Proof Systems for Super-Strict Implication

. This paper studies proof systems for the logics of super-strict implication ST2 – ST5 , which correspond to C.I. Lewis’ systems S2 – S5 freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating STn in Sn and backsimulating Sn in STn , respectively (for n = 2 , . . . , 5). Next, G3 -style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of G3 -style calculi, that they are sound and complete, and it is shown that the proof search for G3 . ST2 is terminating and therefore the logic is decidable.


Introduction
The goal of this paper is to develop proof systems for normal and non-normal logics of super-strict implication (SSI) .SSI is a refinement of C.I. Lewis' strict implication ( ) that has been studied in [4,5].The idea behind SSI is to free strict implication from its paradoxes.For this, A B is defined as true whenever A B is true and, moreover, A is possible. 1 Thus the formulas of Antilogical Antecedent (AA) and Tautological Consequent (TC) are both invalid for .These are also known as the first and second paradox of strict implication.By avoiding these paradoxes, provides a better explication of entailment-related uses of implication than .In this respect, we agree with connexivists [15] and relevantists [1,29], but we disagree with Lewis [11,p. 250], who proposed a formal argument-his so-called independent proof -for the validity of ⊥ B. His argument is based on accepting the following inferential principles: ∧-elimination (∧E), ∨-introduction (∨I), and disjunctive syllogism (DS).These principles are all the ingredients needed for ex falso quodlibet (i.e., the derivation of an arbitrary formula B from a contradiction A ∧ ¬A): Moreover, if we take to be an object language representation of the underlying derivability relation-i.e., if we assume the deduction theorem for -then we must accept the following version of the first paradox of strict implication: (A ∧ ¬A) B. Neither connexivists nor relevantists are convinced by this argument: they reject ex falso quodlibet and at least one of the principles used in Lewis' proof.In particular, connexivists usually reject ∧E or correspondingly the object-language law of Simplification '(A ∧ B) implies A' (SI) 2 ; relevantists usually reject DS.
The rejection of ex falso quodlibet, be it based on the rejection of ∧E or of DS, involves a major departure from classical logic since it means that at least one of ∧, ∨, and ¬ does not satisfy its Boolean semantics.
We believe this is too high a price to pay.We rather prefer to supplement classical propositional logic with an implication connective aimed at expressing entailment-related uses of implication, in the same spirit as Lewis' .It turns out that for the paradoxes of become invalid, since the antecedent ⊥ of the first paradox (AA) is impossible and the antecedent A of the second paradox (TC) is not always possible.What becomes valid, instead, is the negation of the first paradox, ¬(⊥ B) (No Antilogical Antecedent, NAA), and its symmetric counterpart, ¬(B ⊥) (No Antilogical Consequent, NAC).Furthermore, allows us to express as well as the unary modal operators and as follows: A = ( A), A = ¬ ¬A, A B = (A ⊃ B), where ⊃ stands for the material implication.We show that logics of SSI preserve all validities of a semantics for in the following weak sense: if A B is valid (in a semantics for ) then, A ⊃ (A B) is valid (in the corresponding semantics for ), see Corollary 7. As a consequence, logics of can be seen as logics of freed from paradoxical instances.
Moreover, as shown in [4,5], both normal and non-normal logics for are Boethian logics: 3is a non-symmetric implication validating both Aristotle's Thesis (AT) and weak Boethius' Thesis (wBT): But the logics of SSI are not fully connexive, since strong Boethius' Thesis BT (obtained by replacing ⊃ with in wBT), does not hold.In this respect, is similar to Pizzi's consequential implication [20].As the latter, is motivated by the rejection of AA, TC, and the acceptance of the Boethian connexive principles AT and wBT.And as for consequential implication, the stronger BT fails for .However, differs from consequential implication in crucial aspects.First, by the way in which the strict implication is strengthened.Consequential implication, A → B, strengthens the strict implication by imposing that both B ⊃ A and B ⊃ A are true, whereas superstrict implication only imposes that A is true.Second, by the reason for which adding BT is problematic (see Propositions 3.18 and 3.19 from [20] and our Section 5.3).
In previous work, logics for were studied mostly from a semantical perspective: [4] introduced relational semantics for normal logics of and [5] for some non-normal ones.The only proof systems introduced were labelled sequent calculi for normal logics of [4], but no proof system has been introduced yet for non-normal ones and no fully object-language axiomatisation of these logics, be it normal or non-normal has been provided.This paper fills these gaps by introducing a sound and complete Hilbertstyle axiomatisation of the logics of for semantics corresponding to Lewis' systems S2-S5, as well as labelled sequent calculus for each of these logics. 4he underlying idea is this: We already know that (non-normal) relational models provide a semantics for Lewis's non-normal logic S2 for , the logic of all formulas in the language L that are valid in reflexive non-normal relational models.Furthermore, via the standard modal translation of in terms of (defining A B as (A ⊃ B)), Lemmon [9] provided a modal axiomatisation of S2.In this paper we introduce an axiomatic system for the companion super-strict implication logic ST2 of all formulas in the language L that are valid in reflexive non-normal relational models (now used to interpret ).For the completeness results, we rely on the fact that , , and are expressible in terms of and vice-versa: this allows us to use the method of translation and backtranslation developed and used by one of the authors to axiomatise many different conditional logics [23][24][25].That is, we prove the Soundness Theorem of S2 with respect to the class of reflexive non-normal relational models by simulating the system ST2 in the system S2.Vice versa, Completeness is obtained by backsimulating S2 in ST2.
Based on this ground result, we further consider axiomatic extensions of ST2.More precisely, we provide axiomatic systems for the logics ST3, ST4 and ST5, the logics of valid L -formulas over the reflexive non-normal relational models which also are, respectively, transitive, transitive and normal, normal and with an equivalence relation as accessibility relation.The corresponding Soundness and Completeness results are proved similarly via simulation and backsimulation methods using respectively the known results for the corresponding systems of strict implication S3, S4 and S5.
Finally, we introduce the labelled calculi G3.STn for the logics STn (2 ≤ n ≤ 5).We prove that such calculi have good structural properties: heightpreserving admissibility of the rules of weakening and contraction, syntactic admissiblity of cut, and the height-preserving invertibility of all rules.We also provide corresponding Soundness and Completeness results.We prove Completeness by simulating the axiomatic system for STn within the calculus G3.STn.Finally, we prove the decidability of G3.ST2 by showing the termination of proof-search and, as a consequence, we prove that models for ST2 have the finite model property.
The rest of the paper is organised as follows: Section 2 presents the syntax and semantics for logics of and of and shows that they are intertranslatable.In Section 3 we discuss valid and invalid principles for , and introduce an axiomatisation of ST2, which we then prove sound and complete in Section 4, as explained above.Section 5 does the same for the logics ST3-ST5, and discusses the problem of adding BT to normal systems.Section 6 presents labelled sequent calculi G3.ST2-G3.ST5.Section 7 shows that these calculi have good structural properties and Section 8 proves that they Relations between the systems of strict implication (front plane) super-strict implication (two back planes) and the properties of the accessibility relation (r = reflexive, t = transitive, s = symmetric) in normal (top plane) and non-normal (bottom plane) relational semantics are sound and complete and that G3.ST2 is decidable.Finally, Appendix A shows that the deduction theorem for material implication holds in ST2 and S2.We shall use the deduction theorem in our proofs throughout the paper.The paper thus investigates two kinds of proof systems for super-strict implication: the Hilbert-style systems ST2-ST5 (Sections 2-5) and the labelled sequent calculi G3.ST2-G3.ST5 (Sections 6-8).These are independent of each other, and a reader interested in only one of them can read the relevant part without reference to the other part.Nonetheless, we will show some interesting interrelations between the two kinds of proof systems: the Hilbert-style systems will be used in one proof of the completeness of the labelled calculi (Theorem 39) and the latter will be used in one proof of the soundness of the former (Corrolary 40) (Figure 1).

