Connexive Negation

. Seen from the point of view of evaluation conditions, a usual way to obtain a connexive logic is to take a well-known negation, for example, Boolean negation or de Morgan negation, and then assign special properties to the conditional to validate Aristotle’s and Boethius’ Theses. Nonetheless, another theoretical possibility is to have the extensional or the material conditional and then assign special properties to the negation to validate the theses. In this paper we examine that possibility, not suﬃciently explored in the connexive literature yet.We oﬀer a characterization of connexive negation disentangled from the cancellation account of negation, a previous attempt to deﬁne connexivity on top of a distinctive negation. We also discuss an ancient view on connexive logics, according to which a valid implication is one where the negation of the consequent is incompatible with the antecedent, and discuss the role of our idea of connexive negation for this kind of view.


Introduction
According to "the now standard notion of connexive logic" (cf.[59]), a logic L is connexive if it validates the following schemas, where N and > are a negation and an implication, respectively, in the underlying language of L: and, moreover, it invalidates the following one: (Symmetry of Implication) Derivatively, any > that satisfies these requirements is called 'connexive'.Although it can be disputed whether meeting all these conditions is either necessary or sufficient for connexivity, 1 we will assume that it is sufficient and, accordingly, we will call collectively the schemas above 'the connexive schemas'.
The usual procedure to obtain a connexive logic is to take a negation with standard truth and falsity conditions, for example, Boolean negation-as in Angell-McCall's CC1; see [31]-or de Morgan negation-as in Wansingstyle connexive logics; see [41,58]-, and then modify the (standard) truth or falsity conditions of a conditional -as the intuitionistic conditional (see [58]) or the material conditional (see [59])-to validate the schemas above.Nonetheless, another theoretical possibility is to take a conditional with standard truth and falsity conditions -like the extensional or the material conditional-and then modify the (standard) truth or falsity conditions of a negation to validate the schemas above.In this paper we examine this latter possibility, that has not been sufficiently explored in the connexive literature yet.
Thus, just like a > that validates the connexive schemas (in the company of a 'standard' negation) is called a 'connexive conditional' or 'connexive implication', any N that validates the connexive schemas in the company of a 'standard' conditional will be called a 'connexive negation'.True, in many cases further discussion is needed to make sure that the connexive connective, whether a conditional or a negation, belongs in fact to the intended category, that is, that it in fact is a conditional or a negation.We will devote some time to that discussion in due course, in Section 4.
The closest antecedent to the idea that interests us has been the considerations about the account of negation as cancellation, explored in [57] and [46].Nonetheless, such an account of negation does not leave other notions untouched.In particular, the conditional is required to be neither extensional nor material.Actually, what precludes contradictions to imply anything (different from a contradiction) in Priest's proposal is not a distinctive evaluation condition for negation, but either the modified notion of logical consequence or the intensional conditional proposed.
Our general motivation is lead by the validation of the connexive schemas by means of a suitable negation.What is going to count as a negation will not be delivered by a theory of negation occurring in a natural language, but rather by examining the evaluation conditions of the relevant unary connectives.In a way, our project can be seen as more general than the account of negation as cancellation, since it is also disentangled from the particularities of the properties of negation interacting with other connectives. 2he plan of the paper is as follows.In Section 2 we set the technical preliminaries for the rest of the paper.In Section 3, we introduce four families of negations and, in Section 4, we discuss whether our negations can achieve the status of connexive negations.In Section 5 we compare some properties of these connectives with those of negation under the cancellation account.Finally, in Section 6 we examine whether our negations can be used to define compatibility connectives that meet the requirements set out in [45].

Technical Preliminaries
Let NEG be a family of negations, with a negation denoted generically by N i -or even without the subscript if there is no risk of ambiguity-, CON D a family of conditionals, with a conditional denoted generically by > j -again, we might omit the subscript if the context allows so-and CON J a family of conjunctions, with a conjunction denoted generically by k .Let L {N i ,> i } be a propositional language with a denumerable set V ar of propositional variables, and a finite set {N i , > j } of connectives.The set of formulas F ORM is defined on L {N i ,> i } in the usual way.We use upper case letters A, B, C and so on, as meta-variables ranging over arbitrary formulas, and lower case letters p, q, r and so on, as meta-variables ranging over arbitrary propositional variables.
