A logic for evidence and truth ∗

Walter Carnielli1, Abilio Rodrigues2 1CLE and Department of Philosophy - State University of Campinas [email protected] 2Department of Philosophy - Federal University of Minas Gerais [email protected]

Abstract This paper presents a paraconsistent and paracomplete natural deduc- tion system, called the Logic of Evidence and Truth (LETJ ). LETJ is a Logic of Formal Inconsistency and Undeterminedness (LFIU ) that is able to recover classical logic when appropriate. LETJ is anti-dialetheist in the sense that its consequence relation is trivial in the presence of any true contradictions.

1 Introduction

Paraconsistent logics have been assuming an increasingly important place in contemporary philosophical debate. However, in our opinion, there are some aspects of their philosophical significance that are not yet completely under- stood. The distinctive feature of paraconsistent logics is that the principle of explosion, according to which anything follows from a contradiction, does not hold, thus allowing for the presence of contradictions without triviality. Di- aletheism is the view according to which there are true contradictions [cf. e.g. 26]. It is recognized that paraconsistency and dialetheism are not the same thing: the latter implies the former, but one can be paraconsistentist without being dialetheist. However, it seems to us that a deeper philosophical analysis of such a position is still lacking. What exactly does it mean to accept contra- dictions without being a dialetheist? Or rather, is it possible to give an intuitive and plausible reading of a non-explosive negation without any commitment to the truth of contradictory propositions? Our answer here is yes.

∗The first author acknowledges support from FAPESP (Funda¸c˜aode Amparo `aPesquisa do Estado de S˜aoPaulo, thematic project LogCons) and from a CNPq (Conselho Nacional de Desenvolvimento Cient´ıficoe Tecnol´ogico)research grant. The second author acknowledges support from FAPEMIG (Funda¸c˜aode Amparo `aPesquisa do Estado de Minas Gerais, re- search project 21308) and CAPES (Funda¸c˜aoCoordena¸c˜aode Aperfei¸coamento de Pessoal de N´ıvel Superior, grant 3904-14/8). We would like to thank Henrique Almeida, Antonio Coelho, D´ecioKrause, Andr´ePorto and Wagner Sanz for some valuable comments on a previous version of this text.

1 For those who believe in true contradictions, dialetheism does provide a philosophical justification for paraconsistency. If one accepts that reality is in- trinsically contradictory, in the sense that in order to truly describe it (s)he needs pairs of contradictory propositions, then, since reality obviously is not trivial, (s)he needs a logic in which not everything follows from a contradiction, i.e. a paraconsistent logic. However, dialetheism, although a legitimate philosophical position, is not the only way open to provide a philosophical justification for paraconsistent logics. The thesis that reality is (in some sense) contradictory is an old philosophical topic, and it has been a position defended by philosophers such as Heraclitus and Hegel. But there is a strong and widespread reluctance to accepting that there may be entities that disobey the principle of non-contradiction as it is expressed by Aristotle in book IV of Metaphysics (1005b19-21): “the same attribute cannot at the same time belong and not belong to the same subject in the same respect” [1]. It seems to us that the lack of enthusiasm that we sometimes find with respect to paraconsistent logics is due to the awkwardness of the claim that there are meaningful contradictory propositions about reality that are true. On the other hand, it is a fact that contradictions appear in a number of contexts of reasoning, from everyday real-life situations to scientific theories. This is enough to provide at least a pragmatic and metaphysically neutral reason to devise a formal system capable of handling contradictions while avoiding triviality. Our aim here is to work out a third way to paraconsistency, besides meta- physical neutrality and dialetheism. The basic idea is that contradictions that occur in several contexts of reasoning exhibit epistemic character in the sense that their occurrence is related to thought and reason. Let us say that a logic has epistemic rather than ontological character when it is concerned not only with truth, but also with some other concept strictly related to reason. Clas- sical logic is an account of strict truth preservation, maybe the best possible account, but a logic designed to deal with epistemic contradictions will be talk- ing about something other than truth. We need, therefore, a conception of logic according to which logic is not restricted to the idea of truth preservation. This is the case for intuitionistic logic when it is understood as being concerned not with truth but rather with truth attained in a specific way, by means of a constructive proof. We will argue that this is also a way for understanding paraconsistency: a paraconsistent logic may be concerned with a notion weaker than truth, and it is precisely this weaker notion that allows an intuitive and plausible understanding of the acceptance of contradictions in some contexts of reasoning. This paper presents a formal system designed to deal in a non-explosive way with pairs of propositions A and ¬A, and according to which there are no ‘real contradictions’. In other words, there are no pairs of true, contradictory propositions. The thesis that contradictions are yielded by thought and reason, or occur in the process of acquiring knowledge, is not new, and its origins can be traced back at least to Kant in his antinomies of pure reason. But we think that

2 this thesis has not yet been properly developed with respect to a paraconsistent non-dialetheist formal system designed to represent contexts of reasoning in which contradictions have a strictly epistemic character. In the classical account there is a duality between excluded middle and ex- plosion as rules of inference: anything follows from a contradiction, excluded middle follows from anything. Each of these rules correspond to half of the properties of classical negation, and we will see that the rejection of explosion by paraconsistent logics is the mirror image of the rejection of excluded middle by paracomplete logics. Intuitionistic logic is as an example of a paracomplete logic that may be understood as both ontologically and epistemologically moti- vated. The ontological motivation is based on Brouwer’s views on mathematical knowledge and, more precisely, on the concepts of truth and existence of math- ematical objects. We call the epistemological motivation the view according to which classical and intuitionistic logic are not mutually exclusive and may live together as two different but legitimate kinds of proofs, the latter sometimes more informative than the former. Analogously, the rejection of explosion in paraconsistent logics may have an ontological or an epistemological motivation. The ontological motivation boils down to dialetheism. The epistemological motivation depends essentially on a notion weaker than truth to justify the acceptance of (at least some) contradictions, a notion still interesting from the point of view of an analysis of the inferences that are allowed or forbidden in some reasoning contexts. The basic idea is that the acceptance of a pair of contradictory propositions A and ¬A does not need to mean that both are true. Rather, we may understand it as some kind of ‘conflicting information’, namely, that there is conflicting evidence about A. Evidence for A is understood in broad terms as reasons for believing that A is true. Notice that evidence is a weaker notion than truth, in the sense that if one knows that a proposition A is true, one has evidence that A is true, but the converse obviously does not hold. This text has two main parts. The first is more philosophically motivated (section 2), while the second (sections 3 and 4) deals with technical aspects that are intended to fit the philosophical motivations brought out in the first part. Section 2 discusses the duality between paraconsistency and paracompleteness. The main point of that section is to show that each of these concepts may be interpreted ontologically as well as epistemically. We argue in defense of the epistemic interpretation of paraconsistency. Section 3 introduces a paraconsis- tent formal system, that we call the Basic Logic of Evidence, (BLE), designed to express the deductive behaviour of the notion of preservation of evidence. In section 4, the logic BLE is extended to the logic LETJ , that is able to ex- press not only preservation of evidence, but also preservation of truth. This is done by recovering classical logic with respect to formulas that we want to say are true or false. A bivalued complete and sound semantics is presented and some technical results that are intended to capture the intuitive interpretation proposed are also discussed.

3 2 On the duality between paraconsistency and paracompleteness

Let us define two n-ary logical connectives C1 and C2 as dual when

∼C1(A1,A2, ..., An) and C2(∼A1, ∼A2, ..., ∼An) are classically equivalent (∼ meaning classical negation). Thus, ∀ and ∃, ∧ and ∨ are dual to each other, and ∼ is dual to itself. We also say that the corresponding formulas are dual. It is clear then that the formulas ∼(A ∧ ∼A) and A ∨ ∼A (respectively, non-contradiction and excluded middle) are dual. However, this duality may be seen from a different, more fundamental viewpoint, as a duality between rules of inference. The principles of non-contradiction and excluded middle are often presented in books of logic, and philosophy, as fundamental laws of thought and basic tenets of classical logic. However, it is not non-contradiction but rather ex- plosion that is essential to characterize classical negation, and this is a central feature of the classical account of logical consequence. Classical negation ∼ is defined by the following conditions (for classical ∨ and ∧): A ∧ ∼A , (1)  A ∨ ∼A. (2) Condition (1) says that there is no model M such that A ∧ ∼A holds in M. (2) says that for every model M, A ∨ ∼A holds in M. A negation is paracomplete if it disobeys (2), and paraconsistent if it disobeys (1). Notice that each one of the conditions above corresponds exactly to half of the classical semantic clause for negation, M(∼A) = 1 if and only if M(A) = 0. (3) The only if forbids that both receive 1, and the if forbids that both receive 0. Given the classical definition of logical consequence (and the usual meanings of ∧ and ∨), from (1) and (2) above, it follows that for any A and B:

A ∧ ∼A  B, (4)

B  A ∨ ∼A. (5) The inference (4) above is (one version of) the principle of explosion and (5) is (again, one version of) excluded middle. Of course, excluded middle is usually presented as a valid formula or axiom, without the premise B, but this is tanta- mount to the formulation above, which makes it clear that A ∨ ¬A follows from anything. It is easy to see, therefore, that from the point of view of classical logic, the fact that excluded middle is not valid in paracomplete logics and the fact that explosion is not valid in paraconsistent logics are mirror images of each other.

