Reasoning and language at work a critical essay

To summarize, the book is a remarkable source of information, explanations,and inspirations which may be a basic reference for all readers, both novice andadvanced, interested in an insi

Studies in Computational Intelligence 991

Volume 991

Series Editor

Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland

ments and advances in the various areas of computational intelligence—quickly andwith a high quality The intent is to cover the theory, applications, and designmethods of computational intelligence, as embedded in the fields of engineering,computer science, physics and life sciences, as well as the methodologies behindthem The series contains monographs, lecture notes and edited volumes incomputational intelligence spanning the areas of neural networks, connectionistsystems, genetic algorithms, evolutionary computation, artificial intelligence,cellular automata, self-organizing systems, soft computing, fuzzy systems, andhybrid intelligent systems Of particular value to both the contributors and thereadership are the short publication timeframe and the world-wide distribution,which enable both wide and rapid dissemination of research output.

Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago

All books published in the series are submitted for consideration in Web of Science

More information about this series athttps://link.springer.com/bookseries/7092

Enric Trillas · Settimo Termini ·

Marco Elio Tabacchi

Reasoning and Language

at Work

A Critical Essay

Accademia Nazionale di Scienze

Lettere e Arti di Palermo

Palermo, Italy

Marco Elio Tabacchi

Dipartimento di Matematica e Informatica

Università degli Studi di Palermo

Palermo, Italy

Accademia Nazionale di ScienzeLettere e Arti di PalermoPalermo, Italy

ISSN 1860-949X ISSN 1860-9503 (electronic)

Studies in Computational Intelligence

ISBN 978-3-030-86087-5 ISBN 978-3-030-86088-2 (eBook)

or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

of Professors Luis A Santaló (1911–2001) and Abe Mamdani (1942–2010).

This insightful book—maybe short in size but big in ideas, and deep explanationsand inspirations concerning relevant aspects and relations—is concerned with manyissues that are crucial for the human cognition, thinking, and acting, and also relatedissues that are also crucial for some artificial systems mimicking the humans exem-plified by multi-agent systems, and also artificial intelligence-based (AI) systemsthat take science and technology by storm.

Briefly speaking, the authors deal with fundamental aspects of, first, reasoningwhich is the key element of all kinds of systems, both human-centric and artificial, thatare meant for broadly perceived problem solving, notably decision making In suchsystems, we have some premises, for instance some evidence and judgments, and wehave to find conclusions which can then possibly be employed for some purposefulactivities like to find some best option from available or feasible ones Second, sincenatural language is the only fully natural means of articulation and communicationfor the human being, the authors consider natural language, in particular methods,

to represent and then handle its syntax and semantics

What differentiates this book from similar treatises on similar topics is that theauthors, first, consider these above-mentioned topics in a broadly perceived logicalframework Second, to adequately represent an inherent imprecision in the meaning

of linguistic terms and relations, expressed by the humans in the form naturallanguage, they explicitly refer to the pioneering works by the late Lotfi A Zadeh onthe concept of a fuzzy set, then fuzzy logic, and finally computing with words, some-times called computing with words and perceptions Moreover, the book containsextremely valuable references to many concepts and problems considered in variousclassical and extended logics, often using different languages and motivations.More specifically, the authors provide an extremely valuable and insightful expo-sition of, first, some basic more general types of reasoning, that is, deductive, induc-tive, and abductive However, they also refer the reader to various non-standardtypes of reasoning, notably those that have recently appeared, for instance in rela-tions to multivalued, uncertain, temporal, etc., logics As examples, one can citehere defeasible, paraconsistent, probabilistic, or statistical reasoning, to just name

a few Particular emphasis is put on the broadly perceived approximate reasoning,

vii

and the role of fuzzy sets and fuzzy logic has been underlined The authors go evendeeper and discuss commonsense reasoning, notably expressed by using elements

of the fuzzy logic-based paradigm of computing with words, the inclusion of whichcan be decisive for the development and implementation of all kinds of artificialintelligence-based (AI) systems

To summarize, the book is a remarkable source of information, explanations,and inspirations which may be a basic reference for all readers, both novice andadvanced, interested in an insightful and inspiring exposition of the topics covered,notably logics, reasoning, fuzzy logic, natural language, computing with words,commonsense language, and related topics It is highly recommendable!

Warsaw, Poland

May 2021

Janusz Kacprzyk

Wege nicht Werke

(Martin Heidegger)

What follows is not a proper text on fuzzy logic, the basic field of research whichhas seen the now retired first two authors active for around 50 years, and the thirdfor over twenty Instead, it is but a booklet containing a collection of reflections thatfly further from fuzzy logic by continuing where some previous papers by the threeauthors have left

Such reflections are a tentative way to show that fuzzy logic is not only fertiledue to its being relevant in many technologic fields Indeed, it is also as a facilitatorfor building reflections on thinking, language, reasoning, and its mechanization, in

a way that intermingles both ‘scientific’ and ‘philosophical’ aspects It is somethingthat, consequently, can be seen as able to generate a new ‘Humanistic Culture’ for thetwenty-first century, not too far from what prevailed along the seventeenth-centurycentury’s European Enlightening, and including science, as today nobody can doubt

it is a relevant part of culture A possible, innovative field of debate this bookletpresents, especially, to the young scientists and philosophers

A general consensus in the community exists that the good scientists beginreasoning on a delimitated subject, of which some previous knowledge exists, looking

at first for questions that are new as well as good, and subsequently find adequateanswers Such answers are even more satisfactory when their fertility expands tofields different from the one in which the problem was initially posed

The goodness of a question and its fertility are obviously linked One could also saythat, perhaps, the evaluation of a satisfactory judgment cannot but be retrospective: Aquestion ‘was’ a good one if the obtained answers are subsequently demonstrated to

be fertile Just to exemplify, let us recall the questions asked by Einstein on motion,those of Kekulé on the Benzene’s molecule, and Cajal’s ones on the nervous cell.Their answers did all have a noteworthy import in other fields: For instance, Kekulé’sdiscover is one of the bases on which the German Chemical Industry of Colorantsdeveloped

ix

Most of the current technology of information and communication, as well asmany results of the pharmaceutical industry (but those two cases are not exhaustive),come from fertile answers to science’s good questions Currently, the percentage ofGNP devoted to R&D is in direct proportion with the true power of a nation and alldeveloped countries have to do with what is called ‘Politics for science,’ or ‘ScientificPolitics’ which, frankly, sounds horrible.

Such more or less obvious considerations are stated here—at the beginning of thesepages—keeping in mind some remarks done by Isaac Rabi, Nobel Prize for Physics,and the great geometer and thinker Karl Menger They are also tuned with the spiritinforming the book ‘Combining Experimentation and Theory’ which articulates anddevelops an homage to the late Abe Mamdani, just starting from the relationshipsexisting between the two concepts present in the title [1]

We are tempted to state that fruitful new ideas—be they in definitive shape orstill in an informal state—can provoke the asking of unusual, vitalizing questionswhich, when answered, can allow us to see things from a new, different and, in somecases, enlarged perspective ‘Good thinking,’ then, means that it is not enough to

‘think’ and ‘reason’ correctly against an untouchable background of general

presup-positions which cannot be questioned, but it also requires, à la Nietzsche,

systemati-cally, meticulously doubting of what is considered already well known and definitelyassessed Submitting thinking to a rigorous control, pushing it outside the borders ofthe ‘received view,’ must be considered, then, an issue of intellectual hygiene

1

It is just in such sense that this small book is presented as a ‘Critical Essay’, thechoice of terms signaling, respectively, that it raises some doubts even if these do notalways turn into an explicit criticism, and that this is a surface level ‘survey’, with theaim of focusing the attention on some issues more than treating them in detail, and

in a relatively contained number of pages It tries to rethink already known topics

by looking at them from a point of view that is new as well as nạve, the term used

in the same sense in which it characterizes ‘Nạve Set Theory.’ In somewhat a kind

of joke, the authors try to ‘shake before drinking’ what they previously believed aswell known for what concerns reasoning

Notwithstanding, such ‘critical’ approach is not only outward directed againstwhat others express, but as well and mainly inward, against what the authors believe

is an acquired knowledge Years and years of debate among the authors have notproduced any certainty, but in a serious twist for practitioners of fuzziness, a number

of uncertainties It is, partly, due to such uncertainties that the critical approach isnot always explicit in this essay

Einstein once observed that ‘Science comes from refining the usual thinking,’underlining both the important continuity between the known and unknown, andwhere differences reside The same idea underlies the efforts done by deep authorswhen writing ponderous volumes on known subjects with the didactical intention of

showing the multi-facets of the evolution of scientific concepts It is with regret wenote the current trend followed by young (not exclusively young, indeed) researchers,when affording a problem, to concentrate their activity on the scrutiny of a multitude

of very recent papers, while the reading of such kind of comprehensive and reflexivebooks very seldom is part of their engagement

Almost all new findings have deep roots in what past scientists or thinkers wrote;almost nothing, in a sense, is actually completely original since also the most inno-vative ideas spring from a revision of old paradigms Let us also add that althoughwhat appears as totally new seems to be reserved to few authors, experience showsthat often in the ‘antecedents’ one can eventually find some suggestive perspectivealso when doing a traditional, normal, daily research activity on specific questions.Such trend is so widespread that cannot be exclusively attributed to the behaviorand choices of single scientists; it seems fairer to also pin it on a radical change thathas occurred in the organization of scientific research and to the surrendering to thehigh velocity and great pressure the current ‘publish or perish’ paradigm demands

