An Implicit and an Explicit Rationale Expressed by Raphael Lataster
Of an implicit and an explicit rationale expressed by Raphael Lataster in his ‘Case Study, the Death of Herod Agrippa’, the implicit is formally valid while the explicit, allegedly an application of Bayes’ theorem, is formally invalid.
The Formal Validity of Lataster’s Implicit Rationale
The form of Lataster’s implicit rationale is that of a syllogism. It is:
Angels are mythical (non-real/false) as are any reports of their activity.
The death of Herod Agrippa in Acts is a report of angelic activity, namely his being slain by an angel.
The death of Herod as his being slain by an angel in Acts is false, and, specifically, historically false.
The logical form is:
B is a subset of A; C is in B; Therefore, C is in A.
In detail:
B (All reports of an angel or of angelic activity) is a subset of A (mythology/falsity)
C (The report in Acts of Herod Agrippa’s death as slain by an angel) is an element of B
Therefore, C is a an element of A (mythology/falsity)
Lataster’s implicit argument is formally, i.e. logically, valid.
The Logical Form of Bayes’ Theorem
Definitions:
A set or population, P, is composed of two subsets, one of interest, SI, and its complementary subset. The fraction of elements in P that are in subset, SI, is SI/P.
The set, P, is composed of two other subsets, independently of subset, SI, and its complement. These are the subset, X, whose elements possess characteristic X, and its complementary subset. The fraction of elements in P that are in subset, X, is X/P.
These divisions implicitly identify a subset SIX, which is not only a subset of SI, but a subset of X as well. The fraction of elements in SI that are in subset, SIX, is SIX/SI. The fraction of elements in X that are in subset, SIX, is SIX/X.
The nesting of subsets of P is different from the nesting of sets, A, B, and C above. C is completely nested into B, while B is completely nested into A.
X and SI though nested into P, are nested into P independently. They are not nested the one into the other into P. The other subset of interest, SIX, is completely nested into both SI and X.
Due to the nesting arrangements of the subsets of P, we have the following algebraic expression for the fraction, SIX/ SI:
SIX/SI = X/P × P/SI × SIX/X
This equation, which is Bayes’ theorem, is valid because the two P’s cancel out as do the two X’s, leaving the identity, SIX/SI = SIX/SI. We may verbally say of the equation: The probability, that an element of the subset of interest, SI, possesses characteristic X, is determined by means of the equation. This determination is subsequent to knowing the probability that that element, as an element of the population P, possessed characteristic X. In this verbalization, X/P is said to be the prior probability of X and SIX/X is said to be the subsequent probability of X. This is said in the sense that a person learns that an element is an element of the set P, prior to learning that this same element is an element of the subset of interest, SI.
The Sets and Subsets of Lataster’s Explicit Rationale
Lataster indicates that he is going to apply Bayes’ theorem to the death of Herod Agrippa as reported in Acts. In doing so he indicates that deaths reported in Acts is the subset of interest, SI, and that the subsequent probability to be calculated is the fraction of deaths in Acts due to being slain by an angel, SIX/SI. However, he also identifies the individual element in SI, which is of concern, as the death of Herod Agrippa. Further, he identifies the report of Agrippa’s death in Acts as slain by an angel. Consequently, he knows that the probability is 100% for the specific element of SIX, before he begins to use Bayes’ theorem to calculate the probability for a generic element of SIX. The generic element is any element of SIX including the specific element. The generic probability to be calculated is SIX/SI. One may, of course, wish to calculate the generic probability, but not as a moving toward what is already known, such as the probability of a specific element. In the case of Herod Agrippa in Acts, the specific probability is already known to be 1, i.e. 100%.
Lataster’s Explicit Bayesian Rationale
In his alleged application of Bayesian reasoning, Lataster indicates that the population, P, is that of all human deaths and X is the number of human deaths due to being slain by an angel He takes it for granted that the prior probability, X/P, is so small as to be virtually zero, because X is very small and P is very large, relative to one another. He then assumes that X (deaths of persons slain by an angel) is essentially zero, therefore SIX, a subset of X, must also be zero and the report in Acts of Herod’s death by an angel is false.
Lataster’s explicit rationale does not logically conform to Bayes’ theorem. Also, given that X is very small and P is very large relative to one another, thus rendering X/P virtually zero, it does not follow that SIX is zero. Neither does it follow that SIX/SI is zero or virtually zero. Yet, such is Lataster’s explicit rationale.
