Friday, August 16, 2013

“If A, then probably C” entails “Probably, if A then C”

Application



The fact that “If A, then probably C” entails “Probably, if A then C” is useful in a lot of philosophical discussions. Take for example this argument:
  1. If atheism is true, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, atheism is false.
In Does Objective Morality Exist If God Does Not Exist? I basically argued that if atheism is in fact true, objective morality probably doesn’t exist, in which case premise (1) is probably true. I thus used an “If A, then probably C” claim to show that an “Probably, if A then C” claim is true.

I would’ve thought “If A, then probably C” entailing “Probably, if A then C” would be uncontroversial even among internet atheists, but in discussing an argument against the possibility of an infinite past, someone I dialogued with on Facebook claimed the following:
If A, then probably C

does not entail

Probably, If A then C.
If you encounter an internet atheist (or anyone else) who disputes that “If A, then probably C” entails “Probably, if A then C” you can point them to this article which features a mathematical proof demonstrating that “If A, then probably C” does indeed entail “Probably, if A then C.” Fortunately the proof requires nothing more difficult than high school (or middle school) mathematics. Don’t worry if your math is a bit rusty; I’ll give a crash course in some basic probability and set theory.


The General Idea



First, an explanation of what “If A, then C” means exactly. The “If A, then C” material conditional says it is not the case that A is true and C is false (this is often good enough for philosophical arguments, since in a true material conditional, when A is true, C is true as well—because a true material conditional prohibits C from being false when A is true).

The general idea is that “Given A, C is probably true” means it is probably not the case that A is true and C is false. While I think the general idea is somewhat intuitively obvious, this article will mathematically prove the general idea to be true.


Mathematical Background



If you’re already savvy in math (particularly with some basic set algebra and probability theory) feel free to skip this section and go straight to the proof. Otherwise I’ll introduce some basic math stuff so that folks who aren’t quite so math savvy can follow along.

Set Operations



To illustrate some set operations, suppose our “universe” consists entirely of natural numbers 1 through 9. Now let A and B be the following:
A = {1, 5, 9}
B = {1, 5, 7, 8}
C = {2, 3}
SymbolExampleExplanation

(element of)
1 ∈ AFor any set S, x ∈ S means that x is an element of S.

(not an element of)
1 ∉ CFor any set S, x ∉ S means that x is not an element of S.

(intersection)
A ∩ B = {1, 5}Given sets S and T, S ∩ T contains all the elements x such that x ∈ S and x ∈ T.

(union)
A ∪ B = {1, 5, 7, 8, 9}Given sets S and T, S ∪ T contains all the elements x such that x ∈ S or x ∈ T.

(empty set)
A ∩ C = ∅The empty set is a set that doesn’t contain any members.
ξ
(universal set)
A ∪ A’ = ξ
B ∩ ξ = B
ξ is basically “everything” in whatever universe the sets are “talking about,” e.g. if we’re dealing with sets of lowercase alphabets, like {a, e, i, o, u}, the universal set would be the entire lowercase alphabet. Sometimes the universal set is depicted as U or S.
S’
(complement of S)
B’ = {2, 3, 4, 6, 9}The complement of set S, denoted as S or SC or S’ or −S (among other variants), are all the elements x such that x ∉ S and x ∈ ξ.


It should be remembered that the union (∪) is using the “inclusive-or,” and so A ∪ B would include all elements that are in both A and B.

One notable thing is how similar some set operations are to propositional logic:

Set TheoryRough Equivalent in Logic
A ∪ BA ∨ B (“A or B”)
A ∩ BA ∧ B (“A and B”)
A’, −A¬A (“not-A”), alternatively, ~A and −A


So for example, x ∈ (A ∪ B) means that x ∈ A or x ∈ B.

Set Algebra



There are certain equality rules with sets involving stuff like unions and complements. Here’s a sample of some algebraic set rules:

Commutative laws: A ∪ B = B ∪ A  |  A ∩ B = B ∩ A
Identity laws: A ∩ ξ = A  |  A ∪ ∅ = A  |  A ∩ ∅ = ∅
Complement laws: A ∪ A’ = ξ  |  A ∩ A’ = ∅  |  (A’)’ = A
Distributive laws: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)


Probability Symbolism



Probability often uses the language of set theory to symbolize the probabilities of certain events happening. Here a set denotes an event, like getting three or higher when rolling a die, where an event is a set of one or more outcomes. So for example if we let F represent the event of “rolling a four or higher” for a six-sided die, the set of outcomes would look like this:
F = {4, 5, 6}
If we let T be the event of “getting a 3,” T would look like this:
T = {3}
F ∪ T symbolizes all the outcomes that are in F or T, which in this case is rolling a 3 or higher. Pr(F ∪ T) denotes the probability that the outcome will be a member of set F or T. Some basic probability symbolism:

Pr(A) = The probability of A being true; e.g. Pr(A) = 0.5 means “The probability of A being true is 50%.”
Pr(A|B) = The probability of A being true given that B is true. For example:
Pr(I am wet|It is raining) = 0.8
This means “The probability that I am wet given that it is raining is 80%.”
Pr(¬A) = The probability of A being being false (¬A is read as “not-A”); e.g. Pr(¬A) = 0.5 means “The probability of A being false is 50%.”
Pr(B ∪ C) = The probability that B or C (or both) are true.
Pr(B ∩ C) = The probability that B and C are both true.
Pr(A|B ∩ C) = The probability of A given that both B and C are true.


Some alternate forms:

One VersionAlternate Forms
Pr(A) P(A)
Pr(¬A)  Pr(~A), Pr(−A), Pr(AC)
Pr(B ∪ C) Pr(A ∨ B)
Pr(B ∩ C) Pr(B ∧ C), Pr(B&C)
Pr(A|B)Pr(A/B)


The alternate forms can be combined, e.g. an alternate form of Pr(H|E) is P(H/E).

Probability Rules



In addition to the mathematical symbolism, there are also a number of mathematical rules regarding probability. When events A and B have no outcomes in common, i.e. when A ∩ B =∅, events A and B are set to be mutually exclusive or disjoint. For example, “rolling a two or lower” and “rolling a five or higher” are mutually exclusive events for rolling a six-sided die. Two events are said to be independent of each other if the outcome of one does not affect the outcome of the other, e.g. rolling a 6 the first time and rolling a 5 the second time for a six-sided die. Because I think it makes things clearer in what I’ll do later in this article, I’ll use the symbolism ¬A to denote “not-A” rather than A’. With that in mind:

Rule NameRule
Addition rule:Pr(A ∪ B) = Pr(A) + Pr(B) when A ∩ B = ∅
General addition rule:Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B), regardless of whether A ∩ B = ∅
(note that when A ∩ B = ∅, Pr(A ∩ B) = 0)
Complement rule:Pr(¬A) = 1 − Pr(A)
Multiplication rule:Pr(A ∩ B) = Pr(A) × Pr(B) when A and B are independent
General multiplication rule:Pr(A ∩ B) = Pr(A) × Pr(B|A), regardless of whether A and B are independent
(Pr(B|A) = Pr(B) when A and B are independent)


Notice that because of the general multiplication rule (and a bit of simple algebra), this is also true for any events A and B:
Pr(B|A) = 
Pr(A ∩ B)
Pr(A)
And that’s pretty much all the math background you’ll need to follow along.


The Proof



For this to work I’ll break the proof in separate steps. In math and logic, a lemma is a claim that is proved to demonstrate something else later in a proof. For this proof I’ll be using several lemmas.

Recall that the “If A, then C” material conditional means it is not the case that A is true and C is false. Thus the probability that the material conditional is true can be mathematically depicted as this:
Pr(¬(A ∩ ¬C)) = 1 − Pr(A ∩ ¬C)
Which means “The probability of it not being the case that A and ¬C are both true.”

By “If A, then probably C” I mean “Given A, C is probably true,” which in turn means that Pr(C|A) is high. So to show that high Pr(C|A) entails a high Pr(¬(A ∩ ¬C)), I want to prove the following:
Pr(C|A) ≤ 1 − Pr(A ∩ ¬C)
Or equivalently:
Pr(¬(A ∩ ¬C)) ≥ Pr(C|A)
Lemma (1): (A ∩ C) and (A ∩ ¬C) are disjoint (mutually exclusive). We can show that no element in the universe can be a member of both (A ∩ C) and (A ∩ ¬C). Let x be an arbitrary element and let’s suppose x is a member of both (A ∩ C) and (A ∩ ¬C). With a bit of math logic, we show that there can’t be any x such that x ∈ (A ∩ C) and x ∈ (A ∩ ¬C) by assuming there is such an x and deriving an impossibility, like so:
  1. x ∈ (A ∩ C) and x ∈ (A ∩ ¬C)
  2. (x ∈ A and x ∈ C) and (x ∈ A and x ∈ ¬C), from (1) and definition of ∩
  3. x ∈ A and x ∈ C and x ∈ A and x ∈ ¬C, from (2)
  4. x ∈ C and x ∈ ¬C, from (3)
Of course, it’s impossible for there to be an element that is a member of a set and its complement, since (C ∩ ¬C) = ∅. Thus (A ∩ C) and (A ∩ ¬C) are disjoint, i.e. (A ∩ C) ∩ (A ∩ ¬C) = ∅.

