Virtual Colloquium: Ricki Bliss, “Metaphysical Foundationalism and the Principle of Sufficient Reason”
May 19, 2017 — 6:00

Author: Kenny Pearce  Category: General  Tags: , , , , , ,   Comments: 4

Welcome to the final installment of the Prosblogion Virtual Colloquium. Many thanks to all of those who have contributed, both as presenters and commenters.

Our final paper will be “Metaphysical Foundationalism and the Principle of Sufficient Reason” by Ricki Bliss. Dr. Bliss received her PhD from the University of Melbourne in 2012 and is currently Assistant Professor of Philosophy at Lehigh University in Pennsylvania. Her papers on metaphysics have appeared in journals such as Philosophical Review and Philosophical Studies. She is co-author (with Kelly Trogdon) of the Stanford Encyclopedia of Philosophy article on metaphysical grounding and co-editor (with Graham Priest) of Reality and its Structure, forthcoming from Oxford University Press.


Metaphysical Foundationalism and the Principle of Sufficient Reason

Ricki Bliss

The following is the abstract for my paper that I am grateful to have the opportunity to discuss:

The metaphysical foundationalist claims that reality is hierarchically arranged, with maximal chains of phenomena ordered by the grounding relation terminating in contingently existent fundamentalia. Some influential foundationalists claim that there must be something fundamental because being requires a ground or explanation, or because grounding chains that do not terminate are viciously infinitely regressive. Surprisingly, reconstruction of these arguments reveals an enthymematic assumption that makes appeal to a Principle of Sufficient Reason: a principle the foundationalist would not, and should not, accept. I explore three different Principles of Sufficient Reason: two familiar to us from cosmological arguments and one, novel, dependence PSR. I argue that without a PSR, certain of the most influential arguments to the existence of something fundamental do not work; and that with a PSR, certain of the most influential arguments to the existence of something fundamental leave us with a position that is epistemically unstable.

By way of introduction, I would like to say a few things about what has motivated me to write this paper, and what I think some interesting (and important!) open questions are.

Contemporarily, analytic metaphysics seems to be in the thrall of notions of ground, structure and the idea that there is something fundamental. In addition to this, there seems to be a growing tension between those wedded to a Quine-style naturalizing of metaphysics and the ‘neo-Aristotelian’ return to the good old fashioned way of doing it. In theory, a good proper Quinean will hold that the only kinds of arguments we need in defence of fundamentality are arguments from theoretical virtue, whereas the neo-Aristotelian will allow the use of, say, arguments from vicious infinite regress. In practice, however, the contemporary grounding literature and discussions of fundamentality are a mess. Card-carrying Quineans make appeal all over the place to talk of ‘grounds of being’, arguments from vicious infinite regress and the need that there be some kind of ultimate explainers. At the same time, they throw around talk of theoretical virtue with almost no reflection, wheel out their intuitions, and are prone to even suggest that we don’t need arguments in defence of a view as spectacularly ontologically committing and weighty as fundamentality. What is to my mind, however, almost singularly most striking about the state of the contemporary literature is it’s apparent utter blindness to the rich history of debates over the regress problem, the grounds of being, and ultimate explainers in the form of cosmological arguments to the existence of God.

This, then, is my starting point: that so much of the contemporary thinking about fundamentality just is disguised variations on very old, very big and very problematic themes. When properly reconstructed, many contemporary arguments in defence of fundamentality look to be variations on cosmological-style arguments to the existence of an ultimate explainer. The consequences of this for how we understand our commitment to fundamentality, though, are interesting. First of all, it seems that what we need here is some version of other of a Principle of Sufficient Reason. What such a version might be is the central theme of this paper. But in addition to this, clarifying which version of the principle might be in operation is not yet to get us an argument to the existence of something fundamental. In order to have a good proper argument in defence of fundamentality we need to know (i) what exactly it is that the fundamentalia are supposed to explain and (ii) why it is that no dependent entity is up to the task to hand.

