ML: Interval Logic

Introduction

When considering probabilistic and fuzzy logic, we have seen that in logical inference the truth value of a statement or predicate may not yield a fixed value but rather a constraint in the form of an inequality. Therefore, in general, the truth value of $A$ is better characterized not by a single real number but by a pair of numbers $[a_0,a_1]$. In this document, we will explore such a variant of fuzzy logic.

Probabilistic and fuzzy truths

Let us remind you that it is necessary to distinguish between two types of logical statements. The first type is called probabilistic. Such statements relate to future events or events that have already occurred, but the details of which are unknown (and therefore their truthfulness is unknown): "The coin will land heads up", "The count was killed by the butler", "There is a cat in the closed box".

The second type of truth is traditionally called fuzzy. These statements are based on the degree of belonging to fuzzy sets. They are comparative, relative in nature: $B$: "Maria is beautiful", $H$: "The coffee is hot". If $B=0.7$, it means that "Maria is beautiful, but there are even more beautiful women".

For probabilistic logic, the law of excluded middle is typically observed (the event either happens or does not, and there is no third option). For fuzzy logic, this law usually does not hold.

Sometimes, the line between these two types of statements is quite thin. For example, $I$: "the pencil is in the pencil case" can be considered in a probabilistic sense. (thus, $I=1$, if it is indeed there). However, this statement can have fuzzy truth if the pencil is sticking out of the case. For example, if it is sticking out slightly, then $I=0.9$, and if it is almost completely out, then $I=0.1$. In general, these types of truth can be combined (the probability that the pencil is slightly sticking out).

Below, we will consider the second type of truth.

Interval truth

We will characterize the truth of a statement or predicate with a pair of numbers $[a_0,\,a_1]$. We will consider $a_0$ as the degree of certainty in the falseness of statement $A$, and $a_1$ as the certainty in the truth. This means that there are arguments with a cumulative "strength" $a_1$ indicating that $A$ is true and arguments with a "strength" $a_0$ indicating that it is false. It is always $a_0+a_1 \le 1$. The case of equality signifies a completely determined situation: $a_1 = 1-a_0= T_A$, coinciding with the fuzzy logic discussed earlier. In this case, $T_A$ is the usual (in terms of fuzzy logic) degree of truth.

An absolutely true statement is $\mathbb{T}=[0,\,1]$, and a completely false one is $\mathbb{F}=[1,\,0]$. When $\mathbb{U}=[0,\,0]$, the situation is completely undefined (the statement could be either true or false, and we do not know its value). The measure of uncertainty $1-a_0-a_1$ is maximal and equals $1$ for a completely uncertain statement, and minimal ($0$) for a completely determined one.

For example, if we do not know to what extent the pencil is sticking out of the case (or whether it is there at all), then the truth of the statement "The pencil is in the case" would be $[0,\,0]$.

In reality, the truth of most statements will usually be determined. However, uncertainty may arise during logical inference, which can decrease as new information is received.

Logical connectives

Interval truth negation swaps the values of $a_0$ and $a_1$: $$ A=[a_0,\,a_1],~~~~~~~~~~~\neg A = [a_1,\,a_0]. $$ This means that if we had a degree of certainty $a_1$ in the truth of $A$, then with the same degree of certainty, we consider $\neg A$ to be false. The logical connectives for interval truth are defined as follows: $$ \begin{array}{lclcl} [a_0\,a_1] & \,\&\, & [b_0\,b_1] &~~~=~~~& \bigr[\max(a_0,b_0),~\min(a_0,b_0)\bigr]\\ [a_0\,a_1] & \vee & [b_0\,b_1] &~~~=~~~& \bigr[\min(a_0,b_0),~\max(a_0,b_0)\bigr]\\ \end{array} $$ It is easy to see that the introduced operations preserve the property $0 \le a_0+a_1 \le 1$ for composite statements.

