The Bochum Plan and the Foundations of Contra-Classical Logics

The Bochum Plan and the foundations of contra-classical logics

Luis Estrada-González Institute for Philosophical Research-UNAM [email protected]

1 Introduction

Until very recently, most of the more well-known non-classical logics —constructive logics, relevance logics, paraconsistent logics, and so on— were subclassical. Contra-classical logics are logics with valid arguments that are invalid in classical logic (over the same underlying language). Examples of such contra-classical logics are syllogistic, as it validates Subalternation, Darapti or Camestros, or connexive logics, that validate Aristotle’s Theses —the schemas ∼ (A →∼ A) and ∼ (∼ A → A)— and Boethius’ Theses —(A → B) →∼ (A →∼ B) and (A →∼B) →∼(A → B)—. Wansing [48] obtained a connexive logic by tweaking the falsity condition for the conditional in Nelson’s logic N4, with the feature that the converses of Boethius’ Theses also hold. Then Omori [32] used the same idea, tweaking the falsity condition for the conditional on top of LP to get another connexive logic, dLP. After that, he has shown (cf. [31]) that a number of well-known and new paraconsistent and relevant logics can be obtained also by tweaking appro- priately the falsity condition for some connectives while leaving the FDE-like truth and falsity conditions for the remaining ones, in most cases even negation, ﬁxed.1 More recently, Wansing and Unterhuber [52] tweaked the falsity condition of Chellas’ basic conditional logic CK and they obtained a (weakly) connexive logic. Even more recently, Omori and Wansing [35] have put forward a systematization of connexive logics based on certain controlled tweakings in the conditional’s truth and falsity conditions, showing that, in general, tweaking the truth condition has lead only to weak connexivity (Boethius’ Theses hold only in rule form, if at all), whereas tweaking the falsity condition has lead to hyperconnexivity (not only do Boethius’ Theses hold, but also their converses). Let me call that general approach to non-classical logic the ‘Bochum Plan’, in analogy with the Australian, American and Scottish plans for relevance.2

1There are plenty presentations of Belnap and Dunn’s FDE and of González- Asenjo’s/Priest’s logic LP; the reader can choose their favorite ones but the relational ones as found since [13] and [41], respectively, would be particularly helpful. 2If the size of the geographical region alluded has anything to do with the number of proponents of the respective plans, that is left to the reader to guess.

1 However, the scope of the Bochum Plan is far broader than paraconsistency and relevance, as the connexive logics obtained show. Omori and Wansing [34] have generalized that idea of tweaking a falsity condition to get other contra- classical logics, not only connexive logics in the ballpark of Wansing’s. Roughly, the recipe they followed for crafting a contra-classical logic is as follows: - Give the (model-theoretic) semantics for a logic L, in the FDE-like style; - take at least one connective, let its truth condition fixed and tweak its falsity condition by making it almost the falsity condition of some other connective;3 - let the truth and falsity conditions for the remaining connectives fixed. The question I am interested in here is basically the one famously and vividly voiced in [9] in a similar context, namely whether the Bochum Plan is more than an excellent machinery for systematically crafting or modeling crazy logics, or whether it can serve as a semantic project in a more robust way, or a plan properly speaking: a semantic theory that serves to explain the more distinctive features of a (family of) logic(s) and even to make them plausible to some extent. I think that the fact that the Bochum Plan is not only a tool for generating new contra-classical logics but that it also encompasses old ones and helps in suggesting what the source of contra-classicality is in a great number of cases —viz. a very special, deviant understanding of the falsity condition for some connective— speaks for its semantic robustness. That also tells it apart from other machineries where, notwithstanding other virtues, new contra-classical logics appear only as by-products of some other semantic features and there is nothing to be said about well-known cases of contra-classicality.4 My main claim is that, in spite of its impressive achievements, there is much yet to be told if the Bochum Plan is going to serve as an even wider theory of logics. In particular, three objectives should be attained if one aims at such a more general theory of logics: first, if one detaches the Bochum Plan from the idea of leaving the truth conditions fixed, which seems unessential to the Plan although it served excellently to exemplify its power; second, if one frees the Plan from the FDE-like presentation of logics, so more possibilities can be covered; third, if one can provide at least some rough admissibility criteria for the tweakings required by the Bochum Plan. In doing this, one could glimpse to new, more elaborate theories for the meaning of logical connectives. The structure of the paper is as follows. In the next section, I present some basic definitions useful for the rest of the paper. In Section 3 I present some examples of contra-classical logics. In Section 4, I present the basic features of Omori and Wansing’s Bochum Plan, and show how it can be used to produce some well-known contra-classical logics scattered through the literature. In Sec- tion 5 I address some questions raised by the Bochum Plan, leaving for Section 6 the discussion of an issue that suggests that the Bochum Plan should be even more general than it already is, according to the first objective above. Finally, in Section 7 I discuss some prospects for accomplishing the two remaining objectives.

3The tweakings for getting other more familiar non-classical logics are less radical. 4As an example of the latter, see [17].

