Priest on Negation

Abstract

What conception of negation a dialetheist might have, in holding that a statement and its negation can both be true, has been the subject to considerable debate. Several of the issues in play in this area—such as the unique characterization of negation, and the interplay between contrariety and subcontrariety—are broached here by considering some positions taken on them by Graham Priest and assorted critics (in particular, Hartley Slater and Jean-Yves Béziau). Some of the more intricate points, as well as detailed discussions of commentators (including those just mentioned as well as Heinrich Wansing) on Priest are handled in explicitly labelled Digressions and in two Appendices.

Appendices

Appendix 1: Subcontraries and Mere Subcontraries

For the general point to be discussed here, the title could equally well have been (or included) ‘Contraries and Mere Contraries’, but that chosen reflects more specifically some aspects of the discussion of Slater (1995). Let us recall that there are two understandings of the terminology of contraries and subcontraries, distinguished as the ‘say less’ and ‘say more’ interpretations in Humberstone (2003), where numerous textual references and some further discussion may be found.⁶⁴ On the ‘say less’ approach, the most influential advocate for which was perhaps E. J. Lemmon,⁶⁵ semantically formulated, contraries/subcontraries are defined to be incapable of joint truth/falsity, or modulating to a consequence relation relativity, to be contraries/subcontraries relative to $\vdash$ when $\vdash$ is determined by some class of (bivalent) valuations no element of which verifies/falsifies both. This is the usage that has been followed in the present paper.

The ‘say more’ interpretation, understandably, says more, adding a requirement that the class of valuations in question should also provide at least one verifying both of the subcontraries (respectively, falsifying both of them) for them to deserve that description. Both are perfectly legitimate things to mean by the terminology, but the smoothest discussion of the issues with which we have been concerned takes the ‘say less’ approach. Contrariety and subcontrariety are then what are called coercive logical relations in Humberstone (2013b), by contrast with, for example, consistency, understood as the relation holding between A and B precisely when they are not contraries, is what is there called a permissive logical relation. The coercive relations are defined by a requirement—and here we put it syntactically—that something should be provable in the logic to which they are relativized, whereas the permissive relations such as consistency and independence are those obtaining in virtue of something’s not being provable in the logic in question.⁶⁶ (In terms of quantification over valuations in a determining class, the coercive relations involve universal quantification and the permissive relations involve existential quantifications.) Contrariety on and subcontrariety on the ‘say more’ approach are mixed coercive and permissive logical relations, like the relation of unilateral implication (holding between A and B according to $\vdash$ when $A \vdash B$ and $B \nvdash A$), and having selected the ‘say less’ option, we can still make a concession to the alternative option by calling contraries which are not also subcontraries, mere contraries, and subcontraries which are not also contraries mere subcontraries.

Note that the definitions are just given of mere contrariety and mere subcontrariety could equivalently have been put in the following way: mere contraries are contraries which are not contradictories, and mere subcontraries are subcontraries which are not contradictories.⁶⁷ For this reason, it does not matter whether Slater’s objection is put by saying, on the one hand, that LP’s negation turns a formula into a subcontrary but not (in the general case) into one which is a contradictory, or, on the other hand, that the LP’s negation turns a formula into a mere subcontrary, so nothing hangs in this debate on whether Slater had the ‘say less’ or the ‘say more’ understanding of the word subcontrary in mind. Béziau seems to think otherwise, and himself favours the ‘say more’ reading, as is also clear from Béziau (2016). This may be part of why he thinks that Slater’s discussion is confused, since he attributes this interpretation to Slater also (see the Digression below) and it is perhaps also connected with his claim—quoted in the Digression in Sect. 15.3—that intuitionistic negation is not a contrary-forming operator. To illustrate in the latter case, for some choices of A, A and $\lnot A$ will not be mere contraries, in the sense that they are contraries without being subcontraries: for example, take A as $p \wedge \lnot p$. We resume the present discussion after an optional explanation of Béziau’s grounds for attributing the ‘say more’ interpretation to Slater. (Appendix 2 addresses Heinrich Wansing’s similar claim that negation in intuitionistic logic does not form what we are calling mere contraries.)

