When Absence of Evidence is Evidence of Absence

There is a popular and catchy saying which I myself have been caught repeating in the past, but which, for all its intuitive appeal, is false; namely, that the absence of evidence isn’t evidence of absence. Many a new-atheist has repeated the mantra that there is no evidence for God’s existence, insinuating thereby that this absence of evidence is good evidence for atheism. William Lane Craig, a noted philosopher, theologian and tireless Christian apologist has responded as follows:

[Atheists] insist that it is precisely the absence of evidence for theism that justifies their claim that God does not exist. The problem with such a position is captured neatly by the aphorism, beloved of forensic scientists, that “absence of evidence is not evidence of absence.” The absence of evidence is evidence of absence only in cases in which, were the postulated entity to exist, we should expect to have more evidence of its existence than we do.1

He has reiterated as much more informally (but more elaborately) on his podcast, ReasonableFaith, where he says:

The absence of evidence will count as evidence of absence when if the thing existed, then having surveyed the grounds, so to speak, we would expect to see evidence of their existence, and we don’t see it. And so, for example, in the case of fairies, if they existed then we ought to be able to find traces of their existence – their dead bodies when they die, their carcasses, other sorts of remains, little clothing factories where they build their clothes, and we ought to detect them flying about just as we detect dragon flies and bumblebees – but we don’t. So this would be a case where I think the absence of evidence would count as evidence of absence.”2

On this view, the absence of evidence only counts as evidence of absence when we have some reason to expect to see the evidence ex hypothesi. This has enormous intuitive appeal; consider the hypothesis that there is at least one tiger in India. Can the fact that I, sitting in Canada, currently see no tiger really count as evidence that there is not at least one tiger in India? Surely not; presumably because that evidence isn’t expected on the assumption of the relevant hypothesis’ truth. Elliott Sober, reflecting on absence of evidence, notes that in the case of arguments from absence “it is easy to see how each can be turned into a valid argument by adding a premise. The arguments have the form:

I do not have any evidence that p is true.
p is false.

Just add the premise

(P1) If p were true, then I’d have evidence that p is true.”3

This further highlights the fact that it is natural for us to think that absence of evidence is evidence of absence only when we expect the evidence ex hypothesi.

For years I found this response intellectually satisfying, but in recent years I have come to think that it is woefully mistaken. It is true that my failure to observe a tiger in Canada provides no evidence against there being at least one tiger in India, but it is not because I wouldn’t have anticipated seeing a tiger in Canada given that there is at least one tiger in India. All my affection and respect for Craig notwithstanding, if Craig means that absence of evidence E for hypothesis H is only evidence of absence (i.e., not-H) when the probability of E on H is greater than 0.5, then he is, I think, incorrect. In what follows I will try to explain why, as well as explore what to me seem interesting corollaries of Bayesianism.4

John Hawthorne, speaking about probability theory and the fine-tuning argument at a conference back in 2015, warned:

“Human beings, even intelligent human beings, are terrible at reasoning about probabilities. There’s enormous empirical evidence that human beings are terrible at reasoning about probabilities, and so we have to proceed with care.”5

Playfully picking on (presumably) a student in the audience, Hawthorne says: “Justin gave us the kind of awesome sounding principle… [that] if you don’t see something then that can be evidence of its absence only if you expect that you would get evidence were the thing there.”6 Not the cleanest off the cuff articulation, but clearly Hawthorne had in mind the principle for which W.L. Craig advocates. He continues; “that’s wrong… and I can prove to you that it’s wrong.”7 He proceeds to give an illustration using a hypothetical creature he calls a Dynx, where he stipulates that 75% of Dynx are invisible to the naked eye, and the probability that there is a Dynx in a box placed before us is 50%. We open the box, and we see no Dynx. The probability that there is no Dynx given our background knowledge and this new piece of information (namely that we do not see any Dynx) is approximately 57%. You can satisfy this for yourself by simply dividing up the space of possibilities (i.e., ‘seeing a Dynx in the box,’ ‘not seeing the Dynx in the box,’ and ‘there being no Dynx in the box’), eliminating the possibility of ‘seeing a Dynx in the box,’ and then expressing your updated probability assessment accordingly. So, even though we ought not to expect to see a Dynx in the box if there is one in the box, our failure to observe one is still evidence for their being no Dynx. This simple illustration (and others like it) seems to be entirely compelling. What, then, is the genuinely Bayesian determination of evidence?

On the Bayesian theory of confirmation,8 some evidence E will count as evidence for some hypothesis H (given background knowledge B) just in case E (conjoined with B) raises the (prior) conditional probability of H. To put it more formally, E will count as evidence for H just in case: P(H|E&B)>P(H|B). However, [P(H|E&B)>P(H|B)]⊃[P(~H|~E&B)>P(~H|B)]. In other words, if E provides any evidence for H, then ~E provides some evidence against H. It needn’t, of course, be the case that E provides as much evidence for H as ~E does for ~H, but it strictly follows from Bayesianism itself that ~E would be evidence against H just in case E would be evidence for H.

To illustrate with an example, let us take a hypothesis H1: “that aliens exist,” and evidence E1: “I am being abducted by aliens.” Obviously P(H1|E1&B)>>P(H1|B). What is not so obvious is that P(H1|~E1&B)<P(H1|B). The reason it isn’t so obvious is that ~E1 provides negligible evidence for ~H1 (even though E1 would provide compelling evidence of H1). If aliens abduct me, that’s really good evidence that they exist. If aliens do not abduct me that’s really poor evidence that they don’t exist. It may be some evidence, but it isn’t very much evidence.

Not only can the absence of evidence be negligible evidence of absence while the presence of that evidence would be altogether compelling, but the absence of evidence can even be inscrutable evidence of absence while the presence of evidence is scrutable and enormously supportive of the hypothesis in question. Take the example of a miracle, and for simplicity let us use the miracle of the bodily resurrection of Jesus of Nazareth. The bodily resurrection of Jesus, if it did occur, would be relatively good evidence for God’s existence; P(G|R&B)>>P(G|B). However, if Jesus had not been raised from the dead, would that provide any evidence against God’s existence? According to Bayesianism it would, but it seems like it would be not only negligible evidence, but even inscrutable evidence. There is no way one could put a figure (with any justification) on how much more confident it should make us in atheism that some miracle, like Jesus’ resurrection, did not occur. If we could give any estimate of what the probability is that God would perform a miracle when called upon to do so, for instance, then we could make some predictions about how many hospitalized people with terminal diseases (according to medical diagnosis) under observation get better when prayed for. We can’t make these predictions not because there is no actual probability of God doing a miracle, but because we aren’t at an epistemic vantage point from which we can assess that probability with any level of confidence at all.

Further, the evidence may not be merely negligible, but can in special instances be literally infinitesimal (an infinitesimal is a non-zero infinitely small quantity). Consider Hempel’s paradox9 for a moment; any observation of a pink shoe provides some evidence for the hypothesis that all ravens are black. The hypothesis that all ravens are black is logically equivalent to the statement that all non-black things are non-ravens. It follows, therefore, that any observation of a black raven is evidence that all non-black things are non-ravens, and any observation of a non-black non-raven is evidence that all ravens are black. An observation can’t be evidence for one without being evidence for the other precisely because they are logically equivalent statements, at least interpreted at face value; this is just what Hempel called “the equivalence condition.”10 However, it seems as though there are potentially infinitely many things which are non-black non-ravens which, at any moment, we will fail to observe. If this is so, then each of these instances of absence of evidence will count as instances of infinitesimal evidence of absence (or, at least, infinitely many of these instances will count as instances of infinitesimal evidence of absence). One thinks of the infinitely many miracles God could have performed at any given moment (e.g., growing a lost limb, bringing a dead child back to life, parting the Atlantic ocean); is it really the case that every instance of a miracle not happening provides some evidence against God’s existence? If so, and if there are infinitely many opportunities for God to perform a miracle of some kind (in infinitely many of which God decides to perform no miracle), does that not entail that the probability of theism is literally infinitesimal, or else that each instance (or, at least, infinitely many instances) of a non-miracle provides at most infinitesimal evidence against theism? This gets a little tricky, of course, because Bayesian theory isn’t really equipped to deal with cases of what we might call ‘transfinite probabilities,’11 but if we take its implications seriously even in such cases we will plausibly think that at least some things provide literally infinitesimal evidence for a conclusion or hypothesis.

