Four Versions of Open Theism

Open theism has been much-discussed in philosophy of religion and theology circles since the 1995 publication of Pinnock, et al.'s The Openness of God. But in many ways I find that the view is still poorly understood. Critics frequently fail to appreciate that there are several importantly different versions of open theism.

First, we need a working definition of open theism. The core thesis of open theism is that the future is now, in some respects, epistemically open for God. Let's call this the epistemic thesis (ET). In general, the future is epistemically open for God at T with respect to possible future state of affairs X iff for some future time T* neither "X will obtain at T*" nor "X will not obtain at T*" is known by God at T. Whatever is not epistemically open for God at T is epistemically settled for God at T.

In terms of ET, I propose to define as a version of open theism any modification of classical theism to accommodate ET while retaining omnipotence and creation ex nihilo. (The qualifications are intended to exclude process theism.)

Second, to draw the proper distinctions, we need to define one more term. Let us say that the future is alethically open at time T iff with respect to possible future state of affairs X and future time T* neither "X will obtain at T*" nor "X will not obtain at T*" is true at T. Whatever is not alethically open at T is alethically settled at T.

Now, given these definitions, there are four importantly distinct versions of open theism (I'm borrowing here some terminology from my friend and collaborator, Tom Belt):

1. Voluntary Nescience: The future is alethically settled but nevertheless epistemically open for God because he has voluntarily chosen not to know truths about future contingents. Dallas Willard espouses this position.
2. Involuntary Nescience: The future is alethically settled but nevertheless epistemically open for God because truths about future contingents are in principle unknowable. William Hasker espouses this position.
3. Non-Bivalentist Omniscience: The future is alethically open and therefore epistemically open for God because propositions about future contingents are neither true nor false. J. R. Lucas espouses this position.
4. Bivalentist Omniscience: The future is alethically open and therefore epistemically open for God because propositions asserting of future contingents that they "will" obtain or that they "will not" obtain are both false. Instead, what is true is that they "might and might not" obtain. Greg Boyd (and yours truly) espouses this position.

A couple observations before I close.

• Positions (3) and (4) are wholly compatible with a traditional definition of omniscience (i.e., essentially knowing all and only truths). Positions (1) and (2) require some revision of omniscience as traditionally defined (viz., being capable of knowing all truths; knowing all truths that can be known).
• Since knowledge entails truth, if the future is alethically open, then the future must be epistemically open for God. Thus, if the future is alethically open, then it those who hold that the future is epistemically settled for God who must either revise the traditional doctrine of omniscience or run into the incoherence of saying that God knows things that ain't so.

There are, of course, many more important issues to discuss here. But I hope this makes the terrain of the debate a little clearer.

Also posted at Prosblogion.

Joining the "Prosblogion" Team

I've been invited to become a contributor to the Prosblogion blog, a very fine place for discussions in the philosophy of religion. So from time-to-time I'll be posting over there, in which cases I'll either dual-post over here or provide a link for those who might be interested.

In Defense of Conditional Excluded Middle

The rule of excluded middle says that for any proposition P, P is either true or not-true (and not both). The "or" here is exclusive because otherwise P and not-P could be true at the same time, which would violate the law of non-contradiction.

Similar to excluded middle is a controversial rule known as conditional excluded middle (CEM) which may be stated as follows:

(CEM) For all propositions P and Q, if P then (either Q or not-Q).

Again, the "or" is exclusive. Given CEM, either "If P then Q" is true or "If P then not-Q" is true, and not both. We may not know which one is true, but we can be sure that one, and only one, of them is true.

Now I think CEM is true. It seems so intuitively obvious to me that I find it very hard to see how it could be false. Nevertheless, CEM is controversial. David Lewis has offered the following counterexample:

1. If Verdi and Bizet were compatriots, Bizet would be Italian.
2. If Verdi and Bizet were compatriots, Bizet would not be Italian.