Digression. On the ‘say more’ interpretation, subcontraries and contradictories are mutually exclusive relations. Slater (1995) has the example of coming to call red things blue not amounting to changing their colour but only our terminology for attributing colours to them. This is meant to be analogous to coming to calling subcontraries contradictories not actually making them contradictories, so in view of the mutual exclusivity of red and blue as standardly understood, Béziau concludes that Slater assumes as standard an understanding of subcontrary and contradictory one which makes them likewise mutually exclusive, as the ‘say more’ interpretation does. But this may be a purely incidental feature of the example. Here is Béziau making a case for his reading of Slater, from p. 19 of Béziau (2006), beginning with a characterization of the ‘say less’ option:

In this case, confusing subcontraries with contradictories would not be the same as switching red with blue, or cats with dogs, but rather would amount of confusing dogs with canines. Let us call global confusion this kind of error by contrast to the first one that we can call switching confusion. As Slater claims through his red and blue example that paraconsistent logicians are making a switching confusion rather than a global one, it seems implicit that he doesn’t consider that all contradictories are subcontraries, neither do we here.

As I say, the fact that blue and red were chosen may just be for dramatic effect. In any case, although Slater did envisage a switch, calling red things blue and blue things red, the point of the example would have survived if red things came to be called blue but the blue things continued to be called blue. And this would have been more plausible since no party to the present debate could be described as calling contradictories subcontraries in the way that Slater accuses Priest of calling subcontraries contradictories. (The opening sentence of Béziau 1996 seems to offer such a description but this seems to be a matter of not being completely familiar with the ‘calling Xs Ys’ construction in English.) Despite its having (if I am right) no bearing on the merits of Slater’s objection to dialetheism, it would have been interesting to know whether in writing Slater (1995) he had in mind the ‘say less’ or the ‘say more’ interpretation.⁶⁸ End of Digression.

The use of the word ‘mere’ above is a special ad hoc technicalization of the ordinary use. In the everyday sense the dual intuitionistic negation of a formula is not a ‘mere’ subcontrary, if that means there is nothing more informative to be said on the matter, since (as recalled more than once already) it specifically forms the strongest subcontrary of the formula in question (and similarly in the case of intuitionistic negation proper and mere contrariety, interchanging ‘contrary’ and ‘subcontrary’ and ‘weakest’ and ‘strongest’). And in the case of intuitionistic logic itself, we note that there is always something stronger to be said than merely that two formulas are subcontraries, since by the Disjunction Property, one will be provable outright. Really this would be better described as the Subcontrariety Property, since the issue arises in $\vee$-free fragments of the logic (as was noted for LC in the Digression in Sect. 15.3): whenever formulas A, B are subcontraries in the sense there defined according to the fragment-specific consequence relation in question, one or other of them is a consequence of $\varnothing$. Indeed one can supply such an example for a fragment of classical logic too, taking the $\{\leftrightarrow \}$-fragment of $\vdash _\mathsf{CL}$: whenever A and B, constructed using only $\leftrightarrow$ from sentence letters, are subcontraries, at least one of A or B is a classical tautology.⁶⁹ Unlike the intuitionistic case, the Subcontrariety Property for the equivalential fragment of classical logic does not extend to the obvious ternary (or more) generalization of the relation of subcontrariety: for example, p, q and $p \leftrightarrow q$ are ‘ternary subcontraries’ (no Boolean valuation falsifying all of them), no one of which is tautologous. (However, the result does extend to allowing dependence on assumptions: If $\Gamma \vdash _\mathsf{CL} A \vee B$ and the only connective appearing in $\Gamma \,\cup \,\{A, B\}$, is $\leftrightarrow$, then either $\Gamma \vdash _\mathsf{CL} A$ or $\Gamma \vdash _\mathsf{CL} B$. A proof of this, which requires some apparatus going beyond that deployed for the $\Gamma = \varnothing$ case treated in Footnote 69, can be found in Humberstone 2019; see Proposition 1.4(ii) there.)

