Question

What is fuzzy logic?

Answer 1

Fuzzy logic is a generalisation of Boolean logic with truth values between true and false.

Explanation:

In ordinary Boolean logic, propositions are true or false.

In fuzzy logic you could consider propositions to have truth values in the range `[0, 1]` where `0` represents false, `1` represents true and any number in between represents a truth value strictly between false and true.

There are several different systems of fuzzy logic used for different purposes in different areas of mathematics. These different systems have different rules for the truth values of logical operations.

If we write `v(P)` for the truth value of a proposition `P`, then we might use the following rules:

`v(not P) = 1-v(P)`

`v(P ^^ Q) = v(P)*v(Q)`

`v(P vv Q) = v(not ((not P) ^^ (not Q)))`

`= 1-(1-v(P))(1-v(Q))`

These rules correspond to the way that probabilities of two independent events would combine.

What can we do if we don't know whether `P` and `Q` are independent? One option is to introduce ranges rather than single truth values. Another option is to introduce modal logical operators like necessity (`square`) and possibility (`diamond`). Even without 'fuzziness', modal logics can take quite a variety of forms.

Yet another alternative is to introduce the notion of relevance into logic, which typically has the side effect of splitting simple and's and or's into intrinsic and extrinsic conjunctions and disjunctions. Combined with fuzziness, this can have a similar effect to the modal operators.