On the Provable Contradictions of the Connexive Logics C and C3

Despite the tendency to be otherwise, some non-classical logics are known to validate formulas that are invalid in classical logic. A subclass of such systems even possesses pairs of a formula and its negation as theorems, without becoming trivial. How should these provable contradictions be understood? The present paper aims to shed light on aspects of this phenomenon by taking as samples the constructive connexive logic C, which is obtained by a simple modification of a system of constructible falsity, namely N4, as well as its non-constructive extension C3. For these systems, various observations concerning provable contradictions are made, using mainly a proof-theoretic approach. The topics covered in this paper include: how new contradictions are found from parts of provable contradictions; how to characterise provable contradictions in C3 that are constructive; how contradictions can be seen from the relative viewpoint of strong implication; and as an appendix an attempt at generating provable contradictions in C3.


Introduction
The most prominent and familiar systems of non-classical logic, such as minimal logic, intuitionistic logic, substructural subsystems of classical and intuitionistic logic, systems of many-valued and relevance logic etc. are all subsystems of classical logic.However, in the literature one encounters by now also studies on a remarkable variety of contra-classical logics that are orthogonal to classical logic.Among those contra-classical logics are systems of connexive logic, for a survey see [23].A particularly simple and straightforwardly arrived at system of connexive logic is the propositional logic C that was introduced in [21].The logic C is contra-classical as it validates certain non-theorems of classical logic, namely formulas known as Aristotle's theses and Boethius' theses: AT: ∼(∼A → A) AT : ∼(A →∼A) BT: (A → B) →∼(A →∼B) BT : (A →∼B) →∼(A → B).
Moreover, C is a non-trivial negation inconsistent logic.It can be obtained from the system N4 of constructible falsity [1] by taking an alternative account on how the falsity of an implication is supported at a state from a suitable Kripke model.C can also be seen as an expansion of the basic four-valued paraconsistent logic of firstdegree entailment, FDE.An axiomatization of the non-trivial negation inconsistent connexive logic C3 is obtained from an axiom system for C by adding the Law of Excluded Middle, A∨ ∼A.A plea for taking certain non-trivial negation inconsistent logics seriously and for thinking of them as respectable formal systems can be found in [24], where it is pointed out that the development of these logics was driven by motivation completely independent from the aim of defining non-trivial negation inconsistent logical systems.One example are the logics of logical bilattices [2,6,7] that have been applied to logic programming and theories of self-referential truth; other examples are, for instance, the Second-order Logic of Paradox [9,16] and Abelian group logic, see [11,12,15] and references therein.
The triviality of a logic, understood as absolute inconsistency, is certainly too much of inconsistency.If every formula of the language in use is provable, then the logic in question is useless.The contradictory logics C and C3 are non-trivial and, according to Graham Priest [17, p. 178], C is "one of the simplest and most natural" connexive logics.The simplicity of C and C3 together with the motivation of C in terms of constructiveness and of both C and C3 in terms of information states and their possible expansions call for attempts to take the negation inconsistency of C and C3 seriously and to try to obtain a deeper understanding of this remarkable property.The present paper is a contribution to obtaining such an understanding.
The paper is structured as follows.In Section 2, we introduce the logics C and C3 by presenting their Kripke semantics, the axiomatic proof systems (HC, HC3) and the sequent calculi (GC3, G3C3at) introduced in [14,21].Then, in Sections 3.1 of 3, we introduce one possible characterization of provable contradictions in C by showing a necessary condition.This will indicate the interesting feature that provable contradictions in C hide another contradiction within them.In Section 3.2 we give a relative characterization of provable contradictions shared between C and C3.The aim is to obtain a class of formulas so that if a formula A in the class is a provable contradiction in C3, then A is a provable contradiction already in C. In other words, the class assures the conservativity of provable contradictions, thus clarifying partly its relationship with constructivity.We obtain such a class by focusing on the position of propositional variables in a derivation in G3C3at, and by utilizing the particular way the calculus treats non-constructive inferences.Section 3.3 turns the attention to a connective definable in C and C3.The connective of strong implication A ⇒ B, defined as (A → B) ∧ (∼B →∼A), is known to enjoy (in C, C3 as well as N4) the property that . That is to say, provable strong equivalence is a congruence relation, which is not the case for provable equivalence.The connective ⇒ is remarkable in another aspect as well: taking the connective (along with the fusion connective) as primitive in N4 and its extension N3 gives rise to certain substructural logics [18][19][20].The connective is also of interest, because (i) it allows us to obtain a deeper understanding of the notion of contradiction, by taking the support of falsity condition for formulas A ⇒ B into account, and (ii) it enables us to look at → from another perspective.We observe how formulas related to contradiction can be rephrased into strongly equivalent formulas which connect the notion of contradiction with other (sometimes puzzling) forms of inference.We also offer a relativized view of → from the perspective of ⇒, by defining alternative connectives and studying their mutual relationship.
Finally, as an appendix, in Section 5 we suggest a method to generate provable contradictions in C3.Such a method is desirable given that provable contradictions in C3 have usually been constructed manually.Noting that A ⇒ A is a provable contradiction in C3, we use that formula as the initial case to present sequent-calculusstyle clauses which preserve the status of provable contradictions, thereby obtaining what might be called a "calculus of contradictions".We consider several classes with increasing generality, and give evaluations as to their strengths and their limitations.