2 2 Some working deﬁnitions

Let me begin with some useful deﬁnition for the rest of the paper. Let L be a language containing m n-ary connectives {c1, . . . , cm}. Let also Γ and ∆ be sets of formulas of L. A logic L∗ based on a language L∗ ⊆ L is said to be sub-L if and only if ∗ every L -valid argument Γ L∗ ∆ is also L-valid, but there is an argument that is L-valid but not L∗-valid. Said otherwise, and mostly in English, a logic L∗ is sub-L if and only if, over the same language, its valid arguments form a proper subset of those of L. A logic L∗ based on a language L∗ ⊆ L is said to be contra-L if and only ∗ if there is an L -valid argument Γ L∗ ∆ such that Γ 1L ∆. Said otherwise, and again mostly in English, a logic L∗ is contra-L if and only if, over the same language, there are arguments that are valid according to it but invalid according to L. Finally, a logic L∗ is non-L if and only if it is either sub-L or contra-L. A formula A of a logic L is said to be a theorem of this logic if either Γ L A, ∆ or Γ L ∆,A, for any choice of Γ and ∆; an antilogy of this logic is any formula A such that if either Γ,A L ∆ or A, Γ L ∆, again for any choice of Γ and ∆. For simplicity, let me stick right now to the case that will occupy me here, that when L is zero-order classical logic with its usual connectives (two nullary, >, ⊥; one unary, ∼; four binary, ∧, ∨, →, ↔). Also, it will be more convenient if we restrict ourselves just to the SET − F MLA version of classical logic5 and, in particular, if we consider only F MLA cases of contra-classicality.6

3 Examples of contra-classical logics

Aristotelian logic and its medieval successors have two central parts: one of them is the theory of oppositions and the other is syllogistic. This is not the place to even start giving the basics of Aristotelian and medieval logics. Here I will assume the reader’s familiarity with the theory of oppositions and syllogistic. Since there are several good starting points on the topic, I recommend [7] almost randomly. Nowadays, the most natural translation of the syllogistic forms into classical logic is as follows:

Medieval mnemonics English-with-variables First-order language SaP All Ss are P s ∀x(Sx → P x) SeP No Ss are P s ∼ ∃x(Sx ∧ P x) SiP Some Ss are P s ∃x(Sx ∧ P x) SoP Some Ss are not P s ∃x(Sx∧ ∼ P x)

5That is, where arguments are pairs of sets of premises and a single conclusion. 6That is, the limit case of valid arguments where SET is empty.

3 Given this translation, syllogistic gives verdicts concerning the validity of some syllogisms that are inconsistent with classical logic. Consider the inferences called by the medievals Darapti and Camestros, which are, respectively:

All Bs are Cs All Cs are Bs All Bs are As No As are Bs Hence some As are Cs Hence some As are not Cs

Both of these are valid syllogisms. Both are invalid in classical logic. Hence, syllogistic is contra-classical.7 Nonetheless, I said that I am going to restrict myself to zero-order logics, so I will say nothing else about syllogistic; I mentioned it only to show a well-known example of a contra-classical logic, that cannot be dismissed as an eccentricity derived from a too mathematical approach to logic. Having said that, here are some examples of contra-classical logics developed during the 20th and 21st centuries:

Connexive logics ([26], [50], [51]): logics such that ∼(A →∼A) Aristotle’s Thesis ∼(∼A → A) Variant of Aristotle’s Thesis (A → B) →∼(A →∼B) Boethius’ Thesis (A →∼B) →∼(A → B) Variant of Boethius’ Thesis are logically valid in L, and, furthermore, (A → B) → (B → A) Symmetry of implication is not valid in L. Actually, connexive implication is motivated in [25] by reproducing in a first- order language all valid moods of Aristotle’s syllogistic. The name ‘connexive logic’ suggests that systems of connexive logic are motivated by some ideas about coherence or connection between premises and conclusions of valid inferences or between formulas of a certain shape, and in that general, intuitive sense, they are closely related to relevance logics; see for example [45]. Another motivation for connexive logic have been subjunctive and causal conditionals, because it seems that ‘If an object is dropped, it will not hit the floor’ contradicts ‘If an object is dropped, it will hit the floor’. On this, see [27].

Abelian logic ([28], [29]): logics containing the following axiom schema: ((A → B) → B) → A Axiom of Relativity In spite of being a classical contingency, the Axiom of Relativity is closely related to the double negation axiom. Remember that negation can usually be deﬁned as follows:

∼ A =def A → ⊥

7Whether Aristotle held that Camestros is valid is a moot point, but it certainly was regarded as valid in medieval syllogistic.

4 where the nullary connective ⊥ stands, as usual, for an arbitrary falsehood. The double negation axiom can thereby be reformulated in the following way: (Double negation axiom) ((A → ⊥) → ⊥) → A Meyer and Slaney point out that negation can be generalized if an arbitrary proposition, and not only a falsehood, is put in the consequent of the definiens. To this effect, Meyer and Slaney say that “[l]ogic should be in the business of telling us what follows from what, and not of advising us as to which propositions are despised and rejected. Is there a proposition so true that it logically cannot be taken as f [the value false] for some purposes? Our Modest Proposal is that there is not.” [28, 252]. Then, the motivation to define a relative negation in this fashion is precisely that, in principle, any proposition can play the logical role of a falsehood.8 As a result we get:

(Generalized negation) ∼B A =def. (A → B) That is, a conditional A → B is a negation of A relative to B. Then, the relativity axiom can be seen as a generalized version of the double negation axiom and as such, regardless of its contingent character within classical logic, it might as well be considered a logical truth on independent grounds. That a conditional is a generalized negation is not original of Meyer and Slaney, of course. The algebraic counterparts of negations in, for example, Heyting algebras, are usually called ‘pseudo-complements’, and conditionals are ‘relative pseudo-complements’, that is, relative negations. On the other hand, the “extended law of excluded middle” A ∨ (A → B) where the negated disjunct appears as a conditional with arbitrary consequent, although valid in classical logic, is more familiar in constructive and paraconsistent contexts; see [30, Chapter 2]. Originally, Abelian logic was simply the logic RW without the axiom schema of Double Negation Elimination plus the Axiom of Relativity. Humberstone [22] generalized the name to any logic whose implicative fragment included BCI plus Relativity. Although this can distract from Meyer and Slaney’s original motivations for developing —and naming— Abelian logic in the way they did, I prefer to call ‘Abelian logic’ any logic containing Relativity as a valid schema.

