An Amended Modal-Epistemic Argument for God’s Existence

Several years ago I was introduced to a clever and fascinating argument, developed by a philosopher named Emanuel Rutten, which attempts to demonstrate the existence of God from two key premises: (i) that anything which is possibly true is possibly known, and (ii) that it is not possible to know that God does not exist, from which it logically follows that (iii) God exists. The argument has some intuitive appeal to me, though I was initially skeptical about the second premise (skeptical, that is, that the atheist could be persuaded to accept the second premise). I had also heard certain criticisms of the argument which seemed to present nearly insuperable objections to it; although I started working on responses to those objections, I eventually moved on to other philosophical inquiries leaving this argument (and my many notes on it) to gather proverbial dust on my old hard drive. Recently, however, I decided to revisit the argument and use a variation on it in the context of a semi-formal online debate. I was shocked by my interlocutor’s reaction; although he had not been shy about sinking his teeth into every other argument I had presented for theism (from the cosmological argument from contingency, to the transcendental argument from the laws of logic, to a version of the moral argument, to the modal-ontological argument), I received radio-silence when presenting this argument. After several days of him reflecting upon the argument, he eventually rejoined by saying that he couldn’t think of a single criticism, but that he was convinced the argument was bad for some reason he was unable to articulate. This made me want to revisit the modal-epistemic argument for God’s existence and see if it couldn’t be salvaged in light of certain criticisms of which I am aware.

The basic intuition behind Rutten’s argument is that reality’s being intelligible is somehow connected to, and explained by, the existence of a God-like being. This same intuition seems to lurk behind Bernard Lonergan’s argument for God in the nineteenth chapter of his magnum opus, Insight, where he made the tantalizing claim (for which he argued at great length) that “if the real is completely intelligible, God exists. But the real is completely intelligible. Therefore, God exists.”1 There is also a subliminal connection here, I think, even to C.S. Lewis’ argument from reason. The same intuition is also bolstered, to some extent, by Fitch’s paradox, which is a logical proof developed by the philosopher and logician Frederic Fitch in 1963. Fitch was able to prove, using prima facie uncontroversial assumptions, that “necessarily, if all truths are knowable in principle then all truths are in fact known.”2 This philosophical finding was taken to be paradoxical by many, but it sits exceptionally well with the theist who affirms that omniscience is exemplified by God. What these observations show, I think, is that the intuition behind Rutten’s argument is widely shared (at least among theists) and may be well motivated.

The bare-boned sketch of Rutten’s argument can be outlined as follows:

  1. All possible truths are possibly known (i.e., if there are logically possible worlds in which P is true, then there will always be a subset of such worlds in which P is known).
  2. It is impossible to know that God does not exist.
  3. Therefore, God necessarily exists.

It has to be said straight-away that this is an over-simplified formulation of his argument; we will come, in due course, to his more measured articulation of the argument, but the rough sketch provided by this syllogism will help us lay the groundwork for the actual argument.

Rutten stipulates the following relatively modest definition of God, for the purposes of his argument; God is the personal first-cause of the world (where the world is the whole of contingent reality). Since that logically implies that God is incontingent, I will abbreviate this as ‘IPFC.’ He also specifies that, for the purposes of the argument, he means the following by knowledge: “A conscious being… knows that proposition p is true if and only if p is true and the being, given its cognitive situation, cannot psychologically but believe that p is true.”3 More precisely, for any P, if some conscious being B cannot psychologically help believing that P is true, then P satisfies at least one of the following four conditions for B: “(i) The proposition is logically proven; (ii) the proposition is obviously true, i.e. intuitively self-evident; (iii) the proposition is grounded in indisputable experience; or (iv) the proposition is based on indisputable testimony.”4 This makes it obvious that Rutten means that something is known if and only if (a) it is true, and (b) given some conscious being’s cognitive situation, that being, whose cognitive faculties aren’t malfunctioning, cannot psychologically help believing that it is true. In what follows I will refer to this peculiar kind of knowledge as knowledge*, instances of knowing satisfying these conditions as knowing*, et cetera.

The first premise seems to flow directly out of the perennial philosophical commitment to the world’s intelligibility. Arguably, to be intelligible the world has to be the kind of thing which is knowable* in principle (if not always to us, due to some limitations of our cognitive faculties, then at least to some logically possible intellects with different cognitive faculties). This philosophical presumption has, Rutten hastens to note, “led to extraordinary discoveries”5 in science. In fact, it seems to be a fundamental pillar of science itself, for science is predicated on the assumption of the world’s intelligibility. The second premise also seems prima facie plausible; it is, somewhat ironically, appealed to confidently by many agnostics and some atheists.

The argument is, in its rough form, susceptible to a myriad of informative objections. Consider, for instance, the possibly true proposition: “God understands my reasons for being an atheist.”6 The proposition, although plausibly possibly true, is not knowable – for knowledge requires belief, but no atheist can believe the proposition. Similarly the proposition “there are no conscious beings”7 may be possibly true but is also not rationally believable. To avoid these kinds of counter-examples Rutten stipulates that his first premise should only quantify over rationally believable propositions. He thinks it is reasonable to exclude rationally unbelievable propositions, and that this way of restricting his first premise is not ad hoc, for it seems intuitively plausible that all rationally believable possible truths are knowable. Requiring the propositions of the relevant sort to be both (possibly) true and rationally believable navigates the argument away from obvious counter-examples. There are other counter-examples, however, and Rutten discusses some. First, consider a proposition like “‘John left Amsterdam and nobody knows it.’”8 This seems possibly true and obviously unknowable, even though it could be argued to be rationally believable. To deal with objections like this Rutten introduces a distinction between first-order propositions and second-order propositions; first-order propositions, he says, are directly about the world, whereas second-order propositions are about people’s beliefs about the world. Rutten then decides to limit the first premise of his argument to truths expressed by first-order propositions. In this way he blocks cute objections from propositions like ‘there are no believed propositions.’

Then he states his argument9 more formally in the following way (I have changed the wording very little, and added nothing of consequence):

1. If a rationally believable first order proposition is possibly true, then it is knowable* (first premise),
2. The proposition ‘IPFC does not exist’ is unknowable* (second premise),
3. The proposition ‘IPFC does not exist’ is rationally believable (third premise) ,
4. The proposition ‘IPFC does not exist’ is first order (fourth premise),
5. The proposition ‘IPFC does not exist’ is not possibly true (from 1, 2, 3 and 4),
6. The proposition ‘IPFC does not exist’ is necessarily false (from 5),
7. The proposition ‘IPFC exists’ is necessarily true (conclusion, from 6).

The third premise is either true, or else atheism is irrational. The fourth premise is self-evidently true. The fifth premise follows logically from 1,2,3 and 4. Six follows logically from five. Seven follows logically from six. So the key premises are 1 and 2. The first premise is very plausible insofar as its negation would imply that reality is not intelligible, but to deny that reality is intelligible seems absurd. That reality is intelligible (if not to us then at least in principle) seems to be a fundamental commitment of epistemology. However, if reality is intelligible, then for any first-order rationally believable proposition P, if P is possible then P is possibly known*. Can we know this premise in the strong sense of knowledge used within the argument? Maybe (e.g., perhaps it is obviously true, i.e. intuitively self-evident), but that’s also irrelevant; all we need is to ‘know’ it in the more general sense (i.e., having a true justified belief – allowing for whatever epistemology you’d like to use in order to qualify ‘justified’) in order to know (as opposed to know*) that the conclusion is true. 

