# Rescuing Logic From the Abuse of Bayes’ Theorem: Validity, Soundness, and Probability

By R.N. Carmona

In recent years, there has been a surge in the use of Bayes’ Theorem with the intention of bolstering this or that argument. This has resulted in an abject misuse or abuse of Bayes’ Theorem as a tool. It has also resulted in an incapacity to filter out bias in the context of some debates, e.g. theism and naturalism. Participants in these debates, on all sides, betray a tendency to inflate their prior probabilities in accordance with their unmerited epistemic certainty in either a presupposition or key premise of one of their arguments. The prophylactic, to my mind, is found in a retreat to the basics of logic and reasoning.

An Overview on Validity

Validity, for instance, is more involved than some people realize. It is not enough for an argument to appear to have logical form. An analysis of whether it, in fact, has logical form is a task that is seldom undertaken. When people think of validity, something like the following comes to mind: “A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid” (NA. Validity and Soundness. Internet Encyclopedia of Philosophy. ND. Web.).

Kelley, however, gives us rules to go by:

1. In a valid syllogism, the middle term must be distributed in at least one of the premises
2. If either of the terms in the conclusion is distributed, it must be distributed in the premise in which it occurs
3. No valid syllogism can have two negative premises
4. If either premise of a valid syllogism is negative, the conclusion must be negative; and if the conclusion is negative, one premise must be negative
5. If the conclusion is particular, one premise must be particular (Kelley, D.. The Art of Reasoning. WW Norton & Co. 2013. Print. 243-249)

With respect to the first rule, any argument that does not adhere to it commits the fallacy of undistributed middle. Logically, if we take Modus Ponens to be a substitute for a hypothetical syllogism, then undistributed middle is akin to affirming the consequent. Consider the following invalid form:

All P are Q.

All R are P.

∴ All R are Q.

When affirming the consequent, one is saying Q ⊃ P. It is not surprising that these two fallacies are so closely related because both are illegitimate transformations of valid argument forms. We want to say that since all P are Q and all R are P, therefore all R are Q in much the same way we want to infer that P ⊃ Q. Consider the well-known Kalam Cosmological Argument. No one on both sides questions the validity of the argument because validity, for many of us, is met when the conclusion follows from the premises. However, one can ask whether the argument adheres to Kelley’s rules. If one were to analyze the argument closely enough, it is very arguable that the argument violates Kelley’s fourth rule. The issue is that it takes transposing from the fifth rule to fourth rule because the argument does not violate the fifth and therefore, appears valid. However, when restated under the fourth rule, the problem becomes obvious. In other words, the universe is a particular in both Craig’s conclusion and in the second premise of his argument. Let’s consider the KCA restated under the fourth rule:

There are no things that are uncaused.

There is no universe that is uncaused.

∴ All universes have a cause.

Restating it this way appears controversial only because the argument seems to presuppose that there is more than one universe. Two negatives must have properties in common. Put another way, since there are many of all things, then the universe cannot be the only thing of its kind, if we even agree that the universe is like ordinary entities at all. Craig, perhaps unintentionally, attempts to get a universal from a particular, as his argument restated under the fourth rule shows. Given this, we come to the startling conclusion that Craig’s KCA is invalid. Analyses of this kind are extremely rare in debates because most participants do not know or have forgotten the rules of validity. No amount of complexity hides a violation of basic principles. The advent of analytic philosophy with Bertrand and Moore led to an increasing complexity in arguments and for the most part, validity is respected. As shown here, this is not always the case, so a cursory analysis should always be done at the start.

Validity is necessary but not sufficient for an argument to prove effective and persuasive. This is why arguments themselves cannot substitute for or amount to evidence. Soundness is determined by taking a full account of the evidence with respect to the argument. The soundness of an argument is established given that the pertinent evidence supports it; otherwise, the argument is unsound. Let us turn to some simple examples to start.

An Overview of Soundness

“A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound” (Ibid.).

All ducks are birds.

Larry is a duck.

∴ Larry is a bird.

This argument is stated under Kelley’s fifth rule and is no doubt valid. Now, whether or not the argument is sound will have us looking for external verification. We might say that, a priori, we know that there are no ducks that are not birds. By definition, a duck is a kind of bird. All well and good. There is still the question of whether there is a duck named Larry. This is also setting aside the legitimacy of a priori knowledge because, to my mind, normal cognitive function is necessary to apprehend human languages and to comprehend the litany of predicates that follow from these languages. We know that ducks are birds a posteriori, but on this point I digress. Consider, instead, the following argument.

All ducks are mammals.

Larry is a duck.

∴ Larry is a mammal.

This argument, like the previous one, is valid and in accordance with Kelley’s fifth rule. However, it is unsound. This harkens back to the notion that ducks belonging to the domain of birds is not a piece of a priori knowledge. Despite knowing that all ducks are birds, the differences between birds and mammals are not at all obvious. That is perhaps the underlying issue, a question of how identity is arrived at, in particular the failure of the essentialist program to capture what a thing is. The differentialist program would have us identify a thing by pinning down what it is not. It follows that we know ducks are birds because anatomically and genetically, ducks do not have the signatures of mammals or any other phylum for that matter. A deeper knowledge of taxonomy is required to firmly establish that ducks are, in fact, birds.

An exploration of soundness is much more challenging when analyzing metaphysically laden premises. Consider, for example, the second premise of the KCA: “The universe began to exist.” What exactly does it mean for anything to begin to exist? This question has posed more problems than solutions in the literature; for our purposes, it is not necessary to summarize that here. We can say of a Vizio 50-inch plasma screen television that it began to exist in some warehouse; in other words, there is a given point in time where a functioning television was manufactured and sold to someone. The start of a living organism’s life is also relatively easy to identify. However, mapping these intuitions onto the universe gets us nowhere because as I alluded to earlier, the universe is unlike ordinary entities. This is why the KCA has not been able to escape the charge of fallacy of composition. All ordinary entities we know of, from chairs to cars to elephants to human beings exist within the universe. They are, as it were, the parts that comprise the universe. It does not follow that because it is probable that all ordinary things begin to exist that the universe must have begun to exist.

This is a perfect segue into probability. Again, since Bayes’ Theorem is admittedly complex and not something that is easily handled even by skilled analytic philosophers, a return to the basics is in order. I will assume that the rule of distribution applies to basic arguments; this will turn out to be fairer to all arguments because treating premises as distinct events greatly reduces the chances of a given argument being true. I will demonstrate how this filters out bias in our arguments and imposes on us the need to strictly analyze arguments.

Using Basic Probability to Assess Arguments

Let us state the KCA plainly:

Everything that begins to exist has a cause for its existence.

The universe began to exist.

∴ The universe has a cause for its existence.

As aforementioned, the first premise of the KCA is metaphysically laden. It is, at best, indeterminable because it is an inductive premise; all it takes is just one entity within the universe to throw the entire argument into the fire. To be fair, we can only assign a probability of .5 for this argument being true. We can then use distribution to get the probability of the argument being sound, so since we have a .5 probability of the first premise being sound, and given that we accept that the argument is not in violation of Kelley’s rules, we can therefore distribute this probability across one other premise and arrive at the conclusion that the argument has a 50% chance of being true.

This is preferable to treating each premise as an isolated event; I am being charitable to all arguers by assuming they have properly distributed their middles. Despite this, a slightly different convention might have to be adopted to assess the initial probability of an argument with multiple premises. An argument with six individual premises has a 1.56% chance of being true, i.e. .5^6. This convention would be adopted because we want a probability between 0 and 100. If we use the same convention used for simpler arguments with less premises, then an argument with six premises would have a 300% chance of being true. An arguer can then arbitrarily increase the amount of premises in his argument to boost the probability of his argument being true. Intuitively, an argument with multiple premises has a greater chance of being false; the second convention, at least, shows this while the first clearly does not. The jury is still out on whether the second convention is fair enough to more complex arguments. There is still the option of following standard practice and isolating an individual premise to see if it holds up to scrutiny. Probabilities do not need to be used uniformly; they should be used to make clear our collective epistemic uncertainty about something, i.e., to filter out dogma.

Let us recall my negation strategy and offer the anti-Kalam:

Everything that begins to exist has a cause for its existence.

The universe did not begin to exist.

∴ The universe does not have a cause.

Despite my naturalistic/atheistic leanings, the probability of my argument is also .5 because Craig and I share premise 1. The distribution of that probability into the next premise does not change because my second premise is a negation of his second premise. In one simple demonstration, it should become obvious why using basic probabilities is preferable over the use of Bayes’ Theorem. No matter one’s motivations or biases, one cannot grossly overstate priors or assign a probability much higher than .5 for metaphysically laden premises that are not easily established. We cannot even begin to apply the notion of a priori knowledge to the first premise of the KCA. We can take Larry being a bird as obvious, but we cannot take as obvious that the universe, like all things within it, began to exist and therefore, has a cause.