These last two points are not really issues that I take up in this paper. In fact, at the time of writing this paper – which was a few years ago now – I think I had a sense of the importance of these two questions without yet being able to articulate it so clearly. I have since written another paper that does engage with these questions, and I believe the second one is a particularly complex issue. Without engaging with these particular questions, however, there is still much of significance that needs to be explored; and much that is very often taken for granted. What is the relation between self-dependence and necessary being? Can contingent existents be self-explanatory? How does fundamentality intersect with modality? What is the modal status of that which is to be explained by the fundamentalia? Although this paper doesn’t answer any of these questions, I believe it draws to our attention how important it is to ask them. And I believe that the literature on cosmological arguments to the existence of God is an invaluable resource to look to try and start our investigation.


The full paper is available here. Comments welcome below!

Virtual Colloquium: Brandon Warmke, “God’s Standing to Forgive”
April 21, 2017 — 6:00

Author: Kenny Pearce  Category: General  Tags: , , , , ,   Comments: 2

Today’s Virtual Colloquium is “God’s Standing to Forgive” by Brandon Warmke. Dr. Warmke received his PhD in philosophy from the University of Arizona in 2014 and is currently Assistant Professor of Philosophy at Bowling Green State University in Ohio. His work in moral philosophy has been published in journals such as Philosophical Studies, Canadian Journal of Philosophy, Australasian Journal of Philosophy, Pacific Philosophical Quarterly, and Philosophy and Public Affairs.


God’s Standing to Forgive

Brandon Warmke

Consider two cases:

LUCY: I lie to my brother, telling him I bought a gift for our parents when I did not do so. Realizing my guilt, I ask my new plumber Lucy to forgive me for my lie. Lucy forgives me for lying to my brother.
GOD: I lie to my brother, telling him I bought a gift for our parents when I did not do so. Realizing my guilt, I ask God to forgive me for my lie. God forgives me for lying to my brother.

The claim that Lucy could forgive me for lying to my brother will, I think, strike most people as very strange. And yet for many people, it will not seem nearly so strange to think that God could do so. An apparently central tenet of all three Abrahamic faiths is that God can and does forgive human persons for the wrong things they do to one another. But how is this possible? Because I lied to my brother—and not to Lucy—we are inclined to think that Lucy cannot forgive me. She lacks standing to do so. But then why think that God can forgive us for the wrongs we do to others? It is natural to suppose that just like I did not lie to Lucy about the gift, I also did not lie to God about the gift. And so if Lucy does not have the standing to forgive me, how does God? This is the question I wish to explore: how could God have the standing to forgive us for the things we do to one another? Call this the problem of divine standing. In this paper I provide two different solutions to the problem.

I begin by cataloging the various ways that one might have standing to forgive someone for wrongdoing. One has direct standing to forgive a wrongdoer when one is the direct victim of that wrongdoing. One has indirect standing when one is wronged as a result of a wrongdoing to someone else. Controversially, one can possess proxy standing to forgive when one can forgive on behalf of the victim. Also controversially, one can possess third-party standing to forgive a wrongdoer for what she did to someone else.

I then show that none of these individual varieties of standing to forgive explains why God would be able to forgive interpersonal human wrongs. For example, one might argue that when humans wrong one another, both the human victim and God have direct standing to forgive, but for different wrongs. When I lie to you, you can forgive me for lying to you, and God can forgive me for, say, disobeying God. But such a solution would still not secure God’s standing to forgive me for lying to you.

I then develop two different solutions to the problem of divine standing. One kind of solution concedes that God cannot forgive wrongs between human persons because God lacks standing to do so, but argues that this is no problem. There are many things that God cannot do. Just as God cannot, say, keep your wedding vows to your spouse (only you can do that), God cannot forgive you for lying to your spouse (only s/he can do that). This solution also stresses the importance of human forgiveness: because only you can forgive the wrongs done to you, those wrongs will be forgiven only if you do so.

For those who desire for God to be able to forgive us our “trespasses” against others, I develop another solution to the problem of divine standing. On this strategy, when we wrong others: (1) the human victim has direct standing to forgive us for the interpersonal human wrong; (2) God has direct standing to forgive for the wrong against God; and (3) God has third-party standing to forgive for the interpersonal human wrong. In developing this solution, I defend the possibility of third-party standing. I suggest a new strategy for defending third party-forgiveness and show that persons can come to have such standing when they stand in relationships of personal care with both victim and wrongdoer. I conclude that since God stands in relationships of personal care with all of us, this explains why God has standing to forgive us for our wrongs against each other and not just our wrongs against God.


