The conclusion of a valid deductive argument follows logically and inescapably from its premises. Well-structured deductive reasoning with true premises is sound, where one finds the integrity of the result wholly contained within the premises. Nothing more is needed to arrive at the truth. If deductive arguments are so logically unassailable, then why are they not more seductive when it comes to persuasion?  Why don’t arguments from natural theology draw everyone into a firm belief in God? Are they not sound, even though they are undoubtedly valid? When it comes to persuasion, soundness is not always king. We readily reject a sound-argument if we do not believe one or more of the premises. Just because something objectively corresponds with reality doesn’t mean we see it, or even want to. There is a difference between the subjective and objective; between belief and truth. In this post, I examine these concepts in more detail, specifically around deductive reasoning.
The primary contact I have in mind as a topical filter is someone open and unconvinced who finds propositions from natural theology only slightly more plausible than their denial. The reason for this focus is other audiences are relatively uninteresting and uncomplicated. Even a good cumulative-case rarely alters the conviction of those who are sure and confident. Preaching to the choir or debating a hardened skeptic is rarely fruitful. I am also more interested here in rationality than rhetoric. Not to dismiss the art of persuasion, but the objective logical link between our confidence in the conclusion and our acceptance of the premises is more important to me here than how we might be rhetorically swayed.
Let’s start with a way to measure our belief. Probability comes in at least two flavors; objective (statistical) and subjective (epistemic.) Objective-probability is what happens outside an observer, in physical systems where the laws of nature describe repeatable and predictable events, like throwing dice or flipping coins. Subjective-probability relates to what is going on inside an observer’s head; the level of confidence she has in a given belief. It’s objective when we measure the outcome of flipping a fair coin repeatedly. It’s subjective when asked how confident we are “heads” will show on the next flip. The confidence we ought to have and what people do have further delineates the subjective flavor. The former falls into the discipline of epistemology and the latter, psychology. I am more interested here in the former.
Even though the truth-value of a proposition is always binary (true or false), belief rarely is. There are very few things in life where we take an attitude of certainty; instead, we hold most views with a degree of confidence. Subjective probability (SP) is a way of measuring this confidence, ranging from 1 to 0. One is: “I am certain that P is true.” Zero is: “I am certain that P is false.” If you do not know or on balance unsure, then your SP is in the middle at 0.5. We might use the following rubric to describe confidence as an SP:
1.0 – I am sure (true)
0.8 – I am fairly confident
0.6 – I only slightly believe
0.5 – I do not have a belief either way; I don’t know
0.4 – I somewhat doubt
0.2 – I very much doubt
0.0 – I am sure (false)
In the case of flipping a fair-coin, both probability types line up. The objective and subjective probabilities we assign to the outcome of a fair coin landing “heads” are 0.5. However, the two probability types are not identical. For example, I tell you I have a box full of cards with either heads or tails depicted on them and ask you to draw one at random. But just before drawing a card, I invite you to assess your confidence as a subjective probability (SP) in the belief: “You will draw heads.” Of course, you have no idea. You do not believe you will draw “heads” any more than “tails.” I might have put all “heads” in the box or few. The heads-tails distribution is unknown to you. The correct SP is 0.5 because you have no justification for moving up or down without additional information. However, this does not mean as you draw cards from the box; the number of “heads” drawn will approach 50%, as in flipping a coin repeatedly. The objective (statistical) probability in drawing “heads” is unknown because the probability distribution is unknown. SP, however, is rightly 0.5.
Take another example: With an unfair coin, you assess the probability that you will land “heads” on the next toss. Well, again, you have no idea. Perhaps this unfair coin will always land “heads,” or maybe always “tails.” You are unsure, you don’t know, and again your SP is 0.5. But the one thing we do know is that the heads-tails probability distribution is not centered on 0.5 because it is not a fair coin. So the objective probability (OP) of landing a “heads” cannot be 0.5 even though your SP is rightly 0.5. These examples show SP and OP are different concepts. But how do they differ in terms of betting?