An attempt at precisifying something like the ‘mereness’ idea for coercive binary logical relations is set out in pp. 200–204 of my (Humberstone 2013b), which involves passage from such a relation, R, to the relation $\mu (R)$ of standing minimally in the relation R, with $\mu (R)$ defined to hold between A and B when (1) R(A, B) and (2) there is no (coercive) binary logical relation $S \,\subsetneq R$ such that S(A, B).⁷⁰ Thus if R is the relation of subcontrariety, its minimization $\mu (R)$, unlike R itself, never holds between contraries A and B, since if the relation of being contradictories held between them, that would then be a proper subrelation S of R for which S(A, B), violating (2). The discussion of logical relations in Humberstone (2013b) has many features oriented specifically to the setting of classical logic but we can apply this idea to the Disjunction Property, to revert to this slightly misleading label, along with the name $R^{\vee }$ for the relation of subcontrariety, now to be understood as ‘according to the consequence relation of intuitionistic logic’. As observed in the previous paragraph, in this setting subcontrariety is a somewhat degenerate affair, in that there is always something much more informative that can be said, since one or other of the relata is outright (intuitionistically) provable. In terms of the present apparatus this amounts to: $\mu (R^{\vee })\, = \, \varnothing$. The reason is as follows. Writing (respectively, ) for the relation holding between any two formulas the first (respectively, the second) of which is IL-provable, the Disjunction Property means that , so whenever $R^{\vee }(A, B)$ there is a proper subrelation of $R^{\vee }$, whether it be or (or indeed their intersection) which relates A and B; thus we can never have $\mu (R^{\vee })(A, B)$. This example shows that being ‘minimal subcontraries’ does not coincide with being ‘mere subcontraries’ as defined above, since, for example, p and $q \rightarrow q$ are mere subcontraries (subcontraries which are not contraries—equivalently, contraries which are not contradictories) but not minimal subcontraries (since they stand in the relation ). Note that the present example can be understood with either classical or intuitionistic logic as the background setting—the Disjunction Property is not required. (For the intuitionistic case the appropriateness of our official definition of subcontrariety will not be taken for granted in Appendix 2 below.)

Finally, let us note that if one applies the ‘mere’ modifier in connection with contrary or subcontrary-forming operators, an ambiguity arises, as we illustrate in the contrariety case. A 1-ary operator $\#$ may be called a mere contrary-forming operator in order to convey that for every formula A, $\#A$ and A are mere contraries (contraries which are not contradictories). On the other hand one may by this same phrase intend to convey that $\#$ is a contrary-forming but not a contradictory-forming operator: in other words while for every formula A and $\#A$ are contraries, it is not the case that for every formula A, A and $\#A$ are contradictories. (To get back to the first—stronger—sense, replace this last every by any.) It is in this second sense that intuitionistic negation is a mere contrary-forming operator: see the example just before the Digression above. In Appendix 2 we look at the claim that (pace Priest among others), intuitionistic negation forms contradictories. These clarificatory remarks also apply if instead of taking a contrary-forming (or subcontrary-forming or contradictory) operator to be—as we have been presuming—a 1-place primitive or derived connective, but any function from formulas to formulas. In any extension of the normal modal logic KD (sometimes called D), for instance,Otherwise the intuitionist $\Box p$ and $\Box \lnot p$ we can ask whether these are contraries, and answer affirmatively. (The ‘say more’ advocate will have to hedge here and say, for example, that they are contraries in KD but not in its extension KD!,⁷¹ where they are contradictories.) First, we can consider a (partial) function f which assigns to each formula $\Box A$ the formula $\Box \lnot A$. This, in the extended sense of the phrase, would then count as a contradictory-forming (better: contradictory-yielding) operator, but it is what Humberstone (2011a) calls a non-connectival operation on formulas, in that there is no KD-definable connective $\#$ for which $\#\Box A$ is KD-provably equivalent to $\Box \lnot A$. Indeed with A as, e.g. p, $\Box \lnot A$ is a paradigm case of contrariety from the modal square of opposition in some sense in which another mere contrary for all that has been said in this paper, namely $\lnot (\Box p \vee \Box \lnot p)$, is not: see Sect. 3 of my (Humberstone 2005b), for this and references to the work of Robert Blanché, who saw in this last formula together with $\Box p$ and $\Box \lnot p$ a triangle of formulas all mutually contrary in the only sense (as Blanché felt) there is. What is involved here (further elaborated in Humberstone 2005b) may be connected with what Horn (1989) calls polar contrariety; the issue (as with the corresponding issue for subcontraries) has been kept out of sight in the present paper as not being of special significance in connection with Priest’s discussion of negation and the reactions of commentators to that discussion. However, Horn does have a useful discussion, with historical references, to the main subject—‘say less’ versus ‘say more’—on p. 36f. of Horn (1989).