Syntax and Semantics
This Section introduces the syntax and semantics for super-strict implication and states the inter-translatability with strict implication.

Syntax
The language L of modal logics and the language L for logics of superstrict implication are generated by the following grammars (where p is in some non-empty set of sentential variables P, which for simplicity we assume at most countable): Parentheses follow the usual conventions and the modal operators and bind lighter than all other operators.Capital roman letters will be used as meta-variables for formulas (of the appropriate language).The weight of a formula, w(A), is given by the number of binary operators occurring therein.
For both languages, we denote by an arbitrary propositional tautology (for example p ∨ ¬p); ⊥ = ¬ ; and (A ≡ B) = (A ⊃ B) ∧ (B ⊃ A).In the language L , we define A = ¬ ¬A.In the language L , we use the following abbreviation for the inner modality A = ( A), the dual being 5 The formulas that L and L have in common constitute the classical propositional language L (over P).A formula from L will be called classical.

Semantics
We here present Kripke's semantics for non-normal logics.This semantics is based on the fact that non-normal systems, such as S2 and S3, can be consistently extended with the axiom A: we can have models with accessible points where every formula, ⊥ included, is possible.Kripke used this fact to extend the well-known relational semantics for normal modalities with so-called non-normal points where every formula is possible and no formula is necessary, see [2,8,31].We use ℘(W ) to designate the powerset of W and the long arrow notation f : X −→ Y to designate that f is a total function from X to Y .W are the worlds, to be denoted by w, v, u, . ... R is the accessibility relation, and we write wRv for v being accessible from w. Reflexivity means that all worlds are accessible to themselves (wRw).N are the normal worlds with respect to which truth in a model and validity will be defined.V is a usual valuation function, assigning to each propositional variable p a set of worlds V (p)-the worlds where p is true (also called p-worlds, for short).Our relational models are sometimes also called reflexive non-normal relational models.
The models for L and L are in fact the same.However, we need to distinguish the model relation for L , designated by , from the model relation for L , designated by (when necessary).
Definition 2. Truth in a world in a model for L is defined standardly for sentential variables and for the classical connectives ¬, ∧, ∨, and ⊃; for we have: Truth in a world in a model for L is defined as in a model for L for classical connectives and it is defined as follows for : (ii) ∃v ∈ W (wRv and v A) and (iii) ∀v ∈ W (wRv implies v A ⊃ B ) A formula is true in a model if it is true in all normal points of that model; it is valid in a class of models whenever it is true in all models in that class.A logic (over L or L ) is the set of all formulas that are valid in some class of models.In particular, we consider the following logics over L (L ): • S2 (ST2) is the set of formulas that are valid in the class of all models; • S3 (ST3) is the set of formulas that are valid in the class of all transitive models, where a model is • KT (STT) is the set of formulas that are valid in the class of all normal models, where a model is normal if N = W ; • S4 (ST4) is the set of formulas that are valid in the class of all normal transitive models; • S5 (ST5) is the set of formulas that are valid in the class of all normal, transitive and symmetric models, where a model is symmetric if ∀w, v ∈ W (wRv implies vRw).

Intertranslatability
We are now going to show that logics over L are intertranslatable with logics over L by means of two truth-preserving translations • and •.
Definition 4. The translation • : L −→ L is defined by We call A • the backtranslate of A. The backtranslation uses the fact that the models are reflexive and hence serial.A (our A) expresses not only A but also . 6As before, when A is classical, we have A • = A. In particular • = and ⊥ • = ⊥.
The following shows that the translation is semantically well behaved.That is, C and C • express the same proposition: Proof.By induction on the complexity of the formula.It suffices to verify C = (A B), assuming the property holds for A, B (IH).
A similar fact can be proven for •, but we won't need it.
From the previous Lemma, we obtain our weak validity preservation result:

Axiomatic Systems for S2 and ST2
We are now going to introduce axiomatic systems for the logics S2 and ST2.As usual, an axiomatic derivation from the set of assumptions Γ is a sequence of formulas such that each formula is an axiom or a member of Γ or it follows from formulas preceding it by a rule of the system.We use Γ S A to say that the formula A is derivable from the set of assumptions Γ in the system S; we omit the subscript whenever the system is clear from the context.A theorem is a formula derivable from the empty set of assumptions; we write S A when A is a theorem of the system S.