An interesting objection to this suggests that there is not, even potentially, an infinite number of unobserved observables. Given the limited bandwidth of the human body as a kind of measuring apparatus,12 there may be infinitely many different but observationally indistinguishable events. Imagine, for instance, two pairs of pink shoes whose colours or sizes differ by so little as to make it impossible for any human being to tell the difference between them. For any of the attributes assessed by the five senses, there will be limited empirical bandwidth given the human body as a tool of observation. What this seems to entail is that there is not a potentially infinite number of different possible observations, in which case we needn’t concede the absurdity of infinitesimal probabilities. This objection is appreciably practical, but I’m not entirely confident that it settles the matter. After all, I can imagine a human being with “electron-microscope eyes”13 or with any number of other physical alterations which would allow them to observe an apparently potentially infinite number of different events. For any such alteration, I can imagine God miraculously bringing it about that observer S has precisely the alterations necessary to observe some miracle M1 which would have previously been indistinguishable from miracle M2, but is not now indistinguishable from M2 for S. Moreover, I’m not convinced that observational indistinguishability is terribly relevant; there are infinitely many possible pink shoes which I could now be observing, but am not, and even if infinitely many of them would be indistinguishable to me, failing to observe any one provides some evidence against the hypothesis that all ravens are black. So it seems to me that we’re stuck with conceding that at least some things provide literally infinitesimal evidence.

In summary, I think we have seen why the absence of evidence is evidence of absence in all cases except those in which the presence of so-called evidence would do nothing to raise the conditional probability of the hypothesis in question. Thus, my failing to observe a tiger in Canada provides no evidence against the hypothesis that there is at least one tiger in India not because I wouldn’t expect that evidence if there were at least one tiger in India, but because even if I were observing a tiger in Canada it would provide no evidence that there is at least one tiger in India.14 We have also seen that even when absence of evidence is negligible evidence of absence, or inscrutable evidence of absence, or infinitesimal evidence of absence (or any combination of those three), it will still provide some evidence of absence; if E would have been evidence for H, then the absence of E provides evidence against H.

Post Scriptum: I want to thank Tim Blais, Cale Nearing and Sean Boivin who provided me, in discussions subsequent to the original article, with food for thought without which I would never have made the improvements I have lately introduced above.

1 William Lane Craig, “Theistic Critiques of Atheism” The Cambridge Companion to Atheism. Edited by Michael Martin (Cambridge University Press, 2006): 70.

3 Elliott Sober, “Absence of Evidence and Evidence of Absence: Evidential Transitivity in Connection with Fossils, Fishing, Fine-Tuning, and Firing Squads,” in Philosophical Studies 143, no. 1 (2009): 64.

4 As a cautionary caveat lector; though I’m pretty confident that what I’m about to say is correct, I have not taken any class on probability theory (yet); if anyone thinks there’s some subtle mistake somewhere, they are encouraged to share it. I am more than open to updating my views.

8 Elliott Sober, “Absence of Evidence and Evidence of Absence: Evidential Transitivity in Connection with Fossils, Fishing, Fine-Tuning, and Firing Squads,” in Philosophical Studies 143, no. 1 (2009): 66.

9 James Fetzer, “Carl Hempel,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, (Spring 2017 Edition), accessed April 2, 2017. https://plato.stanford.edu/archives/spr2017/entries/hempel/

10 James Fetzer, “Carl Hempel,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, (Spring 2017 Edition), accessed April 2, 2017. https://plato.stanford.edu/archives/spr2017/entries/hempel/

11 If one dislikes this term because they think that probabilities can be no higher than 1, which makes them finite, I would suggest they think about how the conditions I just stipulated could imply that some hypothesis H is infinitely likely without having probability 1. However, if that doesn’t mollify the critic, I could agree to change the term to ‘non-finite’ probabilities.

12 I borrow here from Bas C. van Fraassen, who notes insightfully that “the human organism is, from the point of view of physics, a certain kind of measuring apparatus.” See: Bas C. van Fraassen, The Scientific Image, (Oxford: Clarendon Press, 1980), 17. 

13 Bas C. van Fraassen, The Scientific Image, (Oxford: Clarendon Press, 1980), 17.

14 If one thinks that observing a tiger somewhere raises the conditional probability that one may be observed anywhere then one will reject this conclusion, but they needn’t, in so doing, reject the principle this example is being employed to illustrate.

Some Miscellaneous Reactions to Some of Robert Price’s Points in Favour of Mythicism

In a not so recent debate1 between Bart Ehrman and Robert Price the topic of whether Jesus of Nazareth historically existed was explored. This provides us with one of the first and few high-profile debates with at least one bona-fide scholar where the participants are directly arguing about mythicism. Unfortunately, the debate was a disappointment in several respects in that neither Ehrman nor Price gave performances of the quality many, who were anticipating an outstanding debate, were expecting. However, Price did say a few interesting things which I thought I’d pick up on and say a few words about. This is not intended to be a comprehensive dismantling of Price’s view (I have not the time to be so ambitious), but just intended to provide a registry of some of my miscellaneous reactions to various points.

Price, in his opening speech, provided at least three examples of evidence which may insinuate that one early objection to Christianity was that Jesus never existed. First, he cites a statement which Justin Martyr puts into the mouth of his interlocutor Trypho in his famous Dialogue with Trypho. Second, he cites a statement which Origen is at pains to refute from an anti-Christian polemicist of the second century, Celsus. Third, he calls into evidence the words of 2 Peter 1:16-18 as though they indicate an implicit awareness that there was an allegation already circulating within the first century that Jesus of Nazareth may not have existed at all.

Let us begin with the passage from the Dialogue with Trypho, according to which Trypho, (a Jewish intellectual who, in the dialogue, claims to have been a pupil of Corinthus the Socratic in Argos,2 and may possibly be the second century rabbi Tarfon,3 though that is not widely accepted) makes the following provocative charge:

But Christ—if He has indeed been born, and exists anywhere—is unknown, and does not even know Himself, and has no power until Elias come to anoint Him, and make Him manifest to all. And you, having accepted a groundless report, invent a Christ for yourselves, and for his sake are inconsiderately perishing.”4

Does this passage contain a veiled insinuation that Jesus did not exist? It doesn’t seem so. At very least we gather from the way Justin Martyr proceeds to respond to this comment that he doesn’t have that accusation in mind. Justin promises Trypho that “I will prove to you, here and now, that we do not believe in groundless myths nor in teachings not based on reason, but in doctrines that are inspired by the Divine Spirit, abundant with power, and teeming with grace.”5 However, Justin Martyr goes on to give argument after argument from prophecy to demonstrate that Jesus is a good ‘fit’ for the anticipated messiah of the Tanakh. He never goes on to argue that Jesus of Nazareth existed; he argues on the clear presumption that he and Trypho are agreed that Jesus of Nazareth existed. The likelihood is relatively high that Justin Martyr is writing a largely or entirely fictitious dialogue, but whether it was fictitious or not there is no way to read Trypho’s (alleged) statement as an insinuation that Jesus didn’t exist. That isn’t what Justin Martyr thought the statement insinuated, and it isn’t plausible that a historical Trypho intended to insinuate that the historical Jesus didn’t exist but just let that point drop entirely for the rest of the dialogue with Justin.

My verdict, therefore, is that this provides absolutely no evidence of any early anti-Christian polemic which insinuated that Jesus never existed.

What of Price’s second example, from the second century anti-Christian polemicist Celsus? Well, Price points out that Celsus says: “it is clear to me that the writings of the Christians are a lie and that your fables have not been well enough constructed to conceal this monstrous fiction.”6 However, to read this as a veiled charge that Jesus never existed is implausible for a variety of reasons. First, consider how the passage from Celsus continues: “it is clear to me that the writings of the Christians are a lie and that your fables have not been well enough constructed to conceal this monstrous fiction. I have heard that some of your interpreters…are on to the inconsistencies and, pen in hand, alter the originals writings, three, four and several more times over in order to be able to deny the contradictions in the face of criticism.”7 That is clearly an accusation of embellishment and selective redaction; it is clearly not an accusation of having invented the historical Jesus whole-cloth. Second, consider that Celsus elsewhere argues that Jesus is a bastard child; according to Origen in his Contra Celsus, “[Celsus was] speaking of the mother of Jesus, and saying that “when she was pregnant she was turned out of doors by the carpenter to whom she had been betrothed, as having been guilty of adultery, and that she bore a child to a certain soldier named Panthera.”89 Clearly, however, if Celsus thought that Jesus was born of illegitimate relations between Mary and a Roman soldier named Panthera, then Celsus could not have also believed that Jesus never existed. Those beliefs are so obviously logically incompatible that even an imbecile (as Origen thought) like Celsus could not plausibly have entertained both.

