence of axiomatic schemes, we will denote by S + {A 1 , . . ., A n } the system obtained from S by adding every instance of A 1 , . . ., A n and then closing under the applications of the rules of inference assumed in S. In case n = 1, we will omit the figure brackets.
The first of the systems that we would like to consider is the basic system CK of classical conditional logic, introduced in [3] and defined over L. One variant of a complete axiomatization for CK is given by extending of (A0), (A1), (A5), (MP), (RA ), and (RC ) with the following axiomatic schemes: The addition of (Ax0) to (A0) and (MP) transforms the purely propositional base of the system from Int to the classical propositional logic CL.It is natural to expect that a similar relation holds between IntCK and CK in that the former is the a subsystem of the latter and that IntCK, in its turn, can be transformed into CK by the addition of (Ax0), thus giving us the intuitionistic counterpart of CK.This is indeed the case, as we will show presently.We prepare the result with a technical lemma: Lemma 12.The following statements are true: 1. Every instance of (A2)-(A4), (A6), (RA ), and (RC ) as well as all the theorems and derived rules given in Lemma 4, are deducible in CK.

Every instance of
We sketch the proof in Appendix B.
The relation between IntCK and CK is then analogous to the relation between Int and CL: Proposition 2. The following statements are true for every φ ∈ L: Proof.(Part 1) If φ ∈ L and φ 1 , . . ., φ n = φ is a proof in IntCK, then we can transform it into a proof of φ in CK by replacing every occurrence of (A2)-(A4), (A6), and every application (RA ), and (RC ) by the deductions given in the proof of Lemma 12.1.
(Part 2) If φ ∈ (IntCK + (Ax0)), then we can argue as in Part 1.The only difference will be possible presence of instances of (Ax0) which do not require any additional work.In the other direction, if φ ∈ CK and φ 1 , . . ., φ n = φ is a proof in CK, then we can transform it into a proof of φ in IntCK by replacing every occurrence of (Ax1) by the deduction given in the proof of Lemma 12.2.
Another logic that is very natural to compare with IntCK is the system of intuitionistic conditional logic ICK introduced by Y. Weiss in [15].ICK is defined over the ( )-free fragment of L which we denote by L .One of its complete axiomatizations is obtained by simply omitting (Ax0) and (Ax1) from CK.One (Chellas-style) variant of semantics 3 for ICK can be given, if we replace conditions (c1) and (c2) in Definition (1) by the following condition to be satisfied for every X ⊆ W: We will call the resulting models Weiss models.The satisfaction relation used by Y. Weiss (we will be denoting it by | = w ) is also different from | = in that the inductive clause for is no longer needed and the inductive clause for is given in the following, more classicallyminded 4 version: The relations between ICK and IntCK can be summarized as follows: Proposition 3. We have ICK ⊆ IntCK.However, IntCK extends ICK non-conservatively, in that we have Proof.It is clear that every proof in ICK is also a proof in IntCK.As for the non-conservativity claim, it is easy to see that ¬¬(⊤ ⊥) → (⊤ ⊥) is derivable in IntCK: However, if we consider the Weiss model M = (W, ≤, R, V) where W := {w, v, u}, ≤ is the reflexive closure of {(w, v)}, R := {(w, W, u)}, and V(p) = ∅ for every p ∈ Var, then we see that The question then arises as to how one should interpret this difference between ICK and IntCK ∩ L ; is it due to ICK being incomplete over L , or is the reason that IntCK smuggles in some principles that are not intuitionistically acceptable?The latter answer seems to be favored by the fact that the elimination of double negation is not generally favored by intuitionistic reasoning; moreover, it is clear that the proof of (¬¬(⊤ ⊥) → (⊤ ⊥)) in IntCK essentially uses the principles that are only expressible with the help of the additional connective and can be therefore seen as, loosely speaking, 'impure'.However, the former answer is clearly favored by Theorem 2 of this paper which implies that the standard translation of the questionable formula (¬¬(⊤ ⊥) → (⊤ ⊥)) is a valid first-order intuitionistic principle.Nevertheless, the weight of the latter argument is somewhat diminished by the fact that the proof of Theorem 2 in this paper depends on classical principles.
