Aufklärung. Revista de Filosofia ISSN: 2358-8470 [email protected] Universidade Federal da Paraíba Brasil

López-•Astorga, Miguel TOWARDS AN UPDATED REASONING FORMAL THEORY Aufklärung. Revista de Filosofia, vol. 3, núm. 1, enero-junio, 2016, pp. 11-32 Universidade Federal da Paraíba João Pessoa, Brasil

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How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative ISSN: 2318­9428. V.3, N.1, Abril de 2016. p. 11­32 DOI: http://dx.doi.org/10.18012/arf.2016.28457 Received: 05/03/2016 | Revised: 05/04/2016 | Accepted: 13/04/2016 Published under a licence Creative Commons 4.0 International (CC BY 4.0)

TOWARDS AN UPDATED REASONING FORMAL THEORY

[RUMO A UMA TEORIA FORMAL DO PENSAMENTO ATUALIZADA]

Miguel López­Astorga *

ABSTRACT: In the late twentieth century, ABSTRACT: No final do século XX, a the final version of the mental logic theory versão final da teoria lógica da mente foi was presented. That was a syntactic and apresentada. Ela era constituída por uma formal approach intended to describe and aproximação formal e sintática que explain the human inferential ability. objetivava descrever e explicar a Maybe because of several experimental habilidade de inferências humanas. Tendo results achieved during the last years that em vista vários resultados experimentais have not addressed by the theory, it can be obtidos nos útimos anos que fogem ao thought that its framework is outdated escopo desta teoria, podemos afirmar que today. In this paper, I try to update some ela, hoje, encontra­se desatualizada. Neste particular aspects of the mental logic artigo, objetivamos atualizar alguns theory by taking recent empirical evidence aspectos particulares da teoria lógica da and arguments coming from the mente em relação com evidências specialized literature into account. Such empíricas recentes e argumentos oriundos aspects refer to the symbols that it should da literatura especilaizada. Tais aspectos adopt, its actual essential schemata, and se referem aos símbolos que ela deveria the way in which the theory can deal with adotar, ao esquema essencial e ao modo no denials. qual ela pode conviver com suas negações KEYWORDS: formal schemata; inference; mental logic; reasoning; syntax. KEYWORDS: Esquemas formais; inferência; lógica mental; pensamento; sintaxe.

INTRODUCTION

he proponents of the mental logic theory provided its last Tversion in 1998, in particular, in the book edited by Martin Braine and David O’Brien that very year (Braine & O’Brien, 1998a). It is true that a part of the theory was not finished in that work. That part is the one related to the mental predicate logic (the book basically only * Ph.D. in Philosophy by Universidad de Cádiz (España). Áreas UNESCO: Lógica, Filosofía de la Ciencia. Professor at the Institute of Humanistic Studies “Juan Ignacio Molina,” University of Talca, Chile. m@ilto: [email protected] 12 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 hoy n h esni ht ae ntee uha hs fFodor of the those identify logic as to tries such mental theory theses the logic mental on of the based (1986), case their that, Macnamara or the by is (1975) be spoken reason language the the not the since And should theory. relevant, on this clearly depend Nonetheless, is to issue. authors. it seem important below, often not argued systems and as the logical aesthetic view, in purely use my a to consider in be symbols only However, the to to will appear linked can I is This paper, them theory. of this One in points. But, particular three 2016a). 2015a, 2014a, without Astorga, remains theory logic reviewed, mental be 1998. the since not can modifications said, has or as theory changes work great and, last the this been either this of of course, done all even of theses been to Nonetheless, other added not fundamental simplified. be and while the have can discussed, it of that, they And some theory. field logic them, that mental science a explain the are by cognitive somehow there addressed today in can that said difficulties approaches be as can same of the it done So, number their not data. developed has experimental and it can improved from date, since have circumstance frames updated, that which not This theories, since is rival theory. alternative it obtained the other that been think by have to considered one results been lead the empirical not be of have carried to lot been which have continue a experiments many theory and because out problem the a of is basic this the O’Brien, requirements And 2010), same. (e.g., and Manfrinati, recent & principles, O’Brien relatively 2013; theses, being Li, & them O’Brien published of 2014; and 2009, some written 1998, have theory after the works of adherents the Although present. able and is Braine extent, large are by a book presented to reasoning. the that, text human theory for of final formal account chapters the to or syntactic the way, a of the this shows all In O’Brien of does if account. logic part completed into propositional of important absolutely taken mental the be some most to Yang, to the and referring seem 1998; part However, logic the Braine, 1998). i.e., predicate see framework, O’Brien, mental proposal; & a that Braine, supporting of evidence proposal empirical possible a offers iulLópez­AstorgaMiguel ab h set ocreto mrv r o se .. López­ e.g., (see, lot a are improve or correct to aspects the Maybe at difficulties certain have to appears theory the Nevertheless, Towards an updated reasoning formal theory real rules of the syntax of thought (see, e.g., Braine & O’Brien, 1998b; O’Brien & Li, 2013), i.e., of a deep syntax that is common to all of the languages around the world. In this way, an actual mental logic should not be linked to any language in any way, although such languages are languages so related to philosophy or science as ancient Greek or English. A second point that will be dealt with here is the one of the number of schemata accepted by the theory. The mental logic approach 13 provides seven ‘Core Schemata,’ that is, seven basic formal rules that the 2 human mind can apply very easily. My proposal, nevertheless, is that, if it is assumed that individuals understand what disjunction exactly is, it is clear that two of those rules are really the same. On the other hand, if we actually assume that idea, i.e., that individuals usually understand the general characteristics of disjunction, it can also be stated that a third Core Schema is trivial as well and can be removed. The final result is, in this way, that perhaps there are only five Core Schemata on the human mind. Undoubtedly, this is a very important aspect because, if what is being proposed is, as said, a syntax of thought, that syntax needs to be as simple as possible. And finally, I will comment on the way that a formal theory as the mental logic theory can work with negated propositions. There is little information concerning this point in the original papers and chapters authored by adherents of the mental logic theory. Nonetheless, I will try to develop it with the help of works in which this has been already studied, for example, López­Astorga (2014b, 2016b).

