Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas.

Abstract submitted to Warsaw Workshop on Formal Truth Theories, September 28-30, 2017. Rejected.

Truth as the Admission of Self-Contradiction

I sketch out a definition of truth as the admission of self-contradiction, and thus as the unique referent beyond a noncontradictory system, that which cannot be hidden, which is ultimately obvious, and by which a statement can be taken at face value, understood without any special reading of its context.

In other words, I take the nature of truth to be inherent in the Liar's paradox, "I contradict myself". My thinking here is quite backward! I identify with a State of Contradiction ({$ \bot $}) and then imagine how it gives rise to a State of Noncontradiction (a System). In doing so, I document my exploration of the cognitive limits of my imagination, and perhaps, our imagination.

I relate admission of self-contradiction to Heidegger's "unconcealment" and the orginal Greek term Aletheia, as well as to truth as understood by 4 substantialist theories, 2 deflationary theories, 3 pragmatist theories and 6 formal theories described in the Wikipedia article, in English, on Truth.

A comprehensive theory of truth all but supposes a comprehensive view upon everything and thus a vantage point for absolute truth. In logic, the "set of all sets" is a problematic concept which resembles "everything" but imposes upon it notions of "sets" and "elements". Let us instead consider Everything intuitively, in its everyday sense, and formally refer to it as that which has the following four properties:

Everything has no external context. If you put it in a box, then it includes the box. If you think it, it includes you.
Everything has no internal structure. It can be chaotic or orderly. Thus, all statements are true about everything, for there is no structure to hold onto: Everything is hot, everything is cold, everything is good, everything is bad.
Everything is the simplest possible algorithm, the one which has no filter but accepts all things, whatever we think of. This means that we all have the same Everything, although we may call it by different names, such as Being (all that is), Love (all that is loved), etc.
Everything is a required concept. We all have it, and appeal to it, for example, when we take a stand, which we do with regard to everything. We could not have learned of Everything, because all that we know is bounded, but Everything is unbounded. We cannot rid ourselves of it as a concept. It must have always been with us.

In mathematics, {$ \bot $} is a vantage point from which all statements are true, and is thus a view upon Everything. I try to imagine what such a vantage point might ever yield, and in particular, how might it retreat so as to give rise to a state of noncontradiction? I am only able to suppose that it asks itself, Is {$ \bot $} necessary? Would it be, even if it wasn't? And then it proceeds by dividing itself into perspectives, as with a proof by contradiction, whereby it yields two perspectives: {$ \bot \rightarrow \bot $} (if it exists, then it exists) and {$ \neg \bot \rightarrow \bot$} (if it does not exist, yet even so, it exists).

Next, it might consider, what is it that identifies the original contradiction {$ \bot_{O} $} with the arisen contradiction {$ \bot_{A} $}? We may say that {$ \bot_{O} $} understands (separates itself and its negation into separate statements) and that {$ \bot_{A} $} comes to understand (likewise separates itself and its negation, but within the same statement) and so they understand the same negation {$ \neg \bot \equiv \bot_{N} $}, which separates itself within the same term {$ \neg \bot $}, and which is the contradiction that is understood. And here the initial state of self-contradiction has divided itself into three perspectives - three scopes for self-contradiction - the language ({$ \bot_{O} $}), the statement ({$ \bot_{A} $}) and the term ({$ \bot_{N} $}). As Godel showed with his Incompleteness theorems, if statements determine a language (it is complete), and terms determine sentences (they are consistent), and the language determines its terms (whether to interpret them syntactically or semantically, so that they are able to contradict themselves), then there indeed must be a contradiction somewhere. Or, thinking backwards, a state of contradiction may divide itself into three perspectives, that of a language, a statement and a term.

Such a process of divisions can be described further, but here I simply note - let us say, poetically - that a division of the original contradiction {$ \bot_{O} $} into seven perspectives yields a noncontradictory System given by four corners and three sides of the logical square: {$ \forall x, \forall x \wedge \exists x,\exists x, \exists x \wedge \exists x \neg, \exists x \neg, \exists x \neg \wedge \forall x \neg, \forall x \neg $}. And yet an eighth perspective {$ \forall x \wedge \forall x \neg $} indicates that the entire System is empty, or equivalently, yields our original state of self-contradiction. This eighth perspective may be taken as a unique referent beyond the System which, if referenced, can make the entire System collapse, as an affirmation of self-contradiction.

Let us now consider how divisions of everything are fundamental to our own cognition and how they ground and relate different theories of truth. For issues of existence, we divide everything into two perspectives: "opposites coexist" (so that we can ask questions, does a chair exist or not?) and "all things are the same" (so that we can have answers - if it exists, then it exists, and if not, then not). We conceive of this deep structure through one of four different representations, namely:

"free will" and "fate";
"outside" and "inside" (if we are outside a cup, then there is also an inside, but if we fall inside, then there is only inside);
"theory" and "practice" (in theory, we are distinct from the process we study, but in practice, we become one with the process, reflecting each other, like grist in a mill, or molten bronze in a form);
"same" and "different" (for things to be the same, they must also be different, whereas if they are different, they are simply different).

Issues of knowledge depend on four perspectives: Whether, What, How, Why. Let us illustrate what the levels of knowledge with regard to a water bottle. Our senses present us an image of What the bottle appears to be, in which case we know Something. Whereas our intellect thinks of the bottle as a blueprint for How it is created and used, the principles by which we know Anything. We experience our own life through these two modes, conditionally. And yet we can also conceive of two additional unconditional modes. We can consider Whether a bottle exists if it is hidden and nobody sees it, and so we know Nothing. In order to know Why a bottle exists, we would have to know Everything, for it is associated with absolutely everything.