According to Lewis, it is unclear which statement is correct, yet according to CEM, one must be true. (1) could be true. After all, if Bizet were Italian, he and Verdi would be compatriots. However, (2) could also be true (if Verdi were French). It seems just as likely for Verdi to have been French as Bizet to have been Italian and therefore, neither (3) nor (4) is true.

The standard sort of response to Lewis is to question whether (1) is just as likely to be true as (2) by considering the relative similarity of possible worlds to the actual world. I think that sort of response is unlikely to be effective because our modal intuitions about which worlds are closer to the actual world are likely to be highly variable and unreliable.

Here's what I think the defender of CEM should say: Both (1) and (2) are false.

But wait a minute! Isn't that self-defeating? Am I giving up CEM in the process of defending it? The answer is emphatically no. I can uphold CEM while denying both (1) and (2) because I deny that the consequents of (1) and (2) are contradictories. The denial of the consequent of (1) gives us not (2) but

3. If Verdi and Bizet were compatriots, then it is not the case that Bizet would be Italian.

And the denial of the consequent of (2) gives us not (1) but

4. If Verdi and Bizet were compatriots, then it is not the case that Bizet would not be Italian.

Thus, if both (1) and (2) are false, then it follows that

5. If Verdi and Bizet were compatriots, then it is neither the case that Bizet would be Italian nor the case that Bizet would not be Italian.

But what other options remain? This one:

6. If Verdi and Bizet were compatriots, then Bizet might and might not be Italian.
   

I say that (1) and (2) are false, and (3), (4), (5), and (6) are true. With what justification? It's simple, really. A conditional is strictly true if and only if the truth of the antecedent suffices to guarantee the truth of the consequent. But, for the very reasons Lewis gives, the mere fact that Verdi and Bizet are compatriots suffices neither to guarantee that Bizet is Italian nor that he is not Italian. So both (1) and (2) are false. But (3) and (4) are true, I say, because the mere fact that Verdi and Bizet are compatriots does suffice to guarantee that it is not guaranteed either that Bizet is Italian or that he is not Italian. In other words, I claim that the following square of oppositions holds for subjunctive conditionals:

If this is right, then the top two corners of the square are false just in case both bottom corners are true, and Lewis's counterexample to CEM fails.

Is God "Pure Act"?

According to classical theism (as exemplified by Aquinas), God is "Pure Act". What does that mean, you ask? Good question. The idea goes back to Aristotle, but we'll pick it up with Aquinas.

Very early on in the Summa Theologiae Aquinas says the following:

For motion [motus, i.e., change] is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. Thus that which is actually hot, as fire, makes wood, which is potentially hot, to be actually hot, and thereby moves and changes it. Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially cold. It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e. that it should move itself. (ST I.q2.a3)

The first being must of necessity be in act, and in no way in potentiality. For although in any single thing that passes from potentiality to actuality, the potentiality is prior in time to the actuality; nevertheless, absolutely speaking, actuality is prior to potentiality; for whatever is in potentiality can be reduced into actuality only by some being in actuality. Now it has been already proved that God is the First Being. It is therefore impossible that in God there should be any potentiality. (ST I.q3.a1)

The first quote is from the first of Aquinas' famous "Five Ways". In the second quote Aquinas draws out what he takes to be the implications of the First Way: the "first mover" must be wholly "in act". Hence, the idea that God is "Pure Act".

Aquinas and other medieval theologians clearly took this to be a very important and defining attribute of God. But what does it mean? I must confess that I find the notion rather opaque. Aquinas' discussion in the First Way suggest that "act" and "potency" satisfy at least the following contraints:

1. They are contraries: To be in act in some respect precludes being in potency in the same respect, and vice-versa.
2. Act is causally prior to potency: It takes something in act in the relevant respects to actualize any given potency.
3. No self-actualization: Nothing that is in potency in some respect can actualize itself in that same respect. (This follows from (1) and (2).)
   