The Connexive Logics C and C3
Consider the propositional language L := {∧, ∨, →, ∼} based on a denumerable set At L of propositional variables and define the equivalence connective, ↔, as usual.We use "≡" to denote syntactic identity between L-formulas.If A is an L-formula, at(A) is the set of propositional variables occurring in A. We will often omit outermost brackets of L-formulas.We will first present the connexive propositional logic C semantically.
Intuitively, the non-empty set W is a set of information states, and ≤ is a relation of possible expansion of information states.The function v + maps a propositional variable p to the states in W that support the truth of p, whereas v − maps p to the states that support the falsity of p, and both support of truth and support of falsity persists in the transition from an information state to an expansion of it.In the semantics for C and C3 a state may support both the truth and the falsity of a propositional variable.In the semantics for C3 every state always either supports the truth or the falsity of any propositional variable.
said to be a model based on the frame W , ≤ .The relations M, t | + A (M supports the truth of A at t) and M, t | − A (M supports the falsity of A at t) are inductively defined as follows: The semantic consequence relation between sets of L-formulas and single Lformulas A is defined as follows: Next, we will present axiom systems, HC and HC3, and G3-style sequent calculi, G3C and G3Cat, for the semantically defined connexive logics C and C3.We first introduce HC and HC3.

Definition 3
The axiom system HC is given by the following schematic axioms and inference rule: The axiom system HC3 is obtained from HC by adding the axiom a6 A∨ ∼A.
Provability in the axiom system HC (HC3) and the derivability relation HC ( HC3 ) between sets of L-formulas Γ and single L-formulas A are defined as usual, i.e., we write Γ HC A (Γ HC3 A) if there is a sequence of formulas B 1 , . . ., B n , 0 ≤ n, such that every formula in the sequence B 1 , . . ., B n , A either (i) belongs to Γ , (ii) is an axiom of HC (HC3); or (iii) is obtained by (R1) from formulas preceding it in sequence.If ∅ HC A (∅ HC3 A) we also write HC A (HC3 A).Similar conventions will apply regarding other systems.
Note that if axiom a5 is replaced in HC by ∼(A → B) ↔ (A ∧ ∼B), one obtains the standard axiom system HN4 for Almukdad and Nelson's four-valued constructive logic N4 [1].
It can easily be shown that a negation normal form theorem holds, i.e., every Lformula is logically equivalent with a formula in which the negation sign, ∼, occurs only in front of propositional variables.Moreover, C can be faithfully embedded into positive intuitionistic logic.Also the following results are known, see, e.g., [14].The fact that C satisfies not only the disjunction property but also the constructible falsity property can be seen to indicate that the propositional logic C is more constructive than intuitionistic propositional logic.From that point of view all De Morgan laws, i.e., the left and right directions of axioms a3 and a4, are seen as constructively acceptable as they are valid in C.Moreover, C is natural from the point of view of extending the Brouwer-Heyting-Kolmogorov interpretation of the logical operations to a proof/disproof-interpretation, cf.[22].