But, before doing all of this, it seems to be opportune to explain in AUFKLÄRUNG, ISSN 2318­9428, N.1,ABRIL V.3, DE 2016. p. 11­3 more details which the main theses of the mental logic theory are. This last explanation will be given in the next section, which essentially will describe the 1998 version of the theory. In this regard, to avoid confusions, from now on, I will refer to this later version with the abbreviation ‘98ML’ (1998 Mental Logic). On the other hand, I will call my new version ‘RFT’ (Reasoning Formal Theory). 14 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 al .) sShm nteter.A swl nw,issrcueis: structure its 80; known, well p. is 1998c, As theory. O’Brien, the in & 7 (Braine Schema 98ML is 6.1), to Table according which, and Laërtius, 7.80), (Diogenes Soli of Chrysippus philosopher ( oi”(óe­sog,21b .14.Acereapeo oeSchema Core to of possible example is clear are it A 124). which “the p. is in they 2016b, are occasions (López­Astorga, the they whenever it” all is, do in that restriction apply 79­83), people pp. without that 1998c, results Core schemata O’Brien, of used experimental & number (Braine a are “…are are applicable” there there a that that if is proposes rule mind Schemata theory logical the empirical human Thus, a on that. the that based showing accepts on strictly addition, only schema is 98ML In same rules natural that logic. the of means play last not selection which do this mind evidence, The human in reasoning. the In correct of in rules being them. role the does of rules it all of the that but claims logic, all of 98ML standard all in from valid admit being different inferences not rules makes also formal reasoning with human is accordance that in states Henlé 98ML theory different (1966), later and are this systems Piaget particular, other, these are and of each all abilities Beth course, Of from of reasoning 2011). (1994, that our Rips or as calculus, that (1962), propositional such standard idea of frameworks of those the to all including akin and held rules (1935) to have related Gentzen somehow by that for proposed be, theories system can the That the of schemata. case formal the following claiming works example, approaches several mind been human have the there that that said be can It reasoning. ML89 iulLópez­AstorgaMiguel indemonstrables ou oed Ponens Ponendo Modus roB Ergo ______A B THEN A IF L8i o h nyfra rsnatcter nhuman on theory syntactic or formal only the not is ML98 eae otecniinlaraydtce yteStoic the by detected already conditional the to related ) .. n fthe of one i.e., , ia Philosophorum Vitae άν α πόδ ε ικ τ οι Towards an updated reasoning formal theory

On the other hand, another type of schema is that of ‘Feeder Schemata.’ These schemata “…are used only when their output feeds another schema…” (Braine & O’Brien, 1998c, p. 83). So, “these schemata are only applied when their use can allow drawing conclusions” (López­Astorga, 2016b, p. 124). The two most important Feeder Schemata are the primitive rules that standard propositional calculus assigns to conjunction, i.e., the rules to introduce and remove conjunctions. As is also well known (see, e.g., Deaño, 1999, p. 154), 15 these rules can be expressed as follows: 2