With this in mind, let ξ be the universal set.
A ∩ ξ = A
⇔ A ∩ (C ∪ ¬C) = A
⇔ (A ∩ C) ∪ (A ∩ ¬C) = A
Lemma (2): Since (A ∩ C) and (A ∩ ¬C) and are mutually exclusive, by the rules of probability:
Pr(A ∩ C) + Pr(A ∩ ¬C) = Pr(A)
With those two lemmas in mind, consider this statement:
Pr(C|A) = 
Pr(C ∩ A)
Pr(A)
Now we swap Pr(A) for Pr(A ∩ C) + Pr(A ∩ ¬C), and this is a legitimate move thanks to the equality proved in lemma (2):
Pr(C|A) = 
Pr(C ∩ A)
Pr(A ∩ C) + Pr(A ∩ ¬C)

 
⇔  Pr(C|A) = 
Pr(A ∩ C)
Pr(A ∩ C) + Pr(A ∩ ¬C)
Just to make this easier to read, let’s have x represent Pr(A ∩ C) like so:
Pr(C|A) = 
x
x + Pr(A ∩ ¬C)
Given some value for Pr(A ∩ ¬C), what is the highest Pr(C|A) possible? One hint is this: given some Pr(A ∩ ¬C), when x goes to zero, so does Pr(C|A); a smaller x means a smaller Pr(C|A).[1] So to get the highest Pr(C|A) value given some Pr(A ∩ ¬C), we want x to be as big as possible. Now since this is true:
Pr(A ∩ C) + Pr(A ∩ ¬C) = Pr(A) ≤ 1

    ⇔ Pr(A ∩ C) + Pr(A ∩ ¬C) ≤ 1

    ⇔ Pr(A ∩ C) ≤ 1 − Pr(A ∩ ¬C)
The highest Pr(A ∩ C) (and thus x) can be is 1 − Pr(A ∩ ¬C). So substituting the maximum value for x to obtain an upper limit for Pr(C|A) gives us this:
Pr(C|A) ≤ 
1 − Pr(A ∩ ¬C)
1 − Pr(A ∩ ¬C) + Pr(A ∩ ¬C)

 
⇔  Pr(C|A) ≤ 
1 − Pr(A ∩ ¬C)
1 + [−Pr(A ∩ ¬C)] + Pr(A ∩ ¬C)

 
⇔  Pr(C|A) ≤ 
1 − Pr(A ∩ ¬C)
1 + 0

 
⇔  Pr(C|A) ≤ 
1 − Pr(A ∩ ¬C)
1

 
⇔  Pr(C|A) ≤  1 − Pr(A ∩ ¬C)

 
⇔  Pr(C|A) ≤  Pr(¬(A ∩ ¬C))

 
⇔  Pr(¬(A ∩ ¬C)) ≥  Pr(C|A)
This means that Pr(¬(A ∩ ¬C)) must be at least as great as Pr(C|A), which means Pr(C|A) being high entails Pr(¬(A ∩ ¬C)) being high. This in turn means “If A, then probably C” entails “Probably, if A then C.”

In response, one could attack the relationship between “If A, then probably C” and Pr(C|A). But of course, Pr(C|A) is the probability of C given A. So Pr(C|A) being high means that given A, C is probably true. “If A, then probably C” is saying that given A, C is probably true. Hence, “If A, then probably C” entails a high Pr(C|A), which entails “Probably, if A then C.”





[1] We can prove this more rigorously with some simple calculus. Since Pr(A ∩ ¬C) is constant, we can replace Pr(A ∩ ¬C) in the equation below with k (to symbolize a constant) and take the derivative:
x
x + Pr(A ∩ ¬C)
The calculus would thus be this using the quotient rule:
d
dx
 
x
x + k
   = 
(x + k)(1) − (x)(1)
(x + k
 = 
x + kx
x² + 2k + k²
 = 
k
x² + 2k + k²
Since both x and k symbolize possible probability values, they must each be in the interval [0, 1]. One can see that for all k > 0, all values of x ∈ [0,1] produce a positive value in the derivative above, which means the rate of change is always positive throughout the x ∈ [0,1] interval, which means the largest value in the [0,1] interval will be when x = 1. For when k = 0, the slope will be constant (not changing) for all x > 0. What about when x goes to 0? Obviously we can’t just plug in 0 for x when k = 0, but we can take the limit:
lim
x→0
 
0
x² + 2(0)² + 0²
 


 ⇔  lim
x→0
 
0
x²
 


We can then use L’Hôpital’s rule a couple times:

lim
x→0
 
0
x²
 


 ⇒  lim
x→0
 
0
2x
 


 ⇒  lim
x→0
 
0
2
   = 0
The slope is thus still constant (not changing) for all x ∈ [0,1] when k = 0, and the maximum value will still be found at x = 1.