Finally, what of the words in 2 Peter 1:16-18? They read:

For we did not follow cleverly devised myths when we made known to you the power and coming of our Lord Jesus Christ, but we had been eyewitnesses of his majesty. For he received honor and glory from God the Father when that voice was conveyed to him by the Majestic Glory, saying, “This is my Son, my Beloved, with whom I am well pleased.” We ourselves heard this voice come from heaven, while we were with him on the holy mountain.”
(2 Peter 1:16-18, NRSV).

I consider it obvious that the author gives us an indication of what the allegation of ‘cleverly devised myths’ comes to by the way he responds to the charge. Clearly, however, he spends all his time emphasizing not that he was an eyewitness (or that there were eyewitnesses) of Jesus of Nazareth, but that he was one of many eyewitnesses of the majesty of Christ which was attested to and illustrated by miracles. It is the majesty and/or the miracles which the author believes are being alleged to be cleverly devised myths, not the historicity of the person, Jesus of Nazareth; we know this by inferring it from the way the author responds to the allegations he has in mind.

So, in my opinion, all three of these evidences of some early objection to Christianity to the effect that Jesus of Nazareth did not historically exist are completely bunk.

I want to end this reflection on some points brought out by Price in the debate with a few positive notes. There are some areas where I actually agree with Price over against the majority of New Testament scholars. For instance, Price maintains (and this came out in parts of the debate) that there is no more reason to think that Paul wrote Galatians than there is to think that Paul wrote 1st Timothy. Price’s conclusion is that we have reason to believe that Paul did not write any of the epistles traditionally ascribed to him. My conclusion is that Paul plausibly wrote all of the epistles traditionally ascribed to him. This was somewhat tangential to the debate, but it is a point of interesting qualified agreement nevertheless. More interesting still, Price argued that if we strip away all of the miraculous claims made about Christ, we are left with a first-century Jewish Rabbi about whom nothing would have been worth writing in the first place. He says, at one point, that if Clark Kent existed and superman didn’t, there would be no gradual embellishment of stories about Clark Kent because there would be no reason for anyone to remember any stories about Clark Kent in the first place. There either has to have been something about the Jesus of Nazareth of history which made him worth writing (talking, etc.) so much about in the first place, or else the stories about him were mythological from the beginning.

This, I think, is a very interesting point. If historians are intent on whittling down the Jesus of the Gospels to the point where he was an utterly unremarkable first century Jewish rabbi then there is no explanation for why he caused such a stir in the first place. Obviously most historians will respond, here, by conceding that Jesus claimed to be a miracle worker, and performed exorcism ceremonies in a way which presumed an immense and unprecedented amount of authority for himself. It was his innovative preaching along with what W.L. Craig has called the historical Jesus’ “unprecedented sense of divine authority,”10 which sufficiently explain why there were any stories about him in the first place. So, on the one hand, Price has, I think, failed to take inventory of what most New Testament scholars believe we can say with enormous confidence about the historical Jesus of Nazareth. On the other hand, though, Price does well to remind us that if scholars aren’t careful to preserve something remarkable and unique about the historical Jesus, if they reconstruct only a version of Jesus wholly sanitized by the presumption of naturalism, and about whom there was really nothing terribly special, they may be proverbially cutting the tree branch from which they hang.


1 Anyone interested can find the debate, at least currently, at the following link: https://www.youtube.com/watch?v=oIxxDfkaXVY

2 Justin Martyr, Dialogue with Trypho, Ch. 1, http://www.newadvent.org/fathers/01281.htm

3 Claudia Setzer, Jewish Responses to Early Christians,Fortress Press, 1994: 215.

4 Justin Martyr, Dialogue with Trypho, Ch. 8, http://www.newadvent.org/fathers/01281.htm

5 Justin Martyr, Dialogue with Trypho, Ch. 9, http://www.newadvent.org/fathers/01281.htm

6 Celsus, On the True Doctrine, translated by R. Joseph Hoffman, Oxford University Press, 1987: 37. See: http://www.earlychristianwritings.com/text/celsus3.html

7 Celsus, On the True Doctrine, translated by R. Joseph Hoffman, Oxford University Press, 1987: 37. See: http://www.earlychristianwritings.com/text/celsus3.html

8 Origen, Contra Celsus, Book 1, chapter 32. http://www.newadvent.org/fathers/04161.htm

9 I have written a little bit on this before, a long time ago. Those interested may see: https://thirdmillennialtemplar.wordpress.com/2012/02/13/celsus-attack-on-the-holy-mother/

10 http://www.reasonablefaith.org/does-god-exist-1

Soundness is Neither Necessary nor Sufficient for Goodness

In this (very) short article, I am going to try to explain what makes an argument valid (comparing two views), what makes an argument sound (again comparing two corresponding views) and then I aim to distinguish ‘good’ arguments from either of these. I will attempt to obviate why validity, on either interpretation, will be a necessary but insufficient condition for soundness (on that respective interpretation). It will turn out that soundness (on either interpretation) will not be a sufficient or a necessary condition for goodness, and that validity (on either interpretation) will be a necessary but insufficient condition of goodness. It will also be shown that goodness is neither a necessary nor sufficient condition of soundness. This article can, perhaps, serve as a useful prolegomenon to introductory deductive logic, though its distinctions are themselves somewhat unorthodox and reach beyond the scope of formal logic.


One definition of validity which is relatively common, easily found in most introductory textbooks on deductive logic, is the following:

An argument is valid if and only if it is not logically possible for the premises to be true and the conclusion false.

Let’s explore the dynamics of this definition. It would mean that an argument of the following sort would be considered valid:

  1. All men are human
  2. Socrates is a man
  3. Therefore, Socrates is human.

Clearly, in this argument, it is not logically possible for the premises to (both) be true, and for the conclusion to be false. The same can be said of the following argument:

  1. Bob loves Carroll
  2. If Bob loves Carroll then Carroll loves Joe
  3. Therefore, Carroll loves Joe

This is pretty obviously logically valid. So too, though, is the following argument:

  1. All women are purple.
  2. Socrates is a woman.
  3. Therefore, Socrates is purple.

The reason this argument is valid is that it is not possible for the premises to (both) be true, and for the conclusion to be false. Perhaps the premises and conclusion are all, in fact, false, but in any possible world in which the premises were true, the conclusion would be true. Thus, validity is not concerned with truth so much as truth-preservation. The concern is to ensure that one cannot, in a ‘valid’ argument, move from true premises to a false conclusion. Take the following example as well:

  1. All women are purple.
  2. If all women are purple, then evolution is true.
  3. Therefore, evolution is true.

In this argument, we have a conclusion which is (I presume) true in fact, while the premises are all false. However, the argument is clearly valid as well, since it is not logically possible that the premises be true and the conclusion false. Remember that validity requires nothing more than that it is not possible for both (i) the premises to be true, and (ii) the conclusion to be false.

The difficulty with this account of validity arises when we are confronted with examples of the following variety:

  1. All men are animals.
  2. If all men are animals then Tyrannosaurus Rex makes a good pet.
  3. Therefore, 3+4=7

This argument is logically valid, since it is not logically possible for the premises to be true while the conclusion is false (mostly because it isn’t possible for the conclusion to be false, and it is ‘possible’ for the premises to be true). Such an argument, however, doesn’t have any dialectical appeal. Consider also:

  1. I once drew a square-circle,
  2. If I once drew a square-circle, then I am a married bachelor,
  3. Therefore, I once drew the impossible.

This argument can be tricky; in order to find out whether it is valid we have to ask whether it is possible for both (i) the premises to be true, and (ii) the conclusion false. As it turns out, it is not possible for the premises to be true and the conclusion false, precisely because it is not possible for the premises to be true. Thus, formally speaking, it is a logically valid argument.


