What contradictions say (and what they say not)
Walter Carnielli1, Abílio Rodrigues2
1CLE and Department of Philosophy
State University of Campinas, Campinas, SP, Brazil walter.carnielli@cle.unicamp.br
2Department of Philosophy
Federal University of Minas Gerais, Belo Horizonte, MG, Brazil abilio@ufmg.br
“I myself have always supposed that only statements, and hypotheses insofar as they lead through deductions to statements, could contradict one another. The view that facts and events could themselves be in contradiction seems to me to be a prime example of careless thinking.”
Hilbert, On the infinite
1. Introduction
The aim of this paper is twofold. The first is to present and to defend a view of the nature of contradictions that intends to provide a philosophical justification for logics of formal inconsistency. The view we shall present here is not committed to so- called dialetheism, the thesis according to which there are true contradictions. We want to make clear that the philosophical significance of logics of formal inconsistency, a family of paraconsistent logics that encompasses the great majority of paraconsistent systems developed within the Brazilian and Polish traditions, although not restricted to these (cf. [CarMa01] and [CarCoMa07]), is absolutely independent of one’s beliefs regarding real contradictions1.
1 It might be enlightening to keep in mind the real meaning of the Latin expression contra dicere, which literally means “speak against”.
Our second purpose is to present the outline of an analysis of three concepts central to paraconsistent logics in general; namely, contradiction, negation and consistency. We will show that these notions are equivocal, having more than one sense in informal reasoning and natural language. Specifically with regard to the notion of consistency, we will also argue that this fundamental concept of reasoning can be conceived as a primitive notion and need not be defined in terms of negation.
Paraconsistent logics have been taking an increasingly more important place in contemporary philosophical debate, but there is still some resistance to recognizing their philosophical significance. As we see the problem, this reluctance is primarily related to the awkwardness of the claim that reality, properly speaking, is contradictory (or, equivalently, that there are meaningful contradictory sentences about reality that are true). It is a fact that paraconsistent logics permit extending conventional reasoning to make it able to deal with contradictions. However, the nature of the contradictions that are dealt with by paraconsistent logics is still an open issue.
Once we accept the idea that contradictory information is to be found in a number of contexts of reasoning, the task of the logical system used in such contexts becomes that of separating inferences that are acceptable from those that are not. The crucial strategy is to weaken the logic in such way that the presence of contradictions does not result in triviality, while at the same time preserving desirable features of the notion of logical consequence. This is indeed a difficult and challenging task. However, if we take a look at real situations in which we have to deal with contradictions, we see that the underlying logic does not have to change very much. Virtually all of the inferences allowed before the appearance of the contradiction are still valid. In fact, what does happen seems to be quite simple: the principle of explosion is just not applied to contradictory sentences. The contradiction is put aside, so to speak, awaiting a later decision.
Logics of formal inconsistency have resources for formally representing situations such as these. They internalize the metatheoretical notion of consistency, expressing it in the object language. Hence we can isolate contradictions in such a way that the application of the principle of explosion is restricted to consistent sentences only, thus avoiding triviality.
This is achieved by means of adding to a collection of appropriate axioms and rules already accepted in classical propositional logic a restricted principle of explosion,
oA, A, ¬A ⊢ B
where ‘oA’ means that A is consistent. If A is not consistent, explosion cannot be applied.
There is no commitment to the truth of the sentences that express the contradiction. They may well be taken as a provisional situation that can be decided later, at least in principle. In the framework of logics of formal inconsistency there is a place for contradictions of an epistemic character – we could even say that they are conceived in such a way that perfectly suits the idea that there are no contradictions in space-time phenomena, nor in mathematical objects, although we do reason in the presence of them.
We have presented above a sketch of a somewhat pragmatic argument in defense of logics of formal inconsistency. However, we also want to show that an interest in these logics is not based on pragmatic reasons alone. In section 2 we present a brief analysis of the nature of logic, with the aim of showing that logics of formal inconsistency find their place in the epistemic character of logic as a normative theory of logical consequence. In section 3 we defend a claim about the nature of contradictions that we have to deal with in a number of situations, arguing that they have an epistemic rather than an ontological character. With respect to the existence of real contradictions, although we do not believe in them, we argue that the most reasonable position is to suspend judgment.
The discussion in sections 2 and 3 provides support for the discussion in section 4 of another central issue. There are three concepts that have a special role in paraconsistent logics: contradiction, consistency and negation. In section 4 we try to show that these concepts are equivocal. If we are going to give a precise treatment of them, one that is faithful to the way they are used in informal reasoning, then it must be understood that both negation and contradiction may be conceived from either the epistemic or the ontological point of view, thus justifying at least two versions of both concepts. With regard to consistency, we offer arguments showing that it can be conceived as a primitive notion that is not definable in terms of negation.