We express an expansion of a language by indicating at subscripts the symbols added.For instance, we write L k to denote {N i , > j , k }.The set of formulas is then defined on this expanded language as usual.In many cases below, the exact shape of a language, and therefore of a logic, will be left implicit and will be indicated by the de facto use of connectives.
Let σ : V ar −→ {{1}, {1, 0}, { }, {0}} be a valuation function from the set of propositional variables to the set {{1}, {1, 0}, { }, {0}}.(Thus, propositional variables are interpreted in terms of sets of truth values.)Valuation functions are extended to the set F ORM according to some fixed conditions, depending on the particular connectives one is dealing with.As usual, we write Σ |= L A if and only if for all B ∈ Σ, if 1 ∈ σ(B) then 1 ∈ σ(A).For simplicity, in this paper we use classical logic in the meta-theory.

Definition 2.1. A Dunn atom is an expression of the form
If it is of the first form, we will say that it is a positive Dunn atom; in the latter case, we will say that it is negative.
We observe that valuations can be rewritten in terms of Dunn atoms as follows: (And we will assume that the relation of being a Boolean counterpart is symmetric.)For instance, the following cases-horizontal-wise-are Boolean counterparts of each other: Definition 2.3.A tweaking is a modification in the evaluation conditions of a connective where the only changes are substituting at least one Dunn atom by its Boolean counterpart.
Consider the evaluation conditions for the connectives in FDE, as they will be useful throughout the paper: As an illustration of tweakings, consider negation as evaluated in FDE in the upper left corner and three connectives obtained by changing at least part of its evaluation conditions: For example, the conditional defined through the following evaluation conditions3 is redundant from the classical logic perspective, for at the right of the 'iff' there are Dunn atoms that could be eliminated without loss (in classical logic): Note that already the first clause in the truth condition is redundant from the classical point of view, as it states that one of the options for the A → ϕ B to be true is that the antecedent is false but not true, but these two claims amount to the same thing in classical logic.We present here its truth table for self-containment: Definition 2.5.We will say that a connective is a classically clear case of negation/conjunction/disjunction/conditional if 1. its evaluation conditions are those of negation/conjunction/disjunction/ conditional in the logic FDE; or 2. its evaluation conditions are obtained from these by tweaking them; or 3. its evaluation conditions are classically redundant.
In the first two cases, the evaluation conditions will be called non-redundant. 4or example, the four negations above are clear cases of negations.Note that ∼ and ∼ 3 are de Morgan negation and Boolean negation, respectively.
For definiteness, in this paper we will employ the following conditionals: We refer to → and → m , respectively, as the extensional and material conditional, and to → AM and → W , respectively, as the AM-conditional (for Angell-McCall ) and W-conditional (for Wansing).We call 'extensional' all the usual connectives definable through positive Dunn atoms only.The 'material' conditional is, in a sense, the weakest and most intuitive modification of the extensional conditional that delivers the usually expected properties of a conditional, such as Identity-A > A-or Detachment-If A > B and A, therefore, B-.Note that the extensional conditional is definable as ∼ A∨B, and the material conditional is definable in a similar way but with another negation, ¬ b below, which is not available in the language of FDE.
These conditionals have the following evaluation conditions5 : According to the definitions given above, only the first and second connectives are clear cases of conditionals because the evaluation conditions of → are those of the conditional of FDE, which is definable in that logic in terms of (de Morgan) negation and disjunction, and the evaluation conditions of → m are obtained by tweaking the truth condition of the conditional of FDE.Thus, the evaluation conditions of → and → m are classically non-redundant.We also consider the following negations: Their evaluation conditions correspond, from left to right, to the evaluation conditions of the negations ∼ A and ∼ 3 A presented above, but for reasons that will be given later, we will use the notation of these last tables.These connectives are clear cases of negation because the evaluation conditions of ∼ are those of the negation of FDE, and the evaluation conditions of ¬ b are obtained by tweaking both the truth and falsity conditions of the negation of FDE.Thus, the evaluation conditions of ∼ and ¬ b are classically nonredundant.