SP lines up with betting odds in many cases but not all. In the case of the cards, even though the probability distribution is unknown, a single $1 bet on “heads” for a $2 return is acceptable. After all there are only one of two possibilities regardless of how the distribution is stacked. The same goes for the unfair coin. It may have an affinity for one side, but that doesn’t seem to make a 1:1 bet, a poor one. However, repeatedly betting “heads” on the card example is not at all like flipping a coin. Depending on the distribution, you might end up anywhere from losing all of your bets to doubling them. With a repeated coin-flip, you can only approach breaking even. Knowing more about the probability distribution of the cards would undoubtedly help.
Let’s alter the card example so that instead of “heads” or “tails” on a card, there is a number. I then ask you to assess whether or not you will draw a “seven.” At first, you might think: “I have no idea: 0.5.” But upon reflection, you realize that even though I might have put lots of “sevens” in the box, there are so many other possibilities — as many as there are numbers. So you decide to lower your SP to near-zero. But what changed?  In this case, background knowledge suggests more possibilities. You might, however, move your confidence up if you think I’m trying to make a point by stuffing the box full of “sevens.” But again, that would use background knowledge in the assessment. Somehow, prior probabilistic understanding affects SP. I will go into this in more detail shortly.
Having established some footing for how probability applies to confidence, let’s consider an example using one of the most basic argument-forms called modus ponens: If Annie goes to a movie, then Connie goes with her. Annie went to a movie. Therefore, Connie went with her.
P1: A -> C (reads: “A implies C” or “If A, then C”)
P2: A
ergo, C
In the conditional (A -> C), Annie is the antecedent (A), and Connie is the consequent (C). In the second premise, we assert (A); therefore, (C) obtains. The argument is straightforward, and we immediately see that it is valid. If the premises are correct, then as a sound argument, we are sure in our belief that Connie went to the movies. Right?  Well, it all depends on how certain we are of the truth of the premises. If we are sure that Connie always goes when Annie does (P1), and we are also confident that we saw Annie go to the movies (P2), then we ought to be convinced Connie went as well. But what if we are not so sure? In that case, we must reconsider the argument probabilistically:
P(A & C) = P(C | A) * P(A)
[P(C | A) reads: “probability of C given A” and is equivalent to  P(A -> C)]
The probability Annie and Connie went to the movie together equals the odds Connie went, given Annie went, multiplied by the odds Annie went. P(A & C) is not the same as P(C). They are equivalent when Connie doesn’t go for any other reason other than on the above condition. Let’s say that we are only slightly more confident than not in both P1 and P2 and assign 0.6 as an SP. Using the probability calculus, we see 0.6 x 0.6 = 0.36 — or we ought to slightly doubt Annie and Connie went. At first, this assessment doesn’t seem to add up, given we thought each premise more plausible than not. With a level of confidence in (P1, P2) each > 0.5, shouldn’t we also believe more likely than not Connie went — not the opposite at 0.36?
We have to take into account background knowledge (prior probability) and adjust our view based on the new evidence of this argument. Maybe Connie is always going to the movies without Annie. Perhaps she rarely goes, but when she does it is only with Annie. This background information makes a significant difference in our probabilistic analysis. Conditionalization is the process of updating one’s belief based on new evidence:
P(C) = P(C | A) * P(A) + P(C | ~A) * P(~A)
The probability Connie goes equals the odds Connie goes given Annie does, multiplied by the odds Annie does, plus the odds Connie goes given Annie doesn’t go multiplied by the odds Annie doesn’t go. Let’s say we know on any given night when Annie doesn’t go, Connie is just as likely as not (0.5) to go to a movie on her own. Using the above conditionalization rule on our prior knowledge and new evidence, we get the following:
P(C) = 0.6 * 0.6 + 0.5 * 0.4 = 0.56
This new estimate makes more sense. Given the shakiness of our argument (each premise is barely more probable than its denial at 0.6), we are only slightly more confident compared to some other night where it’s 50-50. What’s interesting is if we know beforehand, Connie never goes without Annie, then the second half of the equation drops to zero, and we are back to our initial assessment of 0.36 — which should now appear more reasonable. If we confidently deny the antecedent  (I know Annie didn’t go last night because she was with me -> P(P1) = 0), then the first part of the equation drops out, and all that remains is our prior probability of 0.5. Hopefully, this is starting to clear things up a bit. 