Lemmon's Axiomatisation
The following Hilbert-style axiomatisation of S2 was given by Lemmon [9]: K is also known as the Distribution axiom, T as the Truth axiom, MP is Modus Ponens for ⊃, rN stands for restricted Necessitation, BR is Becker's rule.S2 was shown to be sound and complete for relational models in L [8].

Super-Strict Axiomatisation
By T* and K* we designate the axioms T and K written for .We can similarly ask for the logic of super-strict implication in relational models (the super-strict companion ST2 to S2).We use a Hilbert-style axiomatisation: AT is Aristotle's Thesis.PA expresses that the conditional implies the possibility of the antecedent.INC has the object language form of a postulate known as Inclusion in belief revision theory and expresses that the conditional implies the strict conditional.SPRES is a form of Preservation called Strong Preservation (in belief revision theory).Together with PA and INC, this expresses the definition of in terms of .AND is a standard axiom of conditional logic which expresses that the consequent of a conditional is closed under conjunction.TID is the identity for , but more importantly it expresses that is possible (in the sense of ).T* expresses that the inner modality implies truth.The rules rLLE and rRW are restricted versions of Left Logical Equivalence and Right Weakening, which are normal rules of conditional logic.BR* can be seen as the translate of the rule BR. 7 The rule rN* is again restricted, to the instances of T*, K*, PA, INC or SPRES.The restriction does not mean that the formulas to which rN* can be applied need to be axioms, they only need to be of a certain form. 8The notion of a derivation is the same here as before.

Principles of Super-Strict Implication
Lemma 10.The following principles are invalid: Proof.To disprove the first eight principles, we use a frame (i.e., a model without valuation) with a normal world w ∈ N such that R(w) = {w}, changing only the valuation for each principle.ID.Let w ¬p.Then p p is false in w, since there is no accessible p-world.C. Assume w p ∧ q.Then p q is true in w, but ¬q ¬p is false in w, since there is no accessible ¬q-world.

SA. Assume w
p ∧ ¬q ∧ r.Then p r is true in w, but p ∧ q r is false in w, since there is no accessible world where p ∧ q is true.TC. ⊥ is false in w since there is no accessible ⊥-world.AA.See TC.SI.Let w p ∧ ¬q.Then p ∧ q p is false in w, since there is no accessible p ∧ q-world.CEM.Let w ¬p ∧ q.Then p q is false since there is no accessible p-world, and p ¬q is false for the same reason.BT.As in ID-p p is false in w, hence there is no accessible world where p p is true.Hence BT is false in w. S. We extend the previous frame to R(w) = {w, v}.Let the valuation be such that w p ∧ q and v ¬p ∧ q.Then we have p q in w.But we do not have q p in w, since there is an accessible q-world which is not a p-world.
ID is a standard principle of conditional logic-the axiom of Identity.C is Contraposition, and SA is the often criticized Strengthening of the Antecedent.TC and AA are the paradoxes of strict implication Tautological Consequent and Antilogical Antecedent, SI is Simplification, CEM is Conditional Excluded Middle and equivalent to the converse of wBT (CwBT), BT is strong Boethius Thesis, and S is Symmetry.
Apart from CEM, BT and S, all these principles are valid for strict implication (replacing by ).Since invalidates TC and AA, it avoids the paradoxes of strict implication.Since invalidates SI, it shares the connexivists' doubts about C. I. Lewis' independent proof.Since invalidates S, it can be thought of as a directional connective, as any implicative connective should be.As David Lewis' counterfactual, invalidates CEM, and thus differs essentially from Stalnaker's conditional.The invalidity of BT is largely due to the fact that BT would imply the possibility of any A B, and hence in particular of A ⊥, which is always false (see below).The remaining invalidities may be motivated as follows: The reaction of the non-monotonic reasoning tradition to strict implication was not that much to criticise TC and AA, but rather SA.Indeed, SA was rejected by the early proponents of this tradition-Stalnaker [30] and Lewis [12].These authors also endorsed unrestricted Right Weakening (RW) as well as Identity (ID), and thus also Simplification (SI).As a consequence, they needed to reject Contraposition (since C + RW implies SA).Super-strict implication shares rejection of SA and C with these accounts. 9Super-strict implication however differs from the non-monotonic reasoning tradition and variably strict conditionals in that ID is invalid and that RW is sensibly restricted, so that the invalidity of SI may equally be explained from the invalidity of ID or the restricted use of RW.
Lemma 11.The following principles are derivable:  Due to the soundness result proven below, these derivabilities are validities, and a standard semantic proof is possible as well (left to the reader). 10* is K for , MI stands for the fact that the conditional (here ) implies Material Implication and MP( ) stands for Modus Ponens for .NAA stands for No Antilogical Antecedent, NAC is the reverse and stands for No Antilogical Consequent, wBT is weak Boethius Thesis, C* is a restricted form of Contraposition (C), ID restricts Identity (ID), together with ID this says that the identity conditional expresses the inner possibility.OR, RM and CM are standard principles of non-monotonic or variably strict conditionals, where OR is also known as Disjunction in the Antecedent, RM stands for Rational Monotonicity, and CM for Cautious Monotonicitiy.
wTR is weak Transitivity, and CTR is known as Cumulative Transitivity. 11 wTR is not to be confused with (strong) Transitivity (TR): By adopting NAA, the super-strict implication rejects the first paradox of strict implication (AA) by accepting the negation, 12 NAC also expresses that there can be no impossible antecedent conditional.Indeed, A ⊥ usually expresses that A is impossible in terms of the outer modality.So that by NAC, has no outer modality.More importantly, adopting NAC-the hallmark of non-vacuism [23]-comes down to treating all impossible antecedent conditionals as false.By wBT combined with AT, the logic of super-strict implication is a Boethian logic, although it is not fully connexive.Interestingly and contrary to strict implication, super-strict implication validates a restricted version of RW (DW) without validating SA, since it invalidates C. Finally, strong Transitivity (TR) is invalid for superstrict implication but derivable for strict implication in S3.
Due to OR, RM and CM, super-strict implication shares important similarities with the non-monotonic reasoning tradition.Indeed, starting with Stalnaker [30] and D. Lewis [12], this tradition adopted the latter two principles as admissible restrictions of the problematic principle SA.The similarity is best seen by considering an equivalent axiomatisation of ST2-call it ST2 -where we replace SPRES by RM and ID. 13 ST2 can be compared to one of the most famous systems of (non-monotonic) conditional logic -the system VW of D. Lewis [12].This system can be axiomatised by MP, the unrestricted versions LLE, RW, and the axioms PT, ID, OR, AND, CM, RM, INC, T*. 14 Thus the only difference is that ST2 has the restricted versions rLLE, rRW, rN*, the restricted ID in the form of ID and TID, and that additionally PA and AT hold, and that OR and CM need not be stated explicitly. 11CTR is sometimes denoted CUT. 12 Interestingly the negation of the second paradox (TC): ¬(A ) (No Tautological Consequent NTC) is invalid due to TID. 13  The essential difference, however, to the non-monotonic reasoning tradition is this: Non-monotonic reasoning does not only reject C to avoid SA all by maintaining RW, it also needs to reject wTR to avoid SA all by maintaining RW and ID (since wTR + ID + RW also implies SA).Super-strict implication takes a different route by maintaining wTR but allowing only restricted forms of ID (TID, ID).Furthermore, although our ground logic for (non-normal) super-strict implication does not adopt unrestricted RW as an axiom, we will shortly see that we can consistently extend the logic of super-strict implication to systems where unrestricted RW holds.
In the following, we establish completeness of the axiomatisation of superstrict implication, using a method developed by Raidl [23,24].The 'neutral conditional' examined in [22,23] is a similar strengthening of an underlying conditional >, namely (A The similarity lies in the fact that our super-strict conditional can equivalently be conceived as The difference is that Raidl uses a semantics where the underlying > satisfies the unrestricted LLE and the unrestricted RW, so that satisfies them as well.These however do not hold unrestrictedly for our (non-normal) super-strict implication.For this reason our Lemma 16, required to prove our completeness result here, needs to be obtained by an intermediary step.