The definition of a sound argument is pretty straightforward: an argument is sound if and only if it is (i) logically valid, and (ii) all of its premises are true. For example,

  1. Socrates was mortal.
  2. Everything that was mortal, was once alive.
  3. Therefore, Socrates was once alive.

In this argument, we find that it is not logically possible for the premises to be true while the conclusion is false, and in addition, we find that both premises are clearly true. Thus, we have a sound argument on our hands. Any argument which is logically valid is sound just in case all of its premises are true. Thus, for example, the following argument is sound:

  1. A tautology is a tautology.
  2. 6-2=4
  3. The sentence ‘is this a question‘ expresses a question.

This exemplifies the problem with the formal definitions of validity and soundness. It shows that one can construct sound and vacuous arguments by simply ensuring that the premises and conclusions are all necessary truths, or at least that the conclusions are necessary truths while the premises are true. In the interest of more off-the-cuff examples, take for instance:

  1. I once wrote this sentence.
  2. If I once wrote this sentence, then I have written at least one sentence.
  3. Therefore, 3+4=7

This argument is both logically valid, and sound, and yet it appears to be a very bad argument. Nobody who didn’t already accept the conclusion could be led by it to accept the conclusion. It is a bad argument, even for those of us who accept the conclusion; if this argument were submitted as our reason for believing the conclusion then our mathematical belief that 3+4=7 would literally be unjustified (a necessary self-evident truth in which we believe can, of course, be unjustified). What all this illustrates is, first, that the formal definitions of validity and soundness are concerned only with truth preservation, and not with the persuasive force of an argument at all. As philosophers who specialize in the study of modal logic often make a distinction between ‘broad’ and ‘narrow’ logical possibility (eg. a square-circle is broadly logically impossible, but narrowly logically possible since there isn’t any purely formal way to obviate a contradiction between the predicates ‘square’ and ‘circle’), so too, perhaps, should we make a distinction between broad and narrow validity & soundness. What we have looked at so far would be the purely formal or ‘narrow’ accounts of validity and soundness. Maybe a ‘broad’ view of validity (which I will henceforth write as ‘validity*’) would be something like: an argument is valid* if and only if i) it is not possible for the premises to be true while the conclusion is false, and ii) the conclusion meaningfully follows from the truth of the premises. This definition of validity* says everything the former one did, with the addition that the premises and conclusion have to be semantically related (i.e., meaningfully related; they have to have something to do with one another). We can correspondingly say that an argument is sound* just in case it is valid* and its premises are true.

Now, validity* and soundness* are not appropriate distinctions in an introductory course on deductive logic, and so are somewhat philosophically unorthodox. However, they are rather useful outside of that narrow context, and in the context of doing philosophy. In philosophy, we don’t just want sound arguments, we want sound* arguments!


Speaking of what philosophers want, there is another issue I wish to examine, which is what makes an argument ‘good’ by philosophical standards. It turns out, I will argue, that neither soundness nor soundness* are necessary or sufficient conditions of ‘goodness’.

I submit that the goodness of an argument consists in two things: i) that the argument is logically valid*, ii) that the accumulated uncertainty of the premises to the argument’s intended audience sets a reasonably high lower bound on the probability of the conclusion. This second criterion is specially crafted to avoid the common mistakes which have, in the past, been made even by some relatively good philosophers like William Lane Craig; namely, the mistake of thinking that premises in a valid argument need be merely each more plausible than their respective negations for the conclusion to follow forcefully. Indeed, the (probability of the) premises of an argument merely set a lower bound on the probability of the conclusion.[1] If that lower bound on the probability of the conclusion is less than or equal to 0.5 then the argument is not compelling. Whether an argument is persuasive or not to some subject is going to depend on their appraisal of the premises, of course, but a good argument will consist of premises which are not merely more plausible than not, but also highly plausible – plausible enough, at least, that the conclusion will also seem highly plausible. This definition obviously subjectivizes ‘goodness,’ making it dependent upon an audience’s appraisal, but that shouldn’t bother us very much because plausibility has to figure into the goodness of an argument in some way, and ‘plausibility’ is already a term of epistemic appraisal.

Consider the following two arguments, both of which are valid and at least one of which is sound. First, the modal ontological argument, which we can roughly reconstruct as:

  1. God possibly exists (i.e., God exists in at least one logically possible world).
  2. If God exists in one logically possible world then God exists in all logically possible worlds.
  3. If God exists in all logically possible worlds then God exists.
  4. Therefore, God exists.

This argument is sound just in case the conclusion is true. However, that doesn’t make it a very good argument in my sense. Indeed, consider its parody:

  1. God possibly does not exist (i.e., there is at least one logically possible world in which God does not exist).
  2. If there is at least one logically possible world in which God does not exist, then there is no logically possible world in which God exists.
  3. If God exists in no logically possible worlds then God does not exist.
  4. Therefore, God does not exist.

At least one of these two arguments is valid, valid*, sound and sound*, but it is arguable that neither of them are good. Goodness, then, consists in more than just soundness*. So, given the way I’ve just outlined things, we can imagine any number of arguments which are good without being sound, sound without being good, valid* without being sound, sound* without being good, but none which are good without being valid*. The goodness of an argument, it seems, is largely in the eye of the beholder; the goodness of a valid* argument is entirely in the eye of the beholder.


[1] See: http://www.reasonablefaith.org/deductive-arguments-and-probability

Grave Findings

The recent opening of the alleged tomb of Christ in the Church of the Holy Sepulchre has attracted worldwide attention as the marble slab overlaying the tomb has been removed exposing it for the first time since 1555 (A.D.). This historic event has served as an occasion for Christians to review or explore the strength of the case for identifying that tomb as the genuine burial place of Jesus. A thought which occurs to me, as I review the evidence for the authenticity of the site, is that the evidence is actually good enough to provide some very moderate but noteworthy evidence for the historicity of Christ.

The historicity of Christ is, of course, not hotly contested among professional historians or academics, but it has gained notable popularity on the Internet among many new-atheists who adopt the ‘mythicist’ view propounded (or defended) by folks like Dr. Robert Price and Dr. Richard Carrier. In fact, as recently as October 26th, Robert Price finally debated Bart Ehrman (Ehrman being one of the preeminent biblical scholars in the world, as well as a staunch agnostic and author of the book “Did Jesus Exist?: The Historical Argument for Jesus of Nazareth”) on the topic of whether there was a historical Jesus of Nazareth. While I haven’t yet seen the debate (because the group uploading the content to youtube is currently still charging money to view it), initial reviews are a little disheartening. The conspiratorial views of the mythicists are a long way off from getting any serious foothold in mainstream academia, but they are (or, at least, seem to be) gaining more ground in the popular culture.

I do not presently have the time, the space, or even the inclination to take a comprehensive approach to dismantling the mythicist’s case, but I do note that, for what it’s worth, the mythicist hypothesis is regarded by academics as on a par with flat earthism, 9/11 conspiracy theories, and young earth creationism (or, as it really ought to be called, young universe creationism). It is a hack conspiracy theory for which no reasonable case can be made (I would invite the skeptic to explore the case(s) presented by Price and Carrier and contrast that(/them) with Ehrman’s work, as well as the work of figures like N.T. Wright). It will be evident to the reasonable person’s satisfaction that there was clearly a historical figure ‘Jesus of Nazareth.’ Mythicism is of fleeting relevance, but the opening of the tomb in Jerusalem gives me an excuse to offer a thought about how the evidence for the veridicality of the site heaps even more evidence against the Mythicist.

As to the Tomb itself, the archaeological community considers it likely to be the burial place of Christ. It fits the description of (along with everything else we’ve learned about) a first-century Jewish tomb. The Biblical accounts say that Jesus’ body was laid in a tomb by Joseph of Arimathea, a well respected and wealthy member of the Sanhedrin (the same council which had been instrumental in condemning Jesus). Some of the Gospels indicate that Joseph of Arimathea had become a disciple of Christ, though only in secret, and the Gospel of John indicates that Joseph of Arimathea and Nicodemus (another Jewish follower of Christ who kept his views secret) worked together to give Jesus a proper burial.

“After these things, Joseph of Arimathea, who was a disciple of Jesus, though a secret one because of his fear of the Jews, asked Pilate to let him take away the body of Jesus. Pilate gave him permission; so he came and removed his body. Nicodemus, who had at first come to Jesus by night, also came, bringing a mixture of myrrh and aloes, weighing about a hundred pounds. They took the body of Jesus and wrapped it with the spices in linen cloths, according to the burial custom of the Jews.” (John 19:38-40)

The tomb itself is an authentic first-century tomb with a disk-shaped rolling stone at its entrance. Although it is true that there were two kinds of tombs with a stone-slab covering the entrance (one kind with a rolling stone, and another with a roughly rectangular stone covering a doorway), and the disk-shaped stone covering is much rarer (and reserved for the wealthy), the Gospels indicate that the tomb of Jesus was found with its stone ‘rolled’ away (Luke 24:4), indicating that it was the rarer variety of tomb in Jerusalem. Some scholars doubt that the actual tomb of Christ had one of the rare disk-shaped stones covering the entrance; Urban C. von Wahlde, for instance, has written an article titled “Biblical Views: A Rolling Stone That Was Hard to Roll.”[1] Nevertheless, I think a stronger case can be made for the disk-shaped stone, especially in light of the case for the authenticity of the tomb safeguarded by the Church of the Holy Sepulchre. Apart from this tomb matching the biblical description, the story of its discovery also lends it immense credibility.

