Paraconsistency and Duality

Home , Intuitionism, Intuitionistic logic

Paraconsistency and duality:

between ontological and 1 epistemological views

1 2 WALTER CARNIELLI AND ABILIO RODRIGUES 2

Abstract: The aim of this paper is to show how an epistemic approach to paraconsistency may be philosophically justiﬁed based on the duality between paraconsistency and paracompleteness. The invalidity of excluded middle in intuitionistic logic may be understood as expressing that no con- 3 structive proof of a pair of propositions A and ¬A is available. Analogously, in order to explain the invalidity of the principle of explosion in paraconsistent logics, it is not necessary to consider that A and ¬A are true, but rather that there are conﬂicting and non-conclusive evidence for both.

Keywords: Paraconsistent logic, Philosophy of paraconsistency, Evidence, Intuitionistic logic 4

1 Introduction 5

Paraconsistent logics have been assuming an increasingly important place 6 in contemporary philosophical debate. Although from the strictly techni- 7 cal point of view paraconsistent formal systems have reached a point of 8 remarkable development, there are still some aspects of their philosophi- 9 cal signiﬁcance that have not been fully explored yet. The distinctive fea- 10 ture of paraconsistent logics is that the principle of explosion, according to 11 which anything follows from a contradiction, does not hold, thus allowing 12 for the presence of contradictions without deductive triviality. Dialetheism 13 is the view according to which there are true contradictions (cf. e.g. Priest 14

1The ﬁrst author acknowledges support from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo, thematic project LogCons) and from a CNPq (Conselho Nacional de Desenvolvimento Cientíﬁco e Tecnológico) research grant. 2The second author acknowledges support from FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais, research project 21308). Both authors would like to thank Henrique Almeida, Antonio Coelho, Décio Krause, André Porto, Wagner Sanz and an anonymous referee for some valuable comments on a previous version of this text.

1 Walter Carnielli and Abilio Rodrigues

15 & Berto, 2013). Paraconsistency and dialetheism are not the same thing: the

16 latter implies the former, but one can endorse a paraconsistent logic without

17 being dialetheist.

18 For those who believe in true contradictions, dialetheism does provide a

19 philosophical justiﬁcation for paraconsistency. If one accepts that reality is

20 intrinsically contradictory, in the sense that in order to truly describe it (s)he

21 needs some pairs of contradictory propositions, then, since reality obviously

22 is not trivial, (s)he needs a logic in which not everything follows from a

23 contradiction, i.e. a paraconsistent logic.

24 The thesis that reality is (in some sense) contradictory is an old philo-

25 sophical quandary, and it has been a position defended by philosophers such

26 as Hegel and, according to some interpreters, also by Heraclitus. This con-

27 fers to contemporary dialetheism a sort of philosophical legitimacy. How-

28 ever, inside and outside philosophy, there is a strong reluctance in accepting

29 that there may be entities that disobey the principle of non-contradiction as

30 it is expressed by Aristotle in book IV of his Metaphysics (1005b19-21):

31 “the same attribute cannot at the same time belong and not belong to the

32 same subject in the same respect” (Aristotle, 1996). Indeed, rejecting any

33 contradiction as false is a basic methodological criterion both in sciences

34 and in philosophy.

35 In order to accept contradictions without endorsing dialetheism, one

36 needs a non-explosive negation not committed to the truth of a pair of propo-

37 sitions A and ¬A. Such a negation can only occur in a context of reasoning

38 where what is at stake is a property weaker than truth, in the sense that a

39 proposition may enjoy that property without being true. We argue in section

40 4 (and also in Carnielli & Rodrigues, 2016a) that the notion of evidence, in

41 the sense of reasons for accepting and/or believing in a proposition, allows

42 an epistemic reading of contradictions that is useful to describe contexts of

43 reasoning where contradictions occur. It is perfectly feasible to imagine a

44 scenario where non-conclusive evidence for both A and ¬A is available,

45 while no evidence for a certain B can be found, thus providing a counter-

46 example to the principle of explosion. The notion of evidence is weaker

47 than truth, since evidence for A does not imply that A is true, thus satis-

48 fying our requirement for a property weaker than truth to be attributed to

49 contradictory propositions.