Monday, July 29, 2013

Debate Round 5: Closing Statements

Home  >  Philosophy  >  Atheism/Theism

Preface



Below is the final round of a debate between me and fellow blogger Potnia Theron (a.k.a. Steven) over the existence of God. The debate thus far:



The Moral Argument



Definition Quibbling



First, a mild quibble: my opponent seems to think that my definition of “objective” was inappropriate in the context of the moral argument, where I stated that moral properties are objective in the sense that the hold independently of human belief and perception of them. But in the context of the moral argument the term “objective” typically means something like this, e.g. speaking in the context of the moral argument, philosopher Robert Adams writes that a moral fact is objective in the sense that “whether it obtains or not does not depend on whether any human being thinks it does”[1], and the Stanford Encyclopedia of Philosophy’s entry on the moral argument speaks of moral properties as “objective in the sense that they hold or not regardless of human opinion.”

Moral Ontology



In my opening statement I discussed moral ontology. I noted that moral properties exist either solely as part of the physical realm or else (to at least some degree) as part of the nonphysical realm. One of these must be true, because if morality exists neither as part of the physical realm nor as part of the non-physical realm, then it follows that morality does not exist as part of reality at all. So if morality exists, some ontological explanation or other must be true.

Properties attached to the physical world likewise have some sort of ontology. For example, certain objects have the property of “redness.” Redness is a physical property and its ontology is fairly well understood (the natural sciences account for how the property exists). The property of moral wrongness however (as in the case of a man stealing a television where the man’s action has the property of moral wrongness) is different. Moral wrongness is a nonphysical property, and as a property of objectively existing oughtness (an action is morally wrong only if one ought not to do it), it cannot be empirically detected; barring the supernatural, the presence or absence of objectively existing oughtness would not affect the physical world at all. The ontology of objective moral properties like moral wrongness is rather curious and cries out for explanation.

I thus put forth what I called the argument from ontological simplicity. Given that all else held constant, the simplest explanation is the best and most probable one, I argued that the simplest ontological explanation leads us to an eternal, transcendent, metaphysically necessary entity that imposes moral duties upon us with supreme and universally binding authority. I further noted that the entity being a personal being most intelligibly accounts for the entity imposing duties upon us and having authority over people. If the entity is a personal being, we end up with an eternal, transcendent, metaphysically necessary being who imposes moral duties upon us with supreme and universally binding authority. We thus end up with a personal Supreme Being.

The other moral argument was noting that moral properties of objectively existing oughtness are objective and non-natural; some moral ontological explanation must account for this, and theism is the best ontological explanation; e.g. moral wrongness is one and the same property as that which God forbids.

I do not know of a better ontological explanation for objective morality than a God-like entity, so I asked for Steven’s alternative ontological explanation. Steven said his view was simpler because “it’s simpler to posit nothing than to posit something.” I thought this was an apposite response, because by my lights Steven scarcely provided an alternative moral ontology; with no metaphysical entities postulated, there is no ontological explanation. A moral ontology should at least posit the existence of moral properties, and it then becomes a question of how exactly these properties exist. One way to answer this question would be to provide an ontological explanation that is as vague and uninformative as possible about the ontology of morality, thereby avoiding the opportunity of describing an obviously inferior alternative to theism.

Was Steven’s answer that bad? Maybe not, but a good ontological explanation should tell us whether moral properties are physical or nonphysical, as well as tell us specifically how these properties exist. For example, moral Platonism tells us that moral values are nonphysical and specifically explains how objective moral values exist (namely, as Platonic objects), thus moral Platonism provides an ontological explanation of objective moral values (though in my opening statement I argued God being the Good is a better ontological explanation). Of course, we need an ontological explanation for objective moral duties too, and one example is divine command theory, which tells us (among other things) that moral wrongness is nonphysical, and it specifically explains how moral wrongness exists (namely, that moral wrongness is one and the same property as that which God forbids). In contrast, Steven’s response does not clearly answer whether moral properties are physical/nonphysical—though from his response I’m guessing the latter—nor does his response specifically explain how objective moral properties exist; his response is vague and largely uninformative on this issue.