Super-contracting logics ([43], [44]): logics validating Super-contraction (A → (A → B)) → B. Super-contraction appeared in the context of investigating what kind of logics trivialize naïve theories (of sets, truth or properties). Super-contraction provided a counterexample to a conjecture by Restall that a logic supports (i.e. not trivializes) a naïve theory if and only if it is robustly contraction-free. 8At an algebraic level, Meyer and Slaney’s main concern was that the relevance logic R seemed suitable to be interpreted with groups, but that was not the case. With the Relativity axiom on top of RW, it is possible to understand negation as a complement and this allows to interpret the resulting logic A using Abelian groups.

5 Demi-negation logics ([21], [22], [23], [34], [36]): logics that validate some schemas of the form c (... ∼∼ Ai ... ∼∼ Aj ...) such that c (... ∼ Ai ... ∼ Aj ...) is classically valid. They can validate, for example, schemas like ∼∼ (A ∨ B) → (∼∼ A∧ ∼∼ B), (A → B) → (∼∼ B →∼∼ A) or ∼∼ (A → A) → B. In [11], [10, Ch. 17], [34] and [36] there are some hints on the kind of phenomena that demi-negation logics can help to study or be applied to, which range from quantum phenomena to the modeling of mixtures of aﬃrmation and denial.

Dialetheic logics: any logic L with contradictory pairs of theorems, that is, theorems Ti and Tj such that one of them is an L-negation of the other. There are too many references to choose even a few; moreover, most of the logics to be discussed below are also dialetheic, so they can be taken as examples of this sort of logics. Nonetheless, it must be said that the most famous logic for dialetheists, González-Asenjo/Priest’s LP, is not dialetheic.

Over-complete logics ([24]): logics where every formula is a theorem, where every argument is valid or where there is a formula which is both a theorem and an antilogy or where every formula is an antilogy. Contrary to common wisdom, those four cases are not equivalent in general.

4 The Bochum Plan

For definiteness and simplicity, let me state the FDE-like evaluation conditions to be used in what follows.9 A model for L is a triple hI, (R2,R3), σi, where I is a non-empty set (of indexes of evaluation); the Rn’s are n-ary relations on I; σ : I × L −→ C (V ), with C (V ) ∈ {P(V ), PP(V ), PPP(V ),...}, is an assignment of (generalized) truth values over V = {1, 0} to index-formula pairs. The idea that a formula A of L is evaluated at i is defined recursively as follows. For every i ∈ I: σ(i, p) ∈ C (V ) 1 ∈ σ(i, >) and 0 ∈/ σ(i, >) 0 ∈ σ(i, ⊥) and 1 ∈/ σ(i, ⊥) 1 ∈ σ(i, ∼A) iff for all j ∈ I with Rij, 0 ∈ σ(j, A) 0 ∈ σ(i, ∼A) iff for all j ∈ I with Rij, 1 ∈ σ(j, A) 1 ∈ σ(i, ◦A) iff for all j ∈ I with Rij, 1 ∈/ σ(j, A) or 0 ∈/ σ(j, A) 0 ∈ σ(i, ◦A) iff for all j ∈ I with Rij, 1 ∈ σ(j, A) and 0 ∈ σ(j, A) 1 ∈ σ(i, A ∧ B) iff for all j, k ∈ I with Rijk, 1 ∈ σ(j, A) and 1 ∈ σ(k, B) 0 ∈ σ(i, A ∧ B) iff, for all j, k ∈ I with Rijk, either 0 ∈ σ(j, A) or 0 ∈ σ(k, B) 1 ∈ σ(i, A ∨ B) iff, for all j, k ∈ I with Rijk, either 1 ∈ σ(j, A) or 1 ∈ σ(k, B) 0 ∈ σ(i, A ∨ B) iff for all j, k ∈ I with Rijk, 0 ∈ σ(j, A) and 0 ∈ σ(k, B) 1 ∈ σ(i, A → B) iff for all j, k ∈ I with Rijk, if 1 ∈ σ(j, A) then 1 ∈ σ(k, B)

9This is not exactly Omori and Wansing’s way of presenting them; I have deliberately chosen a more general form just in case the reader wants more parameters to vary and obtain even more logics, contra-classical or not.

6 0 ∈ σ(i, A → B) iﬀ for all j, k ∈ I with Rijk, 1 ∈ σ(j, A) and 0 ∈ σ(k, B) Let me show you now the Bochum Plan in action just for some contra- classical cases: I am sure the reader can adjust the parameters involved to obtain a lot of already well-known non-classical logics. When the consistency connective ◦ does not belong to the base language, considering its introduction or its deﬁnition using other connectives does not make a big change in most cases below, so it can be safely ignored unless otherwise stated. Also, for simplicity, let me work with C = P(V ) = {{1}, {0}, {1, 0}, ∅} unless otherwise stated, and let the elements of I be sets, in particular i = ∅ and i = j = k.

Example: logics with demi-negation. Let all the evaluation conditions fixed, except the falsity condition for negation, 0 ∈ σ(∼A) iff 1 ∈ σ(A) and replace this condition by the following condition 0 ∈ σ(∼A) iff 1 ∈/ σ(A) then one obtains Kamide’s [23] CP, a logic with demi-negation, which is also dialetheic as it validates both ∼ (A∧ ∼∼ A) and ∼∼ (A∧ ∼∼ A).