The complete paper is here. Comments welcome below!

Yandell on Berkeley and Creation
March 26, 2017 — 10:19

Author: Kenny Pearce  Category: General  Tags: , , , , , ,   Comments: 0

The fourth chapter of Idealism and Christian Theology is “Berkeley, Realism, Idealism, and Creation” by Keith Yandell. This is an interesting paper on Berkeley which, unless I missed something, did not turn out to be about Christian theology at all.

I say purposely that it did not turn out to be about Christian theology, because it sounds at the beginning as if it is going to be. Yandell begins by noting that Berkeley’s position is rare among Christian thinkers (p. 73), and discussing a particular threat to Christianity from those who take the creation of matter to be impossible (p. 73-74). He also briefly discusses the problem of how Berkeley can accommodate certain core Christian doctrines, such as creation and Incarnation, within his idealism (p. 78-79). Now, this paper is a mere 8 pages in length (plus endnotes), so I just mentioned over half of the pages as having something to do with Christian theology. Nevertheless, the paper does not seem to me to be about Christian theology in any significant sense, because the theology (and especially the specifically Christian elements of that theology) are totally inessential to the paper’s central point. Here’s why: Berkeley’s own response to the question of the compatibility of his view with divine creation is, essentially, that the Bible says God created the sun and the moon and the earth and plants and animals and so forth, but it doesn’t say that God created material substrata. So, in other words, there is not special theological problem here: if Berkeley has an adequate analysis of the real existence of ordinary objects, then he can preserve divine creation.* Yandell also mentions the Incarnation, but he says one might worry about how, on Berkeley’s view, we can say that “the Second Person of the Trinity [became] fully human as well as being fully divine, and thus being embodied, crucified, buried, and resurrected” (78). It sounds like what Yandell is worried about here is docetism, the heresy which holds that Christ merely appeared to be embodied and to suffer. But, again, if Berkeley can preserve the claim that human beings are really embodied—or, if you like, that human bodies are real—then it seems there is no special theological problem. (The case would be totally different if we were worried about avoiding Apollinarianism or Nestorianism or something; there might be special theological problems for Berkeley there.)

Indeed, Yandell does not treat these as special theological problems, for when his paper comes to solve them we are merely treated to an account of the existence of objects unperceived by humans. This account, of course, involves God, but it doesn’t seem to me to involve Christian theology in any interesting way. (Yandell goes for a sophisticated version of the divine idea theory somewhat similar to the ‘single-idea’ interpretation proposed by Marc Hight.)

This paper seems to me to be a missed opportunity, in terms of exploration of idealism and Christian theology. Most importantly, Yandell never discusses what looked at the beginning like it was going to be the central issue: why did many philosophers regard the creation of matter by God as a serious problem, and how can Berkeley’s immaterialism be seen as responding to this problem? In response to this, Yandell simply notes that most theists held that God was able to create matter (and had created matter). What reason is there to be dissatisfied with this view? Yandell gives Berkeley’s reasons for being dissatisfied with this view, which is that matter is (according to him) conceptually impossible. But Yandell quotes Berkeley saying that many of his predecessors had thought the creation of matter by God to be impossible, despite believing in matter (and, in some cases, also in God)! The opponents (according to Berkeley) take matter to exist eternally, since it can’t be created. Why do they think so, and how is Berkeley responding? This is not explored.

(Cross-posted at blog.kennypearce.net.)


* The objection raised by Lady Percival, which first led Berkeley to address this problem, was a more serious worry: how can God be said to have created inanimate objects before creating human beings? But that is not the problem of creation discussed by Yandell.