50 Truth is the central notion for classical logic. An inference is classically

51 valid if, and only if, it preserves truth. A logic designed to represent contra-

52 dictions as conﬂicting evidence will not be concerned with preservation of

53 truth but, rather, with preservation of evidence. The situation is analogous

2 Paraconsistency and duality

to intuitionistic logic, when it is interpreted epistemically as concerned not 54 with truth, but with the availability of a constructive proof. Our aim here is 55 to show how such an epistemic approach to paraconsistency may be philo- 56 sophically justiﬁed based on the duality between the rejection of the princi- 57 ple of explosion by paraconsistent and the rejection of excluded middle by 58 paracomplete logics. 59

This text is structured as follows. In section 2 we discuss the duality 60 between excluded middle and the principle of explosion. The invalidity of 61 these inferences are the distinctive features, respectively, of paracomplete 62 and paraconsistent logics. Section 3 shows that the rejection of the principle 63 of excluded middle by intuitionistic logic may be understood both from an 64 ontological and from an epistemic point of view. In section 4, an epistemic 65 3 approach to paraconsistency is defended. 66

2 On the duality between paraconsistency and paracom- 67

pleteness 68

Let us deﬁne two n-ary logical connectives C1 and C2 as dual when 69

∼C1(A1,A2, ..., An) and C2(∼A1, ∼A2, ..., ∼An) 70 are classically equivalent (∼ meaning classical negation). Thus, ∀ and ∃, ∧ 71 and ∨ are dual to each other, and ∼ is dual to itself. We also say that the 72 corresponding formulas are dual. It is clear then that the formulas 73

∼(A ∧ ∼A) and A ∨ ∼A 74

(expressing, respectively, non-contradiction and excluded middle) are dual. 75

However, this duality may be seen from a different, more fundamental view- 76 point, as a duality between rules of inference. 77

The principles of non-contradiction and excluded middle are often pre- 78 sented in books of logic, and philosophy, as fundamental laws of thought 79 and basic tenets of classical logic. However, it is not non-contradiction but 80 rather explosion that is essential to characterize classical negation, and this 81 is a central feature of the classical account of logical consequence. 82

3This text overlaps with other papers of the authors in some points. Parts of section 2 have appeared in (Carnielli & Rodrigues, 2016b). Sections 3 and 4 develop some ideas presented in (Carnielli & Rodrigues, 2015) and (Carnielli & Rodrigues, 2016b).

3 Walter Carnielli and Abilio Rodrigues

83 Classical negation ∼ is deﬁned by the following conditions (for classical

84 ∨ and ∧ ): A ∧ ∼A , (1) 85 A ∨ ∼A. (2)

86 Condition (1) says that there is no model M such that A∧∼A holds in M. (2)

87 says that for every model M, A∨∼A holds in M. A negation is paracomplete

88 if it disobeys (2), and paraconsistent if it disobeys (1). Notice that each one

89 of the conditions above corresponds exactly to half of the classical semantic

90 clause for negation:

M(∼A) = 1 if and only if M(A) = 0. (3)

91 The only if forbids that both A and ¬A receive 1, and the if forbids that

92 both receive 0. Given the classical deﬁnition of logical consequence (and

93 the usual meanings of ∧ and ∨), from (1) and (2) above it follows that for

94 any A and B: A ∧ ∼A B, (4) 95 B A ∨ ∼A. (5)

96 Inference (4) above is (one version of) the principle of explosion and (5)

97 is (again, one version of) excluded middle. Of course, excluded middle is

98 usually presented as a valid formula or axiom, without the premise B, but

99 this is tantamount to the formulation (5) above, which makes it clear that

100 A ∨ ¬A follows from anything. It is easy to see, therefore, that from the

101 point of view of classical logic, the fact that excluded middle is not valid in

102 paracomplete logics and the fact that explosion is not valid in paraconsistent

103 logics are mirror images of each other.

104 It is worth noting that non-contradiction and explosion are not equiv-

105 alent in the following sense: one obtains a complete system of classical

106 propositional logic by adding excluded middle and the principle of explo- 4 107 sion to a system of posititive intuitionistic propositional logic , but the sys-

108 tem so obtained turns out to be incomplete if one changes the latter to non-

109 contradiction.

110 Due to the semantic clause (3) above, a central feature of classical nega-

111 tion is that it is a contradictory-forming operator. Applied to a proposition

112 A, classical negation produces a proposition ∼A such that A and ∼A are

4Positive intuitionistic propositional logic may be deﬁned by the usual introduction and elimination natural deduction rules for ∧ , ∨ and →.

4 Paraconsistency and duality

contradictories in the sense that they can neither receive simultaneously the 113 value 0, nor simultaneously the value 1. 114

In order to give a counterexample to the principle of explosion we need 115 a circumstance such that a pair of propositions A and ¬A hold but a propo- 116 sition B does not hold (¬ being a paraconsistent negation). Dually, a para- 117 complete logic requires a circumstance such that both A and ¬A do not hold 118

(now ¬ is a paracomplete negation). Obviously, neither a paracomplete nor 119 a paraconsistent negation is a contradictory-forming operator, and neither is 120 a ‘truth-functional’ operator, since the semantic value of ¬A is not unequiv- 121 ocally determined by the value of A. Now, the question is: what would 122 be intuitive and plausible justiﬁcations for paraconsistent and paracomplete 123 negations? An answer will be found in reasoning contexts in which nega- 124 tions with such characteristics occur. 125