Turning once more to CK, it has been shown that it corresponds to the basic modal logic K, which is defined over the language L m given by the following BNF: More precisely, consider the translation T r : L m → L defined by the following induction on the construction of φ ∈ L m : It follows from the results of [9] that K is embedded into CK by T r in the sense that, for every φ ∈ L m , φ ∈ K iff T r(φ) ∈ CK.It is natural to expect that T r also embeds some basic intuitionistic modal logic into IntCK and this is indeed the case for the basic intuitionistic modal logic IK introduced independently in [7] and [10].Just like K, IK is defined over L m .One variant of its complete axiomatization can be given by adding to (A0) and (MP) the following additional axiomatic schemes plus a new rule of inference: Again, we prepare the result connecting IK to IntCK with a technical lemma: Lemma 13.The following theorems and rules are deducible in IK for all φ, ψ ∈ L m : Proof.The theorem (t1) and the rule (r1) can be deduced as in K. To obtain the deduction of (r2), one needs to replace the occurrence of (a1) in the deduction of (r1) by the respective occurrence of (a2).Rules (r3) and (r4) are deduced by applying rules (r1) and (r2), respectively plus the definition of ↔.The theorem (t4) can be deduced by applying (nec) to the provable formula ⊤; (t3) is just (a4), the other direction follows by applying (r2) to provable formulas φ → (φ ∨ ψ) and ψ → (φ ∨ ψ).Finally, (t2) can be deduced as follows: We now claim that: Proof.If φ ∈ IK then let φ 1 , . . ., φ n = φ be a deduction of φ in IK.Consider the sequence T r(φ 1 ), . . ., T r(φ n ) = T r(φ).The translation T r leaves intact every instance of (A0) and (MP) and maps every instance of (a1), (resp.(a2), (a3), (a4), (a5)) into an instance of (T1) (resp.(T2), (A6), one half of (A3), (A4)).Similarly, every application of the rule (nec) is mapped by T r into an application of (Nec).Therefore, one can straightforwardly extend T r(φ 1 ), . . ., T r(φ n ) = T r(φ) to a proof of T r(φ) in IntCK by inserting the variants of deductions sketched in the proof of Lemma 4.
In the other direction, let ψ 1 , . . ., ψ n = T r(φ) be a deduction of T r(φ) in IntCK.Consider the mapping T r : L → L m defined by induction on the construction of φ ∈ L: The following can be easily proved by induction on the construction of φ ∈ L m : Claim.For every φ ∈ L m , T r(T r(φ)) = φ.Indeed, both basis and every case in the induction step are pretty straightforward.As an example, we consider the case when φ = ψ.We have then T r(T r( ψ)) = T r(⊤ ψ) = ψ.Claim 1 is proven.
As a further result of the tight connection between IK and IntCK we observe that the countermodels that show in [12, p. 54-55] the mutual non-definability of and in IK can be re-used to show the mutual non-definability of and in IntCK, thus answering the concern expressed in the passage from [4] quoted in the introduction to this paper.
We add that the gap between IntCK and ICK is also mirrored at the level of L m as the gap between the -free fragment of IK and the system HK introduced in [2], which Y. Weiss cites in [16,Footnote 11] as one of the sources for ICK.Indeed, one can easily show both that ICK stands to HK in the same relation ascribed to IntCK and IK in Proposition 4 above, and that ¬¬ ⊥ → ⊥ is in the -free fragment of IK but outside HK .This is all the more surprising in view of some confusing claims made in the existing literature.5

First-order intuitionistic logic
We define the first-order intuitionistic logic FOIL over the langauge L f o , 6 based on a countable set Ind of individual variables and given by the following BNF: where p ∈ Var and x, y, z ∈ Ind.The formulas of the form px, Rxyz, Ox, S x, Exy, x ≡ y, ⊤, and ⊥, will be called atomic formulas, or simply atoms.We will continue to use the abbreviations ↔ and ¬; additionally, we will use (∀x) O φ as an abbreviation for ∀x(Ox → φ).Given a formula φ ∈ L f o , we can inductively define its sets of free and bound variables in a standard way (see, e.g.[5, p. 64]).These sets, denoted by FV(φ) and BV(φ), respectively, are always finite.These notions can be extended to an arbitrary Γ ⊆ L f o , although FV(Γ) and BV(Γ) need not be finite.If x ∈ Ind \ (FV(φ) ∪ BV(φ)), then x is said to be fresh for φ.Given any n ∈ ω and any x 1 , . . ., x n ∈ Ind, we will denote by L {x 1 ,..., Finally, given a φ ∈ L f o , and some x, y ∈ Ind such that y is fresh for φ, we can define the result (φ) y x of substituting y for x in φ simply as the result of replacing free occurrences of x in φ with the occurrences of y.
While the most popular semantics for FOIL is given by Kripke models (see e.g.[5, Section 5.3]), we will use for this logic a slightly more involved but equivalent semantics based on intuitionistic Kripke sheaves.The following definition provides the necessary details: Definition 3. A Kripke sheaf is a structure of the form S = (W, ≤, A, H) such that: • W ∅ is the set of worlds or nodes.
• ≤ is a pre-order on W.
• A is a function, returning, for every w ∈ W a classical first-order model A w = (A w , ι w ) over the vocabulary (sometimes called signature where A w ∅ is the domain and ι w is the function assigning every • Finally, H is a function defined on {(w, v) ∈ W 2 | w ≤ v}, which, for every pair (w, v) in its domain returns a (classical) homomorphism H wv : A w → A v .This function has to satisfy the following additional conditions: -For all w ∈ W, We use S = (W, ≤, A, H) as our standard notation for Kripke sheaves; we will assume that any decorations of S transfer to its components.
Given a Kripke sheaf S, and a w ∈ W, we will call an (S, w)-variable assignment any function f : Ind → D w .In order to determine the truth value of a formula, one needs to supply a Kripke sheaf S, a node w ∈ W and an (S, w)-variable assignment f .With these data, the satisfaction of a formula φ ∈ L f o is given by the relation | = f o which defined by the following induction on the construction of a formula: As usual, it follows from this definition that the truth value of a formula φ ∈ L f o only depends on the values assigned by f to the values of the variables in FV(φ).We will therefore write It is easy to see that | = f o , just like | = before, can be used to introduce the complete set of semantic notions.More precisely, given a Kripke sheaf S, and a w ∈ W, and a tuple (a 1 , . . ., a n ) ∈ A w , we will call the triple (S, w, (a 1 , . . ., a n )) an n-evaluation point, and, given a pair (Γ, ∆) ∈ P(L {x 1 ,...,x n } f o ), we say that an evaluation point (S, w, (a 1 , . . ., a n )) satisfies (Γ, ∆) and write S, w | = c (Γ, ∆)[x 1 /a 1 , . . ., x n /a n ] iff we have: We say that (Γ, ∆) is first-order-satisfiable iff some n-evaluation point satisfies it, and that ∆ first-order-follows from Γ (and write Γ | = f o ∆) iff (Γ, ∆) is first-order-unsatisfiable. We say that Γ is first-order-satisfiable iff (Γ, ∅) is; and if (S, w, (a 1 , . . ., a n )) first-order-satisfies (Γ, ∅), then we simply write S, FOIL is known to be strongly complete relative to the semantic of Kripke sheaves; in other words, we have ), see [8, Section 3.6 ff] for details. 7 However, in this paper we will be mainly interested in things Lemma 16.For every φ ∈ L and for all distinct x, y ∈ Ind the following holds: These lemmas provide a stepping stone for our first result on the relations between T h and IntCK: The proofs of Lemmas 15, 16, and of Proposition 5 can be found in Appendix C. Using the compactness of FOIL, we are now in a position prove one direction of the main result for the present section: Proof.If Γ | = IntCK ∆ then, by Theorem 1, (Γ, ∆) must be unsatisfiable, whence, by Corollary 1, there must exist some Γ ′ ⋐ Γ and some ∆ ′ ⋐ ∆ such that (Γ ′ , ∆ ′ ) is unsatisfiable.But then, again, by Theorem 1 and Lemma 5, we must have But then, trivially, also Example 1.To illustrate the import of Proposition 5, let us consider the intuitionistic meaning of the formula φ := ¬¬(⊤ ⊥) → (⊤ ⊥), whose status as a part of basic intuitionistic logic of conditional is, as we saw in Section 4.1, disputed between ICK and IntCK.Setting ψ := S y∧(∀z) O (Ezy), we see that we must have S T x (φ) = ¬¬∃y(ψ∧∀w¬Rxyw) → ∃y(ψ∧∀w¬Rxyw).By (Th7), we know that T h | = f o ∃yψ, so the whole formula is reduced, modulo T h, to the following theorem of FOIL: ¬¬∀w¬Rxyw → ∀w¬Rxyw.It is clear now that the case of double negation elimination claimed in φ is intuitionistically acceptable and that φ indeed must be accepted as a theorem of basic intuitionistic conditional logic.
