Definable Conditionals

The variably strict analysis of conditionals does not only largely dominate the philosophical literature, since its invention by Stalnaker and Lewis, it also found its way into linguistics and psychology. Yet, the shortcomings of Lewis–Stalnaker’s account initiated a plethora of modifications, such as non-vacuist conditionals, presuppositional indicatives, perfect conditionals, or other conditional constructions, for example: reason relations, difference-making conditionals, counterfactual dependency, or probabilistic relevance. Many of these new connectives can be treated as strengthened or weakened conditionals. They are definable conditionals. This article develops a technique to infer the logic for such definable conditionals from the known logic of the underlying defining conditional. The technique is applied to central examples. The results show that a large part of the zoo of conditionals arises from a basic conditional—a constant nucleus of the different contextual and conceptual variations of variably strict conditionals.


Introduction
Conditionals are notoriously difficult to analyse. Conditionals are natural language sentences of the form 'if A then C', where A is the antecedent and C the consequent. A standard account has however emerged, the so-called possible worlds account (Stalnaker 1968;Lewis 1973b). According to this account, a conditional A > C is true in the actual world   1 However, recent reflections suggest that the defining clause needs to be strengthened by additional conditions. What these conditions are is not settled. Different approaches argue for different conditions (Krzyżanowska et al. 2013;Spohn 2015;Skovgaard-Olsen 2016;Raidl 2019;Rott 2019;Crupi and Iacona 2019). Some of these logics are not worked out yet, or only for specific semantics. To compare them, we need to know how the logic changes with the semantics, and vice versa. The article develops a technique to answer these questions.
The general problem is this: Take a strengthened conditional of the form ϕ ψ in the actual world iff closest ϕ-worlds are ψworlds and X .
Suppose that X is also formulated in terms of closeness. The conditional ϕ ψ can then be rephrased in the language for >, namely as (ϕ > ψ)∧χ , where χ expresses the semantic condition X . The following question arises: 88 E. Raidl -Can we use known completeness results for > to obtain completeness results for ?
The answer is yes and the paper provides a general method. The method also applies to weakened conditionals, where in the defining clause one replaces 'and' by 'or' and thus ∧ by ∨, and more generally to definable conditionals. The idea goes as follows: Redefine > in terms of . This yields a formula α in the language for . We can use this backtranslation to translate axioms for > into axioms for . The backtranslation is a looking glass showing a distorted picture of the logic for >. This picture is a logic for . The plan of the paper is as follows. Section 2 sets the stage with a flexible framework for conditionals. Section 3 introduces the translation between conditional languages. Section 4 defines semantic conditions for a translation to be well behaved. Section 5 introduces the backtranslation and axiomatic conditions for it to be well behaved. Section 6 develops the technique of transferring soundness, completeness and correspondence results. Section 7 applies this to the sufficient reason relation. Section 8 shows how to chain the transfer of results, illustrated by the necessary reason relation. Section 9 applies the method to alternative semantics, exemplified by ranking and belief revision semantics. The Conclusion (Sect. 10) comments on further applications.

Basic Conditional Logic
This section rehearses some known conditional logics and introduces the semantics used for the basic conditional >. The results considered here are known or analogue to famous results in modal logics, and thus proofs are omitted.
The alphabet of our basic conditional language is based on a fixed set of propositional variables Var, classical connectives, ¬, ∧, ∨ and → (the material conditional), and the basic conditional >, as well as the parenthesis ) and (. The set of formulas is defined inductively and is denoted L > . In what follows denotes any classical propositional tautology and ⊥ = ¬ . Let Γ be an axiomatic system in L > , given by a set of axioms and a set of inference rules. We write Γ ϕ iff there is a proof of ϕ in Γ , defined as usual. Possible rules of conditional logic are: 2 ϕ ↔ ϕ (RCEA, LLE) (ϕ > χ ) ↔(ϕ > χ) ϕ ↔ ϕ (RCEC, RLE) (χ > ϕ) ↔(χ > ϕ ) Some well known axioms are: 2 In (X, Y), X refers to Chellas' (1975) notation used here and Y to the KLM-tradition (Kraus et al. 1990).
We say that L ⊆ L > is a conditional logic iff it contains all substitution instances of propositional tautologies (PT) and is closed under Modus Ponens for → (MoPo). We denote L + X 1 + . . . + X n the smallest conditional logic closed under the rules of L, containing the axioms of L, as well as the axioms X 1 , . . . , X n . As an example, the smallest classical conditional logic is CE = RCEA + RCEC + MoPo + PT. The smallest normal conditional logic is CK = CE + Cm + CN + CC.
In what follows, f : X −→ Y indicates that f is a total function from X to Y . To model the logic CE and extensions, The sets [ϕ] M > = {w ∈ W : w > M ϕ} are co-inductively defined.
The truth clauses for propositional variables and the standard connectives are known from modal logic. The truth clause for the conditional says this: the conditional ϕ > ψ is true in w if and only if the set of ψ-worlds is in the ϕ-neighbourhood, or the proposition ψ is an option when supposing ϕ, or the proposition ψ is believed after revising by ϕ. I adopt standard notation and write: M > ϕ iff for all w ∈ W , w > M ϕ. F > ϕ iff for all models M over F, M > ϕ. And for C a class (of models or frames): C > ϕ iff for all X ∈ C, X > ϕ. When the model M is clear from the context, I drop upper indices, writing > instead of > M and [ϕ] > instead of When the language is clear, I drop lower indices. The following was proven by a canonical model construction (Chellas 1975): Theorem 1 CE is sound and complete for the class of minimal frames (models) M, i.e., CE α iff M > α for all α ∈ L > .
For minimal frames, we can translate axioms into properties of the neighbourhood selection function: for ϕ becoming A = [ϕ], ψ becoming B and χ becoming C, ϕ > ψ becomes B ∈ F(w, A). Inner logical operators ( , ⊥, ¬, ∧, ∨, → in the scope of >) are translated algebraically: becomes W , ⊥ becomes ∅, ¬ϕ becomes A = W \ A, ∧ becomes ∩, ∨ becomes ∪ and ϕ → ψ becomes A ∪ B. Outer logical operators are translated in the natural language. For X a conditional axiom, we denote X F or just (x) its corresponding minimal frame property, obtained by the mentioned transformation. The axiom schemes introduced above, have the following corresponding frame properties, quantified over all w ∈ W and all subsets A, B, C of W : "Correspondence" has a precise meaning-the same as in the correspondence theory for modal logic. Let X be an axiom scheme. We write w X iff w ϕ for all ϕ ∈ X. This lifts truth from formulas to axiom schemes. Similarly for models, frames and classes. The correspondence theory for minimal frames says that an axiom scheme is valid in a frame if and only if the frame has the property corresponding to the axiom scheme: Theorem 2 For F a minimal frame, F > X iff F satisfies X F .
From this and Theorem 1, completeness results for extensions of CE follow: Theorem 3 Let X 1 , . . . , X n be axiom schemes (from our list) and X F 1 , . . . , X F n the corresponding frame properties. Then the logic CE + X 1 + . . . + X n is sound and complete for the class C of those minimal frames (or models) which satisfy X F 1 , . . . , X F n .
Proof Soundness follows from Theorem 2. Completeness is proven by re-running Chellas' canonical model construction for Theorem 1, noting that the canonical model M Σ for Σ = CE + X 1 + . . . + X n has the properties X F 1 , . . . , X F n .
For latter sections, we need axioms for the outer and inner modalities. The outer necessity is α ≡ (¬α > ⊥) and the inner necessity is α ≡ ( > α). The outer possibility and inner possibility are defined as duals to and , as usual.
The following are axioms for the inner and outer modalities, which I also reformulate in their modal form: Assume CE + RCM. Then INC follows from CA, given ID. It is known from belief revision as the postulate of Inclusion.
PRES is an instance of CV. It is known as the postulate of Preservation. 3 In CK, the inner necessity is a normal necessity. M follows from CMon. It says that the outer necessity is monotone. C is an instance of CA, it says that the outer necessity is closed under conjunction. The outer necessity is a normal necessity in CE+ID+CMon+CA. P, known as probabilistic consistency, says that the outer and inner modality are consistent. We will also use the following weakening of CC: This axiom is redundant in the presence of CC + ID. Additionally, in a classical conditional logic with Cm and WCC, the axioms CN and ID are equivalent. The frame properties for the new axioms should be clear, and Theorems 2 and 3 extend to these.

