Applying the Processes of Additive Combinatorics and Belief Revision to Address Intractable Disagreements and Cognitive Dissonance

by
Dallas F. Bell, Jr.

1. Additive Combinatorics

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Group theory is the study of groups or algebraic structures in abstract algebra derived from number theory, geometry, and algebraic equations. Permutation groups were developed from the search for general solutions to complex polynomial equations. In group theory, a wreath product is a specialized product of two groups based on a semidirect product.

A semidirect product is a specific method of putting two subgroups together, with one being a normal subgroup. The resultant semidirect product is a generalized product were a new object is added to objects already known. A semidirect product is a Cartesian product as a set with a specific multiplication operation. The Cartesian product is a direct product of sets.

A permutation group is a group whose elements are permutations of a given set and whose group operation is the composition of permutations in that group. Permutation group usually means a subgroup of a symmetric group. Applying a permutation group to the elements being permutated (group action) is made toward the study of combinatorics in mathematics and physics.

Additive combinatorics focuses on the additive properties of the finite subsets of an additive group. That group can be finite or infinite. An infinite group can be reduced to a finite group using the Ruzsa projection (from works by Imre Z. Ruzsa). For example, a set x of infinite integers can imbed x in the finite cyclic group without losing any combinatorial information if one is interested in the sums of no more than two elements of x at a time. Moreover, a finite set x can be modeled in that set by an equal set in a finite group. That projection method has several important uses in additive combinatorics and especially in the structure of belief revision in paraconsistent logic.

2. Belief Revision

Two contradictory propositions, in a real sense, cannot be true at the same time.the Law of Non-Contradiction. But it is reasonable for finite beings to structure a group of inconsistent and non-trivial subset theories in the process of reaching a threshold of evidentiary proof for or against an overarching proposition, as seen in additive combinatorics of finite and infinite groups. Logical systems satisfying that requirement are called paraconsistent logics.

Paraconsistent logic attempts to challenge the logical principle of, ex contradictione quodlibet (ECQ), anything follows from contradictory premises and deals with contradictions in a discriminating manner. The principle of ECQ explosion in most logics can be expressed by the premise (X Λ ¬ X), conjunctive elimination (X), weakening (X V Y), conjunctive elimination (¬ X), disjunctive syllogism (Y), and the conclusion (therefore Y). Paraconsistent logic is logic whose consequence is not deemed to necessarily be explosive and abandons at least one of the common logic processes. Many logisticians choose to reject the disjunctive syllogism. The idea being that, if not X, then X is excluded, so the only way X or Y could be true would be if Y were true. But if X and not X can both be true at the same time, the reasoning fails. (The common symbols in logic for "statements" are p, q, and r.)

In a true sense, both X and not X can't both be true at the same time. However, in reality observations may indicate that subsets of both group X and group not X may be true. Thus, when the finite observations are deemed true for a group they need to be aligned by moving the subset(s) to the appropriate side of the X or not X matrix. This will form a consistent logic that either X is true and not X is false or X is false and not X is true. The steps for addressing an unknown are (1) recognize the possibility of the unknown, (2) discriminate its group by either set X is true or set not X is true, (3) observe, (4) build set X and set not X, (5) and conclude when the threshold has been perceived to have been crossed that either X is true or not X is true.

A group could be an entity defined by its subsets ranging from the characteristics of physics concepts of blackholes to the theological attributes for the infinite God of the first cause of all effects. For example, the group of the existence of blackholes would have the sets of X (blackholes exist from evidence of mathematical proofs by Hawking, etc.) and not X (blackholes do not exist from evidence of violations of laws of quantum physics by Laughlin, Chapline, and professors at Case Western Reserve University, etc.).

In Anton P. Chekhov's play titled Ivanov, the character Anna Petrovna converts to Russian Christian Orthodox from Judaism. Belief revision is usually intended to reflect such models of updating rational beliefs held by cognitive agents. (The preceding definition was excerpted from an email exchange between Graham Priest, author of writings on paraconsistent logic and to be professor at City University of New York in 2009, and Dallas F. Bell Jr. in October, 2008.)

