Fuzzy Theory of Truth

What is a problem for me, maybe for all of us, is how propositions of science become scientific and accepted, given that data and facts are so different, as a genre, from a scientific proposition, so that is such a kind of problem how any facts become relevant for science and how pre-scientific theories become science.

So I propose that a variant-theory of truth, a fuzzy version, could be a possible reply to such problem. Please, read the following short outline.

Suppose a variant to Tarski semantic where a Tarski meta-language shows syntactically “equal to truth” the formal (syntactical) correspondence between an object-language “of facts” and a theoretical language “on facts”. This logical scheme for every empirical control (useful both for dialectic-argumentative refutation and for physical refutation), transforms a semantic correspondence (or the “spiritual” concept of truth) into an unsuspected syntactical comparison of symbols, which could be applied also to scientific theories and to their competition testing.

The correspondence will be, in the variant, a different syntax, that of a vague similarity (fuzzy similarity or fuzzy entropy) between (i) syntagms of variables and terms (symbols of propositions) of facts and (ii) syntagms of variables/terms/proposition of theories incorporated in two compared languages: this is a variant of Popper construction of Tarski semantic theory of truth, because Popper (and Tarski) comparison was a syntax of discrete and percentile comparison, something like a compared measure of percentage of empirical content of object-language and logical content of theory-language (in example: (a) “the snow is white” is partially true if, after an appropriate measure of “percentage of empirical content”, happens of necessity that (b) “the snow is white, but full of stings of grey humus”), while the comparison, in the fuzzy scheme, could be a measure of vague similarity between symbolized facts and theories.

In a simplified scheme, Tarski original scheme is

[50 % T] →L(a) = E (b)

which means: The logical content of (a) is correspondent at 50% (percentage) to empirical content of (b), which is to say that (a) is True at 50%. This syntax implies, directly, the reciprocal refutation [- 50% T] → L(a) = E (b) … Namely, (a) is absolutely false in 50% measure (a non vague measure)!

In the variant scheme the refutation is a little bit complex, but explains better some very common situations of everyday life, like political competitions, religious fights and sport match, and is, maybe, more adequate (even if is not more precise) to empirical contents:

[60% T] (G 20% AND W100 %) → L (a) ≈ E (b).

It means: The fact that the snow is white and full of grey stings is “vaguely” correspondent (similar or “vaguely similar”, see the symbol “ ≈ ”) to a theoretical proposition where the snow is vaguely and truly grey at 20% (if the grey is 20% sufficiently similar to white) and “absolutely” (digitally) white at 100% (independently if the snow is effectively a fact of snow covered by several stings of grey humus, given that we simply suppose the schematic theory that “the snow is white”), so that the proposition (a) is globally and “vaguely” correspondent ( “ ≈ ”) at 60% to empirical content of (b), which is to say that (a) is “vaguely” or “analogically” true at 60%, given that a rough calculation of “vague” similarity grade within grey and white colours could be [ G 20% + W 100% / 2 = 60% ]. NOTE: don’t worry for my arbitrary way of calculating pondered similarity vague grade: is arbitrary this choice to calculate fuzzy entropy and can be improved by mathematicians more expert than me.

Guess following examples of two opposite parties and imagine to shall measure the grade of their opposition.

Let’s say that atheist and theists are opposite in grade of 100% (like for example the concept of existent and inexistent).

Then consider the case of two football teams of the same city (like MILAN and INTER in Italy): they will be opposite in a lower measure: let’s say in a grade comprised between 80% and 60% each other.

Take, then, the example of two different partisan group of same Italian Communist party: they should be contrary (in a political sense) in a grade comprised between 20% and 30%.

Imagine, finally, Mr. Popper, the well-known philosopher, as doubting of his epistemology: Will differ from him in a lower measure?

These examples show, in my opinion, the simpler theory about fuzzy entropy, which concept replies to the dilemmatic philosophical query (dramatically producing great confusion since the times of Hegel): how much two opposite things are similar? Can opposites be similar?

Fuzzy entropy illustrates very clearly a logical theory answering to the problematic inquiry about the evanescent reality of similarity, best called as analogy, within different or opposite objects (no matter if they are tokens, arguments, minds).