Finally, we also consider the following connectives: These connectives have the following evaluation conditions: Only the first connective is a clear case of conjunction, as its evaluation conditions are classically non-redundant.Its evaluation conditions are those of the conjunction of FDE.Even though ∧ AM it is not a clear case of conjunction, one can make a case for its being a conjunction.Its truth condition is that of the extensional conjunction of FDE.On the other hand, A ∧ AM B is untrue if and only if A is untrue or B is untrue.This will be enough for us to treat '∧ AM ' as a conjunction as well. 6s we mentioned in the introduction, the usual procedure to obtain a connexive logic is to take a well-known negation, Boolean negation or de Morgan negation, for example, and then change the evaluation conditions of a familiar conditional to validate the connexive schemas.Consider the approach by Angell [1] and McCall [31].They aim at obtaining a (negation-) consistent connexive logic, and it can be easily seen that they keep Boolean negation untouched.Then, they work with the AM-conditional, but to further avoid contradictions, they also need a special conjunction.Thus, they work with the language L {¬ b ,∧ AM ,→ AM } .On the other hand, Wansing in [58] works on top of Nelson's N4, keeping de Morgan negation and as much as possible of the conditional.Else, working on top of material logic in [59], Wansing keeps de Morgan negation and as much as possible of the material conditional → m .In fact, the W-conditional presented above is like the material conditional, with the exception that the falsity condition of the latter, i.e.
is changed to the following one: Moreover, in Wansing's approach, (negation-)inconsistency is not a problem, only non-triviality is aimed at.

Presenting Classical Negation, Anti-Classical Negation, Falsity Negation and Truth Negation
So much for the preliminaries.In this section, we discuss four families of negation-related connectives in order to approach the topic of negationdriven connexivity.According to [6] and [43], there is no unique unary connective one can expand the language of FDE with, and that can be properly called 'classical negation'.In this context, classical negation can be represented by any of the following sixteen connectives7 : For the sake of simplicity, let us suppose for the moment that the distinctiveness of classical negation, represented here by ¬, is the following truth condition: This is the truth condition of Boolean negation, the connective ¬ b given in the preliminaries and in the tables above.It is is easy to see that all the unary connectives just introduced have this truth condition.But one can overcome the bias towards truth, put forward a dual idea and say that the distinctiveness of classical negation is the following falsity condition: that is the falsity condition of Boolean negation.Then one can obtain sixteen new unary connectives.With the only exception of ¬ b , we distinguish these connectives from the previous ones by inverting the symbol for Boolean negation and putting it backwards, i.e., A, attaching the corresponding subscripts8 : Instead of the evaluation conditions of Boolean negation, one can inquire into the evaluation conditions of de Morgan negation.Recall its truth and falsity conditions: Each condition defines again two groups of sixteen unary connectives.As in the previous cases, in what follows we use ∼ with appropriate subscripts to denote the connectives that have the truth condition of de Morgan negation and , with appropriate subscripts, to denote the connectives that have the falsity condition of de Morgan negation 9 :

Unary connectives with the falsity condition of de Morgan negation
After the study of connectives with the same truth condition as Boolean negation presented in [6], it was just a matter of time to consider connectives with the truth condition of de Morgan negation, and then also to dualize the approach and focus on falsity conditions.This was done independently in [38,39] and [44].In the latter, and although they do not cover the connectives based on falsity conditions, plenty of results about the ∼ i 's added to FDE, K3 and LP are presented.
According to Avron [2, p. 160], the truth condition of de Morgan negation "represents the idea of falsehood within the language", that is, the negation of A expresses that A is false.If this is correct, by parity of reasoning, the falsity condition of de Morgan negation would "represent the idea of truth within the language" in the sense that a formula A expresses its falsity, and when one negates it, one obtains its truth. 10efinition 3.1.We will say that a unary connective N is negative if 1. it has the truth or falsity condition of a clear case of a negation; or, 2. its evaluation conditions have the following structure: where condition 1 and condition 2 are Dunn atoms, either condition 1 or condition 2 is the truth or falsity condition of a clear case of negation, and condition 1 does not contradict condition 2 .