The second premise is plausible given that, for the purposes of the argument, ‘knowledge’ is defined as satisfied just in case at least one of the four stipulated conditions are satisfied. However, God’s non-existence cannot be logically proven (if it can, then obviously this and all other arguments for God’s existence are worthless). On atheism, the proposition that God does not exist is not self-evidently true. On atheism, the proposition ‘God does not exist’ cannot be grounded in indisputable experience. On atheism, the proposition ‘God does not exist’ cannot be believed on the basis of indisputable testimony. It follows that the second premise is true. So, the argument looks sound, at least at first blush.

One immediate reaction to this argument is to suggest that it can be parodied by a parallel argument for atheism by substituting the second premise for: 2.* The proposition “God exists” is unknowable*. However, this is naïve; in at least one possible world in which God exists, plausibly God knows* that the IPFC (i.e., himself) exists, but in no possible world where no IPFC exists can anyone know* that no IPFC exists. As Rutten explains:“on the specific notion of knowledge used for the argument… logical proof, intuition, experience and testimony exhaust the range of knowledge sources, and none of them suffices to know that God does not exist.”10

Years ago now I heard one very interesting objection which I will try to reproduce as fairly as my memory and skill will allow. The objection basically maintains that if God could know* that the IPFC (i.e., God) exists, then it is possible for at least one atheist in at least one logically possible world to know* that the IPFC does not exist. Rutten suggests, in the paper, that “God’s knowledge that he is God – if possible – is an instance of (iii) (or (ii)),”11 meaning that it is either “obviously true, i.e. intuitively self-evident; [or]… grounded in indisputable experience.”12 But what experience could possibly establish the indubitability of being the IPFC? For any experience you can imagine having (if you were God), it seems logically possible that it is the result of an even greater being who created you with the purpose of deceiving you into thinking that you are the IPFC. What about intuitive self-evidence? Well, if it is possible for God to simply look inward and, through introspection, discover his relations (for, to be the IPFC is to bear certain relational properties, such as that of being first-cause), then why can’t there be a logically possible world in which an atheist introspects and discovers that she lacks any relation to an IPFC? If it is logically possible for the IPFC to introspectively survey its own relational properties, then why can’t a logically possible atheist do the same?

I think the best answer to this is to note that it may be possible to introspectively discover at least some of one’s essential properties (as opposed to merely accidental properties). I can know, by rational reflection, that I exist (cogito ergo sum), that I am a thinking thing, that I am either contingent or not omniscient, et cetera. I can also deduce from what I discover as self-evident through introspection that other facts happen to be true, such as that there exists something rather than nothing. So, coming back to God, perhaps God can know by introspection that he is incontingent, personal, and has some uniqualizing properties13 (that is, properties which, if had at all, are had by no more than one thing) etc. – and perhaps that means that he can deduce that he is the only being which could be an IPFC in principle, and that he is an IPFC just in case a contingent world exists. But, he could plausibly know* from indisputable experience (of some sort) that a contingent world exists. Therefore, he could deduce and know* that he is the IPFC. If atheism were true, no being would have, as an essential property, a lack of any relation to an IPFC. Lacking a relation cannot be an essential property, so there’s no reason to think it could be introspectively discovered that one lacks a relational property to the IPFC. Moreover, unless the atheist can actually produce (perhaps with the aid of premises introspectively discovered as self-evident) a logical proof that the IPFC does not exist it seems they cannot know* that no IPFC exists. So while this objection is extremely interesting, I do think that it fails; it is reasonable to maintain that, possibly, God knows* that the IPFC exists, and it does not plausibly follow that an atheist possibly knows* that no IPFC exists.

Another objection might come from considering large facts. Take, for instance, what Pruss has called the Big Conjunctive Contingent Fact (BCCF),14 and let’s take the sub-set of that fact which includes only first-order, rationally affirmable facts (for simplicity, I will abbreviate this as the BCCF*). The BCCF* is plausibly comprised of infinitely many conjuncts, and at least is possibly comprised of infinitely many conjuncts. Is this possible truth, the BCCF*, possibly known? I think it is possible so long as there is possibly a being with an infinite capacity for knowledge (or else, perhaps, an actually infinite number of beings with some finite capacity for knowledge not all of which are such that a discrete set of first-order rationally affirmable truths would have been beyond its ken). But, assuming there cannot be an actually infinite number of beings, doesn’t that presuppose something like theism, by presupposing the possible exemplification of omniscience (here we assume that BCCF*⊃BCCF, and that any being which knows the BCCF* also knows all analytic truths)? After all, the Bekenstein bound15 is generally taken to imply “that a Turing Machine with finite physical dimensions and unbounded memory is not physically possible.”16 However, it seems senseless to suggest that there could be a physical object (like a brain, or some other kind of computer) which is actually infinitely large. Therefore, doesn’t the first premise presuppose something like theism insofar as it presupposes the exemplifiability of omniscience or at least an intellect with an actually infinite capacity for knowledge? That would make the argument ostensibly circular.

First, the IPFC needn’t be omniscient even if it knew the BCCF*. Second, and more importantly, the IPFC isn’t being presupposed to be omniscient, or even knowledgeable enough to know the BCCF*. Third, a being’s being omniscient is necessary but insufficient for the truth of theism. Fourth, I’m not sure whether it is senseless to talk about infinitely large physical objects, or (actually) infinitely many beings, but I am relatively sure that most atheists have a vested interest in allowing for those kinds of possibilities in order to avoid conceding important premises in some (Kalaam) cosmological arguments. So this attempted charge of subtle circularity seems wrong.

[I should grant this this last objection could be accused of being a straw man erected by none other than myself; to that I just briefly want to say that I had originally thought that there may be an objection here, but as I tried to write the objection down clearly it seemed to crumble in my hands. Having already written it out, and having found it interesting to reflect upon it whether or not it is a viable objection at all, I decided to keep it in this final draft.]

I’m sure there are other possible objections which I would have been better able to iterate or anticipate had I done so years ago when this argument, and some objections to it, were still fresh in my mind. However, my sense is that that will do for an introduction to the argument. What makes this argument really exciting, I think, is that it, as Rutten notes, “does not fall within one of the traditional categories of arguments for the existence of God. For it is not ontological, cosmological or teleological. And it is not phenomenological either, such as for example the aesthetic or moral argument[s] for God’s existence.”17 The argument, whether sound or unsound, is doing something genuinely novel, at least for the analytic tradition of the philosophy of religion.