Further, the original Tarski-Popper scheme is a logical theory of truth, which does not explain how both logical propositions and empirical propositions of facts penetrate into the meta-language of Tarski semantic.

This is the problem of the sources of knowledge, which Popper solved denying validity to induction and to privileged sources (as induction, verification, evidence, math, religion and other “authorities”), focusing his epistemology on the main relevance of empirical and logical control of theories. Popper would say that fact and theories are simply conjectures to be tested.

The fuzzy scheme of logical theory of truth has these extra benefits (further to the main result of more adequate schematization of empirical control of truth as correspondence):

1. It rehabilitates “induction”, given that a great collection of tokens can, in fuzzy logic, justify, in a logical sense, the abstraction of an inductive general law, which becomes the set of “propositions on fact” of a pre-scientific theory or the group of the analytical enunciations of particular facts (both incorporated into the “semantic” of empirical controls of variant-Tarski-meta-language). Here the explanation: there is a fuzzy theorem, about “vague similarity” (a general fuzzy entropy scheme of opposition grade) between the PART and the WHOLE, which says that “the PART vaguely contains the WHOLE, in certain measure (the measure of the part)” [P.V.C.W.]. This theorem applied to a great collection of tokens justifies the induction of pre-scientific general propositions (i.e. physical laws to be tested) and the analytical induction of general data from huge collections of particular data;

2. The fuzzy scheme of logical theory of truth and the theorem P.V.C.W. also explains how facts and proposition are creatively found and included into Tarski-Popper empirical controls (into their meta-linguistic semantics): similarity between facts and theories (between math and brute facts) allows the theoretician to discover new propositions (it is the art of scientific discovery, construed as such a kind of art of Topics), discovering and correcting propositions in the light of facts, and viceversa, in a continue internal feedback (dialectic, rhetoric and not simply based on laboratories testing).

2. The theorem P.V.C.W. and the concept of fuzzy entropy rehabilitate topics, dialectics and rhetorics as instruments of invention and comparison of facts and theories. Popper himself said that also non-empirical (or even metaphysical) arguments can control internal consistency of theories and eventual contradictions, which arguments should be the aware or not aware background of non empirical theories. Here the explanation: the mentioned theorem, about “vague similarity” between the PART and the WHOLE, suggests that science is a “PARTial” linguistic praxis of the WHOLE of entire human linguistic praxis, which includes also Religion, History, Economy, Art, and so on…Overall, finding new propositions on facts and new general analysis of particular data is not simply based on induction or on fallible Popper conjectures (of theories and facts based on previous innate knowledge), but also on an argumentative feedback, which is based on primary analogy between facts and theories.

3. Finally, the fuzzy scheme of logical theory of truth offers a wider range of empirical controls of truth of theories, given that this scheme of semantic allows type of “empirical refutation” different from “absolute contradiction” of Aristotelian Principle of non Contradiction (P.N.C.). While P.N.C. (with its companions: Principle of Identity and Principle of Tertium Non Datur) allows to make a choice between two incompatible theories, when one of these presents an internal or external inconsistency (contradiction with its premises or with is empirical conclusions)…what about when the theories are similar at any percentage? And when the internal or external contradiction is not 100%? When opposition to facts or to premises is not 100%?…Well, the refutation should be at any percentage!

4. The fuzzy scheme of logical theory of truth explains how people can understand each other, using a natural language (maybe also an artificial one like math or formalized logic), because if the propositions of a talker were not “partially similar” to counter-arguments of interlocutor (and partially different) there would be not that certain halo of ambiguity, where lies the basic possibility and basic originality of discussions. While using terms and propositions completely incompatibles would not prevent us from the deadlock of an absolute refutation (when any terms are 100% opposite). This happens, I think, also in definitions of symbols of new fields of math (I am not referring to very standardized matters like the derivatives calculus), where many mathematicians use variants symbolization and, anyway, they can talk reciprocally, each other.

I would call number 4) benefit as the language vagueness which appears to be the fundamental source of Philosophy as infinite theorization of new logical/empirical controls.


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