Based on where prior-probability lands on our example, we end up with a range from 0.36 to 0.76. Our assessment of 0.56 reflects an uncertain background where, independent of Annie, on any given night, Connie is just as likely to go as not. Our modus ponens example makes sense and is relatively easy to quantify. But what about common arguments from natural theology? Let’s consider the Kalam Cosmological Argument (KCA) as a conditional.
P1: If a thing begins to exist, then it has a cause. (BE -> C)
P2: The universe (a thing) began to exist. (BE)
ergo: The universe has a cause (C)
The KCA is a valid argument with a relatively modest conclusion (though the conceptual analysis of the “cause” being God is another matter!) Again, we will use our assessments of 0.6 for both premise P1 and P2 and restate the KCA as a probabilistic conditional:
P(C) =  P(C | BE) * P(BE) + P(C | ~BE) * P(~BE)
So what about prior-probability? Here I believe the KCA runs into a problem. What is P(C | ~BE)? The probability the universe has a cause given it not beginning to exist seems very low indeed. With no beginning, it would be an eternal thing without a need for a relevant-cause. The KCA is not interested in some other kind of Leibnizian-cause (reason.) Therefore, it seems appropriate to assign a near-zero value to P(C | ~BE) resulting in P(C) near 0.36. In other words, we ought to somewhat doubt the conclusion at the proposed level of confidence in the premises.
Perhaps other arguments from natural theology fair better than the KCA under this analysis. Let’s take a look at the moral argument (MA):
P1: If God doesn’t exist, then objective moral values do not exist.
P2: Objective moral values do exist.
ergo: God exists
Once refactoring from the more compelling modus tollens form of P1, and then into a probabilistic conditional, we get the following:
P1: If objective moral values exist, then God exists (OMVE -> G)
P2: Objective moral values exist. (OMVE)
ergo: God exists (G)
P(G) = P(G | OMVE) * P(OMVE) + P(G | ~OMVE) * P(~OMVE)
Looking at our prior; what is P(G | ~OMVE)? The probability God exists given objective moral values do not exist, seems highly unlikely to me. As objective moral lawgiver, this doesn’t look like a great-making attribute we can waive. For the Christian theist, a God who is not the locus of moral value has little appeal. Accordingly: P(G) = 0.6 * 0.6 + [very low value] * 0.4 = near 0.36. Once again, we are on the side of doubtfulness.
What’s worse, from a Christian apologist perspective, the KCA and MA are counterproductive at these low levels of confidence. A value of 0.36 indicates we ought to doubt the conclusion more than accept it. If an apologetic contact is hovering slightly above “undecided either way on P1, P2,” then these arguments are unhelpful from my perspective. The reason has to do with the highly-probable material equivalence between the antecedent and consequent in both cases. This equivalence means “if” in P1 can probably be replaced with “if and only if.”
P1: Iff a thing begins to exist, then it has a cause.
P1: Iff objective moral values exist, then God exists.
Given material equivalence, the atheist apologist might favorably reformulate these arguments, even using our above 0.6 confidences:
P1: If a thing doesn’t begin to exist, then it doesn’t have a cause.
P2: The universe did not begin to exist.
ergo: The universe doesn’t have a cause
P(~C) = P(~C | ~BE) * P(~BE) + P(~C | BE) * P(BE)
P(~C) = [very high value] * 0.4 + 0.4 * 0.6 = approaching 0.64
P1: If objective moral values do not exist, then God doesn’t exist.
P2: Objective moral values do not exist.