Now, a final question remains: how exactly does the probability of an argument being sound increase? Probability increases in accordance with the evidence. For the KCA to prove sound, a full exploration of evidence from cosmology is needed. A proponent of the KCA cannot dismiss four-dimensional black holes, white holes, a cyclic universe, eternal inflation, and any theory not in keeping with his predilections. That being the case, his argument becomes one based on presupposition and is therefore, circular. A full account of the evidence available in cosmology actually cuts sharply against the arteries of the KCA and therefore, greatly reduces the probability of it being sound. Conversely, it increases the probability of an argument like the Anti-Kalam being true. The use of basic probability is so parsimonious that the percentage decrease of the Kalam being sound mirrors the percentage increase of the Anti-Kalam being sound. In other words, the percentage decrease of any argument proving sound mirrors the percentage increase of its alternative(s) proving true. So if a full account of cosmological evidence lowers the probability of the Kalam being sound by 60%, the Anti-Kalam’s probability of being true increases by 60%. In other words, the Kalam would now have a 20% probability of being true while its opposite would now have an 80% of being true.

Then, if a Bayesian theorist is not yet satisfied, he can keep all priors neutral and plug in probabilities that were fairly assessed to compare a target argument to its alternatives. Even more to the point regarding fairness, rather than making a favored argument the target of analysis, the Bayesian theorist can make an opponent’s argument the target of analysis. It would follow that their opponent’s favored argument has a low probability of being true, given a more basic analysis that filters out bias and a systematic heuristic like the one I have offered. It is free of human emotion or more accurately, devotion to any given dogma. It also further qualifies the significance of taking evidence seriously. This also lends much credence to the conclusion that arguments themselves are not evidence. If that were the case, logically valid and unsound arguments would be admissible as evidence. How would we be able to determine whether one argument or another is true if the arguments themselves serve as evidence? We would essentially regard arguments as self-evident or tautologous. They would be presuppositionalist in nature and viciously circular. All beliefs would be equal. This, thankfully, is not the case.

Ultimately, my interest here has been a brief exploration into a fairer way to assess competing arguments. All of this stems from a deep disappointment in the abuse of Bayes’ Theorem; everyone is inflating their priors and no progress will be made if that continues to be permitted. A more detailed overview of Bayes’ Theorem is not necessary for such purposes and would likely scare away even some readers versed in analytic philosophy and more advanced logic. My interest, as always, is in communicating philosophy to the uninitiated in a way that is approachable and intelligible. At any rate, a return to the basics should be in order. Arguments should continue to be assessed; validity and soundness must be met. Where soundness proves difficult to come by, a fair initial probability must be applied to all arguments. Then, all pertinent evidence must be accounted for and the consequences the evidence presents for a given argument must be absorbed and accepted. Where amending of the argument is possible, the argument should be restructured, to the best of the arguer’s ability, in a way that demonstrates recognition of what the evidence entails. This may sound like a lot to ask, but the pursuit of truth is an arduous journey, not an easy endeavor by any stretch. Anyone who takes the pursuit seriously would go to great lengths to increase the epistemic certainty of his views. All else is folly.

# On Bayes’ Theorem and Its Proper Application

By R.N. Carmona

There are times in where one’s assumptions fall into question. This could happen in a number of ways, but one of the more common ways is when one’s assumptions are confronted by facts of the matter. I had assumed that Bayes’ Theorem was well understood by my fellow atheists. Unfortunately, that assumption was wrong. It isn’t well understood. Like some of our academic tools (e.g. Bayes’ Theorem; statistics in general) and even entire disciplines (e.g. philosophy), our perception of things can be tarnished by misuse–specifically, the misuse or more accurately, misapplication of religious apologists and theologians. I see that that’s the case with Bayes’ Theorem.