Nothing, Something, Anything, Everything also arise as the structural scopes or selves by which the original {$ \bot_{O} $} retreats from itself, and goes beyond itself, and thus gives rise to itself, and goes into itself, arising as {$ \bot_{A} $}. Truth behaves very differently with regard to these scopes. For example, pragmatically:

"Every medicine has its purpose." There is no way for such a statement to be false because it asserts its own semantics, what is to be meant by "medicine" or "purpose".
"Any medicine has its purpose." This statement requires us to know how to apply this assertion meaningfully to one case at a time, and thus have the knowledge that a doctor might have.
"Some medicine has its purpose." We are now as if participating in a particular conversation, for example, about aspirin, and we may or may not understand each other, as our terms may be quite tentative.
"No medicine has its purpose." Such a statement is purely formal and so will be absolutely right or wrong.

With all of this in mind, we can consider four substantive theories of truth:

Why it is true: consensus - views are compatible with each other - true positions are those which a group will agree upon.
How it is true: construction - a concept is compatible with views - true stands are those which are taken in a given social context.
What is true: correspondence - a view is compatible with concepts - true beliefs and true statements correspond to actual states of affairs.
Whether it is true: coherence - concepts are compatible with each other - a true system of concepts has them support each other.

This foursome has two representations. Idealists think of these levels of knowledge in terms of views for asking questions Why? How? What? and tend to dismiss Whether? Whereas Materialists think them in terms of concepts for stating answers Whether! What! How! and tend to dismiss Why!

This distinction is apparent in two deflationary theories which consider truth as asserting a statement:

A view is defined by performance. Truth is assent to a statement.
A concept is defined by redundancy. Truth is emphasis upon a statement.

We may think of a statement as arising from a view of a concept. Substantive theories base the truth in the relationship between views and concepts by which the statement will be established. Deflationary theories base the truth on the view or the concept of the statement which has already been established. We can think of these as the ways that a statement - an admission - can be made, so that it has the freedom to be an admission of self-contradiction. They are the ways that truth can be ascribed to a statement before it is part of a formal System.

Let us now consider how truth can be defined when a formal System of statements is well defined. Namely, we can negate any level of knowledge, as conditional, so as to refer to the unconditional - the original self-contradiction - which is beyond the System. This yields the concepts:

True - not Whether - it cannot be hidden, like a bottle in a cupboard, so that nobody sees it. In this sense it is unconcealed - aletheia.
Direct - not What - it is not mediated by a representation.
Constant - not How - it does not change.
Significant - not Why - it is not encompassed.

These concepts are, in fact, representations of the nullsome, the division of everything into zero perspectives, as by them we conceive that original self-contradiction beyond the System.

Each of these representations of the nullsome triggers a representation of the division of everything into three perspectives. This threesome is for issues of participation, and defines a three-cycle of taking a stand, following through, and reflecting, as with the scientific method. There are four representations of this threesome, yielding twelve circumstances, much like Kant's twelve categories, but based not on the logical form of a statement, but rather on the following "mind games":

True yields Necessary, Actual, Possible. Consider content (a concept) and its expression (to a view). Then what may be true (unconcealable) is the content, in which case it is necessarily true, as with proof by contradiction. Or what may be true (unconcealable) is the expression, in which case it is Actually true, as when grounds yield consequences. Or what may be true (unconcealable) is the relation between the expression and the content, in which case it is Possibly true, as in the event of consistency.
Direct yields Object, Process, Subject. A Subject's attention may be directed by something else (an Object) or directed by itself (a Process).
Constant yields One, All, Many. If we search for constancy, then we may find One example. Otherwise, All is constantly unconstant. But also, what we select and what we judge must stay the same, and so that must be Multiply constant.
Significant yields Being, Doing, Thinking. If Thinking is significant (unencompassable), then so is the thinker - so is Being. If Being is significant, then so is Doing. If Doing is significant, then so is Thinking.

We can similarly distinguish:

Logic asserts Necessary truth. True statements are unconditional.
Proof theory asserts Actual truth. True statements are provable from true statements.
Model theory asserts Possible truth. True statements have interpretations as true statements

If the original self-contradiction manages to carve out of itself a noncontradictory System, then there may be thought to be a unique self-contradictory referent (the Truth) beyond that System, as all such referents are equivalent, as regards the System. The distinction between the noncontradictory System and the original, self-contradictory Metasystem is thus fundamental. From the System's point of view, we can consider the System as generated by metalingual extensions, but the ultimate extension will be that final referent, the Truth - the (Systemic) admission of (the original) self-contradiction. In this way, we can discuss:

Tarski's semantic theory, whereby true statements are elements of an object language referred to as such by a metalanguage.
Kripke's theory, whereby true statements are generated by metalingual extensions.
Gupta and Belnap's revision theory, whereby true statements are based on circularity within the System, and so their extrasystemic content (their self-contradiction) need not be interpreted.

Finally, pragmatic theories are also compatible with defining truth as an admission of self-contradiction. If the truth is unknowable, and what is knowable is falsehood, as Hocking claims, then we may have that:

Truth is honest self-critique, as says Peirce.
Truth is the limit of self-correction, as says Dewey.
Truth is a confirmed expedient in thinking, as says James, which admits that our thinking is ever vulnerable.

In summary, a study of what we can possibly imagine, and in particular, thinking backward from a state of contradiction, offers a comprehensive perspective upon Everything as needed for absolute truth. But such a perspective suggests that the most basic truth is admission of inherent self-contradiction.

Substantive theories express this when we have views and concepts but not yet statements.
Deflationary theories express this when we have statements but not yet a formal System.
Formal theories express this for statements within a formal System such that truth is the unique referent beyond it.
Pragmatic theories express this as our honesty in our self-critique, our aspirations and our vulnerability.

A diagram that I made for myself...