Wainwright on Berkeley and Edwards
March 16, 2017 — 5:39

Author: Kenny Pearce  Category: General  Tags: , , , , , ,   Comments: 0

The second essay in Idealism and Christian Theology is “Berkeley, Edwards, Idealism, and the Knowledge of God” by William J. Wainwright. The aim of this article is to explore and explain similarities between Berkeley and Edwards in terms of the religious and cultural context in which they wrote, particularly the threat of deism and freethinking to these (relatively) traditional religious thinkers. This is an extremely interesting project, and it is for the most part well-executed, though the brevity of a single paper necessitates glossing over certain details, leaving some points underdeveloped, and so forth.

Wainwright’s central contention, I take it, is that Berkeley and Edwards share a concern with the ways in which God is coming to seem distant in a world governed by mechanistic science. The world is, increasingly, viewed as a grand machine that keeps rolling along without any outside assistance. Berkeley and Edwards regard it as insufficient to reason (as, for instance, Leibniz and Paley do) that behind a great machine there must be a great Engineer, for this may secure the existence of God, but it will not secure the nearness of God to the believer, or God’s immanence in the world. I am not very familiar with Edwards, but Wainwright’s account of Berkeley’s motivations and concerns is certainly sound. For instance, in the conclusion of the Principles Berkeley writes, “to an unbiassed and attentive mind, nothing can be more plainly legible, than the intimate presence of an all-wise Spirit, who fashions, regulates, and sustains the whole system of being” (sect. 151, my boldface) and that God “is present and conscious to our innermost thoughts” (sect. 155). Further, Berkeley tells us that “the main drift and design of [his] labours” was (among other things) to “inspire [his] readers with a pious sense of the presence of God” (sect. 156). Thus, for Berkeley, the mere existence of God is not enough. Similarly, in Alciphron it is said that the divine language argument “proves, not a Creator merely, but a provident Governor, actually and intimately present, and attentive to all our interests and motions, who watches over our conduct, and takes care of our minutest actions and designs throughout the whole course of our lives, informing, admonishing, and directing incessantly, in a most evident and sensible manner” (sect. 4.14). So Wainwright seems to be on firm ground (at least with respect to Berkeley) when he identifies the nearness of God as a key object of concern, and it is easy to see how the Berkeley-Edwards brand of idealism might be thought to do that. This paper is, in my view, quite a welcome addition to the literature. Too often, Berkeley’s religious motivations are treated as an embarrassment, as though the ‘real’ philosophy has been encumbered with a lot of nonsense from which we must separate it if we are to get the value out. Perhaps that may, in the end, turn out to be the case with respect to present-day philosophical value, but if we don’t see Berkeley’s religious vision clearly we’ll never understand his philosophy in the first place and our ‘disentanglement’ will go awry.

Of course, there are also differences between Berkeley and Edwards. Wainwright makes an interesting and plausible suggestion about the source of these differences: Calvinism. (Of course, Calvinism is always at the forefront with Edwards!) Now, I think Wainwright is a little oversimplistic here when he says that “Because Anglicans, like Berkeley, were not [theological determinists], he may have assumed that humanity’s contra-causal freedom required the existence of relatively independent and autonomous choosing substances” (41). Berkeley says almost nothing about human freedom, and what he does say (e.g., in the later sections of Alciphron 7) is pretty ambiguous. The theological debate between Calvinists and Arminians does not exactly track the metaphysical debate between compatibilists and libertarians (though it does track fairly closely), and not all Anglicans were Arminians. Indeed, prior to the Laudian reforms of the 1630s Calvinism had been the dominant view, and Archbishop James Ussher, the primate of Ireland at the time, had vigorously opposed the attempt to impose Arminianism. What was actually going on (several decades later) in the post-Restoration Anglican Communion was more that folks were keeping pretty quiet about the issues in the hope of keeping it from blowing up again. (Civil wars are not fun.) In my previous post I claimed that Berkeley was a latitudinarian. If so, that would explain why he is so carefully ambiguous on these points: part of the latitudinarian strategy was to try to make room for Calvinists and Arminians within the same church.