2. Logical pluralism and the nature of logic
Logical pluralism is a fact. There are a variety of treatments of the notion of logical consequence, and plenty of logics to suit all tastes. Some of them have a better philosophical justification than others, but to ask for the correct logic among them seems to be completely out of the question. In any case, we have more than one criterion for establishing whether or not a given sentence follows from a set of premises.
However, the attempts to justify this plurality of treatments of logical consequence yields some difficulties, the most important being the threat of relativizing truth. The reason is that, in principle, logical consequence has to do with truth preservation. In any account of logical consequence that is worth studying, we cannot make inferences that can lead from truth to falsity. Consider, for example, two different notions of logical consequence, ⊢1 and ⊢2. It can happen that from the same set Γ of premises a given sentence A follows from Γ, and does not follow, depending on the features of ⊢1 and ⊢2. Now, if the sentences of Γ are true, what is to be said about A?
On one hand it must be true, and on the other it could be false. This result looks peculiar, however, since it implies that a given circumstance, expressed by the truth of the premises Γ, does not unequivocally determine the truth of A. This leans dangerously towards assuming that truth is relative, a step we do not want to take.
Provided that the language is the same, if there are different logical consequence relations and all of them are ‘correct’, then the obvious question is this: how we are going to know what logic we should apply in a given situation? It seems that the only available answer is that which follows from what depends on the context: different accounts of logical consequence are designed to deal with different contexts of reasoning that determine which inferences we can or cannot make. This is a reasonable position, but this idea alone is much too pragmatic to have philosophical significance.
After all, the truth of a sentence should not be a matter of a convenient decision between two different logics. What is the feature of logic that allows for different accounts of logical consequence?
In the strict sense, logic is a theory of logical consequence. As such, its task is to formulate principles and methods for establishing when a sentence A follows from a set of premises Γ. The question then becomes: what are the principles of logic about? Are they about reality, thought or language? This is a central question if we are to think about the nature of logic. In other words, the question is whether logic has an ontological, an epistemological or a linguistic character.2
Throughout the history of philosophy, this question has had different answers.
Aristotle’s defense of the principle of non-contradiction as “the most certain principles of all things” (Metaphysics 1005b11) has an ontological character. In this sense, logical principles are about reality, like general laws of nature that apply to any kind of object.
This idea is also generally found in Russell. In Quine, on the other hand, we find a clear rejection of the view that logical principles have to do with the general structure of reality. There is a famous passage in his Philosophy of Logic in which he says that to ask if logic is “a compendium of the broadest traits of reality” is “unsound; or all sound, signifying nothing” [Qn86].
In the modern period, logic has had an epistemological character as the study of the rules of correct reasoning. This idea is fully in accordance with the main feature of modern philosophy, which is concerned above all with epistemology, or, more precisely, with the problem of the justification of knowledge.3
However, we would like here to emphasize the clearly epistemic character of intuitionistic logic, which has its motivation in Brouwer’s views on mathematics as an activity of the human mind. For Brouwer, logic merely represents the operations of thought in constructing proofs4. Note that according to this position logic is therefore also descriptive. This, nevertheless, does not make intuitionistic logic vulnerable to the accusations of psychologism advanced by Frege [Fr64], for at least two reasons. First, it is not the case, as Frege claims, that instead of ‘things themselves’, logic would be interested only in mere representations, since the point is precisely that the things
2 On the problem of the nature of logic, see, for instance, [Ch01], [Tug96], [Po63]. The question of whether logic is about language, thought or reality, is posed and discussed by Tugendhat and Wolf in [Tug96].
3 A good example is the Logic of Port-Royal, [ArNi96] p. 23. It is also clear in Kant, for whom logic was part of a theory of knowledge – see for instance the Introduction of the Jäsche LogicL [Ka92] pp. 527ff.
4 See [Bro07] and the Disputation in [Hey56].
themselves are, in this case, constructions of thought. Second, intuitionistic logic intends to describe correct mathematical reasoning, that is, the way the mind proceeds in the construction of correct mathematical proofs. Accordingly, it has a normative character in the sense that it says how one should reason in order to reason correctly.
We give less emphasis to the conception of logic as a theory about language, although this approach was significant in the twentieth century. A detailed discussion of this issue will not be carried out here, but we do not think that logic is primarily a theory about formal languages and their features. We would just like to remark in passing that it is not clear how language could be responsible for logical pluralism. In this context, language is better seen as an instrument for expressing whatever account of logical consequence one wants.5 Furthermore, language alone has little to say about an issue that we will address in the next section, namely that of whether or not there are contradictions in reality.
However interesting this discussion about the nature of logic might be, what really matters to us is that it indicates why there may be several accounts of logical consequence. The question of the nature of logic at the present time, in light of logical pluralism, has acquired a character that it did not have during the last century. When Quine, for instance, rejects the ontological character of logic, or Frege attacks psychologism, the logic at stake is classical logic. At present, the question is no longer whether or not logic has this or that feature, but has become a question of which logic has this or that feature.
