A definition is a statement of the meaning of a word, whereas a synonym is another word of the same meaning. A synonym may serve as a definition, if the meaning of the synonym is already known, while one is ignorant of the meaning of the word.

It would appear as if a valid definition of probability is ‘the likelihood of an event’. However, likelihood is as vague a term as probability. This can be seen if likelihood is defined as ‘the probability of an event’. Thus, to state that probability is the likelihood of an event, is not to define probability, but to identify likelihood as a synonym for probability.

In contrast, probability can be defined without the synonym of likelihood. Probability is the ratio of the favorable cases of an event to the total of all possible cases or outcomes of an event. What is wrong with this definition? It is based on human ignorance of facts and not upon facts themselves.

An application of this definition will show that it requires a veil of human ignorance. It is said that seven as the sum of two rolled ‘fair’ dice is a probability of one-sixth. However, this requires the human ignorance of the forces to which the dice are subjected in rolling them. The actual forces to which the dice are subjected determine exactly the outcome of the event and the sum evident in that outcome. Given these forces, the outcome will be a sum of seven or a sum of non-seven. The result has nothing to do with other cases or outcomes. Thus, the roll of a pair of dice is always a singular determinate case, having nothing in fact to do with (1) ‘favorable cases’ and (2) ‘all possible cases’. There are no other cases, except in the human imagination. To characterize the singular case as a probability is to place it within human imagination, or more accurately, to veil it in human ignorance.

Does this mean that there is no definition of probability, which is not conditioned by human ignorance? No, the unconditional definition of probability is purely mathematical. It is the ratio of a subset to a set. For example the set of all possible sums of two integers, each of which may be an integer from 1 through 6, is a set of 36 elements. The size of the subset of sevens among this set is 6. Thus the probability of the sum seven for this set is one-sixth. When we play games involving the roll of two dice we typically adopt the relationships of probability of the subsets of this set of 36 as a mutual convention.

It should be evident that when we treat the definition of probability as the ratio of the favorable cases of an event to the total of all possible cases or outcomes of an event, we are proposing an analogy, which is based on human ignorance.

The definitions of random and non-random are also relevant to the meaning of probability.

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Several physical constants are said to be fine-tuned to providing the conditions necessary for life on earth. Is this pro, con, or neutral to the existence of God?

#### A Philosophical argument for God

Everything is explicable in its existence. The things of whose existence we know directly within the scope of human sensation we know through their properties, the intelligibility of which is explained by the natures of those entities. The one intelligible aspect of these entities, that is not explained by their natures, is their very existence. Therefore, there must exist an intelligent agent beyond human experience which explains its own intelligibility and its own existence as well as the intelligibility and existence of all other entities. This intelligent agent’s, this being’s, nature must be To Exist. If it were not, it too would require an intelligent agent to explain its existence. We call this being, God.

#### Relevance of Fine-tuning

Underpinning this argument is: Every intelligent effect is due to an intelligent agent. Fine-tuning of the universe, if the universe is fine-tuned, is an intelligible effect, which requires an intelligent agent.

#### A Philosophical Justification of Fine-Tuning without Positing the Existence of a God

Human artifacts are characterized by order. Human artifacts are the effects of human intelligence. The only intelligent agents, of which we are aware, are humans. Yet, when we observe order in natural things, we gratuitously identify such order as the effect of an intelligent agent. Since this agent, gratuitously generated by our unwarranted anthropocentric extrapolation, is, by definition, beyond the scope of human observation, we identify it as some super human, and call this figment of our extrapolation, God. The rational course is to accept order in nature as given and recognize that it is only the order of human artifacts, which we can identify as intelligible effects of intelligent agents, namely human beings. (This is Richard Dawkins’ argument. The God Delusion, p. 157, “In the case of a man-made artifact, such as a watch, the designer was really an intelligent engineer. It is tempting to apply the same logic to an eye or a wing, a spider or a person,” He claims that the order evident in a biological entity is not due to an intelligent agent, but is due to natural selection, the ordering power of environment.)

#### The Agnostic Assessment

An agnostic accepts neither of these arguments, content that both of their conclusions are equally probable, while neither argument nor probability nudge him toward or away from belief in God.