Nevertheless, Berkeley, while denying the existence of inanimate secondary causes and attributing the causation of sensory ideas to God, tries to carve out some room for genuine, autonomous human agency. Wainwright provides documentation that Edwards (unsurprisingly, for a radical Calvinist) has no such concerns. Indeed, in emphasizing our dependence on God, Edwards (in the quotes provided by Wainwright) appears driven nearly to Spinozism. I expect this issue regarding Edwards will be addressed further in some of the later essays.

An additional interesting point from Wainwright’s essay has to do with the theory of the world as divine language found in both Berkeley and Edwards. I don’t think Wainwright gets Berkeley’s version of that theory quite exactly right, but this is one of my pet issues and I’ll refrain from nitpicking about it here. More importantly: Wainwright notes that Berkeley believes that the status of the world as a language can be established by empirical and philosophical reasoning, and the fact that the world is a language shows that it must have a speaker. Hence the divine language can be used to establish the existence of God. Edwards, on the other hand, seems to take as a starting point a “two books” theology and a principle of typological interpretation. Thus the world, like the Bible, is a communication from God in the form of types and figures in which the presence of Christ must be discerned. This is justified primarily theologically.

I will conclude with one nitpick: Wainwright says that “Recent scholars agree that Berkeley and Edwards arrived at their idealism separately” (48n2). This claim is meant, I suppose, to underline the importance of identifying common contextual factors in order to explain the similar views of Berkeley and Edwards. In support of this claim, Wainwright cites the introduction to the science and philosophy volume of Edwards’ Works. Now this edition of Edwards’ Works was published from 1957–2008 and Wainwright does not indicate when this particular volume was released, so it is not clear what’s meant by “recent.” In any event, Edwards was taught philosophy at Yale by Berkeley’s disciple Samuel Johnson. (Based on the extant correspondence between Berkeley and Johnson, I do not think ‘disciple’ is too strong a word.) I don’t know what the state of the evidence is regarding whether Edwards actually read Berkeley’s works, but there is certainly a vector for indirect influence, at least. In places I took Wainwright to be implying that if we couldn’t uncover some shared contextual factors explaining the similarity of Berkeley’s and Edwards’ views that similarity would have to be regarded as sheer coincidence, and this is much too strong. Nevertheless, this point does nothing to detract from Edwards’ status as an original thinker, or from the interest of Wainwright’s analysis of Edwards’ similarities and differences from Berkeley.

(Cross-posted at blog.kennypearce.net.)

Are Eternalist Worlds Too Valuable?
February 22, 2015 — 10:32

Author: Michael Almeida  Category: General Uncategorized  Tags: ,   Comments: 6

Suppose for the sake of discussion that (1) is true. I have no idea whether there are worlds in which there are just 100 happy people, but it does simplify the discussion.

1. w includes 100 happy people existing for 10 minutes only.

The value of w, I think, is ten times the value of w’ in (2).

2. w’ includes 100 happy people existing for 1 minute only.

Now let w” be exactly like w, but add the fact that w” is an eternalist world. w” includes 100 happy people. There is no time in w” at which it is false that 100 happy people exist.

3. w” includes 100 happy people and it is true at each time t that 100 happy people exist.

Since it is true at each time in w” that a 100 happy people exist (and despite the fact that it is not true at each time that 100 happy people exist at that time), the value of w” should be much higher than the value of w. The value of a world is a function (in part) of the number of happy people existing in the world over time. It doesn’t much matter where in the world they are (spatially or, it seems to me, temporally).

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Mere Addition
February 13, 2015 — 11:35

Author: Michael Almeida  Category: Existence of God General Uncategorized  Tags: , , , ,   Comments: 0

Stephen Grover offers an interesting version of the Mere Addition Paradox (‘Mere Addition and the Best of all Possible Worlds’, Religious Studies, 1999) against Swinburne’s brief argument (The Existence of God, Oxford, 1979, 114 ff.) that there is no best world. Swinburne’s argument goes this way.

… take any world W . Presumably the goodness of such a world.will consist in part in it containing a finite or infinite number of conscious beings who will enjoy it. But if the enjoyment of the world by each is a valuable thing, surely a world with a few more conscious beings in it would be a yet more valuable world W’ . . .  I conclude that it is not, for conceptual reasons, plausible to suppose that there could be a best of all possible worlds, and in consequence God could not have overriding reason to create one.