To illustrate, Aquinas gives one example: something that is actually hot (fire) can make something that is actually cold but potentially hot (a log) to be actually hot.

This suggests that Aquinas has in mind something like one of the following ways of characterizing the act/potency distinction:

A. With Respect to Predication

• substance X is "in act" wrt property Y iff "now exemplifies Y" can be truly predicated of X.
• substance X is "in potency" wrt property Y iff "now exemplifies Y" cannot be truly predicated of X, but in some causally possible future "now exemplifies Y" can be truly predicated of X.

B. With Respect to Perfection

• substance X is "in act" wrt property Y iff X is in a state of perfection wrt Y and has the capacity to perfect that which is in potency wrt Y.
• substance X is "in potency" wrt property Y iff X is perfectible with respect to Y.

Thomist Henri Renard ("The Philosophy of Being"): "Act is a perfection, and potency is capacity for perfection. Act is a reality which perfects, actuates the potency in which it is received, but in no way destroys it. Potency ... is a positive capacity for receiving this perfection."

Definition (A) seems to be a tolerably clear way of articulating the act/potency distinction with respect Aquinas' fire-and-log example: Initially we can truly predicate "exemplifies hotness" of the fire but not of the log. As the fire heats the log, however, the log changes so that we can predicate "exemplifies hotness" of it. (Note that this is an example of deterministic causation in which one existing thing brings about a change in another existing thing.)

Definition (B) is murkier both because of the undefined term "perfection" and because the notion of "potency" occurs in the definition of actuality. Offhand, it's not clear how a log's becoming hot is a reception of a "perfection". For some extrinsic purposes a hot log may be better than a cold one, but there doesn't seem to be anything intrinsically better about hot logs versus cold ones. So how is it that the log itself has been "perfected"? Perhaps heat is supposed to be a "perfection" that comes to reside in the log. Okay, but how exactly is heat a "perfection"? Definition (B) doesn't seem to fit Aquinas' example very well.

One difficulty in making sense of Aquinas' notion of God's "Pure Actuality" is that his clear examples only support something like (A), whereas Aquinas wants something like (B). Another difficulty is that the deterministic implications of (A) work themselves into Aquinas' theology where they create problems.

Principle (3), for example, is plausible when applied to inanimate things like logs, but it seems false when applied to persons. In my view, part of what it means to be a person is to have a power of self-determination, i.e., to be a self-moved mover, a self-actualizer. As a free agent I have the power to be a first cause of at least some of my own actions. For example, I can decide right now to raise my left arm. God, the preeminent agent, is free to create or not, and he is free to decide how to respond to our actions and prayers. In making free decisions we actualize a potency in ourselves. When God chooses to create he actualizes his potency to be a Creator. Or so it would seem.

What's more, for Aquinas, God's "Pure Actuality" entails God's absolute immutability and impassibility, both very difficult doctrines to defend. On the face of it God's freely creating (not to mention the Incarnation) is a change in God: We seem to have a state in which God alone exists followed by a state in which God and creation exist.

I suspect that Aquinas' views on God's absolute immutability and impassibility result from over-reliance on deterministic efficient causal, fire-heating-wood examples of the act/potency distinction, leading to an oversight of the fact that perfection in a person consists significantly in the power of self-determination or agent causation. Furthermore, it is arguable that being in loving relationship is the highest kind of good for a person, in which case perfection in a person also consists significantly in receptivity to another.

What if Aquinas had taken for his model of act/potency something distinctively personal? Take, for example, the act of welcoming. When person A welcomes person B, actively listens to B, etc., A opens himself to be influenced by B. A is being both active and receptive at the same time. If Aquinas had taken something like that for his paradigm of act/potency, then maybe he could have a God who is "Pure Act" in the sense, perhaps, of "Perfect Love" as I have characterized it here, without having to embrace either absolute immutability or impassibility.

Perfect Love and the Trinity

There's an interesting discussion on the Christian doctrine of the Trinity going on at Prosblogion. I figure this is as good a time as any to dust off some speculations of my own on the subject.