#### Probability and the NBA Finals

I happened to watch ESPN’s show, ‘First Take’, this spring when the topic was, ‘What is the probability, P, that the Cavs will be in the 2018 NBA finals?’. Max Kellerman claimed that in answer one cannot simply state a value of probability. One must propose: a numerical probability, R1, of the Cavs’ winning round one of the Eastern playoffs; a numerical probability, R2, of winning round two; and a numerical probability, CF, of winning the Eastern Conference Finals. P would then be the product of these.
P = R1 × R2 × CF

#### Constraint Imposed by an Equation

Of course the algebraic relationship of these four terms, restrained by the equation represents three degrees of freedom. If we are allowed to assign numerical values from 0 and 1 to three of them, we can calculate and are constrained to calculating the fourth. It is apparent that Kellerman thinks there is also a temporal constraint because the temporal order of the series is R1, R2, CF. However, probability is simply a numerical ratio, lacking all units of measure, including units of time. Thus, Kellerman’s constraint is arbitrary. One could exercise one of the three degrees of freedom by assigning a numerical value to P, and thereby answer the topic question without resorting to the equation.

#### Nomenclature

Further, what is being quantified is personal opinion. The probability, P, can be labelled the probability of personal opinion. Given R1, R2, and CF, the equation expresses the probability of personal opinion that ‘the Cavs will reach the NBA finals’. Could we just as well say that the equation expresses or measures the validity of the belief that ‘the Cavs will reach the NBA finals’? Could we just as well say that the equation expresses or measures the truth of the proposition that ‘the Cavs will reach the NBA finals’?

If we can’t because we are merely expressing personal opinion, why should be allowed to make up the numerical values of probability of another equation and claim that, by means of that equation and our made up values, we are not merely expressing personal opinion, but that we are estimating the validity of a belief or estimating the truth of a proposition?

Probability is the ratio of a subset to a set, where the IDs of the elements are nominal. One subset may be labeled true, another false, but these have no real significance, other than complementarity, because every analogy of mathematical probability is merely an illustration of mathematical relationships and not an actual example of mathematical probability. The mathematical relationships of probability are an abstraction of all properties represented by the nominal IDs, except for enumeration.

If one is allowed to assign numerical values to the terms of an equation, the equation constrains this assignment to one less than the total terms, but it does not necessarily identify that one term even if there is only one term to the left of the equals sign.

#### The Weighing of Evidence

One may think that arriving at a numerical value of probability, based on an equation, is evaluating the truth of a proposition, as if the consideration of probability was the basic mode of assessing truth. This bias is reinforced when choosing a numerical value of probability is described as the weighing of evidence pro and con with respect to some proposition. However, deciding whether an element of evidence is to be placed in the pro set (on the pro side of the balance) or in the con set requires some criterion of the nature of the evidence. It too cannot be decided by probability.

#### Conclusion

Personal opinion is personal opinion, whether or not we dress it up in the guise of mathematical probability. The basis of truth is the nature of the object of knowledge, not probability.

From arguments for the existence of God illustrated by human artifacts, such as trains and chains, it would seem that God as first cause is a numerical first of a series of causes.

In accepting for Catholic Stand my essay, “Trains and Chains and God as First Cause”, managing editor Anthony Lane, proposed two objections for my consideration.

Your definition of a cause, on which your argument depends, holds that it explains itself. But St. Thomas’ Second Way argues from efficient causes, stating explicitly that the intermediate causes are effects of prior efficient causes (cf. STh I, Q.2, A.3 resp.), and as such do not explain themselves. St. Thomas puts it “in the world of sense;” i.e., it is an empirical observation rather than an a priori axiom, which pitches the argument at the unbeliever who has not yet conceded an invisible, immaterial order. This, I believe, is why so many apologists invoke chains and trains.

The answers, which follow, would be too lengthy as comments to the essay.

In my words, Tony’s objections are: (1) my claim that causal sequences are irrelevant to proving the existence of God as the First Efficient Cause of the existence of a created being (and all of creation) is ostensibly a contradiction of St. Thomas Aquinas’ Second Way in the Summa Theologica and (2) my definition of a cause, in its inclusion of sufficiency, is gratuitous.

#### Efficient Causality and Sequence

In P1, Q2, A 3, St. Thomas writes:

The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.

One thing to note is that the argument, as stated here, is generic. It makes no clear distinction among types of efficient causes such as that between an efficient cause of local motion and an efficient cause of the existence of an entity. However, God is not a cause at the level of local motion. Therefore, even if it is true that every causal sequence must have a first cause, to concede such would not lead to the specific conclusion of the existence of God.

#### God as the Cause of Existence

In contrast to PI, Q2, Article 3, St. Thomas’s argument for a First Efficient Cause at the level of existence in On Being and Essence, is not about causal sequences.

