Answers.Body: <p>In the bivalued logic case the two variables <span class="math-container">$x_0$</span> and <span class="math-container">$x_1$</span> are discrete with values of either <span class="math-container">$0$</span> or <span class="math-container">$1$</span>, and your four-terms sum is the <a href="https://en.wikipedia.org/wiki/Disjunctive_normal_form" rel="nofollow noreferrer">disjunctive normal form</a> (DNF) of <a href="https://en.wikipedia.org/wiki/Truism" rel="nofollow noreferrer">truism</a> 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 <span class="math-container">$0$</span> to <span class="math-container">$1$</span>, and the graph of your four-terms sum as you showed is a <a href="https://en.wikipedia.org/wiki/Convex_set" rel="nofollow noreferrer">convex</a> pyramid. One consequence is the <em>unique</em> maximum function value of <span class="math-container">$2$</span> when surprisingly both <span class="math-container">$x$</span> and <span class="math-container">$y$</span> take the value of <span class="math-container">$0.5$</span> instead of <span class="math-container">$1$</span>!</p>
  <p>If you replace <span class="math-container">$+$</span> in your fuzzy case with the <em>corresponding</em> Zadeh <span class="math-container">$OR$</span> operator which takes the maximum, the sum of your four-terms would take maximum value back to <span class="math-container">$1$</span> at the four corners and ceases to be a truism formula since the sum is strictly less than <span class="math-container">$1$</span> when either variable is not <span class="math-container">$0$</span> or <span class="math-container">$1$</span>. This suggests fuzzy logic aptly tracks <em>degree of truth</em> 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.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-12-02T05:59:04.183Z
Answers.Text: 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.
Answers.UserDisplayName: Jaume Oliver Lafont
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-12-02T06:56:24.027Z
Answers.Text: 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.
Answers.UserDisplayName: cinch
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-12-02T03:58:46.597Z
Answers.LastActivityDate: 2022-12-02T03:58:46.597Z
Answers.UserDisplayName: cinch
Body: <p>In product fuzzy logic, the <span class="math-container">$AND$</span> operator of two variables <span class="math-container">$x_0$</span> and <span class="math-container">$x_1$</span> is the product <span class="math-container">$x_0x_1$</span>.</p>
  <p>Using the <span class="math-container">$NOT(x)$</span> as <span class="math-container">$1-x$</span>, expressions for the other three minterms are easily obtained.</p>
  <p><span class="math-container">$$\overline{x_0}x_1 = (1-x_0)x_1$$</span>
  <span class="math-container">$$x_0\overline{x_1} = x_0(1-x_1)$$</span>
  <span class="math-container">$$\overline{x_0}\overline{x_1} = (1-x_0)(1-x_1)$$</span></p>
  <p>These four expressions have the property that they sum up to <span class="math-container">$1$</span>, as in bivalued logic.</p>
  <p><span class="math-container">$$ x_0x_1+(1-x_0)x_1+x_0(1-x_1)+(1-x_0)(1-x_1)=1$$</span></p>
  <p>This is not the case when the conjunction is the Zadeh operator <span class="math-container">$min(x_0,x_1)$</span>, where the sum of the four minterms happens to be a square pyramid over the unit cube, neither other possible definitions of fuzzy operators.</p>
  <p><a href="https://i.sstatic.net/x2GQe.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/x2GQe.png" alt="enter image description here" /></a></p>
  <p><strong>Does this property have a name, and any consequences?</strong></p>
  
Tags: <terminology><fuzzy-logic>
Title: Does this property in product fuzzy logic have a name and any consequences?
UserDisplayName: Jaume Oliver Lafont
ThreadText: # Does this property in product fuzzy logic have a name and any consequences?
  
  (Asked by: username_0 on 2022-12-01)
  
  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](https://i.sstatic.net/x2GQe.png)](https://i.sstatic.net/x2GQe.png)
  
  **Does this property have a name, and any consequences?**
  
  **Tags:** terminology, fuzzy-logic
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2022-12-02. Score: 2 | Accepted answer
  
  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](https://en.wikipedia.org/wiki/Disjunctive_normal_form) (DNF) of [truism](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Convex_set) 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.
  
  > **Comments:**
  > **username_0**: 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.  
  > **username_1**: 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.  
  
  ---
  
  