It is up to some extent surprising that many if not most principal researchers havenever published a standalone book on what it is supposed they are leading specialists

of Instead, there is a proliferation of books edited by several of them and collectingchapters around subjects not always concordant

This essay aims not to stop at what seems to be well known; it tries to show thepossibility of embarking into an intellectual journey that can allow to see new thingszigzagging, simultaneously forwards and backwards, from some questions to what

is considered known in the same way in which one discovers new things in a town

by walking back and forth with curious eyes The essay aims to help people, andmainly the younger scientists, to speculatively look for further new horizons; a taskwhich, we want to underline, can actually be carried on only through personal effortand conviction, which can be only triggered and stimulated by others

This small book tries to rethink what is known and following, again, Einstein

in his belief that posing questions is essential for the scientific progress: even in animplicit form, like in this essay, that often leaves to the reader to make such questionsexplicit using their own insight This is a reason for reading it slowly

The text does not search to widening the ‘work’ of its authors Benefiting from thedecades of combined research experience of them—that surpasses 50 years for each

of the elder two—it aims to open a new path toward the clarification of some basicconcepts in thinking and reasoning, as well as that in their possible mechanization.This is a book that, ideally, should be read as if ‘doing nothing,’ quietly and in a kind

of working suspension; turning each paragraph again, on and on It is meant for aSunday afternoon activity!

Just concerning the previously mentioned aspect, let us recall the words of the lateEbrahim (Abe) Mamdani (1942–2010), in the already cited book that is dedicated tohim:

My interest stems from my fascination with the digital computer and how it can be used for simulating behavior.

Such approach persuaded the third, and younger, author that speculations, thespecial kind of conjectures introduced and described by the first author, should beincluded as a first-class member in the toolbox of logic instruments used to make acrack at mechanizing commonsense reasoning.

While the start of such inclusion has pondered the role of speculations when thestructure dealt with is that of a finite Boolean Algebra, we have also laid out somework on expanded structures, such as the Borel Algebra In the general context ofthe construction of algorithmic models that mimic and reconstruct some of the innermechanisms of a working mind, a deductive approach is limiting and does exclude alot of different interpretations and nuances: by implementing, in addition to classicaldeductions, speculations, seen as a back-and-forth process of deducing and abducing,aiming at such integration, and unlocking the key to an important aspect of what iscommonly defined as creative thinking

2

In the last 100 years, the mathematical analysis of reasoning (mathematical logic)was conducted by means of symbolic and discrete models, mainly using algebraicresources and sometimes even (abstract) topological ones This actually implies alimitation John von Neumann regretted at the half of twentieth century, and at whicheven George Boole did not offer an answer, as for instance he freely employed, withhis ‘symbols’, Taylor’s developments in his logical work on ‘Thought.’

We think it could be worthwhile to clarify that von Neumann remarks are done inthe specific context of his work at a particular time He introduces the role of error

in the Theory of Automata; thus,

The subject matter is the role of error in logics, or in the physical implementation of logics—

in automata synthesis Error is viewed, therefore, not as an extraneous and misdirected

or misdirecting accident, but as an essential part of the process under consideration—its importance in the synthesis of automata being fully comparable to that of the factor which

is normally considered, the intended and correct logical structure [2].

However, already in 1949 (the Proceedings will be published 2 years later), hehad spoken specifically of mathematical logic maintaining that:

There exists today a very elaborate system of formal logic, and, specifically, of logic as applied to mathematics This is a discipline with many good sides, but also with certain serious weaknesses … About the inadequacies … this may be said: Everybody who has worked in formal logic will confirm that it is one of the most refractory parts of mathe- matics The reason for this is that it deals with rigid, all-or-none concepts, and has very little contact with the continuous concept of the real or of the complex number, that is, with math- ematical analysis Yet analysis is the technically most successful and best-elaborated part of mathematics Thus, formal logic is, by the nature of its approach, cut off from the best culti- vated portions of mathematics, and forced onto the most difficult part of the mathematical terrain, into Combinatorics [3].

We must add that von Neumann did not contribute to what is considered ‘Logic’from an academic and disciplinary point of view in periods after his remarks quotedabove, although he was certainly able to do so and at the utmost level Instead, heindicated a different path to be followed to come at terms with approximation anderror.

A very meaningful choice, we deem He could have forged and provided newtools by using ‘the technically most successful and best-elaborated part of mathe-matics’ (his words) along the lines followed by the disciplinary tradition of mathe-matical logic In the midst of a furious and creative intellectual climate, we must add,and under the pressure of the urgencies dictated by the realizations of projects likethe building up of electronic computers, he chose to follow another road: showingthat crucial logical issues can be fruitfully pursued in those more general contextsemerging from the questions asked by new fields of investigation like informationprocessing or cybernetics [4, 5]

It can be also added that an important intellectual certification of the reliability

of this line of thought can be attributed to Popper’s position—although no directinfluence can be traced on the developments in which we are interested—when heassert that ’it is always undesirable to make an effort to increase precision for itsown sake—especially linguistic precision—since this usually leads to loss of clarity

… : one should never try to be more precise than the problem situation demands …

an increase in precision or exactness has only a pragmatic value as a mean to somedefinite end’ [6]

The previous passage comes after he has reassured the reader: ’I do not suggest, ofcourse, that an increase in the precision of, say, a prediction, or even of a formulation,may not sometimes be highly desirable.’ For more on this, see [7] Popper’s warningteams with a similar one due to Aristotle in Nicomachean Ethics (1094b), when hestates that the treatment of a discipline ’will be adequate, if it achieves that amount

of precision which belongs to its subject matter’ and adds that ’the same exactnessmust not be expected’ in all arguments or activities

So, the choice of accepting error ’as an essential part of the process under eration,’ as well as the warning against the ’effort to increase precision for its ownsake’ does not suggest relaxing the canons of scientific rigor but instead that a preju-dicially wrong attitude toward the treatment of errors and precision could contribute

consid-to lower them

Actually, such limitations have a number of reasons: the lack of an approach thatlooks inside natural language statements or, usually, to the classes of statements with

the same meaning that logicians call propositions; the idea of not considering its

own linguistic form and components, whose meaning can be drastically modified byslightly varying the linguistic terms in it; the attitude of considering propositions as

‘units’ without parts Those modifications are especially visible when the linguisticterms in a statement are not precise, rigid, bi-valuate, or crisp, but imprecise, vague,flexible, or fuzzy, which is a permanent fixture of natural languages more that anexception

It was fuzzy set theory, introduced by Lotfi A Zadeh after 1965 that, allowing tolook at linguistic terms and connectives as ‘functions,’ permitted to include the above-mentioned flexibility through continuity, a basic concept of mathematical analysis.Such viewing of words as functions meant a real, tangible progress, akin to what wasobtained by exploring the heart’s inner workings using an electrocardiogram instead

of only a stethoscope Zadeh opened a new path toward using mathematical analysis

in the study of commonsense reasoning expressed in a natural language

Among other topics, this functional approach allowed to analyze the validity ofclassical logic laws with fuzzy sets determining, by solving functional equationsand inequalities, which operations of conjunction, disjunction, and negation can orcannot hold

All that is in the backstage of this booklet and did facilitate a direct work withcommonsense reasoning that, involved as it is with natural language, needs to take

into account what is inside the statements, its components, and their variations.

The connection between language and reasoning is so strong that the second can

be seen as ‘Language in Action,’ or ‘Language at Work.’

Sometimes, the close relationship between language and reasoning creeps updirectly in language itself: It is in this sense that it can be noted that Catalan seems to

be the only Latin language in which ‘parlar’ (talking) enjoys the synonym ‘enraonar,’with ‘raonar’ meaning ‘to reason,’ and ‘enraonar’ to enchain reasons [8]

3

Another topic of some relevance, also derived from fuzzy set theory, is the ation of ‘measurable’ linguistic statements, that is, those consisting of words whosemeaning can be represented by means of scalar magnitudes In this context, the

consider-‘measure’ of their meaning is nothing else than the membership function of thefuzzy set whose linguistic label is just the corresponding word It is the concept of

a measure that instills science in the study of meaning since, resuming the words ofLord Kelvin, ‘If you can’t measure it, you are not doing Science.’

Now, and thanks to all that, semantics can start to be a scientific subject, in thesense specified above: and fuzzy logic, or Zadeh’s ‘Computing with Words andPerceptions,’ can become an experimental and theoretic discipline It seems that akind of physics of language and reasoning could be finally approached

It is of some relevance to notice that transforming a statement into a functionrequires to ‘design’ all the measures of meaning concerning the linguistic terms ofwhich the statement consists of This process of design, being these measures notnecessarily unique, should be done with the help of all the contextual informationavailable to the designer, and is a mark distinguishing what is presented here as

‘Language in Action,’ or commonsense reasoning (reasoning as everybody does it),from the previous classical logic approach

Even if it is not considered in this booklet, an important notion in the multifacetedfield in which the presence of a lack of crispness has been studied is the philosophicalconcept of ‘vagueness’ [9, 10].