Translating
This section introduces the idea of translating between conditional languages. Translations are powerful cognitive tools. They allow us to understand another language. In logic, they also provide a powerful methodological tool, since by relating one language to another, they also link different semantics or logics. Modal logic is full of such translations.
For example, the standard translation of modal formulas into first-order formulas, or Kripke's translation of intuitionistic into modal logic. The first links Kripke models to relational structures and thereby builds a bridge between modal and classical worlds. Its methodological power lies in transferring results from first order logic to modal logic, for example compactness, the Löwenheim-Skolem property or recursive enumerability of validities. Similarly, Kripke's translation makes us see intuitionism with modal and ultimately classical eyes. In brief, a translation is a lens through which we see the unknown in the shape of the known, it is also a conversionmachine which transfers known results to unknown domains. This is how we will use translations for conditionals. Translating between conditional languages involves two languages, say L > and L . Here, the language L is exactly like L > , where is a notational variant of >. The shared fragment of the two languages is the classical propositional language, denoted L. However, semantically will be interpreted differently than >. I will provide several examples of such conditionals.
As a first toy example, consider the language L > C , where > C is to be interpreted as the complement conditional. The frames and models for L > C are the same as for L > . The truth clauses are not. In particular, the truth clause for > C differs from the truth clause for >. Truth in a minimal model M for the language L > C is denoted C . Since the truth clauses for propositional variables and classical connectives is defined as usual (as in L > ), I will not repeat them, and I use this short-cut throughout the article:

Example 1
The truth clause for the complement conditional is From the semantic clause one sees that if > C were in the language L > , we could simply define ϕ > C ψ as ¬ϕ > ¬ψ. However, > C is not in the language L > . Thus to state a relation between L > and L > C , we need other resources. This is the role of the translation between conditional languages.
Let θ be a formula of L > and p, q propositional variables. We write θ = θ [ p, q] iff θ has its propositional variables among { p, q}. That is, at least one of p and q must occur in θ and no other variable occurs in θ . We write θ [ p, q] ∈ L > as an abbreviation for θ ∈ L > , p, q ∈ Var and θ = θ [ p, q]. Let ϕ, ψ be formulas of L > and θ [ p, q] ∈ L > , then we write θ [ϕ/ p, ψ/q] for substituting simultaneously ϕ for p and ψ for q in θ , defined as usual.
Definition 3 A translation is a total function •: L −→ L > such that A translation replaces formulas from L by formulas in L > -as if we were translating English to Genglish where, say, we only replace some of the English words by German words. It is only the conditional which makes translating necessary-we are only replacing one particular English sentence form (the conditional) by a German one. This creates recursive echoes. Indeed, the fourth clause is the central one. It says that θ provides the form for the translate of ϕ ψ. Since the translate of classical connectives is fixed, it suffices to state the translate of the conditional, by providing θ . We will say that the translation is induced from One can show, by induction on the complexity of the formula, that if α belongs to L, then α • = α. Classical formulas need no translation. and ⊥ being in L, we also get • = and ⊥ • = ⊥. Note that the identity function id : L > −→ L > fulfils the conditions for a translation. One easily shows, by induction, that it is induced from (ϕ > ψ) • This is not the intended application. However, it will be useful later.
More importantly, we can now rephrase > C in terms of >. For this, we simply read θ [ p, q] off the semantic clause for > C . That is, we consider the translation induced from Here is another example. Let stand for the sufficient reason relation. The corresponding language is denoted L and the truth relation in a minimal model by :