Example: connexive logics. Now let C (V ) \ ∅ and all the evaluation conditions fixed, except the falsity condition for the conditional, 0 ∈ σ(A → B) iff 1 ∈ σ(A) and 0 ∈ σ(B) and replace it by the following condition 0 ∈ σ(A → B) iff 1 ∈/ σ(A) or 0 ∈ σ(B) then one obtains Cantwell’s [8] CN (if ◦, a unary consistency connective is not included in the language) or the Olkhovikov-Omori logic dLP (if the language includes ◦). These connexive logics validate both Boethius’ and its converse together, i.e. Wansing’s Thesis ∼(A → B) ↔ (A →∼B), and are also dialetheic (see [32]).10

Example: logics with informational disjunction. Now let all the evaluation conditions fixed, except the falsity condition for disjunction, 0 ∈ σ(A ∨ B) iff 0 ∈ σ(A) and 0 ∈ σ(B) and replace this condition by the following condition 0 ∈ σ(A ∨ B) iff 0 ∈ σ(A) or 0 ∈ σ(B) Then one obtains logics with informational join, ⊕, like Arieli and Avron’s [2] BL⊃, where ∼ (A ⊕ B) ≡ (∼ A⊕ ∼ B) holds and which contains as theorems

10Richard Sylvan [47] dubbed ‘hyperconnexivism’ the thesis that, in addition to Aristotle’s and Boethius’ Theses, the converses of the latter also hold. In [49], hyperconnexive implication is motivated by introducing a negation connective into Categorial Grammar in order to express negative information about membership in syntactic categories. For example, if an expression of type A does not produce a sentence when combined with a name n, one could say that a non-sentence is produced when that expression of type A is combined with n. Another motivation for hyperconnexive theses has been presented by Cantwell in [8]. He considers the denial of some indicative conditionals in natural language and argues that the denial of, say, the conditional ‘If Oswald killed Kennedy, Jack Ruby did it too.’ amounts to the assertion that if Oswald shoot Kennedy then Jack Ruby did not. This suggests that (A →∼ B) is semantically equivalent with ∼ (A → B).

7 both ((A ⊃ A) ⊕ ∼ (A ⊃ A)) and ∼((A ⊃ A)⊕ ∼(A ⊃ A)), whence is also dialetheic.

Example: multi-contra-classical logics. If one tweaks at least two falsity conditions, for example, that for the conditional as in the dialetheic connexive logics above, that for disjunction as in BL⊃, and the falsity condition for conjunction so that one gets 0 ∈ σ(A ∧ B) iff either 1 ∈ σ(A) and 0 ∈ σ(B), or 0 ∈ σ(A) and 1 ∈ σ(B) then one obtains multi-contra-classical logics, like Francez’s [16] PCON, which inherits the contra-classical features of the logics on which it is based.11 All this works even if the indexes of evaluation are not made irrelevant as in the examples above. If in the truth condition for the conditional one stipulates that k = j but i 6= j, that is, one works with a conditional with positive features of a conditional like that of intuitionistic logic as follows 1 ∈ σ(i, A → B) iff for all j ∈ I with Rij, if 1 ∈ σ(j, A) then 1 ∈ σ(j, B) but the falsity condition is instead 0 ∈ σ(i, A → B) iff for all j ∈ I with Rij, if 1 ∈ σ(j, A) then 0 ∈ σ(j, B) then one gets Wansing’s logic C (see [50]), which is connexive and dialetheic yet distinct from CN and dLP.

5 Philosophical considerations

The Bochum Plan for contra-classicality certainly works. One can obtain in such a way a number of old and new contra-classical logics. At this point, several questions can be asked. First, what is the connection between contra- classicality and the falsity conditions such that tweaking the latter produce the former? Tweaking the falsity conditions seem neither necessary nor sufficient in order to get contra-classical logics. As the history of connexive logic witnesses, most contra-logics have been obtained by tweaking the truth condition instead; see for example the connexive logics so obtained in [1], [25], [39], [40] and [42]. Moreover, not any tweaking in the falsity conditions produces contra-classical logics, not even a large number of such tweakings. Again, assuming that i = j = k, if one tweaks the falsity conditions of all of ∼, ∧, ∨ and → as follows 0 ∈ σ(i, ∼A) iff 0 ∈/ σ(j, A) 0 ∈ σ(i, A ∧ B) iff either 1 ∈/ σ(j, A) or 1 ∈/ σ(k, B) 0 ∈ σ(i, A ∨ B) iff 1 ∈/ σ(j, A) and 1 ∈/ σ(k, B) 0 ∈ σ(i, A → B) iff 1 ∈ σ(j, A) and 1 ∈/ σ(k, B) one obtains Sette’s P1, which is not contra-classical12; in fact, it is paraconsistent at the atomic level and classical for molecular formulas. Nonetheless, remember that it is not any tweaking what does the trick in the Bochum Plan, but only those that make the falsity condition close enough to

11‘PCON’ stands for ‘poly-connexive logic’, because the technique of tweaking the falsity condition is extended to all the other connectives. However, this does not make the logic “poly-connexive”, but rather multi-contra-classical. 12This result can be easily deduced from the work done in [33].