3 Intuitionistic logic: a case of paracompleteness 126

Let us start with intuitionistic negation, which is paracomplete, as men- 127 tioned. There are two different motivations, one ontological, another epis- 128 temological, for saying that in a given context both A and ¬A (¬ being 129 intuitionistic negation) do not hold. 130

We ﬁnd in Brouwer’s writings a conception of mathematical knowledge 131 according to which there cannot be any mathematical truth not grounded on 132 a mental construction. Furthermore, and in accordance with this thesis, the 133 existence of a mathematical object with certain properties can be asserted 134 only if such an object has been so constructed. Mathematics, thus, is in no 135 way independent of thought and mind. This conception is nothing but an 136 idealistic attitude with respect to mathematical objects: truth and existence 137 are conceived on a idealistic basis depending on the human mind. 138

Mathematics can deal with no other matter than that which it 139

has itself constructed. In the preceding pages it has been shown 140

for the fundamental parts of mathematics how they can be built 141

up from units of perception. (...) In the third chapter it will be 142

explained why no mathematics can exist which has not been in- 143

tuitively built up in this way, why consequently the only possi- 144

ble foundation of mathematics must be sought in this construc- 145

tion under the obligation carefully to watch which constructions 146

intuition allows and which not, and why any other attempt at 147

5 Walter Carnielli and Abilio Rodrigues

148 such a foundation is condemned to failure. (Brouwer, 1907,

149 pp. 51, 73-74)

150 The phrase ‘built up from units of perception’ means that the intuition of

151 time is the raw material from which the construction of mathematical objects

152 proceeds. The unfolding of the process of ‘two-oneness’ with respect to time

153 (or put more directly, our intuition of time, in the Kantian sense) is the base

154 of all mathematics:

155 This intuition of two-oneness, the basal intuition of mathemat-

156 ics, creates not only the numbers one and two, but also all ﬁ-

157 nite ordinal numbers, inasmuch as one of the elements of the

158 two-oneness may be thought of as a new two-oneness, which

159 process may be repeated indeﬁnitely; this gives rise still further

160 to the smallest inﬁnite ordinal number ω. (...)

161 In this way the apriority of time does not only qualify the prop-

162 erties of arithmetic as synthetic a priori judgements, but it does

163 the same for those of geometry. (Brouwer, 1913, pp. 127-128)

164 And consequently, about mathematical truth, he says:

165 [T]ruth is only in reality i.e. in the present and past experiences

166 of consciousness (...) expected experiences, and experiences at-

167 tributed to others are true only as anticipations and hypotheses;

168 in their contents there is no truth. (Brouwer, 1948, p. 488)

169 The rejection of excluded middle is in this setting straightforward, since it

170 may be the case that no mental construction of A, nor of ¬A has been ef-

171 fected. Notice, however, that according to this view, there is an identiﬁcation

172 of the notion of truth with a notion of constructive proof in the sense that a

173 (mathematical) proposition is true if and only if a proof of it is in some sense

174 available. If the truth of mathematical propositions is so conceived, then in-

175 tuitionistic logic may still be understood as an account of truth preservation,

176 although an idealist notion of truth. It is worth noting that this conception

177 of mathematical truth is still compatible with a notion of truth as correspon-

178 dence: what makes a mathematical proposition true is some entity that exists

179 in reality, but in this case it is a reality constructed by thought. This is what

180 we call an ontological motivation for rejecting excluded middle.

181 According to Brouwer’s view, the classical and the constructive approaches

182 are two irreconcilable positions in mathematics. But this is not the only way

183 to understand intuitionistic logic. There is a weaker position, whose basic

6 Paraconsistency and duality

idea can already be found in Heyting, that is perfectly compatible with a 184 realist conception of mathematical objects. 185

Here, then, is an important result of the intuitionistic critique: 186

The idea of an existence of mathematical entities outside our 187

minds must not enter into the proofs. I believe that even the 188

realists, while continuing to believe in the transcendent [tran- 189

scendante] existence of mathematical entities, must recognize 190

the importance of the question of knowing how mathematics 191

can be built up without the use of this idea. (Heyting, 1930, p. 192

306) 193

[Brouwer’s program] consisted in the investigation of mental 194

mathematical construction as such, without reference to ques- 195

tions regarding the nature of the constructed objects, such as 196

whether these objects exist independently of our knowledge of 197 5 them. (Heyting, 1956, p. 1) 198

Thus, an investigation of mathematical objects as mental mathematical con- 199 structions does not need to imply that such objects do not exist independently 200 of such constructions, nor that they cannot be investigated by other means. 201

