FUZZY MEASURES OF TRUTH
This short article is about the notion of quantifying truth, if useful to measure an intermediate truth like “x/100%”.
The numerical truth-value in fuzzy logic (a many-valued logic) is the measure of a co-essence or of an undetermined mixture of a value (false, 0) with its opposite (true, 1). It is a complex score of a proposition.
This co-essence is named fuzziness, and it is measured as co-distance of a certain mixed-score, called FIT (the Fuzzy Information unit), from two opposite BIT (1 and 0). A FIT 0.3 is a truth value implying the co-essence of the score 0.3 of trueness with 0.7 of falsehood.
The fuzziness is a sort of numerical evaluation of BITS, namely of those couple (or n-ple) of basic numbers 1 and 0 of Boolean logic (where BITS imply the classical value “true” and “false” of ordinary classical logic).
Fuzziness is a vagueness or indetermination beetwen 1 and 0, which is measured.
Now, the fuzziness, as co-distance of a FIT from two opposite bits can be also displayed as the internal part of a “Boolean Hyper-cube”.
The Boolean hypercube is the three-dimensional version (improperly called “hypercube”, because it is expandable to a multidimensional one) of the normal Boolean square (which contains the table of Aristotelian truth-value, normally opposite each other, as the vertexes 01, 00, 11, 10). Imagine a square with 4 vertexes (01, 00, 11, 10), they are BITS and are all opposite.
Now imagine that a Boolean cube will have the triples 000, 001, 011, etc. as opposite vertexes (if you are familiar with the magic square of opposite and contrary predicates in Aristotle, you can have a picture of Boolean square).
You can now conjecture the Boolean Hypercube with vertexes 0000, 0001, 0011, 0111, ecc.
What does it mean to say that something is .5 true? This is a point not located on of the vertex of the hypercube, but at an internal point of the volume of the Hypercube, at a “certain distance” from more than one vertexes.
There is no statistical or predictive value of trueness, here, but the evaluation of trueness somehow “static” (there is no projection towards the future as in the case of probabilistic predictions.).
A truth-value of .3576 will be “statically”, but “vaguely” and “in a certain, undetermined measure”, lower than .9999.
Those values re-introduce through the expression “in a certain measure” semantics into formal logic. Just because FITS are, if you want, a Tarskian numerical evaluation of other numbers (the BITS).
Those measures are dimensions of meaning and context.
Finally, the same degree of fuzziness of a FIT can be measured (a sort of metalinguistic semantic of semantic), as the degree of vagueness of a set.
The points near to the vertexes or equal to them (.9999 or 1) are less fuzzy than those in the core of the Hypercube: for example .5666, which is fuzzy more than .9999.
The core of the hypercube (0.5) is completely fuzzy (vague, indetermined).
For an introduction to Boolean Hypercube see:
METAPHORS AND THEIR FUZZY LOGIC MODEL
The metaphors are part of quotidian language and I don’t think that fuzzy logic can be a generalization of all MVL logics, maybe there is some paraconsistent language more general than a MVL logic (I suspect it would be something very similar to our quotidian language).
Anyway, the use of numerical values does not restrict the application of fuzzy logic to dimensions of semantics and context which can be straightforwardly quantified, such as temperature, distance, and height, – just because fuzzy logic was expressly introduced, since 1965, to re-connect words (and its metaphorical use) to numbers.
I attach here for you a precious source, the first article of Lofti Zadeh originating all the western math on this important philosophical tool, which all consequences in favor of induction, of solution of strange dichotomies (like i.e. mind/body) are not completely developed. See:https://core.ac.uk/download/pdf/82810078.pdf
We surely know the common sense about “out of sight, out of mind”. It recalls us the first movement of science, outside the limits of our closest experience (the experience of what we have under our noses, literally). It is the same appeal of rhetoric to a local audience to spread the wings of our fantasy toward a wider world, which will be described, rather than from cosmological maps (as in science), from metaphors (an in literature and in the tribunals).
