Consistent Theories in Inconsistent Logics

The relationship between logics with sets of theorems including contradictions (“inconsistent logics”) and theories closed under such logics is investigated. It is noted that if we take “theories” to be defined in terms of deductive closure understood in a way somewhat different from the standard, Tarskian, one, inconsistent logics can have consistent theories. That is, we can find some sets of formulas the closure of which under some inconsistent logic need not contain any contradictions. We prove this in a general setting for a family of relevant connexive logics, extract the essential features of the proof in order to obtain a sufficient condition for the consistency of a theory in arbitrary logics, and finally consider some concrete examples of consistent mathematical theories in Abelian logic. The upshot is that on this way of understanding deductive closure, common to relevant logics, there is a rich and interesting kind of interaction between inconsistent logics and their theories. We argue that this suggests an important avenue for investigation of inconsistent logics, from both a technical and a philosophical perspective.

connexive logics [11,12], Abelian logic [8,9], second-order LP [5,15], and others: a number of examples have been catalogued by Humberstone [6] and Wansing [18]. While there have long been paraconsistent logics, which don't trivialise inconsistent premise sets, logics with inconsistencies in the set of theorems are a newer development, and the program of research into these is still in the early stages.
One particularly interesting aspect of inconsistent logics concerns the same kind of interplay between consistency and inconsistency that has been a hallmark of research in paraconsistent logics. The most natural questions seem to concern just what has to change if we admit inconsistent theorems, and in the comparison with paraconsistent logics this concerns the difference between allowing (non-trivial) contradictory premises and allowing contradictory theorems. One way this distinction comes up concerns how it is that the theorems interact with the set of theories of the logic: that is, when we take the deductive closure of a set of formulas under the logic, what happens with the theorems?
According to the standard, Tarskian, account of consequence relations, the answer is obvious. Since Tarskian consequence relations are monotone, if we take theorems to be those formulas A such that ∅ A, and we take a theory just to be a set such that if ⊆ and A then A ∈ , then every theorem is in every theory. According to this view, the inconsistency of the set of theorems of an inconsistent logic proliferates to any theory the logic touches. On this way of seeing things, inconsistent logics are radical in a sense, as they force inconsistency on everything they are used for -however, this may well be seen as a downside, insofar as it also appears to force inconsistency on any sort of prima facie consistent subject matter to which these logics are applied. Furthermore, this proliferation appears to contradict the usual paraconsistent goal of limiting the spread of inconsistencies as much as possible.
There is, however, an alternate version of the concept of theory, employed in the relevant logic tradition, following [16], which does not require that every theorem belongs to every theory. This notion takes logical consequence to be expressed not by a Tarskian consequence relation but, internally, by provable implications in the logic. Then we require that a theory be a set of formulas closed under those implications which are valid in the logic, and in many logics (e.g. relevant logics) it isn't the case that each formula provably implies every theorem.
On this line, we can distinguish between theories which do, and those which do not, contain all the theorems of the logic. The former are usually called regular theories, and it is a hallmark of such an understanding of 'theory' that some theories are non-regular: they are closed under the logic without having logic as their subject matter, so to speak. With this distinction in mind, we certainly can have logics with inconsistent sets of theorems but which do not require every theory to contain an inconsistency, so long as there are some formulas which do not provably imply inconsistencies.
In this paper, we will present some simple examples of consistent theories couched in inconsistent logics. We'll start abstractly, considering the theories of connexive extensions of relevant logics, initially studied by Mortensen [11]: we'll show that such logics have certain key metalogical properties ensuring that certain theories, generated by collections of implication-free formulas, are consistent. We will then single out those properties in order to obtain a general sufficient condition for the consistency of theories in arbitrary logics. Next, we will use this strategy to obtain some concrete examples of consistent mathematical theories -one relational and one functional -couched in Abelian logic. The upshot is that even in Abelian logic, where inconsistent theorems are especially abundant, we can construct consistent theories by going through, but not including, the theorems. We will conclude by hinting at future directions for this line of research, from both a technical and a philosophical perspective.