A B ______Ergo A AND B

A AND B ______Ergo A

These rules are similar to schemata 8 and 9 of 98ML respectively (see Braine & O’Brien, 1998c, p. 80; Table 6.1). A third type is the one of ‘Incompatibility Schemata,’ that is, of the schemata that “…define contradictions” (Braine & O’Brien, 1998c, p. 83). They “…show that a contradiction has been found and that,

therefore, at least one of the assumptions is false” (López­Astorga, AUFKLÄRUNG, ISSN 2318­9428, N.1,ABRIL V.3, DE 2016. p. 11­3 2016b, p. 125). In this case, an evident and obvious example is the following:

A Not A ______Ergo INCOMPATIBLE 16 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 olw) eetees n9M,te r nyrltdt h former. can the individuals of that to strategy states related theory the only the use restrictions, are certain they with that 98ML, Although of in provides principle is Nevertheless, that the contradiction follows). and principle a Sequitur supposition) if the Quodlibet the and, (i.e., of demonstrate denial to explosion the wished concluding is found, what of contrary nsadr acls nti ae acls otaitosaelne to linked are contradictions plays calculus, contradiction later that this one it the both In about from calculus. different important very standard is is in what role However, its that 6.1). is Table here 81; p. 1998c, O’Brien, atta etllgcter ed itntv ybl htaenot are that point. symbols this with distinctive the deals needs section to next refers theory The language. goals logic any those to mental of linked one a first that the in fact mentioned, As valid here. goals are main nevertheless, that, and consider schemata more more not use logic. does are even standard can theory They individuals the schemata. other that some more those are apply individuals theory, to sophisticated most likely always the the not and is others, than use to sophisticated their and which, essential According schemata, so other is be guaranteed. admits it to Maybe also considered it. theory not of the are view however, that overall add an to have necessary to enough only be can 98ML about said solve. to needs theory Aristotelian the that with problems even certain Irvine finds 98ML and Bolzano and Woods logic compares of on that (2016a) based as characteristic, López­Astorga this such (2004), of logics because paraconsistent fact, to In that akin (1837). a and an it is logic, from make standard This way. from to deduced this 98ML appears distinguishes in by be that work shown point not can as relevant does very because, mind formula simply human this any the evidence, accept the that empirical not What does is premises. it of admit and set incompatibility, given not a does with along theory possible not is assumption iulLópez­AstorgaMiguel eutoa Absurdum ad Reductio hsagmn ssmlrt cea1 f9M seBan & Braine (see 98ML of 10 Schema to similar is argument This l fti ad tsesta ti led osbet drs my address to possible already is it that seems it said, this of All been has what account, into taken are paper this of aims the if But, hti,ta ie otaito,ayconclusion any contradiction, a given that is, that , eutoa Absurdum ad Reductio ie,tesrtg ossigo upsn the supposing of consisting strategy the (i.e., opoeta particular a that prove to xContradictione Ex Towards an updated reasoning formal theory

A PROPOSAL OF SYMBOLS FOR RFT

As also indicated, this is not just an aesthetic problem. A syntax of thought cannot have linguistic influences coming from any particular culture. Nonetheless, the problem is ancient. For example, the Greek logicians tended to use words in their language performing the function of connectives similar to those of current logic. In this way, it is not hard to find that, in sources such as Vitae Philosophorum (by Diogenes 17 Laërtius), Adversus Mathematicos or Pyrrhoneae Hypotyposes (both of them by Sextus Empiricus), the word εì (if) is used to refer to the 2 conditional, the word καί (and) is used to refer to conjunction, and the words ἤτοι…ἤ (either…or) are used to refer to disjunction. Unfortunately, 98ML does something similar, since, although in capital letters, it also resorts to the English words ‘IF’, ‘AND’, and ‘OR’ to refer to the conditional, conjunction, and disjunction respectively. As stated, this does not appear to be adequate, since it seems to provide the existence of certain links between English and the syntax of thought, and this later syntax should be independent of any language, whether it is modern or ancient. But, on the other hand, that seems to be wrong too from another point of view. Many examples of sentences expressing conditional, conjunctive, or disjunctive relationships without using the habitual words (in English, as said, ‘if’, ‘and’, and ‘or’) are to be found in different languages. An obvious English example in this way can be the following sentence:

Whenever I go to the cinema, I eat popcorn AUFKLÄRUNG, ISSN 2318­9428, N.1,ABRIL V.3, DE 2016. p. 11­3