In other words, one may well be a realist about mathematical objects but 202 still have interest in intuitionistic logic as a study of such objects from the 203 viewpoint of mental constructions. 204

Understood in this way, intuitionistic logic is not really about preserva- 205 tion of truth. Rather, it is about preservation of construction, whose spe- 206 ciﬁc features depend on the formal system – indeed, Heyting’s and Kol- 207 mogorov’s logics express different notions of construction. Given the (sup- 208

5It may seem strange that Heyting, a faithful disciple of Brouwer, have endorsed a position that in a certain way contradicts his master. However, the passages quoted above, and the general tone of the papers Heyting (1930) and Heyting (1930b), make it clear that Heyting, at least in his writings, was proposing a position weaker than Brouwer’s, allowing the simultaneous interest of both intuitionistic and classical logic. Perhaps Heyting’s more conciliatory tone could be explained in the light of the conflict between Brouwer and Hilbert, the so-called Grundlagenstreit. This conflict culminated with the exclusion of Brouwer from the board of Mathematische Annalen in 1928. As van Dalen (1990, p. 19) puts it, “the scientific differences between the two adversaries turned into a personal animosity. The Grundlagenstreit is in part the collision of two strong characters, both convinced that they were under a personal obligation to save mathematics from destruction.” Heyting’s papers have been published in 1930, about two years after the exclusion of Brouwer. It is very plausible that Heyting, precisely because he wanted to continue to develop intuitionism, took a more careful and conciliatory position.

7 Walter Carnielli and Abilio Rodrigues

209 posed) soundness of the system, the possession of a proof of A implies the

210 truth of A.

211 This position makes it possible to combine a realist notion of truth with

212 a notion of constructive proof that is epistemic. A view that accepts a non-

213 constructive proof of the truth of a given proposition, but distinguishes such

214 a proof from a perhaps more informative constructive proof, is thus perfectly

215 coherent. According to this view, what is at stake in intuitionistic logic is

216 not an idealist notion of truth that in some way identiﬁes proof and truth, nor

217 some non-realist notion of truth that is constrained from an epistemic point

218 of view. Understood in this way, the claim that in a given circumstance both

219 A and ¬A do not hold (or that both receive the semantic value 0) does not

220 mean that both are not true, but only that there is no constructive proof of

221 them, independently of the question whether any of them may be proved

222 true by non-constructive means. The point is not the existence of the object,

223 but rather the access to the object. Its existence may be guaranteed by a non-

224 constructive proof, although, say, a ‘direct access’ may be provided only by

225 a constructive proof.

226 In fact, Heyting’s remarks quoted above anticipate an approach to intu-

227 itionism according to which classical and intuitionistic logics do not exclude

228 each other. Dubucs (2008, p. 50) points out that today a “peaceful coexis-

229 tence” of intuitionism and classicism has been reached and “[t]imes where

230 controversy was raging are disappearing from collective memory”. Indeed,

231 inside mathematical departments, there is no more a dispute like that which

232 occurred at the beginning of the twentieth century between intuitionism and

233 formalism – the disputes are of a completely different nature. And mathe-

234 maticians today, with some few exceptions, are classical and do not see any

235 problem in applying proofs by cases based on excluded middle. However,

236 of course it is recognized, and taken into consideration, that constructive

237 proofs sometimes are more informative than the classical ones, since they

238 require a witness: a constructive proof of an existential ∃xF x demands the

239 exhibition of an object d that satisﬁes F x.

240 On a familiar example of a non-constructive proof 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.

8 Paraconsistency and duality

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

√ √ √ 2 √ 2 2 is rational or 2 is not rational, it can be proved that there exist√ numbers m and n satisfying the conditions 241 √ 2 √ above. Indeed, suppose that 2 is rational; then√ m = n = 2 completes 242 √ 2 the proof.√ On the other hand, suppose that 2 is not rational, and take 243 √ 2 √ m = 2 and n = 2. The claim is thus proved, but we end up without 244 knowing, after all, which are the numbers m and n that satisfy the required 245 condition. 246

Now suppose that a student of mathematics, sympathetic with a realist 247 conception of mathematical objects, becomes aware of the proof above, say, 248 by reading section 5.1 of van Dalen (2008). (S)he accepts the proof as a 249 proof of the truth of (P) although (s)he is still not able to exhibit the numbers 250 m and n. 251

In this scenario, a non-constructive proof is available, yet a constructive 252 6 proof, in this case essentially more informative, is lacking. 253

This may be represented by the attribution of the value 0 to both P and 254

¬P , but it is important to call attention to the fact that this does not mean 255 that both propositions are false. Rather, this means that neither of them has 256 been constructively proved yet. 257

Our young mathematician then starts working on this problem trying 258 to ﬁnd out numbers that satisfy the required condition. After some√ time, 259 (s)he constructively reaches the following result: consider m = 2 and 260 √ n = log 2(k), for k an odd natural number. It can be proved by constructive 261 n 7 means that m is rational, but both m and n are irrational . 262

√ √ log 2(k) Fact 1 For k ∈ N, 2 is a rational number. 263

6 Aleksandr Gelfond and Theodor Schneider independently proved in 1934√ (see Baker, √ √ 2 1975), while 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. 7We are not claiming that m and n are constructible numbers in the intuitionistic sense. Rather, we are only exhibiting two classical 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.

