viernes, 21 de marzo de 2014

Lógica difusa, tercero excluido, y no contradicción

Respuesta que le daba a la siguiente pregunta de Vishal Yadav en Researchgate:

How important is the Law of excluded middle (LEM) and Law of contradiction (LOC) in fuzzy logic?

The non-availability of these two laws has profound implication for fuzzy logic and serves to distinguish it from crisp set theory. That's mentioned in a book but I want to know how it works.

It is convenient to start by giving some definitions of `borderline' (bear with me for a few moments):

- being in an intermediate position or state (Merriam-Webster Online)
- having some but not all characteristics of something (MWO)
- not quite meeting accepted, expected, or average standards (
- an indefinite area intermediate between two qualities or conditions (The Free Dictionary)

Fuzzy logic is a logic (indeed a family of logics) for borderline cases. If you have a population or universal set, and a property P, you could roughly classify those individuals or elements in three types: those who have property P, those who have property not-P, and those who are borderline.

If you believe that borderline cases are actually P or not-P, only you cannot tell which (but you could if you knew more) then you shouldn't use fuzzy logic. You can use other tools which are fit for that situation, like subjective probability theory.

If you believe that forcing borderline cases into the P and not-P categories might lead you to incorrect conclusions, then you can use some multivalued logic (of which fuzzy logics are a special case).

Then, if you believe it suitable to give a numerical assessment of how close a borderline case is from P in a bipolar scale with extremes `being not-P' and `being P', fuzzy logic is your thing.

If there are borderline cases of P which cannot be reduced by gathering more knowledge, then the LEM and LOC can easily fail. For instance, it needs not be true that every element is either P or not-P: it is either P or not-P or borderline.

In fuzzy logic terms, if an element is borderline then it only has a partial membership in P and not-P. Being fully P and being fully not-P do not exhaust the possibilities, so it is natural that the element can also be a borderline case without full membership in (P or not-P). For that to be possible, the LEM must not be imposed as a law.

That means that, by renouncing the LEM, you are able to model situations which are not covered by bivalued logic.

Similarly, since a borderline element has partial membership in both P and non-P, it is reasonable that it can still be borderline for (P and not-P). What cannot happen is that it had full membership in property (P and not-P).

A logic for borderline cases may not want to use the LNC. [If one regards intuitionistic logic as a `logic for borderline cases', then it still uses the LNC.]

Thus fuzzy logic does not contradict in any way bivariate logic. The LNC in bivariate logic is equivalent to saying "(P and not-P) cannot have truth value 1". It is still true in fuzzy logic that "(P and not-P) cannot have truth value 1". It is not true though that "(P and not-P) must have truth value 0", because those two statements of the LNC can only be equivalent if one previously establishes that the only truth values allowed are 0 and 1, thereby eliminating borderline cases.

Fuzzy logic just makes less assumptions than bivariate logic about what deductions are valid, because it is meant to apply to objects for which some deductions, familiar and very dear in bivariate logic, are simply not valid.

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