“But it is impossible that the act of existing of a thing be caused by a thing’s form or its quiddity, (I say caused as by an efficient cause); for then something would be the cause of itself and would bring itself into existence ‒ which is impossible. Everything, then, which is such that its act of existing is other than its nature must have its act of existing from something else. And since every being which exists through another is reduced, as to its first cause, to one existing in virtue of itself, there must be some being which is the cause of the existing of all things because it itself is the act of existing alone. If that were not so, we would proceed to infinity among causes, since, as we have said, every being which is not the act of existing alone, has a cause of its existence. (translation by Armand Maurer, 1949, p.47)

The crucial clause in this argument is “because it itself is the act of existing alone.” St. Thomas should have ended with that. The next sentence about an infinity of causes is a distraction from the argument. The conclusion is that the existence of any entity, whose quiddity is not its existence, requires as the cause of its existence, the existence of a being, who is solely its act of existing. Then what possible intermediate could there be between these two, the cause and its effect? Surely, not another entity whose quiddity is not its act of existence, i.e. another created entity. Surely, not another entity, who is solely its act of existing, i.e. another God. In the context of efficient causality at the level of existence, why bother stating that the number of intermediates could not be an infinite number, when it can only be zero?

#### Efficient Causality of Motion: An Eternal Universe or Not

I believe Tony gives the answer. St. Thomas is presenting a persuasion relevant to ‘the world of sense’, rather than a logical argument. He is referring to causal sequences at the level of local motion as not being infinite. St. Thomas interprets Aristotle as presenting a persuasion rather than a definitive argument when Aristotle claims that the opposite is true, namely, that the universe of creatures always existed, i.e. may be understood as regressively eternal.

Secondly, because wherever he (Aristotle) speaks of this subject, he quotes the testimony of the ancients, which is not the way of a demonstrator, but of one persuading of what is probable.
Thirdly, because he expressly says (Topic. i, 9), that there are dialectical problems, about which we have nothing to say from reason, as, “whether the world is eternal.” (P1, Q46, A1)

In contrast to Aristotle’s philosophical position of a possibly eternal universe, St. Thomas maintains that whether the universe of creatures always existed cannot be answered definitively by philosophy. However, St. Thomas maintains that it is known from revelation that the universe did have a beginning.

I answer that, By faith alone do we hold, and by no demonstration can it be proved, that the world did not always exist, . . . Hence that the world began to exist is an object of faith, but not of demonstration or science. (P1, Q46, A2)

However, if it cannot be philosophically demonstrated that the world did not always exist, it must be that there can be no definitive philosophical argument against an infinite regression of local motions. In other words, it is not possible to prove philosophically that there is a first cause of local motion, if local motions can be viewed as a series of efficient causes and the universe might be eternal.

#### Sense and Sequence

It is here where Tony has identified the source of the problem, namely, in the world of sense. Human intellectual knowledge is extrinsically dependent upon sensation. Consequently, our thoughts are always particular and serial. To proceed from one intellectual thought to another, we must replace one composite of sensation with another. Human intellectual activity is rational, step by step, serial. Human intellectual knowledge is not intuitive. It requires reasoning.

We know intuitively the two self-evident principles that things exist and are inherently intelligible in their existence. Most of our knowledge is step by step, just as our sensual imagining is serial.

We are inclined to think that the past is a closed system and serial. As closed, we think the past must be finite, because no human intellect can process an infinite set. In this, we take a limitation of the human intellect to be a characteristic of the object of knowledge. In contrast, we imagine that the future as open and, thereby, as a potentially infinite set of finite stages.

#### Time and Eternity

Another problem arises in our appreciation of time and eternity.

The now of time, which we experience, has no duration or quantity. It is always now. Then in what sense is now, time? Time is the quality or condition of mutability. We and all other material things are always changing. One of the most obvious mutations is local motion. But, if time is a quality or a condition, which we experience as a nonextended now, how are we able to identify time as a quantity? Quantitative time is a human thought, a human act. It is the mental comparison of one motion or change with another motion of our choice, preferably some cyclic local motion. The human quantification of time is such a grand utility, that we always think of time as a quantity. However, we do so erroneously, if we think of time as an objective quantity independent of human action. Time is not a quantity. Rather, time is actually a quality, the mutable condition of our existence.

In the condition, which is time, entities are always becoming. Entities that exist in eternity are not subject to change. They are all they can be. They do not exist in or experience time, because they do not and cannot change. They exist fully as they ever will. Just like our experience of our now, their now has no quantitative duration.

#### Persuasion

I believe it is permissible for St. Thomas to have proposed that Aristotle’s argument for an eternal universe of creatures is merely meant to be persuasive. Similarly, I believe it is permissible for me to propose that St. Thomas’s Second Way regarding generic efficient causality as presented in the Summa Theologica is intended to be persuasive, while his argument in On Being and Essence, regarding efficient causality at the level of existence, is intended to be and is philosophically definitive.

The human, step by step thought process, which is dependent upon our serial sensual imagining, colors our appreciation of the very concept of causality, such that we are inclined to talk of causal series of insufficient efficient causes without noticing the self-contradiction in doing so.

#### ‘Insufficient’ Causes or Not

Now, let us consider the objection: Whether the definition of a cause, as requiring sufficiency, is gratuitous in the context of cause and effect.

It is obvious that arguments for a first cause, which employ analogies to the human artifacts of trains and chains as causal sequences, verbally claim to prove the existence of a first cause as the regressive terminal of a finite series of causes. I propose that they do not demonstrate what they claim, even if we accept in principle that there can be no infinite series of finite causes. In fact, what they do demonstrate is that the concept of cause and effect requires that the cause be sufficient.

Let us shorten the train analogy down to the caboose coupled directly to the locomotive. The local motion of the caboose is the effect of the locomotive, which is the cause of the motion of the caboose. Let us add one intermediate cause, namely a unit of rolling stock, a boxcar, between the caboose and the locomotive. In the analogy, the boxcar is labelled an intermediate efficient cause. The motion of the caboose is caused by the motion of the boxcar, whose motion is caused by the locomotive. The boxcar is an intermediate and insufficient cause in a series, which must be finite and terminate in a first cause, the locomotive. Is that a valid interpretation of first cause? No. We could just as well view the boxcar, not as an intermediate and insufficient cause, but as part of the coupling mechanism between the caboose and the locomotive. It is then apparent that the locomotive is the first cause of local motion in this analogy because it is the sufficient cause of the local motion of the caboose.

Let us add 59 more units of rolling stock, intermediate between the caboose and the locomotive. A total of 60 boxcars may be viewed as 60 intermediate and serially insufficient causes of local motion or they could be viewed as one big coupling mechanism between the effect, the motion of the caboose and its sufficient efficient cause, the locomotive. In comparing these two views, it should be apparent that the serial sequence of ‘insufficient’ causes must indeed terminate in a first cause, i.e. first in the sense of sufficiency, and not first in the sense of the terminus of a regressive finite series.

The same conclusion is reached in an analysis of a static or hierarchical sequence of so-called intermediate causes. One such popular analogy, identified as serial causality, is the suspension of a chandelier from a superstructure, nominally the ceiling of a room. If we hang the chandelier directly to the superstructure we have the suspension of the chandelier as the effect and the superstructure as cause of suspension with no intermediates. Consider two other scenarios with intermediates. In one scenario, a rope of 10 feet is placed as an intermediate between the superstructure and the chandelier. In the other scenario, sixty 2-inch chain links are placed as intermediates between the superstructure and the chandelier. In both scenarios, is the superstructure the First Cause of the suspension of the chandelier because, it is numerically prior to an ordered series of insufficient causes whose sum is finite? Or, is it the First Cause because the superstructure is the sufficient cause of the entire suspension? The 10-foot rope is composed of many overlapping fibers. It is hard to imagine the fibers of the rope as a linear series of sequential intermediate, but insufficient causes. In contrast, it is easy to imagine the discretely sequenced sixty 2-inch links as a series of intermediate, but insufficient causes. The lesson is that we should not let our imaginations dictate the conditions and limits of reality.

My judgment is that the analogies of trains and chains demonstrate that the concept of cause and effect requires that there be a ‘first’ cause in the sense of sufficient causality, not that a series of ‘insufficient’ causes plus its terminating cause must be a finite number. Indeed, what these analogies demonstrate is that an ‘insufficient’ cause of an effect is an oxymoron, which may be employed serially for the sake of persuasion outside of a definitive philosophical argument. Its persuasiveness is partially due to the serial nature of the process of human thought, dependent as it is on sensual imagination.

Also, to deny causality to ‘serial insufficient causes’ is not to deny that they are causal factors and, in that sense, secondary causes. The train analogy, in which the each boxcar is identified as an intermediate cause, does not deny that the couplers between the cars are not causal factors. Note that it is precisely in acknowledging the causal insufficiency of the boxcars by labelling them ‘secondary’ causes, that it is apt to identify the locomotive as a ‘first’ cause due to its causal sufficiency.

#### Conclusion

It should be noted that St. Thomas’s Second Way in the Summa Theologica does not imply that intermediate serial efficient causes are insufficient causes as the analogies of trains and chains make them out to be. He implies that causes appear to be serial “In the world of sense”, i.e. to our senses, as Tony noted. The nature of sensation is serial and so we are inclined to attribute our mode of sensation to the nature of causality.

Our love of, as well as, the impressive and indispensable utility of quantification prompts us mistakenly to perceive time fundamentally as a quantity, when it is fundamentally a quality, the condition of mutability. We quantify local motion as the temporal rate of location. Accordingly, we portray efficient causality at the level of local motion as a series of causally insufficient, quantifiable segments (Zeno of Elia, c. 450 BC, similarly proved that local motion must be an illusion. It would take an infinite number of finite segments of time to cross the room)

We judge serially segmented mediation to be a generic characteristic of efficient causality. In doing so, we include efficient causality at the level of existence. However, this contradicts the conclusion that there can be no intermediary, where (1) the effect is the existence of a mutable entity and (2) the cause must be the entity whose nature is its very self-existence, the immutable, “I AM” (Exodus 3:14).

We should be wary of any argument for the existence of God that features an analogy to human artifacts such as trains and chains. St. Thomas presents no such analogies in P1, Q2, A3: Whether God Exists. An elaboration of P1, Q2, A3 does not require any consideration of series.

Richard Dawkins’ most important contribution to our understanding of Darwinian evolution is identifying the role of gradualism as increasing the efficiency of mutation while having no effect on the probability of success of natural selection. He outlined that role in The God Delusion (p. 122), proposing two analogies of gradualism:

(1) “On the summit sits a complex device such as an eye or a bacterial flagellar motor. The absurd notion that such complexity could self-assemble is symbolized by leaping from the foot of the cliff to the summit in one bound. Evolution by contrast goes around the back of the mountain and creeps up the gentle slope to the summit.”
(2) “Theoretically a bank robber could get lucky and hit upon the right combination of numbers by chance. In practice, . . . this (is) tantamount to impossible. . . . Suppose that when each one of the dials approaches a correct setting, the vault door opens another chink, and a dribble of money trickles out. The burglar would hone in on the jackpot in no time.”

On page 121 of The God Delusion, Dawkins labels a large single stage of Darwinian evolution as ‘a single, one-off event’ in contrast to the gradualism of a series of sub-stages, which he describes as incorporating ‘the power of accumulation”.

In 1991 in a lecture on Climbing Mount Improbable at the 4:30 minute, Dawkins gave a numerical illustration of the replacement of a single large Darwinian stage of mutation and natural selection. The large stage was replaced by the gradualism of an equivalent series of Darwinian sub-stages. The single large stage generated mutations based on three mutations sites of six mutations each, thereby defining 216 different mutations. The replacement series consisted of three independent sub-stages. In each sub-stage one mutation site was subjected to Darwinian evolution independently of the other two sites. Each sub-stage defined six mutations. However, the beginning mutation of the second sub-stage was a duplicate of the ending mutation of the first sub-stage and similarly the beginning mutation of the third sub-stage was the ending mutation of the second sub-stage. Consequently, there were 3 x 6 = 18 total mutations defined by the three sub-stages, but only 16 different mutations.

In the single large stage, there were 216 different mutations liable to generation. In the equivalent, series of sub-stages, there were only 16 different mutations liable to generation. That amounts to 200 different mutations of the initial spectrum, which were missing due to the role of gradualism of sub-stages in Darwinian evolution.

Dawkins demonstrated that the role of gradualism in Darwinian evolution is to reduce the spectrum of mutations. The missing mutations cannot be subject to generation due to the gradualism of the series of evolutionary sub-stages. Dawkins’ numerical illustration implies his algebraic equation for the number of mutations missing from the spectrum of evolutionary mutations, which would be liable to generation in a single large, or one-off, stage of Darwinian evolution.

#### Dawkins’ Equation of Missing Mutations

Let M be the number of different mutations defined by a single, one-off Darwinian event, i.e. a large stage of Darwinian evolution. In Dawkins’ illustration, M = 216.

Let M be factored into n sub-stages, each defining the same number of mutations. The number of mutations per sub-stage would be M^(1/n). In Dawkins’ illustration, n = 3, while M^(1/n) = 216^(1/3) = 6.

The total number of mutations defined by the series of sub-stages is n[M^(1/n)]. In Dawkins’ illustration n[M^(1/n)] = 3 = 18. The number of different mutations is n-1 less than the total number. In Dawkins’ illustration the number of different mutations is 18 – (3-1) = 16.

The number of mutations in the single, one-off stage, which are missing from the series of sub-stages is readily calculated using Dawkins’ Equation of Missing Mutations, m:

m = M – n[M^(1/n)] – (n-1)               Dawkins’ Equation of Missing Mutations

In Dawkins’ illustration, m = 216 – 18 – 2 = 200.

In Dawkins’ illustration, only 16 of the mutations defined in the spectrum of 216 mutations are liable to be generated due to the gradualism of the series of sub-stages of Darwinian evolution. This is the basis of the mutational efficiency of Darwinian gradualism, the understanding of which we owe to Richard Dawkins.

#### Two Other Numerical Examples of Dawkins’ Equation

The human genome has roughly 3 billion base-pair sites, while the fruit fly genome has roughly 165 million. Each base pair site represents four possible mutations.

If only 8 base-pair sites of a genome were subject to mutation in a single, one-off stage of Darwinian evolution, these 8 base-pairs would define a spectrum of M = 4^8 = 65536 different mutations. Replacing the one-off stage with n = 4 sub-stages of equal size results in each sub-stage defining M^(1/n) = 16 mutations. However, n – 1 of these are duplicates. We then have the information to calculate the number of missing mutations, m, from the spectrum of 65536, due to the series of sub-stages:

m = M – n[M^(1/n)] – (n-1)                Dawkins’ Equation of Missing Mutations

m = 65536 – 4  – 3 = 65475

There are 65536 different mutations liable to be generated in the spectrum of the one-off stage of Darwinian evolution, affecting 8 base-pair mutation sites of a genome. However, employing Dawkins’ Equation for missing mutations, 65475 of these are missing from the total different mutations which are liable to generation in the equivalent series of four sub-stages of Darwinian evolution. That leaves only 61 of the full spectrum of 65536 as liable to generation in the series of sub-stages. This is the basis of the increased mutational efficiency, which is the role of sub-stage gradualism in Darwinian evolution, as Richard Dawkins has elucidated it.

As another example, let 20 base-pair sites be subject to mutation in a one-off Darwinian stage. Let this be replaced with a series of 4 equal sub-stages. Then,

M = 4^20 = 1,099,511,627,776 different mutations liable to generation in the one-off stage.

Each sub-stage defines M^(1/4) = 1024 mutations, and a total of 4096 for the series.

The total of different mutations liable to generation in the series of sub-stages is 4,093.

The number of different, but missing mutations using Dawkins’ Equation is:

m = 1,099,511,627,776 – 4 – (4-1) = 1,099,511,623,683

The simplest population or set of elements, to which Bayes’ Theorem applies, is one in which the set is divided into two subsets by each of two independent criteria or markers. One marker may be viewed as dividing the set into two rows, while the other marker divides the set into two columns. This results in the set of elements being divided into four quadrants or cells, where each cell is identified by a row marker and a column marker. Each cell is a subset of the total set.

In addition to the four subsets of the total set, which subsets are identified by row and column, another four subsets of the total set are identifiable. Two of these subsets are the sum of each row. The other two subsets are the sum of each column.

In Bayes’ theorem, the row marker is viewed as identifying one row as the Row, and the other row as non-Row. Similarly, the column marker is viewed as identifying one column as the Column, and the other column as non-Column. The specificity of each cell is by its row and column. One specific example from the following table is the cell, Cell(Row, non-Column) or Cell(true, non-our). Probability is the ratio of a subset to a set. Consequently, due to the Bayesian divisions of the set of elements, sixteen probabilities may be identified:
Four probabilities are the ratios of each of the four cells as a subset to the total set.
Four probabilities are the ratios of each of the four cells as a subset to its row sum as a set.
Four probabilities are the ratios of each of the four cells as a subset to its column sum as a set.
Two probabilities are the ratios of each row sum as a subset to the total set.
Two probabilities are the ratios of each column sum as a subset to the total set.

Bayes’ theorem concerns only four of these ratios or probabilities. Bayes’ theorem is an equation for one of these probabilities as equaling an algebraic expression involving three of the other probabilities.

The subset for which Bayes’ theorem is the focus is Cell(Row, Column). The set of focus is Column Sum. The probability of focus is their ratio of subset to set, namely

Cell(Row, Column) / Column Sum.

Let us call this Probability One. Bayes’ theorem permits the numerical calculation of Probability One, if the numerical values of three other probabilities are known. The other three probabilities are:
Probability Two: Row Sum / Total
Probability Three: Cell(Row, Column) / Row Sum
Probability Four: Column Sum / Total

Bayes’ equation is:

Probability One = (Probability Two * Probability Three) / Probability Four

On page 50 of Proving History, Richard Carrier considers a Bayesian population of explanations. The Row is true. The non-Row is non-true. The Column is our. The non-Column is non-our. The four probabilities of Bayes theorem in terms of this particular Bayesian population of elements are:
One: The probability of a true explanation of ours with respect to our total explanations
Two: The probability of a true explanation with respect to total explanations
Three: The probability of a true explanation of ours with respect to total true explanations
Four: The probability of an explanation of ours with respect to total explanations.

Four is, of course, Column Sum / Total. However,

Column Sum = Cell(Row, Column) + Cell(non-Row, Column)

Therefore, Probability Four may expressed as:

Cell(Row, Column) / T + Cell(non-Row, Column) / T Eq. 1

We can multiply and divide the first ratio of Eq. 1 by Row Sum without changing the ratio. This yields:

(Row Sum / Total) * (Cell(Row, Column) / Row Sum)

This is Probability Two times Probability Three.