This general notion when seen from the perspective of measurable words converts

in that of ‘fuzziness,’ [11–14] even allowing to compute how fuzzy, or how crisp, is

a fuzzy set, and thanks to what the second author did introduce already in 1972,jointly with the late Aldo de Luca, under the name of ‘fuzzy entropies’ [15, 16].Among important developments on the subject, we limit here to remember some earlycontributions [17, 18], and interesting related concepts in the setting of MV Algebraspointed out by Di Nola [19] It is a study allowing to say, perhaps paradoxically, thatfuzzy logic can be seen as the scientific study of fuzziness

4

All this opens a door toward the series of reflections of which this essay consists

of and, in the first place, to the introduction of the so called Formal Skeleton ofcommonsense reasoning from which classical logic, or the calculus of rigid state-ments, quantum logic, multiple-valued logics, and fuzzy logic, follow All of them,with their many interesting applications, show the same minimal skeleton of laws

A skeleton that not only allows to define refuting, conjecturing and classifyingconjectures as consequences, hypotheses, and speculations (with these last appearing

by the first time in the logics literature), but also to recognize reasoning as an ential zigzag in search of speculations, similar to a kind of Brownian movementaround the premise, perhaps shaking it thanks to what is known of it, and consisting

infer-of sequentially deducing and abducing A zigzag that sometimes can be cally simulated as it is shown in the booklet, and placing speculation in the center

algorithmi-of reasoning, as well as showing it is in the frontier with directed thinking, and thatreasoning consists in the end in a sequence of trials and errors

Additionally, but important, is that the skeleton shows that most of the knownlogical laws are not always valid in commonsense reasoning, are not universal in itbut are no more than local laws; this ‘locality’ is something that can also be considered

as a new view in the formal study of reasoning

The skeleton permits to note the importance of the transitive law A specificinstance of a law that is not universal in commonsense reasoning but just local.Using transitivity in concourse with the basic laws of the skeleton permits theformal simplification of reasoning as just the zigzag of inference conducting toRefuting and Conjecturing, as well as to analyze the behavior of conclusions whenthe initial information grows, and seeing that only speculations are neither mono-tonic, nor anti-monotonic, but non-monotonic Speculation seems to be equivalent

to non-monotonic logic The skeleton is a minimal support for the formal study ofcommonsense reasoning

After all that, the worrying problem of the breaking of deductive chains is dealtwith A problem that is analyzed starting from the idea of ‘meaning indistinguisha-bility,’ and modeling it by means of the indistinguishability operators, introducedand studied previously in fuzzy logic Such approach seems to model well enough

a linguistic phenomenon that is often due to the limitation of our senses perception,such as the chain of synonyms

Perhaps those roots are not actually so surprising: It is shown in the famous novel

‘The Name of the Rose’ by Umberto Eco that in such time a lot of new ideas thatfurther were successful in Europe were advanced Actually, it was ‘modern thinking’that come into being in that time; in some sense, it can be said that some thinkers ofthat time did shape the future thought in modern Europe

Some scholars have put forward, in fact, the thesis that the ’turn’ toward modernity

was just happening in that period and was interrupted by a catastrophic event (theplague of fourteenth century) which postponed the full realization of this ’researchprogram,’ we could say, for a few centuries This process has been described, in aboth suggestive and very clear way, by Pietro Greco in his monumental work onScience and Europe [20] to which we refer the reader.1

6

The essay’s contents consist in three parts and a final conclusion The first part,influenced by fuzzy logic, reflects on reasoning, and the second, devoted to thebreaking of deductive chains, is not only influenced by fuzzy logic but also byPoincaré’s continua The third part starts with an outlook of fuzzy calculus, butfocalizes in a reflection on the mentioned roots of the essay’s spirit What follows isnot a direct lift from fuzzy logic, but is partially inspired by its study

1 The particular argument referred above is treated in volumes 1 (2014) and 2 (2015).

For what concerns the final conclusions, they are but a closing reflection on allthat are presented and discussed in this booklet The main intention being to suggestthe readers some possible and innovative line of research that concerns language atwork, and reasoning, both in itself and in view of its mechanization Something thatshould encourage the research on computers’ expansion from deducing to ‘thinking’

in, at least, a directed form

What can be done better, today—being research on neuroscience so far away fromour own intellectual and practical capacities—than trying, at least, to help computers

to simulate directed thinking and reasoning as a possible ‘practical use/experimentalverification’ of what is beyond this booklet? This is, in part, a reason for whichsome hints on the computing with conjectures, and especially with speculations, areincluded

It is in such direction that the Afterword Rudolf Seising kindly wrote for thisbooklet outlines the historical path that refers, in its background, to the development

of artificial intelligence In it, language and reasoning acquired the relevance that thenecessity of its computer simulation gives them, step by step with the progress ofdigital computers

In the authors view, this essay reinforces the relevance of fuzzy logic that, inaddition to its fertility with applications in so many technology fields, can also facil-itate a determinant contribution toward foreseeing a new science of language andreasoning, of ‘Language at Work.’

Slightly modifying the words in the Pray of Saint Francis of Assisi, this essayaims at helping to accomplish ‘Where there is darkness let us show light’ for whatconcerns the study and mechanization of commonsense reasoning

At the same time and from another point of view, we must declare (due theintellectual ‘Customs of Reliability’ grounded on Wittgenstein’s words ‘on what wecannot speak we must be silent’) that some ‘sins’ against clarity and the intellectualcourtesy can appear at some points in this booklet, when ideas that are not yet clearand perfectly formed seemed to the authors a nonetheless valuable contribution that

can be suggestive for the readers With Horace, Nihil est ab omni parte beatum.

References

1. E Trillas, P.P Bonissone, L Magdalena, J Kacprzyk, (Eds.) Combining Experimentation

and Theory: A Homage to Abe Mamdani (Springer Berlin Heidelberg, 2012)

2 J Von Neumann, Probabilistic logics and the synthesis of reliable organisms from unreliable

components, in Automata Studies, eds C.E Shannon, J McCarthy (Princeton University

Press, 1956), pp 43–98

3. J von Neumann, The general and logical theory of automata, in Cerebral Mechanisms in

Behaviour—The Hixon Symposium, (J Wiley, New York, 1951)

4 R Seising, M.E Tabacchi, S Termini, E Trillas, Fuzziness, Cognition and Cybernetics:

a historical perspective, in Proceedings of the 2015 Conference of the International Fuzzy

Systems Association and the European Society for Fuzzy Logic and Technology (Atlantis

Press, 2015), pp 1407–1412

5 E Trillas, S Termini, M.E Tabacchi, R Seising, Fuzziness, Cognition and Cybernetics: an

outlook on future In Proceedings of the 2015 Conference of the International Fuzzy Systems

Association and the European Society for Fuzzy Logic and Technology (Atlantis Press, 2015),

pp 1413–1418

6. K Popper: ‘Autobiography’ in Book 1 of the Library of Living Philosopher, ed by P.A.

Schilpp, vol XIV, p 17

7. M.E Tabacchi, S Termini, Back to ‘Reasoning’, in Soft Methods for Data Science, eds.

B.M Ferraro, P Giordani, B Vantaggi,M Gagolewski, M ÁngelesGil, P.Grzegorzewski, O Hryniewicz (Springer International Publishing, 2017), pp 471–478

8. E Trillas, M.G Navarro, An essay on the ancient ideal of enraonar Arch Philos Hist Soft

Comput 1, 1–28 (2015)

9. S Termini, On some vagaries of vagueness and information Ann Math Artif Intell 35(1),

343–355 (2002)

10 M.E Tabacchi, S Termini, Varieties of Vagueness, Fuzziness and a few foundational (and

ontological) questions, in Proceedings of EusFLAT 2011 (Atlantis Press, 2011), pp 578–583

11 M.E Tabacchi, S Termini, Measures of Fuzziness and Information: some challenges from

reflections on aesthetic experience, in Proceedings of WConSC 2011 (2011)

12. M.E Tabacchi, S Termini, A few remarks on the roots of fuzziness measures, in Advances in

Computational Intelligence, proceedings of IPMU 2012, eds, S Greco, B Bouchon-Meunier,

G Coletti, M Fedrizzi, B Matarazzo, R Yager (Springer Berlin Heidelberg, 2012), pp 62– 67

13. E Trillas, D Sanchez, A briefing on fuzziness and its measuring, in IPMU 2012, part II

CCIS, eds, S Greco, B Bouchon-Meunier, G Coletti, M Fedrizzi, B Matarazzo, R Yager,

208, pp 15–24

14. R Seising in Measures of fuzziness and the concept of entropy, in Alla Ricerca!, eds P.

Greco, R Seising, M.E Tabacchi, E Trillas (Doppiavoce, 2015)

15 A De Luca, S Termini, A definition of a non-probabilistic entropy in the setting of fuzzy

sets theory Inf Control 20(4), 301–312 (1972)

16. A De Luca, S Termini, Entropy and Energy measures of a fuzzy set, in Advances in Fuzzy Sets

Theory and Applications, eds Gupta, Ragade, Yager (North-Holland Publ Co., Amsterdam,

1979), pp 321–338

17. E Trillas, T Riera, Entropies in Finite Fuzzy Sets Inf Sci 15(2), 159–168 (1978)

18. B.R Ebanks, On measures of fuzziness and their representations J Math Anal Appl 94(1),

24–37(1983)