Appendix 2: Wansing on Priest on Negation

At the end of the Digression in Sect. 15.3 it was mentioned that Wansing does not share Priest’s view that intuitionistic negation forms mere contraries, preferring to describe it as genuinely contradictory-forming. Here we examine his reasons. For conformity with the rest of the notation used above, ‘$\alpha$’ is replaced by ‘A’ and to avoid a conflict with the ‘(1), (2)’ of Sect. 15.1, that numbering has been replaced by ‘(i), (ii)’ in the following (somewhat ellipsed) quotation from p. 92f. of Wansing (2006), though the unparenthesized ‘1, 2’ remain as in Wansing (2006). The references to Priest are to (1999), which this passage begins by summarizing—essentially the same as the material from Priest (2007), p. 467 (and Chap. 4 of Priest 2006b):

The set $\{A, \lnot A\}$ thus forms a contradictory pair if the LNC and the LEM are provable:

$(i)\,\vdash \lnot (A \wedge \lnot A)$ and $\vdash A \vee \lnot A$.

A reformulated statement of the contradictoriness of $\{A, \lnot A\}$ corresponding to the reading Priest attributes to traditional logic and common sense is:

1. A and $\lnot A$ cannot both be true.

2. One of A and $\lnot A$ must be true.

On another reading, $\{A, \lnot A\}$ is a contradictory pair if

1. Necessarily, if A is true, then $\lnot A$ is false.

2. A and $\lnot A$ cannot both be false.

If the Deduction Theorem is assumed (to avoid using implication), this reading gives

$(ii)\,A \vdash \lnot \lnot A$ and $\vdash \lnot (\lnot A \wedge \lnot \lnot A)$.

If the contradictoriness of $\{A, \lnot A\}$ is expressed by (i), intuitionistic negation fails to be contradictory-forming, and if the contradictoriness of $\{A, \lnot A\}$ is expressed by (ii), intuitionistic negation does give rise to contradictions. As Priest (1999, p. 107) observes, in order to consider the truth conditions of negated statements, we ‘need a definition of falsity. Let us define “A is false” to mean that $\lnot A$ is true. (...) And the present definition is one that all parties can agree upon, classical, intuitionist and paraconsistent’. Since Priest regards intuitionistic negation to be not a contradictory-forming connective but a contrary-forming one, his explanation cannot mean that A is false iff its contradictory is true. Otherwise, the intuitionist as Priest conceives of him or her should not agree upon the definition of falsity. Thus, the notion of falsity in a logic $\Lambda$ is expressed by the negation operation used in $\Lambda$ (assuming that $\Lambda$ comprises only one ‘official’ negation and taking into account that this may not be a contradictory-forming connective and hence not a negation in Priest’s sense). The question thus is: When is $\{A, \lnot A\}$ a contradictory pair?

Now, none of this—Priest’s discussion or Wansing’s development of it—is really methodologically optimal, as Section 1 of the present paper has tried to make clear. The question that should be asked is not what makes $\{A, \lnot A\}$ a contradictory pair, but what makes $\{A, B\}$ in general to a contradictory pair, or more simply put, what it is for A and B to be contradictories. One then applies the general ruling to the special case in which B is $\lnot A$. The most plausible general proposal to see as implicit in the above passage arises by putting ‘B’ in place of all of its occurrences of ‘$\lnot A$’. A further moral of the discussion of the present paper is that by taking the ‘say more’ approach Priest and Wansing make it hard to describe the separate ingredients of their differing proposals permitted on the ‘say less’ approach with the simple vocabulary of contrariety and subcontrariety. (Wansing goes on to consider contrariety specifically in Sect. 3 of Wansing (2006) but this concerns the mixed coercive–permissive relation of mere contrariety rather than just the ‘incompatibility’ conception of contrariety of the ‘say less’ approach.) The first and second conjuncts of (i) are Priest’s chosen ways of formulating the contrariety and subcontrariety conditions on A and B for the case of $B\, =\, \lnot A$, whereas the first and second conjuncts of (ii) are Wansing’s preferred formulations of the same two conditions. Moving to the general level, then, we may say that Wansing’s preference for the contrariety and subcontrariety conditions for A, B according to a consequence relation $\vdash$ is given by:

Wansing contraries iff $A \vdash \lnot B$; Wansing subcontraries iff $\vdash \lnot (\lnot A \wedge \lnot B)$;

Priest contraries iff $\vdash \lnot (A \wedge B$); Priest subcontraries iff $\vdash A \vee B$.