4 It is worth noting that non-contradiction and explosion are not equivalent in the following sense. We get a complete system of classical propositional logic if we add excluded middle and the principle of explosion to a system of posititive intuitionistic propositional logic1, but the system so obtained turns out to be incomplete if we change the latter to non-contradiction. Due to the semantic clause (3) above, a central feature of classical nega- tion is that it is a contradictory-forming operator. Applied to a proposition A, classical negation produces a proposition ∼A such that A and ∼A are contra- dictories in the sense that they can neither receive simultaneously the value 0, nor simultaneously the value 1. In order to give a counterexample to the principle of explosion we need a circumstance such that a pair of propositions A and ¬A hold but a proposition B does not hold (¬ being a paraconsistent negation). Dually, a paracomplete logic require a circumstance such that both A and ¬A do not hold (now ¬ is a paracomplete negation). Obviously, neither a paracomplete nor a paraconsistent negation is a contradictory-forming operator, and neither is a ‘truth-functional’ operator, since the semantic value of ¬A is not unequivocally determined by the value of A. Now, the question is: what would be intuitive and plausible justifications for paraconsistent and paracomplete negations? An answer will be found in reasoning contexts in which negations with these characteristics occur.

2.1 Intuitionistic logic: a case of paracompleteness Let us start with intuitionistic negation, which is paracomplete, as mentioned. There are two different motivations, one ontological, another epistemological, for saying that in a given context both A and ¬A (¬ being intuitionistic negation) do not hold. We find in Brouwer’s writings a conception of mathematical knowledge ac- cording to which there cannot be any mathematical truth not grounded on a mental construction. Furthermore, and in accordance with this thesis, the ex- istence of a mathematical object with certain properties can be asserted only if such an object has been so constructed. Mathematics, thus, is in no way independent of thought and mind. This conception is nothing but an ideal- istic attitude with respect to mathematical objects: truth and existence are conceived on a idealistic basis, both depending on the human mind. Mathematics can deal with no other matter than that which it has itself constructed. In the preceding pages it has been shown for the fundamental parts of mathematics how they can be built up from units of perception. [4, p. 51] [T]ruth is only in reality i.e. in the present and past experiences of consciousness. [5, p. 488]

1Positive intuitionistic propositional logic may be defined by the usual introduction and elimination natural deduction rules for ∧, ∨ and →. Axiomatically, it may be defined by axioms 1 to 8 of Carnielli et al [6, p. 24] plus modus ponens.

5 The rejection of excluded middle is in this setting straightforward, since it may be the case that no mental construction of A, nor of ¬A has been effected. Notice, however that according to this view, there is an identification of the notion of truth with a notion of constructive proof in the sense that a (mathe- matical) proposition is true if and only if a proof of it is in some sense available. If the truth of mathematical propositions is so conceived, then intuitionistic logic may still be understood as an account of truth preservation, although an idealist notion of truth. It is worth noting that this conception of mathematical truth is still compatible with a notion of truth as correspondence: what makes a mathematical proposition true is some entity that exists in reality, but in this case it is a reality constructed by thought. This is what we call an ontological motivation for rejecting excluded middle. In Brouwer’s 1907 dissertation, the classical and the constructive approaches were irreconcilable positions. But this is not the only way to understand intu- itionistic logic. There is a weaker position, whose basic idea can already be found in Heyting [15], that is perfectly compatible with a conception of mathematical objects not committed to Brouwer’s idealism.

[Brouwer’s program] consisted in the investigation of mental mathe- matical construction as such, without reference to questions regard- ing the nature of the constructed objects, such as whether these objects exist independently of our knowledge of them. [15, p. 1]

Thus, an investigation of mathematical objects as mental mathematical con- structions does not need to imply that such objects do not exist independently of such constructions, nor that they cannot be investigated by other means. In other words, one may well be a realist about mathematical objects but still have interest in intuitionistic logic as a study of such objects from the viewpoint of mental constructions. Understood in this way, intuitionistic logic is not really about preservation of truth. Rather, it is about preservation of construction, whose specific fea- tures depend on the formal system – indeed, Heyting’s and Kolmogorov’s logics express different notions of construction. Given the (supposed) soundness of the system, the possession of a proof of A implies the truth of A. This position makes it possible to combine a realist notion of truth with a notion of constructive proof that is essentially epistemic. The latter is no longer a necessary condition for the truth of a mathematical proposition. A view that accepts a non-constructive proof of the truth of a given proposition, but distinguishes such a proof from a perhaps more informative constructive proof, is thus perfectly coherent. Understood in this way, the claim that in a given circumstance both A and ¬A do not hold (or that both receive the semantic value 0) does not mean that both are not true, but only that there is no constructive proof of them, independently of the question whether any of them may be proved true by non-constructive means. The point is no more the existence of the object, but rather the epistemic access to the object. Its existence may be guaranteed by a non-constructive proof, although, say, a ‘direct

6 access’ may be provided only by a constructive proof. From this perspective, not only the rejection of excluded middle, but the whole enterprise of intuitionistic logic, acquires an essentially epistemic character. In fact, Heyting’s remarks quoted above anticipate an approach to intu- itionism according to which classical and intuitionistic logics do not exclude each other, as in the case of Brouwer’s views but, rather, may live together in perfect harmony.

The most striking feature of the present situation of intuitionism is that some sort of peaceful coexistence with classicism has been eventually reached. Times where controversy was raging are disap- pearing from collective memory, and the whole intuitionistic enter- prise nowadays tends to be merely considered one place in the whole landscape of logic, mathematics, and philosophy. [11, p. 50].

2.2 On a familiar example of a non-constructive proof

√ √ log 2(k) Fact 1. For k ∈ N, 2 is a rational number. √ log√ (k) √ √ 2 Proof. Since log 2(k) = log 2(k), from the definition of log, 2 = k, which obviously is rational, since k ∈ N. √ Fact 2. 2 is not a rational number.

Proof. The proof in this case is well known, and it is intuitionistically acceptable, since it employs a reductio ad absurdum that introduces a negation.

√ Fact 3. For k ∈ N odd, log 2(k) is not a rational number.

√ Proof. Suppose, recalling that by hypothesis k is odd, that log 2(k) is rational. √ Then, for some natural numbers a and b, log 2(k) = a/b. It follows that √ a/b √ a ( 2 )b = kb, and thus 2 = kb. Since k is odd, kb is odd, for any b. Now, √ a √ any natural number is odd, or not odd. Suppose a is odd. Then 2 = c 2, √ a for some c ∈ , so 2 6= kb. Now, suppose a is not odd. √ Na √ a Then, 2 = c for some c even. Since kb is odd, for any b, again 2 6= kb. √ Hence, log 2(k) is not a rational number. Let us illustrate this position with a familiar proposition, often mentioned as an example of a non-constructive proof:

(P) There are irrational numbers m and n such that mn is rational.

There is a well-known non-constructive proof of (P), not acceptable to intu- itionists due to an alleged illegitimate use of the excluded middle. From the supposition that √ √ √ 2 √ 2 2 is rational or 2 is not rational,

7 it can be proved that there exist√ numbers m and n satisfying the conditions √ 2 √ above. Indeed, suppose that 2 is rational; then√ m = n = 2 completes √ 2 the proof.√ On the other hand, suppose that 2 is not rational, and take √ 2 √ m = 2 and n = 2. The claim is thus proved, but we end up without knowing, after all, which are the numbers m and n that satisfy the required condition. Now suppose that a student of mathematics, sympathetic with a realist conception of mathematical objects, becomes aware of the proof above, say, by reading section 5.1 of van Dalen [10]. (S)he accepts the proof as a proof of the truth of (P) although (s)he is still not able to exhibit the numbers m and n. In this scenario, a non-constructive proof is available, yet a constructive proof, in this case essentially more informative, is lacking. 2 This may be represented by the attribution of the value 0 to both P and ¬P , but it is important to call attention to the fact that this does not mean that both propositions are false. Rather, this means that neither of them has been constructively proved yet. Our young mathematician then starts working on this problem trying to find out numbers that satisfy the required condition. After√ some time, (s)he √ constructively reaches the following result: consider m = 2 and n = log 2(k), for k an odd natural number. It can be proved by constructive means that mn is rational, but both m and n are irrational3.