Rutten ends his short paper on an optimistic note which may be appropriately appended here, and this is where I will end my short excursus:

As I mentioned in the introduction, I propose to refer to the argument as a modal-epistemic argument. Ways to further improve it may be found, just as has been done with arguments in the other categories. I believe that if this happens, the prospects for the argument are rather promising.”18

1 Bernard Lonergan, Insight: A Study of Human Understanding, Collected Works of Bernard Lonergan, vol. 3, ed. Frederick E. Crowe and Robert M. Doran (Toronto: Toronto University Press, 1992), 695.

2 Brogaard, Berit and Salerno, Joe, “Fitch’s Paradox of Knowability”, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.), URL = <;.

3 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 3.

4 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 4.

5 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 14.

6 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 7.

7 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 8.

8 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 9.

9 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 10-11.

10 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 2.

11 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 5.

12 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 4.

13 Alexander R. Pruss, “A Gödelian Ontological Argument Improved Even More.” Ontological Proofs Today 50 (2012): 204.

14 Alexander R. Pruss, “The Leibnizian cosmological argument.” The Blackwell Companion to Natural Theology, ed. W.L. Craig and J.P. Moreland (2009): 24-100.

15 See: “Bekenstein Bound,” Wikipedia, accessed March 24,2017.

16“Bekenstein Bound,” Wikipedia, accessed March 24,2017.

17 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 28.

18 Emanuel Rutten, “A Modal-Epistemic Argument for the Existence of God.” Faith and Philosophy (2014), 28.

Notes on a Transcendental Argument from Logic

Nearly ever since I was first exposed to transcendental argumentation through listening to that famous debate between Greg Bahnsen and Gordon Stein,1 I have retained the intuition that there is an interesting potential argument from the fact that there are necessary propositions (necessary, that is, simpliciter) to the conclusion that there is a necessary mind. While the analysis of what it means to be a necessary mind will fall short of the God of perfect being theology or classical theism, it will still provide a being which so resembles God that it significantly undermines atheism. This being may not have all the superlative attributes, but it will be a metaphysically necessary immaterial spaceless timeless being with an intellect (and whatever that entails), et hoc omnes intelligunt Deum. However, to avoid the charge of using St. Thomas’ famous phrase in order to paper-over the chasm between my conclusion and full-blown theism, I will state the conclusion more modestly in terms of which the good old reverend Bayes would approve. Enjoy;

1) There are laws of logic.
2) Logical laws are identical to necessary propositions (exempli gratia [P v ~P])
3) Therefore, there are necessary propositions.
4) Propositions are not real entities which exist mind-independently, but are mind-dependent (i.e., there is no proposition for which there is not at least one subvenient mind).
5) A necessary truth is a truth which obtains in all logically possible worlds.
6) Necessary truths are either grounded in at least one contingent mind, or at least one incontingent mind.
7) There are logically possible worlds without any contingent minds.
8) Therefore, there must be at least one necessary mind.
9) If there is at least one necessary mind then it is a being with intellect (plausibly knowing all necessary truths), which is immaterial (spaceless, timeless) in nature.
10) The conditional probability of theism is, ceteris paribus, greater than the conditional probability of not-theism on the condition that there is at least one metaphysically necessary immaterial being with intellect. 
11) Therefore, theism is probably true, 
ceteris paribus.

There are plenty of points at which one could still object to this argument, but it seems to me that most objections are philosophically more costly than the conclusion. One might also just accept the conclusion but deny that, in fact, things really are equal (i.e., cetera non sunt pariba) in this case. For instance, the objector could insist that there are no propositions which are ‘necessary’ in the sense required here (that is, necessary simpliciter – not a merely model-dependent necessity). They might also insist, for some odd reason, that there are not possible worlds without contingent minds, or that those worlds are possible in a merely model-dependent way while other possible worlds are possible simpliciter. That would be pretty wild. Another might argue that the existence of a metaphysically necessary immaterial mind doesn’t raise the conditional probability of theism at all (maybe because the probability of theism is ‘0’ – or because it is ‘1’). Somebody could, of course, deny the major premise, that there are laws of logic. Somebody may also insist that laws of logic are not identical to the propositions which express them (though that seems to reify them so much as to put the objector, for other reasons, in the near occasion of belief in theism anyway). Alternatively one may think that each premise on its own seems more plausibly true than false, but that the collection of them together seems to have a upper-bounded probability of lower than or equal to 0.5, and that would be a principled way to object.

Edit*: it occurs to me that there’s no way of which I’m aware to really set an upper-bound on the probability of a conclusion. What the objector could say, then, is either that the conclusion just seems to be no more likely than 0.5 (notwithstanding the plausibility of the individual premises), or that the premises collectively set a lower-bounded probability on the conclusion of less than or equal to 0.5, in which case the argument fails to be compelling.

To be fair, this argument of mine very likely draws significantly from the influence of James N. Anderson and Greg Welty,2 whose argument seems, to me, much better than what often passes for responsible argument among presuppositionalists (among whom, I should take a moment to clarify, I adamantly do not count myself).

1 For those interested, you can find the audio of the debate, and the transcript (because the audio is really not great) at the following two links: and

2 James N. Anderson and Greg Welty, “The Lord of Non-Contradiction: An Argument for God from Logic” Philosophia Christi 13:2 (2011).

An Amended Minimal Principle of Contradiction

The law of non-contradiction seems self-evidently true, but it has its opponents (or, at least, opponents of its being necessary (de dicto) simpliciter). W.V.O. Quine is perhaps the most well known philosopher to call the principle into question by calling analyticity itself into question in his famous essay “Two Dogmas of Empiricism,” and suggesting that, if we’re to be thoroughgoing empiricists, we ought to adopt a principle of universal revisability (that is to say, we adopt a principle according to which absolutely any of our beliefs, however indubitable to us, should be regarded as revisable in principle, including the principle of revisability). Quine imagined that our beliefs were networked together like parts of a web in that we have beliefs to which we aren’t strongly committed, which we imagine as near the periphery of the web, which are much less costly to change than the beliefs to which we are most strongly committed, which we imagine as near the center of that web. Changing parts of the web nearer to the periphery does less to change the overall structure of the network than changing beliefs at the center of the web. Evolution has, in operating upon our cognitive faculties, selected for our tendency towards epistemic conservatism.

This, he thinks, is why we don’t mind changing our peripheral beliefs (for instance, beliefs about whether there is milk in the fridge or whether a certain economic plan would better conduce to long-term increases in GDP than a competing plan) but we stubbornly hold onto our beliefs about things like mathematics, logic, and even some basic intuitive metaphysical principles (like Parmenides’ ex nihilo nihil fit). Nevertheless, indubitability notwithstanding, if all our knowledge is empirical in principle, then everything we believe is subject to revision, according to Quine. He boldly states:

… no statement is immune to revision. Revision even of the logical law of the excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle?1

This statement is far from short-sighted on Quine’s part. Those who defend his view have suggested that even the law of non-contradiction should be regarded as revisable, especially in light of paraconsistent systems of logic in which the law of non-contradiction is neither axiomatic, nor derivable as a theorem operating within those systems. This is why Chalmers calls attention to the fact that many regard Quine’s essay “as the most important critique of the notion of the a priori, with the potential to undermine the whole program of conceptual analysis.”2 In one fell swoop Quine undermined not only Carnap’s logical positivism, but analyticity itself, and with it a host of philosophical dogmas ranging from the classical theory of concepts to almost every foundationalist epistemological system. The force and scope of his argument was breathtaking, and it continues to plague and perplex philosophers today.