Example 2
The truth clause for the sufficient reason is Thus, the translation for the sufficient reason relation is induced from: The sufficient reason has been suggested in a ranking theoretic framework by Spohn (2012Spohn ( , 2015, and the above conjunctive analysis was coined by Raidl (2018, p. 230). More recently, Rott (2019) analysed the sufficient reason in a belief revision framework. He calls it "difference making conditional" and understands it as a contrastive connective, similar to 'because' or 'since'. In both frameworks, ϕ is a sufficient reason for ψ iff ϕ suffices to make ψ believed, whereas ¬ϕ does not. This analysis can be recovered, if we understand F(w, [ϕ] ) as the set of believed propositions given ϕ. Our analysis of the sufficient reason is more general. First, because Spohn and Rott restrict their analysis to the non-nested fragment of the language. 4 Second, because both impose semantic conditions which are much stronger than those considered here. However, their analysis can be recovered in our framework, as we will see in Sect. 9. The generalisation is also designed to consider sufficient reason relations in a variety of semantic frameworks, for example a probabilistic semantics, where the clause for evaluating ϕ ψ would be: ) < t for some fixed threshold t. This remark holds generally throughout the article, and the flexible semantics was chosen for this purpose.

Embeddings
A translation maps formulas of L to formulas of L > . This mapping should preserve meaning. The initial formula ϕ and the translate ϕ • should express the same proposition. To ensure preservation of meaning, we need a semantic relation between the interpreted models for the basic conditional and the interpreted models for the defined conditional. In  4 Spohn also leaves out the case when ϕ or ¬ϕ are impossible according to the outer modality.
g : N • ≈ M > is an isomorphism between N and M > modulo • iff g : N −→ M and g : W (N) −→ W (g(N)) are bijections and g : N Condition (1) says that an embedding is a function operating on two levels. First, it maps every model N from the model class N for L to a model g(N) from the model class M for L > . The latter model g(N) is a simulator for the former model N. Second, it also maps every world w of N (in N ) to a simulating world g(w) of the simulator g(N). Condition (2) requires that the simulator really simulates the original model modulo the translation. Namely truth in a world w of N is equivalent to truth in the simulating world g(w) of g(N), after having translated the formula under consideration from ϕ to ϕ • . A co-embedding works in the other direction.
Proof The identity function g = id satisfies Condition 1 of the embedding in Definition 4. And it is a bijection on both levels. Condition 2 of the embedding is proven by induction on the complexity of the formula, by proving the property w C α iff w > α • , given M ∈ M. The base case (α = p) follows by definition. In the inductive step, we assume the property for ϕ and ψ (IH-the induction hypothesis) and show it for α = ¬ϕ, ϕ * ψ where * ∈ {∧, ∨, →, > C }. ¬, ∧, ∨ and → are easily verified (using IH). Thus it suffices to prove it for > C : By IH, we have Lemma 1 says that the translation • = • C for the complement conditional is well behaved. The translate α • expresses the same proposition in the language L > as the original formula α in the language L > C . A similar fact holds for the sufficient reason: Lemma 2 For M the class of minimal models and • = • : Proof Again, it suffices to prove the case for ϕ ψ: To close this section, we mention a few simple results for embeddings, co-embeddings and isomorphisms. Obviously, embeddings, co-embeddings and isomorphisms are preserved under sub-classes: Proof Let •: L −→ L > be a translation. We prove (1): ). Thus Condition 1 of Definition 4 remains satisfied for N and g(N ). Condition 2 also remains satisfied for N ⊆ N and g(N ). The proof for (2) is similar and for (3) one notes that restricting a bijection (to a sub-domain and its image) remains a bijection.
The following is also clear (proof omitted): Lemma 4 Let •, N and M > as above.
As a toy example, take the subclass relation N ⊆ M and the identity translation • = id. Then (1) just says that valid formulas in M remain valid in any subclass. More generally, (1) says that when we have an embedding modulo a translation • (N • → M > ), we can read off valid formulas ϕ in N from valid translates ϕ • in M. In this sense, an embedding builds a semantic bridge from the land where L > is spoken (here M) to the land where L is spoken (here N ). Validities of translates ϕ • in M become validities of the original formula ϕ in N . The converse holds for a co-embedding. This is what (2) says. Thus a co-embedding builds the inverse semantic bridge to the embedding. Valid formulas ϕ in N become validities of translates ϕ • in M. Since an isomorphism incorporates both an embedding and a co-embedding, model classes which are isomorphic have the same validities, up to the translation. This is (3). As a consequence, the two model classes have the same logic, up to translation. The translation simply distorts the logic. Section 3 will use this insight to formulate a way to transfer soundness and completeness results from L > to L .

Backtranslation
For the method to work, it will be essential to have a backtranslation • of > into . As the translation, the backtranslation should be semantically well behaved. Additionally, the translation and backtranslation taken together need to be well behaved axiomatically: the backtranslation needs to reverse the translation. This condition will be spelled out here.
First note that the original basic conditional > can be obtained as complement conditional to the complement conditional: This backtranslation is semantically well behaved.

Lemma 5 For M the class of minimal models and • = • C :
Proof Consider g = id. As before, it suffices to verify >: Thus the complement translation can be taken as its own backtranslation.
As backtranslation for the sufficient reason, we take: For this backtranslation to be well behaved, we have to consider stronger than minimal models, namely cm-cn-wcc minimal models. Recall, these models validate the axioms Cm, CN and WCC.