Bayes’ Theorem is just another way to properly confront one’s assumptions. Nat Napoletano stated that “Bayes’ Theorem is a plan for changing our beliefs in the face of evidence.”1  Before we cross that bridge, however, it will be useful to present the theorem in its entirety. I will then explain its variables. Bayes’ Theorem appears as follows:

P( A | B) = [P (B | A) * P (A)] / P(B)]

Prior to proceeding, some definitions are required. As Mario Triola stated:

One key to understanding the essence of Bayes’ theorem is to recognize that we are dealing with sequential events, whereby new additional information is obtained for a subsequent event, and that new information is used to revise the probability of the initial event. In this context, the terms prior probability and posterior probability are commonly used.2

Given this, we learn that the prior probability “is an initial probability value originally obtained before any additional information is obtained” and the posterior probability “is a probability value that has been revised by using additional information that is later obtained.”3 Another way of putting this is that the theorem shows the relation between a conditional probability with its inverse. In other words, this is the probability of a hypothesis given some observation as related to the probability of an observation given some hypothesis. Let’s work with an example. I’ll first state it as a word problem and then I’ll flesh out the variables.

Let’s first assume that there’s an equal chance that Jack or Zack get the job. Then, let us find the probability that either Jack or Zack get promoted some time after being hired. However, we find that since Zack is less suited for the position, the probability of him being promoted is lower. We therefore have the following probabilities.

P(J): the probability that Jack gets the job. (.5)

P(Z): the probability that Zack gets the job. (.5)

P(pr): the probability of promotion. (.55)

P(t): the probability of termination. (.45)

P(pr|J): the probability that Jack is promoted after getting hired. (.8)

P(pr|Z): the probability that Zack is promoted after getting hired. (.25)

P(t|J): the probability that Jack is terminated. (.2)

P(t|Z): the probability that Zack is terminated. (.75)

As you can see, I already assigned probabilities to each case. I want the numbers to be easy to work with, so that this demonstration is clear. So, let us calculate the probability that one of them is promoted regardless of who’s hired.

P(pr) = P(pr|J) * P(J) + P(pr|Z) * P(Z)

P(pr) = (.8 * .5) + (.25 * .5)

P(pr) = .4 + .125 = .525 or 52.5%

The probability that either of them is terminated can be shown in the same way. It can, however, be simplified as follows:

P(t) = 1 – (.525) = .475 or 47.5%

So Bayes’ Theorem comes in when you want to find out either of the following: the probability that termination occurred given that Zack was hired or the probability that termination occurred given that Jack was hired. Alternatively, you can find out the probability of promotion given that either Zack or Jack was hired. Note that this is after the hiring. The equation for the probability that termination occurred given that Zack was hired looks as follows:

P(Z|t) = [P(pr|Z) * P(Z)] / P(pr)

So, P(Z|t) = (.25 * .5)/(.45). P(Z|t) is .278 or 27.8%. Of course, Bayes’ Theorem isn’t used for these kind of cases. This example is simply for ease. As stated earlier, the misapplication of Bayes’ Theorem leads some of us to doubt its utility. Napoletano touched on this exactly. He stated that it has been used by theologians (e.g. Richard Swinburne) to prove the existence of god or to demonstrate that the resurrection actually happened. He does, however, state that “it is difficult to separate out the variables and make them independent”4 in such cases. He also stated the theorem can expose when someone is being irrational. Aside from that, it can expose when someone is being biased–when someone is favoring one hypothesis over the other. In such cases, the theorem is often misapplied.

In medicine, for instance, a correct application will keep one from undergoing a series of medical tests. Like in our example above, doctors will be able to know the probability that you have either x disease or y disease given the symptoms you have. This goes back to showing a relation between the conditional probability and its inverse. This is, in other words, showing the relationship between what we currently know and what we want to know. This is a correct application.

In science, the theorem can be used to arrive at the probability of a hypothesis given observations. In other words, given MOND (Modified Newtonian Dynamics), what is the probability that some phenomena would be the case? Chris Wiggins explains:

One primary scientific value of Bayes’s theorem today is in comparing models to data and selecting the best model given those data. For example, imagine two mathematical models, A and B, from which one can calculate the likelihood of any data given the model (p(D|A) and p(D|B)). For example, model A might be one in which spacetime is 11-dimensional, and model B one in which spacetime is 26-dimensional. Once I have performed a quantitative measurement and obtained some data D, one needs to calculate the relative probability of the two models: p(A|D)/p(B|D). Note that just as in relating p(+|s) to p(s|+), I can equate this relative probability to p(D|A)p(A)/p(D|B)p(B). To some, this relationship is the source of deep joy; to others, maddening frustration.5

Religious apologists have misused the theorem. This is most apparent in their absurd calculations. “Richard Swinburne, for example…estimated the probability of God’s existence to be more than 50 percent in 1979 and, in 2003, calculated the probability of the resurrection [presumably of both Jesus and his followers] to be “something like 97 percent.”6 These numbers are, as Dawkins stated, “not measured quantities but personal judgments, turned into numbers for the sake of the exercise.”7 Put simply, it’s blatant bias.

