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In product fuzzy logic, the $AND$ operator of two variables $x_0$ and $x_1$ is the product $x_0x_1$.

Using the $NOT(x)$ as $1-x$, expressions for the other three minterms are easily obtained.

$$\overline{x_0}x_1 = (1-x_0)x_1$$ $$x_0\overline{x_1} = x_0(1-x_1)$$ $$\overline{x_0}\overline{x_1} = (1-x_0)(1-x_1)$$

These four expressions have the property that they sum up to $1$, as in bivalued logic.

$$ x_0x_1+(1-x_0)x_1+x_0(1-x_1)+(1-x_0)(1-x_1)=1$$

This is not the case when the conjunction is the Zadeh operator $min(x_0,x_1)$, where the sum of the four minterms happens to be a square pyramid over the unit cube, neither other possible definitions of fuzzy operators.

enter image description here

Does this property have a name, and any consequences?

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In the bivalued logic case the two variables $x_0$ and $x_1$ are discrete with values of either $0$ or $1$, and your four-terms sum is the disjunctive normal form (DNF) of truism whose graph is the upper half vertices of the unit cube under your context. While in the fuzzy logic with Zadeh operator case, the two variables are continuous real numbers ranging from $0$ to $1$, and the graph of your four-terms sum as you showed is a convex pyramid. One consequence is the unique maximum function value of $2$ when surprisingly both $x$ and $y$ take the value of $0.5$ instead of $1$!

If you replace $+$ in your fuzzy case with the corresponding Zadeh $OR$ operator which takes the maximum, the sum of your four-terms would take maximum value back to $1$ at the four corners and ceases to be a truism formula since the sum is strictly less than $1$ when either variable is not $0$ or $1$. This suggests fuzzy logic aptly tracks degree of truth of complex composite propositions on a continuous basis based on continuously valued fuzzy inputs and meanwhile it's not the same usual concept of probability. Thus fuzzy logic is used in control applications a lot where controller's complex CNF/DNF condition's degree of truth may be more easy and accurate to estimate than probabilities in many vague cases without many propagated rounding errors.

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  • $\begingroup$ Well, my $x_0$ and $x_1$ are continuous in the interval [0,1], and the truism is not only the upper four vertices of the unit cube, but the upper flat face. Should have added that plot as well and written "continuous" before.$x_0$ and $x_1$... I start with one hyperbolic parabola and write the other three. Another beautiful sum is that of adjacent terms, which leads to ramps in a single variable when the and is the product but not when it is the minimum. $\endgroup$ Dec 2, 2022 at 5:59
  • $\begingroup$ Yes pls add that plot with your Boolean operator applied to continuous variable fuzzy logic case to make it clear and we can see difference with the Zadeh operator more visually. $\endgroup$
    – cinch
    Dec 2, 2022 at 6:56

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