√ √ log 2(k) Fact 4. For k ∈ N, 2 is a rational number.

√ √ Proof. From the law of identity, log 2(k) = log 2(k). Now, from the definition √ √ log 2(k) of log, 2 = k, which obviously is rational, since k ∈ N. √ Fact 5. 2 is not a rational number. Proof. The proof in this case is well known, and it is intuitionistically acceptable, since it employs a reductio ad absurdum that introduces a negation.

√ Fact 6. For k ∈ N odd, log 2(k) is not a rational number.

√ Proof. Suppose, recalling that by hypothesis k is odd, that log 2(k) is rational. √ Then, for some natural numbers a and b, log 2(k) = a/b. It follows that √ a/b √ a ( 2 )b = kb, and thus 2 = kb. Since k is odd, kb is odd, for any b. Now, √ a √ any natural number is odd, or not odd. Suppose a is odd. Then 2 = c 2,

2Aleksandr Gelfond and Theodor Schneider independently proved in 1934 [see 2], while √ √ √ 2 solving Hilbert’s 7th problem, that 2 2 and its square root 2 are both irrational and transcendent, so in principle we know the irrational numbers m and n. Whether or not Gelfond-Schneider’s proof is acceptable by intuitionists is another story. 3We are not claiming that m and n are constructible numbers in the intuitionistic sense. Rather, we are only exhibiting two classic real numbers that fulfil the desired properties. The intuitionistic real numbers and the classical real numbers are not comparable, and to prove that a real number cannot be intuitionistically defined is a hard task.

8 √ a √ a for some c, so 2 6= kb. Now, suppose a is not odd. Then, 2 = c for some √ a b b √ c even. Since k is odd, for any b, again 2 6= k . Hence, log 2(k) is not a rational number. Notice that the proof above depends on the proposition for any natural number x, x is odd or x is not odd, which is valid intuitionistically because the predicate x is odd is decidable – given any natural number, there is a finite procedure that will end up with an answer yes or no. According to the clause for the universal quantifier in the well-known BHK interpretation, a proof of the proposition above is a construction that transforms any given natural number d into a proof that d is odd or d is not odd. This is, therefore, a legitimate use of excluded middle. Once in the possession of a constructive proof, our young mathematician says: ‘now, besides knowing that the proposition is true, I am also able to exhibit numbers that make it true!’ This new scenario is so represented by attributing 1 to P and 0 to ¬P . The rejection of the instance of the excluded middle P ∨ ¬P , from the constructive point of view, was thus a provisional situation. It was not based on the falsity of both P and ¬P but rather on an epistemic viewpoint related to the availability of a constructive, more informative proof. The position that accepts P as true even though it is not constructively proved is an absolutely reasonable position that illustrates the peaceful coexistence between classical and intuitionistic logic mentioned by Dubucs [11].

2.3 Paraconsistency and epistemological contradictions Now, let’s turn to the dual situation, the failure of explosion in paraconsistent logics. A counter-example for the principle of explosion is a circumstance in which both A and ¬A hold but there is some B such that B does not hold. Di- aletheism gives a straighforward answer: both A and ¬A are true, but obviously the world is not trivial. Although the claim that there are true contradictions does not cohere with the idea, central in scientific practice, that avoiding con- tradictions is an indispensable criterion of rationality, from the philosophical point of view, this answer is, in principle, legitimate because the topic of ‘real contradictions’, or contradictions of ontological character, appears in several places in the history of philosophy. But, still, the issue is contentious. The thinkers who most emphatically (and famously) defended the existence of real contradictions in Western philosophy were Heraclitus and Hegel. In both, con- tradictions are related to change, movement: the ongoing motion of reality. So, it is fully feasible to interpret their claims about contradictions in a way that the standard first order formulation of the principle of non-contradiction, the theorem-schema ∀x¬(P x ∧ ¬P x), is not violated. However interesting a discussion on the interpretation of contradictions in Heraclitus and Hegel might be, it will not be developed here. The point that is relevant here is that both philosophers are usually interpreted as having given support to the thesis that there are ontological contradictions. They are taken as the philosophical background that supports contemporary dialetheism: in order

9 to truly describe reality, we cannot dispense with some pairs of contradictory propositions. Dialetheism is what we call an ontologically motivated rejection of the principle of explosion. Nevertheless, the principle of explosion may be also rejected for epistemological reasons. Paraconsistency may be combined with a realist view of truth that both en- dorses excluded middle and rejects the thesis that there are true contradictions. Accordingly, contradictions that occur in a number of contexts of reasoning do not mean that a given proposition A and its negation are true, nor that A is both true and false. Contradictions that occur in empirical sciences deserve special attention. It is routine for physicists to deal with theories that yield contradictions in some critical circumstances or when put together with others theories. As pointed out by Meheus [22, p. vii], however, the fact that “almost all scientific theories at some point in their development were either internally inconsistent [i.e. contradictory]4 or incompatible with other accepted findings” is by no means “disastrous for good reasoning”. In fact, in these cases the gen- eral argumentative framework is already paraconsistent because, obviously, in order to avoid a disaster, the principle of explosion cannot be valid. Further- more, in such contexts it is not the case that all contradictions are equivalent. Some contradictions are, so to speak, impossible to endure, and are a sign that something has gone wrong. Others, even if understood as provisional, have to be dealt with, and in some sense are an essential ingredient of scientific practice. The fact that contradictions are unavoidable in empirical sciences is also pointed out by Nickles [23, p. 2]. According to him, empirical sciences are “non- monotonic enterprises in which well justified results are routinely overturned or seriously qualified by later results. And ‘nonmonotonic’ implies ‘temporally in- consistent’ [i.e. temporally contradictory].” With respect to contradictions, he adds that they are

products of ongoing, self-corrective investigation and neither pro- ductive of general intellectual disaster nor necessarily indicative of personal or methodological failure. In any case we are left with the task of better understanding how inconsistency and neighboring kinds of incompatibility are tamed in scientific practice and the cor- responding task of better modelling idealized practice in the form of inconsistency-tolerant logics and methodologies. [23, p. 2] Of course, it is part of our goal here to contribute to the task of providing a inconsistency-tolerant logic that helps to understand how contradictions are made tractable in sciences. But the point we want to emphasize is that these ‘provisional contradictions’ are not dialetheias. In our view, they may be of ‘different kinds’ in the sense of having different causes. We list some of them: (i) possible limitations of our cognitive apparatus; (ii) failure of measuring in- struments and/or interactions of these instruments with phenomena; (iii) stages in the development of theories; (iv) simply mistakes that in principle could be

4For reasons that will become clear later, we prefer to talk about contradictions rather than inconsistencies in sciences

10 corrected later on. In all these cases, contradictions are related primarily to knowledge and thought. This is what we call epistemic contradictions. Let us represent the fact that some contradiction is accepted in a given context by the attribution of the value 1 to a pair of propositions A and ¬A. Now, a question is: what does it mean to say that both A and ¬A receive the value 1, if 1 does not mean true? The answer must be some property weaker than truth, in the sense that the attribution of such a property to a proposition does not imply the truth of the proposition. A pair of contradictory sentences may be understood by way of a number of concepts that are dealt with in informal reasoning: clashing information, conflicting evidence, incompatible verisimilitude, opposed possibility etc. Among them, the notion of conflicting evidence deserves special attention as particularly promising to paraconsistent logics. Evidence may be understood, as is usual in epistemology, as that which is relevant for justified belief [cf. 16]. So, ‘there is evidence that A is true’ means that ‘there are some reasons for believing that A is true’. From this point of view, the notion of preservation of evidence presents itself as a topic to be further developed in paraconsistency. The idea is that, as much as the BHK interpretation for intuitionistic logic expresses preservation of (some sense of) construction, a set of inference rules and/or axioms that preserve (some sense of) evidence might be also established. This will be done in the section 3 below. Before that, however, let us see an example of a real situation in physics that illustrates what has been said above.

2.4 The special theory of relativity A very good example of a provisional contradiction in physics is the problem faced by Einstein just before he formulated the special theory of relativity. It is well known that there was an incompatibility between the classical, Newtonian mechanics and Maxwell’s theory of electromagnetic field. This is a typical case of two (supposedly) non-contradictory theories that, when put together, yield a contradiction. Classical mechanics gives a description of bodies changing position in space and time. It is intuitively understood, and works very well, with respect to ‘slow objects’ (we will see what it means for an object to be or not to be ‘slow’). Let us recall the example of a train in uniform linear motion with velocity v with respect to the rails and an object o moving inside the train with velocity w with respect to the train (where w and v have the same direction). The velocity w0 of o with respect to the rails is the algebraic sum of w and v.5 The relation between the velocities w0, w and v is given by the theorem of the addition of velocities, w0 = w + v, an elementary result in classical mechanics.

5The examples given here have been adapted from Einstein [12, sections 6 and 7].

11 In the second half of the nineteenth century, the physicist J.C. Maxwell for- mulated the so-called theory of electromagnetic field that gives a unified account of the phenomena of electricity, magnetism and light. According to this the- ory, the velocity of light in vacuum (c) is equal to 300,000 km/sec, and, most importantly, c is independent of the motion of its source. Now let us modify a bit the example above. Suppose that instead of an object moving inside the train, we are concerned with the light emitted by the headlight of the train (and suppose also, for the sake of the example, that the air has been removed). According to classical mechanics, the velocity w of the light with respect to the rails would be the sum of the velocity of the train and the velocity of light: w = c + v, hence, ¬(w = c). On the other hand, according to Maxwell’s theory, the velocity of light does not depend on the velocity of the train: w = c. We have, thus, that classical mechanics and the theory of electromagnetic field ‘prove’ a pair of contradictory propositions, ¬(w = c) and w = c. So, the two theories put together yield a contradiction, and if the underlying logic is classical, triviality follows. In the situation described above, two propositions A and ¬A hold in the sense that both may be ‘proved’ from theories that were supposed to be correct. This fact may be represented by the attribution of the value 1 to both A and ¬A. But clearly, the meaning of this should not be that both are true – actually, we know it is not the case, and nobody has ever supposed that it could be the case. The meaning of the simultaneous attribution of the value 1, as we suggest, is that at that time there was evidence for both in the sense, mentioned above, of some reasons for believing that both were true, because there was evidence that the results yielded by both classical mechanics and the theory of electromagnetic field was true. Classical mechanics is not compatible with Maxwell’s theory because the equations of the latter are not invariant under the so-called Galilean transfor- mations, which in classical mechanics relate the space-time coordinates of two systems of reference in uniform linear motion. By the end of the nineteenth cen- tury, H.A. Lorentz had already presented a group of equations, called Lorentz transformations, and the interesting fact is that Maxwell’s equations are invari- ant under Lorentz transformations.6 Einstein then rewrote Newton’s equations in such a way that the theory so obtained, the theory of special relativity, was fully compatible with the theory of the electromagnetic field. Actually, what Einstein did was to consider that the mass of a body increases with velocity, and it changed the whole thing. From the new equations, a different theorem of addition of velocities can be proved:

0 w+v w = 1+wv/c2 The ‘contradiction’ is now solved (roughly speaking) in the following way: as velocity grows, time ‘slows down’ and ‘space shortens’. So, the relation be- tween space and time that gives velocity remains the same, because both have

6We are not going into the details here. Friendly and accessible presentations of the problem may be found in Einstein [12] and (a more detailed one) in Feynman et al [13, ch. 15].

12 decreased. Newton’s equations work well for ‘slow objects’, that is, objects mov- ing in such a way that the value of wv/c2 may be discarded. Thus, what the special theory of relativity shows is that classical mechanics is a special case of the former. We have just seen a good example of what we call epistemic contradictions. We want to call attention to the fact that the general logical framework Einstein was working in was not classical. He had two different theories at hand, classical mechanics and the theory of the electromagnetic field that, when put together, yielded a non-explosive contradiction. Later, according to the special theory of relativity, the ‘contradiction’ disappeared. Although there was some reasons to believe that both ¬(w = c) and w = c were true, only one, the latter, has been established as true. The value 1 attributed to ¬(w = c) later became 0.

3 A logic of evidence

We turn now to the task of devising a paraconsistent formal system capable of expressing the idea of contradictions as conflicting evidence. As we have seen, ‘evidence that A is true’ is understood as ‘reasons for believing that A is true’, while ‘evidence that A is false’ means ‘reasons for believing that A is false’. We must distinguish the presence of evidence that A is false from the absence of evidence that A is true. Let ‘A is false’ be represented by ¬A. Accordingly: ‘A holds’ means ‘there is evidence that A is true’; ‘A does not hold’ means ‘there is no evidence that A is true’; ‘¬A holds’ means ‘there is evidence that A is false’; ‘¬A does not hold’ means ‘there is no evidence that A is false’. The following four scenarios are possible: 1. No evidence at all: both A and ¬A do not hold; 2. Only evidence that A is true: A holds and ¬A does not hold; 3. Only evidence that A is false: A does not hold and ¬A holds; 4. Conflicting evidence about A: both A and ¬A hold. There are two remarks to be made with respect to the scenarios above. First, it is clear that the formal system we are looking for must be not only paraconsistent but also paracomplete: neither the principle of explosion nor excluded middle should hold. Second, it would be desirable to also have resources for expressing the presence of conclusive evidence for a proposition A, that would be nothing but the truth of A. We will return to this point later, in section 4.2. Let us first see how a formal system capable of expressing the propagation of evidence through implication, conjunction, disjunction and negation may be obtained.

13 Let L0 be a language with a denumerable set of propositional letters {p0, p1, p2, ...}, the set of connectives {¬, ∧, ∨, →}, and parentheses. The set S0 of formulas of L0 is obtained recursively in the usual way. Roman capitals stand for meta- variables for formulas of L0. The logic systems in this paper will be formalized by means of natural deduc- tion rules, instead of axiomatically. Gentzen [14] presented natural deduction systems as formalisms conceived to represent ‘natural’ logical reasoning in a better way than axiomatic systems. Analogously, here the idea is to reflect how people actually, and naturally, draw inferences when the criterion is not preservation of truth but, rather, preservation of evidence. The definition of a derivation D of A from a set Γ of premises is the usual one for natural deduction systems [see 10, pp. 35ff]. A complete natural deduction system for classical propositional logic is ob- tained by adding to the introduction and elimination rules for →, ∨ and ∧ the rules below, explosion and excluded middle: [A] [¬A] . . . . A ¬A B B EXP and PEM. B B

By omitting the rules above and keeping only the introduction and elimination rules for →, ∨ and ∧, one obtains positive intuitionistic propositional logic (PIL). We start with PIL, arguing that it is able to express the notion of preservation of evidence.

3.1 A natural deduction system for preservation of evi- dence It will be seen that the introduction rules for ∧, ∨ and → work well with respect to evidence.7 AB ∧I A ∧ B

A B ∨I A ∨ B A ∨ B

[A] . . B → I A → B 7As usual in natural deduction systems, [A] means that the hypothesis A has been dis- charged (or cancelled). Regarding discharging of hypothesis, see van Dalen [10, pp. 31, 32, 34].

14 First, an evidence-interpretation for the introduction rules for ∧ and ∨ is straight- forward. Indeed, if κ and κ0 are evidence, respectively, for A and B, κ and κ0 together constitute evidence for A ∧ B. Similarly, if κ constitutes evidence that a A is true, then κ is also evidence that any disjunction that has A as one disjunct is true. The rule for → deserves some remarks. When the supposition that there is evidence κ for A leads on to conclude that there is evidence κ0 for B, this is evidence for A → B. So far, so good. However, → does not demand any relation between the contents or meanings of A and B. Hence, if there is evidence for B, we may conclude that there is also evidence for A → B, for any A. Notice that it works analogously to the implication in BHK interpretation: if there is a construction of B, there is a construction of A → B. The elimination rules may be obtained from the introduction rules. Gentzen [14, p. 80] famously remarks that “The introductions represent, as it were, the ‘definitions’ of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions”. Prawitz put this idea as follows, in the so-called inversion principle: Let α be an application of an elimination rule that has B as con- sequence. Then, deductions that satisfy the sufficient condition (...) for deriving the major premise of α, when combined with deductions of the minor premises of α (if any), already ‘contain’ a deduction of B; the deduction of B is thus obtainable directly from the given deductions without the addition of α. [25, p. 33] Let us see how the elimination rule for → (modus ponens) is obtained. The sufficient condition for concluding A → B is a derivation of B from hypothesis A: A . . B The minor premise of →-elimination is a deduction of A: . . A

We have just to put them together to get a derivation of B. Now we reformulate Prawitz’ inversion principle in terms of evidence as follows:

The evidence inversion principle (EIP) Let α be an application of an elimination rule that has B as consequence. Then, any κ that is evidence for the major premise of α, when combined with evidence for the minor premises of α (if any), already constitutes evidence for B; the exis- tence of evidence for B is thus obtainable directly from the existence of evidence for the premises, without the addition of α.

We thus get the respective elimination rules:

15 A ∧ B A ∧ B ∧E A B

[A] [B] . . . . A ∨ B C C ∨E C

A → BA → E B It is clear that the rules above preserve evidence. Let us take a look at → E. Evidence for A → B, we saw above, is obtained when supposing evidence κ for A leads to evidence κ0 to B. Now, if the evidence κ for A is available, the evidence κ0 for B is also available. Up to this point we have what could be called a positive logic of preservation of evidence, which, of course, amounts to the BHK interpretation of positive intuitionistic propositional logic.

3.2 Negation Now we turn to negation. One central and difficult problem for paraconsistent and paracomplete logicians is that of specifying a negation without (some of) the properties of classical negation but that still keeps enough properties to be called a negation. One way to approach this problem is to check whether or not the properties of classical negation fit the intuitive meaning we want to represent by the negation we are looking for. If we do that, we find out that the principle of explosion and excluded middle do not hold from the viewpoint of preservation of evidence. Another property of classical negation that must be invalid is the so-called introduction of negation:

A → B,A → ¬B ` ¬A. Any extension of PIL in which introduction of negation holds is able to prove that for any A and B: A, ¬A ` ¬B, that is, from a contradiction, any negated propositions follows, which is obvi- ously an undesirable result for a boldly paraconsistent system8. Besides, and

8See Carnielli et al [6, p. 14 def. 9] where a distinction between boldly paraconsistent and partially explosive systems is established. In the latter, although explosion does not hold, A, ¬A ` ¬B holds. An example of a partially explosive formal system is Kolmogorov’s ‘logic of judgment’ [see 17].

16 more importantly, introduction of negation does not fit the intuitive interpre- tation in terms of evidence, since there may be a circumstance such that there is evidence for B and for ¬B, and so also for A → B and A → ¬B, but no evidence for ¬A. Now we turn to rules in which the conclusion is a negation of a conjunction, a disjunction or an implication. Notice that these rules cannot be obtained from the rules we already have because we cannot introduce negation, as it is usually done in intuitionistic and classical logic, since introduction of negation does not hold. In general, natural deduction rules for concluding falsities may be obtained in a way similar to the rules for concluding truths. The point is that instead of asking about the conditions of assertibility, we ask about the conditions of refutability [cf. 20]. Thus, as an example, a natural deduction rule whose con- clusion is the falsity of a conjunction is the following

¬A . ¬(A ∧ B) This rule is obtained by asking what would be sufficient conditions for refuting a conjunction. We now apply an analogous idea, asking what would be sufficient conditions for having evidence for the falsity of a conclusion. If κ is evidence that A is false, κ constitutes evidence that A ∧ B is false – mutatis mutandis for B. ¬A ¬B ¬ ∧ I . ¬(A ∧ B) ¬(A ∧ B) Now we ask what constitutes sufficient condition for having evidence that a disjunction A ∨ B is false. What is needed in this case is evidence that both A and B are false. Suppose κ is evidence that A is false and κ0 is evidence that B is false. Hence, κ and κ0 together constitute evidence that A ∨ B is false. So, ¬A ¬B ¬ ∨ I. ¬(A ∨ B) In order to get the introduction rules for a negated implication, we ask what would be sufficient conditions for having evidence that a formula A → B is false. Suppose κ is evidence that A is true and κ0 is evidence that B is false. Then, κ and κ0 together constitute evidence that A → B is false. So, A ¬B ¬ → I. ¬(A → B) The corresponding elimination rules are obtained applying the evidence in- version principle (EIP) mentioned above. The rules so obtained fit well with the notion of preservation of evidence. The elimination of negation with respect to disjunction is: ¬(A ∨ B) ¬(A ∨ B) ¬ ∨ E . ¬A ¬B

17 Suppose κ is evidence that A ∨ B is false. κ must be (or must contain) evidence that both A and B are false. Indeed, a sufficient condition for having evidence that A ∨ B is false is evidence for the falsity of both A and B. Applying EIP, the rule ¬ ∨ I is obtained. By means of analogous reasoning, we obtain the following rules, which we call, respectively, negation elimination with respect to implication and conjunction:

¬(A → B) ¬(A → B) ¬ → E , A ¬B [¬A] [¬B] . . . . ¬(A ∧ B) C C ¬ ∧ E. C

Applying the evidence inversion principle, the rules above are easily obtained. But we claim that they are well suited to the idea of preservation of evidence. Let us take a look at rule ¬ → E. When κ is evidence that a formula A → B is false, κ must also be evidence that the antecedent A is true and the consequent B is false. So, both conclusions may be taken: there is evidence that A is true (left side of the rule) and that there is evidence that B is false (right side of the rule). The rule ¬∧E, although easily obtained from EIP, may be not so clear from the point of view of preservation of evidence. Suppose κ is evidence that A ∧ B is false. Then, κ must contain evidence that A is false or B is false (or maybe both). Now, if evidence that A is false leads to evidence that C is true, and evidence that B is false also leads to evidence that C is true, then, from the supposition that there is evidence that A ∧ B is false, one may conclude that there is evidence that C is true. Lastly, we turn to double negation. Whether or not double negation is valid depends on the notion of evidence that is being accounted for. Indeed, the decision about the validity of double negation determines some features of the specific notion of evidence expressed by the formal system – similar to the fact that decisions taken by Heyting, Kolmogorov and Johansson with respect to the behaviour of negation, while building their systems, resulted in logics that express subtly different notions of constructive proof. We claim, however, that there are reasons to consider double negation as valid in both directions: A ¬¬A DN . ¬¬A A The rules above relate, in a perspicuous way, evidence that a proposition A is true with evidence that A is false. Suppose κ is evidence that A is true. It is reasonable to see κ as evidence that it is false that A is false. Now suppose, conversely, that κ is evidence that it is false that A is false. Again, it is reason- able that κ also constitutes evidence that A is true. We claim, thus, that the validity of double negation in both directions is coherent to a notion of evidence

18 that occurs in informal reasoning and natural language. Notice that it does not mean that such a notion of evidence is the only possible one. Actually, from another viewpoint, it might be plausibly considered that the rules presented here just define a specific notion of evidence. Let us call the logic defined by the rules DN, introduction and elimination for →, ∨ and ∧, plus introduction and elimination for negated →, ∨ and ∧, the Basic Logic of Evidence (BLE).9 We claim that BLE faithfully represents preservation of evidence when the latter is understood as reasons for believing the truth/falsity of propositions. It can also be said that BLE ‘defines’ the deductive behavior of a specific notion of evidence, which is in accordance with the rules as presented. Lemma 7. BLE satisfies the following properties: P1. Reflexivity: if A ∈ Γ, then Γ ` A; P2. Monotonicity: if Γ ` B, then Γ,A ` B, for any A; P3. Cut: if ∆ ` A and Γ,A ` B, then ∆, Γ ` B; P4. Deduction theorem: if Γ,A ` B, then Γ ` A → B; P5. Compactness: if Γ ` A, then there is ∆ ⊆ Γ, ∆ finite, ∆ ` A. Proof. The properties P1, P2, P3 and P5 are direct consequences of the defi- nition of a deduction of A from premises in Γ. The deduction theorem amounts to the rule →-introduction.

Since the properties P1, P2 and P3 hold, BLE is thus a standard logic (cf. Carnielli et al [6] p. 6).