Thus, all the tables given above in this section are tables of negative connectives as they meet these requirements.Now, according to the terminology introduced in the preliminaries, not all negative connectives are (classically) clear cases of negations.There are twelve clear cases of negations, given by ¬ b (or b ), ¬ e (or e ), ¬ 1 , ¬ 2 , e (or ∼ e ), 1 , 2 , ∼ m (or m ), ∼ 1 , ∼ 2 , 1 and 2 .The connectives ¬ b , ¬ e (or e ), e (or ∼ e ) are obtained by tweakings, and the evaluation conditions of the other connectives are classically redundant.11

The Idea of a Connexive Negation
Remember that, under the now standard notion of connexive logic, there is a negative condition to be met to ensure that we are working with a conditional and not with a biconditional.However, that might not be enough, because there are other binary connectives that can be confused with a connexive conditional, in particular, conjunction.Omori has argued in [40] that the alleged conditional in MRS P , studied in [10] and [14], is in fact a conjunction.Therefore, some extra conditions are required in order to properly characterize a conditional.Likewise, there are a few extra negative conditions to be met to ensure that we are working with a negation and not with any other unary connective, especially a unary connective that delivers formulas that are always true.
According to some recent theories (see for instance [41], [47,Ch. 4]), a negation has to form contradictory pairs, i.e., A and NA have to be contradictories.Recall that A and B are contradictories iff 1 ∈ σ(A) iff 0 ∈ σ(B) and 0 ∈ σ(A) iff 1 ∈ σ(B). 12In such an account, a negation has to meet the following negative conditions: (Non-verifier) There is an A such that A |= NA (Non-antiverifier) There is an A such that NA |= A and the following positive conditions, for any formula A: (Marcos [28] considered the negative conditions as minimal requirements for a negation in any logic that is non-degenerate, such as empty or overfilled logics.)As we said above, the negative conditions make sure that negation is not any other unary connective, similarly as the invalidation of Symmetry of Implication is meant to secure that the conditional is not a biconditional.Since the consequence relation is truth-preserving, the negative conditions ensure that we are not speaking of other disguised unary connectives ∓, ± and ± ∓, with the following evaluation conditions: ∈ σ(A) (that is, ∓A is always true, or equivalently it is true under any interpretation.) 13  (Antaut) 1 / ∈ σ(± A) iff 1 ∈ σ(A) or 1 / ∈ σ(A) (that is, ∓A is never true, or equivalently it is not true under any interpretation.) for any i ∈ {1, 0}.In other words, unary connectives that, when they are the main connective in a formula, make it true under any interpretation-this condition is obtained by (Non-verifier)-, or that make it untrue under any interpretation -this condition is obtained by (Non-antiverifier)-or that do not alter its evaluation-this condition is obtained by both (Non-verifier) and (Non-antiverifier). 14 Now, we consider the case where the extensional conditional → appears uniformly in the connexive schemas, i.e.: and the case where the material conditional → m appears uniformly in the connexive schemas: 13 It might be replied that a connective that is always true might be a negation.Such always-true connective has the falsity condition of a negation and satisfies some properties that one can consider as pertaining to any negation.What one would actually require is a condition that ensures that one is not speaking of a unary connective which is just true under any interpretation, but we will not press this point further and assume (Non-verifier) as a necessary condition for a negation. 14In fact, as we are concerned with validating the connexive schemas-and not antiverifying them, making them untrue in all interpretations-we are already assuming (Antaut).
No unary connective presented in Section 3 can validate all the connexive schemas in the company of the extensional conditional and also reject either (Taut), (Antaut) or (Equa).
Proof.For the countermodels below we will ignore 5 and 5 since they do not reject (Non-verifier).
To put it boldly, granting the extensional conditional, no negative connective is a connexive negation.Things are different with the material conditional, as one can validate the connexive schemas using some negative connectives from Section 3 that also reject (Taut), (Antaut) or (Equa).
Proposition 4.2.The only connectives from Section 3 that can stand in place of N to validate the connexive schemas in the company of the material conditional and also reject (Taut), (Antaut) or (Equa) are 8 , 9 , 7 and 8 .
Proof.We provide countermodels to all the connectives that are not 8 , 9 , 7 and 8 ; that these validate the connexive schemas can be easily verified through truth tables.
8 , 9 , 7 and 8 fail to satisfy the positive requirement (EDN)-just consider the case where A = {0}-, yet all of them satisfy (IDN).However, if the problem is to keep both principles of double negation, one can verify that these connectives satisfy at least the following weak versions: Thus, any of 8 , 9 , 7 and 8 can do the job of being a connexive negation with the material conditional.Note that, as we said in the previous section, these connectives are not clear cases of negations.Strictly speaking, they are non-(classically)-clear cases of connexive negations.
