the logic of coherence

Nathan Coppedge, / SCSU 2015/12/05 p.

THE LOGIC OF COHERENCE

NATHAN COPPEDGE

PHILOSOPHY DEPARTMENTSOUTHERN CT STATE UNIVERSITY

UNDERGRADUATE

This paper builds on a dialogue which occurred at Shagaev’s paper on Evolving Systems:

https://www.academia.edu/s/4cd5ae8775


THE LOGIC OF COHERENCE

ABSTRACT: This paper concerns coherence as a specific application which traces its roots to the problem with mathematical coherence. Since the problem is usually defined in mathematical terms, mathematics is assumed to be the main competitor to any coherent system, suggesting that resolving the problem is tantamount to replacing mathematics. However, for obvious reasons, it cannot be taken that seriously. But, the implications are still serious. This paper addresses the functions, background, and falsifiability of this new system of coherence, focusing on its relevance to the logical tradition. The goal of this paper is to provide a rational background for coherence, and specifically its applicability to forming universal knowledge statements. An important part of this will be implicating its equivalence to mathematics balanced with the weaknesses and limitations found in mathematics implicating the need for a different system of knowledge.

KEYWORDS: coherent knowledge, coherent mathematics, objective knowledge, coherence theory of truth, objectivity and subjectivity, objectivist philosophy, coherentism, categorical deductions, philosophical methodology, systematics, application to science, absolute knowledge, absolutism

I. MISSION

In this paper, the object is a system of real knowledge that is unfalsifiable through its universality.

What ultimately qualifies as a system is something contained and yet infinite, logical and yet linguistic, comparative and yet useful. (Systems could exist under different definitions, but they would have different criteria of success than coherence).


II. STRUCTURE

Essential Parts

In Math:1. Counting (infinity)2. Positive numbers.3. Negative numbers.

In Coherency:1. Circle (unity, recursion).2. Quantity (category).3. Quality.4. Exclusivity.5. Polarity.6. Analogy.


III. CHARACTERISTICS

Coherency lies in the analytic tradition, specifically proto-mathematics, set theory, and categorical systems. Its element of analysis fits into the tradition of Aristotelian logic, only with a different assumption network. The goal is to work in parallel with Aristotelian logic, although in practice the two systems may be separate. It is possible that the system branches as early as the choice between causal and non-causal inference. In this way, it is similar to a formal method for finding epiphanies, and bears resemblance to themes from information theory, emergent systems, and game theory. It has a nominal relation to scatter plots, using these as an assistive tool. but with potential relation to applied mathematics such as that found in social science. The most useful aspect of the system is categorical deduction, which has relation to linguistic sentence-forming, auto-generated texts, and analogical constructions. The system was constructed piecemeal from a variety of systems, including Cartesian Coordinates, dimensional and hyperbolic space, analogies, predicate logic, categorical logic and set theory, (Venn diagrams, exclusion proofs), etc. If successful, it would gain a unique status equivalent to ‘the only neutral system besides zero’.

IV. CHARACTERISTIC PROPERTIES

The system has the unique characteristic pattern (AB CD) and (AD CB) when expressed in quadra, differentiated from the analogical form A:B::C:D and A:C::B:D. A just-as symbol can be used in place of the double-conjunction, the difference being that the double-conjunction makes use of a variety of neutral Boolean operators, specifically [is, as is, just as, when, so, as such, as: these are ordered to refer to set levels up to 256 categories].

With somewhat greater difficulty, the system can also be applied to other types of category sets, usually modularizations of four.


The number of deductions (although not its expressed form) can typically be found by the following equations, assuming the conditions of whole values and modularity are met:

Even-numbered modules:2 ^ (M root of C).

Odd-numbered moduless:2 * {2^ (M root of C) - 1}

Where M is a modular value such as four for quadratic sets, and C is the number of categories in all of the lowest subsets of the whole module.

Note that in the case of quadratic modules, the number of deductions is always equal to the square root of the number of categories.

V. PREDICTIVE POWER

P (A+B+C+D) = 1That is, because no other words are assumed to express the same exact type of data. The use of the word ‘All’ for instance, would reflect all uses of the word ‘All’, just as the use of the word ‘Nothing’ would reflect all uses of the word ‘Nothing’. For, if ‘nothing’ has no uses, then suddenly we know everything.

If P(A) = 1, then we know that P (B + C + D) = 0, because that case expresses one in which (A) represents everything in the universe, because it’s opposite must then = nothing or some similar concept.

But, in practice P(A V B V C V D) 1 because even if the first diagonal axis represents everything and nothing, the other axis can represent some other concept.

Thus, because opposites must be positioned at the furthest distance diagonally, the only case in which

P {(A + B) V (B + C) V (C + D) V (D+A)}= 1 is a simple case in which EVERYTHING is ONE HALF, and NOTHING IS THE OTHER HALF.

I.e. because if {(A+B) V (B + C) V (C+D) V (D+A)} contained two 0.5s, then they would not be opposed by 0.5 in those two positions, since opposites are opposed diagonally, so the (0.5) value would already be used up, and the total must (= 1). Either values are equal, or values are inversely proportional. There is no alternative.

But, consider what happens when P (A V B) = 0.75.

Then, if opposites are mutually exclusive, then in the same case, P (C V D) = 0.25, e.g. because C is mutually exclusive with A, and D is mutually exclusive with B.

The result is that for whichever pair does not include (0.75), the probability must (= 0), since the total probability must equal (1).

Thus, the result for a value of 0.75 for any of the categories also results in a simple case, in this case one in which one of the two binary oppositions = nothingness.