In order to explain logical pluralism in a coherent way, while at the same time giving an answer of philosophical significance and avoiding making truth relative, it seems to us that a better alternative is to show that different accounts of logical consequence are related to different aspects of logic, especially its epistemic aspects.
Let us now take a look at the differences between classical, intuitionistic and paraconsistent logics.
We agree that classical logic may be motivated by a realist conception of truth.
Indeed, if truth is independent of mind, and sentences are true or false regardless of the
5 A defense of the thesis that the linguistic aspect is secondary can be found in the Introduction of [Ch01].
knowledge we have of them, bivalence and the excluded middle are very compelling principles. At the same time, non-contradiction also seems to be very compelling as a principle about reality, along the lines defended by Aristotle. In addition, we see the principle of explosion as an even stronger way of expressing the idea that there is no contradiction in reality, since triviality is unacceptable from any point of view. (Notice that the validity of explosion in intuitionistic logic can be plausibly interpreted as saying that there is no contradiction in mathematical objects, even if they are constructions of mind.)
Accordingly, classical logic would be the only account of logical consequence that has a predominantly ontological character. It would in fact be difficult to justify that more than one logic operates on the ontological level. Once we accept that an account of logical consequence has to do primarily with reality, including space-time phenomena and mathematical objects, this account should be unique.
Intuitionistic logic, in its standard formulation, can be taken as a theory of logical consequence of a normative character, when we have as a background the notion of constructive proof expressed by the formalization given by Heyting in [Hey56]. As Raatikainen shows in [Raa04], attempts to identify a notion of truth with an intuitionistic notion of provability have not been successful. The reason is that intuitionistic logic is not about truth, but rather is about reasoning.
With respect to logics of formal inconsistency, at this point our answer should already be clear: they have an epistemic character and are normative in the sense that they tell us how to reason correctly in the presence of contradictions. Like intuitionistic logic, which can also be understood as a description of how the mind proceeds in the construction of correct mathematical proofs, logics of formal inconsistency also combine a normative with a descriptive aspect. It is a fact that in informal reasoning people deal with contradictions in various situations, but in none of them is the principle of explosion applied, regardless of a one's awareness of paraconsistent logics. From this point of view, logics of formal inconsistency are descriptive. They are also normative, of course, since there is no doubt that triviality must be avoided.
Let us return now to the issue of logical pluralism. The relativism that may be associated with the existence of varying accounts of logical consequence has nothing to
do with truth. We may concede that it is related to the context in which we are working, although, at least in the case of intuitionistic and paraconsistent logics, it is not just a matter of pragmatic convenience. It is the epistemic and normative aspects of logic which allow that different contexts determine different treatments of logical consequence.
3. On the nature of contradictions
In the last section we defended the idea that logics of formal inconsistency have an epistemic and normative character, and that their aim is to tell us how to draw inferences in the presence of contradictions. Now we will address the question of the nature of the contradictions we sometimes have to deal with.
Contradictions appear in several situations; for instance, contradictory pieces of information, laws, databases and scientific theories. Notice that in the first three cases, contradictions have a clearly epistemic character in the sense that they are provisional and can possibly, at least in principle, be eliminated by means of further investigation.
Scientific theories, on the other hand, deserve special attention. Here lies an important point about the nature of contradictions; namely, whether or not there are contradictions that have an ontological character.
It is a fact that scientific research deals with contradictions [CoFr03]. In physics, for instance, it sometimes occurs that two different theories, coherent in and of themselves, yield contradictions when put together. Hence, since contradiction implies triviality in the framework of classical logic, it seems obvious that we need a non- explosive logic in order to give a global account of it. It is worth noting in this regard that when the physicist considers two incompatible theories, even if it is a matter of revising one of them, (s)he is already reasoning in a paraconsistent way, since (s)he limits the application of the principle of explosion.
Before addressing the question of the nature of contradictions in scientific theories, we will make a brief excursion into Kantian ‘negative results’ with respect to human knowledge.
Kant, more than two hundred years ago, presented fundamental insights about the limits of human reason that we essentially endorse. Kant’s aim was to give foundations to metaphysics, separating, so to speak, the good metaphysics from the bad.
In brief, he concluded that human reason has limits that should not be surpassed. When such limits are surpassed, we may obtain error and contradiction, which is a sign that something has gone wrong. There is an unbridgeable distinction between reality as it is (things-in-themselves) and reality as it presents itself to us by means of our experience
(phenomena). Hence, and this is the point we want to emphasize, there are aspects of reality that are inaccessible to human knowledge.
Indeed, the path taken by Kant in reaching these conclusions may be questioned.
