2.1 What speaks for connexive implication and co-implication?

Why should one prefer the (hyper)connexive reading of negated implications and co-implications, that is expressed in $\mathbf {2C}$ by

$$\begin{align*}\sim\kern-3pt (A \to B) \to (A \to \sim\kern-3pt B), \quad (A \to \sim\kern-3pt B) \to \sim\kern-3pt (A \to B), \end{align*}$$

being provable and

being refutable, over the Nelsonian (and classical) understanding of negated conditionals, that can be seen in $\mathbf {N4}$ by

$$\begin{align*}\sim\kern-3pt (A \to B) \to (A \wedge \sim\kern-3pt B), \quad (A \wedge \sim\kern-3pt B) \to \sim\kern-3pt (A \to B), \end{align*}$$

being provable?

This question asks for a motivation of (hyper)connexive logics. A detailed motivation goes beyond the scope of the present paper. One reason, see [Reference Wansing59], for considering connexive implication, $\rightarrow $ , and connexive co-implication, , instead of assuming the familiar understanding of negated implications in Nelson’s and other logics, is that one obtains a neat encoding of derivations in the language with $\rightarrow $ and by typed $\lambda $ -terms built up from atomic terms of two sorts, one for proofs and one for refutations (dual proofs), using only

1. (i) functional application and functional abstraction, and

2. (ii) certain sort/type-shift operations that turn an encoding of a dual proof of a formula A ( $\sim\kern-3pt A$ ) into an encoding of a proof of $\sim\kern-3pt A$ (A) and that turn an encoding of a proof of a formula A ( $\sim\kern-3pt A$ ) into an encoding of a dual proof of $\sim\kern-3pt A$ (A).Footnote ⁶

A more recent motivation for first-order connexive logic, in particular for a certain extension of $\mathbf {C}$ to first order, the system $\mathbf {QC}$ , is presented in [Reference Wansing63]. There the axiom $\sim\kern-3pt (A \to B) \leftrightarrow (A \to \sim\kern-3pt B)$ is justified by its use in proving the “drinker truism” and the “dual drinker truism” (cf. the “drinker paradox” in classical logic [Reference Smullyan54]):

(𝔻𝕋) $$ \begin{align} \sim\kern-3pt \exists x(P(x) \to \sim\kern-3pt \exists y P(y)), \end{align} $$

(“It is false that there is someone such that if she drinks, then it is false that someone drinks.”)

(𝔻𝔻𝕋) $$ \begin{align} \sim\kern-3pt \exists x(\forall y P(y) \to \sim\kern-3pt P(x)). \end{align} $$

(“It is false that there is someone such that if everybody drinks, it is false that she drinks.”)

The two formulas $\mathbb {DT}$ and $\mathbb {DDT}$ are valid in $\mathbf {QC}$ and have simple derivations in the sequent calculus G3C for $\mathbf {QC}$ . The provability of $\mathbb {DT}$ and $\mathbb {DDT}$ in the Hilbert-style proof system $\mathbf {HQC}$ for $\mathbf {QC}$ is guaranteed by the axiom $\sim\kern-3pt (A \to B) \leftrightarrow (A \to \sim\kern-3pt B$ ). With it, $\mathbb {DT}$ is provably interderivable in $\mathbf {HQC}$ with $\forall x {(P(x)\!\to \!\exists y P(y))}$ , and $\mathbb {DDT}$ is provably interderivable in $\mathbf {HQC}$ with ${\forall x (\forall y P(y)\!\to \! P(x))}$ .

The above reading of $\mathbb {DT}$ and $\mathbb {DDT}$ is rather non-idiomatic. Usually, the negated existential $\sim\kern-3pt \exists x P(x)$ is read as “No one drinks”. Under that reading, it seems that $\mathbb {DT}$ indeed ought to be valid: No one is such that if she drinks, no one drinks.

Another recent motivation for quantified (hyper)connexive logic is given in [Reference Olkhovikov38] and [Reference Wansing and Weber66].

2.2 On dispensing with $\bot $ and $\top $

Irrespective of whether $A \to \bot $ in $\mathbf {Int}$ is accepted as a negation connective, the presence of $\bot $ leads to a problem with the BHK interpretation of the intuitionistic connectives. Jean-Yves Girard [Reference Girard23] presents the BHK interpretation of $\mathbf {Int}$ as follows (notation and presentation adjusted):

1. 1. for atomic sentences, we assume that we know intrinsically what a proof is (for example, pencil and paper calculation serves as a proof of “ $27 \times 37 = 999$ ”);

2. 2. (a construction) c is a proof of $A \wedge B$ iff $c=\langle d,e\rangle $ , where d is a proof of A and e is a proof of B;

3. 3. c is a proof of $A \vee B$ iff $c=\langle i,d\rangle $ , where $i = 0$ and d is a proof of A, or $i = 1$ and d is a proof of B;

4. 4. c is a proof of $A \rightarrow B$ iff c is a function, which maps any proof of A to a proof of B;

5. 5. no construction is a proof of $\bot $ .

In general, the negation $\sim $ is treated as $A\rightarrow \bot $ . The BHK-interpretation has been criticized for this treatment of negation by Ingebrigt Johansson, Edgar López-Escobar, and others. López-Escobar [Reference López-Escobar28, p. 362 f.] remarks that

By the BHK-clause for negation, a proof of the intuitionistically valid $\sim\kern-3pt (A \wedge \sim\kern-3pt A)$ is a function f, which maps each proof $\pi $ of $A \wedge \sim\kern-3pt A $ to a nonexistent proof $f(\pi )$ of $\bot $ . Since $(A \wedge \sim\kern-3pt A)$ itself has no proof, any function whatsoever proves $\sim\kern-3pt (A \wedge \sim\kern-3pt A)$ , which seems fairly non-constructive. A similar criticism can be found in the work of George Griss. As Thomas Ferguson [Reference Ferguson18, p. 3] explains (notation adjusted), for Griss

López-Escobar [Reference López-Escobar28] suggested to supplement the BHK-interpretation of positive intuitionistic logic with the primitive notion of refutation to give a constructively acceptable interpretation for negation. As a result, and upon disregarding $\bot $ , one obtains a semantics that is sound for $\mathbf {N4}$ . López-Escobar gives the following refutation interpretation of the intuitionistic connectives $\wedge $ , $\vee $ , $\rightarrow $ , and the toggling negation $\sim $ (notation and presentation adjusted):

1. (i) c is a refutation of $A \wedge B$ iff $c=\langle i, d\rangle $ , where $i=0$ and d is a refutation of A or $i=1$ and d is a refutation of B;

2. (ii) c is a refutation of $A \vee B$ iff $c=\langle d, e\rangle $ and d is a refutation of A and e is a refutation of B;

3. (iii) c is a refutation of $A \rightarrow B$ iff $c=\langle d, e\rangle $ , d is a proof of A and e is a refutation of B;

4. (iv) c is a refutation of $\sim\kern-3pt A$ iff c is a proof of A.

In bilateral proof systems with such a pair of derivability relations for $\mathbf {N4}$ , its connexive variant $\mathbf {C}$ , and the bi-connexive logic $\mathbf {2C}$ , a direct interaction between the two relations comes with the primitive toggling negation connective, $\sim $ . The relationship is so tight that in the BHK-style interpretations of $\mathbf {N4}$ , $\mathbf {C}$ , and $\mathbf {2C}$ , proofs of A are seen as identical with refutations of $\sim\kern-3pt A$ and refutations of A are identified with proofs of $\sim\kern-3pt A$ .

Importantly, after discarding $\top $ and $\bot $ we still have access to some connectives similar to two definable negations of $\mathbf {2Int}$ and $\mathbf {2C}$ . We can define surrogates of verum and falsum constants as and and define the new negation and co-negation . These two connectives will play an important role in the paper.

2.3 On dispensing with the toggling negation

If assertion and denial are both primitive speech acts that are on a par, then asserting the negation of A can be achieved by denying A, and denying the negation of A can be brought about by asserting A. Similarly, if proofs and refutations are two primitive kinds of derivations that are on a par, a proof of the toggling negation of A amounts to a refutation of A, and a refutation of the toggling negation of A amounts to a proof of A, cf. the discussion in [Reference Wansing, Piecha and Wehmeier62]. Yet in the bilateral sequent calculi for $\mathbf {N4}$ and its modifications $\mathbf {C}$ and $\mathbf {2C}$ , there is already some interaction between proofs and refutation even without the toggling negation. Is there, then, indeed a need for the toggling negation?