For example, we know, according to Steven’s view, that moral properties supervene on non-moral properties (I agree), but this still doesn’t quite tell us how these mysterious nonphysical properties exist. We also know, according to his view, that moral facts are “fixed independent of what any stance any agent might take towards them,” yet at best this only tells us how mysterious nonphysical moral properties don’t exist rather than how they do exist. Steven does seem to think that objective morality (which presumably includes objective moral values) has “no foundation outside of itself,” but it isn’t clear from this if he accepts moral Platonism (where moral values exist as abstract objects independently of the mind) or some other moral ontology—if indeed he has one to offer at all. If Steven rejects Platonism, how exactly does he think these nonphysical moral properties exist? Alas, no clear answer was provided.

Perhaps he thinks objective moral properties are nonphysical and that they “just exist,” supervening somewhat inexplicably on certain actions in the physical world (like moral wrongness being attached to a man stealing a television) with no further explanation for why these nonphysical properties exist or why these properties are attached to certain actions. Is this simpler? I suspect not. With this view, I tend to visualize nonphysical moral properties as ectoplasmic clouds attached to certain actions in the physical world, and it seems simpler to posit just a single entity grounding all these nonphysical moral properties. This “consolidates” the nonphysical moral properties into a single entity (God being the Good grounds all objective moral values, God being the divine commander grounds objective moral duties e.g. moral wrongness being one and the same property as that which God forbids), and God grounding morality also provides a more intelligible moral ontology of e.g. objective moral duties. Given that some ontological explanation or other is needed for morality to be real, I do not think we can get simpler than a single grounding entity.

The Deductive Moral Argument



The deductive moral argument I gave was this:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
My space was limited in my opening statement so I skipped a few things I perhaps should not have. The first thing is that in the context of the deductive moral argument “God” refers to a personal Supreme Being, and stops short of saying that the Supreme Being is omnipotent (much as my argument from ontological simplicity does). Second (and I left this somewhat implicit) my view is that if no God (no Supreme Being) exists then atheism is the most plausible stance to take on the position of whether there are gods, but if atheism is true objective morality does not exist, and so we have good grounds for accepting the first premise. It should be noted that my opponent is a polytheist and might disagree with me on the probability of “If God does not exist, then atheism is true.”

While two of my moral arguments argue that God grounds morality, the deductive moral argument is an independent argument and makes no such claim.

Objection #1

Steven claims that premise (1) of the deductive moral argument tells us that God is responsible for the distribution of moral values and duties. The fact that this is not so is revealed by considering that even an atheist can agree with premise (1); all that’s needed to accept premise (1) is to believe that objective morality probably doesn’t exist if God does not exist. Many atheists believe this, and none of these atheists believe God is responsible for the distribution of anything. Thus, this objection fails against the deductive moral argument.

However, my other moral arguments do claim that God (in the sense of a personal Supreme Being) grounds objective morality. What about them? Steven seems to think that if God grounds morality then the “pain caused by a rape isn’t what makes rape wrong” and that rape being wrong would have “nothing to do with violating one’s autonomy,” but this doesn’t follow. We can all agree that so cruelly violating someone’s autonomy and inflicting such unwanted pain is something that objectively ought not to be done and that this is what makes rape morally wrong, but notice that this merely pushes the objective oughtness question back a step: we still need an ontology that accounts for the objective oughtness in morality whereby we ought not to inflict such unwanted suffering etc., and theism provides an excellent ontological explanation for objective moral oughtness.

Objection #2

Steven says, “Maverick Christian thinks that objective moral duties exist in every possible world.” I never claimed this (nor do any of my arguments), though I do believe objective moral values exist in every possible world (since God qua the Good grounds objective moral values in all possible worlds). Thus, this objection fails against my moral arguments.

Objection #3

Steven says, “It has been assumed that objective morality needs to have a foundation outside of itself.” It should first be noted that the deductive moral argument makes no claims about moral ontology, and even if morality has no foundation outside itself, that does not attack any premise of the argument. Nor does one need to accept that morality has a foundation outside itself to accept either premise of the argument (e.g. an atheist can believe objective morality has no foundation and accept premise 1). Thus, this objection does not attack my deductive moral argument.