More surprising still is the fact that Quine isn’t alone in thinking that every belief is revisable. Indeed, there is a significant faction of philosophers committed to naturalism and naturalized epistemology, but who think that a fully naturalized epistemology will render all knowledge empirical, and, therefore, subject to revision in principle. Michael Devitt, for instance, defines naturalism epistemologically (rather than metaphysically):

“It is overwhelmingly plausible that some knowledge is empirical, justified by experience. The attractive thesis of naturalism is that all knowledge is; there is only one way of knowing”3

Philosophical attractiveness, I suppose, is in the eye of the beholder. It should be noted, in passing, that metaphysical naturalism and epistemological naturalism are not identical. Metaphysical naturalism does not entail epistemological naturalism, and neither does epistemological naturalism entail metaphysical naturalism. I have argued elsewhere that there may not even be a coherent way to define naturalism, but at least some idea of a naturalized metaphysic can be intuitively extrapolated from science; there is, though, no intuitive way to extrapolate a naturalized epistemology from science. As Putnam puts it:

“The fact that the naturalized epistemologist is trying to reconstruct what he can of an enterprise that few philosophers of any persuasion regard as unflawed is perhaps the explanation of the fact that the naturalistic tendency in epistemology expresses itself in so many incompatible and mutually divergent ways, while the naturalistic tendency in metaphysics appears to be, and regards itself as, a unified movement.”4

Another note in passing; strictly speaking Devitt’s statement could simply entail that we do not ‘know’ any analytic truths (perhaps given some qualified conditions on knowledge), rather than that there are no analytic truths, or even that there are no knowable analytic truths. Quine, I think, is more radical insofar as he seems to suggest that there are no analytic truths at all, and at least suggests that none are possibly known. Devitt’s statement, on the other hand, would be correct even if it just contingently happened to be the case that not a single person satisfied the sufficient conditions for knowing any analytic truth.

Hilary Putnam, unfortunately writing shortly after W.V.O. Quine passed away, provided a principle which is allegedly a priori, and which, it seems, even Quine could not have regarded as revisable. Calling this the minimal principle of contradiction, he states it as:

Not every statement is both true and false”5

Putnam himself thought that this principle establishes that there is at least one incorrigible a priori truth which is believed, if at all, infallibly. Putnam shares in his own intellectual autobiography that he had objected to himself, in his notes, as follows:

“I think it is right to say that, within our present conceptual scheme, the minimal principle of contradiction is so basic that it cannot significantly be ‘explained’ at all. But that does not make it an ‘absolutely a priori truth’ in the sense of an absolutely unrevisable truth. Mathematical intuitionism, for example, represents one proposal for revising the minimal principle of contradiction: not by saying that it is false, but by denying the applicability of the classical concepts of truth and falsity at all. Of course, then there would be a new ‘minimal principle of contradiction’: for example, ‘no statement is both proved and disproved’ (where ‘proof’ is taken to be a concept which does not presuppose the classical notion of truth by the intuitionists); but this is not the minimal principle of contradiction. Every statement is subject to revision; but not in every way.”6

He writes, shortly after recounting this, that he had objected to his own objection by suggesting that “if the classical notions of truth and falsity do not have to be given up, then not every statement is both true and false.”7 This, then, had, he thought, to be absolutely unrevisable.

This minimal principle of contradiction, or some version of it, has seemed, to me, nearly indubitable, and this despite my sincerest philosophical efforts. However, as I was reflecting more deeply upon it recently I realized that it is possible to enunciate an even weaker or more minimalist (that is to say, all things being equal, more indubitable) principle. As a propaedeutic note, I observe that not everyone is agreed upon what the fundamental truth-bearers are (whether propositions, tokens, tokenings, etc.), so one’s statement, ideally, shouldn’t tacitly presuppose any particular view. Putnam’s statement seems non-committal, but I think it is possible to read some relevance into his use of the word ‘statement’ such that the skeptic may quizzaciously opine that the principle isn’t beyond contention after all. In what follows, I will use the term ‘proposition*’ to refer to any truth-bearing element in a system.

Consider that there are fuzzy logics, systems in which bivalence is denied. A fuzzy logic, briefly, is just a system in which propositions are not regarded (necessarily) as straightforwardly true or false, but as what we might think of as ‘true’ to some degree. For instance, what is the degree to which Michael is bald? How many hairs, precisely, does Michael have to have left in order to be considered one hair away from being bald? Well, it seems like for predicates like ‘bald’ there is some ambiguity about their necessary conditions. Fuzzy logic is intended to deal with that fuzziness by allowing us to assign values in a way best illustrated by example: “Michael is 0.78 bald.” That is, it is 0.78 true that Michael is bald (something like 78% true). Obviously we can always ask the fuzzy logician whether her fuzzy statement is 1.0 true (and here she either admits that fuzzy logic is embedded in something like a more conventional bivalent logic, or she winds up stuck with infinite regresses of the partiality of truths), but I digress. Let’s accept, counter-possibly, that fuzzy logics provide a viable way to deny bivalence, and thus allow us to give a principled rejection of Putnam’s principle.

Even so, I think we can amend the principle to make it stronger. Here is my proposal for an amended principle of minimal contradiction:

“Not every single proposition* has every truth value.”

I think that this is as bedrock an analytic statement as one can hope to come by. It is indubitable, incorrigible, indubitably incorrigible, and it holds true across all possible systems/logics/languages. It seems, therefore, as though it is proof-positive of analyticity in an impressively strong sense; namely, in the sense that necessity is not always model-dependent. At least one proposition* is true across all possible systems, so that it is necessary in a stronger sense than something’s merely being necessary as regarded from within some logic or system of analysis.


As a post-script, here are some principles I was thinking about as a result of the above lines of thought. First, consider the principle:

At least one proposition* has at least one truth-value.

To deny this is to deny oneself a system altogether. No logic, however esoteric or unconventional or counter-intuitive, can get off the ground without this presupposition.

Consider another one:

For any proposition* P, if we know/assume only about P that it is a proposition*, then P more probably than not has at least one truth-value.

I’m not certain about this last principle, but it does seem intuitive. The way to deny it, I suppose, would be to suggest that even if most propositions* were without truth-values, one could identify a sub-class of propositions with an extremely high probability of having a truth-value, and that will allow one to operate on an alternative assumption.

[Note: some of the following footnotes may be wrong and in need of fixing. Unfortunately I would need several of my books, currently in Oxford with a friend, to adequately check each reference. I usually try to be careful with my references, but here I make special note of my inability to do due diligence.]

1 W.V.O. Quine, “Two Dogmas of Empiricism,” in The Philosophical Review, Vol. 60, No.1 (Jan., 1951), 40.

2 David J. Chalmers, “Revisability and Conceptual Change in “Two Dogmas of Empiricism”.” The Journal of Philosophy 108, no. 8 (2011): 387.