There are various reasons for dispensing with the toggling negation, but most importantly for us is that we take proofs and refutations to be primitive notions and the toggling negation to be just a device for internalizing refutations within proofs and the other way around. Some authors, instead, take incompatibility to be the prime notion behind the concept of negation, see [Reference Berto9–Reference Brandom11, Reference Price42]. Among these [Reference Price42] is notable in that it is nevertheless quite close to our version of bilateralism and [Reference Berto9] in that it holds negation to be a modal operator: a view that seems to be incompatible with bilateralism. Let us discuss more specific reasons for abandoning the toggling negation.

In [Reference Drobyshevich13] Drobyshevich shows that given a relatively well-behaved semantics, the toggling negation can be introduced conservatively to a bilateral system but argues that this does not constitute a limitation for the bilateral approach. As a demonstration, in the same paper a few examples of non-trivial bilateral systems without the toggling negation are given—both in the form of sequent calculi and via semantics—including one for the bilateral negation-free fragment of $\mathbf {C}$ (confusingly, under the name $\mathbf {P2C}$ as opposed to, say, $\mathbf {PC}$ ). Sara Ayhan [Reference Ayhan6, sec. 3.2] takes an even stronger stance, saying that it is “preferable not to have [the toggling] negation as a primitive connective in the language since, in a way, it stands against the bilateralist idea that refutation (or denial, or rejection, etc.) is a concept prior to negation,” and “incorporating [the toggling] negation would mean to have a primitive connective that is basically expressing exactly what is expressed by our derivability relations.” She also points to the fact that in Nelson’s constructive logics, the toggling negation is non-congruential, namely, that it prevents provable equivalence from being a congruence relation for which a replacement theorem holds.

The toggling negation in $\mathbf {C}$ and $\mathbf {2C}$ is also non-congruential because, as in $\mathbf {N4}$ , for distinct propositional variables p and q, $p \rightarrow p$ and $q \rightarrow q$ are provably equivalent, whereas $\sim\kern-3pt (p \rightarrow p)$ and $\sim\kern-3pt (q \rightarrow q)$ are not. However, one could argue that, exactly from a bilateralist point of view, it is to be expected that provable equivalence fails to be and that provable strong equivalence is a congruence relation. Interreplaceability of A and B in all linguistic contexts requires that A and B are provably strongly equivalent, meaning that A and B as well as $\sim\kern-3pt A$ and $\sim\kern-3pt B$ are provably equivalent, i.e., that A and B are mutually provable and refutable.

Another issue of debate is the nestability of negation versus the non-nestability of speech acts and derivability relations.Footnote ⁷ We remind the reader that two negations $\cdot \rightarrow \bot $ and are definable in $\mathbf {2Int}$ . Ayhan [Reference Ayhan6, footnote 17] remarks that because of this, “an objection coming from a ‘Frege-Geach-point’ angle [ $\ldots $ ], that we need a negation in our language to express it in subclauses of sentences (where an interpretation as refutation would not suffice), does not seem to be a concern” for $\mathbf {2Int}$ . This gives rise to the question whether it nevertheless might be desirable to embed the toggling negation instead of negations $\cdot \rightarrow \bot $ and , and whether it is justified to refer to $\neg A$ defined as and $- A$ defined as as ‘negations’ of A. There is no general agreement about the properties that a one-place connective should have in order to be justifiably called a negation. In [Reference Marcos29] some minimal conditions for a unary connective $\ast $ to qualify as a negation are listed, the chief among them being that (with respect to provability) A cannot be derived from $\ast A$ and $\ast B$ cannot be derived from B for some A and B. In our bilateral setting with both proofs and refutations, we may require that either the above condition is satisfied or that with respect to refutability A cannot be derived from $\ast A$ and $\ast B$ cannot be derived from B for some A and B. According to this, both and can be considered as negations (we make these conditions more formal in Fact 3.5).

Irrespective of such a discussion, we believe that it is interesting to explore what effects dispensing with the toggling negation has for $\mathbf {B2C}$ . What are the results of letting the interaction between proofs and refutations emerge from the meaning of the binary connectives $\to $ and only? That is, we assume the relations of provability and refutability, internalize them by $\to $ and , moreover do without $\top $ and $\bot $ , and do not stipulate a direct back and forth between proofs and refutations enabled by the toggling negation. This approach of “better fewer, but better” allows us, at least, to take a fresh look back at the very connectives we abandoned. As we shall see later in §5 and 6, such a perspective can then be utilized in making an informed choice about restoring some of these connectives in the language, if one so desires. Note that the language of $\mathbf {B2C}$ is in some sense minimal if one i) considers the presence of conjunction, disjunction and implication as essential and ii) takes the desiderata of having proofs and refutations on par with each other very strongly. Observe that having a connective which internalizes the preservation of provability but no connective that internalizes the preservation of refutability can well be construed as a strong preference to provability over refutability. Thus having both implication and co-implication keeps things in perfect balance. Still, it is worth pointing out that as most of the results in our paper are non-definability result, they trivially hold for the co-implication-free fragment of $\mathbf {B2C}$ .

3.1 The natural deduction proof system $\mathbf {NB2C}$

The natural deduction proof system $\mathbf {NB2C}$ for $\mathbf {B2C}$ is obtained as a simplification and modification of the proof system $\mathbf {N2C}$ for $\mathbf {2C}$ from [Reference Wansing59]. Derivations in $\mathbf {NB2C}$ combine proofs and dual proofs (corresponding to refutations), which are differentiated by whether a single line (for proofs) or a double line (for dual proofs) is drawn above a formula. Moreover, a proof may contain dual proofs as subderivations, and a dual proof may contain proofs as subderivations.

First it will be convenient to introduce some notions. By a set-pair we will call a pair of sets of formulas denoted as $(\Delta ;\Gamma )$ . For two set-pairs $(\Delta ;\Gamma )$ and $(\Delta ';\Gamma ')$ we will write $(\Delta ;\Gamma )\leq (\Delta ';\Gamma ')$ if $\Delta \subseteq \Delta '$ and $\Gamma \subseteq \Gamma '$ . In writing set-pairs we will sometimes replace the union sign with a comma and drop curly brackets over finite sets so that, for instance, $(\Delta ,A_1,\dotsc ,A_n;\Gamma ,\Gamma ')$ denotes $(\Delta \cup \{ A_1,\dotsc ,A_n \};\Gamma \cup \Gamma ')$ . We say that a set-pair is finite if both sets in it are finite.

The conclusions of proof and dual proofs depend on finite set-pairs $(\Delta ;\Gamma )$ of premises: a set $\Delta $ of assumptions that are taken to be provable, and a set $\Gamma $ of counterassumptions that are taken to be refutable. Single square brackets [ ] are used to indicate assumptions that may be discharged, and double-square brackets are used to indicate counterassumptions that may be discharged. We usually write $[A]$ instead of and instead of . Note that we allow for the empty discharge of assumptions and counterassumptions. Finally, in the statement of derivation rules dotted horizontal lines are sometimes used: these stand uniformly for either provability or dual provability in each instance of the rule.

Definition 3.1. We consider as a proof of A from $(\{A\}; \varnothing )$ and as a dual proof of A from $(\varnothing ;\{A\})$ . In addition to these stipulations, the system $\mathbf {NB2C}$ comprises the introduction and elimination rules from Tables 2 and 3. In the names of the rules E stands for “elimination from”, I for “introduction into”, p for “proofs”, and dp for “dual proofs”. We write $(\Delta; \Gamma ) \vdash A$ if there is a proof of A from $(\Delta; \Gamma )$ ; and we write $(\Delta; \Gamma ) \vdash ^d A$ if there is a dual proof of A from $(\Delta; \Gamma )$ . We say that a formula A is provable (in $\mathbf {NB2C}$ ) if $(\varnothing ;\varnothing ){\mathbin {\vdash {A}}}$ and refutable (in $\mathbf {NB2C}$ ) if $(\varnothing ;{\varnothing }){\mathbin {\vdash ^{d}}}A$ .

Table 2

Introduction and elimination rules of $\mathbf {NB2C}$ with respect to proofs

Table 3

Introduction and elimination rules of $\mathbf {NB2C}$ with respect to dual proofs

Note that the rules $(\vee Ep)$ and $(\wedge Edp)$ generalize those from [Reference Wansing59]. Such generalized rules can also be found in the term-annotated natural deduction proof system for $\mathbf {2Int}$ in [Reference Ayhan6]. In the latter system the generalized rules are derivable by means of the negation and co-negation operations of $\mathbf {2Int}$ .