Some criticisms made in Sect. 15.1 of Priest’s discussion applies to Wansing’s also: to bring to bear a criterion of contrariety or subcontrariety (or the two of them together, for contradictoriness) on the question of whether something and its negation stand in the chosen relation, the clean approach would formulate the various criteria without deploying that very connective in the formulation, and in fact we went on to prefer formulations purified not only of $\lnot$ but also of $\wedge$ and $\vee$. (For example, for $\vdash$ with $\vee$ in its language the Priest subcontrariety condition will not deliver the intended results unless $A \vee B$’s consequences are the common consequences of A and B, since $\vdash$ may fail to be $\vee$-classical.) But Wansing here is not defending Priest’s approach, so much as looking for ‘Priest-style’ criteria that might be preferable, and, let us recall, specifically noting that his preferred criteria rule that according to the consequence relation $\vdash _\mathsf{IL}$ of intuitionistic logic, since for any formula A, A and $\lnot A$ are Wansing contradictories in the sense of being both Wansing contraries and Wansing subcontraries, even though they are not Priest contradictories because they are not in general Priest subcontraries.⁷²) (As we noted in Appendix 1, intuitionistic subcontrariety à la Priest—purged of any ‘say more’ trappings—is something of a degenerate relation because of the Disjunction Property.) Note that, by contrast with the case of subcontrariety, Wansing contrariety and Priest contrariety coincide for $\vdash \,=\, \vdash _\mathsf{IL}$ and we use the terminology of Wansing contrariety for these equivalent formulations below (so that we can use ‘Wansing contradictories’ for pairs that are both Wansing contraries and Wansing subcontraries).⁷³

Wansing does not say a lot in favour of his subcontrariety condition, which would of course coincide with the disjunctive version against the background of $\vdash _\mathsf{CL}$ rather than $\vdash _\mathsf{IL}$, though certainly it looks like a more direct transcription of—‘2’ under (i) in the quoted passage—‘A and $\lnot A$ cannot both be false’—once $\lnot$ is connected, in the manner Priest suggests, with falsity. Whatever one wants to call it, there is a perfectly good logical relation here, holding—to purify the characterization—between A and B (according to $\vdash$) when the weakest contraries (according to $\vdash$) of A and B are themselves contraries. (Here we rely on the purified—and rather paraconsistency-unfriendly) characterization of contrariety given at the start of the Digression in Sect. 15.3.⁷⁴) There are many families of discernible logical relations of interest each collapsing to a single familiar relation in the setting of $\vdash _\mathsf{CL}$, though leading separate lives in the more discriminating habitat of $\vdash _\mathsf{IL}$, such as the relation of what in Humberstone (2011a), e.g. Remark 4.22.10 on p. 555, is called pseudo-subcontrariety, holding between A and B in that order when $A \rightarrow B \vdash B$. (Formulation purifying away the $\rightarrow$ to get the intended general effect: whenever for all $\Gamma$ if $\Gamma , A \vdash B$ then $\Gamma \vdash B$.) Intuitionistically (i.e. taking $\vdash$ as $\vdash _\mathsf{IL}$) this relation is not symmetric, so the ‘in that order’ is essential. On the other hand any talk of subcontraries with or without the ‘pseudo’ prefix, is inclined to suggest symmetry because of the wording ‘A and B are subcontraries’, so in Humberstone (2001) when A and B stand in this relation we say A anticipates B (according to $\vdash$); this paper looks into several aspects of this relation (see also Humberstone 2011a, p. 625f. for an update). Alternatively, one could symmetrize pseudo-subcontrariety by taking the intersection of the relation with its converse, which amounts to requiring the provability of