Preliminaries
Let's start by fixing some definitions. Definition 2.1 Given a logic L (understood as a set of formulas, and writing L A in place of A ∈ L) with at least the connectives →, ∧, and ¬, define an L-theory to be a set α of formulas in the language such that: Call an L-theory α inconsistent if, for some formula A, A ∈ α and ¬A ∈ α: Given a set X for formulas and a logic L, define the L-theory generated by X as follows: 2 [ Note that we may have non-regular theories which are, nonetheless, closed under the logic (in the sense of the definition). 3 This tracks a distinction, discussed by Meyer, e.g. in [10], between understanding logic as a collection of facts and as a collection of laws. 4 This way of constructing theories has us firmly on the 'laws' side: we can use the theorems of logic as fixing laws for theory-closure, while not requiring that logic as facts get in on the action. Note that this is standard in mathematical and scientific theories: as Meyer and Slaney put it, "there is no more compulsion for 2 Our definition of theory relies on our having ∧ and → in the vocabulary, but we could do away with ∧, requiring just that theories be closed under implications of the form A → B. Assuming that → has enough nice properties, this amounts to treating theories not as filters, as we actually do, but merely as upward closed sets of formulas, where the order on formulas is induced by A ≤ B ⇐⇒ L A → B (or on a congruence on the set of formulas defined in some way or other -see [4,Ch. 2] for salient discussion). In Section 6 we will see how to do away with → as well. 3 While the Tarskian notion of theory is the standard one, it is our contention that L-theories also provide a perfectly cogent, and in some ways preferable, formalisation of taking the deductive closure of a set of formulas. We also take them to be as deserving of the name "theory" -in reference to the informal meaning of the word -no less than any Tarskian theory: in other words, L-theories are theories. 4 A version of this distinction has recently been defended by Beall [2] in his defense of FDE as the correct system of logic.
physicists or gymnasts to assert truths of logic than for logicians to learn gymnastics" [8, p. 277]. We'll go back to this later, but first let's see some examples.
First of all, it is worth noting that every logic L, inconsistent or not, allows for at least one consistent L-theory: Proposition 2.2 For every logic L, the empty theory ∅ is a consistent L-theory.
Proof Both closure conditions for L-theories are inert unless something is already in the theory; furthermore, if the theory contains nothing then it cannot contain a sentence and its negation. Now, this may sound a bit like cheating, and in a sense we can prove that it is. In fact, there is a very large class of inconsistent logics for which the empty theory is the only example of consistent theory. Proof By Weakening, every non-empty L-theory must be regular, so inconsistency of the theory follows from the inconsistency of L.
Logics with this property include, among others, every extension of the implication fragment of intuitionistic logic. That being said, there are plenty of logics in the literature -even inconsistent logics -which reject Weakening, often for principled reasons. The most popular motivation is relevance; another is to preserve the abovementioned distinction between logical facts and logical laws, and between regular and non-regular theories. By the end of this paper, preserving the distinction between inconsistent logics and inconsistent theories might look like yet another reason.

Consistent Theories in Relevant Connexive Logics
As a first substantial example, we are going to show that there is a range of relevant connexive logics which are inconsistent as logics, but have non-empty, consistent theories. 5 Let BA be axiomatized as follows (defining A ∨ B as short for ¬(¬A ∧ ¬B)), following Mortensen [11]: A derivation in this axiom system (and others we consider) is a sequence of formulas each of which is either an axiom, or follows from previous elements in the list by one of the rules. 6 When A is the terminal entry of such a sequence, for a logic L, A is derivable, so L A.
Following [11], it's easy to show the following: To obtain the extension EA, we can add the following axioms and rules (alongside a new propositional constant t): 7 Now, let us build some consistent EA-theories. Hence, no such formula is valid in EA. For the argument for a negated atomic, simply assign the atomic itself 0, so that its negation takes 2.
So EA is an inconsistent logic with consistent theories. In fact, the argument goes through for any system between BA and EA (and, indeed, for some weaker logics, though we won't go into those). This same kind of reasoning generalises to consistent sets of formulas in which → does not appear: Theorem 3.3 Let X be a collection of formulas in the language ∧, ∨, ¬ with a classical model. Then [X) L , for L any sublogic of EA, is consistent.
Proof We use Mortensen's matrix again. Given a classical interpretation of X in {T , F }, assign value 2 to every atomic formula occurring in X which is true in the classical model, and value 0 to every other. By hypothesis, this means that every formula occurring in X (and conjunctions thereof) takes value 2. Note that at least one of A, ¬A must take a value in {0, 1}, but in the target matrix 2 → 1 = 2 → 0 = 0, so one of X → A or X → ¬A will take value 0 for every finite X ⊆ X, and so one will be invalid in EA for every finite X ⊆ X. It follows that no pair of contradictory formulas will appear in [X) EA .
Let's consider properties of EA which elucidate why this result works. In particular, it is known that EA has some nice relevance-adjacent properties: This proves that EA satisfies the Ackermann and variable sharing properties, as discussed in [1]. The latter property means that it is a relevant logic along with all its sublogics; while the former property has the immediate consequence that if we build a theory on implication-free formulas, we won't get any implications out of it: Interestingly, though, we will get some negated implications out of an implicationfree premise set, and in fact some will belong to every nonempty theory (whether consistent or not): Proof The first two points are due to the facts that A ∧ ¬B → ¬(A → B), A ↔ A ∧ ¬¬A, and ¬A ↔ ¬A ∧ ¬A are all valid in EA. The last point is immediate from the first two.
We can avoid this behaviour if we drop (A ∧ ¬B) → ¬(A → B), and in fact this formula is invalidated in a range of subsystems of EA.

A General Result
The proof of Theorem 3.3 only relies on a few particular features of EA, which we may extract to obtain a general consistency criterion.
We will do this in two steps.
Then [X) L is a consistent theory.
Proof By point (3), if we assign to each atomic subformula of a formula in X a value in D , then each formula in X takes a value in D . Take any formula B.
• If both B and ¬B are designated under this assignment, by point (2) one of them must take value in C, and so by point (4) either every A → B or every A → ¬B is going to be undesignated, for every A ∈ X (or conjunction thereof). • If either B or ¬B is undesignated, then the same conclusion follows from point (5).
From this we conclude, by point (1), that either every A → B is invalid in L or every A → ¬B is, for every A ∈ X (or conjunction thereof), Thus, for every B, one of B, ¬B will not appear in [X) L .
Note that D needs to be nonempty for the proof to work. Furthermore, condition (2) works for C empty if and only if there is no x ∈ D such that ¬x ∈ D, which in turn requires by (1) that L be consistent. On the other hand, if C is nonempty then conditions (2) and (4) together entail a failure of Weakening: by Proposition 2.3, this does not limit the scope of the result in the context of inconsistent logics.
The main limitation of this result is condition (3), which forces the language of X to be "positive", so to speak: no connectives taking arguments in D to undesignated values are allowed. However, we can do away with this limitation as follows, matching the generalisation from Proposition 3.2 to Theorem 3.3: Then [X) L is a consistent theory.
Proof The only difference is that D lacks the closure property (3), so we are not guaranteed an appropriate model of X by arbitrarily assigning values in D to atomic formulas. However, point (3') compensates for this by making sure that we can find an assignment such that all formulas in X take value in D , which is exactly what we need.
Note that assuming (3') does not trivialize the result, because there might be no way to interpret X in M, D in a consistent manner; this reflects the fact that [X) L might be an inconsistent theory yet still have non-trivial models.
Examples of inconsistent logics to which these results apply include those with a {¬, ∧, ∨}-fragment which is sound with respect to a matrix which contains the classical two-valued matrix, and a conditional → which is: • sound w.r.t. the Mortensen conditional matrix, e.g. every sublogic of the inconsistent logic CN with strong implication (including the logic C, where → is again taken to be strong implication -see [13] for related discussion); 11 • sound w.r.t. the RM3 conditional matrix, 12 which is one of the possible candidates for second-order LP suggested in [5]; • more generally, sound w.r.t. any implication with a matrix of the following form (where each x may be replaced by any value): In every such logic L, [X) L will be consistent for every X belonging to the {∧, ∨}fragment (by Theorem 4.1), and for every classically consistent X belonging to the {∧, ∨, ¬}-fragment (by Theorem 4.2), taking D = {2} and C = {1}. 13 We might 11 The other implication satisfies Weakening, so it is a non-starter. 12 I.e. the following matrix, with {1, 2} designated: → 0 1 2 0 2 2 2 1 0 1 2 2 0 0 2 13 To be precise, 2 → 0 = 0 ensures (5) is satisfied, while 2 → 1 = 0 ensures (4) is satisfied. even throw → itself into X as well if 2 → 2 = 0 → 2 = 0 → 0 = 2, as is the case for RM3. Note that if we want to consider additional connectives the only conditions we need to concern ourselves with are (3) and (3').

Consistent Theories in First-Order Abelian Logic
Let's consider some concrete examples of theories built on an inconsistent logic where (1) negation and implication both occur in the axioms of the theory and (2) the theory incorporates quantifiers. The incorporation of these two points is compatible with our general results, but it is instructive to see how exactly we can accommodate them.
Abelian logic A is presented in [8] in a primitive language {→, ∧, f} with ¬A := A → f (for an intensional falsity constant f 14 ) and A ∨ B := ¬(¬A ∧ ¬B). Axioms and rules are as follows: We may extend A to a quantificational system Aq by adding the following axioms and rule: From here we can move to a system with identity Aq = via the following axioms: 15 t a term Let also ∃xA := ¬∀x¬A.
14 This is a constant which, like ⊥ in intuitionistic logic, is used to define negation, but unlike that is not required to entail every proposition. In fact, there are very few constraints on how f must behave. 15 Adopting → as the main connective in these last two axiom schemes would make it very hard to have any consistent theory involving identities. The root problem seems to be the behaviour of biconditionals in A, which will be discussed shortly. Aq = has models of the form D, G where D is a domain of individuals and G is a complete abelian l-group G, +, −, 0, ∨, ∧ . 16  This time the source of inconsistency is the contraclassical axiom (Rel). Semantically, formulas are satisfied together with their negation in a model if and only if they take value 0, because all non-negative values are designated and all negative values are undesignated. Hence, a theorem is such that its negation is also a theorem if and only if it takes 0 in each model. In particular, we get the following:

Unlike in relevant connexive logics, inconsistency comes extremely cheap in A.
Here are a few reasons:

Proposition 5.2 Let T be an A-theory.
(1) If T contains A → A for some A, then T is inconsistent. The following result, the idea behind which is essentially the same as in Theorem 4.2, will be crucial in ensuring the consistency of some theories. 16 An abelian l-group is such that: An abelian l-group is said to be complete if every bounded from above [below] subset has a least upper [greatest lower] bound. The reader unfamiliar with these notions may find it useful (and sufficient, for the purposes of this paper) to think of the integers as a paradigmatic example, with a ∨ b := max{a, b} and a ∧ b := min{a, b}. More details can be found in [8]. 17 + provides the interpretation of the fusion connective defined by A • B := ¬(A → ¬B). 18 This can be shown to be the case for every theorem in the {→, f}-fragment.

Proposition 5.3 If a set of formulas X has a linear Aq = -model 19 such that every
A ∈ X takes positive value, then [X) Aq = is consistent. As a special case of this, we get a similar result to the one we obtained for relevant connexive logics:

Corollary 5.4 If X is implication-free and has a classical model, then [X) Aq = is consistent.
Proof Take any (consistent) classical interpretation of the axioms in {T , F }, and build a model on the integers as follows: for every atomic formula p, if it classically takes value T then let it take value 1; otherwise let it take value −1. All axioms will then take value 1, because ∧, ∨, ¬ cannot take us outside of {1, −1}.
Note that this result holds despite A lacking the Ackermann property due to (CI). As an example, we are now going to use Proposition 5.3 to show the consistency of two simple Aq = -theories, one relational and one functional. Let Eq be the Aq = -theory, in the language R , of equivalence relations generated by the following axioms: Let Succ be the Aq = -theory, in the language 0, S , of the successor on the natural numbers with axioms: Note that the models in these proofs are models of A, and are therefore necessarily inconsistent; the point is simply that, by Proposition 5.3, their existence proves the consistency of the theories under consideration.

Not Relying on Implication
There is still an apparent limitation in the very reliance on L-theories as we defined them. By definition, L-theories read valid inferences off valid implications. But in the case where a logic L is given as a set of inferences (rather than a set of theorems), L-theories may be inappropriate if the implication → is not such that L A → B if and only if A L B, where L denotes the consequence relation of L (generalizing the earlier notation). There are certainly cases of this. Consider the logic LP from [14]: while LP (A ∧ ¬A) → B, it is not the case that A ∧ ¬A LP B. 21 Intuitively, it would be inappropriate to close theories under valid LP-implications, as that would entail that according to LP the only inconsistent theory is the trivial one -certainly not an intended or welcome reading.
Still, all is not lost. Define [X) L := {A | ∃ nonempty ⊆ X( L A)}. By asking that be nonempty we recover, at least in principle, the distinction between laws and facts: logical theorems do not get in for free. That being said, Prop 2.3 straightforwardly carries over: Then [X) L is a consistent theory.
Note that these definitions and results do not require anything from L ; in particular, it needs not be reflexive or transitive. 24

