After an illustrious career collaborating with universities and research centres, Enric Trillas remains set on working towards a new experimental science, managing the concepts and tools of computer science, and actually interacting with other disciplines on the way.

Trillas sheds light on his recently translated *The Genesis of Logic *to explore the theoretical promise and real-world applications of fuzzy logic.

Explore Fuzzy Logic and the FMsquare Foundation: researchoutreach.org/fmsquare-foundation-forging-fuzzy-future

Read more on *The Genesis of Logic** *and the *Fuzzy Management Methods* series

Image Credit: Adobe Stock / Badis

Transcript:

Hello and welcome to ResearchPod. Thank you for listening and joining us today.

In this episode, we look at the work of Professor Enric Trillas who, throughout his illustrious career, focused on the mathematics of fuzziness – fuzzy set theory, fuzzy logic, meaning and the formal study of Commonsense Reasoning. Trillas has worked with universities and research centres across Europe, Asia, North and South America, and continues to collaborate with the FMsquare Foundation in their pursuit of fuzzy solutions. In today’s episode, we examine his latest book; ‘The Genesis of Logic’, which was translated by Edy Portmann, available now from Springer Publishers.

After over 50 years of devotion to the mathematical study of Lofti Zadeh’s *Fuzzy Logic,* Trillas’* *The Genesis of Logic looks at a way of dealing with imprecise concepts and words that permeate Natural Language and Commonsense Reasoning in mathematics.

Among the reflections in the book is the identification of a *fuzzy set *with just one of its *membership functions – *as Zadeh and almost all his followers did. This set is, at least theoretically, a relevant drawback since a fuzzy set reflects the ‘extensional’ meaning of its linguistic label, and not a single membership function.

A ‘fuzzy set’s’ membership functions means that ‘a fuzzy set is several fuzzy sets’ – If that sounds confusing, you’re not alone. This confusion between Zadeh’s idea and definition can be resolved when thinking of labels one can attach to the language within the set; a planar magnitude label from which the fuzzy set comes, and an scalar magnitude built up on its base.

That is, theoretically differentiating between the fuzzy set as a meaning reflected by a graph and its membership functions; on a single *basic* magnitude many *scalar* magnitudes can be constructed. To be more precise, those scalar magnitudes can be *designed* according to the available information on how the linguistic label – nothing other than a word – is used in discourse, borrowing here from Wittgenstein and his concept of playing ‘language games’

What’s more, just the fact that a membership function should be designed each time the fuzzy set would be managed, marks a crucial difference between *Fuzzy Logic *and *Logic; *If classical membership is reflected by a single and unique *Characteristic Function *of the set, ‘fuzzy membership’ shows a multiplicity of possibilities for ‘being in’ a fuzzy set, in a flexible ‘fuzzy’ way.

It should also be taken into account that if the classical set’s idea comes from collections of objects, let’s say ‘from boxes with items’, the idea of a fuzzy set is generated by the human trend of assigning extension to imprecise words; if the first is physically generated, the second is mentally generated. Notice that imprecise words allow us to express a lot of information with a bit of words; it is actually difficult to maintain an interesting conversation – with oneself or with other people – without using imprecise words and phrases.

The word ‘design’ shows that Zadeh’s Fuzzy Logic is closer to an Empirical Science, or to Engineering, than to Logic per se, and Fuzzy Logic, Natural Language and Commonsense Reasoning are distinctly interwoven; Science and Technology mainly deal with what is measurable, numerical. It is argued that a new natural science, that is both experimental and theoretical, dealing with language and reasoning of which the mathematical analysis of a complex phrase by a novelist, and the use of ‘and’ in the web are but examples.[t1]

This, and Trillas’ studies on the validity of Logic’s Laws in Fuzzy Logic, was the seed from which this book grew. Some of its roots can be found in former books where recognising Reasoning as a Rule Based activity is held to be a regulated and thought-directed undertaking for the human brain.

If thought can be seen as a brain activity frequently full of images, free of rules and out of language, reasoning is a specialization of thought for answering questions, made under some rules and in close relation with language. Possibly it is acquired when the child learns their first language, and is perfected with education.

The Genesis of Logic concerns taking into account, or leaving out of, logical laws formally modelling specialized types of reasoning, and evolving from a mathematical structure with very few axioms valid in all reasoning, to structures with more axioms and only locally valid for particular reasonings.

Let’s notice that the book’s subtitle, ‘Naïve Reflections’, not only refers to the guise of informality exhibited in presenting personal reflections in the text, but also that it’s written, like books on Classical Sets, by demonstrating ‘Naïve Set Theory’ without necessarily using high-level mathematical knowledge.

* *

The short number of axioms that constitute the so called ‘Skeleton of Reasoning’ was selected by observing the minimal number of hypotheses needed for proving as theorems the old Aristotelian Principles of Non-Contradiction and Excluded Middle. In the Setting of the algebras of fuzzy sets, the author introduced in one case lattices. A lattice structure is normally too rigid for modelling as with the great flexibility shown by Natural Language and Commonsense Reasoning.

Trillas’ set of axioms gives a mathematical structure that escapes from that of lattices. He called it the Skeleton of Reasoning since for each type of specialized reasoning the ‘flesh’ of some more axioms needs to be added. Actually, this Skeleton appears among the axioms presumed in almost all logics excluding self-contradiction and, in particular, in all the algebras of fuzzy sets.

The proof of Non-Contradiction and Excluded Middle by only supposing the Skeleton and Local Transitivity, can conduct to say that Trillas’ Skeleton contains true principles of reasoning. It is to be remembered that Aristotle stated that the principles are not susceptible to be proven, and that both were considered, from very old times, as ‘the principles of Logic’.

Once defined refutation and conjecturing in the setting of the Skeleton, Trillas presents several types of transitivity, and proves that the search for consequences (also called deduction), as well as the search for hypotheses (abduction), are, under local transitivity, nothing else than Conjecturing ‘guess work.’ In addition, according to Trillas it can be presumed that some kind of transitivity should be added to the Skeleton for actually considering it as the full ‘Principle of Reasoning’. It’s his theoretical basement for a Genesis of Logic.

Defining refutation and conjecturing as opposite concepts, according to Trillas, the classification of conjectures takes into account the comparability of the conclusion with the premise (by means of the inference relation). This allows to advance a step up to clarify old questions between deduction and the ‘mysterious’ induction.

If deduction can be seen as forwards inference, and abduction as backwards inference, then there should still be ‘incomparable inference’, that is, the case when the conclusion and the premise are not linked by the inference relation. In this case, in which the conclusion can be seen as inferentially wild, the inference or the reasoning, is according to Trillas just speculating. The conclusion is nothing more than a speculation. Trillas published this recognition, that speculations were firstly considered, already in a paper in the journal ‘Artificial Intelligence’ in 1999.

At its turn, speculations can be classified in two classes by considering that being conjectures are not refutations, and thus its negation is not a consequence of the premise. To Trillas, it only can be either a hypothesis of it, or not comparable with it. In the first case, speculations are called ‘weak’ and in the second ‘strong’ or ‘wild’.

Speculations are the conclusions of inductive reasoning, and, because of a kind of zigzag-reasoning, appear as a kind of Brownian inferential chaining backwards to forwards or from forwards to backwards, ending in a statement not inferentially comparable with the premise. This zigzag-reasoning allows Trillas to see that ‘imagination’ appears with reasoning, whereas ‘cleverness’ comes from the practice of reasoning in specialized environments. But to Trillas both, imagination and cleverness are both typical characteristics of good reasoners.

His Skeleton also allows to analyze which reasonings are monotonic, that is, if when the information grows, when new statements are added as premises, there are no less conclusions than before: The growing of knowledge does thereby not eliminate what was previously concluded. Results from this is that deduction and refutation are monotonic, adduction and conjecturing are anti-monotonic (e.g., some conclusions can disappear). Furthermore, speculating is just non-monotonic, as there is no law asserting that it is either monotonic or anti-monotonic. On that account, speculation appears as the kingdom of induction and non-monotonic reasoning.

All that is, in the book, particularized to the extremely rigid case of Boolean Algebras, to the less rigid case of Ortho-modular lattices, and to the flexible case of Fuzzy Algebras. Accoring to Trillas that is added to the usual mathematical reasoning like the reasoning typical of Quantum Physics, done with imprecise concepts, modeled by Fuzzy Logic.

Let’s notice that in the case of classical sets or, equivalently, so-called Boolean algebras, there is only a self-contradictory element, namely the empty set (i.e., the null element, in a Boolean Algebra). Since contradiction means empty intersection (null conjunction, in a Boolean Algebra), according to trillas, a refutation is a set non intersecting the premise and a conjecture is a set with no empty intersection with it, and a hypothesis is a subset of the premise; and a consequence is a set containing the premise as a subset. Speculations are those sets that are neither contained, nor contain the premise.

All these situations actually exist in some pieces of Language and Reasoning. They are ‘local’ properties corresponding to particular types of reasoning like, for instance, the distributive laws of ‘and’ in relation to ‘or’, and of ‘or’ in relation to ‘and’ in Boolean Algebras. Only Trillas’ Skeleton’s axioms are universal, laws holding everywhere in Reasoning and Language.

As Enric Trillas remarks, his work should be seen as working “towards a new experimental science, managing the concepts and tools of computer science, and actually interacting with other disciplines.” To him, it seems extremely interesting for science and mathematics, to design evermore “experiments within natural language related with the meaning’s representation of larger and more complex linguistic expressions than those currently considered in the applications of fuzzy logic.”

That’s all for this episode, thanks for listening. Links to Enric Trillas research and more on fuzzy logic, as well as the FMsquare Foundation, can be found in the shownotes for this episode. And, as always, stay subscribed to ResearchPod for more of the latest science.

See you again soon.

## Leave a Reply