Lemma 6
For M the class of minimal cm-cn-wcc models and Proof Minimal cm-cn-wcc models validate CE + Cm + CN + WCC (Theorem 3). This logic is equivalent to CE + RCM + WCC + ID (see the remark in Sect. 2, p. 6). It suffices to check the conditional. Assume the property holds for ϕ and ψ (IH). Then we get Thus, using freely RCM (in the last four equivalences): There is also a purely axiomatic sense in which the complement translation is its own backtranslation. It is its own provable inverse. This notion will be essential to transfer completeness.
Let me explain. For a translation • of L into L > and a backtranslation • in the other direction, and α a formula of L , call α •• the twin of α. If α is identical to its twin then they are equivalent in any logic. For our purpose it suffices that α and its twin are equivalent under additional conditions. For our examples, these conditions are rather weak. For the complement translation the conditions are already encoded in CE: Lemma 7 Let • = • C and • = • C the translation and backtranslation for the complement conditional > C . Then CE α •• ↔ α for any α ∈ L > C . 5 Rott (2019, Def2*) uses a similar construction, but not in the form of a translation.
Proof Denote ≡ provable equivalence in CE. It suffices to verify the conditional: Thus, for the complement conditional's translation and backtranslation to be well behaved axiomatically, it suffices to consider a classical conditional logic. For the sufficient reason relation, we need to strengthen CE by: The best way to understand S0 is to look at it in terms of the translation • in the minimal cm-cn-wcc models. ϕ (ϕ ∨ ψ) essentially expresses 6 ¬(¬ϕ > ψ), whereas ¬(¬ϕ (¬ϕ ∨ ψ)) essentially expresses ϕ > ψ. But ϕ ψ essentially expresses this conjunction. Thus there is no loss requiring that ϕ ψ expresses the conjunction of ϕ (ϕ ∨ ψ) and ¬(¬ϕ (¬ϕ ∨ ψ)). Thus S0 arises from the definition of and talks about the internal invisible structure of . For this reason, I call S0 the proper axiom for .
I provide equivalent expressions of S0 and another explanation in Sect. 7. For the moment, it suffices to say that the proper axiom S0 does what it is supposed to do-it is a minimal requirement to be added to a classical conditional logic in order for the translation and backtranslation of the sufficient reason to be well behaved axiomatically: Thus for the sufficient reason's translation and backtranslation to be well behaved axiomatically, we need to consider a classical conditional logic augmented by the proper axiom S0 for .

Transfer of Logic
This section develops the method to derive soundness, completeness and correspondence results for the defined conditional from known soundness, completeness and correspondence results for the defining basic conditional >. The result-transfer uses the way is defined from > and the way > can be backdefined from . Since > is not in the language L and not in the language L > , we can not speak of interdefinability properly. This is why I introduced the translation • and the backtranslation •. Definability is meant modulo translation, and backdefinability is meant modulo the backtranslation. In general, the method is an instance of knowledge transfer. This transfer is obtained by producing an image of what is known via the backtranslation. New insights are gained from this image. Since the following results are generic (they hold in CE), we can apply them in several semantic settings as well as to several definable conditionals (see the next Sections). 7 First we need a syntactic analogue to the semantic notion of an embedding.
Definition 6 Let •: L −→ L > be a translation, and Γ > , Γ axiomatic systems in L > and L respectively.
That the axiom system Γ > simulates the axiom system Γ modulo the translation •, simply means that whenever the second system derives a formula α the first system derives the translate α • . In particular, all axioms of Γ can be simulated by axioms of Γ > and all rules of Γ can be simulated by rules of Γ > . 8 In fact, this suffices to establish that Γ > simulates Γ . I will use this fact in what follows.
Whereas an embedding and a co-embedding build semantic bridges between the semantic region (M) of the land where L > is spoken and the semantic region (N ) of the land where L is spoken, a simulation Γ • ∝ Γ > builds a syntactic bridge between the axiomatic region (Γ ) of the land of L and the axiomatic region (Γ > ) of the land of L > . Derivable formulas α in Γ become derivable translates α • in Γ > . To obtain the inverse syntactic bridge, one considers the converse simulation, namely Γ > • ∝ Γ . Now derivable formulas α in Γ > become derivable backtranslates α • in Γ . We thus have two kinds of bridges, the semantic bridges (embeddings and co-embeddings), allowing to move back and forth between the semantics of the two lands where L and L > are spoken, and the syntactic bridges (the simulations), allowing to move back and forth between the axiomatics of these two lands. The four theorems of this section will use these bridges 7 I used similar ideas in Raidl (2019), without theorising them. 8 Where rule simulation is phrased in the obvious way. to transfer soundness, completeness and the correspondence theory for L > to L .
First, we can transfer a known soundness result for > to .
Theorem 4 (Soundness Transfer) Let N and M > be interpreted model classes and Γ , Γ > systems in L , L > respectively. Assume: Then Γ is sound for N (in L ).
Proof By (2), the assumptions (3) and (4) The proof is easily described as a journey: We start in the axiomatic region (Γ ) of the unknown land where L is spoken, with the assumption that Γ derives α. We take the syntactic bridge that Γ > simulates Γ modulo •. Thus Γ > derives the translate α • . Now we are in the axiomatic region (Γ > ) of the known land, where L > is spoken. We have switched lands but we are still within axiomatics. To move to semantics, we use the soundness assumption. Since Γ > is sound for M, and since Γ > derives the translate α • , we obtain that α • is valid in M. Now we take the semantic bridge back to the unknown land where L is spoken. Since N embeds into M > modulo •, the validity of the translate α • in M means that the original formula α is valid in N . We have landed where we wanted, in the semantic region (N ) of the unknown land of L . In brief: We start in L with a derivability assumption, translate it for L > by taking the syntactic bridge (axiomatic simulation), transform it into a validity (soundness assumption), and translate it back as a validity in L taking the semantic bridge (embedding).
In general, this theorem allows to transfer a known soundness result for a basic conditional > to a defined conditional . In most cases, (1) will be known or easy to figure out, and the form of the translation • in (2) can be read off from the way is defined semantically. Furthermore, (3) will easily be verified. To establish (4), it suffices to check, as mentioned above, that each of the rules and axioms of Γ can be simulated by the rules and axioms of Γ > . This is a purely mechanical task.
Second, we can also transfer a known completeness result: Theorem 5 (Completeness Transfer) With N , M > , Γ and Γ > as above. Assume: Then Γ is complete for N (in L ).
Proof Given (2), the assumptions (3), (4) and (5) The proof reverses the journey of the Soundness Transfer: Instead of starting in the axiomatic region, we start in the semantic region (N ) of the unknown land of L with the assumption that α is valid (in N ). We take the semantic bridge that M > co-embedds into N modulo •. Thus the translate α • is valid in M. We have switched lands, from L to L > , but we are still within semantics. To move to axiomatics, we use the completeness assumption. Indeed, since Γ > is complete for M, and since α • is valid in M, we obtain that Γ > derives the translate α • . Now we take the syntactic bridge back to the unknown land of L . Since Γ simulates Γ > modulo •, Γ derives the backtranslate of α • , that is, it derives α •• . It is here that we need the essential assumption (5) that α and its twin α •• are equivalent in Γ . Now Γ derives the original formula α. In brief: We start in L with a semantic validity assumption, translate it to L > by taking the semantic bridge (co-embedding), transform it into a derivability (completeness assumption), and translate it back as derivability in L by taking the syntactic bridge (simulation). But to obtain derivability of the original formula, we need the assumption that in Γ , a formula is equivalent to its twin.
We can say the same here for (1)-(4) as in the Soundness Transfer. (5) might involve a well chosen Γ , as we saw for the sufficient reason (Lemma 8). For the moment, it is useful to think of Γ as a backtranslate of Γ > .
The third result unites the previous two theorems: Corollary 1 (Adequacy Transfer) With N , M > , Γ , Γ > as previously. Assume