In the case of blackholes, the cognitive agents are the aforementioned physicists, the updating of information are the new discoveries that now seem to contradict the initial beliefs for the existence of black holes (X) or nonexistence of blackholes (not X). Another example could be the rational writer of a science book likely believes that his text is true but also knows that most complex works contain nontruth for a variety of reasons. That writer would be rational to somewhat inconsistently believe that his book is both true to a point and on some level not true at the same time.

There should not be an intractable disagreement (among rational agents) as long as there is not just a simple exclusion of one of the axioms. (The preceding opinion was expressed by Vladimir Voevodsky, winner of the 2002 Fields Medal, in an email exchange with Dallas F. Bell Jr. during October, 2008.) Sadly, some atheists now agree that effects can occur without a cause thereby attempting to illegitimately add to their set of not X for the premise of the existence of God. Such seems to be the true nature of intractable disagreements.

3. Intractable Disagreements

It has been argued that intractable disagreements exist due to third party interjections that unnaturally cause intractable discourse. On a superficial level that involves opposing self-interests, such an explanation may have some merit. We know that there are immutable truths in our value-laden world, e.g. 2 + 2 = 4. If someone elects not to accept the truth that 2 + 2 = 4, then they would have an intractable disagreement with those that do believe the reality. By that simple math example, it is evident that fault lies in the believer that 2 + 2 does not equal 4. The failure is either a lack of information, a deliberate error or some irrational response.

Solipsism is the philosophical position that only one's own experience can be known. The notion is that the self (mind/soul) is the only thing that we can know exists. This pseudo intellectual theory is often used in academia and is easily disproved. If we created all music then we should be able to play all musical instruments, but we can't. If we created all poetry we could write with the same complexity found in all poems, but we can't. If we were all that exists, we would not need language or vocabulary, as used in poems, to communicate with ourselves. Since we humans are not all that exists, (moral) standards are not meaningless and it is important to have goals to search for truth. That quest will likely lead to disagreements.

It is epistemologically important to examine the theological sets for the core of most intractable disagreement. The group of the existence of the God of the first cause of all effects could consist of the following two sets.

Set X (God exists)

--infinite

--omnipresent

--omniscient

--eternity, immutable

--Physical Natural Laws (truth, information, gravity, etc.)

--Natural Laws of Freewill (not murder, not lie, not steal, etc.)

--intellect, purpose, personality, wisdom, grace

--soul, salvation, atonement, justice, love, mercy

--hope

Set not X (God does not exist)

--no infiniteness

--no omnipresence

--no omniscience

--no eternity, no immutability

--no Physical Natural Laws (no truth, no information, no gravity, etc.)

--no Natural Laws of Freewill (no murder, no lying, no stealing, etc.)

--no intellect, no purpose, no personality, no wisdom, no grace

--no soul, no salvation, no atonement, no justice, no love, no mercy

--no hope

Those infinite sets of subsets presented in a finite range reach a threshold of belief from symbiotic logic, observation and experience (including such writings as the Pharisee Josephus's Antiquities of the Jews), and revelation (Bible). Set X is rationally accepted as true and not X can't be rationally accepted as true. Human souls, proven by consciousness etc., can interact with God based on the subsets or attributes of the living creator God. Yet there are many people that have ears but do not hear and have eyes but do not see the evidence (Matt. 13:13-23). They will have an intractable disagreement with believers unless a belief revision is made.

Jesus knew the woman at the well was seeking atoning salvation from the Messiah. When Jesus presented her with the new information that He was the Messiah, she and many others had a belief revision and received salvation. (John 4:4-42). On the other hand, Jesus did not engage in debates initiated by lawyers and Pharisees for the benefit of their belief revision. He defeated their logic for the belief revision of others (Luke 10:23-37). Jesus replied to Satan, identified by Jesus as the father of the Pharisees, in the wilderness by being factual and self-denying according to God's words which glorify the Father God (Matt. 4:3-11). Paul's Sermon at Mars Hill sought to address his listeners that were seeking new information concerning the unknown God. He declared that this was the God of the first cause of all effects (Acts 17:18-34).