3 Louise Antony, “A Naturalized Approach to the A Priori,Epistemology: An Anthology. Second Edition, Edited by Ernest Sosa, Jaegwon Kim, Jeremy Fantl and Matthew McGrath. (Oxford: Blackwell publishing, 2000), 1.

4 Hilary Putnam, “Why Reason can’t be Naturalized,” Epistemology: An Anthology. Second Edition, Edited by Ernest Sosa, Jaegwon Kim, Jeremy Fantl and Matthew McGrath. (Oxford: Blackwell publishing, 2000), 314.

5 Hilary Putnam, “There is at least one a priori Truth” Epistemology: An Anthology. Second Edition, Edited by Ernest Sosa, Jaegwon Kim, Jeremy Fantl and Matthew McGrath. (Blackwell: 2000): 585-594.

6 Auxier, Randall E., Douglas R. Anderson, and Lewis Edwin Hahn, eds. The Philosophy of Hilary Putnam. Vol. 34. (Open Court, 2015): 71.

7 Auxier, Randall E., Douglas R. Anderson, and Lewis Edwin Hahn, eds. The Philosophy of Hilary Putnam. Vol. 34. (Open Court, 2015): 71.

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:


[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.


It was in an article written by Stephen Torre which I read very recently that I was introduced to a very intriguing idea; namely, that tokens, and not propositions, are the fundamental bearers of truth-values. The usual view, of course, is that propositions (whatever one thinks of them) are those things to which the categories/properties ‘true’ and ‘false’ exclusively apply. Tokens, then, merely express truths insofar as (and just in case) they express propositions which are true. On the alternative story, which Torre refers to as the “Token View,” it is tokens which are the fundamental truth-bearers. This alternative story is as indifferent to different theories of truth (eg. correspondence, coherence, pragmatist) as the usual story. Turning to Torre, we read:

“There are different views regarding what the fundamental bearers of truth are. One view is that truth applies fundamentally to tokens. On this view, the predicate ‘is true’ is properly applied only to tokens. Such a view is committed to denying that there are token-independent truths. I will refer to this view as the ‘Token View’. A rival view takes truth to apply fundamentally to propositions. On this view, tokens are true or false only derivatively: tokens express propositions and a token is true iff it expresses a true proposition. This view does allow for the existence of token-independent truths.”[1]

I think it is worth having a bit of fun thinking about what the consequences of this prima facie absurd view would be. As it turns out, the view might have some theologically welcome consequences. For instance, it seems clear that the alleged set-theoretic problems for the doctrine of omniscience are evaporated of significance; even if there is no such thing as the set of all (true) propositions, there is clearly[2] such a thing as the set of all (true) tokens, at least if tokens are created by finitely many minds with finite capacities/abilities.

Tokens, like propositions, may require facts (i.e., extra-mental and extra-linguistic truth-makers), but God could be omniscient either factually (i.e., by having direct unmediated acquaintance with the facts, rather than their representations to the discursive intellect in the form of tokens or propositions), or else God can be token-omniscient. What is it, precisely, to be token-omniscient? Let us stipulate a definition:

G is token-omniscient =def. G knows all true tokens, and believes no false tokens.

Suppose that this view is correct quarum gratia argumentum, and suppose that God’s mental activity produces tokens. In this case it looks as though an old adage of Christian theology is more literally true than it seemed at first glance: to think truly is to think God’s thoughts after Him.

Objection 1: Surely quantification over tokens isn’t problematic unless there are indefinitely many of them. However, it is difficult to imagine a finite mind tokening a truth as of yet not tokened by God, even in any logically possible world. It seems plausible to say, then, that if God tokens any truths then he tokens all truths, but the set of all truths is indefinitely large. Set-theorists have no problem quantifying over infinite sets; the problem was always with quantifying over ‘indefinite’ sets, which are not sets at all. If the set of all true tokens is indefinitely large then the problem recurs.

Response 1: Perhaps we should make a distinction analogous to the distinction between first-order propositions (propositions about the world) and second-order propositions (propositions about propositions about the world), and restrict God’s knowledge to first-order tokens.

G is first-order token-omniscient =def. G knows all true first-order tokens, and believes no false tokens.

God would, of course, still know all first-order tokens about second/third/quadruple/etc-order tokens which occur to finite minds, and that seems sufficient for omniscience.

Objection 2: suppose that (logically/explanatorily) prior to God’s creating anything, He realizes that there are no tokens, and, in realizing this (and being always first-order token omniscient), mentally produces the first-order token T1: “there are no tokens.” This is false, and (being a token), is necessarily false. God would not only not be Token-Omniscient, but wouldn’t even have (all and) only true beliefs.

Response 2: It might not be logically possible for God to token T1, but perhaps it is possible, and inevitable (given the assumptions upon which we are now working in this thought experiment), that God token T2: “There are no (other) tokens” or, rephrased more elaborately, T2’:“there are no tokens other than this one.” Perhaps to avoid self-reference paradoxes we should say of tokens, as I am inclined to say of propositions, that (unless they pick out a universal quality, such as the disjunctive property of being true or false or meaningless) they all come with a caveat de aliis implicite (i.e., with an implicit caveat that they are ‘about’ others). Such stand-alone sentences as “the set of all things I say in this sentence is imponderable” are not true, they are entirely bereft of truth-apt content! Pseudo-meaningful sentence constructions. So also, it seems to me, “this sentence is true” is meaningless, and “there are no sentences” is meaningless. [I am not sure I’m right about this; this is just a knee-jerk reaction on my part to self-reference paradoxes].

What are the downsides of this view (other, of course, than that it seems crazy)? I’m not sure I can think of any unanswerable objections to it, and that alone may make it worth pondering, at least for fun.


Edit: Ok, here’s an obvious objection to Token-Omniscience which I, for whatever reason, didn’t think of previously: suppose I token the following: “I am Tyler.” The token’s content is irreducibly bound up with the sense of the indexical ‘I’ in such a way that nobody distinct from me could recognize that token as true, even if they could have recognized the propositional content to be true. The token, per se, is unknowable to any being distinct from me. Therefore, if tokens are the fundamental truth-bearers, and any more than one being ever uses a personal pronoun to index themselves in tokening a truth, no being can be (first-order) token-omniscient. That seems like a pretty definitive defeater to token-omniscience to me.


[1] Torre, Stephan. “Truth-conditions, truth-bearers and the new B-theory of time.” Philosophical Studies 142, no. 3 (2009): 325-344.

[2] I assume that it is logically impossible to have an actually infinite set of tokens created by finitely many finite minds. This can be challenged, of course, by either insisting that there is no absurdity, contra apparentiam, in positing actual infinities, or else that the absurdities do not arise for tokens. If such suggestions are to be taken seriously, then I would have to weaken my claim here from ‘clearly’ to ‘plausibly,’ but all else would remain the same.