Γ > is sound and complete for M (in
Then Γ is sound and complete for N (in L ).
This uses both previous theorems and thus enacts both journeys-the Soundness and Completeness Transfer. Conditions (3), (4) and (5) essentially say that • and • are semantically and axiomatically well behaved.
The transfer of the correspondence theory is more complicated. Recall, we write M X iff all models in M validate all instances of X. We also write M X ≡ X iff for all M ∈ M and all w ∈ W (M), we have w M X iff w M X . This lifts equivalence of formulas in a model class to axiom schemes. The next central result allows to transform an axiom scheme X holding for the basic conditional > to an axiom scheme X holding for the defined conditional . For this, we first backtranslate the original scheme X into X • = {ϕ • : ϕ ∈ X} and then we transform this into an equivalent "nicer" axiom scheme X . The latter step is useful, since the backtranslate X • often looks ugly.
Theorem 6 (Axiom Transfer) With N and M > as previously, and X , X axiom schemes in L , L > respectively. Assume: The proof of this result makes no assumptions on •. In particular, it does not assume that • and • are provable inverses to each other. When N = M, the last result allows transferring the correspondence theory from > to . This is our last central result: Theorem 7 (Correspondence Transfer) Let F be a minimal frame, M the models over it and M ⊆ M, for M a class of minimal models. Make the assumptions of Theorem 6, with N = M and h = id, i.e.: Proof By assumption we can use Theorem 6 for N = M and M > : In other words, we have transferred the correspondence theory for frames F interpreted in L > to a correspondence theory for the same frames but interpreted in L . It is only in this theorem that we assume N = M , so that N = M; that is, the models are the same, only the truth relations differ. The results of this section are quite general. 9 They allow to generate a sound and complete logic Γ for a defined conditional in a model class N , based on the sound and complete logic Γ > for the defining conditional > in the model class M. Although the first three results appear as trivial as 2+2=4, they provide a powerful method for generating new knowledge on new semantics using old knowledge on known semantics. The more complicated Axiom Transfer says that one can transfer an axiom holding for > to an axiom holding for , using a semantically well-behaved backtranslation • of > into . The Correspondence Transfer says that when the models are the same (only the truth clauses differ), the Axiom Transfer allows obtaining the correspondence theory for from the correspondence theory for >.
The careful reader might have noticed that the Completeness Transfer assumes Γ and • to be given. But how do we find them? I have no answers to these questions, apart from a heuristics that I used in all applications.
To figure out Γ , use the following fixed-point heuristics: 10 Suppose you have found a semantically well behaved backtranslation •.
Step 1: Figure out simple -axioms which are sufficient to prove (5) in the Completeness Transfer (Theorem 5). For this it suffices to take (ϕ ψ) •• ↔(ϕ ψ) as axiom, or an equivalent, obtained by resolving the translations. I called this the proper axiom of . The proper axiom for the complement conditional is (ϕ > C ψ) ↔(¬¬ϕ > C ¬¬ψ). This follows already in CE. The proper axiom for the sufficient reason, given CE, is S0 (Sect. 5). This first step already fixes a minimal part of Γ , which here is typically CE augmented eventually by a proper axiom for , say Γ 0 ⊇ CE.
Step 2: Check which >-axioms are needed to simulate Γ 0 , say Γ 0 > ⊇ CE. This is typically CE augmented by axioms simulating the proper axioms of .
Step 3: Check 9 They make no assumptions on the logic or semantics (except for Theorem 7, assuming Chellas' semantics) and could be generalised to more complex (multi-modal) translations. 10 I am confident that this could be turned into an algorithm for generating Γ . which (additional) -axioms simulate Γ 0 > . Say Γ 1 ⊇ Γ 0 . If these are identical, you are done. Else continue with Step 2, applied to Γ 1 . That is, check, which (additional) >-axioms simulate Γ 1 . Say Γ 1 > ⊇ Γ 0 > . If the latter are identical, you are done. Else continue with Step 3, applied to Γ 1 > . And so forth. In general, one tries to simulate newly appearing axioms, zigzagging between and >, until a fixed point is reached. For the complement conditional, we reach a fixed point in Step 3. The proper axiom (appearing in Step 1) is deducible from CE. Thus no proper axiom needs to be added in Step 1. But CE for > C can be simulated from CE for > (Step 2) and conversely (Step 3). Thus we have reached a fixed point. For the sufficient reason, we will later see, that the proper axiom S0 makes us reach a fixed-point one step later.
The fixed-point heuristics starts with a backtranslation •. But how do we find it? More importantly, is there always such a backtranslation? For the first question, I can only provide another heuristics: Consider the definition of in terms of >, say ϕ ψ is of the form (ϕ > ψ) ∧ α (modulo translation). To define > in terms of (modulo the backtranslation), proceed as follows: Obviously ϕ > ψ could have the form (ϕ ψ) ∨ β, where β should express (ϕ > ψ) ∧ ¬α but in the language . 11 However, sometimes finding such a β directly is not possible. You can then try to strengthen or weaken the component (ϕ ψ), for example to ϕ (ϕ ∧ ψ) or to (ϕ ∨ ¬ψ) ψ or something else (provided these imply ϕ > ψ), and repeat the procedure, i.e., find the β which expresses the complementary case when ϕ > ψ holds. More generally, given the axioms for >, figure out disjoint or covering cases for ϕ > ψ. Express them in terms of . This remains a rather vague heuristics, and I cannot provide more than this in the present article. In brief: be creative.
As for the existence of the backtranslation, I have no answer to this question. If I did, I could eventually either prove that Spohn's (2012) supererogatory reason relation is not axiomatisable or give you its complete axiomatics, to mention just one example of the consequences of the existence or non-existence of such a backtranslation. Luckily, the backtranslation exists for the definable conditionals investigated in this article (and many others-Sect. 10), and it is easy to figure out, as we have seen for the sufficient reason. Thus, the following applications are straightforward implementations of the previous theorems. The essential work will consist in simulating axioms.

Sufficient Reason
We now apply the technique to the sufficient reason relation. We term sufficient reason, any conditional ϕ ψ defined as where > has at least the conditional logic CE + Cm + CN + WCC. Recall, the axiom ID follows. The following is our proposal for a sufficient reason logic: Definition 7 L is a sufficient reason logic iff it is a classical conditional logic and contains the axioms We set S = CE + S0 + S1 + S2.
The smallest sufficient reason logic S is a classical conditional logic, and does not have right weakening (Cm) as an axiom. By classicality, sufficient reasons are insensitive to equivalent rephrasings of the antecedent or the consequent. By S1 there is no sufficient reason for the tautology. S2 says that if ϕ is a sufficient reason for the disjunction ϕ ∨ψ ∨χ then ϕ remains a sufficient reason for the stronger disjunction ϕ ∨ ψ. The most difficult law to understand is S0, explained as proper axiom in Sect. 5. In words: if ϕ is a sufficient reason for ψ then it is a sufficient reason for the weakening ϕ ∨ ψ, and ¬ϕ is not a sufficient reason for the weakening ¬ϕ ∨ ψ, and vice versa.
We may also replace the complicated S0 by equivalent axioms. This provides further insights:
Sa and Sb imply S0 → : chain them.
S implies Sc: Assume ϕ (ϕ∧ψ∧χ). Then (i) ¬(¬ϕ (¬ϕ∨ (ϕ ∧ ψ ∧ χ ))) and (ii) ϕ ϕ (Sb). But (i) implies The axioms Sa and Sb slightly simplify S0. This is certainly the case for Sa, since it says that if ϕ is a sufficient reason for ψ then it is also a sufficient reason for the strengthening ϕ∧ψ as well as for the weakening ϕ ∨ ψ, and conversely. Yet, Sb essentially reintroduces the complication, since (modulo CE) it is an instance of S0, where ψ is replaced by ϕ∧ψ. However, the equivalence allows to compare our S to Rott's stronger difference making logic. Among the axioms assumed by Rott (2019), we find Sa and Sb-his ( 2a) and ( 2b). Furthermore, Rott derives S1 and S2-his (d14) and (d25). Finally, Sc is his derived (d24). I will come back to this relation at the end of the Section. Using Lemmas 2 and 8 and our transfer method, we can now prove:

Theorem 8 S is sound and complete for in minimal cmcn-wcc models.
Proof Let M be the class of minimal cm-cn-wcc models (equivalently cm-id-wcc), and denote Γ > = CE + Cm + CN +WCC. We prove (1) Simulation of the remaining axioms, uses the fact that Γ > = CE + Cm + CN + WCC = CE + RCM + ID + WCC. Denote S0 →1 and S0 →2 the implication of the first and second conjunct.
Completeness and soundness follow from Corollary 1.
Let me exemplify the fixed-point heuristic (end of Sect. 5) with the sufficient reason. The proper axiom for is S0 (Step 1). To simulate S0, we need CE and the > axioms Cm, WCC and CN (Step 2). To simulate CN, we need S1, and Cm requires S2, whereas simulating WCC requires no further axiom (Step 3). Yet, the new axioms, S1 and S2, don't require any new > axioms (Step 4). Thus we have reached a fixed point at Step 4. This yields the axiomatics S. The above Completeness Transfer just reverse engineers the fixed-point heuristic.
As one can strengthen the logic CE + Cm + WCC + CN, one can strengthen the sufficient reason analogue S. We apply Theorem 6, to derive which axiom in L corresponds to which axiom in L > . Furthermore, since the models are the same and only the truth clauses change, we can use Theorem 7 to obtain the correspondence theory for L . Given the original scheme X for > we backtranslate it into X • . Then we apply logical transformations in S to obtain our sufficient reason analogue, denoted X S .
We conclude by Theorem 7, since h = id and with M the models over F.
To explain some of the axioms, some remarks on the outer and inner modality are necessary. In L the new outer possibility is expressed by ♦ S α ≡ ¬α ¬α. This can be seen as follows. In L > the outer possibility of α • is ¬(α • > ⊥). Consider its backtranslation (¬(α • > ⊥)) • . Resolving the backtranslation, we obtain ¬¬(¬α •• (¬α •• ∨ ⊥)). In CE this is equivalent to ¬α •• ¬α •• . But by Lemma 8, α is equivalent to its twin α •• in S. This is how we obtain the new expression for the outer possibility. Dually, the new outer necessity is expressed by S α ≡ ¬(α α). Since α α expresses contingency, α is a sufficient reason for itself exactly when it is contingent. The inner modality S α is now expressed by ¬(⊥ α), by a similar reasoning. Thus ♦ · S α ≡ (⊥ ¬α) expresses compatibility with actual belief.
Back to the axioms in Theorem 9. The first three axioms are conditions on the outer modality. They say the same as the originals: the outer necessity is consistent, monotone and closed under conjunction. If we rephrased them in terms of the new outer-modality, they would be indistinguishable from the originals (with the original outer-modality). We can also rephrase INC S and PRES S with the inner modality (and taking a contraposed version). This would make them more similar to the originals. The remaining four axioms are more difficult to comment on, except for saying that they are the distorted picture one obtains if one looks at the original axioms through the lens of the backtranslation. Propositionally they express the same thing. But literally they do express new properties for the sufficient reason.
We can however say something about the distortion CC S of closure under conjunction CC. In presence of the former, the later becomes derivable: Proof A proof is now in Rott (2019), after I told him about it.
Let me close with a remark on the connection to Rott's axiomatics for his difference making conditional. Given P S and C S , the outer impossibility can be re-expressed alternatively as ϕ ⊥. 12 Then P S is Rott's axiom ( 0), CC S his ( 1), INC S his ( 3) and instead of PRES S he considers the stronger ( 4) with ⊥ (ϕ ∨ ψ) in the conclusion. Both are equivalent if S is monotone and closed under conjunction, which holds by CC S . M S and C S correspond to his ( 5) expressed with the alternative impossibility. His ( 6) is essentially RCEA + RCEC. We saw that Sa and Sb are his ( 2a) and ( 2b) and that he derives S1 and S2. We can conclude that his axiomatics is essentially S augmented by the first 6 axioms in the above table. Thus he considers a stronger sufficient reason.
the basic conditional > (in M). Suppose further that we have another conditional 2 (in N 2 ) definable from >. Finally, suppose that we can also define 2 from 1 . Then, to figure out the logic L 2 of 2 , we need not go through the logic L for >. We can directly use the proven results for 1 . I call this technique chaining. Let me explain by analogy: Say our basic language L > is English. Suppose we know how to translate English into Italian L 1 . We want to translate English into French L 2 . What should we do? Start from scratch, or use our Italian translation? If our translations have no loss of information it seems best to use the Italian translation (from L > to L 1 ) and translate further into French (from L 1 to L 2 ). This is easier than to translate L > into L 2 from scratch. The hard work is to translate English into Italian or into French, since English is from another language family. Translating between Italian and French is much easier, because they are both Roman languages. Thus, once we have translated from English to Italian, it is simpler to translate Italian into French than translating English into French. Stretching the analogy: if we transferred results from English to Italian, we can easily further transfer them to French. Chaining is guided by this idea.
As an example, consider the necessary reason relation ≥, in the language L ≥ and with truth in a minimal model M for L ≥ denoted by ≥ .
Example 3 (necessary reason) The truth clause for the necessary reason is The necessary reason was suggested in a ranking theoretic framework by Spohn (2012Spohn ( , 2015. The necessary reason can be seen as the complement conditional to the sufficient reason (Raidl 2018, 230). The translation is induced from: This translation is semantically well behaved: Proof It suffices to check the conditional (details omitted): This is well behaved from Lemma 5: Since the necessary and sufficient reasons are complements to each other, choosing the conjectured axiomatics for the necessary reason is easy: We take the complement translations of the axioms in S, simplified by substitution of provable equivalents.
Definition 8 L is a necessary reason logic iff it is a classical conditional logic and contains: In CE, the complement translations of the axioms from S are equivalent to the above axioms. That is, although we do not have S0 C = N0, this equality holds in CE. The same holds for the other axioms. Thus S C = N and N C = S. As in the case of S0, we can replace N0 by: Using the complement translation, we can transfer completeness results from S to N: Theorem 10 N is sound and complete for ≥ in cm-cn-wcc minimal models.
Proof Let M be the class of cm-cn-wcc minimal models and S = CE + S0 + S1 + S2. We establish (1) Our natural language analogy was not misleading: translating Italian ( ) into French (≥) really was easier than translating English (>) into French which would have been of the same difficulty as translating English into Italian (Theorem 8).
We obtain the correspondence theory, by chaining translations and results. We know that a basic axiom X corresponds Table 2 Correspondences (for Theorem 11) between necessary reason axioms X N (left), basic axioms X (middle), and frame properties x (right) CMon cmon CV cv to a sufficient reason axiom X S which has a complement (X S ) C which we further simplify in CE, to obtain our necessary reason analogue X N .
Theorem 11 Let F be a minimal cm-cn-wcc frame. Then F X N iff F > X iff F has the corresponding property X F = (x), for the axiom schemes in Table 2.
Proof Let M be the cm-cn-wcc minimal models. We first apply Theorem 6.
(3) M ≥ X N ≡ (X S ) C , since this is how the schemes X N were obtained from X S . The rest results from the correspondence between X S , X and (x) (Theorem 9).
The same remark applies here as after the previous theorem. It was easier to use the previous results for the sufficient reason and transfer them to the necessary reason by taking the complement translation than doing it from scratch.

