Begin with the Parity Principle: If it is almost certain (i.e., if it has probability one) that (a) the basic properties Q and R have exactly the same distribution at t1, and that (b) x is a substance existing at t1 and a member of basic kind K, and no other information is available about x, then the probability that x has Q is equal to the probability that x has R. (Here, I allow such probability values as "undefined", "inscrutable" as well as intervals, vague values.)

For my argument I will need the assumption that it is possible to have indiscernibles--objects that have all the same basic properties. I actually think this assumption is false, but I am hoping that this assumption can be relaxed.

The possibility of indiscernibles and the Parity Principle are going to be the only potentially controversial assumptions in the argument. If you trust me on this point, you can stop reading the argument, and just argue against these two assumptions. Though you might want to read on to see how exactly I understand the "same distribution" condition in the Parity Principle.

The following claim is obviously true: Thesis 1: Given the above information, the probability that x0 has Q at t1 is 1/2 and the probability that x0 has R at t1 is 1/3.

However, the following claim is a consequence of the Parity Principle (argument will be given shortly): Thesis 2: Given the above information, the probability that x0 has Q at t1 is equal o the probability that x0 has R at t1.

It follows that 1/2 = 1/3, which is absurd. Hence, we have to reject the assumption that actual infinities are possible.

Now it's time for the argument for Thesis 2. By the Parity Principle, all we need to show is that the distribution of R and Q at t1 is the same. I haven't yet defined sameness of distribution. The following seems to be the right definition: there is an isomorphism f on the set of substances existing at t1 such that if P is any basic property other than Q and R, x has P iff f(x) has P, x has Q iff f(x) has R, x has R iff f(x) has Q, and if S is any basic relation (with two or more relata), then (x1,...,xn) stand in S iff (f(x1),...,f(xn)) stand in S. But now let's think about what we know about the distribution of R and Q at t1. Almost certainly (i.e., with probability one), it is true that each of the following complex properties is had by infinitely many Ks: P&Q, P&~Q, ~P&Q and ~P&~Q. (It is an easy theorem that the probability that, say, only finitely many Ks have P&Q is zero.) Moreover, the Ks are otherwise indiscernible, and nothing but a K has P or Q. Let f be any one-to-one function defined as follows. If x is not a K, then f(x)=x. If x has both P and Q, then f(x)=x. If x has both ~P and ~Q, then f(x)=x. Moreover, f maps the set of x's that have P&~Q one-to-one onto the set of x's that have ~P&Q. This last condition can be satisfied as both sets are infinite, and at most countably so, and hence have the same cardinality. It is easy to see, using the fact that the Ks are indiscernible except with respect to P and Q, that f is the requisite isomorphism, and hence the sameness-of-distribution point holds.

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Occasionally, I've offered theistic arguments that border on begging the question. Here, for instance, is one that's basically due to Kant, but transposed into an argument in a way that Kant would not approve of:

1. (Premise) We should be grateful for the wondrous universe.
2. (Premise) If something is not the product of agency, we should not be grateful for it.
3. Therefore, the wondrous universe is the product of agency.

Nonetheless, I think there could be something to (1)-(3). Dan Johnson, in the January 2009 issue of Faith and Philosophy has a fascinating little article on the ontological and cosmological arguments. He argues that a certain kind of circularity is not vicious. Suppose that I know p₁. I then infer p₂ from p₁ in such a way that I also know p₂. I then non-rationally (or irrationally) stop believing p₁, but as it happens, I continue to believe p₂. It will then often be the case that there will be a good argument from p₂ back to p₁ (perhaps given some auxiliary premises), and if I use that argument, I will be able to regain my knowledge of p₁. This is true even though there is a circularity: from p₁, to p₂, and back to p₁. Here is an uncontroversial example: I am told my hotel room is 314. I infer that my hotel room is the first three digits of pi. I then forget that my hotel room is 314, but continue to believe it is the first three digits of pi. I then infer that my hotel room is 314.

This same structure may be present in my Kantian argument. By the sensus divinitatis one comes to know that God exists (obviously this is not a Kantian idea!). One infers that the universe is such that we should be grateful for it. One then irrationally comes to be an atheist (again, there is need be no claim that every atheist is irrationally such), but one continues to believe that gratitude is an appropriate response to the universe. And if that belief is sufficiently deeply engrained, one can reason back from it to theism or at least to agency behind the universe.

Now let me move a little beyond the Johnson paper. I think it is not necessary for this structure that the initial knowledge of God's existence come from the sensus divinitatis. Any other way of having knowledge of God's existence will do--say, by argument or testimony. In fact, it is not even necessary for this structure that one oneself ever had the knowledge or even belief that God exists. Suppose, for instance, one's parents knew that God exists (in whatever way), and inferred from this that the universe is worthy of gratitude. They then instilled this belief in one, and did so in such a way as to be knowledge-transmitting. (Surely, knowledge of values can be instilled in such a way.) But they did not instill the belief that God exists (maybe because they thought that the existence of God was something everybody should figure out for themselves). One then knows (1), and can infer (3).