Nevertheless, it is very reasonable to affirm that the problems in Kant’s line of reasoning, especially in his account of space and time, are due to the fact that he was working in the framework of the science of his time (one of Kant’s goals was to provide foundations for Newtonian physics). In any case, the central point is that we believe that Kantian ‘negative results’ express essential features of human knowledge and its limits.
In what follows, with a frankly Kantian inspiration, and based on the account given in [CoKr], we present a view of the workings of scientific theories. Scientific research involves three levels: Reality (with a capital ‘R’), empirical phenomena and scientific theories. Reality represents those aspects of reality that are inaccessible to human reason; this idea, of course, corresponds to the Kantian notion of things-in- themselves. By means of empirical phenomena we establish a mediated relationship with Reality. It is based on this level that theories are elaborated, since it is here that scientific experiments and data collection take place – everything, of course, mediated by perception, our conceptual apparatus and measuring instruments. There is not, nor can there be, unconditioned access to Reality.
The schema presented above is a philosophical view and, as such, cannot be conclusively proved, even if it is correct. Nevertheless, we take it as the best account of the practice of scientific research and the results of the empirical sciences. We also agree that it is a somewhat skeptical view, but we think it is the right measure of skepticism that a thousand years of philosophical reflection suggests as the most plausible position.
Now it may be asked: in the schema above, where do contradictions occur? Of course they occur in the theories, since it is a fact that we deal with contradictory theories (cf. [CoFr03], [CoKr04], [CoKr]). Aside from this, it is very plausible that contradictions also occur in empirical phenomena. Such contradictions would have their origin in our cognitive apparatus, in the failure of measuring instruments, in the interactions of these instruments with phenomena, and even in simple mistakes that can
be corrected later on. Note that in all of these cases contradiction does not have an ontological character.
The central question is whether there are contradictions in Reality (with a capital
‘R’). These would be what we call real contradictions, following da Costa and Krause [CoKr]. However, in order to answer this question once and for all we would have to have unconditioned access to Reality, and according to the schema presented above this is not possible.
We cannot extend the features of empirical phenomena and scientific theories to Reality. The occurrence of contradictions in phenomena and theories does not imply, obviously, contradictions in Reality. Actually, what we have on our hands is a typical philosophical problem that has no prospect of a definite solution, and it is for precisely this reason that it is a philosophical problem! Given that Reality is inaccessible, we cannot, from the fact that a given theory has some internal contradictions, decide whether it is Reality that is contradictory or if such a theory is going to be corrected in the course of further investigations.
It seems to us that claiming that there are real contradictions, in the sense explained above, is pure speculative philosophy in the bad sense of the term, the same kind of speculative philosophy Kant tried to fight against in the eighteenth century. We acknowledge, of course, that we cannot conclusively affirm the opposite, that there are no real contradictions. It would be a refutation of the schema presented above, and we are not so paraconsistent as to endorse simultaneously a view and its refutation.
Nevertheless, we believe that there are no real contradictions for the following reasons. First, in all circumstances in which we have full and unproblematic access to phenomena, there is no sign of contradictions. Assuming that nature is uniform and regular, an assumption that plays an important role in scientific research, it seems that there is no evidence to suppose that there are contradictions in Reality. The second reason is based on the assumption of the consistency of arithmetic. Since arithmetic is an essential part of several physical theories (all of them perhaps), it seems unlikely that, the former being apparently free of contradictions, we would find contradictions in the latter.
At this point we would like to call attention to the fact that logics of formal inconsistency, although neutral with respect to real contradictions, are perfectly well suited to the idea that we do not know whether or not there are real contradictions, despite the fact that we have to deal with contradictions. When a physicist considers two theories to be inconsistent when put together, (s)he is doing exactly the kind of thing that logics of formal inconsistency are designed for – using classical logic in the theories taken separately, but restricting the principle of explosion with respect to the contradiction yielded by combining them together. Thus we affirm that logics of formal inconsistency act primarily within the epistemic domain of logic, without any commitment to the existence of real contradictions.
A question that certainly has already occurred to the reader regards the semantics of logics of formal inconsistency. Indeed, a relevant point is how to provide an interpretation for contradictory sentences.
Like intuitionistic logic, paraconsistent logics were initially introduced exclusively in proof-theoretical terms. Later, bivalued semantics were proposed. As an example, let us consider the logic mbC, a logic of formal inconsistency studied in [CarCoMa07]. mbC admits an adequate semantics that excludes the simultaneous attribution of the value false to a pair of sentences A and not A, but at the same time allows the simultaneous attribution of the value true to a pair of sentences A and not A.
This is done by means of the following clause (¬ is paraconsistent negation):
v(¬A) = 0 implies v(A) = 1.