We make some observations regarding $\mathbf {NB2C}$ .

The logics $\mathbf {C}$ and $\mathbf {2C}$ are non-trivial negation inconsistent logics, see [Reference Niki and Wansing35]. For example, $(A \wedge \sim\kern-3pt A) \rightarrow \sim\kern-3pt (A \wedge \sim\kern-3pt A)$ and $\sim\kern-3pt ((A \wedge \sim\kern-3pt A) \rightarrow \sim\kern-3pt (A \wedge \sim\kern-3pt A))$ are provable in both systems. Meanwhile, in $\mathbf {2C}$ the formulas and are both dually provable. Moreover, $\mathbf {2C}$ has an even more unorthodox property: the co-negation is provable and the negation $A\rightarrow \bot $ is refutable for any formula A. A similar observation can be made regarding $\mathbf {NB2C}$ ; since the empty discharge of assumptions is permitted,Footnote ⁸ we have the following:

Since there are provable formulas in $\mathbf {NB2C}$ (e.g. ), the first item of the last fact implies that $\mathbf {NB2C}$ is negation inconsistent with respect to ${-}$ : there is a formula A such that both A and ${-}A$ are provable. Moreover, it even describes how all provable contradictions of this kind in $\mathbf {NB2C}$ look like; see [Reference Niki and Wansing35] for a similar inquiry. Similarly, the second item implies a kind of incompleteness statement with respect to $\neg $ : there is a formula A (e.g. ) such that both A and $\neg A$ are refutable. From the perspective of classical logic this would also constitute a kind of contradiction albeit of a different kind. Note that these two kinds of contradictions are equivalent if the negation in question is the toggling negation. There is another sense in which $\mathbf {B2C}$ is inconsistent, which is unique to the bilateral presentation of the system; namely

Proof. One can take as demonstrated by the following derivations:

Similarly, one can take .

In $\mathbf {N2C}$ the following dual versions of Aristotle’s and Boethius’ theses are refutable:

1. dAT ,

2. dAT′ ,

3. dBT ,

4. dBT′ .

Moreover, is not refutable in $\mathbf {N2C}$ . We can then say that a logic is dually connexive (w.r.t. $\sim $ ) if its co-implication connective satisfies this non-symmetry of co-implication and if it allows one to derive dAT–dBT $'$ for the derivability relation internalized by co-implication.

3.2 Faithful embedding of $\mathbf {B2C}$ into positive intuitionistic logic

The connexive logic $\mathbf {C}$ is faithfully embeddable into positive intuitionistic logic, $\mathbf {Int^{+}}$ , under a single translation function, $\tau $ . This is possible because the toggling negation toggles between provability and refutability, so that, in particular, a refutation of a formula A is a proof of the toggling negation $\sim\kern-3pt A$ of A. In the absence of the toggling negation, the separate treatment of proofs and refutations requires the use of two translation functions, $\tau ^+$ and $\tau ^-$ .

If $\Delta \subseteq \mathsf {Form}\,\mathcal {L}$ is finite, then $\tau ^*(\Delta ) = \{A^* \mid A \in \Delta \}$ , for $* \in \{+, -\}$ . Taking rules $(\circ Ip)$ and $(\circ Ep)$ for $\circ \in \{ \wedge ,\vee ,\rightarrow \}$ from Table 2 and restricting them to empty sets of counterassumptions and, moreover, restricting $(\vee Ep)$ so that only a single line is permissible above C, results in a notational variant of a natural deduction proof system $\mathbf {NInt^+}$ , for $\mathbf {Int^+}$ . We write $\Delta \vdash A$ if there is a proof of A from $\Delta $ in $\mathbf {NInt^+}$ .

Proof. The directions from right to left are obvious because the rules of $\mathbf {NInt^+}$ are restricted versions of the proof rules of $\mathbf {NB2C}$ . For the directions from left to right, the proof is by simultaneous induction on derivations in $\mathbf {NB2C}$ . As to the induction bases, i.e., the cases $(\varnothing; \{A\}) \vdash A$ and $(\{A\};\varnothing ) \vdash ^d A$ , we have $\tau ^*(A) \vdash \tau ^*(A)$ . Simultaneous induction is needed for the cases of rules that make use of both proofs and refutations. We present two cases for claim (1) and derivations ending in a proof.

. Suppose

By the two induction hypotheses, we have $\tau ^+(\Delta ) \cup \tau ^- (\Gamma ) \vdash \tau ^-(A)$ and in $\mathbf {NInt^+}$ . Since = $\tau ^- (A) \rightarrow \tau ^+ (B)$ , we obtain in $\mathbf {NInt^+}$ :

$(\wedge Edp)$ . Suppose

By the two induction hypotheses, we have $\tau ^+(\Delta ) \cup \tau ^- (\Gamma ) \vdash \tau ^-(A\wedge B)$ , $\tau ^+(\Delta ') \cup \tau ^- (\Gamma ') \cup \{\tau ^-(A)\} \vdash \tau ^+ (C)$ , and $\tau ^+(\Delta ") \cup \tau ^- (\Gamma ") \cup \{\tau ^-(B)\} \vdash \tau ^+ (C)$ in $\mathbf {NInt^+}$ . Since $\tau ^-(A \wedge B)$ = $\tau ^- (A) \vee \tau ^- (B)$ , we obtain in $\mathbf {NInt^+}$ :

The reasoning for claim (2) is similarly straightforward.

The functions $\tau ^+$ and $\tau ^-$ could be used to complement the syntactical embedding from Theorem 3.1, by a semantical embedding along the lines of the semantical embedding of first-order $\mathbf {C}$ into first-order $\mathbf {Int^+}$ in [Reference Wansing, Schmidt, Pratt-Hartmann, Reynolds and Wansing57], in order to obtain an embedding-based completeness proof for $\mathbf {NB2C}$ . The embedding of $\mathbf {B2C}$ into $\mathbf {Int^+}$ clarifies the non-trivial negation inconsistency of $\mathbf {NB2C}$ . Although for every provable formula A, $-A$ is provable and for every refutable formula A, $\neg A$ is refutable, neither is every formula provable in $\mathbf {NB2C}$ nor is every formula refutable in $\mathbf {NB2C}$ . Moreover, under both translations the formula , for example, that is both provable and refutable in $\mathbf {NB2C}$ , is mapped to a theorem of $\mathbf {Int^+}$ :

In the next section we shall present a direct completeness proof for $\mathbf {NB2C}$ . This will demonstrate that our bilateral framework is robust enough to obtain the usual completeness proof via prime theories even in the absence of the toggling negation.

3.3 Semantics

The following semantics for B2C is obtained by simply dropping the missing connectives from the semantics for $\mathbf {2C}$ [Reference Wansing59].

Definition 3.3. A model $\mathcal {M}$ is a structure $\langle I,\leq ,v^{+},v^{-}\rangle $ where (i) I is a non-empty set of states, (ii) $\leq $ is a pre-order on I and (iii) for $\ast \in \{ +,- \}$ , $v^\ast :\mathsf {Prop}\rightarrow 2^I$ is such that $v^\ast (p)$ is upward closed for any $p\in \mathsf {Prop}$ (i.e., $x\in v^{\ast }(p)$ and $y\geq x$ implies $y\in v^{\ast }(p)$ ). Relations $\models ^{+}$ (support of truth), $\models ^{-}$ (support of falsity) between states and formulas are defined by:

Occasionally, we will use $v^{\ast }(A)$ as a shorthand for $\{x:\mathcal {M},x\models ^{\ast }A\}$ ( $\ast \in \{ +,- \}$ ).

As usual, we have the following

Lemma 3.1 (monotonicity)

For any model $\mathcal {M}=\langle I,\leq ,v^+,v^-\rangle $ , formula $A\in \mathsf {Form}\,\mathcal {L}$ and $\ast \in \{ +,- \}$ :

$$\begin{align*}\forall x,y\in I\, (x\leq y\text{ and } \mathcal{M},x\mathbin{\vDash^{\ast}}A\text{ implies } \mathcal{M},y\mathbin{\vDash^{\ast}}A). \end{align*}$$

Next, we establish soundness and completeness of B2C with respect to this semantics. To do so, let us first define the corresponding consequence relations.