The doctrine Trinity, a cornerstone of Christian orthodoxy, may be summed up in the following two propositions:

1. There is only one God.
2. There are three mutually distinct persons (Father, Son, and Holy Spirit), each fully divine.

Now, while (1) and (2) are not blatantly contradictory--they aren't saying that there's one God who is three Gods, or three persons who are one person--their conjunction is not obviously coherent. How can you have three distinct divine persons without having three distinct Gods?

There are several popular models for thinking about the Trinity (e.g., the egg, the triple-point of water, etc.) but all distort the doctrine in one way or another and so fail to make clear how (1) and (2) can be compossible.

I want to propose a different sort of model, one that takes "perfect love" as its starting point (1 John 4:7). Note: What follows is highly speculative.

First, what is perfect love? I suggest the following definition:

Perfect love =def. A cognitive, affective, and volitional state of desiring and actively pursuing the true good of its object grounded in an understanding that it is the true good of it's object.

So understood, perfect love is a 3-term relation (X knows, desires, and pursues the true good Y of object Z), not merely a 2-term relation (X loves Y). Moreover, it is a relation with a threefold nature:

(a) Affective: Perfect love desires the good of its object.
(b) Volitional: Perfect love wills and pursues the good of its object.
(c) Cognitive: Perfect love knows the good of its object.

What constraints does the nature of perfect love place on the X, Y, and Z terms of the relation?

Well, for starters, it seems clear that X must be a person, for the possibility of having affective, volitional, and cognitive states is definitive of personhood.

What about Z? Well, it seems that the fullest expression of perfect love would consist in love of the highest or best type of object. And it seems that persons are categorically better sorts of things than, say, inanimate objects, mere animals, plants, mathematical abstractions, or what have you. Aristotle, for example, argues that persons as such do not essentially lack any of the capacities of inanimate objects, mere animals, or plants, whereas all of those types of things do essentially lack at least some of the capacities of persons (like cognition). Abstractions can be understood by persons, but they cannot understand persons in turn. Finally, some persons are better than others (contrast Hitler with Mother Teresa). So the highest example of perfect love would be perfect love for a perfect person, that is, for a perfectly loving person.

So the highest example of perfect love would seem to be the love of a person by a person. But what about Y? What is in the highest sense the true good of a person? Well, I would propose that the highest good for any person Z is itself another person Y that perfectly loves Z. Think about it. What is the single best thing in life? Isn't it the loving personal relationships we have with others, where love is understood in the fullest sense to have affective, volitional, and cognitive dimensions? And what would be the single worst thing that could even happen to a person? Wouldn't it be to be completely isolated, completely ignored by others? At any rate, this seems plausible.

So what I propose is that if God is, by nature, Perfect Love, then God must be tri-personal, for the highest possible kind of perfect love is the giving by a perfectly loving person of a perfectly loving person to a perfectly loving person. Hence, the one divine nature (Perfect Love) is necessarily tri-personal.

What's more, the other divine perfections may plausibly be argued to flow from God's nature as Perfect Lover. A Perfect Lover perfectly desires and wills the true good of its object (omnibenevolence). A Perfect Lover is able to perfectly pursue the true good of the object (omnipotence). And a Perfect Lover perfectly knows the true good of its object (omniscience).

Now, here's an objection: How can we be sure that the distinction between X and Y is real or metaphysical and not merely conceptual? Why can't X himself be the true good of Z, in which case X=Y, and the phrases "lover of Z" and "the true good of Z" are merely conceptually distinct ways of picking out the same referent (like the "Evening Star" and the "Morning Star", both of which refer to Venus)? In short, why is a third person necessary? I'm not sure how to answer that. Ideas anyone?

Raving Thomists

I just woke up from a dream in which I was being chased by a very persistent Thomist who wanted to sift my soul through some sort of metaphysical meat-grinder. Not wanting to find out what that would do to me, I escaped to Paradise, where I learned that Father Abraham now relaxes in a very large La-Z-Boy recliner.