What is a metaphor? We take the experience of single things in front of our nose and give it a new meaning, “allusive” and conjectural of things more distant from our normal experience (far away in space, in time or distant from our sensibility of Westerners, perhaps of men if we refer to sensibility of animals or extra-terrestrials). More banally, these distant things are often, simply a map of dispersed tokens, so that every general term (a universal term or a concept) is both a metaphor and a theory.
The internal dialectic of a single metaphor between examples and universals (or distant things) is analogy. The analogy is paraconsistent, because it relates, in a way possible or dispositional, what is present and true to what is universally possible and likely. And its aim is to persuade us (inside the field of rhetoric), about reality of what is distant, or make us learn a theory, about a distant universe (inside the field of science), in both cases through that conjecture that is the metaphor.
The relationship between the actual current of the example and the verisimilar of the universal (plus all the intermediate rhetorical figures between example and metaphor) is a dispositional and fuzzy relationship. Analogy is often (not always) fuzzy. Not always, because we have i.e. the proportion (a:b=b:c) as another form of analogy or the Tomistic analogy of Creatures to God (the transcendent term is infinitely not-quasi similar).
Here an explanation of fuzzy nature of analogy and of metaphors, especially inside literature, law, rhetoric and science:
– The use of numerical values does not restrict the application of fuzzy logic to dimensions of semantics and context which can be straightforwardly quantified, such as temperature, distance, and height., – just because fuzzy logic was expressly introduced, since 1965, to re-connect words (and its metaphorical use) to numbers.
– A set of numbers can be evaluated by FITS, but every single FIT is a word (a general term, a universal, a concept, a theory, etc.). For example, a set of different body statures can be evaluated by an Arbitrary Membership Curve (AMC) called “tall”: on the abscissa of a Cartesian graph we have the set of statures, on the ordinates in the same Cartesian plane we have the fuzzy evaluation of membership from 0 to 1.
Now, for example who is 1.54 meters tall is, according to such AMC, .2 tall and, at the same time, .8 short (there is a fuzzy borderline between the two neighboring words & AMC “tall” and “short”).
– The adverb “very” and “a few” can modify the inclination and the base (with respect to the abscissas of the Cartesian plane) of these two fuzzy-neighboring AMC-words “tall” and “short”. The AMC very “tall” will have a stricter base on abscissas and a strong inclination (who is 1.80 meters tall, will be .5 “very tall” and .5 “short”, while he could be .8 tall and .2 short).
Every adjective, verb, new connected proposition to those AMC-words will vary the sense, the meaning and the “proximity” of the description of such statures, enabling us to depict a real man, present in front of our eyes. The complex linguistic statements like “he is humorous man, very tall, coming from San Francisco and allergic to strawberries” is a combination of many vague FITS-word-metaphors-AMC, and the focus and the intersection of such various words and the related AMC will help us, applying all the correct properties (AMC), to visualize such man in front of our mental eyes.
– The different fuzzy FITS, related to the various AMCs which in complex way, describe such man “here and now” (absolutely not “out of sight, out of mind”) can be summed and crossed (as a well ordered group or system of fuzzy sets). So that you calculate, for instance, if the expression “he is humorous man, very tall, coming from San Francisco and allergic to strawberries” is true to a specifiable degree.
Surely we are merely scratching the surface of semantics, but if from such man, who is an organism, you want to infer something about “a community is like an organism”, which is a metaphor starting from this man “here and now”, – you can do it.
– The metaphor will be the procedure of taking the linguistic descriptions of many men (the systems of many fuzzy sets) on a hypothetical abscissa and evaluating it with the AMC “community”, stating which men more or less represent a community, then adding many other adjective, verb and new connected propositions, then conjecturing the analogy of such community with an organism.
The conjectured analogy will be a logic relation of fuzzy possibility, which has surely a certain fuzzy AMCP (Arbitrary Membership Curve of Possibility): for example, such metaphor will be vaguely .2 “possibly” true and .8 possibly false.
– “Possibly” is an adverb which modify the range (inclination and base) of the resultant general (alethic) curve of the entire metaphor.What I have outlined here is called in the scientific literature “computing with words” or “soft computing”.
See this original article: