There is an interesting essay by Bertrand Russell called Logic and Mysticism. He of course admits that many great thinkers used mystical thinking, but he says that logical thinking follows the law of excluded middle (it can either be p or not-p, not both) while an example of a mystical assertion would be like Heraclitus saying “We both are, and are not”. I believe it was Dr. Russell who observed that a prior assumption to the Law of the Excluded Middle is that “Everything that is, is.” Mysticism is categorically illogical because it violates the Law of Excluded Middle.

I would change this claim slightly and say that mysticism (and in general spiritualism) is just the belief that the truth (or reality) can change. The extreme example of this mystical assertion is the Buddhist axiom, one of the distinguishing characteristics of Buddhism, that *everything* is impermanent: Mathematical theories, physical laws, empires, religions, rocks, etc. The Four Noble Truths, the basic truths of Buddhism, are about the *transformation *of suffering. Firstly, everything is impermanent, and more specifically, suffering is impermanent. Suffering is the first truth, and is a necessary part of the other three truths, which end in the cessation of suffering.

The reason the admission of change might be fundamentally mystical is because change is not possible in logic. Logical truth is universal and unchanging, even if it is conditional. A conditional statement has an *ultimate* truth-value. If the ultimate truth of logic were mutable, it would fall into the logical paradox known as the liar paradox. The liar paradox is as follows:

(1) The next statement is false

(2) The previous statement is true

While this statement is not illogical, like the first mystical statement, it is impossible to know its truth-value. This means either that (A) it is not true (p) nor is it false (not p), which would drive it into a clear contradiction instead of a more mysterious paradox or (B) the Liar Paradox would be better described in terms of a kind of permanent uncertainty or *question *tied up with logical understanding and the nature of change. This *question *is unlike other questions in that there is no answer to be discovered, the question is not an appearance of ignorance awaiting new knowledge, it is “real” uncertainty. The only knowledge that can be had is knowledge that it is a real question.

The dimension of time is inextricable in the liar paradox. We cycle back and forth between statements looking to settle on a truth-value. More specifically, suppose what we believe now is that statement (1) is true, we must believe that any future change must be false, but time goes by and we change our minds, entering statement (2). At first it may seem that there has been no change, we still believe statement (1) is true once we arrive at (2). If (1) is indeed any ultimate truth then of course any future realization different from (1) must be false, so we might ask, is (2) different from (1)? What has changed is just the simplest sort of change- the change from one statement to the next. It may be seen as irritating (or humorous) that after we have discovered the ultimate truth that should be the final Word, we find ourselves moving on to some other thought. Only in rare occasions could we possibly avoid this- it may happen that we die the moment we realize the ultimate truth. Perhaps this is why Feyerabend argues that logical proofs are all tragedies. (Conquest of Abundance)

In any case, change is at least a constant appearance both within and without, it a very old idea that the source of change is the opposite of an appearance – it is our very soul; Aristotle mentions the idea in his work “On the soul”. This “soul of change” in the liar paradox might not lead us into uncertainty- perhaps, after arriving at ultimate truth we go on basically without realizing that our previous ultimate realization (1) asserted that any future movement is invalid, yet here you are, in what was the future. Things are different now; we are in a new logical state. If we go on without any examination whatsoever, no uncertainty will arise. We may, on the other hand, find that this ultimate truth we realized is actually weighing us down by making all continued living invalid, a farce. This dissatisfaction could lead us to re-examine logically our previous state (1), in the light of our present state (2). From a purely logical standpoint what this hypothetical person believed in, (1), cannot be true, because he is now in state (2) and if (1) were true then, by (2), (1) must also be false! The simple fact that life goes on has thrown us into a tenacious uncertainty of any logical truth, whatsoever. Unless?

From within logic itself, a logical path has been drawn that leads to *logical *uncertainty of all truth. Now, if we want to keep thinking logically, we improve our logic to include uncertainty. The material implication is the rhetorical offering for logical uncertainty- the “if, then” handles conditional truth. You may not know if the condition is met, but if a condition is met then “something is known”. So truth can be conditional, not ultimate. Unfortunately, statements that include the material implication must have a truth-value *as a whole, *or *ultimately.* This should and does lead to other paradoxes not discussed here (See Suber The Material Implication). The path we have drawn with the liar paradox leads us to question the ultimate truth of any logical statement, including ultimate truth of a conditional statement. An if, then statement as a whole cannot be sometimes true. When i say “If it rains, I get my umbrella” the statement has to be true of false (p or ~p). Either sometimes I don’t get my umbrella when it rains (false) or I always get my umbrella when it rains (true). Equivalent to the idea that the law of excluded middle is true is that a logical statement is populated by a truth-value.