Weird. I'll bet Freud would have fun with that one.

The Paradoxes of Material Disjunction

In a previous post, I commented that truth-functional interpretations of conditionals are bothered by what are known as the "paradoxes of material implication". The problem arises because it is easy to form conditionals that, on the truth-functional interpretation, come out as true when, intuitively, they aren't true. What I want to point out now is that an exactly parallel problem afflicts truth-functional interpretations of disjunctions.

Recall that a disjunction is just an either-or proposition. Taking the "or" in the usual inclusive sense, a disjunction says, at a minimum, "Here's a set of options. At least one of these is true." For example, "Either the stoplight is red, green, or yellow" gives us a set of three options {red, green, yellow} and tells us that at least one of them obtains. It doesn't tell us which one obtains, nor does it tell us that only one obtains, but it does tell us that it's false that none of them obtains.

Now, on the standard truth-functional construal, a disjunction is true if and only if at least one of the options or "disjuncts" is in fact true. But it is not hard to form disjunctions for which this is counterintuitive. We'll call these the "paradoxes of material disjunction". Consider the following:

Either 2+2=4 or water is wet.
Either grass is green or the Eiffel Tower is in London.
Either the stoplight is red or it is green.

The problem with the first two is that the disjuncts are completely irrelevant to each other, yet each has at least one true disjunct. It would be very odd for anyone in normal discourse to utter either disjunction, unless they were being sarcastic or just plain silly. So when asked, "Is it true that either grass is green or the Eiffel Tower is in London?" most people would probably say, "Huh?" We might wonder, then, whether we should regard that disjunction as true. Perhaps we should conclude, instead, that it is meaningless and thus has no truth value. I'm not claiming to have shown that this is the proper response, but it seems at least arguably a plausible reaction.

The problem with the last one is that the disjuncts are obviously incomplete—the stoplight could be yellow (or off, for that matter). Now here I think that many people would respond by saying that "Either the stoplight is red or it is green" is just plain false. It's not hard to imagine a person saying, "Not necessarily. It could be yellow." What this suggests is that what makes a disjunction true is not simply that one or more of the disjunctions is in fact true, but also that there be no other (relevant) alternatives. In other words, "Either p or q" is true iff at least one of the alternatives must be true. We can accommodate this in parallel with C.I. Lewis's 'strict implication' by introducing a notion of 'strict disjunction'. On this construal, a disjunction is strictly true iff necessarily, at least one of the disjuncts is true.

Like strict implication, strict disjunction is not truth-functional. It does a good job of squaring with our intuitions on the latter two disjunctions above: both come out as false. It does not, however, explain our puzzlement with "Either 2+2=4 or water is wet." The latter comes out as true according to strict disjunction because 2+2=4 is a necessary truth, but it still sounds odd.

Are Disjunctions Truth-Functional?

A 'disjunction' is an either-or proposition. It has the form "Either A or B or ...", where the terms A, B, etc. are called 'disjuncts'. The simplest type of disjunction has only two disjuncts: Either A or B. Taking the "or" here in the usual inclusive sense, what this says is simply "Here are the possibilities (A,B); at least one of these is true."

Now, disjunctions are often considered to be 'truth-functional' compounds, i.e., their truth value is supposed to be determined entirely by the truth values of the disjuncts. Thus, "Either A or B" is true if A is true and B is false, B is true and A is false, or both A and B are true; and "Either A or B" is false if A and B are both false.

But it can be argued that disjunctions (or some of them at any rate) are not truth-functional. Consider the following argument patterns:

Either A or B
Not-A
Hence, B

If Not-A then B
Not-A
Hence, B

Since the second argument is obviously valid (modus ponens), if the first is to be valid, then "Either A or B" must say at least as much as the conditional "If Not-A then B". Conversely, since the first argument is obviously valid (disjunctive syllogism), if the second is to be valid, then "If Not-A then B" must say at least as much as the disjunction "Either A or B". Accordingly, it is natural to equate "Either A or B" with "If Not-A then B". Given that equation, disjunctions will be truth-functional if and only if the corresponding conditional is truthfunctional. It is arguably the case, however, that conditionals are not truth-functional.