Interval logical AND, OR, just like in Boolean algebra, are symmetric, associative, and distributive.
They also satisfy De Morgan's laws and double negation: $$ \neg(A\,\&\,B) = \neg A\vee \neg B,~~~~~~~~~~~\neg(A\vee B) = \neg A\,\&\, \neg B,~~~~~~~~~~~~\neg(\neg A) = A. $$ Additionally, absorption laws hold true: $$ A\vee A=A,~~~~~~A\,\&\,A = A,~~~~~~~A\vee (A\,\&\,B)=A,~~~~~~~A\,\&\, (A\vee B)=A. $$ However, the laws of the excluded middle are violated: $$ \begin{array}{llll} A\vee \neg A = \bigr[\min(a_0,a_1),\,\max(a_0,\,a_1)\bigr] ~~~~\neq~~~ \mathbb{T}=[0,1].\\ A\,\&\, \neg A = \bigr[\max(a_0,a_1),\,\min(a_0,\,a_1)\bigr] ~~~~\neq~~~ \mathbb{F}=[1,0], \end{array} $$ which is typical for many-valued logics. The closer the measure of truth $a_1$ or falseness $a_0$ is to one (almost true or almost false statement), the "better" the laws of the excluded middle are satisfied.

Truth tables

For statements that are definitely false $\mathbb{F}=[1,\,0]$ and definitely true $\mathbb{T}=[0,\,1]$, and for any arbitrary statement $A=[a_0,\,a_1]$ we have: $$ \mathbb{F}\,\&\, A = \mathbb{F},~~~~~~~~~~~\mathbb{T}\,\&\,A = A,~~~~~~~~~~~\mathbb{F}\vee A = A,~~~~~~~~~~~\mathbb{T}\vee A = \mathbb{T}. $$ The undefined statement $\mathbb{U}=[0,0]$ with $\neg \mathbb{U}= \mathbb{U}$ adds to them: $$ \mathbb{U}\,\&\, \mathbb{F} = \mathbb{F},~~~~~~~~~~~\mathbb{U}\,\&\,\mathbb{T} = \mathbb{U},~~~~~~~~~~~ \mathbb{U}\vee \mathbb{F} = \mathbb{U},~~~~~~~~~~~\mathbb{U}\vee \mathbb{T} = \mathbb{T}, $$ which is equivalent to truth tables in Lukasiewicz's three-valued logic.

Let's interpret this table if we limit ourselves to three-valued logic. The undefined statement $\mathbb{U}$ can turn out to be both true and false, but we don't "yet" know this value. Therefore, for example, $\mathbb{U}\vee \mathbb{F}$ will be true if $\mathbb{U}$ is true and false if $\mathbb{U}$ turns out to be false. Therefore, the result of the operation is undefined: $\mathbb{U}$. At the same time, $\mathbb{U}\vee \mathbb{T}$ will be true, regardless of the value of $\mathbb{U}$.

Let's note the projection properties for any arbitrary statement $A=[a_0,\,a_1]$: $$ \mathbb{U}\,\&\, A = [a_0,\,0],~~~~~~~~~~~\mathbb{U}\vee A = [0,\,a_1]. $$

Geometric representation

Since the truth of statements is determined by two numbers, they can be represented by points on the plane inside the triangle with vertices $\mathbb{F},\,\mathbb{U},\,\mathbb{T}$:

The value $a_0$ is plotted along the $\mathbb{U}-\mathbb{F}$ axis, and the value $a_1$ is plotted along the $\mathbb{U}-\mathbb{T}$ axis (as shown in the first figure above). On the $\mathbb{F}-\mathbb{T}$ line are the fully determined probabilities for which $a_0+a_1=1$.

If the segment $A-B$ between two statements forms a smaller angle with the $\mathbb{U}-\mathbb{F}$ axis, then geometrically, the points of disjunction (OR) and conjunction (AND) lie at the vertices of a rectangle, as shown in the second figure. If the segment $A-B$ forms a smaller angle with the $\mathbb{U}-\mathbb{T}$ axis, then $A\vee B$ equals the statement located above on the diagram, and $A\,\&\, B$ equals the one below.

A partial order can be introduced on the set of statements using the following definition: $$ A\leq B ~:~~~ (a_0 \geq b_0)\,\&\,(a_1 \leq b_1). $$ In the first figure, all statements greater than $A$ are located in the upper shaded region, and all statements less than $A$ are in the lower region.

Since the order has an upper boundary $\mathbb{T}$ and a lower boundary $\mathbb{F}$, it forms a distributive lattice. As an example, such a lattice is represented in the third figure for six statements, where $S=[1/2,1/2]$, $S_1=[1/2,0]$, $S_1=[0,1/2]$. It is worth noting that in such diagrams, if moving from node $B$ upwards along the edges leads to node $A$, then $A\geq B$, and $B \leq A$. If there is no such path, then the nodes are not ordered (such as $\mathbb{U}$ and $\mathbb{S}$). The edges are drawn minimally (if there exists $X$ such that $A\leq X\leq B$, then no edge is drawn between $A$ and $B$).