Still, don’t I assume that objective morality needs to have a foundation outside itself in my argument from ontological simplicity? Not quite. I gave an argument that some sort of moral ontological explanation has to be true if morality exists (and Steven never disputed this point), and it just so happens that the simplest ontological explanation appears to give us a God-like entity.

Objection 4:

In criticizing the first premise, Steven says, “God’s non-existence would not be a sufficient reason to give up objective morality because it is more obvious that morality is objective than that objective morality depends upon God’s existence.” Again, the first premise does not say that morality is dependent on God; as a material conditional it merely says “It is not the case that God does not exist and objective morality exists are both true.” But what if “objective morality exists” is more obvious than “It is not the case that God does not exist and objective morality exists are both true”? All this really implies is that the second premise is more obvious than the first, and this is quite compatible with the claim that both premises are probably true.

The LCA



The LCA was as follows.
  1. If the contingent universe has an explanation of its existence, that explanation is an eternal, transcendent, metaphysically necessary, personal cause.
  2. The contingent universe has an explanation of its existence.
  3. Therefore, the explanation of the contingent universe’s existence is an eternal, transcendent, metaphysically necessary, personal cause.
Steven says:
For, however we characterize God’s “explanation” of this contingent universe—whether it be as an event, a state of affairs or an action—it will itself be contingent, thus belonging to the contingent universe it’s explaining.
Not quite, because in my opening statement I defined the contingent universe as (roughly) the totality of all contingent things (as rocks, trees, and galaxies), whereas God being the personal cause of the universe is more of an action. By asking “Why does the contingent universe exist?” I’m asking the question, “Why do contingent things exist at all?” Steven himself admits, “Of course, we can simply shift to speaking of purely contingent things, which excludes contingent events, states of affairs or actions involving necessary beings,” apparently unaware that contingent things was what I was talking about all along.

Apart from this misconstrual, Steven does not seem to dispute any premise of the argument; indeed he seems to think it is essentially sound. He does say the LCA doesn’t get us God. Right; the LCA by itself doesn’t get us to God (as we have defined him in this debate) but I’m making a cumulative case here and the LCA does give us some an entity with some of the key attributes of God.

Simplicity



In my opening statement I noted that ceteris paribus, the simplest explanation is the best and most probable one. I also noted that the simplicity leads us to a God that is omnipotent and omniscient. This was not disputed in Steven’s rebuttal.

Steven does say that simplicity isn’t the only earmark for truth; that beauty is another. One problem with this claim is that old cliché that beauty is in the eye of the beholder. Steven says that
no description of this ‘personal cause’ fills me with the sense of awe and admiration as that of the greatest beings in existence working with one another to bring the universe into existence.
As for me, no description of a polytheistic explanation fills me with a sense of awe and elegance as the following. God, the greatest being in existence, is an infinite power that all other power ultimately derives from (both physical and volitional; e.g. God delegated some of his power to us humans when he gave us power over our own actions). God is also the locus of morality, and as an omniscient being God knows everything. Some of our beliefs are properly basic, and theists believe that God designed us (by evolution or otherwise) in such a way that when our cognitive faculties are functioning properly we intuitively apprehend certain elementary truths about logic, mathematics, and morality. Our knowledge of such basic truths ultimately originates from God himself. By my lights, there is a kind of elegant simplicity in the theory that all power, knowledge, and goodness ultimately originates from some ultimate source, and that God delegated some of his power to us so that we have the ability to follow his will or defy it.

I suspect “beauty” is too subjective to be a good earmark for truth (witness my and Steven’s differing tastes for beauty), unless perhaps it is metaphorical for e.g. how well a theory ties in with background knowledge, the breadth of its explanatory scope, the depth of explanatory power, and so forth. Moreover, while it may well be that a polytheistic explanation fills Steven with a sense of awe and admiration, to think this constitutes an earmark for truth strikes me as more a fallacy of wishful thinking than an appeal to some philosophically sound truth-conducive virtue.

Evil



Steven’s arguments are perhaps not quite arguments from evil but they at least live in the same neighborhood. One argument relied on the belief that if God is real, nothing would be ultimately unfair in any possible world. But this doesn’t follow from the type of God I am arguing for in this debate, since while the definition of God being used in this debate says that God is omnipotent and omniscient in the actual world, no claim is made about how powerful God is in other possible worlds. The theist could believe that God is omnipotent in the actual world but there are also possible worlds where God is incapable of e.g. preventing a Boltzmann brain from living a merely ephemeral existence.

Another argument relied on the assumption that allowing a child to suffer purely for someone else’s mistake is child abuse. I pointed out this was mistaken with the counterexample of leaving Billy suffer a headache so I could save two women from dying, and thus while I would be permitting Billy to suffer purely for someone else’s sake this would not constitute child abuse.

