Prudence, Prejudice, and Probability

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One form of modern relativism states that to understand, “You need to weigh out the probabilities.” (at minute 45:30) From this perspective, we can surmise, but we cannot know definitively. This essay proposes that probability applies in the absence of definitive knowledge (prudence and prejudice), but does not apply to what is definitively knowable (God exists).

The Background

Consider three words and their complements:

Truth and falsehood

Knowledge and ignorance

Certitude and doubt

All three refer to what is known as well as to the human knower. Truth is the conformity of human judgment to reality. Knowledge is of reality. Thus, both of the first two words are primarily characterized by reality, i.e. as the known. In contrast, certitude is a characteristic of the knower. As a characterization of and by the knower, certitude is subjective. I can have my certitude and you can have your certitude. They may differ, but as certitude, they are equally valid. This is not the case with truth and knowledge. Truth and knowledge can be viewed as objectively independent of us as knowers.

It can be dangerous to take, as a next step, the use of the word, probability, as a synonym for personal certitude. This subtly changes the meaning of personal subjective certitude. Instead of my discussing my personal certitude of the truth of a proposition, I discuss the probability of its being true. I switch the focus from my subjective qualitative judgment to what appears to be a mathematical assessment and thereby an objective assessment.

Human Virtues

Knowledge is a human virtue, a habit which enables us to seek the truth.  However, as human beings we don’t merely seek knowledge, we also seek to act upon it. To act upon knowledge well is to exercise the virtue of prudence. It should be noted that no human can act with complete knowledge. Thus, the virtue of prudence enables one to act with a mixture of knowledge and ignorance in seeking a goal.

Acting on the same bases of knowledge and ignorance, one person will prove consistently to make prudential decisions while another’s decisions prove consistently to be imprudent. In the former instance, the course of action achieves the intended goals. In the latter, it does not.

We are all faced with making prudential decisions in our daily lives. In what type of instance should prudential judgments be entrusted to a prudent person and in what type would it be more prudent to rely on algorithms based on mathematical probability? As a rule of thumb, we should rely on the prudential judgments of individuals (including ourselves) where the action directly involves humans and on algorithms where the action primarily involves non-human processing. Examples of the former would be governance, livelihood, schooling, and matters of friendship and family. Examples of the latter would be clinical medical trials and manufacturing. Coaching decisions in professional sports would be an example of a mixed bag.

Algorithms employing the principles of probability are suitable to the exercise of prudence because decisions of action are necessarily made with limited knowledge. i.e. with some ignorance. Whereas the virtue of prudence is concerned with action, the virtue of knowledge is primarily concerned with the natures of things. Consequently, a mathematical probability is not typically suited to gaining knowledge because mathematical probability is purely logical. It is an abstraction from all the properties of things except for enumeration. Enumeration is common to all entities irrespective of their natures, while it is their natures which render them knowable in themselves.

Mathematical Probability Is Purely Logical

One demonstration that mathematical probability is purely logical is the comparison of material density to mathematical probability. Let the material density of rotten oranges in a population be 1 rotten orange per 10 oranges, 1 R / 10 O’s. The corresponding probability of a rotten orange is 1/10, which is purely a number, having no further qualification. The material natures of the elements of a subset and a set are entirely irrelevant to the value of probability, but essential to the concept of material density. In probability, it is not the natures of material things, but their assignment to nominal sets and nominal subsets that matter (pun intended).

Another demonstration of the purely logical character of probability is that the probability relationships among the following sets are identical: three paper clips and seven envelopes, three elephants and seven giraffes, and three giraffes and seven envelopes. The IDs of elements, subsets, and sets, vis a vis probability, are purely nominal. (See end Note 1)

A third demonstration is that material analogies are illustrations, not examples, of mathematical probability. First, consider a case in which an analogy does appear to be an example, i.e. where there is a one to one correspondence between a concept and its material equivalent. Mathematical probability is the ratio of a subset to a set. The illustration of a probability of 1/52 would be the shuffling of a deck of playing cards with the top card turned face up. The shuffling and turning of the top card would materially correspond to random selection resulting in a probability of 1/52, where the top card is the material subset and the entire deck is the material set.

In a second case, shuffling again corresponds to random selection from a set. The sequence of the deck is the subset. Its probability is 1/52! The deck as a sequence corresponds to the subset. The set, from which selection was made and to which the denominator of probability corresponds, is a set of 52! subsets of playing cards. However, such a set can have no material existence. At 8.06 × 10^67, it is too large to be material. It can only be a logical set. Yet, the probability of 1/52! is just as legitimate a probability as 1/52.

The Anthropic Principle and Probability

It is the third demonstration above, which negates any argument invoking the anthropic principle where the argument is based on rejecting probability as being relevant because the calculated value of probability is too close to zero. The following quotation is an example of rejecting probability as an explanation of the fine-tuning of the environment for life on earth, where the rejection is based on the numerical value of probability:

“Most reasonable and responsible individuals would not attribute this to random occurrence (because the odds are so overwhelmingly against it), and so, they look for another explanation”

The statement is illogical. Probability is or is not an explanation independently of its numerical value between zero and one.