As it is known at least since the works by Dunn [7] -see also [8,49,53]-, the double negation principles are relatively strong, so demanding them for a connective to be a negation might be too much.We agree, the negative requirements suggested by Marcos or even Contraposition in Dunn's kite: would be enough.Nonetheless, several of our unary connectives satisfy the stronger requirements, and those that do not, satisfy the weaker versions that we have just mentioned-even Contraposition-so we do not think it is necessary to consider weaker principles here.Of course, that would be interesting in other contexts where the expressible unary connectives are weaker than those presented here.Finally, it goes without saying that Sylvan (see [57, p. 312]) considered the possibility of requiring that a negation satisfied other principles, among them the connexive schemas; our negations would meet this requirement by default.
Let us make clear our position about the previous result.That certain unary connectives are not (classically)-clear cases of negations is not a negative result in itself.We have shown that the previous connectives meet the minimal requirements for being a negation.If we wanted to say something more about the nature of such connectives, especially by comparing them to other connectives, we would need to be more cautious about the underlying notion of logical consequence.Commonly, the standard properties of a connective depend heavily on the consequence relation adopted; this is particularly important to determine whether there are certain relations holding among connectives and to evaluate whether a connective is similar to another.For example, it has recently been argued-see [9], [15]-that certain connectives, like transplication 15 , are closer to its intended connective-in this case, an implication-when the entailment relation is not Tarskian, in particular, when it is non-transitive, because with a Tarskian consequence relation it seems more a conjunction.
In general terms, the idea is that, in order to clarify the nature of a connective by comparing it with others, we probably would need to verify whether a clear case of a given connective-say, conjunction-entails or is entailed by a non-clear case of that connective.Until now, we assumed the usual Tarskian consequence relation.But this is not necessary.Let us suppose that the underlying consequence relation is p-consequence: Then, one can make sense more easily of some of the previous unary connectives as negations, as some of them entail or are entailed by a clear case of negation.A more detailed and systematic description of the interaction of different consequence relations, connectives, and evaluation conditions would be needed, though. 16We leave this point aside and continue using the usual Tarskian consequence relation.
Let us note in passing that these unary connectives do not exclude a connexive conditional, in the sense that there are languages containing any of these four negative connectives and either Wansing's conditional or Angell-McCall's conditional that validate the connexive schemas.Here we summarize the combinations that deliver the validity of the connexive schemas: 15 A connective introduced by that name has been introduced in [5] through a table structurally identical to the following one: Nonetheless, this connective appeared earlier, in [48], and it is nowadays more well-known as the de Finetti conditional, see [9] and the references therein. 16A detailed discussion of this issue can be found in [17].
Definition 4.1.A negation N is connexively stable with respect to a family of connexive conditionals for any Since it is already assumed that > i is a connexive conditional, there is no need to mention the Non-Symmetry condition. 17 With a toy but still illustrative example, consider the case where the family of connexive conditionals has only two members: Wansing's and Angell-McCall's.Then, only 9 is connexively stable with respect to that family of connexive conditionals.For this reason we express our preference for the negation 9 .
The results in this section suggest the following picture about connexivity: it depends on both negation and the conditional, not only on one of them.This is not news.What is new is the right emphasis on both.In the usual approaches, the conditional plays the most important role; it almost seems that what is at the heart of connexivity is the conditional.And in this context, that view seems to be supported by Proposition 1.The extensional conditional is not regarded by many as a real conditional, and it would be no wonder that, as per Proposition 1, none of the negations produce connexive validities because we did not have a conditional to begin with, it was merely the extensional disjunction in disguise.
Nonetheless, we have also shown that, if some specific negations do not accompany a conditional better than the extensional conditional, it might not validate the connexive schemas, either.The example in this case is the conditional by Angell-McCall, that validates the connexive schemas only with some negations.The reply here is that the Angell-McCall conditional is rather a biconditional.Thus, in the end it would not be better than the extensional conditional, so the proof that our negations do not deliver the connexive schemas when combined with the Angell-McCall conditional carries little weight.We do not agree with the claim that the Angell-McCall conditional is not a conditional, but let that pass.