We list some simple properties of the two syntactic consequence relations:

A bi-theory (in B2C) is a set-pair $(\Gamma ^+;\Gamma ^-)$ such that for any formula A and $\ast \in \{ +,- \}$ we have

$$\begin{align*}(\Gamma^+,\Gamma^-)\mathbin{\vdash_{{\mathbf{{B2C}}}}^{\ast}}A \implies A\in\Gamma^\ast. \end{align*}$$

A bi-theory $(\Gamma ^+;\Gamma ^-)$ is prime if $\Gamma ^+,\Gamma ^-$ are both non-empty and the following constructive disjunction and conjunction properties are satisfied:

$$\begin{align*}A\vee B\in\Gamma^+\implies A\in\Gamma^+\text{ or }B\in\Gamma^+;\qquad A\wedge B\in\Gamma^-\implies A\in\Gamma^-\text{ or }B\in\Gamma^-. \end{align*}$$

Proof. We consider the case of $\ast =+$ , the other case is similar.

Take X to be the set of all expressions of the form $A^+$ and $A^-$ , where A is a formula and fix some enumeration $\{ x_i\mid i\in \mathbb {N} \}$ of elements of X. Put $\Delta ^+_0:=\Gamma ^+$ , $\Delta ^-_0:=\Gamma ^-$ and for any $i\in \mathbb {N}$ :

$$\begin{align*}(\Delta^{+}_{i+1};\Delta^{-}_{i+1}):= \begin{cases} (\Delta^{+}_{i},B;\Delta^{-}_{i}), & \text{if } x_i=B^+\text{ and } (\Delta^{+}_{i},B;\Delta^{-}_{i})\mathbin{\nvdash_{{\mathbf{{B2C}}}}^{+}}A;\\ (\Delta^{+}_{i};\Delta^{-}_{i},B), & \text{if } x_i=B^-\text{ and } (\Delta^{+}_{i};\Delta^{-}_{i},B)\mathbin{\nvdash_{{\mathbf{{B2C}}}}^{+}}A;\\ (\Delta^{+}_{i};\Delta^{-}_{i}), & \text{otherwise}. \end{cases} \end{align*}$$

Then put $\Delta ^\ast = \bigcup _{i\in \mathbb {N}}\Delta ^{\ast }_{i}$ for $\ast \in \{ +,- \} $ . We show that $(\Delta ^+,\Delta ^-) $ is the required bi-theory. That $(\Gamma ^+;\Gamma ^-)\leq (\Delta ^+;\Delta ^-) $ is trivial by the construction.

A simple induction shows that $(\Delta ^{+}_{i};\Delta ^{-}_{i})\mathbin {\nvdash _{{\mathbf {{B2C}}}}^{+}}A$ for any $i\in \mathbb {N}$ and hence that $(\Delta ^+;\Delta ^-)\mathbin {\nvdash _{{\mathbf {B2C}}}^{+}}A$ due to compactness.

Suppose $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{{\mathbf {B2C}}}^{+}}B$ and $B\notin \Delta ^+$ . Then there is $i\in \mathbb {N}$ such that $B^+= x_i$ and $(\Delta ^{+}_{i},B;\Delta ^{-}_{i})\mathbin {\vdash _{\mathbf {B2C}}^{+}}A$ . By monotonicity we have $(\Delta ^+,B;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}}^{+}}A$ hence by the cut we have $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}}^{+}}A$ , which contradicts the assumption. So $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}}^{+}}B$ implies $B\in \Delta ^+$ . Similarly, $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}}^{-}}B$ implies $B\in \Delta ^-$ .

That $\Delta ^+$ ( $\Delta ^-$ ) is non-empty then follows from the fact that () is provable (refutable) in $\mathbf {NB2C}$ using (dual) cut.

Finally, suppose $B\vee C\in \Delta ^+$ , $B\notin \Delta ^+$ and $C\notin \Delta ^+$ . Again, there are $i,j\in \mathbb {N}$ such that $B^+=x_i$ , $C^+=x_j$ ,

$$\begin{align*}(\Delta^{+}_{i},B;\Delta^{-}_{i})\mathbin{\vdash_{{\mathbf{{B2C}}}}^{+}}A;\qquad (\Delta^{+}_{j},C;\Delta^{-}_{j})\mathbin{\vdash_{{\mathbf{{B2C}}}}^{+}}A. \end{align*}$$

Then the following derivation

shows that $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}}^{+}}A$ . Similarly if $B\wedge C\in \Delta ^-$ , $B\notin \Delta ^-$ and $C\notin \Delta ^-$ , then there are $i,j\in \mathbb {N}$ such that $B^-=x_i$ , $C^-=x_j$ ,

$$\begin{align*}(\Delta^{+}_{i};\Delta^{-}_{i},B)\mathbin{\vdash_{\mathbf{{B2C}}}^{+}}A;\qquad (\Delta^{+}_{j};\Delta^{-}_{j},C)\mathbin{\vdash_{\mathbf{{B2C}}}^{+}}A. \end{align*}$$

And hence the following derivation shows that $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}}^{+}}A$ :

The canonical model (for B2C) is $\mathcal {M}_{\mathbf {B2C}}=\langle I_{\mathbf {B2C}},\leq ,v^{+}_{\mathbf {B2C}}, v^{-}_{\mathbf {B2C}}\rangle $ , where

• • $I_{\mathbf {B2C}}$ is the set of all prime bi-theories for $\mathbf {B2C}$ ;

• • $\leq $ is the order relation on set-pairs defined above;

• • $(\Gamma ^+;\Gamma ^-)\in v^{+}_{\mathbf {B2C}}(p)\text { iff } p\in \Gamma ^+$ for $p\in \mathsf {Prop}$ ;

• • $(\Gamma ^+;\Gamma ^-)\in v^{-}_{\mathbf {B2C}}(p)\text { iff } p\in \Gamma ^-$ for $p\in \mathsf {Prop}$ .

Lemma 3.3 (truth)

For any $(\Gamma ^+;\Gamma ^-)\in I_{\mathbf {B2C}}$ and any formula A we have:

$$\begin{align*}\mathcal{M}_{\mathbf{{B2C}}},(\Gamma^+;\Gamma^-)\mathbin{\vDash^{+}}A\text{ iff } A\in\Gamma^+;\qquad \mathcal{M}_{\mathbf{{B2C}}},(\Gamma^+;\Gamma^-)\mathbin{\vDash^{-}}A\text{ iff } A\in\Gamma^-. \end{align*}$$

Proof. By induction on the complexity of A. The basis of induction follows from the definitions. Let us consider the cases of conjunction and co-implication, the other two cases are dual.

$(\wedge ,+)$ . By the induction hypothesis it is enough to show that

$$\begin{align*}A\wedge B\in\Gamma^+\text{ iff } A\in\Gamma^+\text{ and }B\in\Gamma^+. \end{align*}$$

To show that, it is enough to use $(\wedge Ip)$ and $(\wedge Ep)$ .

$(\wedge , -)$ . It is enough to show that

$$\begin{align*}A\wedge B\in\Gamma^-\text{ iff }A\in\Gamma^-\text{ or }B\in\Gamma^-. \end{align*}$$

For the right-to-left direction use $(\wedge Idp)$ , the left-to-right direction follows from the definition of a prime bi-theory.

. It is enough to show that

Suppose , $(\Gamma ^+,\Gamma ^-)\leq (\Delta ^+,\Delta ^-)$ and $A\in \Delta ^-$ . Then , $A\in \Delta ^-$ and one application of shows that $B\in \Delta ^+$ .

For the other direction assume that . If, additionally, $(\Gamma ^+;\Gamma ^-,A)\mathbin {\vdash _{\mathbf {B2C}}^{+}}B$ , then by we have , hence by the definition of a prime bi-theory. This contradicts the assumption. Hence $(\Gamma ^+;\Gamma ^-,A)\mathbin {\nvdash _{\mathbf {B2C}}^{+}}B$ and by the extension lemma there is $(\Delta ^+;\Delta ^-)\in I_{\mathbf {B2C}}$ such that $(\Gamma ^+;\Gamma ^-)\leq (\Delta ^+;\Delta ^-)$ , $A\in \Delta ^-$ and $(\Delta ^+;\Delta ^-)\mathbin {\nvdash _{\mathbf {B2C}}^{+}}B$ . The latter implies that $B\notin \Delta ^+$ and concludes the proof.