There are good reasons to deny that Swinburne’s argument shows anything like there is no best world. Still, the argument does not suffer from the Mere Addition Paradox (MAP).

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Evil and Compatibilism
February 8, 2015 — 11:33

Author: Michael Almeida  Category: Concept of God Existence of God Free Will General Problem of Evil Uncategorized  Tags: , , , ,   Comments: 17

There is widespread belief that compatibilism + theism cannot offer a credible solution to the logical problem of evil. Why does anyone believe that? I think they’re reasoning this way: if compatibilism is true, then, necessarily, God can actualize a morally perfect world. That’s of course true, and it entails that the free will defense fails. But then they reason, if, necessarily, God can actualize a morally perfect world, then, necessarily, God does actualize a morally perfect world. It is then observed that, obviously, there is evil. So, compatibilism + theism is incoherent; it cannot solve the logical problem.

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Adams on Creating the Best
January 27, 2015 — 23:21

Author: Michael Almeida  Category: Christian Theology Existence of God General  Tags: ,   Comments: 25

Robert Adams famously argued that an unsurpassable being need not actualize the best possible world. Adams urges that he does not believe that there is a best world, but assumes there’s one for the sake of argument.

I think it is fairly plausible to suppose that God could have created a world that would have the following characteristics: (1) None of the individual creatures in it would exist in the best of all possible worlds. (2) None of the creatures in it has a life which is so miserable on the whole that it would be better for that creature if it had never existed. (3) Every individual creature in the world is at least as happy on the whole as it would have been in any other possible world in which it could have existed. (‘Must God Create the Best’, PR, 1972)

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Handling the infinities in Pascal’s Wager
January 23, 2014 — 12:01

Author: Alexander Pruss  Category: Afterlife Atheism & Agnosticism General  Tags: , , , , ,   Comments: 24

In its classical formulation, Pascal’s Wager contends that we have something like the following payoff matrix:

God exists No God
Believe +∞ a
Don’t believe -b c

where a,b,c are finite. Alan Hajek, however, observes that it is incorrect to say that if you don’t choose to believe, then the payoff is finite. For even if you don’t now choose to believe, there is a non-zero chance that you will later come to believe, so the expected payoff whether you choose to believe or not is +∞.

Hajek’s criticism has the following unhappy upshot. Suppose that there is a lottery ticket that costs a dollar and has a 9/10 chance of getting you an infinite payoff. That’s a really good deal intuitively: you should rush out and buy the ticket. But the analogue to Hajek’s criticism will say that since there is a non-zero chance that you will obtain the ticket without buying it—maybe a friend will give it to you as a gift—the expected payoff is +∞ whether you buy or don’t buy. So there is no point to buying. So Hajek’s criticism leads to something counterintuitive here, though that won’t surprise Hajek. The point of this post is to develop a rigorous principled response to Hajek’s criticism entailing the intuition that you should go for the higher probability of an infinite outcome over a lower probability of it.

A gamble is a random variable on a probability space. We will consider gambles that take their values in R*=R∪{−∞,+∞}, where R is the real numbers. Say that gambles X and Y are disjoint provided that at no point in the probability space are they both non-zero. We will consider an ordering ≤ on gambles, where XY means that Y is at least as good a deal as X. Write X<Y if XY but not YX. Then we can say Y is a strictly better deal than X. Say that gambles X and Y are probabilistically equivalent provided that for any (Borel measurable) set of values A, P(XA)=P(YA). Here are some very reasonable axioms:

  1. ≤ is a partial preorder, i.e., transitive and reflexive.
  2. If X and Y are real valued and have finite expected values, then XY if and only if E(X)≤E(Y).
  3. If X and Y are defined on the same probability space and X(ω)≤Y(ω) for every point ω, then XY.
  4. If X and Y are disjoint, and so are W and Z, and if XW and YZ, then X+YW+Z. If further X<W, then X+Y<W+Z.
  5. If X and Y are probabilistically equivalent, then XY and YX.

For any random variable X, let X* be the random variable that has the same value as X where X is finite and has value zero where X is infinite (positively or negatively).