4 A logic of evidence and truth

The logic BLE presented above, although able to express preservation of evi- dence, is not able to express preservation of truth. The sketch of a semantics presented in the section 3 attributes 1 and 0, respectively, to the presence and absence of evidence for a given proposition A. Up to this point we do not yet have any means to express that A is true (or false), which would be the same as saying that its truth value has been conclusively established. But what hap- pens in real-life contexts of reasoning is that we deal simultaneously with truth and evidence, that is, with propositions that we take as well established (i.e. conclusively established as true or false in that context) and others for which only non-conclusive evidence is available. On the other hand, classical logic gives a very good, maybe the best, account of truth preservation and truth as understood from a realist and non-dialetheist viewpoint. Thus, the problem we have in our hands could be solved if we were able to restore classical logic for some propositions, those that we want to say are either true or false. If classical logic holds with respect to them, we

9It is worth noting that BLE can easily be proved to be equivalent to Nelson’s logic N4 [cf. 24, p. 133]. Besides, BLE is also equivalent to the propositional fragment of refutability calculus presented by L´opez-Escobar [20].

19 may represent/express (with respect to them) the relation of truth-preservation. What we need is to add to BLE the means to recover the properties of classical negation. Or, more precisely, we need to restore the validity of explosion and excluded middle with respect to those formulas for which we want to recover classical logic. In what follows, we will see how this can be done.

4.1 Logics of Formal Inconsistency and Undeterminedness Logics of Formal Inconsistency (from now on, LFI s), are a family of paracon- sistent logics that have resources to express, inside the object language, that a formula A is (in some sense) consistent. This is done by means of a unary propositional connective: ◦A means that A is consistent. The family of LFI s incorporate a great number of different paraconsistent logics, including several formal systems studied in Carnielli et al [6]. In LFI s, negation is explosive only with respect to formulas marked with ◦ (i.e. consistent formulas):

(a) For some Γ, A and B:Γ, A, ¬A 0 B; (b) For every Γ, A and B:Γ, ◦A, A, ¬A ` B. A logic in which (a) and (b) hold is said to be gently explosive [cf. 6, pp. 19-20]. The idea of expressing a metalogical notion within the object language is not new. It is found in the Cn hierarchy introduced by da Costa [7]. In C1, the so- called notion of the ‘well-behavior’ of a formula A is expressed by A◦. However, ◦ an important difference is that in C1, A is an abbreviation of ¬(A∧¬A), which makes the ‘well-behavior’ of A equivalent to saying that A is non-contradictory. On the other hand, in LFI s the consistency of A may not be definable in terms of the non-contradictoriness of A. That is, it may be the case that ◦A and ¬(A ∧ ¬A) are not logically equivalent. Notice that ◦A and A◦ should not be confused. Once we break up the equivalence between ◦A and ¬(A ∧ ¬A), some very interesting developments become available. Indeed, ◦A may express notions other than consistency as freedom from contradiction. We have some good reasons for adopting a consistency connective in a para- consistent logic. First of all, in any paraconsistent logic with a few logical resources it cannot be the case that all contradictions are logically equiva- lent, otherwise the principle of explosion holds. Suppose for any A and B, A ∧ ¬A `a B ∧ ¬B. Then, A ∧ ¬A ` B ∧ ¬B, and by conjunction-elimination, A ∧ ¬A ` B. Notice that this result, rephrased contrapositively, states that if a logic is paraconsistent, then it has some pairs of non-equivalent contradic- tions. This nonequivalence between contradictions is actually a desired result, which fits the idea that in real-life contexts of reasoning some contradictions are more relevant than others. Indeed, in such contexts, information that contra- dicts a proposition that has been conclusively established as true is immediately rejected as false. Hence, it seems natural to have a connective that is able to distinguish ‘different kinds’ of contradictions, separating the contradictions that do not lead to explosion from those that do.

20 The basic idea of restricting the validity of the principle of explosion may be generalized. The validity of some inference rule (or axiom) may be restricted in such a way that some ‘logical property’ does not hold unless some information is added to the system. In particular, excluded middle may be restricted in a way analogous to the restriction imposed to explosion.

(c) For some Γ, A and B:Γ,A ` B, Γ, ¬A ` B but Γ 0 B; (d) For every Γ, A and B: if Γ,A ` B and Γ, ¬A ` B, then Γ, 9A ` B, where 9A means that A is determined. A Logic of Formal Undeterminedness (LFU ) is a logic such that the properties (c) and (d) hold. In other words, excluded middle holds only for formulas marked with 9, which we call deter- mined.10 We have seen above that it cannot be the case that all contradictions are equivalent in a paraconsistent logic. Analogously, in a paracomplete logic in which cut and ∨I hold (BLE for example), it cannot be the case that all in- stances of A ∨ ¬A are equivalent, otherwise excluded middle is generally valid. The proof is as follows. Let A be any theorem. So, A ` A ∨ ¬A. Now, if for any A and B, A ∨ ¬A `a B ∨ ¬B, by cut it follows that A ` B ∨ ¬B. Since A is a theorem, we get ` B ∨ ¬B.

4.2 LETJ : a logic of evidence and truth The basic ideas of LFI s and LFU s may be combined in an LFIU – a Logic of Formal Inconsistency and Undeterminedness. In a context that is at the same time paraconsistent and paracomplete, we may recover at once explosion and excluded middle with respect to a given formula A, and hence the properties of classical negation with respect to A. Since consistency and determinedness will be recovered at once, we change the symbol 9 to ◦. An LFIU is obtained by adding the following inference rules to the logic BLE:

[A] [¬A] . . . . ◦AA ¬A ◦A B B EXP ◦, PEM ◦. B B ◦ ◦ The logic defined by the addition of EXP and PEM to BLE we call LETJ . The language L1 of LETJ is obtained from the language L0 of BLE by adding to the set of connectives the unary connective ◦. The definition of the set S1 of formulas of L1, and the other definitions, are analogous. As well as BLE, LETJ has properties 1 to 5 of lemma 7, and is also a standard logic. The name LETJ stands for ‘logic of evidence and truth based on positive intuitionistic proposi- tional logic’.11 Notice that the rules above may be considered elimination-rules for ◦, but there is no introduction-rule for ◦ (we will return to this point soon).

10The notion of LFU s is introduced in Marcos [21]. 11 ◦ In LETJ , the rule EXP plays the role of axiom bc1, ◦A → (A → (¬A

21 From the outside the system, ◦A may be understood as saying that the truth value of A has been (or may be) conclusively established. Thus, A ∧ ◦A means that A is true;

¬A ∧ ◦A means that A is false. Now we have resources to express not only that there is evidence that A is true/false but also that A has been established (by whatever means) as true/false: ◦A ∧ A and ◦A ∧ ¬A respectively. Notice that how the truth or falsity of a proposition is going to be established is not a problem of logic. Truth comes from outside the formal system. On the other hand, if the truth- value of A has been conclusively established, it cannot be the case that there is still conflicting evidence for A and for ¬A. If A has been established as true, any evidence for ¬A is cancelled (mutatis mutandis when A is false).

4.3 A semantics for LETJ 12 Now, we present a complete and sound bivalued semantics for LETJ .

Definition 8. A semivaluation s for LETJ is a function from the set S of formulas to {0, 1} such that:

1. if s(A → B) = 1, then s(A) = 0 or s(B) = 1; 2. if s(A → B) = 0, then s(B) = 0; 3. s(A ∧ B) = 1 iff s(A) = 1 and s(B) = 1; 4. s(A ∨ B) = 1 iff s(A) = 1 or s(B) = 1;

5. s(A) = 1 iff s(¬¬A) = 1; 6. s(¬(A ∧ B)) = 1 iff s(¬A) = 1 or s(¬B) = 1; 7. s(¬(A ∨ B)) = 1 iff s(¬A) = 1 and s(¬B) = 1;

8. s(¬(A → B)) = 1 iff s(A) = 1 and s(¬B) = 1; 9. s(◦A) = 1 implies [s(¬A) = 1 iff s(A) = 0].

Definition 9. A valuation for LETJ is a semivaluation for which the condition below holds:

rightarrowB)), in the LFI s presented in Carnielli et al [6]. 12As far as we know, bivalued non-truth-functional semantics for paraconsistent logics were presented for the first time by da Costa and Alves [9] where we find a sound and complete semantics for da Costa’s C1. Loparic [18] presented a bivalued sound and complete semantics for da Costa’s logic Cω.

22 (Val) For all A1...An and for all B not of the form C → D: if v(A1 → (A2 → ... → (An → B)...)) = 0, then there is a semivaluation s such that for every i, 1 ≤ i ≤ n, s(Ai) = 1 and s(B) = 0.