9 Walter Carnielli and Abilio Rodrigues

√ √ 264 Proof. From the law of identity, log 2(k) = log 2(k). Now, from the √ √ log 2(k) 265 deﬁnition of log, 2 = k, which obviously is rational, since k ∈ 266 N. √ 267 Fact 2 2 is not a rational number.

268 Proof. The proof in this case is well known, and it is intuitionistically ac-

269 ceptable, since it employs a reductio ad absurdum that introduces a nega-

270 tion.

√ 271 Fact 3 For k ∈ N odd, log 2(k) is not a rational number.

√ 272 Proof. Suppose, recalling that by hypothesis k is odd, that log 2(k) is ra- √ 273 tional. Then, for some natural numbers a and b, log (k) = a/b. It follows √ √ 2 a/b b b a b b 274 that ( 2 ) = k , and thus 2 = k . Since k is odd, k is odd, for any 275 √b. Now,√ any natural number√ is odd, or not odd. Suppose a is odd. Then a a b 276 √2 = c 2, for some c, so 2 6= k . Now, suppose a is not√ odd. Then, a b a b 277 2 = c for some c even. Since k is odd, for any b, again 2 6= k . √ 278 Hence, log 2(k) is not a rational number.

279 Notice that the proof above depends on the proposition for any natural num-

280 ber x, x is odd or x is not odd, which is valid intuitionistically because the

281 predicate x is odd is decidable – given any natural number, there is a ﬁnite

282 procedure that will end up with an answer yes or no. According to the clause

283 for the universal quantiﬁer in the BHK interpretation, a proof of the proposi-

284 tion above is a construction that transforms any given natural number d into

285 a proof that d is odd or d is not odd. This is, therefore, a legitimate use of

286 excluded middle.

287 Once in the possession of a constructive proof, our young mathematician

288 says: ‘now, besides knowing that the proposition is true, I am also able to

289 exhibit numbers that make it true!’ This new scenario is so represented by

290 attributing 1 to P and 0 to ¬P . The rejection of the instance of the excluded

291 middle P ∨ ¬P , from the constructive point of view, was thus a provisional

292 situation. It was not based on the falsity of both P and ¬P but rather on

293 an epistemic viewpoint related to the availability of a constructive, more

294 informative proof. The position that accepts P as true even though it is not

295 constructively proved is an absolutely reasonable position that illustrates the

296 peaceful coexistence between classical and intuitionistic logic.

10 Paraconsistency and duality

4 Paraconsistency and epistemological contradictions 297

Now, let’s turn to the dual situation, the failure of explosion in paraconsis- 298 tent logics. A counter-example for the principle of explosion is a circum- 299 stance in which both A and ¬A hold but there is some B such that B does 300 not hold. What would a such counter-example be? Dialetheism offers a 301 straightforward answer: both A and ¬A are true, but the world is not trivial. 302

Although the claim that there are true contradictions does not cohere with 303 the idea, central in scientiﬁc practice, that avoiding contradictions is an in- 304 dispensable criterion of rationality, from the philosophical point of view, this 305 answer is, in principle, defensible. The topic of ‘real contradictions’, or con- 306 tradictions of ontological character, appears in several places in the history 307 of philosophy. But, still, the issue is contentious. In Western philosophy, 308 the thinkers who famously defended that reality is (in some sense) contra- 309 dictory were Heraclitus and Hegel. However, for them contradictions are 310 related to change and to movement: the ongoing motion of reality. So, it is 311 not impossible to interpret their claims about contradictions so that the stan- 312 dard ﬁrst-order formulation of the principle of non-contradiction (namely, 313 the theorem-schema ∀x¬(P x ∧ ¬P x)) is not violated. 314

However interesting a discussion on the interpretation of contradictions 315 in Heraclitus and Hegel might be, it will not be developed here. The relevant 316 point here is that both philosophers are usually interpreted as having given 317 support to the thesis that there are ontological contradictions. They are taken 318 as the philosophical background that supports contemporary dialetheism: 319 in order to truly describe reality, we cannot dispense with some pairs of 320 contradictory propositions. Dialetheism is what we call an ontologically 321 motivated rejection of the principle of explosion. Nevertheless, the principle 322 of explosion may be also rejected due to epistemological reasons. 323