The story of the tomb’s discovery is ancient history, but it is extremely interesting. William Lane Craig, speaking casually (and excitedly) on his podcast recently recounts the following:

“Scholars believe that the The Church of the holy Sepulchre has a very credible claim to be on the site of the actual tomb of Jesus, and this is based on a couple of very interesting facts about its discovery. In the year 326 (this is just one year following the council of Nicea that was convened by the emperor Constantine and then promulgated the famous Nicene creed – the following year) Constantine’s mother, Helena, went on a pilgrimage to the Holy Land, for the purpose of finding relics from the time of Christ and when Constantine’s mother came to Jerusalem she asked the residents of Jerusalem where the tomb of Jesus had been… The people in Jerusalem at that time pointed her to this site where a pagan temple now stood and they said the tomb of Jesus was on this site and this pagan temple was built over it. Well, Helena ordered the temple to be razed and the earth to be excavated [to] get rid of this pollution of paganism. Now, what was interesting about the site identified by the residents of Jerusalem at this time is that the site lay within the walls of Jerusalem. If you look at where the Church of the Holy Sepulchre is, it’s inside the city walls, but the Gospels state that Jesus was crucified and buried outside the walls of the city; they would never allow a crucifixion site and burial of unclean corpses to be going on inside the Holy City, it had to be outside the walls, and so it was odd that the residents of Jerusalem would point Helena to a site inside the city walls. Well, as it turned out many centuries later archaeologists excavating the city discovered that the original walls of Jerusalem were more narrowly constrained in that the site that the residents of Jerusalem pointed Helena to actually lay outside the original walls of Jerusalem. They had been later expanded. … The second thing that’s interesting… is that when they began to excavate the site and remove the earth they dug down and… lo and behold they excavated a tomb exactly where the residents of Jerusalem said that it would be. Now what’s interesting is that this Pagan temple stood on that site since it was built by the emperor Hadrian in A.D. 110. Now, since Jesus was crucified around A.D. 30, that means that the memory of this temple being on the site of Jesus’ tomb goes back to within just 80 years of the crucifixion and burial of Jesus, well within the time that historical memory might be preserved. And so there’s a very very good chance that this is the very tomb in which Joseph of Arimathea lay the corpse of Jesus of Nazareth.”[2]

The fact that the site originally identified was identified within the walls of Jerusalem (to the best of everyone’s knowledge), and that it came to light only centuries later through archaeological discovery that it was actually outside the original walls of Jerusalem, gives this site immense plausibility. Being originally inside the walls lowered the conditional probability of its being authentic (though the fact that there was a tomb there fitting the description of the biblical tomb and that it was identified by the residents of Jerusalem as the spot, raised the conditional probability of it being the authentic tomb). However, once it was discovered that this tomb was, in fact, outside of the walls of Jerusalem in place at the time of Jesus’ burial, that greatly raises the conditional probability of its being authentic. It is not merely that the tomb resides outside the original walls which is relevant for the conditional probability assessment here, it is that it was identified first as the tomb and was later discovered that it lay outside of the original walls of Jerusalem. That discovery raises the conditional probability tremendously. To formalize this a little bit:

Pr(A|B&W) < Pr(A|B&~W)

Pr(A|B&D&~W) >> Pr(A|B&~W)

Where A means the tomb is authentic, B stands for our background knowledge, W stands for ‘the site of the tomb is located within the city walls’ and D stands for ‘discovering after the fact that ~W.’ In the words of the archaeologist Dan Bahat, “we may not be absolutely certain that the site of the Holy Sepulchre Church is the site of Jesus’ burial, but… we really have no reason to reject the authenticity of the site.”[3] In fact, the discovery after the fact (in conjunction with the other properties which fit the description of the tomb from early sources) raises the probability of this being the authentic tomb highly enough that we can say it provides evidence that there was an authentic tomb. This entails that there was a place where the historical Jesus of Nazareth was buried, and so a historical figure, ‘Jesus of Nazareth.’

Pr(J|B&D*) > Pr(~J|B&D*)

Where J stands for ‘Jesus of Nazareth existed,’ B stands, once again, for our background knowledge and Dstands for ‘the case for the authenticity of the tomb in light of the discovery that it lies outside of the original city walls.’

This case isn’t compelling. It’s just something to think about… Also worth thinking about, depending upon how strong you think the case for the Shroud of Turin is, is the following report from the U.K. branch of EWTN.[4] I leave that here, without endorsing any of it, for those of you who may be interested in a pretty far-fetched but provocative suggestion.


[1] http://members.bib-arch.org/publication.asp?PubID=BSBA&Volume=41&Issue=2&ArticleID=10

[2] http://www.reasonablefaith.org/opening-the-tomb-of-jesus

[3] http://news.nationalgeographic.com/2016/10/jesus-christ-tomb-burial-church-holy-sepulchre/

[4] https://www.ewtn.co.uk/news/latest/astonishing-discovery-at-christ-s-tomb-supports-turin-shroud

Math, therefore God?

“Monsier! (a+bn)/n=x, donc Dieu existe; répondez!”[1]

Thus (allegedly) spoke the mathematician Leonard Euler when, at the invitation of Russian Empress Catherine the second, he confronted Denis Diderot in a (very short) debate on the existence of God. Diderot, who was not very good at math, was dumbstruck; he had absolutely no idea how to even begin responding to such an argument. In fact, he couldn’t even understand the argument, and Euler knew it! The court laughed him literally out of town (he promptly asked the Empress for leave to return to France). The formula, of course, is entirely meaningless, and may have been sleight of hand on Euler’s part (making his argument mathemagical rather than mathematical). Additionally, the anecdote has survived only in sparse notes (of dubious historical relevance) here and there with probably varying degrees of accuracy, so it is anyone’s guess what Euler actually meant. This amusing anecdote does, however, invite us to think about what arguments there could be, in principle, from mathematics for the existence of God.[2] Without offering much commentary on how promising these arguments are, I want to distinguish three viable (or, at least, viably viable) types of arguments which could be constructed.

The Argument from Mathematical Beauty

Although the formula Euler originally spouted off didn’t signify anything of mathematical (or philosophical) consequence, the beauty of Euler’s equation, eiπ + 1 = 0, gave rise to the apocryphal anecdote that Euler argued “eiπ + 1 = 0, therefore God exists.” There is (mathematicians tell us) a sublime mathematical beauty in this equation, and there is no obvious or intuitive reason why it is true. What is so special about this equation? One savvy commentator I ran across online put it so nicely I feel compelled to quote him:

“It’s a sort of unifying identity in mathematics, containing each of the fundamental operations (additive, multiplicative, exponential) and each of the fundamental constants (e, i, pi, 1, 0) combined in a theorem that united trigonometry, analysis, and algebra and geometry. It’s really an amazing identity, and the proofs for it are diverse and fascinating…”[3]

It has, thus, been called the origin of all mathematics. Keith Devlin is purported to have said:

“like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.”[4]

Its elegance cries out for an explanation, but that explanation has proved so elusive that a desperate appeal to God begins to look almost reasonable, even to (some) mathematicians.

What should we make of this sort of argument? It seems on its face to be about as prima facie (in)admissible as any other argument from beauty. However, this argument may have more to recommend it than meets the eye. In particular, mathematical beauty has an uncanny predictive ability, at least in the sense that the more beautiful the mathematical formula, the more likely it is to describe the fundamental structure of the real world. Robin Collins has noted, for instance, that:

“To say that the beauty of the mathematical structure of nature is merely subjective, however, completely fails to account for the amazing success of the criterion of beauty in producing predictively accurate theories, such as Einstein’s general theory of relativity.”[5]

John Polkinghorne, in a lecture I recently had the pleasure of listening to (via podcast), said something similar though with less economy of words:

“It isn’t just [to satisfy] an aesthetic indulgence that theoretical physicists look for beautiful equations; it is because we have found, time and again, that they are the ones which actually do describe… a true aspect of the physical world in which we live. I suppose the greatest physicist I’ve known personally was Paul Dirac, (who held Newton’s old chair… in Cambridge for more than 30 years, who was one of the founding figures of quantum theory, [and] unquestionably the greatest British theoretical physicist of the twentieth century) and he made his great discoveries by a relentless and highly successful lifelong quest for mathematical beauty. Dirac once said ‘it is more important to have beauty in your equations than to have them fit experiment.’ Now he didn’t mean by that that it didn’t matter at the end of the day whether your equations fit the experiments (I know no physicist could possibly mean that), but what he meant was this: ok, you’ve got your new theory, and you use the solution and you find it doesn’t seem to fit what the experimentalist is telling you – now there’s no doubt that’s a setback, but it’s not absolutely necessarily fatal. Almost certainly, you will have solved the equations in some sort of approximation, and maybe you’ve just made the wrong approximation, or maybe the experiments are wrong (we have [known that] to happen even more than once in the history of physics – even in my lifetime I can think of a couple examples of that), so at least there’s some sort of residual hope that with a bit more work and a bit more luck you might have hit the jackpot after all. But, if your equations are ugly, there’s no hope. The whole 300-year history of theoretical physics is against you. Only beautiful equations really describe the fundamental structure of the world. Now that’s a very strange fact about the world… What I am saying to you is that some of the most beautiful (mathematical) patterns that our pure mathematical friends can think up in their studies just thinking abstractly… are found actually to occur, to be instantiated, in the structure of the world around us.”[6]

So mathematical beauty satisfies the empirical desideratum of predictive power in the sense that the more beautiful the mathematical expression, the more likely it is to describe reality.