$$((A \rightarrow B) \rightarrow B) \wedge ((B \rightarrow A) \rightarrow A),$$

well known as the definiens in a definition of $A \vee B$ in the intermediate logic LC, mentioned already in the Digression in Sect. 15.3, though for this point one really must give the fundamental reference, Dummett (1959b). According to Proposition 1.1 in Humberstone (2002) two formulas A and B are implicit implicational converses in intuitionistic logic, in the sense that there are formulas $C \rightarrow D$ and $D \rightarrow C$ respectively equivalent to A and B, if and only if the conjunction inset above is intuitionistically provable. (Alternatively put: iff A and B are mutual pseudo-subcontraries according to $\vdash _\mathsf{IL}$; for $\vdash _\mathsf{CL}$ we can just say ‘are subcontraries’—i.e. replace the above formula with $A \vee B$, which indeed we can already say at the level of the intermediate $\vdash _\mathsf{LC}$, in view of the above definability point.) But let us return to Wansing’s discussion of contradictoriness, to see him dismissing three other candidate characterizations (Wansing 2006, p. 83).

Among the various classically equivalent Priest-style formulations of Aristotle’s characterization of contradictoriness, there are three pairs of conditions that may be set aside:

• (a) $A \vdash \lnot \lnot A$,   $\lnot A \vdash \lnot A$

  (Necessarily, if A is true, then $\lnot A$ is false. Necessarily, if A is false, then $\lnot A$ is true.)

• (b) $\lnot A \vdash \lnot A$,   $\lnot \lnot A \vdash A$

  (Necessarily, if $\lnot A$ is true, then A is false. Necessarily, if $\lnot A$ is false, then A is true.)

• (c) $\vdash \lnot (A \wedge \lnot A),\quad \vdash \lnot (\lnot A \wedge \lnot \lnot A)$

  (A and $\lnot A$ cannot both be true. A and $\lnot A$ cannot both be false.)

In (a) and (b) the condition $\lnot A \vdash \lnot A$ holds because of a general property of derivability. In (c), one condition is an instantiation of the other. We may ask whether or not the remaining classically valid and classically equivalent pairs of conditions, exhibiting in addition to $\lnot$ at most $\wedge$ and $\vee$, hold in various non-classical propositional logics.

The reference to ‘the remaining classically valid and classically equivalent pairs of conditions’ is perhaps to those collected in a table on the following page of Wansing (2006), though it could be interpreted quite generally as referring to arbitrary further pairs of conditions. As before, abstracting from the above formulations to the underlying binary relations involved in A and B’s being contradictory according to the various proposals, we get general conditions on A and B which are all classically equivalent, which, for the special case on which the above passage concentrates in which B is $\lnot A$, conditions are not just equivalent but classically valid (satisfied for arbitrary A, according to the consequence relation of classical logic, that is). And for a clearer focus on the issues, it pays to consider separately the contrariety and subcontrariety components in each case, and do so in the generalized setting—which is to say as conditions on arbitrary A and B, rather than just taking B as $\lnot A$. This removes any inclination to think of the second condition for (c) as a redundant special case (substitution instance) of the first, since we now see that Table 1’s contrariety and subcontrariety conditions as independent (still reading $\vdash$ as $\vdash _\mathsf{IL}$), and in particular we can have the contrariety condition satisfied for a particular A, B while the subcontrariety condition is not. For example, take $A\, =\, p$ and $B\, = \, \lnot p \wedge q$. Thus, in the terminology introduced above, these formulas are (intuitionistically) Wansing contraries but not Wansing subcontraries. (Conversely, keeping A as here and changing $\wedge$ to $\vee$ in B gives us a pair of Wansing subcontraries—though not Priest subcontraries, for the reason given in Appendix 1—which are not Wansing contraries.)