It has been said that Aristotelian logic is the calculation of distinction (Kauffman). C.S. Pierce invented a notation for logic with only one symbol – the circle – that took advantage of logic as distinction. In a plane, a circle was a distinction being made between the inside and outside of the circle. Any logical statement could be represented with multiple circles embedded in other circles. Equivalent logical statements could be transformed into each other by cancelling out circles within circles- a “not-not” cancels itself out. This symbolism for logic was a powerful argument that logic could be reduced to just the key operation “not.” [1]

[Enter here about how using the “not” requires assumptions about the “operating table” (see Foucault Order of things)]

With more complicated logical statements, this calculation becomes harder to swallow. Generally, mathematicians like to cut out as much as possible, so they are talking about something very small. Carefully taking the negation of a statement reveals that the more we distribute or embed the negation into our statement, the more gets cut out, disqualifying Pierce’s notation.

For example take the logical statements

(A) ~[q->(p AND r)]

(B) [q AND ~(p AND r)]

and

(C ) q AND ~p OR ~r

Even though “logically” these statements are said to be equivalent, they are actually different.

In English, (A) is the set of all things that are NOT the statement “If I have to go then I am going home and I am eating cake”, which includes you, me, my community, ?

However the set of all things that are (B) “I have to go and NOT the statement “I am going home and I am eating cake” includes less than the previous statement[2], but still more than the next statement:

(C) “I have to go and I am not going home or I am not eating cake.”

The question is expansive and inclusive; it is the opposite of the “not” operation. The following example is merely a mathematized play to show the power of questions. Maybe a “?” would push *out* the “not” one level in the statement. Lets see if it can help us out of the Liar Paradox.

(1): The next statement is false

(2): The previous statement is true

On our path to total uncertainty we eventually return to (1) with a “?” operation.

(1) becomes The next statement is false**?**

Which translates into

~(The next statement is true)**?**

Which in English means “Anything but “the next statement is true.”” Now, to see how this disentangles us from the Liar Paradox, observe:

Equate “next” with “previous” and call it “<->”, since they perform the same function of moving us to the “other” state.

(1)?: “Anything but “The <-> statement is true.””

(2): “The <-> statement is true.”

We start with “anything but “the <-> statement is true.”” While we can literally choose anything to be true, generally a person would use his intelligence to choose. However time continues and later we enter (2) “the <-> statement is true”, which simply means we continue believing what we chose to believe in “(1)?”. Time shifts again and we pick another thing to believe. Etc.

The main difference is here we avoid contradiction and are able to assign truth-values to the statements. “(1)?” does not assert “The <-> statement is false”, it asserts *something else. *“Anything but “The <-> statement is true.”” It is possible for “The <-> statement is true” to be true as well.

Questioning is an intention that leaves you open to experience, it takes you out of analytical calculations and begins empirical work. Thus a logic with the “?” is a combination of analytic and synthetic reasoning, where where each type of reasoning is represented in the notation.

Language is capable of suggesting more than it is capable of literally expressing with the question. Mathematics can only suggest “real” numbers because there are more of them than can be literally expressed. The difference being that “real” numbers have garnered some authority in being “real”, while questions, a non-specialized tool of the entire range of humanity, does not necessarily have authority of its own, but depends often on other things to have authority. Questions are often used to persuade “away from” a position by suggesting no true answer is possible or desirable, e.g. the “rhetorical question”. Questions however, have a power in themselves; they invite consideration of many possibilities at once. It is just the power to suggest many potential answers at once in a more simple way than real numbers suggest inexpressible quantities. The innate power of questions, same as the real numbers, is a power of *suggestion*.

You can find about the power of questions in my essay on math and poetry here:

Click to access MRftAoCs_Nightingale2014-05.pdf

[1] The notion of difference is highly related to the Law of Excluded Middle. If ~p were not different from p, pV~p (thats “p or not p”) wouldn’t hold.

[2] (in that now “I don’t have to go” has now been excluded from (A))

Kevin Barnhurst

said:Compelling! I especially like the last paragraph on questions.

LikeLike

cageyb2

said:Having read your recent work, I wonder if your use of “logic” and “logical” overstates or at least is too general, there being multiple logics?

LikeLike