. Again, it is enough to show that

The left-to-right direction is obtained easily using . For the other direction we assume . As before, if $(\Gamma ^+;\Gamma ^-,A)\mathbin {\vdash _{\mathbf {B2C}}^{-}}B$ then an application of gives us a contradiction with . Then $(\Gamma ^+;\Gamma ^-,A)\mathbin {\nvdash _{\mathbf {B2C}}^{-}}B$ and applying the extension lemma we obtain $(\Delta ^+;\Delta ^-)\in I_{\mathbf {B2C}}$ such that $(\Gamma ^+;\Gamma ^-)\leq (\Delta ^+;\Delta ^-)$ , $A\in \Delta ^-$ and $B\notin \Delta ^-$ .

Theorem 3.2 (soundness and completeness)

Where $(\Gamma ;\Delta )$ is a set-pair, $A\in \mathsf {Form}\,\mathcal {L}$ and $\ast \in \{ +,- \}$ :

$$\begin{align*}(\Gamma^+;\Gamma^-)\mathbin{\vdash_{\mathbf{{B2C}}}^{\ast}}A\text{ iff } (\Gamma^+;\Gamma^-)\mathbin{\vDash_{\mathbf{{B2C}}}^{\ast}}A. \end{align*}$$

We are now ready to clarify the status of $\neg $ and ${-}$ in $\mathbf {B2C}$ as negations. First, for a unary connective $\theta $ put $\theta ^{(0)}A:=A$ and $\theta ^{(n+1)}A:=\theta \theta ^{(n)}A$ where $n\geq 1$ . The following is established by a straightforward application of the completeness result above.

Fact 3.5. For every $n\in \mathbb {N}$ the following conditions hold (where $p\neq q$ ):

$$ \begin{align*} & (i) && (\neg^{(n+1)}p;\varnothing)\mathbin{\nvdash_{\mathbf{B2C}}^{+}} q;\qquad && (ii) && (\neg^{(n+1)}p;\varnothing)\mathbin{\nvdash_{\mathbf{B2C}}^{+}}\neg^{(n)}p;\\ & (iii)\quad && (\varnothing;\varnothing)\mathbin{\nvdash_{\mathbf{B2C}}^{+}} \neg^{(n+1)}p;\qquad && (iv)\quad && (\neg^{(n)}p;\varnothing)\mathbin{\nvdash_{\mathbf{B2C}}^{+}}\neg^{(n+1)}p;\\ & (v) && ({-}^{(n+1)}p;\varnothing)\mathbin{\nvdash_{\mathbf{B2C}}^{-}} q;\qquad && (vi) && ({-}^{(n+1)}p;\varnothing)\mathbin{\nvdash_{\mathbf{B2C}}^{-}}{-}^{(n)}p;\\ & (vii)\quad && (\varnothing;\varnothing)\mathbin{\nvdash_{\mathbf{B2C}}^{-}} {-}^{(n+1)}p;\qquad && (viii)\quad && ({-}^{(n)}p;\varnothing)\mathbin{\nvdash_{\mathbf{B2C}}^{-}}{-}^{(n+1)}p. \end{align*} $$

Conditions $(i)$ – $(iv)$ are natural variations of requirements for a minimal negation put forward in [Reference Marcos29] (under the name mid-negation, where mid stands for “minimally decent”), while conditions $(v)$ – $(viii)$ are their natural duals.

4.1 Non-definability

Before we introduce these new connectives via natural deduction rules it is imperative to show that they are not already definable in B2C via a formula. The argument will be semantical, so by a connective $c(p_1,\dotsc ,p_n)$ here we will understand a pair of support of truth and support of falsity conditions of the form

$$ \begin{align*} \mathcal{M},x\mathbin{\vDash^{+}}c(A_1,\dotsc,A_n)&\text{ iff } S^+(x,A_1,\dotsc,A_n);\\\mathcal{M},x\mathbin{\vDash^{-}}c(A_1,\dotsc,A_n)&\text{ iff } S^-(x,A_1,\dotsc,A_n). \end{align*} $$

Then by $c(p_1,\dotsc ,p_n)$ being definable we will mean that there is a formula $B(p_1,\dotsc ,p_n,q_1,\dotsc ,q_m)$ (where $q_1,\dotsc ,q_m$ are parameters) in the given language such that

$$ \begin{align*} \mathcal{M},x\mathbin{\vDash^{+}}B(A_1,\dotsc,A_n,q_1,\dotsc,q_m)&\text{ iff } S^+(x,A_1,\dotsc,A_n);\\\mathcal{M},x\mathbin{\vDash^{-}}B(A_1,\dotsc,A_n,q_1,\dotsc,q_m)&\text{ iff } S^-(x,A_1,\dotsc,A_n) \end{align*} $$

holds for any model $\mathcal {M}=\langle I,\leq ,v^+,v^-\rangle $ , any $x\in I$ and any formulas $A_1,\dotsc ,A_n$ . In particular, we will sometimes talk of definability in classical models, when we restrict our attention to those models.

To proceed we list the support of truth and falsity conditions for these connectives; later in the section we will provide them with natural deduction rules and prove the corresponding completeness results. Then assuming $\mathcal {M}=\langle I,\leq ,v^+,v^-\rangle $ to be a model put

As we discussed at the beginning of the section, the constant $\mathbf {b}$ turns out to be definable in B2C. In fact, we have already established this in Fact 3.3.

We proceed to the non-definability results. The main technical lemma is the following:

Note that the claims of Lemma 4.2 do not hold for $\mathbf {2Int}$ and $\mathbf {N4}$ .

Next, we shall check that the choice of having both implication and co-implication in $\mathcal {L}$ is a reasonable one. This will be established by a McKinsey-style argument [Reference McKinsey31] showing that implication and co-implication are inter-independent in the absence of the toggling negation, even when $\bot $ and $\top $ are present.

Proof. (i) Let $I:=\{x,y\}$ and $\leq :=\{\langle x,x\rangle , \langle y,y\rangle \}$ and $\mathcal {M}:=\langle I,\leq ,v^{+},v^{-}\rangle $ be such that $\langle v^{+}(p),v^{-}(p)\rangle \in \{\langle I,I\rangle ,\langle I,\{x\}\rangle , \langle I,\emptyset \rangle , \langle \emptyset ,I\rangle \}$ for all $p\in \mathsf {Prop}$ . Then we have the next tables of $\langle v^{+}(A), v^{-}(A)\rangle $ for each formula A in : note that $\top $ and $\bot $ are assigned the pairs $\langle I,\emptyset \rangle $ and $\langle \emptyset ,I\rangle $ , respectively.

Thus any formula in has one of the four pairs of values. Now suppose that $\langle v^{+}(p),v^{-}(p)\rangle =\langle I,\{x\}\rangle $ and $\langle v^{+}(q),v^{-}(q)\rangle =\langle \emptyset ,I\rangle $ . Then and , but $\langle \{y\}, I\rangle $ does not belong to the collection of pairs. Therefore it cannot be defined in .

(ii) This time, let $v^{+},v^{-}$ be s.t. $\langle v^{+}(p),v^{-}(p)\rangle \in \{\langle I,I\rangle ,\langle I,\emptyset \rangle , \langle \{x\},I\rangle , \langle \emptyset ,I\rangle \}$ for all $p\in \mathsf {Prop}$ . Then we obtain the following tables for $\langle v^{+}(A), v^{-}(A)\rangle $ for each formula A in $\mathcal {L}^{\to }_{\{\top ,\bot \}}$ .

Thus any formula in $\mathcal {L}^{\to }_{\{\top ,\bot \}}$ has one of the four pairs of values. Now suppose that $\langle v^{+}(p),v^{-}(p)\rangle =\langle \{x\},I\rangle $ and $\langle v^{+}(q),v^{-}(q)\rangle =\langle I,\emptyset \rangle $ . Then $v^{+}(p\to q )=I$ and $v^{-}(p\to q)= \{y\}$ , but $\langle I,\{y\}\rangle $ does not belong to the collection of pairs. Therefore it cannot be defined in $\mathcal {L}^{\to }_{\{\top ,\bot \}}$ .

4.2 Natural deduction rules and completeness

We are ready to introduce natural deduction rules corresponding to the new connectives introduced in this section. These rules are listed in Tables 4 and 5.

The subsystems/expansions of $\mathbf {B2C}$ defined with some of these rules will be denoted similarly to their linguistic counterpart: e.g. $\mathbf {B2C}^{\to }_{\{\top ,\bot \}}$ is a system in $\mathcal {L}^{\to }_{\{\top ,\bot \}}$ obtained by removing $\to $ -related rules from $\mathbf {B2C}$ and adding the $\top $ - and $\bot $ -related rules from Table 4.