19. A Di Nola, MV algebras in the treatment of uncertainty In: Fuzzy Logic, Proceedings of

the International Congress IFSA, Bruxelles 1991 Löwen P., Roubens E., Eds., (Kluwer, Dordrecht, 1993), pp 123–131

20 Pietro Greco, ‘La Scienza e l’Europa’, in five volumes L’Asino d’Oro edizioni, Roma, 2014-2019

Part I Reasoning

1 Introduction 3

2 Trying to Pose the Main Problem: Measurable Meaning 9

3 A Formal Skeleton of Commonsense Reasoning 17

4 Examples on the Analysis of Statements Trough Design 31

5 The Problem of Monotonicity and the Skeleton 41

6 Conclusions for Part I 45

Part II From a Perceptive Point of View 7 Introduction 51

8 The Linguistic Continua 55

9 Quasi-transitivity 59

10 Context, Inference’s Safety, Proving and Further Comments 65

11 Conclusions for Part II 69

Part III Fuzzy Calculi, the Context and Historical Roots 12 An Overview of the Fuzzy Calculi 79

13 Common Reasoning in a Computational Context 83

14 Looking for Some Historical Roots 87

The End: General Conclusions 97

Afterword 103

xix

Enric Trillas born in 1940 in Barcelona, earned his

Ph.D in Sciences from the University of Barcelona,and was Full Professor at the Technical Universi-ties of Catalonia (of which he was Vice-Rector), andMadrid From 1983 to 1996, he hold several posi-tions concerning the scientific and technological activ-ities in the Spanish Government Notably, he wasPresident of the High Council for Scientific Research(CSIC), Director General of the National Institute forAerospace Technology (INTA), and Secretary General

of the National Plan for R&D From 2006 to 2018, hewas Researcher at the (now closed) European Centrefor Soft Computing (ECSC) in Mieres, Asturias, Spain,and then Emeritus Professor at the University of Oviedo,Asturias, Spain He is also Distinguished VisitingProfessor at the Universidad Nacional de Córdoba inArgentina, and holds the honorary doctorate (doctorhonoris causa) of two Spanish universities, the Univer-sidad Pública de Navarra, and Universidad de Santiago

de Compostela He is the author and the editor of morethan a dozen of books and volumes in Catalan, Spanish,and English and published more than 400 papers in jour-nals, conference proceedings, and volumes He super-vised 23 Ph.D dissertations and delivered courses andlectures at universities and research centers in Germany,Italy, France, Argentina, Chile, and the USA He trans-lated three books into Spanish He is recognized as theinitiator in Spain, and a pioneer in Europe, of—first—Karl Menger’s probabilistic metric spaces, and then ofLotfi A Zadeh’s fuzzy Sets and fuzzy logic He had

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served for years in many editorial boards of tional journals and international program committees ofconferences He received a dozen of official distinctionsfrom Spain, Catalonia, Italy, and Peru, and was awardedmedals and honors exemplified by the Kampé de FérietMedal, the European Fuzzy Pioneer Award, the IEEEFuzzy Pioneer Award, the Outstanding ContributionsAward of the International Fuzzy Systems Association(IFSA), and the Honorary Membership at the EuropeanSociety of Fuzzy Logic and Technology He is Member

interna-of the Accademia Nazionale di Scienze, Lettere e Arti diPalermo (Italy) He retired in August, 2018, and inten-sively continued research activities until May, 2021.This essay somehow subsumes his life as a scientificresearcher, as he finally moves to different intellectualactivities

Settimo Termini born in 1945, was Full Professor

of Theoretical Computer Science at the University ofPalermo from 1990 until his retirement in 2015, andalso Full Professor of Cybernetics at the University ofPerugia from 1987 to 1990 From 2002 to 2009, he wasDirector of the Institute of Cybernetics ’Eduardo Caian-iello’ of the CNR (Consiglio Nazionale delle Ricerche)

in Naples, where he was a researcher from 1969 to

1987 A theoretical physicist by training, his researchinterests mainly concern problems related to the pres-ence of uncertainty, approximation, and vagueness ininformation sciences In this context, he had introducedand developed the theory of measures of fuzziness,looking as well for their epistemological aspects Inthe last fifteen years, he has also been concerned withthe analyses of the existing connections between theproductive economic model of a country and the level

of ’basic’ research He is Fellow of the InternationalFuzzy Systems Association (IFSA), Vice President ofthe Accademia Nazionale di Scienze, Lettere e Arti diPalermo, and President of the Marina Diana MercurioAssociation He is a member of the Editorial Board ofthe journal ’Lettera Matematica.’ Among his works: ’Adefinition of a non-probabilistic entropy in the setting

of fuzzy sets theory’ (Information and Control, 1972)and the volumes: Aspects of Vagueness, Kluwer 1984(edited with Heinz J Skala and Enric Trillas); Imag-ination and Rigor, Springer 2006; Contro il declino,

Codice, 2007 (with Pietro Greco); Memoria e Progetto,GEM 2010 (edited with Pietro Greco) have been themost influential.

Marco Elio Tabacchi born in 1971, is a fuzzy logician,

aquatint printmaker and transdisciplinary researcherwhose scientific interests lie at the border betweenlogic, data analysis, and cognitive sciences He holds

a Ph.D in Physics from the Université Paris-Sud XI(now Université Paris-Saclay) in France He has been avisiting academic at the University of Surrey and Impe-rial College, UK Currently, he is the Scientific Directorand Chief Technological Officer at the Istituto Nazionale

di Ricerche Demopolis in Italy, an Associate Researcher

at Dipartimento di Matematica e Informatica at sità degli Studi di Palermo, Italy, adjunct at Accademia

Univer-di Belle Arti Univer-di Palermo for etching and litographycourses, and mentor of the startup PDP Tanuki, SmartDevices developer Previously he designed and devel-oped fuzzy-based clinical diagnosis support systems

He is a co-editor in Chief of the ’Archives for SoftComputing’ journal and is on the Editorial board ofthe ’Lettera Matematica’ journal He also serves as anambassador at the FMsquare Foundation in Switzerland,and has been a member of the board of the AssociazioneItaliana di Scienze Cognitive for two terms He is theauthor or co-author of more than 120 scientific papersand a co-editor of two Springer’s volumes: ’DesigningCognitive Cities’ and ’Fuzziness and Medicine: Philo-sophical Reflections and Application Systems in HealthCare.’ He holds two national patents on fuzzy softwaremethodologies He has served as a chair and reviewerfor a number of national and international conferences,among them many editions of WILF, IFSA-EUSFLAT,and AISC

If you change the way you look at things,

the things at which you look change.

(Max Planck)

In the intention of the authors, this booklet does not want to be a textbook; instead,

it contains some reflections that aim to guide ‘Computing with Words and tions’, Zadeh’s final view on his Fuzzy Logic, towards its future as a new science

Percep-of both Language and Reasoning Towards an experimental and theoretical scienceconcerning the Natural Phenomena of Language, Thinking and Reasoning, in rela-tionship with Neurosciences, whose possibilities are foreseen by the authors Theyare alas not competent enough for trying to initiate here such new science, even if, nodoubt, willing to encourage younger researchers to embark in such a new ‘scientificadventure’

1.1 Fuzzy systems are usually seen as sets of linguistic rules with imprecise terms

typical, for instance, of Fuzzy Control Rules unsophisticated as linguistic statementsthat, usually, deal with the (measurable) variables of a dynamic physical artifact,system, or machine, translating into a linguistic description the knowledge on the

“machine” inner workings already acquired by some expert [1]

Nevertheless, and since its inception, the interest of Fuzzy Set Theory goes furtherthan these systems of conditional statements whose practical interest in view of itsapplication to technological problems do not deserve any kind of debate At the endwere just thanks to these industrial applications to machines’ control that the famousname ‘Fuzzy Technology’ was coined and conducted Fuzzy Logic to its ‘fame’ inthe Media

Actually, Fuzzy Set Theory tries to represent Natural Language’s statementscontaining precise and imprecise, certain and uncertain linguistic terms, as well asits theoretical analysis mainly concerning its use in Ordinary Reasoning in order toarrive at some conclusion Since such statements permeate Language and OrdinaryReasoning, Fuzzy Methodologies are relevant for that endeavor and can be seen asHuman-Centric

Translating statements into formulas and the consequent computerization ofCommonsense Reasoning, constitute the original, true goal of Fuzzy Logic To reach

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

E Trillas et al., Reasoning and Language at Work, Studies in Computational

Intelligence 991, https://doi.org/10.1007/978-3-030-86088-2_1

3

this target it needs to count using some form of symbolic ‘calculus’ It is the translation

of statements into ‘fuzzy formulas’ that allows its semantic analysis by mathematicalmeans, in the same way it is usually done in Physics This is the way we intend FuzzyLogic in this essay, and why most of the well analyzed mathematical instrumentsoffered by fifty-five years of theoretical study will be overlooked here

In our view, Fuzzy Logic is a new step ahead in the long way initially paved byLeibniz with his ‘Do not discuss, compute!’, and Boolean Calculus as the first step inthis way for what concerns precise statements, even affected by a kind of uncertaintyable to be modeled by a probability [2]

1.2 If the currently known Fuzzy Algebras can allow some computing by copying

reasoning, nevertheless they don’t fulfill all the laws of a Boolean Algebra that is theindisputable structure modeling predicate calculus, classical logic They just satisfysome of them, since its totality can’t be always presumed in language It is, forinstance, the case of the commutative law of the linguistic conjunction ‘and’, which

in language is not a universal but a local law; usually, in language, time or sequencingreally matters, and have a powerful effect on the meaning of statements

Analogously, the supposed strong character attributed to linguistic negation, ‘not(not p) = p’, is just a local law in language For saying nothing on the usuallypresumed associative law of conjunction: from the ancient Greek’s Dodona oracle’s

‘Ibis redibis nunquam peridis’ moving commas trick on, different subdivisions of

the same phrase can easily change the meaning of statements

Most of the classic theoretical results that were obtained with, for instance, theso-called Standard Algebras of Fuzzy Sets [2], strongly depend on the laws satisfied

by the t-norm T modeling ‘and’, the t-conorm S modeling ‘or’, and the always strongnegation function N modeling ‘not’ Consequently, they are not universal but justlocal results; only hold provided the corresponding and contextual meanings of thelinguistic connectives can allow presuming the involved laws

Actually, such connectives, ‘and’, ‘or’, ‘not’, are not ‘logical constants’ in bothNatural Language and Ordinary Reasoning, as they were classically considered in

‘Logic’ Neither Ordinary Reasoning, nor Language, can be fully and rily analyzed only through Logic, as its too rigid tools are insufficient for such ananalysis [3]

satisfacto-1.3 As such, it is required that statements should be suitably represented by means

of mathematical formulas, and that is the reason for Algebras of Fuzzy Sets beingmandatory But since all such algebras are necessarily just local, the laws actuallysatisfied by each linguistic term, including the connectives, can be known only fromthe context in which the statement is inscribed

Hence all the mathematical functions in a formula representing a linguistic ment should be ‘designed’ The word ‘design’ marks one of the several differencesbetween the ‘logical’ (in the traditional sense) and the ‘fuzzy’ approaches; perhapsthe most visible one [1] ‘Design’ represents, up to some extent, a realistic and closerlook at the language used for both reasoning and expressing it

state-For instance, the possible validity in language of the statement p= p · q + p · q’

is proven immediately in the case p and q belong to a Boolean Algebra, as by thedistributive law its second member coincides with p · (q+ q’), and this with p · 1 = p

A coincidence that is not guaranteed if p and q are not in a Boolean Algebra: it holdsneither in an Orto-lattice, nor in a De Morgan Algebra; in the first due to the lack

of the distributive law, in the second due to the lack of the law q+ q’ = 1 In DeMorgan algebras, the former law is just locally satisfied by its Boolean elements; it

is not a generally valid equality How its validity can be analyzed in the setting oflanguage? It is possible through fuzzy logic by:

• First, and once the universes X and Y of discourse at which p and q apply areknown, by designing suitable membership functions f: X → [0, 1] for p, andg: Y→ [0, 1] for q

• Second, analyzing the laws that can be presumed by the involved connectives

‘and’ (·), ‘or’ (+), and ‘not’ (‘) in each of their apparitions

• Third, choosing suitable functions T1, T2, S, and N, for mathematically senting such connectives, and provided such functions do exist (since its existence

repre-is not always guaranteed)

• Fourth, checking that the formula:

f(x) = S (T1(f (x), g (y)), T2(f (x), N (g (y))),

holds for all x in X, and y in Y

An alternative way can consist in previously solving the functional equation

a= S (T1 (a, b), T2 (a, N (b))), for all a, b in [0, 1], in the unknowns S, N, T1

and T2, and checking if their solutions agree with the context in question

What, in any case, will result proven from the formula’s validity is that the formulacan be used in the context in which p and q appear Obviously, the solutions of thefunctional equation will show the contextual laws that are necessary to count withfor it to be valid

In language, statements are almost always ‘situational’—that is, dependent and purpose-driven For instance, the statement ‘Kate is a nice girl’,changes its meaning according to the fact that in the context in which the phrase

context-is uttered beautifulness matters or not, as well as if the purpose of the utterance

is to flatter the person, or just to be merely descriptive of the character In eachpossible case, the membership function G of ‘nice girl’ will be different and, thus,the consequent mathematical representations of the statement G (Kate), will notcoincide

In general, elemental statements ‘x is P’ should be mathematically represented

in the form fP (x), with suitable membership functions fP designed not only byconsidering object x and predicate P, but also the context surrounding both, and thepurpose managed in it for predicating P of x The same predicate P in a universe ofdiscourse X can be represented by several membership functions

1.4 At this point a personal reflection on the future of Fuzzy Logic can be pertinent.

The many results obtained in the setting of the so called ‘Standard Algebras’ of FuzzySets, that is, when the operators T, S, and N reflecting the linguistics connectives

‘and’, ‘or’, and ‘not’, are a continuous t-norm, a continuous t-conorm, and a StrongNegation Function, respectively, did mainly serve to manifest the ‘locality’ of theclassical laws that were analyzed with them Among these laws, the one called

‘perfect repartition’ is outstanding, since its study did show that it can hold withoutpresuming duality [2,4] Consequently, duality is also a law appearing as local,notwithstanding it is usually taken for granted as a universal law

Such studies mainly show that if almost all classical laws can hold with fuzzy sets,most of them are only of a local validity, depending on the properties that can berecognized in the corresponding praxis, the reality at which the linguistic discourserefers to It does not seem that a ‘universal algebra’ covering all discourse can exist,which implies the necessity of knowing more on both Natural Language and OrdinaryReasoning, as they are so strongly dependent on language [2,5 8] Those studiesare now appearing as a ‘controlled experimentation’ from which ‘good lessons’ can

be learned, and that conducted to what will be Chap 3 even if not only

1.5 Up to now, neither language nor ordinary reasoning are well enough

domes-ticated in scientific terms The road to this domestication can be followed only

by abiding to the usual rules of science: first observation, and then, controlledexperimentation

Hence, and provided Fuzzy Logic or Computing with Words and Perceptions isthe mean to surpass the limits of application to this specific kind of problems, themove toward a wider view becomes necessary: toward an experimental science oflanguage and reasoning in which mathematical models can play an important role

A move that can convert Fuzzy Logic in a kind of ‘Physics of Language andReasoning’,1 , 2 in which ‘meaning’ should play a central role, since Semantics iscrucial to understand linguistic statements, and membership functions do measuremeaning Like true Physics deals with matter and energy, and whose dynamics areessential

All in all, if Language and Reasoning are but Natural Phenomena, their studycorresponds to, and requires, a Natural Science of Language and Reasoning Further

in this Essay, which actually hides a proposal to go further than CwW, we will returnagain to this point: at what this booklet tries to look, and from its border with FuzzyLogic, is Language at Work or Commonsense Reasoning [5,6,9],

Because there is no University-specific teaching of Fuzzy Logic, with professorswhose specialty is just that, if Fuzzy Logic will not convert to such a ‘kind of Physics’,

1 It is interesting to observe that the reference to physics is a somewhat leading idea for establishing

a reference frame Not for imposing exactly the same sort of ‘mathematization’ that occurred in physics (something we have criticized for what has been done in Economics), but for the crucial reference to the ‘natural phenomena’ one must always have in mind.

2 For the same use of this metaphor in a slightly different context, see: [ 10 ]

it is destined to disappear as a scientific discipline and will perhaps just survive forsatisfying the needs of some applications.

The actual fact is that today Fuzzy Logic is not studied in itself and ‘for thehonor of mankind’, as the mathematician Karl G J Jacobi said of mathematics, but

is applicable to all discipline With Virgil, in the Georgics, Felix qui potuit rerum

cognoscere causas.

References

1 E Trillas, S Guadarrama, Fuzzy representations need a careful design Int Jour of General

Systems 39(3), 329–346 (2010)

2 E Trillas, On the Logos (Springer, 2017)

3 E Trillas, Divagaciones sobre pensar y razonar (In Spanish) (EUG, 2021)

4 E Trillas, Reconsiderando el concepto de conjunto borroso (In Spanish) (EUL, 2020)

5 E Trillas, El desafío de la creatividad (In Spanish) (EUSC, 2018)

6 E Trillas, Reflexiones sobre la génesis de la Lógica (EUSC, 2021)

7 E Trillas, C Alsina, Standard theories of fuzzy sets with the law (f · g’)’ = g + f’ · g’ Int.

Jour Approximate Reasoning 37(2), 87–92 (2004)

8 E Trillas, C Moraga, S Guadarrama, S Cubillo, E Castiñeira, Computing with antonyms, in

Forging New Frontiers I, ed by M Nikravesh et altri, pp 133–153 (2003)

9 E Trillas, Narrar, conjeturar y computar: El pensamiento (In Spanish) (EUG, 2020)

10 M E Tabacchi, S Termini, Theory of Computation, Fuzziness and a physics of the immaterial.

In Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology Atlantis Press (2013)

Trying to Pose the Main Problem:

Measurable Meaning

2.1 We have shown the necessity to reconsider the concept of ‘meaning’ In this

framework ‘meaning’ should be a situational concept requiring to be bounded byspecific definitions rendering it applicable to our needs Membership functions should

as well be seen as ‘measures’ of meaning, which would give them a meaning as well.The basic idea for this process comes from the following: in language notonly precise words but also imprecise words, the connectives and the semantichedges, are endowed with meaning Meaning, semantics, is the way of relating state-ments with the reality at which a discourse, uttered or written, refers to Withoutmeaning a statement escapes comprehension For instance, how we can relateformula a2 + b2 = c2 with triangles, without identifying what a, b, and c refer

to, what these letters mean?

The difference between the case of precise and imprecise words P, lies in therespectively rigid or flexible character shown by their action on the elements inthe universe of discourse; in short, on how the property P names varies along X.Such variation is, in principle and in the language, empirically captured through theknowledge of when

‘x is less P than y’ for all pair of elements x and y in X;

that is, x shows less than y the property named P that is manifest in the elements ofX

By symbolically writing such conditional statement as x <Py, the binary relation

<P ⊆ X x X can be taken into account It generates a graph (X, <P) that will be

called the ‘primary or qualitative meaning’ of P in X, and, also, the fuzzy set P with

linguistic label P Notice, thus, that a fuzzy set is but a graph reflecting the behavior

of P on X; hence, in the main fuzzy sets just represent linguistic entities Entitiesexisting and usually managed in language that, as such, can be submitted to the sametroubles with perception that language can suffer For instance, to manage well thefuzzy set labeled ‘young’ in the universe of Londoners, it is not only necessary toperceptively recognize that there are some Londoners deserving to be predicated asyoung (John is young, Sally is young, etc.), but also that ‘Sally is less young than

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

E Trillas et al., Reasoning and Language at Work, Studies in Computational

Intelligence 991, https://doi.org/10.1007/978-3-030-86088-2_2

9

John’, ‘Peter is less young than Eddy’, etc., for knowing that the word ‘young’ notonly applies effectively to Londoners, but does it with variable intensity On thecontrary, a word like ‘odd’ not only effectively applies to natural numbers, but does

it uniformly, with the same intensity at each number of which it applies and with no

intensity at the others; numbers are either completely ‘odd’, or are completely ‘notodd’

The only properties presumed for the binary relation <Pare [1 3],:

(1) Reflexivity, x <Px for all x in X, guaranteeing that <Pis not empty, and,(2) Transitivity, x <Py & y <Pz= > x <Pz, guaranteeing that if x is less P than

y, and y than z, then x is less P than z

Such linguistic entity’s representation, the graph (X, <P) or fuzzy set P, can be

seen reduced to a set when <Pcoincides with<−1

p , that is, if a graph’s arc connectswhatever elements x and y, also another arc connects y and x In such case, whatreally matters is the subset of X whose elements are fully connected among them Inaddition, the relation ‘equally P than’ can be identified with<p ∩ <−1

p ==p; since

it is=P= <P, the graph is (X,=P), and the word P is precise or rigid in X Theonly relevant part in the graph is the former set of fully connected elements, the setspecified by the precise word P in X; all of them are equally P than the others.Hence, sets are a particular case of fuzzy sets: those appearing when the linguisticlabel is precise; this is the sense in which sets (crisp sets) are seen as ‘degenerate’fuzzy sets

Once the concept of primary meaning is defined as the empirical relation <P, itshould be said that a word P is meaningful when <P is not coincidental with theset {(x, x); x∈ X}, that it is bigger When the contrary is true P is meaningless, ormetaphysical, in X

2.2 The meaning of a word P, imprecise in X, is not totally captured by the graph

or fuzzy set P; usually a lot of contextual nuances cannot be reflected in the graph.

Essentially it lacks ‘extension’, that is, knowing up to which degree each x in X is

P, how much x satisfies the property named by P It needs to give a ‘measure’ ofthe strength of P at each element x; for instance, within the fuzzy set of the youngLondoners, it still lacks to know up to which extent a Londoner can be qualified asyoung To such an end, let’s define what follows:

A mapping mP: X→ [0, 1] is a measure of the meaning of P in X [1] providedthe following three properties are verified:

This definition is worthy of some comments:

(a) Measures grow in value when elements grow under < ;

(b) Minimal elements in the graph, have the smallest value’s measure;

(c) Maximal elements have the biggest value’s measure;

(d) Nevertheless, these three laws do not suffice, in general, to specify a measure,usually it lacks information from the context on which P is used and, for thisreason, it is said that each mPreflects a ‘state’, that on which P is considered

in X

Each scalar magnitude (X, <P, mP) is called a ‘total or quantitative meaning’ of

P in X; hence, even provided the primary or qualitative meaning can be consideredunique, the quantitative meaning is not unique In this form, meaning is seen as ascalar magnitude in which the fuzzy set is its qualitative under-laying magnitude.The only exception to the non-uniqueness of the total meaning, appears with therigid words In this case, all elements in the (classical) set specified by P, totallyfulfill the property named P, thus, it can be accepted that they show measure 1, and,consequently, the others do not verify at all the property and show measure 0 Thereciprocal is obvious Thus, the set specified by P is mP−1(1), and its complement

in X is mP−1(0)

Both sets constitute a partition of X provided no one of them is empty, in whichcase the other is X The empty set has the measure mempty (x) = 0 for all x in

X, and the total set or universe X, has the measure muniverse (x) = 1 for all x in

X It is obvious that any other fuzzy or classical set with a measure mP, verifies

mempty≤ mP≤ muniverse

2.3 Let’s show a very simple example in the universe X = [0, 1], with the word

B= big In this case, it is obvious that <big is just the order≤ of the unit interval(x <B y⇔ x ≤ y), and that there is a unique minimal, 0, and a unique maximal,

1 Hence, the measures of the meaning of ‘big’ in [0, 1] are all the mappings

mB: [0, 1]→ [0, 1], that are non-decreasing and verify mB (0) = 0, and mB (1)

= 1 There is a great amount of them, and to specify a single one more information

is needed

For instance, were such information that the measure is lineal, that is, the variation

of ‘big’ is so, from mB(x) = ax + b, and the two border conditions, it will follow

a= 1, and b = 0: The only lineal measure is mB(x) = x Nevertheless, were the

information that mBis a quadratic function, mB(x) = ax2+ bx + c, what follows is c

= 0, and a + b = 1, and there is a multitude of such measures, mB(x) = ax2+(1−a)x,

depending on the parameter a Then, to specify one of them it lacks some additionalcondition; were it, for instance, that the curve y= mP(x) should pass by the point(0.5, 0.6), from 0 6= 0.25a + 0.5(1–a) = 0.5–0.25a, it will follow a = −1/25 and themeasure will be -x2/25+ 16x/25 In any case, once a suitable membership function

fBis designed and before fixing it, fBshould be submitted for testing against morecontextual information Notice that a= 1, b = c = 0, are suitable values giving thequadratic measure x2, the square of the linear one

How a measure of ‘young’ = Y, can be defined in the universe of Londoners?Provided young refers to chronologic age, and provided k is the maximum age

of current Londoners, it is obvious that mY (x) = Age (x)/k is such a measure,since the minimals (age equal zero) have zero measure, the maximals (age k)

measure one, and since it is x <Y y ⇔ Age (x) ≤ Age (y), x <Y y implies mY

(x)≤ mY(y) This measure’s values are not only 0 or 1, it agrees with the cise character of young; for instance, if k = 100 and John is 45 years old, then

On the contrary, if ‘young’ refers to attributes different from chronological age,

as x wishes new experiences, likes theatre, has a fluent and interesting conversation,etc., then the set of ‘young olders’ can be not empty at all but a crisp part of theNursing Home

In the back of any meaning for a word there are some ideas for using it in a givensituation That is like Wittgenstein’s: ‘The meaning of a word is its use in language’,

in his ‘Philosophical Investigations’

2.4 Let’s stop for a while in a topic showing how the former definition allows to

model what can be called the “social component” of meaning; that is, why and howmeaning varies through the use of words by people To such end a simplified casewill suffice [2]

Suppose that two people use the same word P but with two different meanings(X, <Pk, mPk), k= 1, 2 A ‘conversation’ using P will be impossible whenever theintersection<1

be represented by<common

P =<c

P, and that is supposed contained in both<1

Pand<2 P

provided their intersection is not empty

In this form, the graph (X, <c

p) allows both people to understand the same for

P, and for what concerns its extent it suffices to consider common measures like

mcP(x)= T (m1

P(x), m2P(x)) for all x in X, and T= min, or T = prod for instanceand depending, respectively, on the possible properties: P and P~P, or P and P < P.Obviously, these arguments do not essentially depend on the number of people,and, in general it can be said that such social employment reduces both the primarymeaning and its measure up to a tighter use of the words

2.5 Let’s offer a comment concerning the design of rules representation, a problem

that, for instance, is considered as basic in Fuzzy Control (and elsewhere) [1]

A rule is, in the simplest case, a conditional statement of the form ‘If x is P, then

y is Q’, with x in a universe of discourse X, y in a universe Y, and P and Q acting,respectively, on X and Y

The problem is now reduced to search for a Fuzzy Relation,J: [0, 1] x [0, 1]→ [0.1], with which the rule can be represented by the formula,

J

fP(x), fQ(y), for all x in X, and y in Y,

being fP a measure of the meaning of P in X, and fQone of Q in Y, that is, the

respective membership functions of the fuzzy sets P in X, and Q in Y These two

membership functions should be designed, analogously to the former example with

B= big, accordingly with the contextual information available from the behavior of

P in X, and Q in Y; supposing that both are designed and fixed, let’s concentrate inhow to find J