This transmission can be mediated by the wider culture, too. Culture can transmit knowledge, whether scientific or normative, and arguments can work at a cultural level. It could be that a theistic culture where the existence of God was known grew into a culture where (1) was known. The knowledge of (1) can remain even if the culture non-rationally rejects the existence of God (as American culture has not done, and might or might not do in the future). And then the individual can acquire the knowledge of (1) from the culture (we don't need to attribute knowledge to the culture if we don't want to; we can just talk of knowledge had by individuals participating in the culture), and then infer (3).

I think there are probably many consequences of theism that are embedded in the culture, from which consequences one can infer back to theism. If the participants in the culture knew theism to be true when these consequences were derived, then it is perfectly legitimate to reason back from these consequences to theism.

I never was sure exactly what this blaspheming referred to. Anyway, whatever it is, it's something that, according to Jesus, a person will not be forgiven for. If a person will not be forgiven for it, then it follows that God will not forgive the person for it. Then I thought of this argument:

1) Necessarily, the Greatest Conceivable Being (GCB) will forgive all sins.
2) Possibly, God, as conceived of by Jesus, will not forgive all sins.
3) So, God, as conceived of by Jesus, is not the GCB.

Regarding (1), it seems to me that any being that will forgive all sins is greater than one that will not. I take it that A can forgive B even if B has not repented in any way. My only concern is that maybe the GCB's being perfectly just prohibits him from forgiving all sins. But I can't see why this is so. It seems that a parent can both justly punish a child and forgive her child. Any thoughts?

Carnap's prior probability measure is best seen as a measure for the probability of claims made by sentences of a truth-functional language with n names, a₁,...,a_n, and k unary predicates, Q₁,...,Q_k. Let N be the set of names, Q the set of predicates and T the set {True, False}. Call the language L(Q,N). Say that a state s is a function from the Cartesian product QxN to T, and let S be the set of all states. There is a natural way of saying whether a sentence u of L(N,P) is true at a state s. Basically, you say that the sentence Q_i(a_j) is true at s if and only if s(Q_i,a_j)=True, and then extend truth-functionally to all states.

There is a natural probability measure on S, which I will call the "Wittgenstein measure", defined by P_W(A)=|A|/|S| for every subset A of S, where |X| is the cardinality of the set X. This probability measure assigns equal probability to every state. Given a probability measure P on states, we get a probability measure for the sentences of L(Q,N). If u is such a sentence, define the subset u^T={s:u is true at s} of S. Then, we can let P(u)=P(u^T). The Wittgenstein measure does not allow induction. Suppose that we have three names, and two predicates, Raven and Black. Our evidence E is: Raven(a₁), Raven(a₂), Raven(a₃), Black(a₁) and Black(a₂). Then, P_W(Black(a₃)|E)=1/2=P_W(Black(a₃)), as can be easily verified, because all states are equally likely, and hence the state that makes all the a_i be black ravens is no more likely than the state that makes all the a_i be ravens but with only a₁ and a₂ black.

So, Carnap wanted to come up with a probability measure that allows induction but is still fairly natural. What he did was this. Instead of assigning equal probability to each state, he assigned equal probability to each equivalence class of states. Say that s~t for states s and t if there is some permutation p of the names N such that s(R,p(a))=t(R,a) for every predicate R and every name a. Let [s] be the equivalence class of s under this relation: [s]={t:t~s}. Let S* be the set of these equivalence classes. Then, if s is a state, we define: P_C({s})=1/(|[s]||S*|). In other words, each state in an equivalence class has equal probability, and each equivalence class has equal probability. If A is any subset of S, we then define P_C(A) as the sum of P_C({a}) as a ranges over the elements of A.

The merit of Carnap measure is that it assigns a greater probability to more uniform states. Thus, P_C(Black(a₃)|E) should be greater than 1/2 (I haven't actually worked the numbers).

Problem 1: Carnap measure is not invariant under increase of the number of predicates. Intuitively, adding irrelevant predicates to the language, predicates that do not appear in either the evidence or the hypothesis, should not change the degree of confirmation. But it does. In fact, we have the following theorem. Let u be any sentence of L(Q,N). Let Q_r be Q with r additional predicates thrown in. Let u_r be a sentence of L(Q_r,N) which is just like u (i.e., u_r is u considered qua sentence of L(Q_r,N)).

Theorem 1: P_C(u_r) tends to P_W(u) as r tends to infinity.

In other words, as one increases the number of predicates, one loses the ability to do induction, since P_W is no good for induction. The proof (which is non-trivial, but not insanely hard) is left to the reader.

Problem 2: Let d be a sentence of L(Q,N) saying that indiscernibles are identical. For instance, let d_ij be the disjunction ~(Q₁(a_i) iff Q₁(a_j)) or ... or ~(Q_k(a_i) iff Q_k(a_j)), and let d be the conjunction of the d_ij for all distinct i and j.

Theorem 2: P_C(u|d)=P_W(u|d).