Part 3 is trivial, and an easy induction on n ∈ ω also yields us Part 4.
The set of all global choice functions will be denoted by G.It is easy to see that an analogue of Lemma 18 can be proven for global choice functions: Lemma 19.Let (Γ, ∆) ∈ W c .Then the following statements hold: there exists an F ∈ G such that F ↾ S eq(Γ, ∆) = f .
The global choice functions are the basis for another type of sequences, that, along with the standard sequences, is necessary for the main model-theoretic construction of the present section.We will call them global sequences.A global sequence is any sequence of the form (F 1 , . . ., F n ) ∈ G n where n ∈ ω (thus Λ is also a global sequence with n = 0).Given two global sequences (F 1 , . . ., F k ) and (G 1 , . . ., G m ), we say that (G 1 , . . ., G m ) extends (F 1 , . . ., F k ) and write (F 1 , . . ., F k ) ⊑ (G 1 , . . ., G m ) iff k ≤ m and F 1 = G 1 , . . ., F k = G k .Furthermore, we will denote by Glob the set n∈ω G n , that is to say, the set of all global sequences.
The final item in this series of preliminary model-theoretic constructions is a certain equivalence relation on L. Namely, given any φ, ψ ∈ L and any (Γ, ∆) ∈ W c , we define that: For any φ ∈ L, we will denote its equivalence class relative to ∼ by [φ] ∼ .
We now proceed to define a particular sheaf S c which can be seen as induced by the model M c of Definition 2. Definition 5. We set S c := (Glob, ⊑, A, F), where: • For every F ∈ Glob, A F = A = (A, ι), i.e. every global sequence gets assigned one and the same classical model A. As for the components of A, we set: ι(p) := {β ∈ S eq | p ∈ π 1 (end(β))}, for every p ∈ Var.
Proof.Assume the hypothesis; we may also assume, wlog, that, for some k ≤ m < ω, F and Ḡ are given in the following form: In this case, we also get the following representation for F F Ḡ: Part 1 now easily follows from (def1-F F Ḡ) and Lemma 19.4.As for Part 2, we observe that, if α ∈ S eq, then Now Part 1 and Lemma 19.5 together imply that α ≺ F F Ḡ (α).By Definition 4, this means that end(α) ≤ c end(F F Ḡ(α)), or, equivalently, that π 1 (end(α)) ⊆ π 1 (end(F F Ḡ(α))).
Lemma 21. S c is a Kripke sheaf.
Proof.It is clear that Glob is non-empty and that ⊑ defines a pre-order on Glob.It is also clear that A = (A, ι), as given in Definition 5, makes up a classical model, and that, for any given F, Ḡ ∈ Glob such that F ⊑ Ḡ, we have F F Ḡ : A → A.