Although this sentence does not include the word ‘if,’ there is no doubt that it indicates a conditional relationship between going to the cinema and eating popcorn. In fact, an alternative way to express it meaning the same could be as follows:

If I go to the cinema, then I eat popcorn 18 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 ih ieo h ybladcnb ald‘osqet)i reor true. true are is ‘consequent’) called be or can true and the is on symbol ‘antecedent’) placed is the called too. (which happens be of one other can side on the and by placed right expressed is symbol is (which what the them then of of happens, one side by left expressed is the what if that, indicating symbols the propositional ‘¬,’ following: the symbol standard are the properties in its by and it RFT for to stood to propose I be attributed that could one which and the calculus, from as different such very systems and logic logic later standard of this symbols e.g., RFT. the between or (see, 98ML use confusions valid to that be added cause ones be to could those must ‘A,’ as it from such And examples B’ 2015a). to López­Astorga, or 2015b). is, 1998a; ‘A O’Brien, that López­Astorga, deriving & rule, Braine allows introduction 2014; by that disjunction O’Brien, the rule given consider the it e.g., not of do (see, material they interpretation Likewise, the Megara the accept of i.e., RFT For way, conditional, Philo logic. same the standard the of of in have interpretation operators nor, not the 98ML do RFT, as neither of example, properties hence of and either. same 98ML, symbols appropriate the be of the to operators necessarily appear the of not that does use is them) reason the of The since all However, symbols, of new them. (or create logic has or standard invent already to necessary this logic not although is standard that, it think case, might the one truly course, is Of language. particular a from iulLópez­AstorgaMiguel hssmo eest h atta h w etne ikdb it by linked sentences two the that fact the to refers symbol This ● sentences two between relation every represents symbol This ● not is RFT and 98ML in meaning whose denial, the from apart So, taken words of not and symbols, of use the advises this of all So, ojnto:‘∙’ Conjunction: ‘:­‘ Conditional: Towards an updated reasoning formal theory

● Disjunction: ‘:’ This symbol indicates that at least one of the sentences linked by it is true.

As it can be noted, these symbols are not directly related to any word in natural language, whether that language is English or any other one. They just try to reflect the three basic relationships between 19 sentences that exist in the syntax of thought. And a very important point in this regard is that, in addition to the fact that what is expressed in that 2 syntax can be expressed in very different ways by means of different languages, as commented on, it can expressed in several ways in the same language as well. Therefore, giving further details about those symbols may not be appropriate. Perhaps what does be important is to include one more symbol to denote deduction or derivation relation. That symbol can be ‘:­:’, and its meaning can be that the right formula can be drawn from the left formula or formulae. This can be enough as far as the symbols of RFT are concerned. From now on, they will be used in this paper, whose next section addresses the problem of the number of Core Schemata in 98ML. The five Core Schemata of RFT As said, 98ML has seven Core Schemata. However, in my view, all of them are not equally basic. On the one hand, two of them can be considered to be the same. One of these two schemata is Core Schema 5 in Braine and O’Brien (1998c, p. 80; Table 6.1), and can be expressed

with RTF symbols as follows: AUFKLÄRUNG, ISSN 2318­9428, N.1,ABRIL V.3, DE 2016. p. 11­3

[(A : B); (A :­ C); (B :­ C)] :­: C

And the other one is Core Schema 6 in Braine and O’Brien (1998c, p. 80; Table 6.1), and, in RFT symbols, could be similar to this one: 20 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 nBan n ’re 19c .8;Tbe61,adcudb expressed be could and 7 6.1), follows: Table and as 6, 80; symbols 4, p. RFT 3, in (1998c, 1, O’Brien Schemata and be Braine would which in now, Schemata of Core five virtue only by therefore, And, well. as too. true is true it conclusion, the true, is is antecedent (B), Ponens i.e., Ponendo that the Modus premise, that that first So, provides the true. clear schema of is previous is antecedent B) the or the (A of of disjuncts disjunct premise two second the second of the one that least given at whenever true is B) formally be can and 6.1), Table way: this 80; Schema in p. Core symbols (1998c, RFT is in That O’Brien expressed ML. and 98 Braine of Schema in Core 2 another reject to leads seems also 5 Schema Core So, assumed, (C). as she) to if, equivalent (or fact But, he in C). trivial. is means, : clearly C) (C really : 6, (C disjunction that Schema (B what knows C); Core :­ understands of (A B); virtue what individual : by not, [(A that as is just, to such schema is set use that C)] premises if a :­ hence from Indeed, derive removed. people can Core be individual that that can an evident 98ML central is and of a it 5 that disjunction, relationships, Schema thought, disjunctive is of different the syntax for syntax correctly a stand understand that is truly there of not that it do element stated, assume formulae As we the two conclusion. If the those on schemata. is that formula (:­:) thought the deduction be of and can symbol semicolons, the of by side separated right are which premises, iulLópez­AstorgaMiguel hs ehv oebscsna ftogt ic tincludes it since thought, of syntax basic more a have we Thus, : (A as such sentence a that knowing is disjunction Understanding C :­: B} C]; :­ B) : {[(A implies disjunction all know individuals that idea the Nevertheless, the contain brackets square the formulae, two last the in Obviously, D) : (C :­: D)] :­ (B C); :­ (A B); : [(A hc s ssi,Cr cea7o 8L (C), 98ML, of 7 Schema Core said, as is, which , Towards an updated reasoning formal theory