Table 4

Natural deduction rules for $\top $ , $\bot $ , $\mathbf {n}$ and $\sim $

Table 5

Natural deduction rules for strong implication and strong co-implication

We proceed to extend our completeness result to the new connectives. Suppose $\mathcal {L}_i$ is a language such that $\mathcal {L}\subseteq \mathcal {L}_i\subseteq \mathcal {L}_{all}$ and $\mathbf {B2C}_i$ is its corresponding expansion.

We have to modify the notion of a prime bi-theory. Note that for B2C itself there is a prime bi-theory $\langle \Gamma ^+,\Gamma ^-\rangle $ such that both $\Gamma ^+$ and $\Gamma ^-$ coincide with the set of all formulas. This is consistent with the semantics since we can have support of truth and support of falsity for all formulas in a given state. This is no longer the case once we introduce one of the constants $\top $ , $\bot $ or $\mathbf {n}$ . On the other hand, the presence of $\Rightarrow $ or forces us to adopt properties similar to those of the constructive properties for conjunction and disjunction.

Then a bi-theory (for $\mathbf {B2C}_i$ ) is a set-pair $(\Gamma ^+;\Gamma ^-)$ such that for any formula A and $\ast \in \{ +,- \}$ we have

$$\begin{align*}(\Gamma^+,\Gamma^-)\mathbin{\vdash_{\mathbf{{B2C}}_i}^{\ast}}A \text{ implies } A\in\Gamma^\ast. \end{align*}$$

A bi-theory $(\Gamma ^+;\Gamma ^-)$ is prime (for $\mathbf {B2C}_i$ ) if $\Gamma ^+,\Gamma ^-$ are both non-empty; $\Gamma ^+$ satisfies the constructive disjunction property, $\Gamma ^-$ satisfies the constructive conjunction property and, additionally:

1. 1. $\Gamma ^+\neq \mathsf {Form}\,\mathcal {L}_i$ in case $\bot \in \mathcal {L}_i$ or $\mathbf {n}\in \mathcal {L}_i$ ;

2. 2. $\Gamma ^-\neq \mathsf {Form}\,\mathcal {L}_i$ in case $\top \in \mathcal {L}_i$ or $\mathbf {n}\in \mathcal {L}_i$ ;

3. 3. $A\Rightarrow B\in \Gamma ^-$ implies $A\rightarrow B\in \Gamma ^-$ or in case $\Rightarrow \in \mathcal {L}_i$ ;

4. 4. implies or $A\rightarrow B\in \Gamma ^-$ in case .

The canonical model (for $\mathbf {B2C}_i$ ) is defined exactly as for B2C.

First, we see how this modification changes the proof of the extension lemma.

Proof. We consider the case $\ast =+$ as an example. So suppose $(\Gamma ^+;\Gamma ^-)\mathbin {\nvdash _{\mathbf {B2C}_i}^{+}}A$ and from there $(\Delta ^+,\Delta ^-)$ is obtained like in the proof of Lemma 3.2. That is $\Delta ^\ast = \bigcup _{i\in \mathbb {N}}\Delta ^{\ast }_{i}$ for $\ast \in \{ +,- \} $ , where $\Delta ^+_0=\Gamma ^+$ , $\Delta ^-_0=\Gamma ^-$ and for any $i\in \mathbb {N}$ :

$$\begin{align*}(\Delta^{+}_{i+1};\Delta^{-}_{i+1}):= \begin{cases} (\Delta^{+}_{i},B;\Delta^{-}_{i}), & \text{if } x_i=B^+\text{ and } (\Delta^{+}_{i},B;\Delta^{-}_{i})\mathbin{\nvdash_{\mathbf{{B2C}}_i}^{+}}A;\\ (\Delta^{+}_{i};\Delta^{-}_{i},B), & \text{if } x_i=B^-\text{ and } (\Delta^{+}_{i};\Delta^{-}_{i},B)\mathbin{\nvdash_{\mathbf{{B2C}}_i}^{+}}A;\\ (\Delta^{+}_{i};\Delta^{-}_{i}), & \text{otherwise}; \end{cases} \end{align*}$$

and $\{ x_i\mid i\in \mathbb {N} \}$ is some enumeration of the set X of all expressions of the form $A^+$ and $A^-$ ( $A\in \mathsf {Form}\,\mathcal {L}_i$ ).

All of the properties established in Lemma 3.2 are proved in exactly the same way, so we consider them to be already established. We show the additional properties.

1. Suppose $\mathbf {n}\in \mathcal {L}_i$ and $\Delta ^+=\mathsf {Form}\,\mathcal {L}_i$ . Then $\mathbf {n}\in \Delta ^+$ and by $(\mathbf {n} Ep)$ we have $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}_i}^{+}}A$ , which contradicts the already established. The case $\bot \in \mathcal {L}_i$ is treated similarly.

2. Suppose $\mathbf {n}\in \mathcal {L}_i$ and $\Delta ^-=\mathsf {Form}\,\mathcal {L}_i$ . Then $\mathbf {n}\in \Delta ^-$ and by $(\mathbf {n} Edp)$ we have $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}_i}^{+}}A$ . The case $\top \in \mathcal {L}_i$ is similar.

3. Suppose $\Rightarrow \in \mathcal {L}_i$ . Assume additionally that $B\Rightarrow C\in \Delta ^-$ but $B\rightarrow C\notin \Delta ^-$ and . Then there are $i,j\in \mathbb {N}$ such that $x_i=B\rightarrow C^-$ , and

Then the following derivation

shows that $(\Delta ^+;\Delta ^-)\mathbin {\vdash _{\mathbf {B2C}_i}^{+}}A$ , which gives us a contradiction. The case of is considered similarly.

Lemma 4.4 (truth; for expansions)

Fix some expansion $\mathbf {B2C}_i$ where $\mathcal {L}\subseteq \mathcal {L}_i\subseteq \mathcal {L}_{all}$ . For any $(\Gamma ^+;\Gamma ^-)\in I_{\mathbf {B2C}_i}$ and any $A\in \mathsf {Form}\,\mathcal {L}_i$ we have:

$$\begin{align*}\mathcal{M}_{\mathbf{{B2C}}_i},(\Gamma^+;\Gamma^-)\mathbin{\vDash^{+}}A\text{ iff } A\in\Gamma^+;\qquad \mathcal{M}_{\mathbf{{B2C}}_i},(\Gamma^+;\Gamma^-)\mathbin{\vDash^{-}}A\text{ iff } A\in\Gamma^-. \end{align*}$$

Proof. We consider the case of $\Rightarrow $ , the rest is either similar or simple.

$(\Rightarrow , +)$ . It is enough to show that for $(\Gamma ^+;\Gamma ^-)\in I_{\mathbf {B2C}_i}$

$$\begin{align*}&A\Rightarrow B\in\Gamma^+ \text{ iff }\forall(\Delta^+;\Delta^-)\geq (\Gamma^+;\Gamma^-)\ \big((A\in\Delta^+\text{ implies }B\in\Delta^+)\text{ and }\\&\quad (B\in\Delta^-\text{ implies } A\in\Delta^-)\big). \end{align*}$$

The left-to-right direction follows easily from the definition of $\leq $ using rule $(\Rightarrow Ep)$ . For right-to-left, assume that $A\Rightarrow B\notin \Gamma ^+$ . Suppose also that both of the following conditions hold:

$$\begin{align*}(i)\quad (\Gamma^+,A;\Gamma^-)\mathbin{\vdash_{\mathbf{{B2C}}_i}^{+}}B; \qquad (ii)\quad (\Gamma^+;\Gamma^-,B)\mathbin{\vdash_{\mathbf{{B2C}}_i}^{-}}A. \end{align*}$$

Then by $(\Rightarrow I)$ we conclude that $(\Gamma ^+;\Gamma ^-)\mathbin {\vdash _{\mathbf {B2C}_i}^{+}}A\Rightarrow B$ , thus $A\Rightarrow B\in \Gamma ^+$ , which contradicts the assumption. Then either $(i)$ or $(ii)$ does not hold. In the former case by the extension lemma we obtain $(\Delta ^+;\Delta ^-)\in I_{\mathbf {B2C}_i}$ such that $(\Gamma ^+;\Gamma ^-)\leq (\Delta ^+;\Delta ^-)$ , $A\in \Delta ^+ $ and $B\notin \Delta ^+$ ; in the latter case by again the extension lemma we find $(\Delta ^+;\Delta ^-)\in I_{\mathbf {B2C}_i}$ such that $(\Gamma ^+;\Gamma ^-)\leq (\Delta ^+;\Delta ^-)$ , $B\in \Delta ^- $ and $A\notin \Delta ^-$ , as required.

