Excerpt from Ref. 1.

This section discusses what I call the initial probability, but the more technical terminology is the naïve probability or prior probability. This probability is just a basic probability not modified by conditions. It is called the initial probability because it is the first probability calculation. Subsequently it may be adjusted to produce the final probability. There are two different types of basic probabilities which I call the extont and nonextont probability. They are based on the two different types of finite probability sample spaces. Extont probability as explained in Section 4.3.1 is based on a sample space that consists of outcomes that existed. Nonextont probability as explained in Section 4.3.2 is based on a sample space that consist of outcomes that did not necessarily existed, but could have occurred.

The extont probability is based on a sample space that consists of a set of outcomes that occurred or are expected to have occurred. The extont probability is calculated by dividing the number of actual outcomes that qualify as matching by the number of all the outcomes that could have occurred or are expected to have occurred. It is the probability that a certain item is a specific unique item out of possible set of items which does contain that specific unique item. In the following discussion are examples of extont probabilities.

Consider the case where there are 10 balls in a bucket labeled 1 through 10. So these 10 possible outcomes all exist. Suppose you randomly select a ball out of the bucket, the extont probability that it is ball #5 is 10%. In this example there is 100% certainty that ball #5 and the other 9 balls do exist in the bucket so the probability is the extont type.

Consider the case where you see two identical ossuaries both inscribed with Jesus son of Joseph and you know for sure one of them contains the Gospel Jesus. In this case the extont probability that one of them contains the Gospel Jesus is 50%.

The extont probability is the probability of one specific outcome divided by the sum of the probability of all the possible outcomes which do exist or occurred. So the extont probability is a percentage that the probability for one outcome is of a total probability. If the total probability is a measure of all the outcomes that exist or occurred, then the extont probability is a direct measure the % chance of a identifying a specific outcome. Thus, only with extont probability is it appropriate to state the odds such as 2:1 for something being true or 1:2 against something being true.

The nonextont probability is based on a sample space that consists of a set of outcomes that could occur, but did not necessarily occur. The nonextont probability is calculated by dividing the number of potential outcomes that qualify as matching by the number of outcomes that could have occurred. Since the nonextont probability is not based on a sample space of outcomes that existed or occurred, there is no direction relation of the nonextont probability to the chance of something being true. In the following discussion are examples of nonextont probabilities.

Assume no special natural phenomenon occurring. Consider you are investigated a coin and flipped it many times and found it to land 50% of the times head and 50% tales. Then your friend came along and said by his supernatural powers he could make it land heads every time. So you flip it once and it lands head. The chance for this is a nonextont probability of 0.500. So is 0.500 nonextont probability enough evidence to be convinced that your friend is supernaturally causing the coin to land heads? Obviously not because a nonextont probability of 0.500 for an event means the event is just about what is expected to occur in a perfectly random world. What if you flipped it again and it lands head a second time in a row? The chance for this set of events is a nonextont probability of 0.250. Three times would be 0.125 etc .... The chance is one divided by the number of the possible permutations which keep growing as you do more coin flips. We often observe many events with probabilities quite low and do not assume the match was not just a random occurrence.

The following is an example of the value of a nonextont probability that the whole world can appreciate. Plate tectonics or the movement of the continental plates was inferred by probability before there was any known evidence that the continental plates were moving. I have read this was accomplished by calculating a nonextont probability of ~0.000001 (not a extont probability) for the match between the shapes of the continents in how they would have fit together as Pangaea, the original super continent. This is a one time event that has no freedom for biased selection of opportunities for matches, but there would be some subjectivity in interpreting how well the contour of the edges matched. In fact this was the initial clue that got the scientist looking for more evidence for continental plate movement and they sure found plenty of corroborating evidence.

So the nonextont probability is just the probability of a match occurring by random. The value for the nonextont probability is not directly related to something being true. However, the smaller the nonextont probability for a certain hypothesis, the greater the chance for that hypothesis being false. Also, nonextont probability argument strengths can be compared directly be comparing the nonextont probability values.

For both the extont and nonextont probability, the evidence is evaluated based on how well the evidence matches with the item the theory is trying to identify. For the extont probability, the probability calculation is based on a sample space of outcomes that existed or occurred or are expected to have existed or occurred. For the nonextont probability the probability calculation is based on a sample space of outcomes that could have occurred, but did not necessarily occur. This difference makes the meaning of the value for the extont probability quite different from the meaning of the value for the nonextont probability. As explained in Section 4.3.1, the value for the extont probability can be directly related to the chance of a hypothesis being true. As long as the extont probability is less than 0.50 there is no justification for claiming the hypothesis is true. The more the extont probability is over 0.50 and closer to 1.00 the more justification for claiming the hypothesis is true or the greater the argument strength. As explained in Section 4.3.2, the value for the nonextont probability is not directly related to the chance of a hypothesis being true. However, the smaller the nonextont probability for a certain hypothesis, the greater the chance for that hypothesis being false. Thus, the higher the extont probability value, the stronger the argument that something is true and the lower the nonextont probability the stronger the argument that something is false.