Now, if the P(A + B + C + D) = 1, then if (A, B, C, D) are opposites that are mutually exclusive,

there is little alternative to the determination that P(A V B V C V D) 0.25 in the case of a quadratic arrangement. The only alternative is an equality in each diagonal axis, creating a series such as (10, 40, 10,


40) or (15, 35, 15, 35). These sets are completely identical to ( 0.25) except that they vary in an axis-to-axis relationship, which, if the terms are seen to be exclusive, can be justified expressly in terms of universal coherence.

Thus, it could be predicted that the probability of a given category within a set is always equal to 1 / C, where C is the total number of categories within the set, plus a universal qualifier for the difference between one and another axis of the set, expressed by its degree of exclusion.

It might be supposed that cases are valid where diagonal zed opposites are equal by not = 0.25 probability. However, these cases are circumvented by data exclusivity.

In practice, the calculation is simpler, since in knowledge terms each category is equally-well-founded so long as each term represents the best concept of exclusion.

Because the sets are exclusive, P(A V B V C V D) 0.25, and so the general rule for determining probability is 1/ C.

VI. EXAMPLES

Here are a variety of examples showing the system.

The simplest way to see it is that good is to bad as X is to Y, when X and Y are opposites. The deduction reached then says that good X must be correlated with bad Y and vice versa. That’s the key insight for the general form is due in part to Avi Sion, who said:

"If an affirmation is true, then its denial is false; if the denial is true, then the affirmation is false"1

Sion‘s method is intentionally creative, but he did not apply it to the most general case. Here are some examples of the application of the method to the most general case, as expressed in the earlier logic:

‘Something right is nothing wrong.’‘Something bad is nothing good’

‘Bad solutions are good problems’‘Bad problems have good solutions’

‘Visual imagination is blindness to a lack of inspiration’‘A lack of visual inspiration implies a blind imagination’

‘Infinite sun means less water’ (e.g. because water evaporates).‘Infinite water means less sunlight’ (e.g. because water is deep and therefore dark).

‘Infinite destruction limits constructions’ (e.g. because everything except the construction of destruction is destroyed).‘Infinite construction means there is finite destruction’ (e.g. because there can be nothing destroyed except destruction).

But examples also exist that are less obvious:

‘Ugly stoics are beauty-sensitive.’ (e.g. because they are ugly).‘Ugly sensitive people are beautiful stoics.’ (e.g. because they are sensitive to what they are not).

1 This is mentioned in passing, with little elaboration, in Avi Sion, Future Logic, Chapter 3.4. Axioms of Logic, B. The Law of Contradiction..


The limit of the applications is essentially the limit on the total number of words, word combinations, and capacity to realize opposites. If it is assumed that there is a word for every concept, or that every concept could have a word, and that it is possible to find an approximate opposite for every concept, then the system has at least approximate importance in the field of all conceivable ideas.

VII. FALSIFIABILITY

Rejecting naïve realism, non-polar opposites, and non-opposites, then given the CCS, we get a fair system.

The system rejects comparisons that do not concern real opposites. Thus, there is no way to compare black person to white person except by assuming that they are opposites, when in fact they are not: both terms fall under the category ‘human’ so they cannot be polar opposites.

Thus, these (what are called ‘non-polar’ opposite pairs) like Dog and Cat are NOT valid under this system, if both terms clearly fall into the same category, such as animals, or wisdom, etc.

Terms like Love and War are valid opposites, if they are assumed to be opposites, and the same might go for the above.

In other words, the system assumes that the concepts used in the comparisons are valid opposites, even if they are not.

It is for the person or computer using the system to determine if opposite comparisons are valid. This is a cursory procedure, and can be edited independently of the functioning of the deductions.

The terms, by being opposite, should be opposite and exclusive. Polar opposites are used as a definition of exclusive opposition.


VIII. BACKGROUND

Alan Hájek, who earned his PhD. in philosophy at Princeton in 1993, argues that “philosophy has a variety of heuristics” (1). And goes on to say that “such heuristics can enhance one’s ability to make creative contributions to philosophy.” (1).

As recently as 1972, Karl Popper did not even consider philosophical objectivity (in other words, progress) possible. He writes:

“From a rational point of view, we should not ‘rely’ on any theory, for no theory has been shown to be true, or can be shown to be true.” (Popper 21).

This is a strange absoluteness to hold, which must be self-contradictory.

Typology, which is perhaps the closest previous system for realizing coherence outside of mathematics, emerged as recently as the 1870s (145 years ago) with Charles Peirce. (“This theory of meaning (‘speculative grammar’) was to provide foundations for his writings in logic” --- Oxford Companion to Philosophy).

Nicholas Rescher’s book (1973), dates work on coherence somewhat more recently.2

A system of coherent logic can borrow some part of one or all of the systems mentioned under

characteristics.

IX. DEFININING COHERENT OBJECTIVITY

2 Rescher’s work is largely unhelpful, because his book falls into a probablistic reasoning mindset, thus falling prey to the mathematical compromises mentioned later.


“To speak of empirical determination independently of any reference to general laws means to use a metaphor without cognitive content” ----Carl Hempel

Objectivity is “[A theory] that various kinds of [judgment] are respectively objective, i.e. pertain to objects…” [Oxford Companion to Philosophy].

Coherence Theory: “A theory of truth according to which a statement is true if it ‘coheres’ with other statements.” [OCP].

In philosophy, objects can be anything material, and perhaps anything real. But this says nothing inherently about subjective reality (the reality of personal experiences or imaginative truth), while subjective reality is not treated as an object. The system does not say that it is impossible to treat subjects as objects. It simply does not define that it is necessary.

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the logic of coherence

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