Paraconsistency may be combined with a realist view of truth that, at 324 the same time, endorses excluded middle and rejects the thesis that there are 325 true contradictions. Accordingly, contradictions that occur in a number of 326 contexts of reasoning do not mean that a given proposition A and its nega- 327 tion are true, nor that A is both true and false. Contradictions that occur in 328 empirical sciences deserve special attention: it is routine for physicists to 329 deal with theories that yield contradictions in some critical circumstances or 330 when put together with others theories. As pointed out by Meheus (2002, 331 p. vii), however, the fact that “almost all scientiﬁc theories at some point in 332 their development were either internally inconsistent [i.e. contradictory] or 333 incompatible with other accepted ﬁndings” is by no means “disastrous for 334

11 Walter Carnielli and Abilio Rodrigues

335 good reasoning”. In fact, in these cases the general argumentative frame-

336 work of science is already paraconsistent because, obviously, in order to

337 avoid a disaster, the principle of explosion cannot be valid. Furthermore, in

338 such contexts it is not the case that all contradictions are equivalent. Some

339 contradictions are, so to speak, impossible to endure, and are a sign that

340 something has gone wrong. Others, even if understood as provisional, have

341 to be faced and in some sense are an essential ingredient of scientiﬁc prac-

342 tice.

343 The fact that contradictions are unavoidable in empirical sciences is also

344 pointed out by Nickles (2002, p. 2). According to him, empirical sciences

345 are “nonmonotonic enterprises in which well justiﬁed results are routinely

346 overturned or seriously qualiﬁed by later results. And ‘nonmonotonic’ im-

347 plies ‘temporally inconsistent’ [i.e. temporally contradictory].” With re-

348 spect to contradictions, he adds that they are “products of ongoing, self-

349 corrective investigation and neither productive of general intellectual disas-

350 ter nor necessarily indicative of personal or methodological failure.” (Nick-

351 les, 2002, p. 2) These ‘provisional contradictions’ are not dialetheias. In

352 our view, they may be of ‘different kinds’ in the sense of having different

353 causes. We list some of them: (i) possible limitations of our cognitive ap-

354 paratus; (ii) failure of measuring instruments and/or interactions of these

355 instruments with phenomena; (iii) stages in the development of theories;

356 (iv) simple mistakes that in principle could be corrected later on. In all these

357 cases, contradictions are related primarily to knowledge and thought. This

358 is what we call epistemic contradictions.

359 This idea of epistemic contradictions ﬁts well with the conception of em-

360 pirical theories as tools to solve problems and not correct descriptions of the

361 world. These two approaches are discussed by Nickles (2002). The concep-

362 tion of theories as tools “give more attention to local problem solving and

363 the construction of models of experiments and of phenomena than to grand

364 uniﬁed theories” (Nickles, 2002, p. 2). Clearly, problems of consistency are,

365 so to speak, more serious with the representational view of theories, since

366 the latter requires that such a representation be correct (i.e. true). It is likely

367 that some scientiﬁc theories are not only tools but may be taken as descrip-

368 tions of some group of phenomena or some ‘fragment’ of the world, and

369 thus can be considered representations, in the realist sense. But in contem-

370 porary science, a grand uniﬁed representation of the world is completely out

371 of question. If this non-representational view of scientiﬁc work is accepted,

372 and we think that it is much more plausible to understand contemporary

373 science in this way, then contradictions that occur in the empirical sciences

12 Paraconsistency and duality

must be considered epistemically and not ontologically. Thus, contradic- 374 tions in empirical sciences are not even potential candidates for dialetheas. 375

They do not suggest the existence of ‘contradictions in the world’. Instead, 376 all available evidence indicates that such contradictions are epistemic. 377

Let us represent the fact that some contradiction is accepted in a given 378 context by the attribution of the value 1 to a pair of propositions A and ¬A. 379

Now, a question is: what does it mean to say that both A and ¬A receive 380 the value 1, if 1 does not mean true? The answer must be based (as men- 381 tioned in section 1) on some property weaker than truth, in the sense that 382 the attribution of such a property to a proposition does not imply the truth 383 of the proposition. A pair of contradictory sentences may be understood 384 by way of a number of concepts that are dealt with in informal reasoning: 385 clashing information, conﬂicting evidence, incompatible verisimilitude, op- 386 posed possibilities, etc. Among them, the notion of conﬂicting evidence 387 deserves special attention as particularly promising to paraconsistent logics. 388