$(\Rightarrow ,-)$ . Similarly, it is enough to show that

$$ \begin{align*} & A\Rightarrow B\in\Gamma^- \text{ iff } && \forall(\Delta^+;\Delta^-)\geq (\Gamma^+;\Gamma^-)\ (A\in\Delta^+\text{ implies }B\in\Delta^-)\text{ or }\\ & && \forall(\Delta^+;\Delta^-)\geq (\Gamma^+;\Gamma^-)\ (B\in\Delta^-\text{ implies } A\in\Delta^+). \end{align*} $$

If $A\Rightarrow B\in \Gamma ^-$ , then according to the definition of prime bi-theories (in case $\Rightarrow \in \mathcal {L}_i$ ) we have $A\rightarrow B\in \Gamma ^-$ or . As already established, the former implies the first disjunct and the latter implies the second disjunct of the condition on the right-hand side.

Suppose $A\Rightarrow B\notin \Gamma ^-$ . If, additionally, $(\Gamma ^+,A;\Gamma ^-)\mathbin {\vdash _{\mathbf {B2C}_i}^{-}}B$ , then by $(\Rightarrow Idp)$ we conclude that $(\Gamma ^+;\Gamma ^-)\mathbin {\vdash _{\mathbf {B2C}_i}^{-}}A\Rightarrow B$ , which contradicts our assumption. Then applying the extension lemma we obtain $(\Delta ^+;\Delta ^-)\in I_{\mathbf {B2C}_i}$ such that $(\Gamma ^+;\Gamma ^-)\leq (\Delta ^+;\Delta ^-)$ , $A\in \Delta ^+$ and $B\notin \Delta ^-$ , as required.

Similarly $(\Gamma ^+;\Gamma ^-,B)\mathbin {\vdash _{\mathbf {B2C}_i}^{+}}A$ cannot be the case by $(\Rightarrow Idp)$ and the assumption, so one application of the extension lemma gives us $(\Delta ^+;\Delta ^-)\in I_{\mathbf {B2C}_i}$ such that $(\Gamma ^+;\Gamma ^-)\leq (\Delta ^+;\Delta ^-)$ , $B\in \Delta ^-$ and $A\notin \Delta ^+$ . This concludes the proof.

Theorem 4.2 (soundness and completeness)

Fix some expansion $\mathbf {B2C}_i$ where $\mathcal {L}\subseteq \mathcal {L}_i\subseteq \mathcal {L}_{all}$ . Suppose $(\Gamma ;\Delta )$ is a set-pair in $\mathcal {L}_i$ , $A\in \mathsf {Form}\,\mathcal {L}_i$ and $\ast \in \{ +,- \}$ . Then

$$\begin{align*}(\Gamma^+;\Gamma^-)\mathbin{\vdash_{\mathbf{{B2C}}}^{\ast}}A\text{ iff } (\Gamma^+;\Gamma^-)\mathbin{\vDash_{\mathbf{{B2C}}}^{\ast}}A. \end{align*}$$

6.1 Half-toggling connectives

Semantically, the characteristic property of the toggling negation is that both $x\models ^{+}{\sim }\kern-3pt A\text { iff } x\models ^{-}A$ for all A, and $x\models ^{-}$ ${\sim }\kern-3pt A\text { iff } x\models ^{+}A$ for all A. One way to liberalize it is then to require only one of the equivalences.

Let us call a unary connective ${\sim }_{*}$ half-toggling if it shares the condition for the support of truth or the support of falsity with the toggling negation: i.e., it satisfies either $\mathcal {M},x\models ^{+}{\sim }_{*}A\text { iff } \mathcal {M},x\models ^{-}A$ for all A, or $\mathcal {M},x\models ^{-}{\sim }_{*}A\text { iff } \mathcal {M},x\models ^{+}A$ for all A.Footnote ¹¹ When we want to be more specific, the former will be called positively half-toggling and the latter will be called negatively half-toggling. Proof-theoretically, we may understand these by the admissibility of the next pairs of rules, respectively.

For some immediate examples, the toggling negation is a half-toggling connective, and negation and co-negation in $\mathbf {2Int}$ are each negatively and positively half-toggling. In addition, the variants of the $\mathbf {FDE}$ -negation discussed in [Reference Omori, Wansing and Bimbó40] are all positively half-toggling.Footnote ¹² In particular, among them is a connective named ${\sim }_{4}$ , which has the next support of truth/falsity condition.

The connective ${\sim }_{4}$ in addition satisfies Belnap’s criterion [Reference Belnap and Ryle8] that an operator in the bilattice FOUR must be monotone with respect to the informational ordering (Scott’s thesis). Therefore there are good reasons to think that it is a legitimate connective. In terms of natural deduction, ${\sim }_{4}$ can be defined by adding to the rules for positively half-toggling connectives the next $0$ -premise rule.

This consideration informs us about the acceptability of adding strong implication (along with strong co-implication) to $\mathbf {B2C}$ as well, for it will allow us to define ${\sim }_{4}$ .

Proof. The following shows that $A\Rightarrow \mathbf {b}$ serves the purpose. (See Lemma 4.1 for a definition of $\mathbf {b}$ .)

Note that the proof above only uses strong implication. One may also obtain a negatively half-toggling connective by means of strong co-implication alone, using .

On the other hand, if we stay in $\mathcal {L}$ then we are saved from half-toggling connectives. In fact, the addition of $\mathbf {n}$ to the language does not affect the undefinability of such connectives.

Proof. The argument is like for Proposition 4.1: we offer countermodels for both positively and negatively half-toggling connectives. Let $I:=\{x,y\}$ and $\leq :=\{\langle x,x\rangle , \langle x,y\rangle ,\langle y,y\rangle \}$ . Let $\mathcal {M}_{1}:=\langle I,\leq ,v^{+}_{1},v^{-}_{1}\rangle $ be s.t.

$$ \begin{align*} \langle v^{+}_{1}(p),v^{-}_{1}(p)\rangle\in\{\langle I,I\rangle,\langle I,\{y\}\rangle,\langle I,\emptyset\rangle,\langle \emptyset,I\rangle,\langle \emptyset,\{y\}\rangle,\langle \emptyset,\emptyset\rangle\}, \end{align*} $$

for all $p\in \mathsf {Prop}$ . Note that $\mathbf {n}$ is assigned the pair $\langle \emptyset ,\emptyset \rangle $ . Then we obtain the following tables.

So the set $\{\langle I,I\rangle ,\langle I,\{y\}\rangle ,\langle I,\emptyset \rangle ,\langle \emptyset ,I\rangle ,\langle \emptyset ,\{y\}\rangle ,\langle \emptyset ,\emptyset \rangle \}$ is closed under the operations. Now if a positively half-toggling connective is definable, then $v^{+}_{1}(A)=\{y\}$ must be possible; a contradiction. Next, Let $\mathcal {M}_{2}:=\langle I,\leq ,v^{+}_{2},v^{-}_{2}\rangle $ be a model such that

$$ \begin{align*} \langle v^{+}_{2}(p),v^{-}_{2}(p)\rangle\in\{\langle I,I\rangle,\langle I,\emptyset\rangle,\langle\{y\},I\rangle,\langle\{y\},\emptyset\rangle,\langle \emptyset,I\rangle,\langle \emptyset,\emptyset\rangle\}, \end{align*} $$

for all $p\in \mathsf {Prop}$ . Then we obtain the following tables.

Hence the set $\{\langle I,I\rangle ,\langle I,\emptyset \rangle ,\langle \{y\},I\rangle ,\langle \{y\},\emptyset \rangle ,\langle \emptyset ,I\rangle ,\langle \emptyset ,\emptyset \rangle \}$ is closed under the operations. Now if a negatively half-toggling connective is definable, then $v^{-}_{2}(A)=\{y\}$ must be possible; a contradiction.

Remark 6.1. If we restrict our attention to classical models, then a different picture emerges. The formula in $\mathcal {L}_{\{\top ,\bot \}}$ then has the next forcing condition in a model $\mathcal {M}$ where $I=\{x\}$ .