Equivalent Semantics
We now apply the method to a well known fact, that different semantics may share the same logic. Here we consider so-called set selection models, and other semantics, based on ranking functions and belief revisions. These are used to exemplify the Completeness Transfer to alternative semantics. Known or obvious results are omitted.
We here want to model the logic CK = CE+Cm+CC+CN. By Theorem 3, we could use cm-cc-cn minimal frames. But a simpler semantics exists: f is a standard frame and V : Var −→ ℘ (W ). The truth clause for > in a standard model M is: In a standard frame, the selection function selects a set (proposition), whereas in a minimal frame, the selection function selects a neighbourhood (set of propositions). Accordingly, we call f a set selection. Set selections and standard frames were used by Stalnaker (1968) and Lewis (1971). If we assume f (w, A) ⊆ A in a standard frame, we may say that f (w, A) selects closest A-worlds.
The following result was proven by Chellas (1975): Theorem 12 CK is sound and complete for standard frames (models).
A Lewisean model is a standard model with the first four properties above. By our remarks, the logic V = CK + ID + CA+CMon+CV is sound and complete for Lewisean models. This is Lewis' weakest logic. Because the identity function id : L > −→ L > is a translation, the method can be applied to cases where we stay in the same language, but change the models. Conditions (2), (4) and (5)  A ranking function represents an agent's degrees of disbelief. By definition, the degree of disbelief is a natural number (including zero) or infinity, the tautology is not disbelieved (has rank zero), the contradiction is maximally disbelieved (has rank infinity) and the rank of a disjunction (alias union) is the minimal rank of its disjuncts. Call a proposition crazy if it has rank infinity-it is treated as if it were a contradiction. The conditionalisation of κ by A is defined by Definition 11 Let W be a non-empty set. W , (κ w ) w∈W is a ranking frame iff each κ w is a ranking function. R = W , (κ w ) w∈W , V is a ranking model iff W , (κ w ) w∈W is a ranking frame and V : Var −→ ℘ (W ). Truth in a ranking model R for L > is denoted > R , with classical connectives as usual and: If we rephrase the defining clause in terms of belief, the clause says that the conditional holds in world w according to the agent whose doxastic state is represented by the ranking function κ w if and only if the agent believes the consequent given the antecedent or considers the antecedent to be crazy. Friedman and Halpern (2001) proved: Theorem 14 VP = V + P is sound and complete for ranking models.
Proof By our method. VP is sound and complete for (p)-Lewisean models M (Correspondence Theory for standard frames). Denote R the set of ranking models. Show: For the converse, see the proof of Theorem 3.6 in Raidl (2019). Thus R ≈ M (Theorem 13). Therefore VP is sound and complete for R (Corollary 1).
Here we used class point equivalence as a special case of an embedding to show that (p)-Lewisean models and ranking models are equivalent semantics and have the same logic VP.
Our second example is based on belief revision: Definition 12 Let W be a non-empty set. F = W , * is a belief revision frame iff * is a collection of functions with classical connectives as usual and: A belief revision frame is a notational variant of a minimal frame by the equation F(w, A) = * w A. Notational variants are point equivalent! However, the interpretation is another one. * w is a oneshot belief revision function on propositions. It associates to every proposition A a set of propositions * w A, namely those the agent believes after revising by A (according to * w ). Actual belief, say B w , is implicit. We can recover it, assuming that revising by the tautology is inert: B w = * w W . Call this belief inertia. The inner modality α := > α then expresses actual belief. The clause for the conditional says that ϕ > ψ is accepted by the agent (in state * w ) iff she believes the proposition [ψ] after revising by the proposition [ϕ]. 14 How do the AGM "belief revision postulates" relate to axiom schemes for conditional logic, when we interpret the former as frame properties? Let * : ℘ (W ) −→ ℘ (℘ (W )) be a one-shot belief revision (dropping the world index, for simplicity). Actual belief is B = * W .  To simplify notation, let us define κ(C|A) * = κ(C|A) if κ(A) < ∞ and else = ∞. Then the truth clauses for the conditional in ranking semantics (6 R ) is equivalent to: Example 4 (sufficient reason) Let R be a ranking model and B be a belief revision model. Truth for the language L in these models is denoted R and B . The truth clause for the sufficient reason is respectively Using Theorems 9 and 8 , and our results for ranking and belief-revision semantics, we obtain completeness results for in these alternative semantics: Corollary 2 defined as sufficient reason in the semantics on the left (or >-logic in the middle) has the sound and complete -logic on the right, as shown in Table 3.
Proof By chaining our results.
The result also shows that the logic for the sufficient reason is the same in ranking semantics as in full belief revision models. Furthermore, contrary to a remark made by Spohn (2012, p. 118), the sufficient (and necessary) reason can be developed in belief revision semantics, as in other frameworks.
We can do the same for the necessary reason: Example 5 (necessary reason) Let R and B as in Example 4. Truth for L ≥ in these models is denoted ≥ R and ≥ B and: Again, by chaining, we get: Corollary 3 ≥ defined as necessary reason in the semantics on the left (or >-logic in the middle) has the sound and complete ≥-logic on the right, as shown in Table 4.
This section showed that the sufficient and necessary reason relation can be developed in alternative semantics, illustrated by set-selection semantics, ranking semantics and belief revision semantics. Our method allowed to transfer completeness and soundness results to these settings. Other alternative semantics, for which there is a completeness result or for which such a result can easily be figured out, can be treated similarly. For example, V is also sound and complete for Leitgeb's (2012) Popper measure semantics and thus the sufficient and necessary reason can be developed there-their logic is the same as for Lewisean models.
the probabilistic-causal conditional of van Rooij and Schulz (2019). The neutral, doxastic, metaphysic and counterpossible conditionals were devised to treat the problem of vacuously true conditionals with impossible antecedent. The sufficient reason, the dependency, the evidential, the simple and the probabilistic-causal conditional were devised to capture relevance. Comparing the logics of these conditionals, also allows to test these new conditionals against possible counterexamples within the context for which they were developed.
Our investigation shows that the study of conditionals is not as messy as it might appear at first glance. Many conditional constructions can be obtained as definable conditionals from a basic conditional. As a consequence, many conditional constructions which attempt to solve one or the other problem of the Lewis-Stalnaker account by strengthening or weakening the basic conditional, can be analysed in a uniform fashion. The basic conditional constitutes a shared nucleus. And I would like to close with a conjecture: The strengthenings or weakenings of that nucleus are contextually variable and express implicit presuppositions. The nucleus however, represents a psychological, linguistic or conceptual constant.