The problem concerns what are known as the 'paradoxes of material implication'. ('Material implication' refers to the equation of "Either A or B" with "If Not-A then B", coupled with a truth-functional interpretation of each.) On the truth-functional interpretation of conditionals, a conditional is true whenever the antecedent is false and whenever the consequent is true. But this leads to counter-intuitive results. For example, it means that the following conditionals are true:

If the Eiffel Tower is in London, then the moon is made of cheese.
If the geocentric model of the solar system is correct, then water is wet.
If grass is green, then water is wet.

But many people would hestitate to call these conditionals 'true'. The problem is that the antecedent isn't relevant to the consequent. Strengthing the conditional to avoid such counter-intuitive cases, however, makes it no longer truth-functional.

For example, C.I. Lewis tried to get around the paradoxes of material implication by introducing what he called 'strict implication'. On his view

If p then q ≡ Nec(Either not-p or q),

where "Either not-p or q" is understood truth-functionally. Now, while this move doesn't avoid all of the counter-intuitive paradoxes, it does yield the result that all of the odd conditionals above are false. The result, however, is that "If p then q" is no longer truth-functional, for its truth value is no longer simply a function of the truth values of p and q. Rather, introducing the modal notion of 'necessity' makes the truth-value of "If p then q" be a function of the world-relative truth values of p and q across a domain of possible worlds.

So, the upshot is this: There are motivating reasons for interpreting (at least some) conditionals in a non-truth-functional way. If, therefore, (some) disjunctions are logically equivalent to any of those non-truth-functional conditionals, then those disjunctions are not truth-functional either. Furthermore, if all disjunctions are (by DeMorgan's laws) equivalent to conjunctions, then we get the further, and very surprising, result that (some) conjunctions are not truth-functional.

The Dialectic of Liberty and Security

I was reminded today of some insightful thoughts from the Maverick Philosopher:

Liberty and security stand in a dialectical relation to each other in that (i) each requires the other to be what it is, and yet (ii) each is opposed to the other. ...

Ad (i). Liberty worth having is liberty capable of being exercised fruitfully and often. Liberty in this concrete sense requires security to be what it is. My liberty to leave my house at any time of the day or night would be worth little or nothing if I were to be mugged every time I stepped over the threshhold. On the other hand, a security worth having is a security that makes possible the exercise of as much liberty as is consistent with the liberty of all. The security of a prison or of a police state is not a security worth having. A security worth having, therefore, requires liberty to be what it is.

Ad (ii). Nevertheless, liberty and security oppose each other. The security of all requires limitations on the liberty of each. For if the liberty of each were allowed untrammelled expression, no one would be secure in his life and property. Thus security opposes and limits liberty. Equally, liberty opposes and limits security. The right to keep and bear arms, for example, poses a certain threat to security, as everyone must admit whether liberal, conservative, or libertarian. The question is not whether it poses a threat, but whether the threat it poses is acceptable given the desirability of the liberty it allows.

Ad (i) + (ii). The situation is complex. Liberty requires the very security that it limits, just as security limits the very liberty that it requires. It follows that any attack on our security is also an attack on our liberty. It seems to me that this is a point that liberals and leftists do not sufficiently appreciate, and that some of them do not appreciate at all. The 9/11 attack on the Trade Towers did not merely destroy the security of those working in them, it also destroyed their liberty, while impeding to greater and lesser degrees the liberty of all the rest of us. But it must also be said that any restriction on our liberties also negatively affects the value of our security – a point conservatives need to bear in mind.

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In Defense of Prior's 'Peircean' Tense Logic

I just finished a paper defending what philosopher Arthur Prior called the "Peircean" system of tense logic over against the rival "Ockhamist" system.

