This is the continuation of a previous post. In order to deal with more in-depth concepts, I will write for an audience that has read that post. However, a quick recap would not be untoward.
In the previous post, we defined two new truth values, but before them, let's look at the original two. The first, True, applies to statements of which it can be truly said that they are found True, and it can only be falsely said that they are found False. The second, False, applies to statements of which it can be truly said that they are found False, and only falsely said that they are found True. If this seems circular, it is. It's exactly as circular as saying that the traits of a positive number can be indicated by the fact that multiplying a positive by a positive yields a positive, while multiplying a positive by a negative yields a negative (and similarly, that the traits of a negative number can be shown by the fact that a negative multiplied by a positive yields a negative, whilst a negative multiplied by another negative yields, oddly, a positive). These somewhat obliquely stated definitions also serve to indicate that a statement which is found alternately both True and False is belongs neither to the set of properly True statements nor to the set of properly False ones, and similarly that a statement which cannot be found to be either True or False at all belongs to neither set.
Now, let us formally define our two posited complex truth values that supplement True and False. The first we have called Mu, due to the fact that asking whether the statement valued Mu is true or false is a question which ought to be unasked, though the coincidence between that word's first letter and the initial letter of 'maybe' certainly was not a dissuading factor, nor was the fact that its name is the first two letters of 'multiple'. The value Mu applies to statements which can by found to be True when treated as such, and which can equally be found to be False when treated as such. Mu does not fit into the previous paragraph's analogy of integers, because while the statements it applies to are not without disproof nor without proof, likening it to zero would press its etymology beyond the breaking point, for it does not have the properties of that integer, either. Mu's mathematical analogue is best understood in relation to its opposite, Nix.
Just as Mu's name comes from the answer to beautiful koans, the name of Nix, as its opposite, has no such beautiful origins. Though I'm sure if these ideas catch on, some Jungian or Thelemite will come along and replace the middle letter of Nix with a 'y' or an 'o' so that they can wax poetic, its name was chosen simply because, just as V (vrai, French for 'true') and F are one degree of freedom apart in the way they are articulated, so are M and N. A statement's value is Nix if attempts to assess the statement's truth value which start by assuming it to be True find it False, and attempts which start by assuming it to be False find it True. In the previous article, the simplest Empedoclean paradox was used as an example: "This statement is false." If you assume that the statement as a whole is True, you can derive from this truth that what the statement says must be true, and therefore that the statement is False. In mirrored fashion, if you start with the idea that the statement is False, you must interpret that what the statement says must be false in order for it to be false, and thus you arrive at the notion that the statement is True. No, the statement is Nix.
We had best save the mathematical analogy of these two values for the next post in this series, for I fear we have gotten far to abstract for anyone but the logician to see value in continuing forward. Let us then veer away from complex Boolean algebra towards real-life examples of these.
In the previous post, we used the example of Cindy, who has no patience for people who have no patience for her but gets along fine with people who expect the best in her (sound like anyone you know?). We said that, "Cindy is hard to get along with," was Mu, given that the way acceptance or negation of the statement affected one's actions made it self-fulfilling. Similarly, there were studies done where classes of students were given fake aptitude tests where the teachers (sometimes just the teachers, sometimes the students as well) were informed that they were gifted and talented. Whichever students were given those results performed remarkably the rest of the time they were with that teacher. In this case, "Student P has great potential," is a way of expressing the Mu statement. Now let's look at some other places where Mu turns up.
One great place to look for Mu is in the rules of particular perceptual sets or paradigms. For instance, both "The external world is more real than our minds; when we perceive, we perceive it, mediated through our minds," and "The world is what attention is directed to; no external world need be postulated, for we unconsciously account for continuity when we look back at what we've previously attended," are Mu. Same with the statements, "The intricacy of the universe indicates a transcendental being either created it at one point or tailored it to Its liking as time passed," and "The intricacy of the universe indicates that it is transcendent to any being." The same holds for theories of personality, "The personality consists of an aggregation of complexes distributed over unconscious, preconscious, and conscious levels as it unfolds over time," and "The personality has a fiery component (will), a watery component (emotions), an airy component (cognition), an earthy component (sensations and patterns of muscular tension), and an etheric component (the intuition)," are both Mu.
The truth value of these set of statements may have something to do with the way psychology has been treated by the 'hard' sciences. In various essays on the philosophy of science, psychoanalytic theory is disparaged as unfalsifiable pseudoscience and sent away on the same barge that carries away astrology. All of this is done from a paradigm where true and false are the only possibilities, and it makes sense that supporters of such an ontology would find Mu statements anathema. Rather than admitting that there may be another set of statements besides those true or false from all points of reference, we have the game rule that anything that can be shown false at all is false indeed, and any instance of seeing it as true is a result of deception or self-delusion. All this in order to preserve a binary logic.
That's not to say that recognizing some Mu statements for what they are is incompatible with an ultimate worldview where properly phrased statements about the universe are either true or false. For instance, consider the statements "The Sun revolves around the Earth," and "The Earth revolves around the Sun," are both Mu statements. For a long period in human history the first was believed true, then for a shorter period it came to be believed false while the second was believed true, but as our understanding of the relativity of motion developed, we came to realize that the order of things can be just as accurately described from the reference frame of the Earth or that of the Sun. We could even take a reference frame where the Earth is quite stationary, and all the stars revolve around it in the span of a day, and not have an inaccuracy.
What we do for the sake of practicality is to use whichever perspective is convenient to our purposes, but we have discovered a deeper truth in this process: "The motion of the Sun and Earth is such that it can be described by order O from reference frame R and by order P from reference frame S." No one talks like that, though, and with the lack of actual change created by other methods like General Semantics and E-Prime to transform our language to make it accord better with reality, it may be more convenient and accomplish many of the same goals to simply add another truth value to our ontology.
By now you may see the use of Mu in philosophy and interpersonal interaction, but what about Nix? Surely it's confined to logical paradoxes like "The set that contains all sets that do not contain themselves contains itself." Surely we can avoid worrying about Nix and enjoy our lives to the fullest. Perhaps. But consider those experiences that can flit through your mind which vanish as soon as you start thinking about them. Think about how thinking is just another mode of experience, and that it often does quite poorly at describing events that are not mediated through thought. Recall how the ancient sages often speak in paradoxes in order to get across a message that transcends thought. "The name that can be named is not the eternal name."
Next time we'll look at recursion in Grelling's paradox, interpersonal interaction, and paradigms of belief to see why they have Mu statements in common, and we'll start to explore mathematical analogies for all four truth values.