8 the classical falsity condition of some other connective. Although ‘close enough’ is vague, in the present case one can assume the limit case when such a tweaked falsity condition is identical to the (classical) falsity condition of some other connective. It is in this case of tweaked falsity conditions where contra-classicality enters the scene: the tweaked falsity condition endows the connective with properties of some other connective, and hence validates things that it does not validate in classical logic. Second question: even if there is a connection between tweaking the falsity conditions and contra-classical logics in general, does the Bochum Plan shed some light on speciﬁc cases of contra-classicality? For example, how is tweaking the falsity conditions for the conditional related to the commonly believed source of connexivity, the connection between the antecedent and the consequent of the conditional? If, unlike connexive logic, the logic so produced does not have an antecedently given philosophical purpose, can the generation procedure provide some insight on its philosophical import? Leaving the evaluation conditions as they are do not give rise to connexivity. As I have said regarding the previous concern, one might consider that the truth conditions are wrong, as they fail to capture the intended connection between antecedent and consequent in true or valid conditionals. Or else, the falsity conditions might be blamed for the usual ones might fail to capture when the connection between antecedent and consequent is absent. Both are live options, as the literature shows. (Whether it is possible to have a non- dialetheic connexive logic by tweaking the falsity condition of the conditional, or by a suitable tweaking of both evaluation conditions, remains to be shown.) In [11], [10, Ch. 17], [34] and [36] there are some hints on the kind of phenomena that demi-negation logics can help to study or be applied to, and the role of tweaking the falsity condition for negation is more or less clear in producing such a kind of logics. And although the philosophical understanding of tweaked conjunction or disjunction has been scarce, especially in the light of the more mathematical studies of bilattices and their generalizations, one can consult again [36] for some possible ways to understand the connectives so generated in terms of, say, acceptance and rejection. Another question is how can one reply to those saying that the connectives so tweaked are no longer the intended connectives. One initial reaction towards the notions involved in contra-classical logics is distrust. Consider an hypothetical contra-classical logic validating the following schema:

(A ∨ B) → A

Without any further information as to why we should think of this occurrence of ‘∨’ as representing any kind of disjunction, we should probably not think much of this as a genuine contra-classical logic. A similar skepticism might be extended to any logical notion involved in the characteristic theorems of contra-classical logics. Consider for example the (positive) conditions for a connexive logic. If a biconditional is put in them instead of a conditional, all the connexive theses become theorems of classical

9 logic. The same happens with Relativity: putting a biconditional instead of a conditional in it, one gets a classical theorem. In fact, if one turns all the conditionals in the usual implicative fragments for Abelian logics into biconditionals, one gets classical theorems. Of course, the negative condition for connexive logics is there to say that the conditional is not a biconditional. In that case, the negation sign can be blamed by saying that it is not negation, but a connective with the following evaluation conditions:

• ∼A is true if and only if A is false; otherwise, ∼A has the same value as A.

And then the critic can reply that this way of presenting things is misleading, for ∼ A would be the constant value 1, and so not a negation at all, as the connexive theses can be rewritten with > instead of ∼ C, for any C, and that is perfectly classical.13 Considerations like the above have led some authors to suggest a stricter definition of contra-classicality. Humberstone [22] for example has suggested that a logic L∗ is profoundly contra-classical (as opposed to being merely su- perficially contra-classical) if there is no translation τ : L∗ −→ L such that for ∗ every L -valid argument P L∗ Σ, τ(P) CL τ(Σ) is classically valid too. There are at least two problems with this approach. The first one is how to interpret a translatability result. Is the lesson to be drawn from it that the signs employed in contra-classical logics are misguiding and that they represent other connectives in disguise? The second problem, closely related to the previous one, is why classical logic should play such a special role in determining the profound contra-classicality of a logic and, in general in studying the properties of a non-classical logic. A different reaction towards contra-classical logics is charity. If a logic is weird but it nonetheless contains a bunch of otherwise recognizable principles, both positive and negative, about the usual connectives, it seems unlikely that these features are there merely by chance. In those cases, a creative, more charitable stare towards the formalism might be needed; not everything should be left to the founding parents of the logics nor everything should be judged empty or meaningless without having tried really hard to get something from the apparent madness. Of course, life is short and one should try to avoid dead ends. However, in logic it is sometimes extremely difficult to know when something is a dead end, because the dead can resurrect eight centuries later, as connexive logics themselves. Thus, a different approach would be taking a view of the logical notions as independent as possible of any particular logic, take the logical notions appearing in contra-classical logics at face value and then judge whether they fit to an acceptable degree the prior theory of the logical notions.

13This of course would miss the point of most connexive logics, as their proponents in general do not want to have every ∼C as a theorem.

10 I think charity is the kind of approach underlying the Bochum Plan. Omori and Wansing are well aware of the problem of the meaning of connectives, and hint at a solution when investigating whether the sign ∼ in Kamide’s logic CP is a negation at all:

The pressing question is probably the following: is ∼ a negation at all? (. . . ) [T]here is a sense in which ∼ is a negation: the truth condition is exactly the one for the negation in FDE. However, there is another sense in which ∼ is not a negation: the falsity condition is more like that for “affirmative” operators. In the end, one’s answer heavily depends on the account of negation one is ready to accept. (. . . ) One of the related questions that is crucial is: what is negation? We believe that the following answer (. . . ) will be almost universally accepted: Negation is in the first place a phenomenon of semantical opposition. Of course, the next question will be to ask what a semantical opposition is. (. . . ) If one follows the idea that semantic opposition should be understood in terms of truth and falsity, then there seem to be three choices: • require that ∼A is true iff A is false; • require that ∼A is false iff A is true; • require that ∼ A is true iff A is false, and that ∼ A is false iff A is true. (. . . ) Among these three choices, only the first choice [according to which negation represents “the idea of falsehood within the language”] justifies to claim that the unary operation ∼ in CP is a negation. (. . . ) If one follows the idea that semantic opposition should be understood in terms of truth and untruth, then there seems to be only one way to go: • require that ∼A is true iff A is untrue; But this path will not lead us to claim that the unary operation ∼ in CP or IP is a negation. [34, p. 815f]

Let me explore further the idea that one can rely in the FDE-like truth

11 condition to identify a logical connective as the intended one.14 In the field of proof-theoretic semantics, there is an idea traceable back to Gentzen, that it is the introduction rule of a connective what determines its meaning (see [18, p. 84]). There are well-known methods to translate truth conditions and introduction rules, on the one hand, and falsity conditions and elimination rules, on the other, and vice versa. Thus, a rationale for keeping the truth conditions fixed and varying the falsity conditions, besides the evident technical efficiency, can be provided by borrowing to some extent the Gentzenian idea from proof- theoretic semantics. Thus, according to the model-theoretic Gentzenianism that could be attached to the Bochum Plan at this stage of the discussion, it is its truth condition what determines the meaning of a connective, and the falsity condition can be tweaked without real semantic loss or without altering the identity of such a connective.15 In this sense, and for some cases, one can alleviate the doubts on whether the logical notions appearing in contra-classical logics could be seriously regarded as the intended logical notions and explain in some ways the important departures from the more usual cases.