Semantic consequence in LETJ is defined as follows. We say that a valuation v is a model of Γ (v  Γ) if for all B ∈ Γ, v(B) = 1; v  A means that v(A) = 1. Γ  A means that for every valuation v, if v is a model of Γ, then v(A) = 1. The connectives of LETJ are not compositional, in the sense that the seman- tic value of a molecular formula is not functionally determined by its structure and the semantic values of its component parts. It is clear when we take a look at the clauses for negation – for example, when v(A) = 1, v(¬A) may be 1, as well as 0 – and also implication. The implication is defined by clauses 1 and 2 of definition 8 plus condition Val of definition 9. When v(A) = v(B) = 0, v(A → B) may be 1 or 0. We give below two examples in order to illustrate how this semantics works, especially the condition Val.

Example 10. A ∨ (A → B) is invalid in LETJ .

A 0 1 B 0 1 0 1 A → B 0 1 1 0 1 A ∨ (A → B) 0 1 1 1 1 s1 s2 s3 s4 s5

Notice that the subformula A → B receives 0 in the semivaluation s1. The latter is not excluded by condition Val (that is, it is a legitimate valuation) because there is a semivaluation s such that s(A) = 1 and s(B) = 0, namely, s4.

Example 11. A → (B → A) is valid in LETJ . A 0 1 B 0 1 0 1 B → A 0 1 0 1 1 A → (B → A) 0 1 1 0 1 1 1 s1 s2 s3 s4 s5 s6 s7 The table above shows how condition Val excludes the ‘undesired semivalua- tions’, when it is the case. The semivaluations s1 and s4 do not satisfy condi- tion Val because there is no semivaluation s such that s(A) = 1, s(B) = 1 and s(A) = 0. Therefore, s1 and s4 are not valuations. So, the formula receives 1 under every valuation (namely s2, s3, s5, s6 and s7) and is thus valid.

4.3.1 Soundness

In order to prove that LETJ is sound with respect to the semantics presented above, we show that for every derivation D with premises in Γ and conclusion A,Γ  A holds. The proof is based on the proof presented by van Dalen [10], and is carried out by induction on the length of derivation D.

23 Theorem 12. If Γ ` A, then Γ  A. Proof. Base: if D has one element A, it means {A} ` A. So, A ∈ Γ, and Γ  A. Inductive step. For each rule, the inductive hypothesis says that there are sound derivations for the premise(s). Then, we show that the derivation obtained by the application of the rule is sound. As an example, we show that the rule EXP ◦ is sound. By the inductive hypothesis: there are derivations D, D0 and 00 0 00 D , respectively of Γ,A ` C,Γ , ¬A ` C and Γ ` ◦A, such that Γ,A  C, 0 00 0 00 0 Γ ,B  C and Γ  ◦A (Γ, Γ and Γ contain precisely the premises of D, D 00 0 00 and D ). Let ∆ be the union of the sets Γ, Γ and Γ . So, ∆,A  C, ∆, ¬A  C and ∆  ◦A. ∆ contains the premises of the new derivation obtained by the ◦ application of EXP whose conclusion is C. Now we show that ∆  C. Suppose ∆ 2 C. Then, for some valuation v, v  ∆ and v 2 C. Since v  ◦A, either v  A or v  ¬A, and we get a contradiction in a few steps.

4.3.2 Completeness

The reader finds a Henkin-style completeness proof for da Costa’s logic Cω in Loparic [18]. Like LETJ , Cω is an extension of positive intuitionistic logic (PIL). The bivalued semantics for Cω and LETJ have in common clauses 1 to 4 of definition 8 plus condition Val of definition 9. In order to get a completeness proof for LETJ , we will slightly modify Loparic’s completeness proof for Cω.

Theorem 13. If Γ  A, then Γ ` A. Proof. Suppose Γ 0 A. An A-saturated set ∆ (so that A/∈ ∆) is obtained from Γ by means of a Lindenbaum construction in the usual way. The following propositions, corresponding to clauses 1 to 9 of definition 8 plus condition (Val) of definition 9, can be proved easily: (10) if A → B ∈ ∆, then A/∈ ∆ or B ∈ ∆;

(20) if A → B/∈ ∆, then B/∈ ∆; (30) A ∧ B ∈ ∆ iff A ∈ ∆ and B ∈ ∆; (40) A ∨ B ∈ ∆ iff A ∈ ∆ or B ∈ ∆; (50) A ∈ ∆ iff ¬¬A ∈ ∆;

(60) ¬(A ∧ B) ∈ ∆ iff ¬A ∈ ∆ or ¬B ∈ ∆; (70) ¬(A ∨ B) ∈ ∆ iff ¬A ∈ ∆ and ¬B ∈ ∆; (80) ¬(A → B) ∈ ∆ iff A ∈ ∆ and ¬B ∈ ∆;

(90) ◦A ∈ ∆ implies ¬A ∈ ∆ iff A/∈ ∆;

0 (V al ) If ∆ is A-saturated and A1 → (A2 → ... → (An → B)...) ∈/ ∆, then 0 0 there is a B-saturated set ∆ such that ∆ ∪ {A1,A2, ...An} ⊂ ∆ .

24 Consider now the characteristic function v of the set ∆: in view of clauses 10 to 0 0 9 and V al , v is a valuation for LETJ and v is a model for ∆. Since Γ ⊆ ∆, v is also a model for Γ. But v(A) = 0, so Γ 2 A. By contraposition, completeness is thus obtained.

4.4 Some facts about LETJ

The logic LETJ has some interesting features that fit the intuitive interpretation presented here. We show some of them below. Fact 14. A bottom particle ⊥ (that is, a formula that by itself trivializes the system) is definable in LETJ .

Proof. Define ⊥ def= ◦ A ∧ A ∧ ¬A. It can be proved in few steps that ⊥ ` B, for any B.

Fact 15. LETJ has no trivial models; hence, LETJ does not prove ⊥. Proof. The semantic clause 9 relates ◦ and ¬ in such a way that trivial models are excluded. In other words, in LETJ there is no valuation v such that for every formula A of L1, v(A) = 1. As a consequence, given soundness, LETJ does not prove ⊥, as defined in fact 14. The above result is the paraconsistent counterpart of the usual consistency- proofs. Since in paraconsistency logic triviality is not equivalent to freedom from contradiction, the relevant result, tantamount to proving consistency (as freedom from contradiction) when the underlying logic is classical, is to prove that ⊥, or whatever would imply triviality, is not a theorem.

Fact 16. ¬(A ∧ ¬A) and A ∨ ¬A are logically equivalent in LETJ . But neither of them is logically equivalent to ◦A. Proof. Since DN (double negation) holds, A ∨ ¬A is equivalent to ¬A ∨ ¬¬A. The latter, in its turn, is proved equivalent to ¬(A ∧ ¬A) by means of the rules ¬ ∧ I and ¬ ∧ E. In LETJ , ◦A ` ¬(A ∧ ¬A). Suppose ◦A. So, A ∨ ¬A. The latter implies ¬A ∨ ¬¬A, which is equivalent to ¬(A ∧ ¬A). To see that neither ¬(A∧¬A) nor A∨¬A implies ◦A, take v(A) = 1, v(¬A) = 0 and v(◦A) = 0. It is worth noting that the non-equivalence above fits the intuitive interpretation suggested. It may be the case that ¬(A ∧ ¬A) holds even if ◦A does not hold. There may be a situation such that there is some non-conclusive evidence for the truth of A but no evidence for the falsity of A. In such a scenario, we have v(¬(A ∧ ¬A)) = 1, although v(◦A) = 0, since the evidence available is non- conclusive. On the other hand, if v(◦A) = 1, there are only two possibilities: either v(A) = 1 and v(¬A) = 0, or v(A) = 0 and v(¬A) = 1. Notice, as well, that the non-equivalence above depends essentially on clause 9 of definition 8. Fact 17. According to the intuitive interpretation suggested here for ¬ and ◦, LETJ trivializes in the presence of a true contradiction.

25 Proof. Suppose A is a true proposition. This is represented in LETJ by A∧◦A. So, a true contradiction would be represented in LETJ by (A∧¬A)∧◦(A∧¬A). The latter implies ¬A. Applying ∨I we get ¬A ∨ ¬¬A, that is equivalent to ¬(A ∧ ¬A), and triviality follows.

In view of the fact 17, we say, therefore, that LETJ is anti-dialetheist in the sense that it does not tolerate true contradictions.