Interestingly I think this kind of consideration can motivate a scientist (and perhaps even a die-hard empiricist, and/or a naturalist) to believe in the objectivity of aesthetic properties. In fact, unless they find a plausible evolutionary account for why our brains should be calibrated so as to recognize more beauty in the abstract mathematical equations which, it turns out, describe reality, than we find in other equations, there will be a residual mystery about the eerie coincidence of mathematical beauty and accurate mathematical descriptions of physics. An eerie coincidence the queerness of which can perhaps be mitigated by admitting the objectivity of aesthetic qualities.

However, the puzzling queerness of that eerie coincidence can only be (or can most plausibly be) ultimately alleviated if the universe is seen as the product of a (trans-)cosmic artist. If behind the fundamental structure of the universe there lies an intellect with aesthetic sensibilities (in some sense), then that would explain why the world showcases the mathematical-aesthetic qualities it does at the level of fundamental physics even when there is no (obvious?) reason why it should have. That, though, begins to look quite a lot like Theism.

The Argument from the Applicability of Mathematics

This segues into the next kind of argument from mathematics, which concerns the applicability of mathematics to accurate descriptions of the fundamental structure of the physical world. For the purposes of this argument beauty is entirely irrelevant. What is surprising, and in need of an explanation (according to this argument), is that the physical world would turn out to be describable in the language of mathematics (and here we are not simply referring to the basic truths of arithmetic, which are true across all logically possible worlds). William Lane Craig has become the most well-known proponent of this argument, and his articulation of it is relatively succinct.

“Philosophers and scientists have puzzled over what physicist Eugene Wigner called the uncanny effectiveness of mathematics. How is it that a mathematical theorist like Peter Higgs can sit down at his desk and by pouring over mathematical equations predict the existence of a fundamental particle which experimentalists thirty years later after investing millions of dollars and thousands of man-hours are finally able to detect? Mathematics is the language of nature. But, how is this to be explained? If mathematical objects are abstract entities causally isolated from the universe then the applicability of mathematics is, in the words of philosopher of mathematics Penelope Maddy, “a happy coincidence.” On the other hand, if mathematical objects are just useful fictions, how is it that nature is written in the language of these fictions? In his book, Dr. Rosenberg emphasizes that naturalism doesn’t tolerate cosmic coincidences. But the naturalist has no explanation of the uncanny applicability of mathematics to the physical world. By [contrast], the theist has a ready explanation. When God created the physical universe, he designed it on the mathematical structure he had in mind. We can summarize this argument as follows:

  1. If God did not exist, the applicability of mathematics would be a happy coincidence.
  2. The applicability of mathematics is not a happy coincidence.
  3. Therefore, God exists.”[7]

I am not sure of this argument’s philosophical quality, since it seems to me that it may be a metaphysically necessary truth that a logically possible world be amenable to mathematical description of some kind. For instance, it certainly seems true that whatever the geometry of space happens to be, there’s no necessary fact of the matter, but it also seems true that if the geometry of space isn’t Euclidean, it may be hyperbolic, or elliptic, (or maybe something else, je ne sais quoi) but it has got to be something, and what it happens to be may, therefore, not cry out for any more explanation than any other quaint contingent fact about the world.[8] However, maybe I’m mistaken about this; maybe the argument is, in fact, just as viable as other teleological or ‘fine-tuning’ arguments are.

Argument from Mathematical Truth

Finally, the third kind of argument I can think of would go something like this: mathematical truths, like all truths, have truth-makers. These truth-makers will have to be metaphysically necessary on pain of mathematical truths being contingent – but it seems obvious that mathematical truths are necessary truths, that they hold across all logically possible worlds. Now, Nominalism about mathematical objects is incompatible with the commitments we just outlined (unless one adopts Nominalism about modal properties as well), and so seems implausible (or, at least, less plausible than it otherwise would have been in virtue of this incompatibility). Platonism also, however, seems to be problematic. Between Platonism and Nominalism, there is a wide range of views including Divine Conceptualism (according to which mathematical objects exist as necessary thoughts in the necessary mind of God), Theistic Activism, Scholastic Realism[9] and many others besides. In fact, a variety (and perhaps a majority) of the accounts of abstract objects on offer today presuppose the existence of God in different ways.

This opens the way to at least two arguments we could construct for the existence of God. First, we could argue that one of these accounts in particular is most plausibly correct (such as Greg Welty’s Theistic Conceptual Realism),[10] and work our way up from there to the implication that God exists. Second, we could take the disjunction of all the accounts of abstract objects which require the existence of God and argue that (i) if any of them is correct then God exists, but (ii) it is more plausible than not that at least one of them is correct, from which it follows (iii) it is more plausible than not that God exists.

So, there we have it, three kinds of arguments from mathematics for the existence of God; a transcendental argument (from beauty), a teleological argument (from applicability), and an ontological argument (from necessity). Could there be others? Maybe, but I suspect that they will all end up falling into one or another (or maybe at least one) of the general categories I tried to outline here. I admit that I didn’t outline them as general categories very well, but that exercise will have to wait for another day when I have more time to blog to my heart’s content.

As a quick post scriptum; if Euler had any substantive argument in mind and wasn’t merely mocking Diderot for his lack of mathematical aptitude, which of these three kinds of arguments would he most likely have had in mind? It’s hard to say, of course, but my best guess is that if he had anything in mind at all, it would fall into the third category. He may have been thinking that the fact that mathematical and purely abstract (algebraic) ‘structural’ truths exist at all requires some explanation, and this explanation must be found in God. This is just a guess, and I make no apologies for it – I am happy to think that Euler was just teasing Diderot, but I am equally happy to entertain the thought that if Diderot had not immediately asked to leave (because of his embarrassment), Euler may have been able to elucidate his point.

[1] Gillings, Richard J. “The so-called Euler-Diderot incident.” The American Mathematical Monthly 61, no. 2 (1954): 77-80. http://www.fen.bilkent.edu.tr/~franz/M300/bell2.pdf

[2] Notice that these are not to be confused with mathematical arguments per se; they are merely arguments from mathematics, in the same way as you might have arguments from physics (the argument from cosmological fine-tuning, the Kalam, etc.) for the existence of God which are not intended to be scientific proofs of God’s existence, but scientifically informed philosophical proofs/arguments for God’s existence.

[3] Russel James, Why was Euler’s Identity Supposed to be a Proof for the Existence of God, https://www.quora.com/Why-was-Eulers-identity-supposed-to-be-a-mathematical-proof-for-the-existence-of-God; Note that he finishes the quoted paragraph with the words “but It has nothing to do with god whatsoever.” I have left this out not because I think he is wrong, or to misrepresent his position, but because it has nothing to do with the formula and everything to do with the propositional attitude he adopts with respect to the question of whether the formula is any kind of reason to think there is a being like God.

[4] Paul J. Nahin, Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills, (Princeton University Press, 2011), 1. https://books.google.co.uk/books?id=GvSg5HQ7WPcC&pg=PA1&redir_esc=y#v=onepage&q&f=false

[5] Robin Collins, The Case for Cosmic Design, (2008), http://infidels.org/library/modern/robin_collins/design.html

[6] John Polkinghorne, Science in the Public Sphere, http://www.veritas.org/science-public-sphere/

[7] William Lane Craig, Is Faith in God Reasonable? William Lane Craig vs. Dr. Rosenberg, http://www.reasonablefaith.org/debate-transcript-is-faith-in-god-reasonable

[8] I am really, honestly, no more sure of this counter-argument than I am of the argument. For those interested, please do check out the debate between Craig and Daniel Came on the Unbelievable? Podcast, which you can also find here: https://www.youtube.com/watch?v=nn4ocx316dk

[9] J.T. Bridges defends this view: https://www.youtube.com/watch?v=eFU1BKxJf1k

[10] See: Greg Welty, “Theistic Conceptual Realism,” in Beyond the Control of God: Six views on the Problem of God and Abstract Objects, ed. Paul Gould, (New York: Bloomsbury Academic, 2014), 81-96.