Thus, when we condition on the identity of indiscernibles, Carnap measure collapses to Wittgenstein measure. But Wittgenstein measure is worthless for induction. And often the identity of indiscernibles holds. For instance, suppose we have a₁,a₂,a₃ as our individuals, and our evidence is this: a₁,a₂,a₃ are each a raven, a₁ and a₂ are black. So far so good, we can do induction and we get some confirmation of a₃ being black. But suppose we also learn that identity of indiscernibles holds for these three ravens. Then we lose the confirmation! And we might well learn this. For instance, we might learn that exactly a₁ and a₃ are male, and exactly a₁ and a₂ each have an even number of feathers, and that means that identity of indiscernibles holds.

Moreover, I think most of us have a background belief that our world has such richness of properties that, at least as a contingent matter of fact, the identity of indiscernibles holds for macroscopic objects. If so, then Carnap measure makes induction impossible for macroscopic objects.

Sketch of proof of Theorem 2: Let D be the set of states at which identity of indiscernibles holds. Thus, D is the set of states s with the property that if a and b are distinct, then there is a predicate R such that s(R,a) differs from s(R,b). Observe that if s is any state in D, then |[s]|=n!, where n is the number of names. For, any permutation of the names induces a different state given the identity of indiscernibles, and there are n! permutations. Therefore, P_C({s})=1/(n!|S*|). Hence, P_C({s}) has the same value for every s in D. Therefore, P_C({s}|D)=1/|D|. But, likewise, P_W({s}|D)=1/|D|. The Theorem follows easily from this.

Remark: Theorem 2 gives an intuitive reason to believe Theorem 1. As one increases the number of predicates while keeping fixed the number of names, a greater and greater share of the state space satisfies the identity of indiscernibles.

On the other hand, it would be disproportionate for me to achieve G for myself at the expense of your suffering E (in the Mt Everest case, after all, E is weeks of hard labor, frostbite, danger, etc.). Thus, ~P(E,G,you,you,I). (The utilitarian, on the other hand, thinks that P(E,G,x,y,z) iff P(E,G,x',y',z') for any x,y,z,x',y',z'. But she's wrong about that.) Interestingly, it would also be disproportionate for me to achieve G for you at the expense of your suffering E (without your permission, that is--I shall take it for granted in all the discussion that there is no consent).

Let "p(x,y,z)" be the "proportionality standard" for permitting burdens to x that are the cost of benefits to y as produced by agent z. We can think of p(x,y,z) as the set of all pairs (E,G) such that P(E,G,x,y,z). Any such pair (E,G) is said to be "allowed" for (x,y,z). We can then compare proportionality standards by stringency. A simple and precise way is to say that p(x,y,z) is at least as stringent as p(x',y',z') iff anything that is allowed for (x,y,z) is allowed for (x',y',z'). However, it may be necessary to modify this, as there may be some pairs (E,G) that need to be special-cased. So, let's say that p(x,y,z) is at least as stringent as p(x',y',z') iff typically a pair (E,G) allowed for (x,y,z) is allowed for (x',y',z').

I think that thinking about the Mt Everest case suggests that if x, y and z are mutual strangers, then p(x,y,z) and p(x,x,z) are much more stringent than p(x,x,x) or p(x,y,x)--it is much easier to justify my imposing a burden on me than for another to justify imposing that burden on me.

Now, here is where the philosophy of religion comes in. Question: Where do p(x,y,God) and p(x,x,God) rank in stringency?

I have the intuition that p(x,y,God) and p(x,x,God) are no more stringent than the fairly lax p(x,y,x) and p(x,x,x). In other words, it's no harder for God to be justified in achieving a good (for myself or another) at an expense to me than it would be for me to be justified in achieving a good at that expense to me. While I think what I said is plausible, I am a little more comfortable with this in the case of p(x,x,God). (p(x,x,z) are the standards for z's paternalism with respect to z, and I think God has the right to be the ultimate paternalist.)

Intuitively, p(x,x,x) is a fairly low standard. After all, the costs of climbing Mt Everest are very, very high, and yet p(x,x,x) justifies these costs for the sake of goods that are not, perhaps, incredibly great. If p(x,x,x) is also the standard for deciding in theodicy whether God was permitted to achieve a good at the expense of a certain cost, then that should make theodicy easier than if the relevant standard were like the more stringent p(x,x,z) or p(x,y,z) (where z is a stranger to x).

But what I don't at present have is much in the way of an argument that p(x,y or x,God) is like p(x,y or x,x), apart from hearing at the back of my mind a maxim that God is closer to us than we are to ourselves. If you can supply an argument or refutation, I will be grateful.

Starting salary £36,715 to £43,840 a year (potential progression on performance once in post to £49,342 a year).

Joining the Department of Philosophy in September 2010, you will have a PhD in Philosophy and will be able to teach ethics and/or the philosophy of religion to Masters level. Your research will be in one or more of the Department's areas of research specialisation: metaphysics and epistemology, the philosophy of language and mind, and ethics and ethical theory. You will have outstanding potential in teaching and research, and be able to contribute efficiently to administration. Informal enquiries may be directed to Professor Alex Miller, Head of Department at a.miller@bham.ac.uk or 0121 414 7539.

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