Note that v(¬A) = 0 only specifies a sufficient condition for attributing the value 1 (i.e.
true) to A, and that v(A) = 1 is a necessary condition for v(¬A) = 0. So, although they cannot be both false, they may be both true. It is clear that we have a situation dual to intuitionistic logic, where there may be a stage t of a Kripke model such that both A and not A have the value 0 in t (i.e. false, or not forced in t if one prefers, but changing the terminology does not change the essential point). In Kripke models for intuitionistic logic, true and false may be read as proved and not yet proved. Analogously, regarding the above-mentioned bivaluation semantics that allows both A and not A to be true in mbC, the value true may be read as taken to be true, or perhaps, better, as there is some evidence that it is true. Thus when we attribute the value true to a pair of sentences A
and not A, this does not need to be interpreted as if we were really claiming that both are true, in the strong sense that there is something in reality that makes them true. Rather, A and not A are both being taken or interpreted as true, waiting for further investigations that would decide the issue, discarding one of them. Hence, the value true that in bivaluation semantics can be attributed to a sentence and to its negation does not mean any commitment to the truth of a contradiction.
Before closing this section, we would like to make some remarks about a position in paraconsistency that fundamentally differs from our own. According to a common understanding of so-called dialetheism, a philosophical view defended in a number of books and papers by Graham Priest and his collaborators (cf. [PrBe10]), there are true contradictions; that is, sentences A such that A and not A are both true.
Claiming that there are true contradictions, however, is virtually the same as saying that there are contradictions with ontological character. Indeed, dialetheism maintains that contradictions also occur in the empirical world (cf. [Pr06]).
As we have tried to make clear in the previous sections, we accept contradictions of an epistemic character, but we do not find it necessary to believe that there are real contradictions. However, independently of the personal beliefs of the authors of this paper, the most reasonable position would be to suspend the judgment with respect to the existence of real contradictions. With respect to dialetheias, however, since the contradictions that we do accept have an epistemic character, we consider it inappropriate to speak of true contradictions. The central point we want to emphasize once more is that logics of formal inconsistency are not only neutral with respect to these issues, but have additional resources. Such resources give them the ability not only to express a number of positions, from the ontological to the epistemic views on the character of contradictions, but also to fully recover classical logic where appropriate.
4. On the multiple senses of contradiction, consistency and negation
4.1. On the senses of contradiction
Hegel is a philosopher invariably mentioned when it comes to contradictions. He was perhaps the thinker who most emphatically addressed the issue, stating that both thought and the world are inherently contradictory. However, the meaning of
‘contradiction’ in Hegel is equivocal.
Explaining any concept of Hegel’s system in a few words, without appealing to jargon, is not an easy task. It is perhaps appropriate to begin with a warning. We will not become involved here in the details of the idiosyncratic Hegelian terminology, but will simply try to make clear the essential points needed to grasp the ambiguity in the meaning of contradiction in Hegel.
We must first return to the Kantian ‘negative results’ mentioned in section 3.
Kant’s analysis establishes an insurmountable gap between subjectivity and objectivity.
Scientific knowledge depends on our cognitive apparatus; we know nothing about the objects of experience apart from the conditions by which they are given to us. As an immediate consequence, unconditioned knowledge is not possible.
In opposition to Kant, Hegel believes that unconditioned knowledge should be possible. A necessary condition for attaining it is to overcome the gap between subject and object. In Hegelian terminology, nothing can be left ‘outside’ the absolute, otherwise the absolute would not be really absolute. The point is that the truth must be the whole truth, the absolute truth, and hence must encompass the whole of reality in its temporal development, together with subjectivity, in a unity. But what could this unity be? We can safely say that it includes reality in all of its constituent moments, opposing forces and conflicts, and collapses the distinction between subject and object6.
6 The idea that there are opposing elements existing simultaneously in a whole is found in many places in Hegel’s writings. The following passage from [Heg77], Sec.20, illustrates this interpretation and the weaker sense of contradiction: “The True is the whole. But the whole is nothing other than the essence consummating itself through its development. Of the Absolute it must be said that it is essentially a result, that only in the end is it what it truly is; and that precisely in this consists its nature, viz. to be actual, subject, the spontaneous becoming of itself. Though it may seem contradictory that the Absolute should be conceived essentially as a result, it needs little pondering to set this show of contradiction in its true light.”
The point is that in Hegel a contradiction also means a mere opposition that is a result of conflicts between different things or moments of the same thing. Clearly, this is not a contradiction in the sense expressed by the Aristotelian claim that “the same attribute cannot at the same time belong and not belong to the same subject in the same respect” (Metaphysics 1005b19-21) and expressed in first order logic by the schema
∀x¬(Px ∧¬Px) for a given predicate (property) P.