ergo: God doesn’t exist
P(~G) = P(~G | ~OMVE) * P(~OMVE) + P(~G | OMVE) * P(OMVE)
P(~G) = [very high value] * 0.4 + 0.4 * 0.6 = approaching 0.64
So if this all seems a bit strange, don’t take my word for it. Let’s see what the experts have to say. William Lane Craig used to say that if we take a valid deductive argument and believe each premise to be more plausible than its denial (confidence in P1, P2 each > 0.5), then we ought to accept the result. Craig’s view doesn’t follow from the above analysis. Tim McGrew recently corrected Craig on this matter by stating that to guarantee the conclusion is more probable than not, the conjunction of the premises must be more probable than not. For the KCA, the product of the confidence in P1 and P2 would need to be > 0.5, for example, SP > 0.71 for each would barely do. Though this puts a higher burden on the apologist than I initially thought necessary, my analysis agrees with McGrew’s correction. Craig agreed with McGrew as well and has since said so on his website.(1)
McGrew and DePoe propose an approach whereby the sum of the uncertainties in (P1, P2) is used as a lower bound on the probability of the conclusion. In their paper on the topic (2), they use a few esoteric examples to invalidate other strategies. However, I found their strategy mostly unhelpful. By treating the uncertainties as a lower bound, we are potentially left worse-off solving the credibility problem than before. Taking the above KCA at (P1, P2) = 0.6, the lower confidence bound is 1.0 – ((1.0 – 0.6) + (1.0 – 0.6)) = 0.2. That’s: “I very much doubt the KCA.” So for the purposes herein, looking at those contacts hovering just above 0.5-uncertainty, all we can say is that such an argument cannot be any worse than very poor. I understand this is merely a lower bound, but that hardly makes the strategy helpful in terms of persuasion even though the lower-limit might be raised using other means — like a complex cumulative-case.
Consider this before we wholesale abandon classical apologetics: First, some arguments routinely have a high confidence level in one of the two premises. Take the causal principle in the KCA: “Things that begin to exist, have a cause.” I find this to be near-certain, and it’s denial ridiculous — all deception from pop-scientists like Lawrence Krauss notwithstanding. If we recalculate using P1 = 0.9 and P2 at our original 0.6, then our confidence in the conclusion ought to be > 0.5. Some classical arguments, like the KCA, might have one highly-confident premise, even for the kinds of contacts we are considering here. The MA, on the other hand, is not so fortunate. Untenable as I believe it is, there are many worldviews where objective moral values are thought to exist apart from God. P1 is far from guaranteed for many agnostics. Ironically though, Craig says the MA has been more effective in his apologetic efforts than the KCA. (3) This anecdote leads me to another consideration.
This entire analysis looks at deductive reasoning, ideally and objectively. How background-knowledge and arguments from natural theology might entail and intertwine within the psychology of any given contact is practically impenetrable. People willingly accept weak arguments and reject strong ones. I know someone who found the KCA compelling as a young believer only to dismiss it later when their desire to go their way made following God inconvenient. How one’s will relates to all of this is another matter entirely. There is a wide gap between sound apologetic arguments and persuasion, and this objective rubric is not going to bridge it. That, however, does not mean apologists ought to ignore the quality of their approach. Is it a sound apologetic-practice to leave one persuaded in a conclusion that is objectively unwarranted based on their confidence in P1, P2? Do the ends justify the means if we are confident in the truth? I have to say no — other apologists may disagree. The fact remains, many of the classical arguments from natural theology have a suboptimal logical form when dealing with the honest agnostic only slightly convinced in the premises.
In light of this analysis, we might want to reconsider our apologetic-style. Inferences to the best explanation and other forms of abductive reasoning do not suffer from the same problem as some of the above classics. The historicity of Christ and the Resurrection based on generally agreed upon historical facts is a good example. Design arguments, where we consider the best explanation for information in nature; the primacy of information over matter and discussions around the Arche (the ultimate foundation of reality) being a mind versus non-mind (material) are all good candidates. It might be purely coincidental, but these kinds of arguments have always resonated more with me than some of the classics.
In conclusion; an objective assessment of subjective-confidence in deductive arguments from natural theology shows some to be problematic for the Christian apologist. The Kalam Cosmological Argument and the Moral Argument are two examples. Any deductive reasoning from natural theology that can be logically-refactored as modus ponens is potentially problematic. If your apologetic contact is hovering just above 0.5-uncertainty, such arguments are potentially counterproductive. Even though the vast majority of listeners will not consider any of this, that doesn’t negate our responsibility as apologists to put forth solid reasoning. If we are going to propose cases (like the KCA and MA), we will need to ensure higher confidence-levels on the premises if we want them to be honestly persuasive. Other abductive arguments and inferences to the best explanation do not suffer from the issues raised and are worth considering within the context of our approach to apologetics.
(3) – recent interview with Ben Shapiro on his Sunday Special