6 More on the meaning of logical connectives

I have by no means addressed all the questions that can be raised against the Bochum Plan and, although up to this point one can tackle with relative success the concerns about it, more serious ones can be formulated. So far, all the examples of contra-classical logics within the Bochum Plan have been obtained by generalizing Wansing’s original idea to other connectives. Does this mean that it is a constitutive part the Bochum Plan to tweak only the falsity conditions? If the answer is aﬃrmative, as it might be suggested by the examples and the model-theoretic Gentzenianism, the Bochum Plan might face a problem. Con- sider transplication, a fantastic connective introduced by Stephen Blamey (see [6]) deﬁned by the following truth table16:

14This does not mean that Omori and Wansing think that this can be done for all connectives. Actually, they are presenting this as an option to make sense of negation in Kamide’s logic, but they also point out that there are other views according to which that would not be a negation at all, and they seemingly are endorsing none but merely landscaping. How to make a choice will be the subject of the remaining of the paper. 15Of course, this just indicates what can be borrowed from proof-theoretic semantics, as the analogy is not perfect. To name but one very important difference, that the introduction rules determine the meaning of connectives does not imply that elimination rules can be arbitrarily tweaked. 16It is equivalent to a connective known as the ‘De Finetti conditional’ in other logical enterprises; see [19]. I am resisting this label because in all fairness De Finetti left unspecified some valuations for his connective, although scholars have filled in the gaps to make it equal to transplication.

12 AB A →B B 1 1 1 1 ∅ ∅ 1 0 0 ∅ 1 ∅ ∅ ∅ ∅ ∅ 0 ∅ 0 1 ∅ 0 ∅ ∅ 0 0 ∅ Note that using the machinery of the Bochum Plan, one is working with C (V ) \{1, 0} and the evaluation conditions are as follows: 1 ∈ σ(A →B B) if and only if 1 ∈ σ(A) and 1 ∈ σ(B) 0 ∈ σ(A →B B) if and only if 1 ∈ σ(A) and 0 ∈ σ(B) that is, the typical truth conditions of a conjunction and the typical falsity conditions of a conditional. Were the Gentzenian story right, at least for conditional, transplication would be a conjunction, not a conditional. Actually, there would be some reasons to think this is the case. Consider a conjunction with its usual truth and falsity conditions: 1 ∈ σ(A ∧ B) if and only if 1 ∈ σ(A) and 1 ∈ σ(B) 0 ∈ σ(A ∧ B) if and only if 0 ∈ σ(A) or 0 ∈ σ(B) If, moreover, logical consequence is deﬁned as

(T-consequence) Γ A if and only if 1 ∈ σ(A) whenever for every B such that B ∈ Γ, 1 ∈ σ(B) then A →B B a` A ∧ B, and this seems evidence enough to hold that transplication is a conjunction and not a conditional. Moreover, both (A →B B) A and (A →B B) B, on the one hand, and (A ∧ B) A and (A ∧ B) B hold; but (A →B B) →B A and (A →B B) →B B, and (A ∧ B) →B A and (A ∧ B) →B B do not hold. This strongly suggests that transplication is not a conditional at all, and the alleged failure of Simpliﬁcation is simply due to the fact that the transplication arrow is just another conjunction. In fact, Rossi, Egré and Sprenger [19] have shown that transplication plus T-consequence will fail at least one of the following: — Detachment, A, A → B B — Self-identity, A → A — Non-symmetry (A → B 1 B → A) or Non-entailment of conjunction (A → B 1 A ∧ B). Nonetheless, they have also shown that transplication plus TT-consequence, that is

(TT-consequence) Γ A if and only if 0 ∈/ σ(A) whenever for every B such that B ∈ Γ, 0 ∈/ σ(B)

13 satisfies Self-identity, Non-symmetry and Non-entailment of conjunction, although it still fails Detachment. Let me grant for the sake of the argument that Detachment is so central to conditionality that if a connective satisfies a bunch of other conditional-ish properties but fails Detachment, it is not really a conditional. This result for transplication resembles the situation of the conditional in González-Asenjo’s/Priest’s LP. Beall has stressed several times (see for example [3], [4]) that even if Detachment is invalid for the conditional in LP, it is default valid in the sense that A, A → B B ∨ (A∧ ∼ A) holds, that is, either Detachment holds or the antecedent is a formula with the value {1, 0}, which arguably is not the case in most situations. The second disjunct internalizes in the conclusion, so to speak, the structure of truth values into the object language. In the case of transplication, the underlying structure of truth values is C(V ) \{1, 0}, and one actually has that A, A → B B∨ ∼ (A∨ ∼ A) holds, that is, either Detachment holds or the antecedent has the value ∅. Thus, the moral is that if one wants to judge a connective and tell it apart from others using not only its truth and falsity conditions, but also appealing to some pre-theoretical properties that the connective is supposed to meet, the underlying notion of logical consequence plays a pivotal role.17 Therefore, in order to accommodate cases of contra-classical logics like one containing transplication, one cannot rely on the Gentzenian model-theoretic view according to which it is the truth condition which is meaning-determining. In trying to accommodate transplication, saying that the meanings of some connectives are determined by the truth conditions and the meanings of other connectives are determined by the falsity conditions is at best uninformative, unless there is a principled way of putting some specific connectives in one bag and others in the remaining one.