CS1 = ¬¬A :­: A CS2 = [(A : B); ¬A] :­: B CS3 = [¬(A ∙ B); A] :­: ¬B CS4 = [(A : B); (A :­ C); (B :­ D)] :­: (C : D) CS5 = [(A :­ B); A] :­: B 21 Where ‘SC’ means ‘Core Schema,’ and the number following refers to the new number that corresponds to the schema in the RFT 2 version. These would hence be the Core Schemata of RFT, which clearly show that this later theory is heir to different frameworks and approaches. Of course, as stated, it is basically a new version of 98ML, which is in turn means that it takes important and relevant elements of standard propositional calculus and Gentzen’s (1935) system. But there is also an evident relationship to a much older logic: Stoic logic. True, four of the five Core Schemata of RFT come directly from this later logic. CS1 refers to the double negation, and that an expression such as (¬¬A) is equivalent to (A) was an idea already held by Stoicism, which spoke about the ὑπερ ἀποφατικόν (Diogenes Laërtius, Vitae Philosophorum 7.69; Barnes, Bobzien, & Mignucci, 2008, p. 102). Secondly, CS2 is one of the versions of Modus Tollendo Ponens, another ἀναπόδεικτος (indemonstrable) also attributed to Chrysippus by Diogenes Laërtius (Vitae Philosophorum 7.81; see also, e.g., fragment 9.7 in Boeri & Salles,

2014, pp. 216­217 & 228­229; López­Astorga, 2015c, pp. 3­4). On the AUFKLÄRUNG, ISSN 2318­9428, N.1,ABRIL V.3, DE 2016. p. 11­3 other hand, CS3 is a Chrysippus’ indemonstrable too, in particular, a version of Modus Ponendo Tollens (e.g., Diogenes Laërtius, Vitae Philosophorum 7.80; fragment 7 in Boeri & Salles, 2014, pp. 216­217 & 228­229; López­Astorga, 2015c, p. 6). And, finally, it has already said twice that CS5 is Modus Ponendo Ponens (see also, e.g., the same fragment in Boeri and Salles, 2014, or López­Astorga, 2015c, p. 3). So, it is obvious that the greatest debt owed by RFT is a debt contracted with Stoic philosophy (as far as 98ML is concerned, a study 22 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 adt nesadfrpol,adti safc htater uha RFT as such theory a that equally fact not a are is this (1847) been and DeMorgan’s people, has that for discussion show understand this to that results for hard fact, such important that In is is What here. (2014b). paper it López­Astorga is discuss by That theory. not addressed models already will mental I the and with that, consistent correct results are experimental them, certain present to authors according the work, Campitelli that & In Girardi, (2014). Pereyra Bolzán, (e.g., Crivello, science Razumiejczyk, Macbeth, cognitive of literature the 2015d). in 2015b, found 2015a, 2014b, easily López­Astorga, the models be empirical of mental can the proponents those the and theory interpret 98ML by only between proponents comparisons such out is Furthermore, how results. carried of here regardless studies theory, interesting models the mental such is in If paper. obtained what this of information considered, aims the are from of away theses aims us general move the account, would is explaining the into framework Nonetheless, this factors taking forms. theory logical pragmatics inferences their and This not make sentences and the beings 2014). of human meaning 2012, that semantic proposes Johnson­Laird, 2015; & and Johnson­Laird, & semantic Byrne & Orenes, (e.g., Trafton, Lotstein, theory Khemlani, Khemlani, models & 2015; Khemlani, works mental Johnson­Laird, Goodwin, from 2015; the come Johnson­Laird, negations theory, 2009; address Johnson­Laird, can rival it a way the supporting to respect with RFT RFT approach. later RFT, this by from assumed different in be this can out perspective results in carried a its Stoic below, work help from shown lot as can made a but, the are researches no researches are recent there Such and relatively know, way. I Nevertheless, negations. as with framework regard. far deal as this can that, it later how is is truth theory The this the for important aspect between another relations indemonstrables the of iulLópez­AstorgaMiguel N DENIALS AND hs ae ihrslsvr eeatfrm ol sta of that is goals my for relevant very results with paper a Thus, complete to used be can that results experimental the Actually, a efudi óe­sog,21c.However, 2015c). López­Astorga, in found be can Towards an updated reasoning formal theory needs to take into account. One of those results is that DeMorgan’s law denying a disjunction is very easy for individuals. That law can be expressed in RFT symbols in a following way:

¬(A : B) = ¬A ∙ ¬B 23 Indeed, the experiment proposed by Macbeth et al. (2014) 2 included sentences with thematic content matching those formal structures and they noted that people really are aware of the equivalence. According to López­Astorga (2014b), a possibility is that 98ML assumes the equivalence as a schema, and I agree with that. But this raises a problem and has a consequence. The problem is that, as indicated by López­Astorga (2014b, p.27), if admitted, it would be necessary to make a decision on which type of rule this schema is. There are different possibilities, but the most adequate seem to be it is a Core Schema or a Feeder Schema. In my view, although people often understand the equivalence, individuals only make cognitive efforts when necessary. For that reason, in principle, it appears to be a better option to assume the law as a Feeder Schema that is applied only when its use can enable to draw further conclusions. Nevertheless, to add this law as a Feeder Schema can really mean to add two schemata, since the equivalence can be understood in the two directions. Thus, the final result would be hence that RFT has four

Feeder Schemata, which, in the symbols of this later system, would be AUFKLÄRUNG, ISSN 2318­9428, N.1,ABRIL V.3, DE 2016. p. 11­3 these ones:

FS6 = (A; B) :­: (A ∙ B) FS7 = (A ∙ B) :­: A FS8 = ¬(A : B) :­: (¬A ∙ ¬B) FS9 = (¬A ∙ ¬B) :­: ¬(A : B) 24 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 te ceaa hyhv etitosadaentawy applied. always not are and restrictions have on, commented they as admits, schemata, 98ML although other since, RFT, of schemata basic can follows: as which former 98ML, the the of of that symbols 10 assumed the Schema be in is expressed could RFT be it B), of of way, 10 Schema : this Schema Incompatibility In ¬(A Incompatibility only it. of into removing version allows premise simple and a second 98ML, it the makes turn transform in can which it since superfluous, ‘inconsistency.’ symbols, RFT in denied and, one: above that this mentioned That clear Schemata. indicated, it Incompatibility As framework. make way. later right and a this in RFT, them for the interpret problem extend in to tend greater a people FS9 al. a not et and to are Khemlani justify disjunctions FS8 of one that of and this in confirmed as inclusion example, were such for (2014) Circumstances too, al. (2014). studies highlight et other to Macbeth important in by also supported achieved is that results it law DeMorgan’s the Maybe the correctly. that of interpret versions to two the use are people FS9 and FS8 and above, iulLópez­AstorgaMiguel n h e ceaaidctds a ol etefnaetlor fundamental the be could far so indicated schemata ten the And >< :­: ¬A) (A; = IS10 and trivial schema this makes FS9 seen, easily be or can it ‘contradiction’, As ‘incompatibility’, means ‘><’ Where >< :­: ¬B)] ∙ (¬A B); : [(A two proposes 98ML consequence. a raises also this said, as But, indicated Schemata Feeder the are FS7 and FS6 noted, be can it As ‘Feeder.’ to here refers ‘F’ Obviously, Towards an updated reasoning formal theory

Likewise, they can depend on individuals’ sophistication level. However, this does not mean that there are not further difficulties. The experimental results concerning denied disjunctions can be assumed by RFT without great problems, but the case of denied conjunctions is different. As is well known, there is another law proposed by DeMorgan:

¬(A ∙ B) = ¬A : ¬B 25 2 The problem with this law is that Macbeth et al. (2014) carried out experiments on it as well and the results were not positive. They noted that people tend not to understand correctly this law. Likewise, Khemlani et al. (2014) came to similar results, another important datum obtained with their research being that individuals often interpret ¬(A · B) as (¬A · ¬B), i.e., in exactly the same manner as ¬(A : B). But, as far as this point is concerned, papers such as those of López­Astorga (2014b, 2016b) are very interesting. His position is initially that, given that 98ML has no predictions about DeMorgan’s laws, results such as those do not have an influence on its fundamental and basic theses. Nevertheless, López­ Astorga also proposes that it is very possible that people do tend to interpret ¬(A · B) and ¬(A : B) in a similar way, that is , as (¬A · ¬B) (see López­Astorga, 2016b, pp. 132­133). In my view, this later idea can be assumed without difficulties, since, if we review the sentences used by both Macbeth et al. (2014) and Khemlani et al. (2014) in their experiments, we can see that such sentences are very odd in English (and

probably in any language that they are translated into). AUFKLÄRUNG, ISSN 2318­9428, N.1,ABRIL V.3, DE 2016. p. 11­3 For example, one of those of Macbeth et al. is “IT IS NOT TRUE THAT: LONDON IS A CITY AND AFRICA IS A CONTINENT” (Macbeth et al., 2014, p. 141; Table 2; capitals in text. See also López­ Astorga, 2014b, p. 25). Obviously, this sentence is hard to understand because, although it really has the formal structure ¬(A · B), it is not a habitual sentence in daily life conversations, and what it actually seems to mean is that neither London is a city nor Africa is a continent. In this way, its real structure appears to be (¬A · ¬B). Something similar 26 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 (¬A eidcnucin r eyodi al ie eody hl tis so not it is ¬(A while it as B), such Secondly, : expressions life. ¬(A understand as appropriately daily such to expressions important in two interpret odd for correctly to least to very referring necessary at languages are natural that in conjunctions seem expressions not denied the does said, as it Firstly, And reasons. RFT. for problem important 133). p. 2016b, (López­Astorga, false not eodpeiei hc ti ttdta n ftecnucsreally conjuncts the ¬(A of a interpret one includes to possibility that also the stated blocks It which is conjunction. true, it is denied should and which happens a that people in only in premise theory, not that, second is appears the criticism it possible to schema, this the according Nonetheless, against conjunctions. response that, denied López­Astorga’s interpret means correctly That CS3 and indicated, Braine argument. that as understand in be, 4 would his RFT, Schema which 6.1), as against Table in such 80; schema p. criticism a (1998c, O’Brien includes possible and 98ML if a that, is account criticism into takes htwa ttuymasi htBbwr ete elwsitnrblue nor shirt yellow thought (¬A neither easily is wore be structure formal can Bob its it that that fact, i.e., is Monday, In means on is. pants truly meaning and it true people López­ what its for also which that odd see clear appears 4; not sentence p. is the wore 2014, it Again, he al. 126). and et p. shirt Khemlani yellow 2016b, a (see Astorga, wore them Monday’ he of on that One pants denied (2014). ‘Bob blue al. one: et this to Khemlani akin by was used sentences the with happens o cetsmlrshmt novn etne ihtesrcue¬(A structure But, the does with FS9. it sentences conjunctions, and denied B). involving of schemata FS8 interpret case similar the as adequately accept in happen not such to not does schemata tend that that people assumes given that it given disjunctions, So, denied literature. denied the conjunctions with from denied work with to work rule to no schema has a theory have (CS3). the does while it because, disjunctions, so is this iulLópez­AstorgaMiguel · oee,b hta tmy h rt sta tde o eman seem not does it that is truth the may, it as that be However, hrfr,LpzAtrasie per omk es.H even He sense. make to appears idea López­Astorga’s Therefore, utemr,a hw,RTcnieseprclrslscoming results empirical considers RFT shown, as Furthermore, B,snetescn rms hw htoeo h ojnt is conjuncts the of one that shows premise second the since ¬B), · B swell. as ¬B) · ) And B). · )as B) · Towards an updated reasoning formal theory

Finally, the remaining connective, the conditional, is not a problem when denied. Khemlani et al. (2014) carried out experiments in this regard and find that, while people can deny conditionals in two different ways, as ¬(A :­ B) and as (A :­ ¬B), which, as indicated by them and López­Astorga (2016b), was already researched in works such as, e.g., Byrne and Johnson­Laird (2009) and Handley, Evans, and Thompson (2006), they always understand that the denial means that scenarios in which the antecedent is true and the consequent is so as well are not 27 possible. Their experiments tried to detect which the scenarios that people consider to be consistent with a denied conditional are, and the 2 two majority response patterns were these ones:

● Three scenarios: A and ¬B, ¬A and ¬B, and ¬A and B. ● Just one scenario: A and ¬B.