Someone could say that the Wansing conditional in fact shows that what is at the heart of connexivity is the conditional, because it is stable under changes of negation.This is a moot point, though.The Wansing conditional internalizes, so to speak, a particular conception of negation-at least, of the negation of a conditional-in its falsity condition.Thus, it is not entirely clear that in this case the conditional is more important than negation.What would be in need of explanation is why such a conception of negation internalized in the falsity condition is connexively stable under changes of negation.We admit not having an answer to that.

Our Connexive Negations and the Negations as Cancellation Account
A rather different approach to obtaining a connexive logic by looking at the negation, instead of the conditional, has been associated with the cancellation account of negation.The cancellation account of negation is the view according to which the pair A, N A has no content, since the contents of A and NA "cancel" or "erase" each other.(See [50][51][52].)In all fairness, nobody knows what 'content' amounts to here, but let us assume that the contents of A and NA cancel each other.If one couples the cancellation account of negation with the idea that a valid implication is one in which the content of the consequent is included in the content of the antecedent, and makes explicit that all propositions have some content-as the ancient and medieval thinkers might have assumed (see [55])-then neither A can imply NA nor vice versa, since the consequents in those implications cancel the contents of their antecedents.For the same reason, if A implies B, NB cannot be part of the content of A. Thus, one arrives at the connexive schemas.Among the distinctiveness of the cancellation account of negation is the invalidation of Explosion (A NA) > B, and Simplification, (A NA) > A or (A NA) > NA, because, as explained above, the antecedents cancel themselves, so there is nothing that implies the consequent.Additionally, the invalidation of Simplification is motivated by the fact that in the presence of (B2e), Simplification leads to inconsistency.Wansing and Skurt [60, p. 482] also suggest that the cancellation account of negation can motivate Abelard's First Principle, N ((A > B) (A > NB)) and Aristotle's Second Thesis, N ((A > B) (NA > B)).That is so because the former involves the negation of instances of Simplification, ((B NB) > B) ((B NB) > NB)), and dually, the latter involves the negation of instances of Addition, both of which are rejected in the cancellation account, as we have remarked above. 18owever, in [60] the authors pointed out several conceptual problems with the cancellation account of negation in the Routleys' version, and have argued that it is better not to consider it a conceptual basis of connexive logic.They observe that if the cancellation account of negation were correct, A NA would have no content, but then the external N in N (A NA) would be operating upon no content at all.However, this goes against the assumption that all negations operate over some content.Moreover, N (A NA) could not cancel A NA because A NA would cancel itself, so there would not be anything to cancel to begin with.
Our proposal is that one can obtain connexive logics by modifying the evaluation conditions of negations in such a way that, coupled with the evaluation conditions of a standard conditional, would deliver the validation of the connexive schemas, and that one can do so without assuming that such connexive negations represent some cancellationist ideas.Thus, in what follows we examine to what extent our connexive negations give similar results to the Routleys' idea of connexive negation.In particular, we want to know whether we can detach the idea of connexive negation from the failure of Simplification and the validation of Abelard's First Principle and Aristotle's Second Thesis. 19  Remark 5.1.Simplification is valid with either the material or the extensional conditional, Angell-McCall's conjunction ∧ AM or extensional conjunction ∧ e , and 8 or 9 : 20 However, by using the connexive negations 7 and 8 one obtains different results.All instances of Simplification are valid just with the material conditional: lead to the rejection of Addition A > (A⊕B) and B > (A⊕B) (see [60, p. 480]).Nowadays, it is widely agreed that connexivity has nothing to do with the failure of Simplification, much less of Addition, connexive logic being characterized as we reported at the beginning of Section 1. 19 These principles tend to divide connexivists.While in Angell-McCall's CC1 Simplification and Addition are invalid but Abelard's First Principle and Aristotle's Second Thesis are valid, in Wansing's C, Simplification and Addition are valid, but Abelard's First Principle and Aristotle's Second Thesis are not. 20In the following we use the notation (A {∧ AM ∧e} N { 8 , 9 } A) > {→,→m} N { 8 , 9 } A to indicate that from the connectives that appear between the formulas one can uniformly choose any of them.For instance, if one chooses 8 to appear in the antecedent, it also needs to appear in the consequent.