Another Perspective of the Set and Subsets of Bayes’ Theorem to Aid in Its Understanding
Consider another perspective of identifying the set and subsets of Bayes’ theorem in order to understand its formal logic. Of the subsets of set P, are two, SI and X which share an overlap, subset SIX. Subset SIX is fully contained in subset SI and is fully contained in subset X. These relationships of set P and its subsets are expressed by Bayes’ theorem:
(SIX/SI) / (SIX/X) = (X/P) / (SI/P)
Multiplying both sides by SIX/X yields Bayes’ theorem in a common format,
SIX/SI = ((X/P) × (SIX/X)) / (SI/P)
In this format, Bayes’ equation lists four ratios, each ratio is that of a subset to a set, i.e. a probability. In the order listed in the equation, Bayes’ theorem indicates that SIX is a subset contained in SI, X is a subset contained in P, SIX is a subset contained in X, and SI is subset contained in P.
From this perspective, the logic and utility of Bayes’ theorem should be clearer. Given the numerical value, X/P, the generic probability for a group, Bayes’ theorem enables the calculation of the numeric value, SIX/SI, the probability for a specific subgroup (the numeric values of two other probabilities must also be given).
The probability for the specific subgroup is the fraction of SI that is SIX. For Lataster’s ‘Case Study’ the specific subgroup, SI, is the set of all deaths reported in Acts. SIX/SI is the fraction (or percentage) of the set of all deaths reported in Acts as slayings by angels, one of which is the death of Herod Agrippa. Strangely, in Lataster’s ‘Case Study, the Death of Herod Agrippa’, Herod Agrippa’s death is known to be reported in Acts as slain by an angel prior to invoking Bayes’ theorem for calculating SIX/SI, the fraction of all deaths reported in Acts that are reported therein as slayings by angels.
Insufficient Information to Employ Bayes’ Theorem
Lataster claims that there is insufficient information to employ Bayes’ Theorem to his ‘Case Study’ of the death of Herod Agrippa in Acts. Therefore he employs Bayesian reasoning which he claims is the basis for his explicit rationale.
In Lataster’s ‘Case Study’, the numerical value of the specific probability, SIX/SI, cannot be calculated because not all of the numerical values of the other three probabilities of Bayes’ theorem are known.
The specific probability, SIX/SI, calculated by Bayes’ theorem is the answer to a verbal question. Generally, the verbal question is, “What is the probability that an element in the specific subset, SI, possesses characteristic, X?”. That verbal question in the context of Lataster’s ‘Case Study’ is this, “What is the probability that a death reported in Acts (all deaths reported in Acts is the specific subset, SI) is reported therein as a death due to being slain by an angel (being slain be an angel is characteristic, X)?” Consequently, Lataster’s question is, “What is the probability that the death of Herod Agrippa, as it is reported in Acts, is reported therein as a death due to being slain by an angel? This is the question which Lataster notes that he lacks sufficient information by which to calculate the answer by employing Bayes’ theorem. He, therefore, resorts to his explicit rationale, which he identifies as ‘Bayesian reasoning’. If Lataster had sufficient information to employ Bayes’ theorem, he could calculate the probability that any death, and specifically the death of Herod Agrippa, as reported in Acts is reported as a death due to being slain by an angel. To propose the calculation of this probability is, of course, ludicrous because Lataster has already read the report of Herod’s death in Acts, in which Herod Agrippa’s death is reported as due to being slain by an angel. Lataster’s problem is that he doesn’t fully understand the algebra, i.e. the formal logic of Bayes’ theorem.
Conclusion
Of an implicit and an explicit rationale expressed by Raphael Lataster in his ‘Case Study, the Death of Herod Agrippa’, the implicit rationale is formally valid while the explicit rationale, allegedly an expression of the reasoning inherent in Bayes’ theorem, is formally invalid.
His explicit rationale is illogical within the context of Bayes’ theorem, because Bayes’ theorem is irrelevant to his rationale. His explicit rationale is also mathematically illogical as non-sequitur.
However, given Raphael Lataster’s keen interest in truth and understanding, it should be just a matter of time before he has studied Bayes’ theorem sufficiently to understand its algebra, i.e. its logical form.
A Bayes’ Theorem Sampler
Given:
The fraction of the world population with blue eyes, X/P
The fraction of the world population that is the population in Sweden, SI/P
The fraction of the world blue-eyed population that is in Sweden, SIX/X
Question:
What is the probability that a person in Sweden has blue eyes?
Answer:
SIX/SI
Thanks to Bayes’ theorem!
Note: The blue-eyed population of Sweden, SIX, is a subset not only of the world population of blue-eyed persons, X, but also a subset of the population of Sweden, SI.
Common verbalization: Given the probability that someone in the world has blue eyes, one may determine, using Bayes’ theorem, the probability that some specific someone, or anyone, in Sweden has blue eyes.
theyhavenowine 