Perhaps it would be child abuse if one had no morally sufficient reason for allowing a child to suffer, and not knowing of a morally sufficient reason for God to suffer gives the argument from evil some plausibility. But a big problem, as I pointed out in my rebuttal, is that due to the chasm between God’s infinite mind and the finite human mind, it is plausible that if God and evil were to co-exist we would not (fully) know why God allows evil any more than an ant knows why the sun gives off heat. So while our ignorance of a morally sufficient reason for God to allow evil gives the argument from evil a superficial plausibility, it doesn’t quite work on a deeper intellectual level. The consequences of the infinite chasm between our minds and God’s may be unfortunate (e.g. we plausibly wouldn’t understand all the reasons why God allows evil), but they would be real if God existed. For good or ill, a comprehended God is not God.





[1] Adams, Robert M. The Virtue of Faith (New York: Oxford University Press, 1987), p. 105.

Monday, July 22, 2013

Debate Round 4: Answers

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Preface



Below are my answers for my opponent in the “Answers” round of a debate between me and fellow blogger Potnia Theron (a.k.a. Steven) over the existence of God. The debate thus far:



Question #1



You wrote:
God's allowing a child to suffer purely for someone else's sake would constitute child abuse because—unlike you in your scenario—God could ensure that the child ultimately benefits from her suffering.
To some degree I take what I said back about that, because I don’t know whether God could do this without also eliminating some greater good.
To incorporate this into your analogy, it'd be like being perfectly able to prevent both Billy's suffering and save those women, but nevertheless ignoring the child. Obviously, I'd understand prioritizing those women's lives, but can you imagine someone ignoring a child in agony when he need only walk to the medicine cabinet and get some aspirin? Sounds like abuse to me. What do you think?
It depends whether that person leaving the child in agony has a morally sufficient reason for doing so. If not, then I think it would be child abuse or at least a cruel act. Maybe you think there can be no morally sufficient reason, but I can think of a few conceivable scenarios. Suppose the boy has a health condition such that, if the person were to give the boy aspirin, the boy would later suffer even worse than what he’s experiencing now. Or perhaps you can’t think of any morally sufficient reason beyond benefiting the child, in which case I could again think of counterexamples. Suppose for example the person doesn’t give the boy some aspirin because he is needed immediately to deliver injections that will save thirty people, and if he paused to give the boy some aspirin, the thirty people would die in horrible agony.

I think we agree that it wouldn’t be child abuse to allow the child to suffer if there were a morally sufficient reason. If a person allows something as horrible as child agony to happen and we can’t think of a morally sufficient reason, at face value we think the person is doing something morally objectionable, and so it’s understandable why some people think up the argument from evil. But a big problem, as I pointed out in my rebuttal, is that due to the chasm between God’s infinite mind and the finite human mind, it is plausible that if God and evil were to co-exist we would not (fully) know why God allows evil any more than an ant knows why the sun gives off heat. So while our ignorance of a morally sufficient reason for God to allow evil gives the argument from evil a superficial plausibility, it doesn’t quite work on a deeper intellectual level. The consequences of the infinite chasm between our minds and God’s may be unfortunate (e.g. we plausibly wouldn’t understand all the reasons why God allows evil), but they would be real if God existed. Inevitably, a comprehended God is not God.

Question #2



You wrote:
While there may be a possible world in which God counter-balances the unfairness of a Boltzmann brain's life-span with an after-life (or something else), the possible worlds my argument references consist in the events of the coming into, and passing out of being of a Boltzmann brain. There's no room in such worlds for compensation because only two events happen in them. Do you agree that such worlds are possible? If not, why not?
I don’t think such a world is possible for two reasons. First, because I believe in the Plantingan God (omnipotent, omniscient, and morally perfect in all possible worlds) and I’m not sure God wouldn’t provide an afterlife for Boltzmann brains. Second, if such a world existed there would also need to be another event: God (or some other personal cause) creating the universe. I don’t believe it’s metaphysically possible for physical universes to exist without being created by some transcendent personal cause (though explaining why I think this would involve an argument I did not get to in my opening statement).

As for whether there are possible worlds where a Boltzmann brain pops into being and neither God nor anybody else provides an afterlife for it, my answer is I’m honestly not sure. You’ve said that such an event happening is unfair but it’s unclear to my why that it so. Existence and life is a gift. If a small child receives $50 for his birthday and not $5 million, that his wealthy father could have given him more than $50 is insufficient for saying his father treated him unfairly. We might say it is better to have a lengthy life rather than a short one, and perhaps this is so, but when receiving a small gift (short life) rather than a larger one (long life), this doesn’t seem quite sufficient to fit the category of “unfair.”