We wish to call attention to the fact that the above interpretation, which identifies a weaker notion of contradiction in Hegel, is closely related to the basic idea of possible-translations semantics (and in particular to society semantics) introduced in [Car90] and treated in detail in [CarCoMa07].
Indeed, the internal logic of a group of individuals is not necessarily the same as the logic of individuals taken separately. There are different standards of rationality with respect to individuals on the one hand and groups on the other. Non- contradictoriness, for instance, when applied to groups, differs from that applied to individuals, precisely because among the individuals (members of the group) there may be conflicting beliefs, opinions, judgments, etc. Individuals very often represent contrary forces within a society. The whole society, however, is subject to stricter standards of rigor.
As a very simple example, suppose a group of referees has to decide upon a selection of papers to be published. Only some of them can be selected, and with respect to each paper the whole group must take a position A or not A, but not both. Clearly, in the case of two different referees that defend respectively positions A and not A for a certain paper, we may well say that there is a contradiction between them, but this is clearly a weaker sense of contradiction. The example above shows that having more than one notion of contradiction can be a useful tool for expressing a number of features in different contexts of reasoning.
4.2. On the senses of consistency
In logic books we find two notions of consistency, simple and absolute [Hu73].
Simple consistency is defined in terms of negation. A deductive system S is simply consistent if and only if there is no formula A such that S ⊢ A and S ⊢ ∼A. It is the
same as saying that system S is not contradictory. The other notion of consistency, known as absolute consistency, says that a system S is consistent if and only if there is a formula B such that S ⊬ B. Clearly, the latter is tantamount to saying that S is not deductively trivial, while the former is tantamount to saying that S is non-contradictory.
Classically, since the principle of explosion holds, both notions are equivalent.
Of course, in logics of formal inconsistency these two notions do not coincide, since the point is precisely to allow some contradictions (i.e. simple inconsistency) without triviality (maintaining absolute consistency). However, as has been noted elsewhere [Car11], in natural language, as well as in informal reasoning, we may say that there are a number of different notions of consistency.
First of all, notice that absolute consistency definitely does not match any informal or intuitive notion of consistency. As pointed out earlier, there are situations where we have to deal with contradictions, and this is perfectly rational and acceptable.
Triviality, on the contrary, is unacceptable in any situation.
There are intuitive notions of consistency that do not correspond exactly to the notion of simple consistency. We sometimes use consistency in a sense that is not necessarily related to or defined in terms of negation. Let us look at an example.
Moral dilemmas are a typical situation where a person has to deal with inconsistency, but not necessarily with a formal contradiction. A moral agent may believe that (s)he has moral obligation to perform two actions, A and B, but cannot do both, because they are mutually exclusive. We may well say that the belief of the agent is inconsistent, in a very reasonable sense of inconsistency.
Let us take as an example the well-known dilemma, posed by Sartre, of the man in occupied France who on the one hand wants to fight the Nazis but on the other must take care of his mother [Sar78]. He believes that each alternative is a moral obligation, and that doing one implies not doing the other. However, he may well believe that he has both obligations, while not believing a formal contradiction in the strict sense, i.e.
not believing that he has and does not have the obligation of doing something.
In order to obtain a formal contradiction, we need some other principles of deontic logic. Let A and B be, respectively, ‘fight the Nazis’ and ‘take care of his
mother’. Although both are moral obligations, it is not possible to do both. The premises are the following: OA, OB, ¬◊(A ∧ B).
We also need the following principles if we wish to derive a contradiction:
(i) (A → B) → (OA → OB) (ii) OA →¬O¬A
The second one seems indisputable, but we cannot say the same with respect to the first.
A contradiction may be obtained in few steps:
1. OA premiss
2. OB premiss
3. ¬◊(A ∧ B) premiss
4. (A →¬B) 3, classical modal logic 5. OA → O¬B 4, (i), modus ponens 6. O¬B 5, 1, modus ponens
7. ¬OB 6, (ii), classical propositional logic 8. OB ∧¬OB 7, 2, classical propositional logic
Thus, without accepting (i) and (ii), the agent has no reason to reach a contradiction, even if (s)he believes OA and OB. Nevertheless, it seems to us that we can safely affirm that the belief of the agent in both OA and OB is inconsistent, and that this is a case of inconsistency that is not necessarily committed to negation.
The fact that consistency does not necessarily depend on negation, as illustrated by the above example, shows that consistency may be conceived as a primitive concept meaning already conclusively established as true (or false) – by whatever means, depending on the subject matter.
Consistency, in this sense, should certainly be a notion independent of model- theoretical and proof-theoretical means. Hartry Field (cf. [F91]) famously argued that it would be possible to treat a notion of logical consistency as a primitive metalogical notion, echoing some of the ideas of Kreisel and Etechmendy. Field had in mind a notion of consistency which could not be reduced to semantic consistency or syntactic consistency, but apparently his efforts were not successful, as Krzysztof Wojtowicz
points out in [Woj01] (Field's attempts essentially amount to abbreviations for statements about the consistency of some metatheoretical sentences).