In the first place, since J tries to represent an If/then statement p < q (If p, then q),with p= x is P, and q = y is Q, it should verify the known Modus Ponens’ condition,

(p · (p < q)) < q,

stating that q can be effectively reached from both p ‘and’ (·) p < q Consequently,

an operation T does exist such that

T(fP(x), JfP(x), fQ(y) ≤ fQ(y), for all x and y,

a condition that provided T is a continuous t-norm can be proven equivalent to

at the end, reflects the flexibility of the imprecise terms

Thus, the first question is to reach J, and it only can come from how the conditionalstatement ‘If p, then q’ is understood in the corresponding problem; that is what

p < q means in it For instance, provided it were understood as ‘p and q’, as it is oftenthe case in Fuzzy Control, what is required is finding an operator T*, be a t-norm ornot, such that

J

fP(x), fQ(y) = T∗

fP(x), fQ(y), for all x and y,

that is, J= T* In this case, it can be said that p < q is recognized as a conjunction

p · q, that the statement ‘if p, then q’ has the symbolic proto-form p · q; has themeaning of ‘p and q’

In general, to reach J is necessary to previously recognize, from its meaning, towhich symbolic proto-form can it be associate, something requiring the best possibleknowledge on the problem’s context T* is not always a commutative operation and,for instance, if product is commutative, it is very easy to obtain non-commutativeoperations as, for instance, T* (x, y)= x2y that only commutes when x= y

As another possibility, were the symbolic proto-form p’+ q, that is, understanding

p < q like in Boolean Algebras, as the unconditional statement ‘not p or q’, then itwill be J (a, b)= S*(N*(a), b), for operators S* representing the disjunction ‘or’, andN* representing the negation ‘not’ Operators that will be, or not, a t-conorm and astrong negation function depending on the laws that can be contextually recognizedfor both linguistic connectives

There are more possible proto-forms, such as those also appearing in BooleanAlgebras as p < q ⇔ p’ + q if p · q = 0, and p · q otherwise, or in the Logic ofQuantum Physics, p’+ p · q (Sasaki hook), and q + p’ · q’ (Dishkant hook), etc.What in all cases should be checked is the existence of an operation T with which Jverifies the (first) Modus Ponens inequality

To sum up, the sequential process to be followed for designing a rule is:

(1) Design of all the membership functions intervening in the rule;

(2) Search of a suitable proto-form for the rule, provided it exits;

(3) Transform this proto-form by means of suitable functions representing theinvolved connectives, up to have a function J;

(4) Checking that J verifies the Modus Ponens inequality; and

(5) Submitting J to checking it against contextual additional information, beforefixing it

Nevertheless, it should be pointed out that:

(a) The functions JTdon’t properly correspond to a (fuzzy) proto-form Not alwaysthe meaning of p < q is so simple to come from a symbolic proto-form Hence,there are cases in which, to made explicit the meaning of p < q, a differentpath should be followed, and that can come from, for instance, observing that

JT(a, b)= 1 ⇔ a ≤ b, and that with a, b in the set {0, 1}, J reduces to expressthe characteristic function of ‘not p or q’, the typical Boolean implication.(b) Since in Boolean Algebras it is

p’ · (p · q)= (p’ · p) · q = 0 · q = 0, it is clear that in those situations inwhich p’ is impossible, the proto-form p’+ p · q can be simplified up to theproto-form p · q, as if it were p’= 0 Notice that p’ = 0 just implies p = 1when the negation is strong (p= (p’)’ = 0’ = 1); something that is not sure if,for instance, the negation is just ‘intuitionistic at p’, that is, p” < p, in whichcase is p’ < p”’⇔ 0 < p”’ that is always a correct formula

2.6 Before ending the Chapter let’s present a way, called the ‘Extension Principle’

(EP) that permits to extend numerical functions and operations from the real line tothe fuzzy field The root of EP lies in the following [1,4],,

Given a function F: X → Y, for each subset A in X, it can be obtainedits image in Y by F, F (A) = {F (a); a ∈ A} ⊆ Y Let’s call Fˆ the function{0, 1}X → {0, 1}Y defined, between the power sets, by Fˆ (A) = F (A) Let

it be fA the characteristic or membership function of A; then, the membership

or characteristic function of Fˆ (A), is:

fF∧(A)(y) = Sup{fA(X); F (X) = Y},

since it is fFˆ(A)(y)= 1 ⇔ It exists x in A such that F(x) = y, and it equals 0 otherwise.Changing to the fuzzy subsets of X, that is changing the power set{0 1}X by the set of possible membership functions of such fuzzy sets,

or ‘fuzzy power set’, [0, 1]X, the former formula can be generalized to extend F up tofuzzy sets For instance, if F: [0, 10]→ [0, 1} is F(x) = 1−x/10, then the membershipfunction g (x)= x/10 in [0, 1][0, 10}, extends to the membership function in [0, 1],

F∧(g)(y) = Sup{g(x); F(x) = y} = Sup[x/10; 1 − x/10 = y} = 1 − y, for all y

in [0, 1]:

In this way, ‘small’ in [0, 10] extends to ‘small’ in [0, 1]

Easy examples show that a fuzzy set can be extended up to a crisp set Forinstance, the fuzzy set in the finite universe {1, 2, 3, 4} given by the member-ship function g= 1/1 + 0.4/2 + 1/3 + 0.7/4, extends, by means of the functionF: {1, 2, 3, 4}→ {a, b, c], given by:

F(1) = F (2) = a, F (3) = F (4) = b,

to Fˆ (g)= 1/a + 1/b, that is, to the crisp set {a, b}

Consequently with all that, the EP allows that all operation in the real line,

*: R x R→ R, can be extended to fuzzy sets in [0, 1]Rby the formula

(g ∗ h)(z) = Supz= x ∗ y{min (g (x), h (y)},

that permits to generate the so-called Fuzzy Arithmetic allowing to operate with

‘imprecise numbers’ like ‘around 3’, and showing, for instance, that ‘around 3’ plus

‘around 4’ is but ‘around 7’, as well as that their ‘product’ is ‘around 12’

2.7 To conclude this Chapter, a reiterated comment concerning the authors’ current

view on Fuzzy Logic and Computing with Words In our opinion both have, at least,the great merit of having shown the first successful way to use some of the resources ofMathematical Analysis to deal with imprecise or vague words, and the mathematicalmodeling of sequences of precise and imprecise words, linguistic statements, andsome aspects of Commonsense Reasoning

Curiosity on such modeling produced a series of studies conducting to realizethat the typical logical view is insufficient for dealing with Commonsense Reasoning.That most of the laws given for granted in Logic can’t be considered always valid, butonly locally in some parts of the discourse or, better, of the linguistic argumentation

on which Reasoning consists in

This implies the necessity of carefully designing the mathematical tation of the linguistic terms appearing in the statements on which Reasoning isexpressed [5] An erroneous design risks representing a problem different from thecurrent one, a problem which is not the true one And, furthermore, that the necessity

represen-of such design moves Fuzzy Logic’s praxis towards an Engineering Art grounded

on Science; something probably signaling the direction that the computerizing ofCommonsense Reasoning will have to follow in the near future Dealing with thenatural phenomena of Language and Reasoning—to survive, and with the words ofHans Christian Oersted in the 19th Century-Fuzzy Logic should place itself in an

“anticipatory consonance with Nature”

All that suggested the simple but not simplistic view of what follows

References

1 E Trillas, On the Logos (Springer, 2017)

2 E Trillas, Narrar, conjeturar y computar: El pensamiento (In Spanish) (EUG, 2020)

3 E Trillas, Reflexiones sobre la génesis de la Lógica (EUSC, 2021)

4 E Trillas, Reconsiderando el concepto de conjunto borroso (In Spanish) (EUL, 2020)

5 E Trillas, S Guadarrama, Fuzzy representations need a careful design Int Jour of General

Systems 39(3), 329–346 (2010)

A Formal Skeleton of Commonsense

Reasoning

3.1 After referring several times to Commonsense or Ordinary Reasoning, let’s

devote a few pages to present a (minimal) mathematical model of it that can beseen as the ‘Skeleton’ of Reasoning, since it is defined by a set of few, simple lawsappearing in the models of particular and specialized modes of reasoning like, forinstance: Boolean Algebras for the reasoning with precise concepts; Orto-modularlattices for the reasoning with the concepts of Quantum Physics; and also in the socalled Algebras of Fuzzy Sets for the reasoning with imprecise concepts, and amongthem De Morgan-Kleene algebras All these models have interesting applications[1,2]

Such skeleton is supposed to support a set of linguistic statements p, q, u, v, etc.,related between them by the conditional statement ‘If p, then q’, symbolized by

p < q Once the statements p, q, and p < q, are given or presumed, the antecedent p

is called the premise, and the consequent q the conclusion Reasoning, or inference,always starts from a given premise and is done by means of the structure < endowswith the set of statements

Reasoning can be seen as a specialization of the Natural Phenomenon of

‘Thinking’, and directed to attain a goal q from the previous or initial information

p Both the initial information and the goal, or conclusion, are supposed expressed

by linguistic statements Usually, the conclusion is not known, it should be searchedfor, and it is not necessarily unique Each of the different ways that can be followedfor going from p to q are what will be called a ‘process of reasoning’ Notice howdifferent is the situation with that of studying a theorem in a mathematics textbook

in which there is not only explicit the premise p and the conclusion q, but also allthe steps conducting deductively from p to q, the proof of q from p Usually, only p

is known, and one should find both q and a way to reach it from p, to prove q

3.2 It is obvious that, given two linguistic statements whatsoever u and v, and for

what concerns its relation with < , there can only exist one of the following fourpossibilities:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