Difurcating ‘Knowing P’ and ‘Knowing P to be true’

William Lane Craig has argued that the difference between ‘Billy the Hippo is fat’ and ‘the proposition Billy the Hippo is fat is true’ is one of semantic ascent. One semantically ascends when moving from a claim of the first kind to a claim of the second kind, and, conversely, semantically descends when going from a claim of the second kind to a claim of the first kind. He says:

I could say “Hitler was a really bad man.” Or I could ascend semantically and say “It is true that Hitler was a really bad man.” Do you see the difference between the first order and the second level claim? And that second level claim doesn’t need to be made—I can just say “Hitler was a really bad man,” and I can make that affirmation sincerely and so forth without ascending semantically to saying, therefore there is a proposition which has the value true.[1]

I used to think this was right (though I’ve always been a little suspicious of it), but I think I’ve come across a reason to think that it may be wrong. I think that going from ‘Hitler was a really bad man’ to ‘it is true that Hitler was a really bad man’ is not merely a semantic ascent; it also involves the acquisition of new semantic content. Strictly speaking, it expresses a new proposition altogether. I will argue in what follows that one can know that Hitler is a bad man without knowing that the proposition ‘Hitler is a bad man’ is true, and one can also know that the proposition ‘Hitler is a ban man’ is true without knowing that Hitler is a bad man. Perhaps Craig could say that just because two propositions are related in this kind of ‘semantic order,’ knowing one needn’t entail knowing the other. This, in fact, is what he should say; he should deny that the two expressions have all and only the same semantic content. Maybe he does deny this (I don’t know), but I will try to argue that he (and we all) should.

One can know that Hitler is a bad man without knowing that the proposition ‘Hitler is a bad man’ is true because a necessary condition on knowledge is belief, but there is no psychologically necessary connection between knowing that Hitler has the property of being a bad man, and knowing that a proposition (about Hitler’s being a bad man) has the property of being true. After all, when one knows that the proposition “Hitler is a bad man” is true, one doesn’t necessarily believe that the proposition that “the proposition that “Hitler is a bad man” is true” is true. In fact, if that were psychologically necessary then it would follow that by knowing any one proposition one would, of psychological necessity, know infinitely many propositions, and this is absurd. Ergo, one can know that the content expressed by a proposition is true without knowing that the proposition itself has the truth-value ‘true.’

One can also know that a proposition is true without knowing its content. For instance, suppose that there were a being who only ever proclaimed true things (and suppose you were aware of this being having that quality).[2] Suppose further that this being proclaimed to you that proposition P is true. Even if you had only a vague understanding of the content of proposition P (maybe it’s some proposition about quantum mechanics, or actuarial mathematics, or epistemology, or whatever field you may be least familiar with), you could know that it is true. In fact, even if you had no way of making heads or tails of proposition P, you could know that it is true. Therefore, one can know a proposition to be true without believing in its content, and thus without ‘knowing’ its content.

There may be a problem here; we can legitimately wonder whether to know that the proposition P is true requires being properly acquainted with P, and whether to be ‘properly acquainted‘ with P requires an understanding of its content. After all, when I see a man in the distance (who is a stranger to me, but whose name is actually ‘Bill’) break a window, I may know that somebody has broken the window, but I do not know that Bill has broken the window. In the same way I can perhaps know that a proposition is true (namely whatever proposition was uttered by the infallible proclaiming being), without knowing that the proposition (proclaimed by the infallible proclaiming being) is true.

How can one be properly acquainted with a proposition, so as to be able to know whether it is true? If one maintains that one must know what the content of the proposition is in order to be properly acquainted with it, then it will turn out that it may not be possible to know that the proposition “Hitler is a bad man” is true without knowing that Hitler is a bad man. If, on the other hand, it suffices to be able to indicate or ‘pick out’ P from other propositions in some way, such as by saying ‘that proposition’ or ‘the proposition just uttered’ or, perhaps, by repeating/recreating the same combination of sounds/scribbles used to express the proposition in the first place, then obviously one can know a proposition P to be true without knowing its content.

In everyday life we find ourselves doing this all the time. We may hear a professor relate a proposition to us, for instance, and without as of yet having understood it, we know (in the sense of having a true and appropriately justified belief) that it is true. The same may happen in a court of law where an expert witness submits testimony which we can know to be true, even if we haven’t understood that to which the expert is testifying. We also often know propositions to be true, such as “E=MC2,” which few of us have any genuine understanding of. This goes to show that one needn’t be very well acquainted with a proposition in order to know that it is true. Propositional acquaintance comes in degrees, and all one needs in order to know that a proposition is true is enough acquaintance to ‘pick out’ that proposition by some means. If this is right, then to be properly acquainted will require very little, and too little to present any challenge to my argument.

Where does this leave us with respect to Craig’s claim that to move from a claim like that Hitler is a bad man, to a claim like that the proposition Hitler is a bad man is true, is to merely to ascend semantically? It will depend on what Craig means, precisely, by semantic ascent/descent; it should be rejected in light of my observation just in case it does not allow one to maintain i) that the propositional contents in any two propositions related in this ‘order’ of semantic ascent/descent are not identical with each other, and ii) that one can know a truth of either kind without knowing a truth of the other kind. In short, I am calling for a semantic bifurcation of propositions related to each other in a semantic order of ascent/descent.


[2] Thanks to Dr. Brian Leftow, who shared this thought in conversation. I had written in an essay that a being was propositionally omniscient if and only if it knew all true propositions and believed no false ones, but later in the same paper I defined propositional omniscience again as follows: a being B is propositionally omniscient if and only if for any proposition P, if P has the property of being true then B knows that it is true, and if P does not have the property of being true then B does not believe it. This clumsy mistake of mine led to more careful reflection on precisely this point; knowing that P is true is not the same as knowing P.

A Scientistic argument for Determinism, and some related thoughts

I would like to write a little bit, today, about Determinism. First, I want to try to give another argument for determinism which occurred to me recently (though I think it is a very poor argument, but may be worth mentioning if for no other reason than that it is some kind of argument for determinism). Following this I wish to draw on a thought experiment presented by Alexander Pruss to show that libertarian free will can be consistently combined with physical determinism.

What arguments are there for determinism? Let us take determinism to be the thesis that for any even E, E either follows of causal necessity from some prior (or posterior) event(s), or else from E every event follows of causal necessity. To avoid trouble, let us stipulate that no two events are both simultaneous and non-identical (i.e., events are complete states of affairs at a moment). Obviously the reason I used ‘causal’ necessity in the definition, as opposed to logical necessity, is that at any time t1, plausibly there is a future-tense (or past-tense) fact about any time t1+n (where n can be negative), so that at least one proposition at any time (and thus for any E) will logically entail every other proposition at every other time. Even the libertarian accepts that, so we should be careful not to conflate that with determinism.

I have said previously that I can think of one argument for a modest kind of determinism which would still be strong enough to rule out libertarian free will; 1) that human beings are entirely material entities, 2) that all material entities are governed entirely by deterministic physical laws, and therefore 3) human beings are determined to act and think exactly as they do act and think. I mentioned that this argument seems implausible to me for two reasons; first, that human beings are not plausibly entirely material entities,[1] and second that the laws of physics are not actually deterministic.[2] However, notice that this argument, even if it were sound, would not go as far as to entail that determinism per se is true (but only that physical determinism is true), nor would it give us any justificatory reason(s) for believing that determinism is true. Additionally, the restriction to physical determinism may actually undermine determinism per se. On determinism per se, even the universe is deterministically caused to begin to exist (assuming it does so), but on physical determinism there is no physical determinant responsible for the beginning of space, time, energy and matter. Physical determinism would, then, imply that materialism (and anything like it) is false, or that determinism per se is false. That’s a hard bullet for the champion of scientism to bite.