Soundness and Completeness of ST2
We here prove soundness and completeness of the system ST2 in our (reflexive non-normal) relational models used to interpret the language L .[23]).But the latter formula is an instance of PT in L .Hence it is derivable in S2.
AT. Suppose χ is an instance of AT.The translate is of the form ¬( (A ⊃ ¬A) ∧ ¬ ¬A).¬ ¬A ∨ ¬A is PT in L .(A ⊃ ¬A) ⊃ ¬A is PT.Hence ((A ⊃ ¬A) ⊃ ¬A) by rN.Therefore (A ⊃ ¬A) ⊃ ¬A by K. Since ⊃ contraposes, we get ¬ ¬A ⊃ ¬ (A ⊃ ¬A).By PT, we obtain the translate of AT.

T*. The translate of T* is ( ⊃ A) ∧ ⊃ A. From ( ⊃ A) we obtain ⊃ A by T, and hence A.
Derivations of length n + 1. Suppose the property (if ST2 χ then S2 χ • ) holds for all derivations of length n or smaller, our induction hypothesis (IH), and assume the derivation of χ is of length n + 1.Then there are two possibilities.Either χ is one of the axioms of ST2, or it is obtained by one of the rules of ST2 (MP, rLLE, rRW, BR*, or rN*).In the first case, the reasoning is as above.Thus let us consider the second case.
. holds by T. We have (( ⊃ ¬A) ⊃ ¬A).Thus by BR ( ( ⊃ ¬A) ⊃ ¬A).Since ⊃ contraposes, we have ( ( ⊃ ¬A) ⊃ ¬A) ⊃ ( A ⊃ ¬ ( ⊃ ¬A)) as PT.Applying rN, K, the previous result and MP, we obtain ))) is PT.Hence rN, K, the previous result and MP deliver ( ⊃ ( (A ⊃ B) ∧ A ⊃ ¬( ( ⊃ ¬A) ∧ ))).Suppose θ is an instance of INC.We need to find a derivation of χ )) ∧ .This works by a similar reasoning as for PA.We have ) by applying rN to T (and then contraposing with PT, using rN K).Using this result, PT, K and MP, we obtain ( ⊃ )). Suppose θ is an instance of SPRES.We need to find a derivation of χ . This works by a similar reasoning as for the previous ones.By rN applied to T, we obtain ( ⊃ ).By PT, rN, and BR, we obtain ( ¬A ⊃ ( ⊃ ¬A)).Thus by PT, rN, K and MP, we also obtain ( ¬A ⊃ ( ( ⊃ ¬A)∧ )).Since ⊃ contraposes, and by rN, K and MP we get (¬( ( ⊃ ¬A) ∧ )