The extont and nonextont probability types are the two basic type of probabilities so they cover the two basic ways of probabilistic analysis. Both are important and are suited for addressing different issues. If you are trying to identify something that you know exists, then the extont probability is appropriate because you can calculate the chance of the thing you found is that unique thing of interest. For a sample space of things that exist, one can also calculate a nonextont probability; however, it would not be as useful as the extont probability because the value of the nonextont probability cannot be directly related to the chance that you have actually identified the unique item of interest. If you are trying to check if something exist, then the nonextont probability is more appropriate because it does not assume the thing exists. It is a calculation for the chance of the thing found matching the item of interest. If there is a low nonextont probability of the thing not existing, then an inference can be made that the thing must exists. An extont probability cannot be calculated for the chance of

finding something existing whose probability for existence is unknown, because the total probability for the sample space would not be related to probability for something existing. However, it would be conservative to assume the item of interest does exist and calculate a probability assuming it is part of a sample space of other similar items that are known to exist. Thus, one could calculate a conservative estimate for the extont probability for identifying the item of interest. One could assume the probability for existence which is the Fgj factor mentioned in Section 5.1.1; however, the Fgj value could be arbitrary. The true strength can be no greater than that determined by a correct conservative extont probability calculation.

Inductive reasoning uses probabilities and logic so the arguments typically never have 100% certainty. Deductive reasoning just uses logic so the arguments typically have 100% certainty. However, for any argument about something in reality being true, there is usually always some uncertainty. Since the nonextont probability is not directly related to something being true, the question is how can it be used to determining if something is true. Well the key theory used in science for determining if something is true is proof by elimination (PE). This logical concept is explained fully in Ref. 4 and 20. If there is a theory that describes a certain reality and all possible hypothesis for explaining that certain reality are false except for one hypothesis, then PE implies that this one non-false hypothesis is true. For example, if there were 10 different possible hypothesis for explaining a certain event and it was shown that 9 out of the 10 were implausible, implying they were false, then there would be a logical case that the one remaining plausible hypothesis is true.

Science typically does not prove anything true directly. It only shows things false directly by showing an explanation has a low probability so it is implausible. The way to show something true through science is to show all other possible explanations false implying the one remaining explanation must be true. The is called proof be elimination. In dealing with nonextont probabilities there is no definite way to define the % chance of something being true. It is just the chance of something occurring by random. The smaller the chance for something happening, the more likely a theory that it happened by chance is false.

As explained in Ref. 4, in my experience scientist are not even interested in dismissing that a match occurred by random unless the nonextont probability is less than 0.01. A significant match would be less than 0.0001 and a compelling match would be 0.000001. If you want to see an objective way of determining this nonextont probability threshold read my explanation in Ref. 4 Section 2.2.5. It basically depends upon the amount of potential for theories inferred true to contradict that you are willing to accept.

For example, in the coin flipping example in section 4.3.2 of just two heads in a row, obviously the 25% chance of this occurring by random does not mean that the chance of supernatural involvement is 75% or that there is a 75% chance of somehow a double headed coin was snuck in. If you considered 25% chance as the threshold, then if you reran the hypothetical example of two flip coins many times, you would conclude 25% of the time the supernatural intervened and 75% of the time it did not. This shows if you want an approach the does not make false conclusions or contradictions, then you would use very low probability thresholds for determining something false in order to get the point of determining something true by the process of elimination.

If you show an explanation has a logical contradiction, then consider it as having a zero probability for being true and 100% probability for being false. Contradiction cannot be true. Scientist do not like approaches that are likely to produce contradictions because they know that two things that contradict cannot both be true. So they use low probabilities threshold for determining something false, so that way they will not mistakenly infer something is true that is false. This applies to nonextont probabilities, the threshold is different for extont probabilities.

If you are 100% sure that your objective analysis has considered all possible explanations and you have correctly shown that all possible explanation except one has a low nonextont probability, then the nonextont probability is a

measure of how much you have reduced the risk of being wrong that the one remaining explanation is the true explanation.

In Matthew 12:25-29 the Gospel Jesus uses the idea of proof be elimination to respond to the Pharisees who claimed that Jesus cast out demons by using Satan’s power. Jesus explains that a partner of Satan would not cast out Satan’s demons; thus, Jesus power must not be from Satan.