Here’s a more ambitious argument for determinism:

  1. Determinism is a necessary presupposition of the scientific method.
  2. The scientific method is the only, or in any case the best, avenue to genuine discovery (i.e., the finding of truth, since a discovery of something false is not a genuine discovery).
  3. Therefore, the presupposition of determinism is a necessary condition of the only, or in any case the best, avenue to genuine discovery (i.e., to the truth).
  4. For any P, if P is a presupposition necessary for the only, or in any case the best, avenue to genuine discovery, then P ought to be believed.
  5. Therefore, determinism ought to be believed.
  6. For any P, if P ought to be believed then P is true (i.e., nothing untrue ought to be believed).
  7. Therefore, determinism is true.

We might call this a presuppositionalist argument for determinism. If it were sound then it would provide us with a good reason to believe that determinism is true.[3]

Is this argument any good? Unsurprisingly, I think not. To start off, the first premise seems dubious, especially in light of the same points I made in response to the last argument for determinism which I examined – namely that quantum mechanics may not be deterministic (and yet clearly indeterministic theories of quantum mechanics are scientific, whether or not they are possibly true), and even Newtonian mechanics is certainly not deterministic (and yet, again, is clearly scientific, regardless of whether it is true, or even possibly true – scientific theories can suggest metaphysical impossibilities without ceasing to be scientific). The second premise is also problematic in my view, since it seems to me to simply enunciate the prejudice of scientism, which we have no good reasons for accepting, along with very good reasons for rejecting. So, I outright reject both of the first two premises of this argument.

I also think there are significant problems with the sixth premise which, even though I accept it, seems dubious on the assumptions of determinism and scientism. If determinism is correct, that seriously threatens the possibility of genuine ethics, including the ethics of belief, and if scientism is true then we have no good reason for believing that there are no false beliefs which we ought to adopt (for instance, if scientism and determinism are true, maybe I ought to believe that I am free in a morally relevant sense, even though, in fact, I am not and cannot be – or, paradoxically, if determinism/scientism are true, then, possibly, I ought not to believe that they are true). The whole reason for thinking that a belief ought to be believed if and only if it is true is based on a kind of metaphysical conception of truth on which truth, beauty and goodness are, we might say, ‘natural siblings.’ This makes perfect sense on the Christian way of seeing things, as well as many (perhaps most) other worldviews, but it does not make much sense on materialism or naturalism (which scientism enjoins on us). I’m not even sure it makes much sense on any non-materialist, but yet deterministic, view of the world (like the Calvinist worldview).[4]

Returning to the topic of physical determinism, I would now like to talk about an illustration I found in Pruss’ writing which helps to show that physical determinism is logically compatible with libertarian free will. Pruss uses the image of a cannonball flying through the air to clarify the difference between the Principle of Sufficient Reason (PSR) and what he calls the “Hume-Edwards-Campbell Principle” (HECP). According to the HECP, if each member of an infinite set could be explained in terms of the preceding member(s) then (i) every member of the set would be explained, and (ii) the set itself would stand in need of no additional explanation. The HECP is sometimes used as a response to cosmological arguments from contingency, for obvious reasons. Hume, for instance, writes:

Add to this that in tracing an eternal succession of objects it seems absurd to inquire for a general cause or first author. How can anything that exists from eternity have a cause[?]… In such a chain, too, or succession of objects, each part is caused by that which preceded it and causes that which succeeds it. Where then is the difficulty? But the whole, you say, wants a cause. I answer that the uniting of these parts into a whole, like the uniting of several distinct countries into one kingdom or several distinct members into one body, is performed merely by an arbitrary act of the mind and has no influence on the nature of things. Did I show you the particular causes of each individual in a collection of twenty particles of matter, I should think it very unreasonable, should you afterwards ask me, what was the cause of the whole twenty. This is sufficiently explained in explaining the cause of the parts”[5]

This principle is, I believe, demonstrably wrong for several reasons, though my favorite demonstration is provided by Pruss who proves that an infinite series of successive explanations is logically equivalent to one great big viciously circular explanation. However, my interest here is not to find out whether the HECP is correct, but how thinking through the HECP can help make clear how physical determinism is compatible with libertarian free will.

To illustrate the difference, let’s imagine that there were a cannonball flying through the air in a logically possible world where there was no time at which the cannonball was not flying through the air. Every point in time at which the cannonball was in a certain place, going a certain speed in a particular direction, those facts could all be explained by pointing to facts about where a cannonball was (how fast it was going, and in what direction it was moving) at a preceding point in time (at least presuming the regularity of its motion, which is to say that its motion is governed by certain laws). For Hume (et al) it would make no sense to ask for an explanation over and above this for the fact that there is a cannonball flying through the air; we can explain why the cannonball exists, why it is moving as fast as it is, and why it is going in this direction rather than that direction, all by referring to facts about the laws governing its movement along with the fact that it existed, where it was, how fast it was moving, and in what direction it was going at some previous time. Where the HECP makes any further explanation unnecessary, the PSR demands that there be an explanation for why there is a cannonball at all, for the PSR demands that there be an explanation of any contingent fact.

Keeping this distinction in mind, let us imagine a logically possible world W which had no beginning, but was just stretched out temporally infinitely in its past, and in which physical determinism is true. At any point in time in W, W could be said to have existed for an infinite number of some unit of temporal length (hours, days, milliseconds, etc.), call this unit T, so that it had no beginning in the sense that there is no first T in W.[6] Now, for any state of affairs in W picked out by any time tn, all the facts about that state of affairs can be explained by the facts which obtain in W at a slightly earlier time tn-1. So, for any state of affairs at any time in W, there is an adequate explanation for that state of affairs in terms of some other state of affairs at another time which deterministically brings it about. In W, the HECP is satisfied by the facts we have laid out, whereas the PSR requires a deeper explanation for the existence of W, and for contingent facts obtaining in W. The PSR reminds us that such explanations are possible, and this will help us to see that libertarian free will possibly coincides with physical determinism.

Bear in mind that all we need to do in order to demonstrate the compossibility of two propositions is to show that there is a logically possible world out there which satisfies both propositions. In W, physical determinism is satisfied. If in/at W there is at least one libertarian-free act (or, technically, even just one libertarian-free agent), then the compossibility of libertarian free will and physical determinism will have been logically demonstrated. Clearly, however, it is logically possible that the existence of W is explained by the voluntary election of a libertarian-free divine agent (i.e., God). If God, in a libertarian-free capacity, chose to create such a world, then the world and all of its happenings would ultimately be explained in terms of God’s acting freely to create it. Thus, physical determinism is clearly logically compatible with libertarian free will. This is because God is, ex hypothesi, not a material entity. Suppose, further, that people are not merely material entities (i.e., the mind is immaterial), but that epiphenomenalism is true of all embodied people, and the mind, which persists after bodily death, becomes libertarian free once freed of the body. So long as this is logically possible its very possibility goes to show that physical determinism is demonstrably logically compatible with libertarian freedom.