Proposition 3 The connexive logics C and C3 are decidable.
Provable equivalence fails to be congruence relation in HC and HC3.(Note, for instance, that As in HN4, provable strong equivalence is a congruence relation in HC and HC3, where the strong equivalence connective is defined as usual, i.e., A ⇔

Proposition 4 The set {A | ∅ HC3 A} is closed under the rule A ⇔ B / C(A) ⇔ C(B).
We now present the sequent calculi G3C for C and G3C3at for C3, where uppercase Greek letters stand for finite, possibly empty multisets of formulas, A, Γ and Γ, A stand for {A} Γ , and Δ, Γ stands for Δ Γ , and where is multiset union.Sequents are expressions of the form Γ A, and we write A instead of ∅ A.

Definition 4
The sequent calculus G3C is defined by the following rules.
Gem-at then we obtain the calculus G3C3at.
By induction on the complexity of A one can show that A, Γ A and ∼A, Γ ∼A are derivable in G3C and G3C3at for any L-formula A. Moreover, the following results are known, cf.[14].

Proposition 5 (i) The rules of weakening and contraction
For G3C3at the disjunction property and the constructible falsity property fail.Moreover,

Proposition 6
The excluded middle rule is admissible in the propositional logic G3C3at.

If a sequent Γ
A is provable in a sequent calculus, i.e., derivable from Ax1 or Ax2, or both, we write Γ A. To every finite set of formulas Γ , there corresponds a unique multiset (with no element of Γ occurring more than once).If Γ is such a set, let Γ be a conjunction of all formulas from the corresponding multiset.Conversely, to every finite multiset Γ , there corresponds a unique set, which we will also denote by Γ .

Proposition 7
Let Γ be a finite set of L-formulas and let ∅ = ( p → p), for some fixed propositional variable p.The following equivalences hold.

(i) Γ HC A if and only if in G3C Γ A. (ii) Γ HC3 A if and only if in G3C3at
Γ A.

Observations on Provable Contradictions in C and C3
Both logics C and C3 are negation inconsistent, i.e., there are L-formulas A such that both A and ∼ A are provable.We will refer to such formulas as provable contradictions.An example of a contradiction provable in both HC and HC3 is given with the following pair of formulas: ).We will give other examples in what follows.Although the last formula is an instance of AT, this is not essential to the provability of contradictions in HC and HC3.Since there exist negation consistent connexive logics, it is clear that the provability of the connexive principles themselves is not bringing about the negation inconsistency of C and C3. 1 As N4 is negation consistent, the provability of contradictions in HC and HC3 can be attributed to axiom a5, against the background of the other axioms, modus ponens, and the definition of derivability.As has been shown in [25] for the {∼, →}-fragment of HC (or HC3 for that matter), giving up either the contraction or the monotonicity (weakening) axiom of positive intuitionistic logic does not suffice for regaining negation consistency.We now take a closer look at the provable contradictions of C and C3. 1 Recently, in [5] the remarkable observation has been made that a connexive implication, → c , can be defined within intuitionistic propositional logic by setting (notation adjusted), where ¬ is intuitionistic negation.123

An observation on contradictions hidden within contradictions
Our first investigation concerns what kind of information we can extract from a provable contradiction.We shall observe that each provable contradiction 'hides' another contradiction within it.We begin with introducing a preliminary notion.Definition 5 Let s = A 1 , . . ., A n C be a sequent.Then A 1 , . . ., A n are antecedent parts, ap, of s and C is a succedent part, sp, of s.Moreover, -if A ∧ B is an ap (sp) of s, then so are A and B; -if A ∨ B is an ap (sp) of s, then so are A and B; Then the next observation indicates how we can uncover additional information contained in a provable contradiction.