As for the conditions imposed by Definition 3 on the functions of the form F F Ḡ, it is clear from Definition 5 and from our convention on the superpositions of empty families of functions that (1) for any F ∈ Glob we will have in It remains to establish that, for each pair F, Ḡ ∈ Glob such that F ⊑ Ḡ, the function F F Ḡ is a (classical) homomorphism from A to itself.The latter claim boils down to showing that the extension of every predicate P ∈ Σ is preserved by F F Ḡ. In doing so, we will assume that, for some appropriate k ≤ m < ω, F, Ḡ, and F F Ḡ are given in a form that satisfies (def-F), (def-Ḡ), (def1-F F Ḡ), and (def2-F F Ḡ).The following cases have to be considered: Case 1. P ∈ {O, S }.Trivial by (def1-F F Ḡ) and (def2-F F Ḡ). Case 2. P = p ∈ Var.If α ∈ A is such that α ∈ ι(p), then, by Definition 5 and Lemma 20, we must have all of the following: (1) α, F F Ḡ(α) ∈ S eq, (2) π 1 (end(α)) ⊆ π 1 (end(F F Ḡ(α))), and (3) p ∈ π 1 (end(α)).But then clearly also p ∈ π 1 (end(F F Ḡ(α))), whence, further, F F Ḡ (α) ∈ ι(p), as desired.

Conclusion, discussion, and future work
We have shown that IntCK is indeed the correct version of basic intuitionistic conditional logic in the sense outlined in the opening paragraphs of this paper.Thus, Theorem 1 shows that IntCK is basic in the sense that it is strongly complete relative to a (suitably defined) universal class of Kripke models; Theorem 2 then shows that IntCK is intuitionistic in the sense that it is strongly complete relative to an intuitionistic reading of the classical semantics of conditional logic.Finally, IntCK is fully conditional in that it features the full set of conditional connectives { , } which are not definable in terms of one another.It seems that the construction of S c used in the proof of Theorem 2 is relatively novel, since similar results for intuitionistic modal logic are proved by other methods; in particular, [12] proceeds proof-theoretically whereas [6] uses the method of selective filtration forming a countable chain of finite models.The latter method was not very convenient to use in the case of conditional logic since one has to keep a supply of counterexamples distinguishing modal accessibility relations induced by formulas that fail to be provable complete.
Our answer to the question of what is the basic intuitionistic conditional logic is still open to criticism, mainly in relation to the intuitionistic component of our claims.We would like to briefly mention here two possible counter-arguments.First, despite the fact that Theorem 2 shows that the reasoning given in IntCK is but a subsystem of the first-order intuitionistic reasoning and can be embedded into the latter by the same sort of a standard translation that is also appropriate in the classical case, the fact that our proof of Theorem 2 is itself decidedly classical, diminishes the foundational importance of this result in the eyes of an intuitionist.Secondly, the theory T h used in this result is open to doubts as to whether it smuggles in too much of a classical set-theoretic principles to be acceptable for an intuitionist.
As for the second concern, we note that (Th1)-(Th5) are clearly harmless principles typical for two-sorted formulations of FOIL, and (Th12) is a form of extensionality axiom; the latter is generally uncontroversial and present in every known form of constructive set theory.Finally, (Th6)-(Th11) are particular forms of comprehension.Even though the question about the intuitionistically acceptable amount of comprehension is definitely open and contested, the comprehension principles given in (Th6)-(Th11) all seem to be very tame in that they only use the formulas with guarded quantifiers over the object sort.It seems reasonable to expect, therefore, that they will be acceptable under any of the existing accounts of intuitionistc set theory.
As for the first concern, however, we can only acknowledge it as a drawback of our work; to do better in this respect, one should rather prove Theorem 2 in the spirit of [12,Ch. 5], and we hope that we will be able to publish in the near future some sort of continuation to the present paper in which we will close this gap.
Another major direction for future work is to extend the methods and results of this paper to the treatment of conditionals in constructive logics with strong negation, for example, to Nelson's logics N3 and N4, and to the negation-inconsistent connexive logic C introduced by H. Wansing in [13].Among these three systems, C looks, perhaps, the most promising one, given that this subject already has seen its first rather intriguing steps in [14], and the methods of the current paper seem to open a way to a considerable refinement of these first results.
In case our formulas get too long, we will be replacing then with their labels, writing e.g. ( 123) → (124) instead of φ → ψ in case φ did occur earlier as equation ( 123) and ψ as equation (124).Case 2. φ = ψ χ.We consider the following deductions from premises.