Thus by AND and rRW again, we get ( (A ⊃ B) ∧ A ≡ (A ⊃ B) •• ∧ A •• ). But by applying rN* to PA, INC, and SPRES, we respectively obtain ((A B) ⊃ A), ((A B) ⊃ (A ⊃ B)), and (( (A ⊃ B) ∧ A) ⊃ (A B)). By AND and rRW, we conclude (A B ≡ (A ⊃ B) ∧ A). By using our previous result, AND and rRW deliver (A B
By T*, we can now conclude that χ is provably equivalent to it's twin: Lemma 16. (Twin Equivalence) For all χ ∈ L , we have ST2 χ ≡ χ •• .
Proof.By Lemma 15, T* and MP.This Lemma was also used in [23][24][25] and can be obtained directly when LLE and RW are unrestricted in the logic.It is because these principles are restricted here, that we needed to pass through first proving that any formula is strictly equivalent to its twin.
From what we have shown, we obtain our desired completeness result: Theorem 17. ST2 is complete for reflexive non-normal relational models in L .
Proof.Suppose A. Thus A • by Lemma 6. Hence S2 A • by the known completeness of S2 for relational models in L [8].Therefore ST2 A •• by Lemma 14. Hence ST2 A by Lemma 16.
The curious reader might wonder about the following: (1) Is a direct completeness proof possible?(2) Given the known decidability of S2 [18], is ST2 also decidable? 16The answer to both questions is yes.
For (1) we should note that a direct completeness proof would roughly amount to using similar translation mechanisms but now on the semantic side.Overall, the canonical model construction would be messier, 17 and it would only be instructive, if we had no completeness proof for S2.
For (2) one can simply use the translation and simulation results.If there is a procedure that decides whether or not S2 A in finitely many steps, then there is such a procedure that decides whether or not S2 B • and hence whether or not ST2 B, since we have established that ST2 B iff S2 B • .

Stronger Systems
As one can extend the system S2, we can similarly extend the system ST2 all by keeping the tight relation between the two logics in the sense that they are logics for the same semantics but once interpreted in terms of the strict implication and once interpreted in terms of the super-strict implication.

Consider the following rule and axioms
S3 is obtained from S2 by removing K and replacing the rule BR by the corresponding axiom BRA (so that rN now applies to the axioms PT, T and BRA).KT is obtained from S2 by removing the rule BR and replacing the rule rN by the unrestricted version N. S4 is obtained from KT by adding 17 For example, for the normal case, instead of constructing an accessibility relation out of formulas in maximal consistent theories (aka canonical worlds) by wRv iff {A : A ∈ w} ⊆ v, one would need to construct the relation through wRv iff {B : ¬(¬B ¬B) ∈ w} ⊆ v, see footnote 6, and then prove analogues to the truth lemma, and the determination lemma, which involves proving several analogue preparatory results as for the canonical Kripke model construction.
the axiom 4, and S5 is obtained from S4 by replacing 4 by the axiom 5. 18  Note that S5 extends S4 which extends S3 which extends S2. (see Figure 1) 19

Extensions of ST2
Consider the following rules and axioms:  18 Lemmon's axiomatisation of S4 (replacing rN by N in S3) is equivalent to our S4: From BRA and T we obtain K, from BRA, K and N we also obtain 4; conversely, K and 4 imply BRA. 19In S3, BR can be simulated (by BRA, MP), K is derivable (by BRA, T, PT, K), and rN applied to K can also be simulated.In KT, BR can be simulated by K, N, MP. 20 In ST3, BR* can be simulated (by BRA*, MP) and K* is derivable (by BRA*, T*).Thus AND is also derivable (using K*, INC, PA, SPRES), and rN* applied to K* can also be simulated.In STT as in the stronger systems ST4 and ST5, BR* and rN* can be simulated.In ST5, 4* is derivable.Note that ST4 could equivalently have been obtained from ST3 by extending rN* to N. Furthermore, in ST2 and all stronger systems, the deduction theorem for ⊃ holds (the proof is similar as for ST2-see appendix A).
Proof.For ST3: ST2 is the super-strict companion to S2 (Theorem 13, 17, and the soundness and completeness of S2 for reflexive non-normal relational models).The result then follows by Theorem 19, since BRA* = BRA • , and because S3 is sound and complete for transitive reflexive non-normal relation models in L .Same reasoning for ST4 and ST5, starting from the previous fact that STT is the super-strict companion to KT, and noting that 4* = 4 • and 5*=5 • .

What About Strong Boethius Thesis?
We already proved that strong Boethius Thesis (BT) is invalid (Lemma 10).One might thus ask, whether the addition of BT to a system of super-strict implication is at all possible.
The question may at first seem parallel to the one raised by Pizzi and Williamson [20] for their consequential implication →.These authors show that in normal systems of consequential implication with Strong Boethius Thesis (BT), the addition of MP for → makes the following derivable (see their Proposition 3.18): And, if in normal systems with BT, we add the principle MI for → ((A → B) ⊃ (A ⊃ B)), then the following is also derivable (their Proposition 3.19):