Table 15.1 Classically equivalent contrariety and subcontrariety conditions

Table 15.1 separates out the general components in play in the passage from Wansing (2006) last quoted. Let us consider the various conditions with special reference to the choice of $\vdash$ as $\vdash _\mathsf{IL}$, so as to connect with Wansing’s proposal that $\lnot$ is a contradictory-forming operator in intuitionistic logic. The first point that jumps out is that the three contrariety conditions are all equivalent, in the sense that for any A, B, any one of these conditions is satisfied (with $\vdash$ as $\vdash _\mathsf{IL}$) if and only if any other is satisfied. We noted this already before in connection with the contrariety conditions for (a) and (c), temporarily baptized Wansing contrariety and Priest contrariety, so Wansing’s own favoured proposal, combining Wansing contrariety with Wansing subcontrariety (the subcontrariety condition for (c) in Table 15.1) is in fact equivalent, in the intuitionistic case, to proposal (c), which Wansing writes ‘may be set aside’. for the reason that ‘in (c), one condition is an instantiation of the other’. Wansing was there, of course, looking at the special case in which B was $\lnot A$, but we can see in Table 15.1 that the same holds in the general case: the subcontrariety condition is the special case or ‘subschema’—though of course in this case not generally provable—of the contrariety condition, in that every instance of the former is an instance of the latter (though not conversely). But this is not an objection to the characterization of contrariety and subcontrariety in this way, since we would simply be saying that formulas are subcontraries (according to $\vdash _\mathsf{IL}$) iff their negations are contraries—an entirely familiar state of affairs: think of the Square of Opposition. In the present case we can equally say, as we can there, that on proposal (c) formulas are contraries iff their negations are subcontraries, since the outer negations allow for double negation replacements—or indeed any classically sanctioned (propositional) replacements which would otherwise raise an intuitionist’s eyebrow—in their scopes (by Glivenko).

That same consideration reminds us that while Wansing’s conception of contrariety coincides with the Priest’s conception, his subcontrariety condition differs from Priest’s disjunctive conception—a point in its favour in view of the degenerate nature of the latter, even though it is Priest’s rather than Wansing’s notion that amounts in the present setting to subcontrariety according to the consequence relation $\vdash _\mathsf{IL}$, as this was given above in ‘connective-free’ terms (last recalled in Footnote 34). Wansing’s version can be thought of as replacing the reference to $A \vee B$ in Priest’s with its double negation—intuitionistically equivalent to $\lnot (\lnot A \wedge \lnot B)$. (Of course, the positions are here described in ‘say less’ terminology rather than that Priest and Wansing themselves officially adopt.) There is no need to exclude it on the grounds that the subcontrariety half follows from the contrariety half of the condition, since, as we have seen, this is not so in general for the two forms as defining as binary relations, neither of the binary contrariety and subcontrariety relations given in (c) of Table 15.1 is included in the other. And this is just as well, since it coincides with what Wansing wants to say in the intuitionistic case.

The subcontrariety conditions recorded for (a) and (b) are not, by contrast with the corresponding contrariety conditions, equivalent to each other (taking $\vdash$ as $\vdash _\mathsf{IL}$), and are evidently close relatives of the pseudo-subcontrariety conditions mentioned above.⁷⁵ There is no reason why they should not be given consideration as subcontrariety-like conditions, preferably (again) in a form purified at least of the presence of $\lnot$. This multiplicity of options arises for the same reason as we noted in Sect. 15.4 that when the true and the false no longer partition the field, ‘what might otherwise be taken to be mere reformulations now emerge as non-equivalent possibilities’. In the case of subcontrariety in IL the situation is even more complicated, since we don’t just have to distinguish ‘at least one of A, B is true’ (the $A \vee B$ version of subcontrariey) from ‘not both A, B are false’ (the Wansing version, under (c) in Table 15.1), but we have also these further options (using $(A \rightarrow B) \rightarrow B$, $\lnot A \rightarrow B$, etc., once we are considering, as with Wansing subcontrariety, departures from the abstract characterization mentioned in Footnote 34 and earlier). These are, of course, repercussions of the generally greater discriminatory power of deductively weaker logics—the subject of Humberstone (2005c)—though Humberstone (2011a) does suggest some guidance in looking for the most direct intuitionistic analogues of classical connectives in the shape of their topoboolean incarnations in the Kripke semantics (making the intuitionistic compound true at a point in a model if the corresponding classical compound is true at all accessible points, that is). See, for example, p. 786ff. of Humberstone (2011a), where this is applied in the case of exclusive disjunction (${\veebar }$, say), a particularly significant connective for the present discussion since the Lemmon-style logical relation (along the lines of $R^{\vee }$ in Appendix 1) $R^{{\veebar }}$, induced by this connective in the classical setting is the relation of being contradictories.⁷⁶