This shows that treating consistency as independent from model-theoretical and proof-theoretical means is not so obvious; the only decent mathematical way seems to be to treat the question more geometrico, properly axiomatizing the notion of consistency as a primitive one, but making it relate to negation and other logical connectives in an appropriate way, and this is precisely what the logics of formal inconsistency do.
4.3 On the senses of negation
A point that sometimes causes uneasiness at first sight is that logics of formal inconsistency have more than one negation. This is strange, one might say, because it seems that a negation without the features of classical negation would not properly be a negation. Our reply is that this uneasiness is mistaken. It is quite possible that this uneasy feeling may be caused by a realist conception of truth, connected to the meaning of classical negation, even if one is not completely aware of it. However, we can safely affirm that there are at least two senses of negation, one ontological and one epistemic, corresponding to the two aspects of logic we tried to distinguish in section 2.
We do use, in informal reasoning, a negation that is weaker than the classical one and that arguably has an epistemic character. It happens, for instance, when we say not A, but are not sure about how much we are denying – or, in other words, when we are not completely sure that A is consistent (or solid, well-established, etc.). We write
¬A where ¬ means paraconsistent negation.
On the other hand, suppose that A is not the case has already been confirmed, and we can safely say that it has been well established. We affirm not A, but now this is done in a stronger sense, employing a stronger negation. We are now making an assertion that intends to say something in the framework of classical logic, and we write
~A where ~ means classical negation.
What occurs, in this case, is that in conclusively establishing ~A we have also established the consistency of A. When we write ~A we also mean that it is not the case that we have A. On the other hand, in writing ¬A we still leave open the possibility that
the conclusion may be revised, i.e. that we are not completely convinced that A is not the case. Notice that it is not by chance that epistemic notions (such as to be convinced, to establish conclusively, etc.) occur here.
Thus it seems clear that classical and paraconsistent negation have, respectively, an ontological and an epistemic character. Of course, the distinction we have made in section 3, namely, that a contradiction may have an epistemic or an ontological character, also naturally applies to the negation used to express the contradiction.
Even if one admits that negation is unique from an ontological viewpoint (a reasonable position, since it is hard to conceive that there may be more than one ontology), from the epistemic point of view we have at least one additional negation, precisely the negation that occurs in contradictions that have an epistemic character.
Furthermore, nothing prevents us from having still more negations.
Now we may well take consistency as a primitive operator, o, which is fully justifiable once we acknowledge that consistency does not coincide with triviality, that there is more than one sense of consistency, and that consistency is not always characterized in terms of negation. We may also take as primitive a paraconsistent negation, which, from the epistemological point of view, antecedes classical negation.
With all of this, we can then define classical negation in the following way:
~A ≝ oA ∧¬A.
Note that this philosophical view fits in well with the practice of scientific research.
To sum up, suppose that we have some grounds, working with some data and previous results, for considering that not A should be the case. But we still have doubts with respect to the previous results; in other words, we have not yet found that A is consistent and so write ¬A. Once we establish, by whatever means, that not A is indeed the case, we can now use the classical, stronger negation to affirm not A, i.e. ~A. We have here a primitive notion of consistency plus a paraconsistent negation that has an epistemic character, and, by combining them, we obtain through further investigations a negation with ontological character, viz. classical negation. This is completely in agreement with the spirit of logics of formal inconsistency.
The fact that negation is an ambiguous notion that sometimes has a weaker sense is not fully understood, and occasionally causes mistaken reactions against
paraconsistency. In [Sla95], Slater argues against Priest’s paraconsistent system LP [Pr79], but his criticism supposedly holds for paraconsistency in general. He appeals to the traditional notions of sentences being contradictories or subcontraries, claiming that paraconsistent negations are not “real” negations.
We say that two sentences A and B are subcontraries when they cannot be both false but can be both true. Of course, if paraconsistent negation were only a way to express subcontrariness, it would hardly be of any philosophical significance. Beziau, in [Be02] and [Be06], has worked out an effective logical and philosophical defense of paraconsistent logics against Slater’s arguments, but we think that there are still some remarks to be made on this issue from the point of view of logics of formal inconsistency.
A very important idea for logics of formal inconsistency is that by means of further investigation we can go from a paraconsistent negation to a strong one, a negation that has all the properties of classical negation. A contradiction may have a provisional character, as we have argued in section 3. When we are able to express the notion of consistency in the object language, we can define classical negation through the notions of consistency and paraconsistent negation, recovering classical logic.