Evidence may be understood, as is usual in epistemology, as what is rele- 389 vant for justiﬁed belief (cf. Kelly, 2014). In this sense, ‘there is evidence 390 that A is true’ means that ‘there are some reasons for believing that A is 391 true’ (cf. (Kelly, 2014) and (Achinstein, 2010)). Thus, evidence may be 392 non-conclusive, and there may be evidence for the truth of A even if A is 393 not true. Conﬂicting evidence occurs when one has, at the same time, rea- 394 sons for accepting A and reasons for accepting ¬A, both non-conclusive. 395

The notion of preservation of evidence, thus, presents itself as a topic to 396 be further developed in paraconsistency. The idea is that, as much as the 397

BHK interpretation for intuitionistic logic expresses preservation of (some 398 sense of) construction, a set of inference rules and/or axioms that preserve 399 8 (some sense of) evidence can be established. Let us see an example of a 400 real situation in physics that illustrates what has been said above. 401

The special theory of relativity 402

A good example of a provisional contradiction in physics is the problem 403 faced by Einstein just before he formulated the special theory of relativity. 404

It is well known that there was an incompatibility between the classical, 405

Newtonian mechanics and Maxwell’s theory of electromagnetic ﬁeld. This 406 is a typical case of two (supposedly) non-contradictory theories that, when 407 put together, yield a contradiction. 408

8A formal system designed to express preservation of evidence can be found in Carnielli and Rodrigues (2016a).

13 Walter Carnielli and Abilio Rodrigues

409 Classical mechanics gives a description of bodies changing position in

410 space and time. It is intuitively understood, and works very well, with re-

411 spect to ‘slow objects’ (we will see what it means for an object to be or not

412 to be ‘slow’). Let us recall the example of a train in uniform linear motion

413 with velocity v with respect to the rails and an object o moving inside the

414 train with velocity w with respect to the train (where w and v have the same 0 415 direction). The velocity w of o with respect to the rails is the algebraic sum 9 0 416 of w and v. The relation between the velocities w , w and v is given by the

417 theorem of the addition of velocities,

0 418 w = w + v,

419 an elementary result in classical mechanics.

420 In the second half of the nineteenth century, the physicist J.C. Maxwell

421 formulated the so-called theory of electromagnetic ﬁeld that gives a uniﬁed

422 account of the phenomena of electricity, magnetism and light. According to

423 this theory, the velocity of light in vacuum (c) is equal to 300,000 km/sec,

424 and, most importantly, c is independent of the motion of its source.

425 Now let us modify a bit the example above. Suppose that instead of an

426 object moving inside the train, we are concerned with the light emitted by

427 the headlight of the train (and suppose also, for the sake of the example, that

428 the air has been removed). According to classical mechanics, the velocity w

429 of the light with respect to the rails would be the sum of the velocity of the

430 train and the velocity of light: w = c + v, hence, ¬(w = c). On the other

431 hand, according to Maxwell’s theory, the velocity of light does not depend

432 on the velocity of the train: w = c. We have, thus, that classical mechan-

433 ics and the theory of electromagnetic ﬁeld ‘prove’ a pair of contradictory

434 propositions, ¬(w = c) and w = c. So, the two theories put together yield a

435 contradiction, and if the underlying logic is classical, triviality follows.

436 In the situation described above, two propositions A and ¬A hold in

437 the sense that both may be ‘proved’ from theories that were supposed to be

438 correct. This fact may be represented by the attribution of the value 1 to

439 both A and ¬A. But clearly, the meaning of this should not be that both are

440 true – actually, we know it is not the case, and nobody has ever supposed

441 that it could be the case. The meaning of the simultaneous attribution of the

442 value 1, as we suggest, is that at that time there was evidence for both in

443 the sense, mentioned above, of some reasons for believing that both were

9The examples given here have been adapted from Einstein (1916, sections 6 and 7).

14 Paraconsistency and duality

true, because there was evidence that the results yielded by both classical 444 mechanics and the theory of electromagnetic ﬁeld was true. 445

Classical mechanics is not compatible with Maxwell’s theory because 446 the equations of the latter are not invariant under the so-called Galilean 447 transformations, which in classical mechanics relate the space-time coor- 448 dinates of two systems of reference in uniform linear motion. By the end 449 of the nineteenth century, H.A. Lorentz had already presented a group of 450 equations, called Lorentz transformations, and the interesting fact is that 451 10 Maxwell’s equations are invariant under Lorentz transformations. Ein- 452 stein then rewrote Newton’s equations in such a way that the theory so ob- 453 tained, the theory of special relativity, was fully compatible with the theory 454 of the electromagnetic ﬁeld. Actually, what Einstein did was to consider 455 that the mass of a body increases with velocity, and it changed the whole 456 thing. From the new equations, a different theorem of addition of velocities 457 can be proved: 458