We conclude with some general remarks on the subject of Wansing contradictories (in intuitionistic logic). Let us denote the set of Wansing contradictories of a formula A by WCtd(A), recalling here the precise definition:

$$\textit{WCtd}(A) = \{B :\, \vdash _\mathsf{IL} \lnot (A\wedge B) \wedge \lnot (\lnot A \wedge \lnot B)\}.$$

As one expects, the relation here is symmetric: for any A, B, $A \in \textit{WCtd}(B)$ if and only if $B \in \textit{WCtd}(A)$. Another expectation—cf. the title of Cresswell (2008)—might be that contradictories are unique, i.e. that for any A, all formulas in WCtd(A) are (IL-)equivalent, but it is easy to see that this expectation is not fulfilled for Wansing contradictories, since, for example, WCtd($\lnot p$) contains both p and $\lnot \lnot p$. A certain interest accordingly attaches to the question of how much variation one might expect to find within WCtd(A) for a given A.

For example, one can ask whether there is in general for any given formula, strongest or weakest Wansing contradictories in intuitionistic logic, and for assistance in pursuing that question, make the observation that the set of Wansing contradictories of a formula WCtd(A), coincides with the set of formulas which are classically equivalent to $\lnot A$. If A is, for example, ${\lnot p \vee q}$ WCtd(A) contains $\lnot (\lnot p \vee q)$, $p \wedge \lnot q$, and (infinitely) many other non-IL-equivalent formulas. Relatedly, one could explore the inverse image under $\lnot$ of a formula C in intuitionistic logic, where this is understood along the same lines as the corresponding notion in (classically based) modal logic, for which (Humberstone 2013a) introduces the notation, for a monomodal logic S, ‘$\Box ^{-1}[C]^{S}$’ to denote the set $\{B\text {: } \vdash _{S} \Box B \leftrightarrow C\}$, called the inverse image under $\Box$ of the formula C, according to S. For present purposes, one would be interested in the set $\lnot ^{-1}[C]^{\textit{IL}}$, comprising those formulas whose negations are intuitionistically equivalent to C. This will be empty if C has no equivalent with $\lnot$ as main connective, but in the simple case of $C = \lnot p$ contains not only p and $\lnot \lnot p$, already mentioned, but also such things as $p \wedge (q \vee \lnot q)$, not equivalent to either of them. More generally a little calculation reveals that for any formula B:

$$\lnot ^{-1}[\lnot B]^{\textit{IL}}\, = \,\{A\text { : } A \dashv \vdash _{\mathsf{CL}} B\}.$$

We see that there is no IL-strongest formula in general in such a set, since given any formula B not containing the sentence letters $q_1,\dots q_n,\ldots$ the sequence:

$$ \begin{array}{cc} &{}B\\ &{}(q_1 \vee \lnot q_1) \wedge B\\ &{}(q_2 \vee \lnot q_2) \wedge ((q_1 \vee \lnot q_1) \wedge B)\\ &{}(q_3 \vee \lnot q_3) \wedge \big ((q_2 \vee \lnot q_2) \wedge ((q_1 \vee \lnot q_1) \wedge B)\big )\\ &{}\vdots \end{array} $$

contains ever stronger formulas, as long as B is IL-consistent. There is a similar sequence of ever weaker formulas to establish the dual claim, obtainable from the above sequence by replacing all occurrences of $\wedge$ with $\rightarrow$ (and replacing the proviso on B with: ‘as long as B is not IL-provable’). To tie this up with our discussion of Wansing contradictories, and clarify the recent use of the word ‘relatedly’, note that for any formula C,

$$\textit{WCtd}(C) = \lnot ^{-1}[\lnot \lnot C]^{\textit{IL}}$$

and thus

$$\textit{WCtd}(C) = \{A\text { : } A \dashv \vdash _\mathsf{CL} \lnot C\}.$$

Such a characterization tells immediately, for example, that the set of Wansing contradictories of any given formula is closed under $\wedge$ and $\vee$. But with this we conclude our reflections on (some of the strands in) Wansing’s discussion of Priest on negation.