It is known that people will have the tendency to defend their belief status quo unless their needs and aspirations are factored into the argument. A recent paper suggests that people assign significant future value to victory at auctions over bids of other people even at a loss but do not behave this way when the opposition is a nonhuman computer.

Benefits and features favoring belief revision should be made in debate. It is known that if their belief status quo is attacked, they will retreat into an impregnable defensive posture. This will make belief revision a less likely outcome and make an intractable disagreement a more likely outcome, even if the new information causes cognitive dissonance. It is the experience of Phillip Johnson, Berkeley law professor/author and seasoned debater, that logical argument does not convince anyone to change a core position. The goal is to persuade those open to persuasion. (Phillip Johnson's opinion was expressed in an email exchange with Dallas F. Bell Jr. in November, 2008.)

4. Cognitive Dissonance

Cognitive dissonance describes a psychological state where beliefs are at odds. The stress of this circumstance motivates people to resolve the inconsistent issue. Social psychologists specialize in the application of cognitive dissonance for individuals in society. (This sentiment was expressed by Paul Bloom, coauthor of cognitive dissonance studies at Yale University, in an email exchange with Dallas F. Bell Jr. during October, 2008.)

Children and monkeys have been studied to find how they rationalize their choices to avoid cognitive dissonance. In one study, monkeys were given a choice of either of three M&M colored candies; one red, one green, and one blue. If the monkey chose red over blue, the monkey was given another choice between blue and green. Almost two-thirds of the time the blue M&M that was rejected the first time was rejected again and the green M&M was chosen. The belief was that this choice was due to not wanting to think that a wrong decision had been made with the first choice to reject the blue M&M.

The interpretation of the monkey study with M&Ms has been clouded by the Monty Hall game. In this game, one of three doors is chosen for a prize by the player. One of the remaining two doors will be opened by the host to reveal a lesser prize. A second choice is made by the player of whether to keep the door chosen first or to switch it for the remaining door. The odds for the best prize favor switching to the last door but most people seem to stay with the original door chosen even thought they have less odds of receiving the best prize.

Cognitive dissonance can be avoided by new information, such as statistical evidence. A method was developed to make better group decisions called Delphi. Three to five heterogeneous experts in different fields make a decision for a course of action. Unfortunately, congresses are groups of elected officials with no expertise in the areas they are expected to make decisions about and expectedly make numerous laws that harm societal efficiency. The jury system is closer to reflecting a Delphi process. (The information on the Delphi method was excerpted from an email exchange with Murray Turoff, Delphi coauthor and U. S. defense analyst, and Dallas F. Bell Jr. in October, 2008.)

The 1908 book, Orthodoxy, by Gilbert K. Chesterton links Christianity to promotion of the governmental system of democracy. Chesterton explains how a person can come to believe the Christian faith and does not waste the effort of attempting to show why it is to be believed. He believed it is not an arbitrary truth from outside the boundaries of human experience. Instead, Christianity is the answer to natural human needs.

5. Conclusion

Additive combinatorics can be used as a tool for groups. Belief revision is a natural process for acceptance of new information. When new information is rejected intractable disagreements can result as seen in the fictional story of Dostoevsky's The Grand Inquisitor (found in the work The Brothers Karamazov) where Jesus was arrested and confronted by the anti-Christ priest who is a latter day Pharisee. Richard Dawkins' algorithm in his book, The Blind Watchmaker, makes the same 'Faustian' (from a work by Johann W. Goethe) error as Wolfram's algorithm to disprove the existence of creative intellect and purpose (God). Humorously, their algorithms required each of their purposeful intellects to be created. They hear but do not understand and they see but they do not perceive (Is. 6:9). That circumstance causes an unnecessary state of cognitive dissonance.

King Solomon wrote, "that which has been is now, and that which is to be has already been, and God requires that which is past because He will judge the righteous and the wicked" (Eccl. 3:15-17).