Observation 1 If in G3C,
A and ∼ A, then there exists an ap B of A, such that B provably implies a contradiction, i.e., B → C and B →∼C, for some C with at(C) ⊆ at(A).

and clearly at(D) ⊆ at(B ∨ C).
If A is an implication B → C, we have: The cases of negated conjunctions, negated disjunctions, and negated implications are analogous because A iff ∼∼A.If A is a negated negation ∼∼B, we may apply the induction hypothesis because ∼∼∼B iff ∼B.
Example 1 In the sequent calculus G3C, we have By Definition 5, (A∧ ∼A) is an ap of ((A∧ ∼A) → A) and ∼((A∧ ∼A) → A), and obviously, A∧ ∼ A provably implies a contradiction such that the required condition on atomic formulas is satisfied.
The formula ∼A is an antecedent part in its leftmost occurrence of the sequents ∼ A →∼ (A →∼ A) and ∼ (∼ A →∼ (A →∼ A)), and ∼ A provably implies a contradiction such that the required condition on atomic formulas is satisfied since The example shows that in G3C, any negation ∼A provably implies a contradiction, and indeed, in G3C any formula provably implies a contradiction because there exist provable contradictions and the weakening rule is an admissible rule of G3C.

Conservative Classes and Provable Contradictions
Despite the fact that C3 validates additional contradictions in comparison to C, the latter system already appears to capture many important instances.These two kinds of provable contradictions can be seen as non-constructive and constructive, respectively.In order to understand the relationship between constructivity and provable contradictions, it would be desirable to have a method to distinguish them in C3.For classical logic, Glivenko's theorem [8] gives a way to check if a classical theorem is an intuitionistic theorem, by looking at its form alone (e.g., whether it is of the form ¬A, where ¬A is the intuitionistic negation of A.) Our goal here is to obtain a similar criterion, by providing classes of formulas which tell that formulas of certain forms are provable contradictions already in C, if they are so in C3.In other words, such provable contradictions of C3 are obtainable constructively.In what follows, we are working towards Corollaries 1 and 2, where such results are obtained.
The main idea is first to convert a derivation in G3Cat to a derivation in G3C with extra formulas in the antecedent, and then point out that the extra formulas are in fact eliminable in some cases.The following classes help us to achieve this by classifying formulas in a derivation.

Definition 6
We define classes α, . . ., of formulas by the next clauses: Equivalently, we can also define the same classes by simultaneous induction: The classes α, β and γ , δ each form a pair, in the sense that the negation of a formula in one class brings it to the other class.The class is then a superclass of the first pair.
Another key element in the definition is the restriction on the form of implication in each class, which plays an essential role in the proofs to follow.
We are interested in the cases where the classes and δ each contain the formulas respectively in the antecedent and the succedent of a sequent, because we can then obtain the next lemma.Proof Assume that a derivation of Γ C is given.We shall check that if the lower sequent satisfies the condition, then so do the upper sequents.Suppose that the step in question is an instance of L∧.If the step is an instance of R∧,

A, B,
then it must be the case that A ∧ B ∈ δ.So either A ∧ B ∈ α or A, B ∈ δ.In the former case, A, B ∈ α and so A, B ∈ δ as well.
If the step is an instance of L∨ or R∨, the arguments are similar to those for L∧ and R∧.If the step is an instance of L→, If the step is an instance of R→, Proof Suppose G3C3at Γ ∼ C, where Γ ⊆ and C ∈ γ .Then G3C Γ, p 1 , . . ., p n ∼C for some p 1 , . . ., p n , by keeping, when constructing the derivation in G3C, the left premise of Gem-at for each application of the rule in the derivation in G3C3at.The admissibility of Wk is also used to make sure that the premises are in the right shape.Then note that each p i is in the class .Hence by Lemma 1 the derivation of Γ, p 1 , . . ., p n ∼C satisfies the condition that the sequents occurring in it have their antecedents in and the succedents in δ.Now since p i / ∈ δ, all p i must have been introduced as contexts.It is therefore possible to construct a new derivation almost identical to the old derivation, except that the p i are not introduced as contexts, and also similarly absent at each step in the derivation.In particular, all p i introduced as contexts disappear in the endsequent of the new derivation; thus we obtain the derivation for Γ ∼C in G3C.
This result shows that γ gives (via Proposition 7) a syntactic class of formulas which have the same status concerning provable contradictions in HC and HC3.