Fact 18. The logic LETJ does not have any theorem of the form ◦A. Proof. The strategy of this proof is to show that ◦A is independent of the rules and therefore cannot be a consequence of the rules of LETJ . Consider for this purpose the following alternative semantics: for the connectives ∨, ∧, → and ¬, the classical conditions over {0, 1}; for ◦, v(◦A) = 0 for any value of A. According to this semantics, no rule of LETJ yields a conclusion with value 0 if all premises have value 1. On the other hand, a formula ◦A receives 0, for any A. Hence, such a formula is independent of the rules, and cannot be proved in LETJ . This result is in accordance with the intuitive idea that the attribution of ◦ to a formula A may be done only from outside the formal system. It is the user of the system that establishes in which circumstances a formula may be marked with ◦. It is not a weakness, but rather a philosophical position about the limits of formal systems. What constitutes evidence for a given proposition A, and whether or not such evidence is conclusive and A may be established as true, are problems that depend on the specific area of knowledge being dealt with. These problems must be kept outside of the formal system.

Fact 19. If ◦A1, ◦A2, ◦A3... hold, we get classical logic with respect to all formulas that depend only on A1, A2, A3... and are formed with →, ∧, ∨ and ¬.

Proof. Let A1,A2, ...An be any formulas and B = φ(A1,A2, ...An) be any com- pounded formula formed from one or more formulas A1,A2, ...An through ¬, →, ∧ and ∨. We prove below that if ◦A1, ◦A2, ... ◦ An hold, then the rules [B] [¬B] . . . . B ¬B B0 B0 and . B0 B0 hold. Therefore classical logic holds for all such formulas. Base: B = Ai (for any i). The proof is straightforward. Inductive step: Case 1. B = ¬C. We prove PEM. Inductive hypothesis is the rule below: [C] [¬C] . . . . 0 0 B B . B0

26 In order to prove that [¬C] [¬¬C] . . . . B0 B0 B0 holds, we need only an application of DN. The proof of EXP is left to the reader. Case 2. B = C ∨ D. We prove EXP. The inductive hypothesis is: C ¬C D ¬D and . B0 B0 ¬(C ∨ D) ¬(C ∨ D) ¬ ∨ E ¬ ∨ E ¬C ∧ ¬D ¬C ∧ ¬D ∧E ∧E [C]1 ¬C [D]1 ¬D 0 i.h. 0 i.h. B B C ∨ D ∨E, 1 B0 The proof of PEM, as well as the proofs of cases 2 and 3 (respectively ∧ and →), are left to the reader. A corollary of fact 19 is that once a formula A is marked with ◦, no contradiction is allowed with respect to formulas with ¬, ∨, ∧ and → that depend only on A. However, from ◦A it does not follow that any formula that depends only on A is also marked with ◦. Again, the point is that the attribution of ◦ to any formula can be done only from outside the formal system. It must always be a proposition added to the theory at stake by means of non-logical means. Notice that the above result is different from the so-called ‘propagation of consistency’ that holds, for instance, in da Costa’s system C1. In LETJ , the fact that ◦A and ◦B hold does not imply that ◦(A ∗ B) hold, ∗ ∈ {∧, ∨, →}, but only that there is no valuation v such that v(A ∗ B) = v(¬(A ∗ B)) = 1. It is also worth mentioning that fact 19 is tantamount to a very perspicuous derivability adjustment theorem (DAT ). The purpose of a DAT is to establish a relationship between two logics in the sense of restoring inferences that are lacking in one of them.13 The basic idea is that we have to ‘add some infor- mation’ to the premises in order to restore the inferences that are otherwise lacking. The general form of a DAT is the following:

For all Γ and B, there is a ∆ such that: Γ `L B iff Γ, ∆ `L∗ B. Suppose that the logic L above is classical logic and L∗ is some non-classical logic. So, the DAT says that classical logic may be recovered with respect to L∗, but does not say how it can be done. More precisely, it says that there exists a set ∆ that contains the ‘information’ (i.e. the propositions) that must

13As far as we know, DATs were proposed for the first time by Diderik Batens, one of the main researchers in the field of paraconsistency [see 3]. A seed of this idea, however, may be found in da Costa [7] and da Costa [8] [cf. 6, p. 23].

27 be available in order to recover classical consequence, but says nothing more specific about these propositions. The result proved above, on the other hand, shows that the ‘information’ needed for a DAT in LETJ may be the atomic propositions that occur in Γ, marked with ◦. More precisely, let ∆0 be the set of the atomic propositions that occur in the formulas of Γ. The set ∆ we need is {◦p : p ∈ ∆0}. Of course, this is not the only set that allows recovering classical consequence. But, in our view, it seems to be very natural that atomic propositions are those that are first marked with ◦, as they are conclusively established as true or false in the respective context of reasoning.

Fact 20. A compound formula A is contradictory in a valuation v (i.e. v(A) = 1 and v(¬A) = 1) only if at least one atom p that occurs in A is contradictory in v. Proof. Suppose there is a valuation v such that v(A) = v(¬A) = 1. By induction on the complexity of A, we prove that there is at least one atom p that occurs in A such that v(p) = v(¬p) = 1. Base case: A = p. Clearly, v(A) = v(¬A) = v(p) = v(¬p) = 1. Inductive step. Case 1: A = ¬B. Inductive hypothesis: if v(B) = v(¬B) = 1, then there is a p in B such that v(p) = v(¬p) = 1. Suppose v(¬B) = v(¬¬B) = 1. So, by semantic clause 5, v(¬B) = v(B) = 1 and apply inductive hypothesis. Case 2: A = B ∨ C. Inductive hypothesis: if v(B) = v(¬B) = 1, then there is a p in B such that v(p) = v(¬p) = 1; m.m. for C. Suppose v(B ∨ C) = v(¬(B ∨ C)) = 1. So, by semantic clause 7, v(¬B) = v(¬C) = 1. Hence, v(B) = v(¬B) = 1 or v(C) = v(¬C) = 1. Now, apply inductive hypothesis. We leave the remaining cases to the reader. Notice, however, that the converse does not hold. There may be a contradictory atom p in a formula A without A being contradictory. Let A be the formula p ∨ q and consider the valuation v such that v(p) = v(¬p) = 1, v(q) = 1 and v(¬q) = 0. In this case, v(p ∨ q) = 1 but v(¬(p ∨ q)) = 0.

5 Final remarks

Let us recall the passage from Dubucs [11] quoted in section 2.1, according to which intuitionistic and classical logic have reached a sort of peaceful coexis- tence. We strongly suggest that it is time to recognize that paraconsistent and classical logic may live together in harmony, and do not need to be seen as ir- reconcilable positions. Indeed, in real-life contexts of reasoning this coexistence between paraconsistent and classical logic already obtains. The basic idea that makes possible the development of LFI s is to restrict some logical property to propositions that are distinguished from others. LFI s

28 do that with respect to explosion, but we have seen how this idea may be extended by LFIU s. The logic LETJ presented here is both paraconsistent and paracomplete and, we claim, gives a sustained analysis of reasoning in contexts with simultaneously an excess and lack of information. The path that leads us to the logic LETJ , in which contradictions may be understood epistemically as conflicting evidence and the notion of truth may be recovered from outside the formal system, has been inspired by an analysis of real situations of reasoning in which contradictions occur. ◦ may be considered kind of an epistemic operator that has to do with the justification of a proposition A. The fact that LETJ has no theorem of the form ◦A fits the idea that the particular way of establishing propositions as true (or false) is not a problem of logic. We end up by returning to the title of this paper: a radical epistemic ap- proach to paraconsistency. The approach proposed here is epistemic because it understands contradictions as conflicting evidence, and evidence is of course an epistemic concept. Besides, and mostly important, it is radical because true contradictions are not tolerated: they imply triviality as much as they do in classical logic. It is worth emphasizing, however, that our position is not reluc- tant or ‘sitting on the fence’: it does not intend to be metaphysically neutral, actually, quite the contrary. By means of a formal system that, according to the intuitive interpretation suggested, becomes trivial in the presence of true contradictions, a clear and explicitly anti-dialetheist view on contradictions is assumed (cf. fact 17). Notwithstanding, we believe that no philosophical, a priori argument can conclusively confirm or reject the claim that reality needs pairs of contradictory propositions in order to be truly described. In fact, real contradictions seem to be quite impossible, but of course we cannot, and are not intending to, prove that. The intuitive interpretation of the formal system LETJ would fail if someday it were ‘proved’ that real contradictions do exist. But in such an improbable scenario, a considerable part of science, and also philosophy, would fail together.

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