However, there may be another way in which libertarian free will is compatible with physical determinism, at least on the B-theory of time. Suppose that there is a set of physical states of affairs P, consisting of {P1, P2, P3… Pn}. Now, suppose that any Pn+1 follows from Pn of causal necessity (for closed physical systems).  Every physical state of affairs in P is causally explained by some other physical state of affairs in P. Nevertheless, it is logically possible that the sufficient reason for a state of affairs in P involves the fact that a libertarian-free agent in that world makes a libertarian-free decision Fn at some time tn. Here, we might schematize this relationship as follows:

P1 → P2 → P3 → P4
↑↑↑↑       ↑↑↑↑
F1           F2

So, although P2 is physically-causally explained (i.e., HECP explained) by P1, P1 and P2 may only be sufficiently explained (i.e., PSR explained) by appeal to F1 (which itself is sufficiently explained just in case it is logically possible that facts about libertarian-free acts can be sufficiently explained).[7] It may seem strange to talk about libertarian-free acts which occur, in some sense, independently of their space-time context (for, if they occurred within that context, then physics, as we’re imagining it, would provide the context and impetus for the decision, along with determining the decision), but certainly that’s no stranger than thinking of God’s choices as libertarian free even though they are independent of any space-time context.

There is also, perhaps, a stranger way in which we can conceive of this relationship of free choices in a space-time context and a physically deterministic world. I should note that I’m not entirely sure whether this is coherent (it may run into unforeseen problems which more extensive analysis could tease out), but my suspicion is that it is coherent (and, therefore, logically possible). We might imagine that the context in which a libertarian free choice is made is physically under-determinative, but that, once a free decision is made, the result is that the world is supplied physical properties which make that decision appear physically determined. Here we have to imagine that free decisions occur with a limited space-time context (an under-determinative one), and that backwards causation is possible (i.e., events from the future can cause things in the past). Then, we might imagine that even though a temporally antecedent state of affairs P1 causally determines that P2 occurs next, a person’s free choice after the time at which P1 is the case, and before the time at which P2 is the case, is the sufficient reason P1 has the causally determinative features it has for bringing P2 about. On this view, a libertarian free agent makes a decision in light of an under-determinative slice of P1, and their making a decision has temporally backwards-reaching effects which supply P1 with all the physical features necessary for it to deterministically bring P2 about. On this view, a libertarian free decision can be the sufficient reason why P1 deterministically brings P2 about, even though P2 is HECP explained adequately in terms of P1 alone. This view is strange only because we generally think of causal sequences as parallel with temporal sequences, but, at least on the B-theory of time, there is no reason causal antecedence and temporal antecedence need to go hand-in-hand; my free decision may (atemporally) cause features of the past, and maybe those features physically-deterministically cause events in the future.

The Temporal Sequence: P1 → Fn → P2

The (atemporal) Causal Sequence: P1* → Fn → P1 → P2

In conclusion then, we still have no good arguments for believing either in determinism per se, nor in physical determinism. Moreover, even if physical determinism were true, we would have, it seems, no good reasons to doubt the fact that we are libertarian free, at least if we accept the possibility of temporally backwards causation (and, therefore, the B-theory). This can more easily be seen when we distinguish the HECP from the PSR, and note the two different levels of explanations which satisfy them. The PSR needn’t be true, but explanations of the kind it demands, if even possible, carve out a space for libertarian-free decisions even in a physically deterministic world.

[1] For further reading on this point, see Koons, Robert C., and George Bealer, eds. The Waning of Materialism. Oxford University Press, 2010.

[2] I cited the Copenhagen theory of quantum mechanics, as well as John Norton’s now famous example of a ball on a dome (in the comments section, in response to a reader), which illustrates that even Newtonian mechanics is not entirely deterministic. I could easily have added (though I did not think to) that Newton’s laws were all stipulated for closed systems anyway, and it is no part of those laws as such to stipulate that the physical universe as a whole is a closed system, so that his laws cannot imply physical determinism. Newtonian physics did not preclude God’s intervention in the world, for instance, and this is precisely why Newton was not being inconsistent when he maintained both that his laws were true, and that God occasionally intervened in the physical world (for instance by providing the planets with an extra ‘push’ every now and again). This demonstrates clearly that Newton’s laws, even if they were deterministic for closed systems (which the ball-on-dome example disproves), wouldn’t come anywhere near to entailing physical determinism.

[3] Not all sound arguments are good arguments, for the soundness of an argument is neither a necessary, nor sufficient, condition of the goodness of an argument (just as the goodness of an argument is neither necessary nor sufficient for soundness). I will discuss this distinction in more detail in an upcoming post. For now, however, observe that if this argument were sound, then it would give us good reasons for accepting its conclusion, or at least for accepting premise 5.

[4] Calvinism requires a compatibilist view of free will and determinism in order to allow normative statements about what one ought or ought not to believe, but I’m not convinced that such accounts are even coherent. In fact, I am convinced they are not.

[5] Pruss quoted a passage from Hume, but I have provided a more extended excerpt of the same passage from Hume. David Hume, “Dialogues Concerning Natural Religion” in Modern Philosophy: An Anthology (Second Edition), Edited by Roger Ariew and Eric Watkins (Indianapolis: Hackett Publishing Company, 2009), 622.

[6] At first I thought, in passing, that if somebody had trouble with the idea of an infinite past I could just say that there was a world W* in which temporally backward causation (i.e., causation from future events to past ones) is the only kind of causal relation which obtains, and where every event is deterministically caused by some posterior event, and although the world has a temporal beginning, it has no temporal end. However, one should only be concerned with actually infinite regresses of past events if one is i) an A-theorist, or ii) worried about infinite chains of causes. If one is an A-theorist, they will not likely accept the possibility of backward causation anyway, and if one is, like me, worried about infinite chains of causes, then they will have the same problem with W* as they had with W. If you, like me, do have a problem with accepting that W is logically possible then either suspend your modal suspicions here for the sake of argument, or just notice that any length of time can be infinitely subdivided, so that over any measurable length of time it is logically possible that an infinite number of causes are at play just in case it is true that there is no particular time tn at which no cause can logically possibly obtain.

[7] I strongly believe they can be sufficiently explained, and this is because I adamantly reject the assumption that all explanations can be reduced to, or expressed by, entailments. However, I will leave off giving an account of this highly contentious position for now; the reader who disagrees with me is invited to accept the weaker conditional claim that if facts about libertarian free actions could be sufficiently explained, then any combination of libertarian free acts might figure into a sufficient explanation for precisely why the physical states of the universe are precisely as they are. However, notice that, for the purposes of my argument, the PSR needn’t be true, it just needs to be possible that there be an underlying explanation which goes beyond the demands of the HECP.