We can multiply and divide the second ratio of Eq. 1 by non-Row Sum without changing the ratio. This yields:

(non-Row Sum / Total) * (Cell(non-Row, Column) / non-Row Sum).

We now have all of the terminology to verbalize Bayes’ theorem in the manner of Carrier, in his example of page 50 of Proving History.

A lucid verbalization of Bayes’ Theorem for the set of explanations defined by Carrier is: In contrast, Carrier’s verbalization of Bayes’ Theorem from page 50 of Proving History is obscure: Probability is the ratio of a subset to a set. When a set is composed of discrete elements such as explanations, the probability is the ratio of the number of elements in the subset to the number of elements in the set.

In the lucid version of Bayes’ theorem, each of the five verbal probabilities is of the form: ‘the probability of an element of a subset with respect to the elements of a set’. This has the same meaning as the form ‘The ratio of the elements of a subset to the elements of a set’. For each of the five probabilities, the subset numerator and the set denominator of the ratio are clearly identified in the lucid verbalization of Bayes’ theorem.

Carrier’s verbalization uses the words, typical and expected, as synonyms for probability and the word, atypical, as a synonym for improbability. Also, the only evidence is the data, the elements of the Bayesian data set. Carrier has identified the elements of the set as explanations. Therefore, Carrier’s verbalization uses the words, evidence and explanations, as synonyms.

In Carrier’s verbalization of Bayes’ theorem, if probability, typical, expected and atypical are replaced with ratio, it is difficult, but not impossible to determine the numerator and the denominator of each ratio specified. However, these ratios are not valid expressions of the probabilities of Bayes’ theorem. Thus, Carrier’s verbalization is invalid.

Consider Carrier’s verbal expressions of the five ratios, each purportedly a probability specified by Bayes’ theorem.

First: “The probability our explanation is true”.
This literally designates the ratio of our total explanations to total true explanations. However, neither our total explanations nor total true explanations is a subset of the other. The ratio verbalized is not a probability.
Also, “The probability our explanation is true” cannot refer to or be a test of a particular explanation of ours, because probability refers to sets of elements, not to a particular element. If the numerical value of Bayes’ equation were 1, then each particular explanation of ours would be true, because all are true. However, this is an inference from a calculated result, not the implication of Bayes’ theorem in its algebraic expression, which accommodates any numerical value from 0 to 1.

Second: “How typical our explanation is”.
This literally designates the ratio of our total explanations to total explanations. Our total explanations is a subset of total explanations. That ratio is Probability Four above. However, Probability Four is the denominator of Bayes’ theorem, not an expression in its numerator as it is in Carrier’s verbalization.

Third: “How expected the evidence is, if our explanation is true”.
As noted, the only evidence is the elements of the data set. The first phrase of this quotation, “How expected the evidence is”, would then read, ‘The probability of an element’. Every probability is the probability of an element of a subset. Thus, “How expected is the evidence” can be replaced with ‘the probability’. We then have ‘the probability, if our explanation is true’. This is of the form, ‘the probability if A is B’. ‘A is B’ means A is a subset of B, which alludes to the probability of A with respect to B. The entire expression, “How expected the evidence is, if our explanation is true”, is, thus, reduced to ‘the probability our explanation is true’, which is identical to the expression of the First consideration. The ratio verbalized is not a probability.

Fourth: “How atypical our explanation is”.
This literally designates the ratio of non-our total explanations to total explanations. This ratio verbalized is a probability.

Fifth: “How expected the evidence is, if our explanation isn’t true”.
In line with the rationale above in the Third and First considerations, this can be reduced to ‘the probability of our total explanations with respect to total non-true explanations’. However, neither our total explanations nor total non-true explanations is a subset of the other. The ratio verbalized is not a probability.

Another thing to note is that in the lucid verbalization, only the second of the probabilities in the numerator of Bayes’ theorem identifies a subset as ‘our’. Yet, in Carrier’s verbalization, each of the two expressions for probability (one referred to as a typicality, the other as an expectation) in the numerator of Bayes’ theorem identify a subset as ‘our’.

There is a marked contrast between the clarity of specification of the five ratios of probability in the lucid verbalization of Bayes’ theorem and the obscurity of the ratios in Carrier’s verbalization. Three out of Carrier’s five verbal ratios are not probabilities.

Bayes’ theorem is not the equation for ‘The probability our explanation is true’, in Carrier’s words. Rather, Bayes’ theorem is the equation for ‘Among our explanations, the probability that an explanation is true’. It is the ratio of our true explanations to our total explanations.