(A → B) ≡ (A ≡ B) (Collapse to ≡)
Based on these results, Pizzi and Williamson argue that it is difficult to interpret BT in normal systems of consequential implication.The situation is different for super-strict implication.First, note that in normal systems of super-strict implication, the addition of MP for (MP( )) causes no problem.MP( ) is already derivable in ST2 (Lemma 11) and thus remains derivable in normal systems.Second, (1) for , which is just Symmetry (S), is invalid in non-normal semantics of , and remains invalid in normal semantics. 22Third, (2) for is invalid in non-normal semantics of , since although wBT is valid (Lemma 11), the converse ¬(A ¬B) ⊃ (A B) (CwBT) is invalid, since the equivalent CEM is invalid (Lemma 10).Again CEM, and thus (2), remains invalid in normal semantics. 23Fourth, (3) for is invalid in non-normal semantics, since is clearly distinct from classical equivalence ≡, due to the modal assumption, and this remains so in normal semantics.So no problems such as (1), ( 2) or (3) appear in normal systems of , as long as we do not add BT.
The reason to reject BT for is not that much that it creates a collapse similar to (1)-( 3) above, but that the acceptance of BT would amount to accepting that any super-strict implication A B is possible, simply because C D implies C (due to PA), so that (A B) ¬(A ¬B) would imply (A B).This creates a contradiction, not a collapse.
Lemma 21.Adding BT to STT is inconsistent.
Proof.Adding BT to a system of strict implication means to add the validity (A B) for any A, B (by PA).In particular (⊥ ⊥).This creates a contradiction.Indeed, ¬(⊥ ⊥) by NAA.Thus in normal systems ¬(⊥ ⊥) due to N (or RW), so that ¬ (⊥ ⊥) by duality.This is the real reason why adding BT makes no sense.However, we may consider the following sensible weakening of BT which we call possibilistic Boethius Thesis: pBT This is valid in our normal semantics: Lemma 22. pBT is valid in reflexive normal relational models.
Proof.wBT is valid in non-normal semantics, according to (Lemma 11 and Theorem 17).Hence wBT is valid in normal semantics.Hence the translate wBT -is valid, according to .Now consider a world w in a reflexive normal model where (A B), according to .Thus (A B) • in w, according to (Lemma 6).But ( (A B) • ∧ wBT • ) = pBT • .Hence, overall, w pBT • .Therefore w pBT (Lemma 6).Since w was an arbitrary world in a reflexive normal model, we can conclude that pBT is valid in reflexive normal relational models.
We conclude: although it makes no sense to add BT to a normal system of super-strict implication, due to inconsistency, we can add the sensible weakening pBT.Especially to reflexive normal systems, where that addition is in fact redundant, since pBT is already ensured there.

Labelled Sequent Calculi
Starting from the labelled calculi for normal logics of SSI introduced in [4] and from the calculi for Lewis' systems introduced in [19] we are now going to introduce labelled sequent calculi for the logics ST2 − ST5.First of all we extend the language with a denumerable set W of so-called world-labels, to be denoted by w, v, u, . . ., and with the two predicates N and R of arity 1 and 2, respectively.The labelled language L W is the smallest set such that: is obtained by replacing in E each occurrence of the label u with an occurrence of the label v.The weight of an L W -formula E, w(E), is given by positing that, if E is w : A for some A ∈ L , then it is equal to that of A; else it is 0.
A sequent is an expression of the shape Γ ⇒ Δ where Γ is a finite multiset composed of labelled formulas, normality atoms, and relational atoms; Δ, instead, is a finite multiset composed only of labelled formulas.Substitution is extended to multisets and sequents by applying it componentwise.
The rules for in labelled calculi can be obtained as meaning explanations from the truth conditions for , cf. [14].In particular, the right-to left clause if w ∈ N and ∃v ∈ W (wRv and v |= A) and ∀u ∈ W (wRu implies is expressed by the following right rule: where the variable condition on u encodes the existential quantification of the semantic clause.Analogously, the left-to-right truth clause is expressed by the following left rule 24 : The rules of the labelled calculi G3.ST2−G3.ST5 are presented in Table 1.In particular G3.ST2 contains the initial sequents, all logical rules, and Ref as its only non-logical rule.G3.ST3 is obtained by adding the non-logical rule Trans to G3.ST2.G3.ST4 is obtained by adding the non-logical rule Norm to G3.ST3.Finally, G3.ST5 is obtained by adding the non-logical rule Sym to G3.ST4.We use G3.STn to denote an arbitrary calculus among them.The calculi G3.ST4 (G3.ST5) can be obtained by simply dropping normality atoms from rules L/R , thus obtaining the calculus G3SS.S4 (G3SS.S5) from [4].
A G3.STn-derivation of Γ ⇒ Δ is a finite tree of sequents whose root is Γ ⇒ Δ, whose leaves are initial sequents, and which grows according to the rules of G3.STn.The height of a derivation is the number of nodes of one of its longest branches.The expression G3.STn (n) Γ ⇒ Δ means that there is a G3.STn-derivation of Γ ⇒ Δ (of height n).A rule is G3.STnadmissible if its conclusion is G3.STn-derivable whenever its premisses are G3.STn-derivable, and it is hp-G3.STn-admissible if it is admissible and its conclusion has a derivation bounded by the height of the derivations of its premisses.In the rules in Table 1 Γ and Δ are called contexts; the formulas displayed in the conclusion (premisses only) are called principal (active).
Proof.We just have to apply bottom-up the two rules for w : A and then the lemma follows by the induction hypothesis (IH, for short).
Lemma 24.(Substitution) The following rule of substitution is hp-G3.STnadmissible: By induction on the height of the derivation D of Γ ⇒ Δ.We start by applying Lemma 24 to the premiss(es) of the last rule instance in D if it is by a rule with a variable condition clashing with some label in Π, Σ; then we apply IH to the (substituted) premiss(es).Finally, we obtain Π, Γ ⇒ Δ, Σ by an instance of the last rule applied in D.
Corollary 26.The following rules are admissible in G3.STn: The admissiblity of rules L/R is shown by the following derivations, where A is expressed by A, and where we used the known fact that sequents of shape Γ ⇒ Δ, w : are derivable ( was defined as p∨¬p) Proof.The proof is by induction on the height of the derivation of Γ ⇒ Δ.The propositional cases are standard and can thus be omitted.All nonlogical rules as well as rule L w.r.t.both premisses and rule R w.r.t.its right premiss are height-preserving invertible because of the hp-admissibility of weakening.Finally, the height-preserving invertibility of R w.r.t.its left premiss can be proved by the same strategy adopted for the propositional rules, possibly applying an hp-admissible instance of the rule of substitution.

Theorem 28. (Contraction)
The following rule of contraction is hp-G3.STnadmissible: By induction on the height of the derivation D of Π, Π, Γ ⇒ Δ, Σ, Σ.We assume that one of Π and Σ is a singleton and the other is empty.We have three cases according to whether (i) none, or (ii) exactly one, or (iii) both instances of the formula we are contracting are principal in the last rule inference in D which will be denoted by Rule.
In case (i) we permute the instance of contraction with that of Rule thus getting one (two) instance(s) of contraction that is (are) hp-admissible by IH.In case (ii) we start by applying hp-invertibility w.r.t. the non-principal instance of the contraction-formula to the premiss(es) of the last rule instance Rule in D, then we apply an (hp-admissible) instance of substitution and the inductive hypothesis to it (them).We conclude by an instance of Rule.Case (iii) arises only when two instances of wRw are principal in an instance of Trans.In this case we apply the inductive hypothesis twice to the premiss of Trans and we are done.
Theorem 29.(Cut) The following rule of Cut is G3.STn-admissible: Proof.We use the standard Dragalin-style proof: we consider an uppermost instance of Cut and we have a principal induction on the weight of the cut-formula w : A and a sub-induction on the sum of the heights of the derivations of the premisses of the instance of Cut under consideration (cutheight, for short).It is convenient to divide the proof into three exhaustive cases according to whether (i) some premiss of Cut is an initial sequent; (ii) the cut-formula is not principal in the last step of the derivation of some premiss of Cut; (iii) the cut-formula is principal in the last step of the derivations of both premisses.In case (i) it is easy to show that also Π, Γ ⇒ Δ, Σ has a cut-free derivation.In case (ii) we can permute the instance of Cut upward in the derivation of the premiss where the cut-formula is not principal, thus obtaining a derivation with an instance of Cut that is admissible by the sub-induction hypothesis.If we are in case (iii) and the principal operator of the cutformula is a propositional one, then the proof is standard and can thus be omitted.
The transformation showing the admissibility of Cut for the final case of a cut-formula of shape w : B C principal in both premisses is given in Figure 2.For reasons of space, we consider instances of rules L and R without contexts (if we have contexts, their duplications are handled in the final step by contraction).In the transformed derivations we have some instances of Cut that are admissible by IH (those marked with ( †) have lesser cut-height, and those marked with ( ‡) have a cut formula of lower weight).

Soundness
The soundness of G3.STn will be proved by introducing a notion of validity for sequents that simulates the semantic notion of consequence and then by showing that each rule of G3.STn preserves validity w.r.t.models for STn.Definition 30. 1.Given a model M = W, R, N, V a M-realisation is a function σ : W −→ W mapping each world-label to a world of the model M.

The truth of a L
fact that all formulas in Γ are true under σ implies that some formula in Δ is true under σ.If the construction ends we a derivation otherwise we obtain an infinite tree.In the latter case, by König's Lemma, the tree has an infinite branch.It is easy to see that this branch is saturated and, therefore, it allows to extract a countermodel.
Theorem 43.Given a saturated branch B in a proof search tree for Γ ⇒ Δ built according to the rules of system G3.STn, we can extract a countermodel M for the endsequent that is based on a model for STn.
Proof.Given a saturated branch B from Γ ⇒ Δ, we define the following countermodel: M = W, R, N , V such that: • W is the set of all world labels occurring in ↑ Γ∪ ↑ Δ.
• N is the set of all labels w such that N w ∈↑ Γ.
• wRu if and only if wRu occurs in ↑ Γ.
• V(p) is the set of all worlds w such that w : p occurs in ↑ Γ.
Notice that V is well defined by the saturation condition Ax and by the saturation condition Ref we know that the accessibility relation is reflexive.We define the realisation ρ such that ρ(w) ≡ w.We claim that: The proof, which is by simultaneous induction on the weight of A, is omitted for the sake of brevity.
To conclude, we have to argue that the model defined above is a model for STn.This is an immediate consequence of saturation under the non-logical rules of G3.STn.

Theorem 44. (Completeness) For every formula A (and w.r.t. relational models in the appropriate class):
A if and only if G3.STn Nw ⇒ w : A Corollary 45.The rule of Cut is semantically admissible.
We have thus shown that the procedure given in Def.42 outputs a countermodel for each underivable sequent and a derivation for each derivable one).We are now ready to prove that G3.ST2 is decidable by showing how to modify the procedure given in Def.42 in order to avoid the construction of infinite branches in a proof search tree for an underivable sequent.Given that a branch of a G3.ST2 proof search tree can be infinite because of an unbounded number of application of the rules Ref, L , and R , we have to show how to bound the number of applications of these rules.
As a consequence, the tree of successors is finitely branching.It remains to show that in every branch the length of a chain of labels is finite.This depends on the fact that the relation defined by the tree of immediate successors in a proof search is intransitive.In particular, a label sees only its immediate successors and itself (by reflexivity).Therefore the length of a branch is determined by the number of modal operators occurring in a formula.
Theorem 48.Every chain of labels in a branch in a proof search for the sequent Γ ⇒ Δ is finite.
Proof.Given a chain of labels in a branch in a proof search for the sequent Γ ⇒ Δ and a label u in the chain, every non-trivial immediate successor of u is introduced (bottom-up) by the application of one of either L or R to a formula B labelled by u.However, since every label sees only its immediate successors, every label introduced by the analysis of u : B will label only formulas of lesser weight.Since by definition the weight of each formula is finite, the chain is finite.
Theorem 49.The proof search for a sequent Γ ⇒ Δ in the system G3.ST2 terminates.
Proof.The proof is immediate because in every branch the number of labels generated is finite.
Proof.By Theorem 43 we can extract a countermodel out of a saturated branch and by Theorem 49 we know that every saturated branch can be made finite, so we get the finite model property and the decidability of the system.

Conclusion
In this article, we considered a strengthening of strict implication by imposing that the antecedent is possible: A B iff (A ⊃ B) and A. The connective is known as 'super-strict implication'.Logics of super-strict implication are Boethian and paradox-free versions of Lewis' logics of strict implication that have been introduced semantically in [4,5].In this paper we provided two proof-theoretic characterisations of normal and non-normal logics of super-strict implication-in a Hilbert-style proof system and in a labelled sequent calculus.The logics ST2-ST5 investigated here can be thought of as super-strict companions to C. Lewis' logics S2-S5 of strict implication.First, we showed that our Hilbert-style axiomatisations of ST2-ST5 are sound and complete for the respective semantics by simulating and backsimulating complete axiomatic calculi for the strict implication companions S2-S5.Next, we showed that our labelled sequent calculi have good structural properties: all rules are height-preserving admissible, the structural rules of weakening and contraction are height-preserving admissible, and the rule of cut is syntactically admissible.We also showed that each of these calculi is sound and complete and that the calculus expressing ST2 admits for terminating proof-search and is hence decidable.We further proved that strong Boethius thesis (BT), the headache of consequential implication [20], cannot be added to a normal system of super-strict implication without creating inconsistency.However, a sensible weakening of BT-by the possibility assumption of the antecedent conditional-termed 'possibilistic BT (pBT)' here, is valid in normal reflexive semantics.
There is an obvious way to strengthen super-strict implication .Namely by imposing that the negation of the consequent is also possible: A B iff A B and ¬B.This connective has been investigated semantically [4] (termed strong super-strict implication) and axiomatically [27] (termed implicative conditional ). 25 There are some interesting similarities, as well as differences.Both connectives invalidate the paradoxes of strict implication-Antilogical Antecedent (AA) and Tautological Consequent (TC).But they do so in different ways.Whereas validates the negation of the first paradox of strict implication NAA (¬(⊥ B)), also validates the negation of the second paradox NTC (¬(A )).Both connectives are weakly transitive (wTR), validated weak Boethius Thesis (wBT) and Aristotle's thesis (AT), but invalidate the principles of Right Weakening (RW), Identity (ID), Simplification (SI), and Strengthening the Antecedent (SA), as well as strong Boethius Thesis (BT).But, contrary to , also validates Contraposition (C) and Aristotle's Second Thesis.Whereas is expressible in the language L of super-strict implication, it is not always expressible in the language L of strong super-strict implication.In [27] a Hilbert-style axiomatisation of was given and shown to be sound and complete for reflexive normal Kripke models.Yet non-normal versions of are also available [5].There remain thus at least three open questions for , inspired by the present work: (1) Is it possible to provide an axiomatisation of without using ?(2) What about axiomatisations of for non-normal semantics?(3) How do systems for behave when we add BT or sensible weakenings? 25 Also suggested by Priest [21], Gomes [6], and Lenzen [10].
In the future, we plan to the results presented here to the superstrict companion of Lewis' system S1 and to give a proof-theoretic characterisation of logics of super-strict implication by means of an internal sequent calculus such as hypersequents or nested sequents.

Definition 1 .
(Models) A relational model M is a tuple W, R, N, V , where W = ∅, R is a reflexive relation over W , N ⊆ W , and V : P −→ ℘(W ).
From the first, we get ¬(A ¬B) by wBT.Together with A C this yields A ∧ B C by RM. wTR.Suppose A B and B C. Thus A, (A ⊃ B) and (B ⊃ C).The two last yield (A ⊃ C) by AND and rRW.Together with A this yields A C. CTR.Assume A B and A ∧ B C. Thus (A ⊃ B), and (A ∧ B ⊃ C) by INC.Hence (A ⊃ C) by AND and rRW.From A B we also obtain A. Together with (A ⊃ C), this yields A C by SPRES.
SPRES and ID are derivable in ST2.Conversely, SPRES is derivable by RM and ID and the remaining axioms: From RM we get (A ⊃ B) ∧ A ⊃ (A (A ⊃ B)), using LLE applied to the PT equivalence A ≡ (A ∧ ).But (A (A ⊃ B)) ⊃ A by PA and A ⊃ (A A) by ID.Furthermore (A (A ⊃ B)) ∧ (A A) ⊃ (A B) by AND and rRW.Chaining the results, we obtain (A ⊃ B) ∧ A ⊃ (A B). 14INC is redundant, as it follows from ID, RW, OR.Readers might be more acquainted with the axiomatisation for VW where instead of INC and T* we have MI.The two axiomatisations are equivalent: INC and T* imply MI; MI implies T*.We use the axiomatisation with INC and T* for comparative reasons.

Figure 2 .
Figure 2. Transformation for principal cuts with cut-formula w : A B Similarly from B C we obtain (B ⊃ C).This together with (A ⊃ C) yields (A ∨ B ⊃ C) by AND and rRW.Together with (A ∨ B) we obtain A ∨ B C by SPRES.RM.Suppose A C and ¬(A ¬B).From the first we obtain A and (A ⊃ C) (cf. the proof for OR).From (A ⊃ C) we obtain (A ∧ B ⊃ C) by rRW.Contraposing SPRES, from ¬(A ¬B) we obtain ¬ A or Cumulative Transitivity Proof.We use the deduction theorem for ⊃ in ST2 (Appendix, Lemma 52), MP and other classical reasoning, often without saying (e.g.⊃ contraposes and is transitive).K*.Applying rN* to a formula of form K*, we obtain ( (A ⊃ B) ⊃ ( A ⊃ B)).By T* we thus get (A ⊃ B) ⊃ ( A ⊃ B).This is K*.C*.Suppose ¬B and A B. The second implies (A ⊃ B) by INC.Thus (¬B ⊃ ¬A) by rRW.Together with ¬B this yields ¬B ¬A by SPRES.¬ (A ⊃ ¬B).Since A, we must have ¬ (A ⊃ ¬B).That is (A ∧ B).This together with (A ∧ B ⊃ C) delivers A ∧ B C by SPRES.CM.Suppose A B and A C.

Table 1 .
Rules of the calculi 4. A sequent Γ ⇒ Δ is STn-valid if it is realised by each M-realisation withM a model for STn.