• |= (A {∧
The same schemas, but with the extensional conditional, are invalid, though.Just take A to be neither true nor false, { }: Finally, the problem of inconsistency mentioned above for Simplification in the light of (B2e) arises for certain combinations of conditionals, conjunctions and negations.
Remark 5.3.Finally, for any of our negations, the truth of NA does not exclude the truth of A, nor vice versa, since NA is true in the same conditions under which A is true, i.e., {1} or {1, 0}.This also has consequences on the validity of Explosion.Explosion is valid with either Angell-McCall's conjunction or extensional conjunction, the extensional conditional and the negations 8 and 9 : In all other cases, it is invalid:

Compatibility and Connexivity
According to a traditional view that can be traced back to Chrysippus (see [54] and the references therein), the connexive schemas arise from certain (in)compatibility relations between the antecedent and (the negation of) the consequent.In such a view, an implication is valid if and only if the negation of the consequent is incompatible with the antecedent.Let • denote a (binary) compatibility connective, such that (A • B) is read, for example, as "A is compatible with B", "A does not exclude B", "the realizability of A does not exclude the realizability of B" or "the truth of A does not exclude the truth of B".Then, according to the above, It is easy to check whether the compatibility connectives defined in this section meet Pizzi's requirements.Whereas the property of (Modal compatibility) had some purpose for Pizzi's discussion on consequential implicationan implication → ci interpreted in modal terms that validates so-called '(Weak) Boethius' Thesis' ((A → ci B) → m ¬ b (A → ci ¬ b B))-we leave it out in the consideration of our demodalized languages.The results are summarized in the following table: Thus, all the previous connectives pass the Pizzi test for being compatibility connectives.

Conclusions
In this paper we considered a common view of connexive logic according to which the validation of the so called "connexive schemas" (Aristotle's Thesis and its variant, and Boethius' Thesis and its variant) and the invalidation of Symmetry of Implication constitute sufficient conditions for a connexive logic and, derivatively, to have a connexive conditional.The standard way to obtain a connexive conditional is by taking a known negation and then modifying the conditional to validate the connexive schemas.We explored the converse situation: obtain a connexive negation by taking a known conditional-the extensional conditional or material conditional in this case-and then modifying the negation in such a way as to validate the connexive schemas.Our exploration led us to consider the cancellation account of negation, and compare its properties with our connexive negations.To round our analysis of the connexive negations, we connected the connexive schemas with the notion of (in)compatibility; we defined some connectives of compatibility by means of our connexive negations and discussed its adequacy with respect to some criteria given by Pizzi.
Finally, this sort of investigation opens some avenues for further research.One of the most obvious is whether the satisfaction of the connexive schemas is a sufficient condition for a non-classical negation be taken as connexive, or whether the shift of focus to negation makes necessary to consider some other conditions, even detached from conditionals.Another topic for future work is approaching the issue from the modal theories of negation; for example, using those in [3,4,29,[34][35][36][37].Finally, one could employ other semantic frameworks already used to study connexivity, for example, relating semantics (cf.[23][24][25][26][27]), and try to adapt them to focus connexivity on negation.Negation in general remains understudied in the relating framework, and this could be a good topic for further testing of the machinery.
, in company of 8 , 9 , 7 or 8 , and ∧ e or ∧ AM .With respect to the conditional →, the argument is valid in company of the negations 8 or 9 , and ∧ AM .However, it is not valid using 7 or 8 , and ∧ e or ∧ AM -just consider the case when σ(A) = { }. |= 9 ((A → m B) {∧ AM ,∧ e } (A → m 9 B)).All other cases are rendered invalid: 8 , 8 , 9 } A leads to inconsistency-following the standard proof by the use of modus ponens.However, something similar does not happen by using the extensional conditional, since it is possible to find countermodels to (A {∧ AM ,∧ e } N { 7 , 8 } A) → A-just consider the case when σ(A) = { }.The extensional conditional in the company of either 8 or 9 , and ∧ AM or ∧ e in the schemas above leads to inconsistency, notwithstanding.Remark 5.2.With respect to Abelard's First Principle and Aristotle's Second Thesis, only the former is valid with either Angell-McCall's or extensional conjunction, material conditional, and the negation 9 .