This abuse of the theorem is likely behind people’s suspicion of the theorem. Whether or not a theist or an atheist can use Bayes’ Theorem to arrive at a probability of god’s (non-)existence or of the resurrection is perhaps an open question. There are, however, accurate historical applications. It must first be pointed out that Bayes’ Theorem cannot be used in isolation. “The only way to argue for a claim is to look at the evidence and compare that claim to alternative claims.”8 More importantly, however, assumptions reduce prior probability. Rather than reduce their prior probability, religious apologists and theologians inflate it. This is perhaps at the core of people’s suspicions. Your suspicion, however, isn’t in Bayes’ Theorem itself but rather in how it is being applied. It must be noted that a reduction in the prior implies a reduction in the posterior. Richard Carrier explains:

[I]f the evidence doesn’t match your claim so well (as theism always doesn’t, e.g. the Problem of Evil, or to continue the last analogy, the fact that when miracle claims are properly investigated they never turn out to be genuine), you can invent “excuses” to make your claim fit that evidence better. This means, adding assumptions to bolster your claim. But what Bayes’ Theorem teaches us is that every such assumption you add reduces the probability that your claim is true, rather than increasing it (as that excuse was supposed to do). Because adding assumptions to any claim C reduces the prior probability of C, and any reduction in the prior probability of C entails a reduction in the posterior probability of C, which is simply the probability of C. This is because the prior probability of several assumptions is always necessarily lower than the prior probability of any one of them alone, a basic fact of cumulative probability, which when overlooked produces the Conjunction Fallacy.9

When it comes to the existence of god or the occurrence of a miracle or a purported past occurrence like the resurrection, a neutral probability of .5 can serve as our prior. An atheist will no doubt assume that the prior probability of Jesus’ resurrection is much lower. A theist will assume it’s much higher. If, however, we set our assumptions aside, our posterior probability also goes unaffected. This will allow the numbers to do the work, so to speak. Though it remains difficult to separate the variables and make them independent, our results aren’t laden with bias. Ultimately, we can do closer analysis of Richard Swinburne’s misapplication or Richard Carrier’s application of Bayes’ Theorem. Such analyses are, however, cumbersome and for our purposes, unnecessary.

Ultimately, if you have any suspicions, I must reiterate, they do not lie with the theorem itself but rather with the way it is being applied. We must always mind the source. Not in the sense that we’re dismissive in an ad hominem or perceived guilt by association sort of manner, but in the sense that we’re acquainted with their biases. If, for instance, s/he is a believer in god, if it turns out that the calculations are suspicious or even dubious, we must single out the bias. There is, for instance, no way that the probability of the resurrection of Jesus is “something like 97 percent.” This is prima facie ludicrous and tenuous. It, of course, must be questioned. Common sense dictates that if the probability of his resurrection is that high, the probability of all miracles, regardless of which religious adherent makes the claim, is comparably high. This certainly isn’t the case, especially when we account for the dubious epistemic support for such claims (e.g. eyewitness testimony). I repeat, this irresponsible and to be blunt, non-academic misapplication must be questioned. We, however, must learn to do so responsibly and that isn’t possible when we’re directing our suspicions at the theorem rather than the theorist.

Works Cited

1 Napoletano, Nat. “Bayes’ Theorem for Everyone 01 – Introduction”YouTubeYouTube, LLC. 29 Jan 2012. Web. 1 Mar 2015.

2 Triola, Mario F. “Bayes’ Theorem”University of Washington. ND. Web. 1 Mar 2015.

3 Ibid. [2]

4 Ibid. [1]

5 Wiggins, Chris. “What is Bayes’s theorem, and how can it be used to assign probabilities to questions such as the existence of God? What scientific value does it have?”Scientific American. 4 Dec 2006. Web. 1 Mar 2015.

6 Ibid. [5]

7 Ibid. [5]

8 Carrier, Richard C. “If You Learn Nothing Else about Bayes’ Theorem, Let It Be This”Freethought Blogs. 26 Apr 2014. Web. 1 Mar 2015.

9 Ibid. [8]