Corollary 1 Let HC A and A ∈ γ . Then A is a provable contradiction in HC if and only it is so in HC3.
Example 3 We have seen that ( p∧ ∼p) → p is a provable contradiction already in HC.Correspondingly, since p ∈ β, ∼p ∈ α and so p∧ ∼p ∈ .Also p ∈ γ and thus the whole formula is in the class γ .On the other hand, we can show ( p ↔∼p) → p is a provable contradiction only in HC3.Correspondingly, ( p →∼p) / ∈ and so the whole formula is not in γ .
One shortcoming of the above corollary is that we have to assume the derivability of A. In order to obtain a result without such an assumption, we shall define another family of classes from α, . . ., by changing the positions of propositional variables.

Definition 7
We define classes α , . . ., of formulas by the next clauses Again, we also give the same classes in terms of simultaneous induction: The idea of the classes is analogous to the one we saw earlier; the main difference is to which classes propositional variables belong.Now we obtain a class for which the provability of contradictions in HC coincides with the one in HC3.

Corollary 2 For A ∈ γ ∩ δ , A is a provable contradiction in HC if and only if it is so in HC3.
Proof Suppose HC3 A∧ ∼A.Then HC3 A and HC3 ∼A.Now, A ∈ γ implies ∼A ∈ δ, and so by Observation 2 (along with Proposition 7) we infer HC ∼A.Similarly, if A ∈ δ , by Observation 3 it holds that HC A. Hence HC A∧ ∼A.

Example 4
It is easy to check that γ ∩δ has an element, as p ∈ γ ∩δ .Moreover, there are provable contradictions satisfying the criterion; as p∧ ∼p ∈ and p∧ ∼p ∈ it holds that ( p∧ ∼p) → p ∈ γ ∩ δ .
It is also possible to note that γ = δ , because (a) (∼p → p) → p ∈ γ but is not a member of δ ; and (b) ( p → p) → p ∈ δ but is not a member of γ .
It might be suggested that a similar result could be obtained using only one class, by considering A such that A∧ ∼A ∈ δ or A∧ ∼A ∈ δ .We can however show that no such A exists.

Observation 4
The following relationships obtain among the classes.
Proof Here we look at (i); the proof of (ii) is analogous.We show the claim by induction on the complexity of formulas.For propositional variables, we have p / The argument for δ is analogous.

Contradiction and Strong Implication
In order to understand provable contradictions in C and C3, it is crucial that their implications are understood.In these logic, implications can be seen from a relative perspective, with strong implication as another reference point.
Strong implication can also convey more information about contradictions, since it takes support of falsity into account.This makes us to question what should be a support of falsity for contradictions.If we take a contradiction to be A∧ ∼ A, then its negation, namely ∼ (A∧ ∼ A), holds if and only if A∨ ∼ A holds; so the falsity is not always supported in C under this account.This, however, is not the only conceptualization.For instance, consider the formula A ∧ (A ⇒∼ A): this formula arguably also represents a state of contradiction, because if it is provable, then so are A and ∼A.Nonetheless, this latter formula has a different status on its support of falsity, as It therefore becomes apparent that we have to consider multiple notions of contradiction.One idea to acquire insights into them is by looking at formulas that are strongly equivalent to them.We will establish some equivalences which we believe are significant in the understanding of contradictions as well as of strong implication.The equivalences can be rather hard to interpret, but we shall attempt to either offer a tentative explanation, or relate them to relevant notions.
Firstly, we look at the formulas A∧ ∼A and A ∧ (A ⇒∼A).