The anthropic principle is said to be based on the probability of several physical constants, the numerical values of which make human life possible on earth. Personally, I have not measured any one of these. Consequently, any conclusion of an argument based on their numerical values, I would have to take on faith in other humans. I am ready to place my faith in others, if it is prudent to do so or if there is no other recourse.

However, I am confident that any philosophical conclusion which could be drawn from an anthropic principle based on faith in others, can be drawn from my own personal experience. That is the beauty of philosophy. It is based on common experience. Philosophy does not depend upon unusual or technical experience. Consider this observation as the basis for an anthropic principle: I have noticed that my breathing organs work well in air, but not in water, while I live in air and not in water. Yet, water is necessary to quench my thirst and to keep me alive. Where I find air and where I find water are well suited to human life.

The Jargon of Prejudice vs. the Jargon of Truth/Belief in the Application of Bayes’ Theorem

Prejudice is the judgment of the presence of an unobserved property, X, based on the presence of observed property, A. Let Set A be the set of elements possessing observed property, A. Let Subset X be the subset of elements of A, which possess the unobserved property, X. Then we can quantify prejudice X as the ratio of Subset X to Set A, i.e. the probability of unobserved X given observed A.

Given (1) the numerical quantification of prejudice X based on the observation of A, (2) another numerical factor, and (3) the presence of observable property, B, we can employ Bayes’ Theorem to quantify prejudice X with respect to B. Given the proper information, the reiterative application of Bayes’ Theorem enables the successive quantification of prejudice X with respect to observed C, D, etc.

Rather than employing the jargon of prejudice, we could quantify ‘belief in’ or ‘the truth of’ unobserved or unverified X, given the ‘truth of’ i.e. the observation of A. That quantification would be the probability of X with respect to A. In this jargon, the successive iterations of Bayes’ Theorem quantify the ‘validity’ of  belief in, or the ‘truth’ of, X given the truth of observed B, C, D, etc.

I prefer the jargon of prejudice because prejudice implies deliberate ignorance of property X, rather than its indirect verification. In contrast, the jargon of truth/belief implies that the algorithm is seeking to verify the truth of X or validity in the belief of X. However, Bayes’ Theorem cannot achieve verification or validity with or without iteration, except in cases where the probability is zero or one.

Bayes’ Theorem is thus well suited to the exercise of the virtue of prudence because prudence concerns taking a course of action without full knowledge. It is also well suited to quantifying prejudice because prejudice requires the preservation of ignorance. It is not well suited to any instance where the presence of property X, or if you will, the truth of X as a proposition, can be directly verified.

Proofs of the Existence of God

The conclusion of each proof of the existence of God is: There must exist a being beyond human experience whose nature is To Exist. Its nature and act of existing are identical. This conclusion is reached in order to explain the existence of the entities within human experience. In the words of St. Thomas Aquinas, it is in the conclusion of an argument that both the nature and the existence of a being are coincidently realized:

“(T)here must be some being which is the cause of the existing of all things because it itself is the act of existing alone.” (On Being and Essence, translation by Armand Maurer, 1949, p.47)

We have this same concept of God from revelation. When Moses asked God to identify himself, Moses received the reply: “I am who I am.” He said further, “Thus you shall say to the Israelites, ‘I am has sent me to you.’” Exodus 3:14. Similarly, when Jesus of Nazareth claimed to be God: Jesus said, “Very truly, I tell you, before Abraham was, I am.” John 8:58. In common speech in recognizing the existence of God as creator, we are expressing the conclusion of an implicit rationale that there must be a being whose nature is to exist, in order to explain the existence of things which we directly experience.

Conclusion

Typically, determining the truth of a proposition entails an examination of the nature of things directly. It does not involve probability, which is purely logical and is suitably employed in matters of prudence, prejudice, and, in some cases, of the ignorance of scientific details, which details cannot be directly assessed.

Note 1:

Although the syllogism is purely logical in form, its premises must be in accord with the natures of the ‘sets’ discussed. Example: Every fox is human. John is a fox. Therefore John is human. The form is logical, but the major premise is naturally false. The syllogism presumes that set composition is in accord with the natures of things. It matters whether or not foxes are humans. The conclusion of the syllogism is intended to be a definitive statement about the nature of something. In contrast, probability does not concern the natures implied by the ID’s of sets. Example: For a population of foxes and humans, the probability of a fox is 75%. John is a member of the population. The validity of the truth that ‘John is a fox’ is 75%. It doesn’t matter whether John is a fox or a human. Probability requires ignorance of the natures of things, except coincidently for values of zero and one.

Note 2:

The mathematical concept of probability is not solely expressed in terms of discrete elements and discrete sets and subsets. An extremely popular expression of probability is as a wave, specifically, a mathematical wave oscillating between the values of zero and one, thereby transiently assuming in a continuum all the values of probability over the range of definition of probability. When the wave is terminated, it collapses to a value of either zero or one, which are the endpoints of the range of probability. On January 16, 2016, on national television, the common method of generating such a mathematical probability wave failed. The official had to toss the coin a second time to generate the wave, which collapsed to a value of one. The Arizona Cardinals consequently received the ball to begin overtime. They scored a TD on that drive and won the playoff game over Green Bay.

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