Evidently, these results are important for the mental models theory and have a clear meaning under its framework. But what is interesting for us in this paper is that they are also important for RFT and have a clear meaning under its framework too. Based on them, something is obvious: as indicated, if we deny a sentence such as (A :­ B), there is no doubt that that means that, if A happens, B cannot occur as well. And this is absolutely coherent with what has been said on the conditional above. If (A :­ B) stands for the situation in which, when A happens, B must happen too, ¬(A :­ B) stands for the situation in which, when A happens,

¬B must happen too. And from this perspective, it can be claimed that, in AUFKLÄRUNG, ISSN 2318­9428, N.1,ABRIL V.3, DE 2016. p. 11­3 RFT view, it does not matter whether the denial of the conditional is interpreted as ¬(A :­ B) or as (A :­ ¬B). It always means that B will never be possible if A is true. Thus, it can be stated that, both from ¬(A :­ B) and A, and from (A:­ ¬B) and A, ¬B can derived. It is true that, while the second case, i.e., that in which the premises are (A :­ ¬B) and A, is a clear application of Modus Ponendo Ponens, the second one appears to be more problematic. In fact, it could be thought that maybe it would be necessary a new schema that allowed 28 AUFKLÄRUNG, ISSN 2318­9428, V.3, N.1,ABRIL DE 2016. p. 11­32 n htdrv Bfo ( ­B n sa vdn s of use evident an is A and B) :­ theory, ¬(A the Ponens from complicate Ponendo ¬B unnecessary also derive can can it that schemata However, and more ¬B). :­ that (A into thought B) be :­ ¬(A example, for transforming, a eepesdi ifrn as o xml,teei odutthat doubt no (A from is A there draw be example, to would RFT enables For of only ways. that schemata not different the FS7 modification in that also acknowledged expressed additional is be be must only can consequent it that the the be regard, that would needed is this conclusion In obvious false. the (A), antecedent ieie aervee h rbe ftednasto sfra this as far As too. information denials new the consider of problem not And, laws. the do DeMorgan’s reviewed to hand, related have that I other as likewise, hand, such the one literature the on the on from and, 98ML, coming of lot have schemata logical I a the a language. that seem that same problem a the problem in underlying with the ways dealt thought different as also in of well expressed syntax as be a can world, connective about the simpler around speak formalization new language we the of a every when problem symbols i.e., the here the above, addressed and have explained I and ten 98ML. the of presented is version result RFT The account. of into findings schemata empirical those of some taking of experimental that outdated. as interesting such is anymore. approaches very them syntactic ignore 98ML the cannot and and ML however, 98 years, new last view, the found during my results has In predictive science great power. a Cognitive have explanatory RFT the about by of paper and complemented theses this ML98 general course, in the of presented as and, characteristics it alone 98ML reject as cannot such We systems considered. be still can mind C B) :­ well. ¬(A as from trivial ¬B is also because but said explicitly A, and not is B) not this :­ CS5 And (A that A. from and doubt B no derive is to there enables Likewise, only trivial. is because said explicitly iulLópez­AstorgaMiguel ONCLUSIONS nti a,ti ae a enitne oudt htter by theory that update to intended been has paper this way, this In human the on logic mental a of idea the that obvious is hence It o.I h odtoa sfleadw aethe have we and false is conditional the If too. · ) u loB n hsi not is this And B. also but B), Modus Towards an updated reasoning formal theory point is concerned, the only complicated case is, as explained, that of conjunction. Nevertheless, as also shown, its problems do not really have an influence on the theory, since it has a schema to be applied to denied conjunctions, and the sentences with denied conjunctions used in the experiments reported in the literature are often very odd. In any case, I am aware that, of course, the proponents of 98ML may not agree with the updates introduced in this paper to the theory. Thus, they may even consider RFT to be a different framework from 29 98ML. That is the reason why I have preferred to use different names to 2 the two versions. In my opinion, both systems are in essence the same and, as said, RFT is just an update of 98ML. It acknowledges, as also mentioned, the heritage it received from the Stoics in Ancient Greece, from calculi such as that of Gentzen (1935), and from approaches such as those of Beth and Piaget (1966) and Henlé (1962). RFT is neither one of these frameworks, but it has elements of all of them. And this is also exactly the case for 98ML, whose differences with RFT are not radical or basic. So, I insist, as I understand it, the latter is only a version of the former.

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