The point of the above axioms is to avoid having to take expected values where there are infinite payoffs in view.

Theorem. Assume Axioms 1-5. Suppose that X and Y are gambles with the following properties:

  1. P(X=+∞)<P(Y=+∞)
  2. P(X=−∞)≥P(Y=−∞)
  3. X* and Y* have finite expected values

Then: X<Y.

It follows that in the lottery case, as long as the probability of getting a winning ticket without buying is smaller than the probability of getting a winning ticket when buying, you should buy. Likewise, if choosing to believe has a greater probability of the infinite payoff than not choosing to believe, and has no greater probability of a negative infinite payoff, and all the finite outcomes are bounded, you should choose to believe.

Proof of Theorem: Say that an event E is continuous provided that for any 0≤xP(E), there is an event FE with P(F)=x. By Axiom 5, without loss of generality {XA} and {YA} are continuous for any (Borel measurable) A. (Proof: If necessary, enrich the probability space that X is defined on to introduce a random variable U uniformly distributed on [0,1] and independent of X. The enrichment will not change any gamble orderings by Axiom 5. Then if 0≤xP(XA), just choose a∈[0,1] such that aP(XA)=x and let F={XA&Ua}. Ditto for Y.)

Now, given an event A and a random variable X, let AX be the random variable equal to X on A and equal to zero outside of A. Let A={X=−∞} and B={Y=−∞}. Define the random variables X1 and Y1 on [0,1] with uniform distribution by X1(x)=−∞ if xP(A) and X1(x)=0 otherwise, and Y1(x)=−∞ if xP(B) and Y1(x)=0 otherwise. Since P(A)≥P(B) by (7), it follows that X1(x)≤Y1(x) everywhere and so X1Y1 by Axiom 3. But AX and BY are probabilistically equivalent to X1 and Y1 respectively, so by Axiom 5 we have AXBY. If we can show that AcX<BcY then the conclusion of our Theorem will follow from the second part of Axiom 4.

Let X2=AcX and Y2=BcY. Then P(X2=+∞)<P(Y2=+∞), X2* and Y2* have finite expected values and X2 and Y2 never have the value −∞. We must show that X2Y2. Let C={X2}=+∞. By subdivisibility, let D be a subset of {Y2}=+∞ with P(D)=P(C). Then CX2 and DY2 are probabilistically equivalent, so CX2DY2 by Axiom 5. Let X3=CcX2 and Y3=DcY3. Observe that X3 is everywhere finite. Furthermore P(Y3=+∞)=P(Y2=+∞)−P(X2=+∞)>0.

Choose a finite N sufficiently large that NP(Y3=+∞)>E(X3)−E(Y3*) (the finiteness of the right hand side follows from our integrability assumptions). Let Y4 be a random variable that agrees with Y3 everywhere where Y3 is finite, but equals N where Y3 is infinite. Then E(Y4)=NP(Y3=+∞)+E(Y3*)>E(X3). Thus, Y4>X3 by Axiom 2. But Y3 is greater than or equal to Y4 everywhere, so Y3Y4. By Axiom 1 it follows that Y3>X3. but DY2CX2 and X2=CX2+X3 and Y2=DY2+Y3, so by Axiom 4 we have Y2>X2, which was what we wanted to prove.

Philosophy of Religion and Apologetics
April 12, 2012 — 13:05

Author: Kenny Pearce  Category: General  Tags: , , , ,   Comments: 17

Philosophy of religion, as practiced by religious believers, is often confused with apologetics. (Perhaps it is even so confused, on occasion, by some of its practitioners.) Indeed, if we use the term ‘apologetics’ more broadly, to include not just the giving of an apologia (defense) of religion, but of just any belief system, then we could say that philosophy in general is often confused with apologetics. This is, I think, a serious mistake. The philosopher, qua philosopher, is up to something quite different than the apologist, qua apologist. The ‘qua’ clauses are necessary, because of course the same person may engage in both philosophy and apologetics and, as will emerge, it is even possible to do both at the same time, but as activities they have fundamentally different aims. I will try, in this post, to clarify this difference and explain why it matters.

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