Some Considerations in Favor of Moral Realism

I was pulled into a facebook discussion today about moral realism. I decided I should use that as an excuse to write a short blog-post outlining some philosophical considerations which, I think, should lead us to affirm moral realism with confidence.

First, if there are properly basic beliefs which are not analytic, it seems that the belief in moral realism will definitely be properly basic. If we adopt a purely pragmatic account of epistemic justification then it also seems as though moral realism will be preferable to its negation. In fact, on a variety of epistemologies it looks as though moral realism fares pretty well. What kind of epistemology would justify moral anti-realism? I think the epistemologies which come to mind seem more philosophically suspicious than their competitors. So, unless we have a defeater for moral realism, we seem to be well within our rights to accept moral realism.

What might that defeater be? I suppose one could argue that if Naturalism is true, the moral anti-realism is true, but Naturalism is true, therefore et cetera. However, what reason do we have for believing that Naturalism is true? In all my time as a philosopher I have still yet to hear an even half-way decent argument for Naturalism. I would invite Naturalists to offer arguments here, but experience and my gut both tell me that most Naturalists have, at best, a vague sense that Naturalism seems right, and a poorly thought out set of reasons for thinking that metaphysical naturalism is true. Nevertheless, I remain open to incoming arguments, should anyone wish to present them. I should note, however, that even on Naturalism one should do whatever they can to make room for moral realism, for instance by trying to work out an account of Moral Naturalism.[1]

Second, there is a reductio ad absurdam we can run against arguments for moral anti-realism, which, if I recall correctly, W.L. Craig has presented. The idea goes like this: any argument you could give against moral realism can be parodied with near perfect parity into an argument against belief in the noumenal external world apprehended through the empirical senses. In the case of the external world we apprehend through the five senses that there is such a thing, which seems mind-independent and experience-independent. We have, of course, never verified that the external world is there absent any experience at all, and this is why we occasionally run into philosophers who adopt subjective idealism and deny that there is any such thing as a mind/experience-independent world. We are in a similar position with respect to our meta-ethical beliefs. In our moral experience we apprehend (through experience) that there are moral duties, values and facts which appear to be as objective as anything else we apprehend by experience. We naturally conclude that we encounter, in and through our moral experiences, a moral reality, a world of objective moral facts. In fact, our belief in moral realism is closer in kind to our belief in the noumenal external world than are our beliefs in mathematical facts or modal facts. The latter are the result of the operations of pure reason, whereas the former are the deliverances of experience.

There is, we think, something objectively real about the rock we touch, but this judgment is as much an intellectual knee-jerk reaction as it is possible to conceive. We have the same kind of intellectual knee-jerk reaction when it comes to moral realism, and the reaction comes with just as much force. It takes equally extreme cunning to convince ourselves to believe in moral nihilism as it does to fool ourselves into accepting subjective idealism. Both are just forms of skepticism. In fact, if one puts the arguments down on paper and compares them it will become obvious that there is no reason to deny moral realism which won’t count as an equally good reason to deny the noumenal external world. The subjective idealist won’t be impressed with this reductio, but most people will be.

Third, we can argue in the spirit and fashion of G.E. Moore, whose famous response to the skeptic was “I have a hand!” G.E. Moore’s point was that he would always be more sure that he had a hand than he could be that any argument for skepticism was sound. He might think that all the premises seem true, and agree that the argument seems logically valid, but he would deny that this gives him good enough reason to think that such an argument is sound. The credence which an apparently sound argument for skepticism provides would always, according to Moore, be outweighed by the credence given by experience for the proposition that he has a hand. No argument for skepticism, however good, will justify embracing skepticism, because no argument can make skepticism more plausible than things like ‘that I have a hand.’

To illustrate this point with an analogy, let’s use a logical argument for being skeptical of logical entailment. Suppose a teacher tells her elementary students that they will have a surprise quiz next week. Susie, a young student and budding logician, figures that the quiz wouldn’t be a surprise if it were on Friday, since they would have gone all week without it and would, therefore, be expecting it on Friday. She concludes that the surprise quiz can’t take place on Friday. However, she reasons that since Friday has been logically eliminated, the quiz cannot take place on Thursday either, since, if it hadn’t occurred until Thursday, but the quiz can’t possibly be a surprise Friday, then it can’t be a surprise Thursday either. She continues this process of elimination and determines that there is no day next week on which it is possible to have a surprise quiz. She has not made any obvious logical error in her reasoning, and yet just imagine her surprise when she has a quiz on Tuesday! Therefore, logical reasoning doesn’t always lead from true premises to true conclusions, even if it starts off with true premises and at each step the logical structure of the argument is impeccable.

What are we to make of such an argument? Well, we could come up with very clever responses, but Moore’s point is that even if we weren’t clever enough to discern where the reasoning is going wrong, we would (and should) still not accept that the argument justifies skepticism about logic! I am, and always will be, more sure of modus ponens than I can be that an argument for logical contradiction is sound. This is Moore’s point, and it translates well to the issue of moral realism/anti-realism.

The Atheist philosophy Louise Antony put it nicely when she said “Any argument for moral scepticism will be based upon premises which are less obvious than the existence of objective moral values themselves.”[2] If she is right then it seems like the Moorean response works as well here as it does anywhere.


One popular objection to the second argument I presented, which I hear surprisingly often, is that in our empirical experiences we can achieve a reasonable consensus about what the physical world is really like, but our moral experiences are less conspicuous, less clear, ‘fuzzier,’ and are less conducive to creating consensus about the fabric and structure of moral reality. This, it is suggested, gives us at least one reason to think that we should place more confidence in our empirical experience of the noumenal external world than we should place in our moral experience. Our moral experiences are more suspicious because they are a great deal vaguer than our empirical experiences.

I have two responses to this. First, although not all scientific matters can be adjudicated by empirical experiments either (think, for instance, of empirically equivalent scientific theories in equally good scientific standing, such as the neo-Lorentzian view of relativity, and the standard view of relativity), I will grant (and not just for the sake of argument) that scientific consensus is more easily reached than moral/ethical consensus. However, I want to note in passing that nearly everyone (who is some kind of moral realist) agrees that it is wrong to (without a justifying reason) kick a pregnant woman in the stomach repeatedly for amusement. There is, I think, a great deal more moral consensus than people typically imagine, but I digress. Having admitted this point of disanalogy, it still looks to me as though moral realism is so nearly as well justified as belief in the noumenal external world that the disanalogy makes no practical difference; moral realism ought still to be believed in the absence of a defeater, or else, on pain of inconsistency, we will be putting our belief in the external world in the very near occasion of philosophical abandonment.

My second response is to say that I think this objection confuses moral realism with particular meta-ethical accounts, or normative accounts. The arguments presented are not suggesting that we should be able to discover, through moral experience, what is objectively morally right (or wrong) with as much clarity and consensus as we discover what is objectively scientifically right (or wrong). What is being claimed is merely that in our moral experience we are as sure that we are being confronted with some set of objective moral facts as we are that, in our empirical experience, we are being confronted with an external world. The point here is that moral experience leads us to be confident not in any particular moral theory (eg. Utilitarianism, Egoism, Deontology, etc.), but in moral realism itself, a presumption which all moral theories share in common, and on which their coherence depends.


[1] See James Lenman, “Moral Naturalism,” inThe Stanford Encyclopedia of Philosophy ed. Edward N. Zalta, (2014). http://plato.stanford.edu/archives/spr2014/entries/naturalism-moral/

[2] https://www.youtube.com/watch?v=B6WnliSKrR4

Banach-Tarski paradox, א Infinities, Infinitesimals, and the A-theory

I will offer an analysis of what is going wrong with the Banach-Tarski paradox suggesting that points, construed as infinitesimal surface areas, are nothing more than mathematically useful fictions. I will suggest that infinitesimals raise the same kinds of modally-prohibitive paradoxes in metaphysics as positing actually infinite quantities does (and for the same or similar reasons), and then consider an argument against the A-theory (in most of its forms) which can be purchased from these insights. I will then scout out some philosophical avenues available to the A-theorist.

The Banach-Tarski paradox is a famous mathematical paradox according to which it can be proved that if you divide the surface area of a sphere into little bits, and simply rearrange the bits appropriately, you can reconstruct two spheres each with the same surface area as the original sphere. In layman’s terms, you can prove (something just a shocking as) that 1=2.[1] To explain how it works, it may be worth calling to mind the various paradoxes associated with actual infinities.