Observation 5
The following equivalences hold.
For the first equivalence, we note that the right-hand side (RHS) is equivalent to the conjunction of It is straightforward to see that the left-hand side (LHS) implies both of the conjuncts; conversely, we can observe the RHS implies the LHS because the latter conjunct is equivalent to For the second equivalence, the RHS is equivalent to the disjunction of: - Thus the equivalence holds in HC3.
(ii) The outline is analogous to (i).Since the negation of the RHS is provable in HC, the negation of the LHS has to be provable as well; this can be checked easily.
The above provable strong equivalences suggest that (if a contradiction is seen as a peculiarity), then assuming A is decidable (i.e., A∨ ∼A) and then deriving one of the disjuncts (∼A) from the other (A), is somehow considered to be peculiar.This might mean that assuming A∨ ∼A (w.r.t.⇒) commits one to the position that A and ∼ A are unrelated, which precludes (in non-peculiar circumstances) the possibility to infer a disjunct from the other.
We may further note that we can replace the RHS of the above equivalences with (A∨ ∼A) ⇒ (A ⇒ (A∨ ∼A)), which is an instance of the positive paradox of material implication.Hence the addition of the schema A ⇒ (B ⇒ A) to the axiomatic system for C and, a fortiori, C3 results in a trivial system.
Next, we consider the formula A ⇔∼A.In C3, the formula can be seen as expressing the state of a contradiction, because both A and ∼A are derivable from it.On the other hand, the same does not hold in C, and this gives a motivation to consider a rephrasing of the formula to have a better understanding of it.

Observation 6
The following equivalences hold.

Lemma 3
The following formulas are provable.
Proof (i) follows using A∨ ∼ A in the left-to-right direction.For (ii), the RHS is equivalent to , which immediately justifies the implication.(iii) follows using A∨ ∼A for the left-to-right direction, and B∨ ∼B for the right-to-left direction.For (iv), the LHS is equivalent to Proof We have to show The first equivalence follows from (i) and (ii) of the previous lemma.The second equivalence follows from (iii) and (iv) of the same lemma.
A yet different formula strongly equivalent to weak implication is ((A ⇒ B) ∨ B)∧ ∼(A ⇒ (B∧ ∼B)), which is obtained by simply dropping ∼(A ⇒ B) from the formula given in [14] seen above: it turns out upon inspection that the negation does not play any role for establishing the equivalence.
The formula ((A ⇒ B) ∨ B)∧ ∼((A ⇒ ∼B) ∨ (B ⇒ ∼B)) gives the idea that weak implication consists of two parts, each telling what → conveys about the support of truth and falsity.We see that the conjuncts are largely symmetric, once we recall that B ⇒ ∼B is weakly equivalent to ∼B in C3.The difference in the falsity condition of the two formulas is the only asymmetry between them.Thus from the viewpoint of strong implication, the asymmetry may be understood as the key characteristic of weak implication.At the same time, it gives rise to the naïve question of what happens when we symmetrify the formula completely.
We can think of three types of connectives defined by a modification of the formula above. - We will not suggest that each of these connectives makes sense as an 'implication'.They can nonetheless give certain comparative insights into → and ⇒.They also allow us to question whether there is a reason to prefer defining weak implication over other connectives, if one takes strong implication as the starting point.The connectives are also not unrelated to our main interest, because it is possible to have a provable contradiction with respect to them.Observation 9 For i ∈ {1, 2, 3}, there is a formula A → i B such that HC3 A → i B and HC3 ∼(A → i B). (i) → 1 -→ 3 all satisfy Aristotle's theses.(ii) → 2 satisfies Boethius' theses, and for i ∈ {1, 3}, → i satisfies a variant form Proof For (i), we note that Aristotle's theses hold for →, and it is clear from the diagram that the theses for → 1 -→ 3 are implied by it.
As clarified in [18][19][20], it is possible to define in N3 and N4 a connective A * B := ∼(A → ∼B)∨ ∼(B → ∼A) so that the following equivalence holds: That is to say, * in N3 and N4 is residuated with respect to ⇒.
It would be desirable if an analogous result holds for ⇒ in C with respect to some connective.We shall show, however, that this cannot be done, because introducing such a connective leads to triviality in any extension of C. Observation 12 There is no connective * definable in L satisfying Proof Assume * to be definable.Then since HC A * B ⇒ A * B, it must be that HC A ⇒ (B ⇒ A * B).So as an instance, we have But then for one of the conjuncts we have: That is to say, we have Now because B ⇒ B is a theorem of HC, the second disjunct of the premise and consequently the whole premise are also theorems.Hence HC A for any A. Therefore by reductio, the statement holds.