E Trillas et al., Reasoning and Language at Work, Studies in Computational

Intelligence 991, https://doi.org/10.1007/978-3-030-86088-2_3

17

(1) u < v, but not v < u; (2) v < u, but not u < v; (3) u < v and v < u, written

u∼ v ⇔ v ∼ u; (4) Neither u < v, nor v < u, u < / v and v < /u, written u ♦ v ⇔ v ♦ u

In (1) it is said that conclusion v is a consequence of u; in (2) that conclusion v

is a hypothesis for u; in (3) that u and v are inferentially equivalent; in (4) that u and

v are inferentially orthogonal, or non-comparable The main goal of reasoning, orinference, is to reach either a consequence, or a hypothesis from a given premise.The processes that can be followed for reaching such an end are discussed next.The structure with which < endows the set of statements is supposed to verify thefollowing six axioms, or universal laws:

(1) For all p is p < p, that is, relation < is reflexive;

(2) Denoting by u’ the linguistic negation of u, If p < q then q’ < p’, shortly written(p < q) < (q’ < p’), axiom or law assuring that the negation reverses < ;(3) No statement verifying p < p’ will be accepted as a premise

(4) p and p < q facilitate q, shortly written(p · (p <q)) < q, axiom or law called

Modus Ponens;

(5) p· q < q and p · q < q, axioms called of monotony of the conjunction ‘and’,

symbolized by the dot (·);

(6) p < p+ q and q < p + q, axioms of monotony of the disjunction ‘or’, symbolized

by the cross (+)

It should be noticed that six axioms is a very small number of universal laws, andthat any other law that will be need to consider will be just a ‘local’ law, that is, onlyverified by some subset of statements This is the case, for instance, of the transitivelaw ((p < q) · (q < r)) < (p < r), that can’t be universally presumed for all triplet (p,

q, r) of statements in language and, consequently, in ordinary reasoning

Notice that if < is universally transitive, then∼ is also so and, since ∼ is alreadyreflexive and symmetric (p ∼ p, p ∼ q ⇔ q ∼ p, respectively), it results that ∼

is an algebraic equivalence whose classes, [p] = {q; p ∼ q}, can be seen as the

‘propositions’ considered by Logic Nevertheless, ordinary reasoning mainly dealswith statements and not with propositions because transitivity usually only holdslocally in some triplets of statements

The local character of a law is not seen here as a ‘logical failure’ of CommonsenseReasoning, but rather as a matter of fact It is considered that the model should beadapted to the underlying reality, but not the reality to the model What is mathemat-ically regimented or modeled is Reasoning, and Logic is seen here as the collection

of all models for the diverse types of Reasoning

3.3 Given p, it is said that r is a refutation of p whenever p < r’, that is, when the

negation r’ of r is a consequence of p In such case, and provided < is locally transitivewhere it were suitable, is p♦ r In fact, if it were p < r, from axiom 3, is r’ < p’,and provided transitivity holds in the triplet (p, r’, p’) it will follow p < p’ that isforbidden by axiom 3 Were it r < p and provided < is transitive in the triplet (r, p, r’),

it will follow r < r’ that shows r can’t be taken as the premise for p; hence it should

be p♦ r Under transitivity, at least local, refutations are orthogonal to the premise

It is analogously said that u is a conjecture from p, whenever u does not refute p,that is, if p < /u’.

Obviously, given p and u, the second only can be either a refutation of p, or aconjecture from p Given a premise p (such that p < / p’), and independently of theformer six axioms, another statement whatsoever u only can be a refutation of p or aconjecture from p; there are no more possibilities and from this point of view it can

be reasoning identified with conjecturing and refuting Notice that laws 1 and 3 statethat a premise should be a consequence of itself, and also a conjecture of itself.Under transitivity, any consequence is a conjecture In fact, if p < r and it were p

< r’, since the first implies r’ < p’, transitivity in the triplet (p, r’, p’) will imply theabsurd p < p’ Hence, if it is p < q it should be p < /q’ Under transitivity, at leastlocal and where suitable, reaching conjectures or conjecturing is more general thanreaching consequences or deducing

Analogously, if h is a hypothesis for p, h < p, and h is not self-refuting, h < / h’, h

is a conjecture from p In fact, were it p < h’, if < is transitive in the triplet (h, p, h’)

ot will follow h < h’ Hence it should be p < /h’ Under transitivity, at least local andwhere suitable, reaching hypotheses or abducing is a particular case of conjecturing.Notice that with transitivity and in the case p < q of a consequence, it is notpossible to have q < q’, since p < q and local transitivity implies p < q’ when

it is p < /q’: Consequences can’t be self-refuting like hypotheses can actually be.Even more, no two consequences can be contradictory: If p < q, p < q*, and it wereq* < q’, since it is q’ < p’, transitivity will imply p < p’ in two steps Under transitivitythere are not contradictory consequences like it is a common experience that thereare contradictory hypotheses

3.4 Thus, without at least local transitivity, it neither can be presumed that

conse-quences and non self-contradictory hypotheses are conjectures, nor that refutationsare orthogonal to the premise, nor that two consequences are not contradictory Itcan be the surprising case of not counting with the possibility of conjecturing thatsome statement can be either a consequence or a hypothesis

Hence, the reasoning conducted without transitivity is very weak, do not allow toomuch safe conclusions, and we will say it is ‘not regular’ As such, regular ordinaryreasoning is what follows the six former laws plus transitivity, at least locally andwhere it can correspond

Let us give proof that regular ordinary reasoning is ‘good’ reasoning, i.e that

it verifies the two old Aristotelian ‘Principles’ of Non Contradiction and ExcludedMiddle These Two principles follow as a consequence of the seven formal laws of

‘regular reasoning’: In the regular skeleton they abandon the character of ‘principles’Aristotle chose for them, and assume that of ‘theorems’

Theorem of Non Contradiction Under at least local transitivity, for any statement

p, it is p · p’ < (p · p’)’ That is, p · p’ is always a self-contradictory statement and itcan’t be taken as a premise

Proof From p · p’ < p follows p’ < (p · p’)’ From p · p’ < p’, and transitivity inthe triplet (p · p’, p’, (p · p’)’) implies p · p’ < (p · p’)’•

Theorem of Excluded Middle Under at least local transitivity, for any statement p,

it is (p+ p’)’ < ((p + p’)’)’ That is (p + p’)’ is always a self-contradictory statementand it can’t be taken as a premise

Proof From p < p+ p’ follows (p + p’)’ < p’ From p’ < p + p’, transitivity inthe triplet ((p+ p’)’, p’, p + p’) implies (p + p’)’ < p + p’ that, at its turn, implies(p+ p’)’ < ((p + p’)’)’•

These two theorems actually reveal that the seven laws of regular ordinaryreasoning reflect pretty well the best kind of ordinary reasoning, the one with whichpeople tries to wisely choose their daily actions In fact, the theorem of Non Contra-diction allows to avoid considering any process of reasoning that starts from a premiselike p · p’, a kind of statement that really quells any reasoning concerning p Forinstance, if I am in a shop with the intention of buying something, ‘I bought it and Ididn’t bought it’ is a paralyzing information; it is for nothing

Any self-contradictory statement is, as starting information, a paralyzing one;when the aim is acting, self-contradictory statements are for nothing

Hence, the Skeleton is not related with those logics that work with contradictions, like the Paraconsistent[3], or some of the so-called MaterialisticLogics appearing in some models of the Marxian thought [4] and confusing the nega-tion q’ with an antonym qa, with which it is presumed that q is a non-regular statementwith a single antonym qataken inferentially equivalent with q’ It is in this sense thatthe theorem of Non-Contradiction shows a margin for reasoning and, consequently,for acting Such theorem actually shows that the ‘Regular Skeleton’, with the sevenaxioms, is a good enough model for the best possible Ordinary Reasoning

self-Notice that p+ p’ is not paralyzing but, in the shop’s example, ‘Either I buy this

or I don’t’, simply shows something sure and, in this sense, it is also for nothing Inaddition its negation (p+ p’)’ is self-contradictory and, thus, inferentially impossible

3.5 Some remarks on the former two theorems are suitable.

In Ortho-lattices and consequently in Orto-modular lattices and in BooleanAlgebras, it is:

u < u’= > u · u = u < u’ · u = 0, that is, u = 0: Its only self-contradictory element

is the lattice’s minimum 0, the ‘nothing’ Analogously, it is:

u’ < u= > u’ · u’ = u’ < u · u’ = 0 = > u ‘ = 0 = > u = 1, the only elementsuch that u’ < u is the lattice’s maximum 1

Hence, the former theorems are reduced, thanks to the additional laws holding inOrtho-lattices, to p · p’= 0, and to p + p’ = 1

Even more, because in Ortho-lattices and also in De Morgan Algebras the laws

of duality hold, it is

(p + p’)’ = p’ · (p’)’ = p’ · p = p · p’,

and, analogously,

(p · p’)’ = p’ + (p’)’ = p’ + p = p + p’,