Just as I have previously concluded from a different perspective of his verbalization: “Carrier’s actual use of his terminology does not merely obscure, but totally obliterates the algebraic and intentional meaning of Bayes’ theorem.”

Equation 1 expresses relationships among ratios based on the following table. a / (a + c) = [a / (a + b)] * [(T) / (a + c)] * [(a + b) / T]             Eq. 1

Equation 1 is Bayes’ theorem, where a, b, c and d individually are positive or zero. The equation would obviously have no utility, if we knew the numerical values of a and c. Its utility lies in the ability to calculate the value of the ratio a/(a + c), without knowing the values of ‘a’ and ‘c’, in the invent that we do know the numerical values of the ratios: a/(a + b), T/(a + c), and (a + b)/T.

In an attempt to employ Bayes’ theorem in a ‘Case Study, the Death of Herod Agrippa’, Raphael Lataster proposed a mathematical problem where (a+b)/T may be considered to be virtually zero because T is much larger than (a+b). This implies that ‘a’ is virtually zero with respect to T. Based on this implication and allegedly on Bayes’ theorem, Lataster argues that if ‘a’ is virtually zero when compared to the size of T, then the ratio, a/(a + c), is also virtually zero. This would be true, if the ratio, (a + b)/T, was an essential factor in the algebraic expression of a/(a + c), i.e. in Eq. 1, Bayes’ Theorem.

Lataster’s argument views Eq. 1 as having the form,

a / (a + c) = Factor1 * Factor2* [(a + b) / T]                          Eq. 2

The argument is that a/(a+c) is virtually zero, because it is the product of factors, one of which is virtually zero, namely (a + b)/T. This would be a valid argument, if T were not a factor within the product, Factor1*Factor2. However, T is such a factor.

Factor1 * Factor2 = {a /[(a + b) * (a + c)]} * (T)                     Eq. 3

Consequently Eq. 1, which is Bayes’ theorem, can be expressed as Eq. 4, in which a/(a+c) is not a function of T and is not a product of which (a+b)/T is a factor.

a / (a + c) = [a / (a + b)] * [1 / (a + c)] * (a + b)                      Eq. 4

Also, from an examination of the table, it is apparent that the values of ‘a’ and ‘c’ in column One are independent of the values of ‘b’ and ‘d’ in column Two. Therefore, the value of a/(a+c) does not depend on the value of T. That a/T is virtually zero, is irrelevant to the value of a/(a+c).

Consequently, in his case study of the death of Herod Agrippa, not only is Lataster’s conclusion, that a/(a+c) is virtually zero, non-sequitur from his premise that a/T is virtually zero, but his argument is not based on Bayes’ theorem.

Notes:
1) In Lataster’s nomenclature, P(h/b) is the ratio, (a+b)/T, in Eq. 1 of this essay.

2) For a critique of Lataster’s expression of Bayes’ theorem, which is that of Richard Carrier, see the essays, “Verbalizing Bayes Theorem” and “Carrier’s Explanation of Bayes’ Theorem is False”.

3) Adopting Lataster’s nomenclature in his verbal expression of Bayes’ Theorem: The table is a table of explanations; Row One is true explanations; Row Two is non-true explanations; Column One is Our explanations and Column Two is other than our explanations.
a / (a + c) = [a / (a + b)] * [(T) / (a + c)] * [(a + b) / T]             Eq. 1
Using Lataster’s nomenclature and verbalizing Eq. 1, i.e. Bayes’ theorem, yields:
Among all of our explanations, ‘(a+c)’, the probability that any one, ‘a’, is true
EQUALS
[among all true explanations, ‘(a+b)’, the probability that any one, ‘a’, is ours]
TIMES
[one over the probability that among all explanations, ‘T’, any one, ‘(a+c)’, is ours]
TIMES
[among all explanations. ‘T’, the probability that any one, ‘(a+b)’, is true]

4) Probability is the ratio of a subset to a set. Bayes’ theorem permits the numerical calculation of the fraction of all our explanations, which are true, or synonymously, the probability that among our explanations any one is true. This probability concerns only ‘our explanations’, namely ‘(a+c)’ and ‘our true explanations’, namely ‘a’. Consequently, this probability is independent of all other explanations in the total set, because the quantities, ‘a’ and ‘c’, are independent of all other explanations in the set. In contrast, Lataster claims that he is determining the validity of our explanations on the basis of other explanations in the set, which claim is a contradiction of Bayes’ theorem. See Bayes, Baseball and Bowling, which illustrates the parallel error of claiming that the validity of bowling scores can be determined on the basis of baseball scores using Bayes’ theorem.