You can download the paper here (100kB, PDF).

I'll be reading this paper at the group meeting of the Philosophy of Time Society at the Pacific APA conference in March.

For some more background on the Peircean / Ockhamist tense-logic debate, follow this link to an earlier blog post on the topic.

What's Wrong with Hume's Fork

In Section 4 of his Enquiry Concerning Human Understanding, David Hume makes a famous distinction between "matters of fact" and "relations of ideas"

All the objects of human reason or enquiry fall naturally into two kinds, namely relations of ideas and matters of fact. The first kind include geometry, algebra, and arithmetic, and indeed every statement that is either intuitively or demonstratively certain. That the square of the hypotenuse is equal to the squares of the other two sides expresses a relation between those figures. That three times five equals half of thirty expresses a relation between those numbers. Propositions of this kind can be discovered purely by thinking, with no need to attend to anything that actually exists anywhere in the universe. . . . Matters of fact . . . are not established in the same way; and we cannot have such strong grounds for thinking them true. The contrary of every matter of fact is still possible, because it doesn't imply a contradiction and is conceived by the mind as easily and clearly as if it conformed perfectly to reality. That the sun will not rise tomorrow is just as intelligible as - and no more contradictory than - the proposition that the sun will rise tomorrow.

Note Hume's claim that "All the objects of human reason or enquiry fall naturally into two kinds, namely relations of ideas and matters of fact." This claim is known today as Hume's Fork. Relations of ideas are subject to strict demonstration or proof, but imply nothing at all about what exists. Matters of fact, on the other hand, are not subject to strict demonstration or proof but do tell us something about what exists. So far so good.

But what about Hume's Fork itself? It too is an object of "human reason or enquiry". Accordingly, it too must either be a relation of ideas or a matter of fact. Which is it?

It does not seem to be a relation of ideas, since it's denial does not obviously entail a contradiction. Nor does it seem to be a matter of fact, since Hume presents the Fork as though it were an a priori truth knowable independently of experience. Hume's Fork is, therefore, a prima facie counterexample to itself. Given the importance of the Fork in Hume's thought, this suggests that there's something amiss at the very foundations of Hume's philosophy.

Limits vs. Limit Cases

I just finished reading Barry Miller's The Fullness of Being, a very stimulating book. Miller argues against Russell, Frege, Quine, et al. that existence can be predicated of individuals and develops an interesting metaphysical account of the relation between an individual and its existence.

Anyway, at one point Miller argues for a sharp distinction between limits and limit cases. Both are endpoints of a series, but the difference is that limits are themselves part of the series, whereas limit cases stand outside the series.

For example, according to physics, the maximum speed is the speed of light, 300,000 km/s. That is a limit. We have a series of speeds in which the uppermost speed, the speed of light, is itself part of the series. On the other end of the spectrum, we have 0 m/s. Now we might think that this is the minimum speed, but we would be wrong. 0 mph is not a speed at all, but the absence of speed. It's a limit case, not a limit. (Miller, p. 137)

Consider again a series of regular polygons with an increasing number of sides: triangle (3), square (4), pentagon (5), hexagon (6), ... , dodecagon (12), ..., chiliagon (1000), ..., etc. The triangle is the lower limit of the series, because there can be no polygons with fewer sides. And, of course, the triangle is itself part of the series. On the upper end, however, there is no limit, but there is a limit case: the circle. Circles are not polygons, and so they lie outside the series. (Miller, pp. 138-139)

One last example. Consider a series of lines of decreasing length: 1 in, 0.5 in, 0.1 in, ..., etc. There is no lower limit here because for any line we can always cut it in two to make a shorter line. But there is a limit case: a point, which is not a line at all. (Miller, p. 139)

Mathematicians, of course, like to use the word "limit" to cover both limits and limit cases, but I think Miller's made an interesting and potentially useful distinction nonetheless.