A better objection, it seems to me, is that a perfectly good God qua being perfectly good would ensure the Boltzmann brain person continues to exist for all eternity, whether via the afterlife or something else. Maybe, but I’m not sure. It largely depends on what God views as good, and his ways may be slightly different from my own, and there might be infinitely many aspects of reality that factor into a decision like that (and this might include some goods I am not aware of). If I were to believe there is no possible world where a perfectly good God would allow that though, then since I believe in the Plantingan God, I would believe there is no possible world where God allows a Boltzmann brain to pop into being like that without also giving the Boltzmann brain an afterlife. For what it’s worth, if I were agnostic about the Plantingan God existing I would also be agnostic about whether such a world is possible.

All that said, I’m not arguing for the Plantingan God in this debate. The sort of being I am arguing for does not commit one to believing that God is sufficiently powerful in all possible worlds to stop events like that (Boltzmann brains popping into being for a short time and then ceasing to live with no afterlife) from happening. I argue that God is omnipotent in the actual world but I stop short of claiming that God is omnipotent and omniscient in every possible world. This would allow for the possibility of Boltzmann brains popping into being with no afterlife, since the theist could believe that in some possible worlds God is incapable of doing anything about it.

Question #3



Recall that the position I’m arguing for is that there is a being with the following characteristics in the actual world:
  1. omnipotent
  2. omniscient
  3. morally perfect
  4. metaphysically necessary
While I am to argue that God’s existence is metaphysically necessary (i.e. that he exists in all possible worlds), it is not part of the above description that God has all of these attributes in every possible world. To illustrate, I believe Michael Jordan played great basketball in the actual world, but in believing this I need not believe that Michael Jordan played great basketball in every possible world that he exists (there are some possible worlds where Michael Jordan was unable to do so due to a disability). Similarly, the theist could believe in the God of the above description without also believing that God has the property of omnipotence in every possible world, even though such a theist would believe God is omnipotent in the actual world.

To recap for the folks reading this, the second argument you offered was this:
  1. If God exists, then nothing is ultimately unfair in any possible world.
  2. But, there is some possible world where something is ultimately unfair.
  3. Therefore, God doesn't exist.
In your question #3, you said:
Finally, with respect to my second argument, you say "The justification for the first premise seems to rely on the assumption that if God exists he is omnipotent and omniscient in all possible worlds." But, it's unclear to me why my justification relies on this assumption.
Perhaps I should clarify my reasoning then. You defended premise (1) by saying, “if God existed, no gratuitous evil could obtain, and if something was ultimately unfair, it'd be gratuitously evil.” This seemed to be an allusion to the argument from evil that posits God as an omnipotent, perfectly good, and omniscient entity. When you said that in any possible world, nothing would happen unless God allowed it, this seemed to be alluding to God’s omnipotence. So it seemed to me that your defense of the first premise was relying on the assumption that if God exists in any of these worlds he is omnipotent and omniscient. I thus pointed out that while I was arguing that God is omnipotent and omniscient in the actual world, the position I was defending in this debate does not claim that God is omnipotent and omniscient in all possible worlds.

You say that, “God only needs finite power and knowledge to be in control of what happens in any given possible world.” You then ask if this has “changed your mind at all.” I agree one could formulate the argument and defend premise (1) in such a way that doesn’t require God being omnipotent in all possible worlds, but if God is so powerful that literally nothing can happen in the universe unless he allows it (and if he knows about all such events happening so he can prevent events that he doesn’t like) it does seem like we have at least a sort of quasi-omnipotence and quasi-omniscience, though again I agree that one doesn’t need to suppose that God has omnipotence and omniscience in every possible world to defend premise (1).

With that said, this hasn’t changed my mind about the success of the argument. Remember, my initial objection involved noting that the theist (of the sort of God I was describing) need not believe that God is omnipotent and omniscient in every possible world. Similarly, the theist could believe there are possible worlds where God’s power is nowhere near quasi-omnipotence, perhaps even possible worlds where God’s power is much closer to that of an ordinary human being. So I don’t think this works as a successful argument against the sort of God I am arguing for in this debate, because the theist could still believe that there are some possible worlds where God is incapable of preventing ultimately unfair events from happening, even though the theist would believe God is omnipotent and omniscient in the actual world.