Hence it makes no sense to suppose that paraconsistent negation is a kind of subcontrary operator (otherwise classical negation would not be definable). We conclude that if Slater were aware of the philosophical motivations and the resources of logics of formal inconsistency, he would not have supposed that paraconsistency is grounded in a mere confusion between contradiction and subcontrariety. Moreover, he would not have mistakenly considered that LP and logics of formal inconsistency are in the same ballpark.7
7 It is in fact a common mistake, especially among Anglo-American-Australian philosophers, to take the logic LP, the paraconsistent logic endorsed by Graham Priest, as the unique and paradigmatic approach to paraconsistency. Another example is the paper by Bromand [Bro02], according to which paraconsistent logics employed by dialetheism fail as a solution to semantic paradoxes. Regardless of whether Bromand is right or wrong, his criticism is limited to a specific approach. Logics of formal inconsistency have the resources to address the problem in a significantly different way. It is possible, for instance, to add the T-schema and a truth predicate to Peano Arithmetic, obtaining a theory with a semantically closed language but not trivial as we defend elsewhere [CarRod13].
5. Conclusions and final comments
Mathematicians deal with lack of information, for there are many unsolved mathematical problems. For instance, some time ago we did not know the status of the claim that ‘there exist irrational numbers a and b such that ab is rational’. There is a simple well-known classical mathematical proof, but it is not acceptable to intuitionists due to an inherent use of the excluded middle. More recently, another simple constructive proof has been spotted: take a =√2 and b = log92, which are constructively seen to be irrational; it can be also constructively checked that ab= 3, so the claim is constructively proved. The intuitionist’s rejection of the respective instance of the excluded middle was thus a provisional situation.
In empirical sciences on the other hand, although obviously many things are not known, the researcher sometimes deals with too much information, and perhaps even with contradictory information. Thus (s)he sometimes may have to provisionally consider two contradictory claims, one of which in the due time will be rejected.
Both intuitionistic and paraconsistent logics may be conceived as normative theories of logical consequence with epistemic character. Moreover, both are descriptive: the first, according to Brouwer and Heyting, intends to represent how the mind works in constructing correct mathematical proofs, while the latter, we argue, represents how we draw inferences correctly when faced with contradictions.
In fact, it is not surprising that we find a kind of duality in the motivations for intuitionistic and paraconsistent logics. In [BrCar05], Brunner and Carnielli showed that dual-intuitionistic logics are paraconsistent, and hinted at similar results in the other direction. Nevertheless, the dual of Heyting’s well-known intuitionistic logic gives rise to a new paraconsistent logic, that is, one that is not a familiar paraconsistent logic.
These results, in any case, show an intrinsic relationship between both paradigms, but there is of course much to be investigated in this regard.
In this paper, we have tried to make clear the central features and the philosophical motivations of logics of formal inconsistency. Like intuitionistic logics, logics of formal inconsistency work primarily with the epistemic aspects of logic. In section 4 we showed that negation, contradiction and consistency, concepts central to
paraconsistency, can be interpreted in different ways. Again, this opens many possibilities of investigation. Conceptions of consistency as coherence of a group of statements, or as continuity in time of holding a group of statements, could also be studied from a paraconsistent point of view. It is worth noting that the ideas presented here indicate that there is much more to be explored in Hegel than just the misleading idea of true contradictions.
In the previous sections, we have provided a philosophical analysis of the logics of formal inconsistency, and these efforts are plainly justifiable in light of the plethora of potential applications of paraconsistency. In conclusion, we would like to make some comments on the applications of paraconsistent logics.
Although classical first-order logic is a very powerful tool for modeling reasoning, the fact that it does not handle contradictions in a sensible way, due to the principle of explosion, is an essential drawback. In theoretical computer science, for example, facts and rules of knowledge bases, as well as integrity constraints, can produce contradictions when combined, even if they are sound when separate, and this is also the case for description logics. For this reason, the development of paraconsistent tools has turned out to be an important issue in working with description logics and expandable knowledge bases, as well as with large or combined databases (see [CarMaAm00] and [GrSu00]).
The new approach to consistency (and inconsistency) granted by the logics of formal inconsistency has also generated a good deal of interest in the field of inferential probability and confirmation theory. Because of the well-known debate in the philosophical literature on the long-standing confusion about probability when confronted with confirmation (see e.g. [Fi12] and [Fi06]), notions such as coherence, credence, individual consistent or coherent profiles versus group profiles, etc. (see [Fi07]), can be fruitfully approached from the point of view of the logics of formal inconsistency, although there is still a long way to go.
Acknowledgements
We are indebted to Décio Krause and Marcelo Coniglio for remarks and comments on previous versions of this work. The first author acknowledges the support of FAPESP
(Thematic Project LogCons2010/51038-0, Brazil), The National Council for Scientific and Technological Development (CNPq, Brazil) and CPAI-UnB. The second author acknowledges support from the Universidade Federal de Minas Gerais (edital 12/2011).
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