7 Further directions for the Bochum Plan

There is also the option of answering in the negative to the question whether it is a constitutive part the Bochum Plan tweaking only the falsity conditions. In that case, one looses the already tiny connection with the Gentzenian semantic idea, and a principled way of saying what tweakings are admissible would be even more pressing. Not without a touch of irony, a solution to this difficulty can be found exam- ining yet another idea inspired by Gentzen. Many proposals about the meanings of logical connectives are separatist, which means that the conditions fixing the meaning of connectives should not involve any other connectives from the object language, as in the original proposal by Gentzen. Clear examples of this are the usual sequent-calculi definitions of connectives and also the FDE-like definitions. This contrasts with interactionist proposals, where “[c]onnectives interact

17And this is a recurrent lesson in many semantic projects. For one of its more recent appearances in proof-theoretic semantics, see [12].

14 with one another in a way that defeats natural separation of the role of connectives.” [46, p. 93] Examples of this are the Hilbert-Bernays axiomatic presentations of logics when axioms are understood as the main meaning-determining devices in a logic. Nonetheless, a further assumption underlying separatist proposals in practice is uniformity: not only the conditions ﬁxing the meaning of connectives should be given independently of any other connective from the object language, but these conditions are of the same kind for all connectives. In proof-theoretic proposals, for example, when introduction rules are taken as meaning-determining, this is so for all connectives, and the same goes for the case when elimination rules play that role; or when both structural and operational rules in a sequent calculus are regarded as meaning-determining, this is so for all connectives. Therefore, a nice feature of the Bochum Plan with its tweaking strategy is that it highlights the uniformity assumption and thus can be isolated and examined. What if the meaning of a given connective is determined by its truth condition, but the meaning of another is determined by its falsity condition, and yet another by both? As we have seen with the case of transplication, in the non-uniform separatist view of the meaning of connectives that can be as- sociated to the Bochum Plan, considering the expected satisfaction or failure of certain properties could also play a crucial role, not entirely left to the pair of truth and falsity conditions. That would provide the required principled way of associating the meanings of some connectives with their truth conditions, and the meanings of others with their falsity conditions. All those are interesting options that should be explored. Nonetheless, I will sketch here what I think that is a very live option and that can give the Bochum Plan a wider room for maneuver. FDE-like evaluation conditions are obtained from the usual classical evaluation conditions by either relaxing the functionality of semantic assignments or allowing certain set-theoretic construc- tions to do the job of (generalized) truth values. But in general, there is more structure attached to a connective than this. Or, more precisely, certain presentations of the connectives might highlight some kind of structure that is kept covered in other presentations. Thus, if certain typographically identical formal signs share a meaning or not, or if they have anything to do with their intended non-formal or pre-theoretical counterparts, which is the very issue at stake in contra-classical logics and is even more pressing than in other non- classical cases, are important issues that cannot be settled in the most usual theories of the meanings of connectives, of which the Bochum Plan in the presentation of Omori and Wansing could be regarded as one of the most reﬁned model-theoretic versions. What is required is, I think, a structural preservationist program that takes into account the varieties of structures in logic, a program like the one announced by Hjortland just a couple years ago:

In logic the mathematical structure is not equations, but there are a number of other candidates. For the proof theoretically minded, it might be derivational structure, but it could equally well be al-

15 gebraic structure, model theoretic structure, etc. Indeed, logical theory is rife with examples of structure preservation. (. . . ) On the structuralist account the identiﬁcation of logical connectives across theories is a matter of structure preservation. I think that has some plausibility in the above cases. The reason why the Strong Kleene ¬-expression is considered the negation counterpart of classical logic is precisely because it preserves the classical semantics in special cases. The reason why the intuitionist conditional is identiﬁed as the counterpart of classical material implication is because it is, essentially, the classical truth condition generalized to a set of valuations. [20, p. 484]

Structural preservationism is a reﬁned version of semantic minimalism, the idea that not every element related to a connective –for example, the evaluation conditions, the exact number of truth values and their relations, the notion of logical consequence, all the valid arguments on which it appears, and so on– contributes to determine its meaning. This opposes semantic maximalism, the idea that every such element contributes to determining the meaning of the connective. Thus far, the discussion has been mostly in proof-theoretic terms, with Paoli as the main champion of minimalism after having introduced the distinction (see [37], [38]) and with few scattered attempts in model-theoretic terms (see [5], [14], [15]). I say that structural preservationism is a reﬁned version of minimalism because again not every element related to a connective contributes to determine its meaning, but unlike traditional versions of minimalism, in structural preservationism there are several ways to isolate min- imal meaning-determiners, and many of them are mixed in nature, combining model-theoretic, proof-theoretic, order-theoretic and algebraic components, just to name a few. n In classical logic, for every connective sign c there is a family EC n of sets i ci n of evaluation conditions such that every such set characterizes ci in classical logic. Conceptually, these sets of evaluation conditions are important because n each of them stresses particular features of ci , that are in general not reducible to each other except in the case of classical logic. Moreover, each of those sets of n evaluation conditions highlights some connections of ci with its intended non- n formal or pre-theoretical counterpart, c. Whether in non-classical contexts ci will be considered a formal or theoretical counterpart of c will depend not only on the set(s) of conditions of evaluation used to characterize it, but also on the h+ set of positive properties attached to c, P n , and the set of negative properties ci j − n not possessed by c, P n , that c will be expected to satisfy or fail, respectively. ci i To put this in less abstract terms, consider again the conditional, with its very well-known table in classical logic:

16 AB A → B 1 1 1 1 0 0 0 1 1 0 0 1

It can be given the FDE-like evaluations, of course, but also the following one, which stresses the ordering-inducer feature of a conditional independently of the number of truth values or other connections they might have: 1 ∈ σ(A → B) if and only if σ(A) ≤ σ(B) 0 ∈ σ(A → B) otherwise. Moreover, there is the following presentation of its evaluation conditions which stresses that a conditional is a relative-identity of the consequent, that is, that a conditional is always like its consequent except in one case: σ(A → B) = 1 if σ(A) = σ(B) = 0 σ(A → B) = σ(B) otherwise. Parts of these evaluation conditions can be tweaked, or they might include explicit features that are irrelevant or redundant in classical logic, so they might be sources of new and old non-classical logics, too. To give just an example, consider these two further ways of describing the classical truth table for the conditional: • If σ(A) 6= 0, σ(A → B) = σ(B), and if σ(A) = 0, σ(A → B) 6= 0 • If σ(A) = 1, σ(A → B) = σ(B), and if σ(A) 6= 1, σ(A → B) 6= 0 Given that 1 and 0 are collectively exhaustive and mutually exclusive values in classical logic, any of these two evaluation conditions characterize the classical properties of the conditional. But they can become radically apart in many- valued contexts. If the second occurrence of ‘6= 0’ in the first of these conditions is changed to ‘= 1’, then one has the conditional as in LP. If the second occurrence of ‘6= 0’ in the second of these conditions is changed to ‘= ∅’, then one has transplication. Thus, the overcoming of the bias towards truth can be accompanied by an additional overcoming of the bias towards any definite value, be it truth or falsity, in favor of exhibiting in different ways their possible interactions.18 If I were to say what the Bochum Plan is, not in the sense of how it has been presented so far, but what are its essential features, I would say that it comprises the following two parts: (BP1) the formulation, for a language L with a collection of connectives C, a model-theoretic semantics S with the following features:

n h+ j − • for each c ∈ C, a family P of pairs of sets {P n ,P n }; informally, a i ci ci n family of sets of properties that ci could satisfy and could fail, respectively, in order to be the intended connective c. 18Of course, values are still important in that it they are the semantic values in this approach. What I am saying is that, pending further argument, none of them is logically more important than the other.

17 n • for each c ∈ C, a family EC n of non-empty sets of evaluation conditions i ci n such that each ECkcn ∈ EC characterizes ci in the base logic. i

n (BP2) the generation of logics by fixing an ECkcn ∈ ECc for a connective i i n ci and substitute the elements of ECkcn in a way that one could still match i n h+ j − c with a pair {P n ,P n }; informally, this is tweaking some given evaluation i ci ci conditions for a connective while it still is the intended connective. Note that in this way one can take the connectives in contra-classical logics at face value but not naïvely, and that this could alleviate the doubts of the skeptic about contra-classical logics, because, first, the evaluation conditions for n a connective ci are taken from a base logic that might be in a better conceptual h+ j − position; second, the tweaking has to match a pair {P n ,P n }. These two steps ci ci n are designed to ensure that ci is still the intended connective c, or at least that n ci is still in the ballpark of the typographically identical connective in classical logic, even if it does not have all the features it possesses in classical logic. Let me compare more clearly this approach with Humberstone’s. For him a connective, say, the conditional in an Abelian logic, is not a conditional but a biconditional because it can be translated in that way while keeping the validity of the arguments in which it appears in Abelian logic. In the approach underlying the Bochum Plan, if such a connective satisfies enough of a certain family of conditional-ish properties (and fails enough of non-conditional-ish properties), and moreover there are conditional-ish evaluation conditions for it, then it can be safely taken as a conditional.

8 Conclusions

I started with the question whether the Bochum Plan is more than an excellent machinery for systematically crafting extremely non-classical logics, or whether it can serve as a semantic project in a more robust way. The Bochum Plan is not only a tool for generating new non-classical logics, but it also encompasses old ones and helps in suggesting what the source of non-classicality is in a great number of cases: the Plan suggests that a deviant understanding of the falsity condition for some connective is at the core of many cases of contra-classicality, by far the most challenging form of non-classicality nowadays. I think that speaks for its robustness. What I wanted to do here was to have a better understanding of the Bochum Plan, and moving it to the next level, ﬁrst by detaching from it the idea of leaving the truth conditions ﬁxed, which seems unessential to the Plan but suggested that the truth conditions have some special semantic status; second, by freeing the Plan from the FDE-like presentation of logics, so more possibilities can be covered; and thirdly, by probing at least some rough admissibility criteria for the tweakings required by the Bochum Plan. Putting this to work is left for another occasion; my aim here was merely to show that there are further ways worth exploring. In the journey, several implications for the philosophy of logic were

18 hinted at, most obviously on the topic of the meaning of logical connectives. This also shows that the Plan is fruitful in connections with greater topics and that it deserves much more examination.

Acknowledgments I want to thank Hitoshi Omori, Matthias Unterhuber, Paul Egré, Walter Carnielli, Charlie Donahue, Francesco Paoli, Elisángela Ramírez- Cámara, Nissim Francez, Heinrich Wansing, Raymundo Morado, Axel Barceló, Marcello D’Agostino and Hykel Hosni for their comments and suggestions to other works closely related to this one. Also, I want to thank the audiences at Logic in Bochum IV, the IIF-UNAM Colloquium, the Logic Seminar of the University of Milan. This work was partially done with the support of the PA- PIIT project IN403719 “Intensionality all the way down. A new plan for logical relevance”. I also want to thank the CLE e-Prints Editors for their assistance.

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