Concluding Remarks
In this paper, we have made various observations concerning provable contradictions in C and C3.They in particular include characterizations for provable contradictions, given in terms of necessary conditions (Observation 1, Corollary 5) and sufficient conditions (Corollary 2, 5, Observation 15).We also touched upon the understanding of weak implication from the perspective of strong implication.
Needless to say, our observations are but initial steps towards a fuller comprehension of the negation inconsistency of the systems.We may not be at the stage yet where it can be described what exactly provable contradictions share, based on the philosophical interpretation of C and C3.Nonetheless, our investigation suggests that there are at least two main paths to proceed to the above stage.It depend on whether we focus on: -the internal structure of provable contradictions, to discover what kind of information (e.g.another contradiction/constructivity) they possess; or, -the outward appearance of provable contradictions, to discover new interpretations by seeing them from a different perspective using e.g.equivalent formulas or translations.If this distinction is sensible, then we may expect that any definitive characterization of provable contradictions in C or C3 offers an insight into how these two sides are related.Potentially, the same perspective can be fruitful for the investigation of other non-trivial negation-inconsistent logics as well, such as the ones listed in [24].

5 Appendix: Generating Provable Contradictions in C3
In studying provable contradictions, it is desirable to have a method to find a large class of such formulas from some simple instances.In this respect, strong implication in C3 has a favourable property, because the formula A ⇒ A is provably contradictory in it.This enables us to construct more complex provable contradictions of the form A ⇒ B, through which we can observe under which kinds of inferences provable contradictions are closed, 3In what follows, we shall use proof-theoretic clauses in the style of the sequent calculus Sn4 [10] originally devised for N4. 4 In this setting, given finite multisets Γ, Γ , Δ, Δ , expressions (we shall abuse the language and call them sequents) of the form Γ ; Γ ⇒ Δ; Δ denote the formula ∼Γ ∧ Γ ⇒ ∼Δ ∨ Δ (where ∼Γ = {∼C : C ∈ Γ }).In particular, both Γ Γ and Δ Δ are non-empty.In a sequent, the semicolon, ; , separates two places on the left, respectively the right hand side of ⇒.As the denoted formula indicates, in each case the first place is reserved for negated formulas.
We shall define there kinds of classes of provable contradictions with increasing generality.Let us start with a class containing clauses for conjunction, disjunction and negation, as well as clauses which resemble structural rules.Proof By induction on the depth of derivation.For (Ax−), it is straightforward to check that HC3 ∼A ∧ ∼Γ ∧ Δ ⇒ ∼A∨ ∼Δ∨ Γ .Moreover, the negation of the formula follows from the provability of (∼A → A) ∨ (A → ∼A) in HC3.The case for (Ax+) is analogous.