0 w+v w = 1+wv/c2 459

The ‘contradiction’ is now solved (roughly speaking) in the following way: 460 as velocity grows, time ‘slows down’ and ‘space shortens’. So, the relation 461 between space and time that gives velocity remains the same, because both 462 have decreased. Newton’s equations work well for ‘slow objects’, that is, 463 2 objects moving in such a way that the value of wv/c may be discarded. 464

Thus, what the special theory of relativity shows is that classical mechanics 465 is a special case of the former. 466

We have just seen a good example of what we call epistemic contradic- 467 tions. We want to call attention to the fact that the general logical framework 468

Einstein was working in was not classical. He had two different theories at 469 hand, classical mechanics and the theory of the electromagnetic ﬁeld that, 470 when put together, yielded a non-explosive contradiction. Later, according 471 to the special theory of relativity, the ‘contradiction’ disappeared. Although 472 there was some reasons to believe that both ¬(w = c) and w = c were true, 473 only one, the latter, has been established as true. The value 1 attributed to 474

¬(w = c) later became 0. 475

10We are not going into the details here. Friendly and accessible presentations of the problem may be found in Einstein (1916) and (a more detailed one) in Feynman, Leighton, and Sands (2010, ch. 15).

15 Walter Carnielli and Abilio Rodrigues

476 5 Final remarks

477 In the classical account there is a duality between excluded middle and ex-

478 plosion as rules of inference: anything follows from a contradiction, ex-

479 cluded middle follows from anything. Each of these rules correspond to

480 half of the properties of classical negation. The rejection of explosion by

481 paraconsistent logics is the mirror image of the rejection of excluded middle

482 by paracomplete logics. We have considered here two basic motivations for

483 paraconsistency and paracompleteness, one ontological, the other epistemo-

484 logical. Just as excluded middle is rejected by intuitionistic logic by reasons

485 that may be ontological or epistemic, explosion is rejected by paraconsistent

486 logics by reasons that may be ontological or epistemic. The ontological rea-

487 sons consider that what is at stake is truth. This is the case for dialetheism,

488 and also for intuitionistic logic from the viewpoint of Brouwer’s idealism.

489 The epistemic approach, that we ﬁnd much more plausible and promising,

490 holds that what is at stake is not truth, but the availability of evidence, in

491 the case of paraconsistency, or the availability of a constructive proof, in the

492 case of intuitionistic logic.

493 There is a sort of pragmatic argument in defense of paraconsistent logics

494 that goes as follows. It is a fact that people reason and make inferences in

495 contradictory contexts without trivialism. So, we need a logic able to give

496 an account of such contexts; and so, paraconsistent logics are useful and

497 deserve to be investigated. But a question remains: what is the nature of the

498 contradictions handled by paraconsistent logics? We endorse the view that

499 all contradictions are epistemic. We believe that this is the best (maybe the

500 only) path available to provide a plausible understanding of paraconsistency.

501 References

502 Achinstein, P. (2010). Concepts of evidence. In Evidence, explanation, and

503 realism. Oxford University Press.

504 Aristotle. (1996). The complete works of aristotle. Oxford University Press.

505 Baker, A. (1975). Transcendental number theory. Cambridge University

506 Press.

507 Brouwer, L. (1907). On the foundations of mathematics. In Collected works

508 vol. I. (ed. A. Heyting). North-Holland Publishing Company (1975).

509 Brouwer, L. (1913). Intuitionism and formalism. In Collected works vol. I.

510 (ed. A. Heyting). North-Holland Publishing Company (1975).

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Brouwer, L. (1948). Consciousness, philosophy and mathematics. In Col- 511

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Carnielli, W., & Rodrigues, A. (2015). Towards a philosophical understand- 514

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Carnielli, W., & Rodrigues, A. (2016a). A logic for reasoning about evi- 516

dence and truth: an epistemic approach to paraconsistency. Submitted 517

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Carnielli, W., & Rodrigues, A. (2016b). On the philosophy and mathe- 519

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539

Meheus, J. (2002). Preface. In Inconsistency in science (ed. J. Meheus). 540

Dordrecht: Springer. 541

Nickles, T. (2002). From Copernicus to Ptolemy: inconsistency and method. 542

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http://plato.stanford.edu/archives/sum2013/entries/dialetheism/546

547 van Dalen, D. (1990). The war of the frogs and the mice, or the crisis of 548

the Mathematische Annalen. The Mathematical Intelligencer, 12(4), 549

17 Walter Carnielli and Abilio Rodrigues

550 17-31.

551 van Dalen, D. (2008). Logic and structure (4th ed.). Springer.

552 Walter Carnielli

553 CLE and Department of Philosophy - State University of Campinas

554 Brazil

555 E-mail: [email protected]

556 Abilio Rodrigues

557 Department of Philosophy – Federal University of Minas Gerais

558 Brazil

559 E-mail: [email protected]