Consider what it would be like to count upwards from -7 to infinity and stop only once you’ve arrived. Even if given an infinite amount of time you would never arrive, because no finite additions can sum up to a transfinite quantity. Subtract infinity from infinity, and what do you have? You have zero, but you also have infinity, and you also have 18.9801 (and every other real number); all of these are not just legitimate answers, they are mathematically correct answers. However, clearly 18.9801 is not equal to either zero, infinity, or anything else! Have a (Hilbert) hotel with an infinite number of rooms, all of which are occupied, and you want to check in an infinite number of new guests? No problem, just move every person from the room they are in (n) to the room with a room number equivalent to two times the original room’s room number (2n). Done; you’ve managed to move people around in such a way as to create an infinite number of vacant rooms without asking anyone to leave. Most of us (who are interested in this sort of thing) know the myriad paradoxes which arise from postulating even the possibility of an actual infinity. It seems relatively philosophically secure that there cannot be an א number of things (where א represents the first transfinite number, not to be confused with ∞ which symbolizes infinity taken as a limit rather than a quantity). If there are philosophically sophisticated caveats then so be it, but the point will remain that there are plenty of examples of things for which having an א number of them is clearly (broadly logically) impossible.

Let’s return, for a moment, to Hilbert’s Hotel, because it’s a particularly useful illustration. Suppose that the guest in room 3 checks out, while all the (infinitely many) other rooms remain occupied. The desk clerk decides that they want every room occupied, so they ask each person in room n (where n>3) to move one room over; that is, from room n to room n-1. That will fill up room 3, but the process will also leave no room empty because there is no room number n for which there is not an occupied room n+1. This works equally well for two dimensional shapes, such as circles; remove one ‘point’ from the circumference of a circle and you may have an infinitesimal gap, but simply move every other point along the circumference over (uniformly) by an infinitesimal amount and, voila, the gap is plugged and there will be no new gap. The trick in the case of the Banach-Tarski paradox is to apply the same reasoning to three-dimensional objects. For the best explanation of this paradox I’ve ever seen, (especially for readers who aren’t familiar with it, please make your life better and) check out Vsauce.

Alexander Pruss has noted on his blog that this result “is taken by some to be an argument against the Axiom of Choice.”[2] However, he argues that you can get the same paradoxical result in similar cases (and even in the same case) without the axiom of choice, so that the axiom of choice should be cleared of all suspicions. I agree (though I’m certainly no expert). Richard Feynman is purported to have said, upon being shown the proof, that “it’s fine you can do it with ‘continuous spheres’, since there’s no such thing. The important thing is you can’t do it with oranges, because oranges are made of a finite number of indivisible parts.” I think he is wrong about oranges (being actually comprised of indivisible finite parts, at least if the ‘parts’ are extended in three spatial dimensions), but his sentiment is appreciably insightful nonetheless.

The problem with the paradox, in my submission, is that it divides the surface of the sphere up into points. However, points on a sphere, like points on a line segment, are infinitesimals. This is precisely why (Aristotelians) say that line segments are not composed of points the way walls are composed of bricks, but, instead, points act as the limits between which a line segment is continuously extended. An infinitesimal is a quantity which is infinitely small. It is non-zero, but it is also smaller than any finite quantity. Sure infinitesimals are useful for doing things like infinitesimal calculus, developed by one of my all time favorite philosophers Gottfried Wilhelm Leibniz, but they remain, I believe, nothing more than useful fictions. To borrow a phrase from W.L. Craig;

“They are akin to ideal gases, frictionless planes, points at infinity, and other useful fictions employed in scientific theories.”[3]

If we are to accept the possibility of infinitesimal quantities in reality, then we will quickly run into paradoxes like the Banach-Tarski paradox (which, quite apart from being obnoxious to the rational intellect, seems to violate the law of conservation of matter and energy). Positing infinitesimals is just as paradoxical as positing sets of actually infinitely many discrete things (where ‘things’ is an ontologically loaded term). I am suggesting that infinitesimals are just as paradoxical as actual infinities, and, at bottom, for the same reason(s). In fact, I have this intuition that every argument for thinking that there cannot be any actual infinities (as opposed to potential infinities, where ‘infinity’ merely acts as a limit), admits of a parody for an argument against the existence of infinitesimals. I’m not sure I can rigorously prove it, but I think it’s very plausible.

It seems to me that there’s something conceptually parasitic about infinitesimals relative to infinities. They each conceptually supervene on each other symmetrically. To visualize this symmetry, consider plotting the function ƒ(x)=  1/x which will look like this:


The distance between the curved line and the x-axis (i.e., y=0) as x approaches (positive or negative) infinity is shrinking (or, at least, its absolute value is shrinking), and approaching an infinitely small non-zero measure. When X is infinite, the absolute value of the y-axis coordinate of the curved line (i.e., the distance between the curved line and it’s asymptote, here being the x axis) is infinitesimally small. This example helps to illustrate the point that the concept of an infinitesimal is bound up with the concept of infinity, so that in the absence of one the other would be inconceivable. That at least motivates the suspicion that if one turns out to be metaphysically impossible, so will the other.

What relevance does this have for the philosophy of time? Well, consider that on the A-theory there is such a time as the present. How long does the present last? What, precisely, is its magnitude, its duration? Let’s consider the following argument:

  1. If the A-theory is true, then the present is either infinitesimal in duration, or it is finite in duration.
  2. The present cannot be infinitesimal in duration.
  3. The present cannot be finite in duration.
  4. Therefore, the A-theory is false

Premise 3 can be established with Leibniz’ argument against the (logical) possibility of a physically indivisible element, or ‘atom’ (in the etymologically literal sense). For anything extended in three-dimensional space, however small, it will always be logically possible for me to divine it in two, even if I am physically incapable of doing so (due to some constraint, such as not having the appropriate equipment for the job, or maybe not even being able to develop any tool which could do the job). Physical impossibilities are not (all) logical impossibilities, and logically there is no constraint on how many times I could divide an object extended in space. To say that there is an object extended in space which is not logically possibly divided up into smaller constituent pieces is, according to Leibniz, incoherent. The exact same argument, mutatis mutandis, works against there being chronons (i.e., atomic chunks of time).

The denial of premise 2 is absurd given our observations that positing infinitesimals leads to modally unconscionable paradoxes like Banach-Tarski.

Ways out: I see four ways, not all of them equally viable, for an A-theorist to escape the conclusion of this argument.

First, they could challenge premise 3 on the grounds that, if there are chronons, then by definition they are entities which cannot be physically divided. The suggestion would be that the prima facie absurdity of a Chronon de dicto doesn’t entail the impossibility of a chronon de re. This dangerously dislocates rational intuition from epistemic reliability, but I can imagine extreme empiricists embracing this response.

Second, they could challenge premise 2 by arguing that positing any more than one real infinitesimal of any kind might be problematic, but that there’s no way to derive similar paradoxes from positing a maximum of one infinitesimal. In other words, perhaps paradoxes involving infinitesimals only arise when there are n infinitesimals, where n ∈ ℕ, and n>1. Multiply an infinitesimal by any natural number, or even a transfinite number, and you will still get an infinitesimal result, so it seems harder to show that from one infinitesimal you could derive some kind of contradiction.

Quick thought: Perhaps if there are rules/axioms such as (i) no infinitesimal can be larger or smaller than any other infinitesimal, but (ii) anything (other than 1) to the power of itself is larger than itself, you could derive a contradiction by taking an infinitesimal X, running it through Xx=Y, and then asking whether Y is larger than X, or the same size (it appears to be both). However, I don’t have the kind of facility in mathematics to be able to produce a rigorous proof that even a single infinitesimal would lead to some kind of contradiction or unconscionable paradox. Moreover, it isn’t entirely clear to me what relevance that kind of mathematical paradox would have for the metaphysical consideration at hand. In any case, the second challenge to premise 2 cannot be lightly dismissed.

Third, one could adopt a really wild philosophy of time, such as the Apresentism I wrote about in the last post (thus denying the first premise).

Fourth, one could deny the first premise by adopting what has been called a non-metric view of the present. This is the view preferred by William Lane Craig.[4] I have more than expended my allotted time for blogging and casual writing today, so I will leave this post here for now. I may return to the idea of non-metric present in the (near) future in another post.

[Ha, I don’t presently have time to write more. Get it?]

[1] For fun, check out and try to find the mistake in the following mathematical proof that 1=2 here: https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html

[2] http://alexanderpruss.blogspot.ca/2013/06/the-banach-tarski-paradox-and-axiom-of.html

[3] William Lane Craig, “Response to Greg Welty,” in Beyond the Control of God: Six Views on the Problem of God and Abstract Objects, ed. Paul Gould (A&C Black, 2014), 102.

[4] See: Craig, William. “The extent of the present.” International Studies in the Philosophy of Science 14, no. 2 (2000): 165-185.