Similarly, the formula gives:

Now it is also easily checked that $(A\to B)\land ({\sim }_{*}B\to {\sim }_{*}A)$ defines strong implication when ${\sim }_{*}$ is positively half-toggling, and defines strong co-implication when ${\sim }_{*}$ is negatively half-toggling. (Note that classical models always support the falsity of $A\Rightarrow B$ /the truth of .)

6.2 The toggling connective

Our second concern is the definability of the toggling negation. We shall first observe that the language does not define the toggling connective. We utilize the fact that the languages $\mathcal {L}_{\{\top ,\bot \}}$ and have no difference in classical models in their expressiveness.

Proof. We shall use the same model as for Proposition 4.1, except that for all $p\in \mathsf {Prop}$ :

$\langle v^{+}(p),v^{-}(p)\rangle \in \{\langle I,I\rangle ,\langle I,\{x\}\rangle , \langle I,\{y\}\rangle , \langle I,\emptyset \rangle , \langle \{x\},I\rangle ,\langle \{x\},\{y\}\rangle ,\langle \{y\},I\rangle ,\langle \emptyset ,I\rangle \}.$

Then we obtain the following tables for $\langle v^{+}(A), v^{-}(A)\rangle $ for each formula A in $\mathcal {L}_{\{\top ,\bot \}}$ .

Hence the set of values are closed under the operations. Now if the toggling negation ${\sim }$ is definable, then $\langle v^{+}(p),v^{-}(p)\rangle =\langle \{x\},\{y\}\rangle $ implies $v^{+}({\sim }\kern-1.5pt p)=\{y\}$ and $v^{-}({\sim }\kern-1.5pt p)= \{x\}$ , but $\langle \{y\},\{x\}\rangle $ does not belong to the collection of pairs. Therefore ${\sim }$ cannot be defined in $\mathcal {L}_{\{\top ,\bot \}}$ .

Let us next consider languages including $\mathbf {n}$ . We have already seen that $\mathcal {L}_{\mathbf {n}}$ does not define half-toggling connectives; so a fortiori it does not define the toggling connective. As for , the situation is rather different from that of . Recall that half-toggling connectives are definable already in the latter language. We may then apply a technique of combining the support of truth condition of one connective and the support of falsity condition of another connective, in order to define the toggling negation (cf. e.g. [Reference Arieli and Avron3, Reference Odintsov, Skurt and Wansing37]).

Proof. We claim that the formula defines the toggling negation. Indeed:

Hence if one wishes to keep clear of the toggling negation (in the present setting), then it is necessary and sufficient to give up either the neither constant or the strong implication/co-implication pair.

6.3 Indirectly half-toggling connectives

It is possible to come up with a notion of toggling that is less direct than half-toggling. We shall call a unary connective ${\sim }_{*}$ indirectly half-toggling if it satisfies either:

$$ \begin{align*}& \mathcal{M},x\models^{+}{\sim}_{*}A\text{ iff }\forall{y\geq x}\exists{z\geq y}\ \mathcal{M},z\models^{-}A, \text{ or }\\&\mathcal{M},x\models^{-}{\sim}_{*}A\text{ iff }\forall{y\geq x}\exists{z\geq y}\ \mathcal{M},z\models^{+}A. \end{align*} $$

As before, we will call the former the positive and the latter the negative form of indirect half-toggling. This notion clearly coincides with that of a half-toggling connective in a classical model, which provides a potential reason to be sceptical of the acceptability of these connectives for those who take a more strict stance towards the separation of proof and disproof.

It is straightforward to check that positive/negative indirectly half-toggling connectives are definable in $\mathcal {L}_{\{\top ,\bot \}}$ via the formulas in Remark 6.1, namely and . In comparison, once $\top $ and $\bot $ are discarded, we can avoid such formulas, again even in the presence of $\Rightarrow $ and .

Proof. Let $I:=\{x,y,z\}$ , $\leq :=\{\langle x,x\rangle ,\langle x,y\rangle ,\langle y,y\rangle ,\langle z,z\rangle \}$ . Let $\mathcal {M}:=\langle I,\leq ,v^{+},v^{-}\rangle $ be s.t. $\langle v^{+}(p),v^{-}(p)\rangle \in \{\langle I,I\rangle ,\langle I,\{y\}\rangle ,\langle \{y\},I\rangle \}$ for all $p\in \mathsf {Prop}$ . Then we obtain the following tables for $\langle v^{+}(A), v^{-}(A)\rangle $ for each formula A in .

Now if ${\sim }_{*}$ s.t. $\mathcal {M},w\models ^{+}{\sim }_{*}A\text { if and only if } \forall {w'\geq w}\exists {w"\geq w'}(\mathcal {M},w"\models ^{-}A)$ is definable, then for $v^{-}(A)=\{y\}$ we would need $v^{+}({\sim }_{*}A)=\{x,y\}$ , but this is not possible. Similarly, if ${\sim }_{*}$ s.t. $\mathcal {M},w\models ^{-}{\sim }_{*}A\text { if and only if } \forall {w'\geq w}\exists {w"\geq w'}(\mathcal {M},w"\models ^{+}A)$ is definable, then for $v^{+}(A)=\{y\}$ we would need $v^{-}({\sim }_{*}A)=\{x,y\}$ , but this is again impossible.

Remark 6.2. Connectives satisfying one of the next conditions can be used to define positive/negative indirectly half-toggling connectives (when both of them exist), respectively by $\dagger _{+}\dagger _{-}A$ and $\dagger _{-}\dagger _{+}A$ .

$$ \begin{align*} \mathcal{M}, x\models^{+}\dagger_{+}A\text{ iff }\forall{y\geq x}\ \mathcal{M},y\not\models^{-}A,\text{ or }\mathcal{M},x\models^{-}\dagger_{-}A\text{ iff }\forall{y\geq x}\ \mathcal{M},y\not\models^{+}A. \end{align*} $$

The converse is generally not the case, as can be shown by an argument similar to the proof of Theorem 4.1. However it is not difficult to see that they are definable from an indirectly half-toggling connective once we are in $\mathcal {L}_{\{\top ,\bot \}}$ . So the definability of a pair of indirectly half-toggling connectives and the pair $\dagger _{+},\dagger _{-}$ turns out to be equivalent, when it comes to the languages under consideration.

Proof-theoretically, $\dagger _{+}$ and $\dagger _{-}$ have an advantage over an indirectly half-toggling connective in that they have a simpler presentation in terms of the admissibility of pairs of rules: respectively,

Then we may observe a behaviour similar to positive/negative half-toggling, but with respect to $\dagger _{+}A\to A$ or . For the former:

(where ). In particular, if one of the following:

is satisfied, then ${\sim }_{*}A$ becomes positively/negatively half-toggling in each case.

6.4 The indirectly toggling connective

It is of course possible to think of a connective which relates to indirectly half-toggling connectives in the same way the toggling negation relates to half-toggling connectives: i.e., the connective defined by:

$$ \begin{align*} &\mathcal{M},x\models^{+}{\sim}_{*}A\text{ iff }\forall{y\geq x}\exists{z\geq y}\ \mathcal{M},z\models^{-}A, \text{ and }\\&\mathcal{M},x\models^{-}{\sim}_{*}A\text{ iff }\forall{y\geq x}\exists{z\geq y}\ \mathcal{M},z\models^{+}A. \end{align*} $$

We shall call this connective the indirectly toggling connective, for which we have the next observation.

6.5 Summary

We end up with Table 6, which exhibits the hierarchy of languages for models/classical models. Table 7 displays more in detail possibilities of defining types of toggling connectives under various languages.

Table 6

Hierarchy in constructive (left) and classical (right) settings

Table 7

Definability of toggling connectives

An emerging picture is that a bilateralist who wishes to include only $\top $ , $\bot $ or only $\Rightarrow $ , can adopt a criterion that half-toggling/indirectly half-toggling is acceptable while other types of toggling are not. Alternatively, one can be permissive and allow both half- and indirectly half-toggling connectives in order to include all of $\top $ , $\bot $ and $\Rightarrow $ , . Or, one can accept the indirectly toggling connective along with the neither constant. Even with such liberal attitudes, however, one must at least make a choice between the neither constant and strong implication/co-implication, so as to avoid the toggling negation. Finally, one can remain strict and reject all four kinds of toggling, which supports the position of $\mathbf {B2C}$ that does not accept any of $\mathbf {n}$ , $\top $ , $\bot $ or $\Rightarrow $ , .