Discussion
These results give us the grist for some comments on under what circumstances a theorist may give a consistent theory despite employing an inconsistent logic. In particular, so long as the theorist keeps to discussing formulas expressed in a classically behaved fragment, then -if they start out consistent -there is a vast range of inconsistent logics which will not force them to become inconsistent.
As already mentioned, one way to think about this is the following: if a theorist sticks to using logic as a set of laws, without committing themselves to the facts of logic, then the inconsistency of logic need not infect their theory. A major advantage of the above understandings of theories is precisely that they make such distinctions possible.
The point can also be made in terms of a distinction between the logical and extra-logical content of theories. If we understand logical content to be expressed by implication (as fixing logical validity in the object language), then it makes sense for a theory concerned primarily with non-logical content to not contain any implications among its axioms, and it makes sense for such a theory to not inherit the logic's inconsistency. Furthermore, the limitation on the language of X does not prevent us from expressing any putative implicational axioms using a material implication defined as A ⊃ B := ¬A ∨ B -which may well be more appropriate for some nonlogical subject matter -although how satisfying the result would be may depend on the theory. More generally, we may identify the logical content of a theory with its inclusion of logical theorems; if a theory has no substantial logical content in this sense -which, following Meyer and Slaney, is quite natural for most mathematical 22 This is shorthand for: if , B admit an assignment where all the elements of take values in D and B takes value in C, then L B. 23 What was (5) in Thm 4.1 simply follows from soundness here. 24 This is to say, we need not assume that L is a Tarskian consequence relation. It may be any relation between collections of formulas and formulas satisfying the constraints mentioned in the theorem. and empirical theories -then there is no need for the logic's inconsistency to enforce the same status on the theory. Now, under a somewhat instrumental view of logic, the advantage of this is clear enough: we can use inconsistent logics to reason about various phenomena without thereby being forced into adopting dialetheism, i.e. the existence of true contradictions, concerning those phenomena. Still, a commitment to inconsistencies appears to not be avoided under the fairly common view that we are in fact committed to every logical truth determined by the logic we are using. However, there are two ways in which the above considerations remain pertinent even on this view.
First, even if an inconsistent logic is identified as "the" correct logic -either as the one true logic, or as the correct logic for a particular subject matter -it may still be the case that no inconsistent non-logical theory is accepted. This may not save one from dialetheism but it strongly limits the scope of the inconsistency, insofar as it can be limited to the purely logical level.
It may be objected that if an inconsistent logic is taken to be the one true logic, then it will be topic-neutral, and thus there are always going to be some non-logical theories which nevertheless are made inconsistent by the logic. Maybe so; the point is that in principle it may be open to maintain that the theorist should strive to formulate all their theories in such a way that they remain consistent, so that the inconsistency does not spread beyond the logical. This may go together with a view of inconsistencies as only being inevitable when we work at a level of absolute generality. We may call this a purely logical dialetheism, and take the associated research program to be one of finding consistent formulations for all scientific theories. 25 If this sounds far-fetched, note that this is exactly the goal of classical logicians! The second consideration is more tentative. If we take the distinction between logical facts and logical laws very seriously, it may be possible to reject logical facts altogether. They may be treated as a ladder to access the one true collection of logical laws, but not be taken themselves as being a genuine part of the logic. 26 Under this view, "inconsistent" logics would not really be inconsistent after all, and sure enough contradictions would be nowhere to be found even in applications as long as we are careful with the selection of theories.
Obviously both considerations would need a lot more fleshing out, but we hope this brief sketch nevertheless suffices to show how our technical results may pave the way towards a substantial broadening of the philosophical space concerning the role of inconsistent logics.

Conclusion
In this paper we have shown that it is perfectly possible for inconsistent logics to support consistent theories, as long as they forgo Weakening. Whenever logical inconsistency arises from the contraclassical behavior of implication, consistency of the theory can often be achieved by simply not having any implications occur in the axioms; but we have seen that, even in a logic as inconsistency-happy as Abelian logic, this is hardly a necessary condition for a theory not to inherit the inconsistency of a logic.
From a technical perspective, the very existence of such theories opens the door to a general study of the way the inconsistency of a logic relates to the inconsistency of theories built on it. Much research in paraconsistent logic has gone into the ways inconsistencies spread within theories; the kind of research this paper is pointing towards, and making a very modest first contribution to, extends the paraconsistent perspective by also looking at the ways in which inconsistencies spread from logics to theories.
From a more philosophical perspective, one upshot is that the adoption of an inconsistent logic needs not commit us to any non-logical inconsistent theories. The belief in a strong metaphysical commitment of true theories and the rejection of true contradictions appear to be together compatible with inconsistent logics, as long as we treat them as a collection of laws rather than of facts. Furthermore, even someone who is very committed to a view of logic as a set of a priori truths may still be interested in the distinction between inconsistency at the purely logical level, and inconsistency at the level of mathematical or empirical theories.