✅❎**Non-axiomatic truth-values.** Tangrui(Tory) Li, tuo90515@temple.edu --- In general binary mathematical logic, there is no truth-value that is both true and false, but can such a black and white judgment really cover even our daily life? The answer is "**NO**". Even if you insist on using such binary judgments, there are still some propositions that cannot be measured in terms of "absolute true" or "absolute false". Suppose there are the following three propositions, please understand them as in natural language way. 1) Proposition \( X \): "Apples are red 🍎"; 2) Proposition \( Y \): " \( A \) is an apple"; ✅ 3) Proposition \( Z \): " \( A \) is red". ✅ Assuming that that both propositions \(Y\) and \(Z\) are absolute true, then the object \(A\) can be used as an evidence to support proposition \(X\) (BTW, it can also support "everything in red is an apple", we will see it in my further updates). But at this time, if two other propositions with absolute truth-values are given, say: 1. Proposition \( \alpha \): "\(B\) is an apple"; 2. Proposition \( \beta \): "\(B\) is not red". 🍏 Then $B$ becomes a counterexample of proposition \(X\). So, what is the truth-value of \(X\)? In this case, only the absolute truth-value is used to obtain the non-absolute truth-value. One mathematical logic justification is that the proposition "apples are red 🍎🍏" is false (true-value 0), but this may become unreasonable when more evidences are considered. For example, if there are 99 objects that are both apples and red (🍎 99%), and there is only one object that is an apple but is not red (🍏 1%). Although there is a counterexample at this point, as a human being, I generally say "apples are mostly red" instead of "apples are **NOT** red". Perhaps some mathematical definitions can be used, such as "the proposition is true when the positive examples are in the majority, otherwise it is false", then when there are 51 positive examples (🍎 51%) and 49 negative examples (🍏 49%), this judgment will be true, but as a human being, I generally say "I don't know the color of apples", that is, the proposition itself has no meaning. Therefore, a better approach is to change the description of the true-value from a binary one to a continuous one from 0 to 1 to describe the "proportion of positive evidences to all evidences", namely **frequency**. Of course, a proposition with more evidences has a higher degree of confidence, so there is another part for this truth-value specification determined by the total number of examples, namely **confidence**. $$ T=(frequency,confidence) $$ $$ frequency=\frac{w^+}{w^++w^-} $$ $$ confidence=\frac{w}{w+k} $$ $$ w=w^++w^- $$ In which \(w^+\) is the number of positive evidences and \(w^-\) is the number of negative evidences. The \(k\) in the confidence is a hyperparameter used to make sure confidence is between 0 and 1, which means the stability of the history, compared with a limited future. You may find that the truth-value of a proposition will be an average value accumulated over a long period of time, which does not seem to apply to time-sensitive propositions, such as "today is a rainy day 🌧". Its truth-value of a proposition needs to be bound to a timestamp ⏲️, so it will not be accumulated all the time. If the truth-value of this proposition is similar in multiple time stamps, then the timeliness of this proposition will be gradually canceled, and it will be called a relatively "eternal" proposition.