Definition 8 (CL1)
The cases for (LC−), (LC+), (RC−), (RC+) are immediate.For (LW−), there are cases depending on the position of the additional instance of A. If it appears only on the antecedent, then the case is simple.If it appears in the succedent, then that HC3 ∼A ∧ ∼Γ ∧ Γ ⇒ ∼Δ ∨ A ∨ Δ readily follows from the I.H.The negation also follows from the I.H. in addition to ∼A → ∼A.The cases for (LW+), (RW−) and (RW+) are similar.It is therefore desirable to expand the class to capture this and other related formulas.This can be achieved by focusing on formulas of the form ∼A ⇒ A and A ⇒ ∼A.For these types of formulas, it is possible to formulate more general clauses of implication, and weakening.
Validity of a formula A in a C-model, respectively C3-model, (M | A) and validity of A on a frame (F | A) are defined in the usual way.This means that ifM = W , ≤, v + , v − is a C-model, respectively C3-model, then M | A iff for every t ∈ W , M, t | + A,and F | A holds iff M | A for every model M based on F. A formula is C-valid, respectively C3-valid, iff it is C-valid, respectively C3valid on every frame.Support of truth and support of falsity for arbitrary formulas are persistent with respect to the relation ≤ of possible expansion of information states.That is, for any C-model, respectively C3-model,

Proposition 2
The logic C satisfies the disjunction property and the constructible falsity property.If HC A ∨ B, then HC A or HC B. If HC ∼ (A ∧ B), then HC ∼ A or HC ∼ B.

Lemma 1
Suppose G3C Γ C, where Γ ⊆ and C ∈ δ.Then for any sequent Γ D in the derivation of Γ C, it holds that Γ ⊆ and D ∈ δ.
Γ C A ∧ B, Γ C L∧ Then by I.H.A ∧ B ∈ , and so either A ∧ B ∈ α, A ∧ B ∈ β or A, B ∈ .In the first two cases A, B ∈ α or A, B ∈ β and so A, B ∈ in all cases.

Observation 2
and B ∈ δ.But we also have A ∈ and B ∈ δ in the former case.If the step is an instance of L∼∼, A, Γ C ∼∼A, Γ C L∼∼ then either ∼∼A ∈ α, ∼∼A ∈ β or ∼A ∈ .It is easy to see that A ∈ in the first two cases.In the last case, either ∼ A ∈ α or ∼ A ∈ β or A ∈ .Thus A ∈ in this case as well.If the step is an instance of R∼∼, Γ A Γ ∼∼A R∼∼ then ∼∼A ∈ δ and so either ∼A ∈ γ or ∼∼A ∈ α.In each case, it holds that A ∈ δ.If the step is an instance of L∼∨, ∼A, ∼B, Γ C ∼(A ∨ B), Γ C L∼∨ then either A, B ∈ β, A, B ∈ α or A, B ∈ .Hence in any case, ∼A, ∼B ∈ .If the step is an instance of R∼∨, Γ ∼A Γ ∼B Γ ∼(A ∨ B) R∼∨ then either A, B ∈ γ or A, B ∈ β.Hence ∼A, ∼B ∈ δ. 123 If the step is an instance of L∼ ∧ or R∼ ∧, the arguments are similar to the cases for L∼ ∨, R∼ ∨.If the step is an instance of L∼ →, ∼(A → B), Γ A ∼B, Γ C ∼(A → B), Γ C L∼→ then A ∈ α and B ∈ ; so A ∈ δ and ∼B ∈ .If the step is an instance of R∼→, A, Γ ∼B Γ ∼(A → B) R∼→ then A ∈ and B ∈ γ or B ∈ β.In either case, ∼B ∈ δ.Let Γ ⊆ and C ∈ γ .Then G3C3at Γ ∼C implies G3C Γ ∼C.

Lemma 2 Observation 3
Suppose G3C Γ C, where Γ ⊆ and C ∈ δ .Then for any sequent Γ D in the derivation, it holds that Γ ⊆ and D ∈ δ .Let Γ ⊆ and C ∈ δ .Then G3C3at Γ C implies G3C Γ C. Proof The lemma is shown analogously to Lemma 1. Then the observation follows by noting (a) that G3C3at Γ C implies G3C Γ, ∼p 1 , . . ., ∼p n C for some p 1 , . . ., p n ; and (b) ∼p cannot appear as a succedent formula in the latter derivation.

Fig. 1
Fig. 1 Relationship among the arrows
the argument is similar: Assume, w.l.o.g., that B and ∼ B. By the induction hypothesis, there is an ap D of B, such that D → E and D →∼ E, for some E with at