Gwiazda on Swinburne: God's Properties

Tweet

This post is part of my series on Jeremy Gwiazda's criticisms of Richard Swinburne's argument for the existence of God. For the necessary introduction and background, see here.

To recap, Swinburne's argument takes a Bayesian form. He says that the probability of God's existence (h) given certain empirical evidence (e), as well as background tautological evidence (k), is high. Or, at least, higher than any alternative explanations of the same empirical evidence. The requisite probability is derived using Bayes' theorem, as follows:

• Pr(h|e&k) = Pr(e|h&k) Pr(h|k) \ Pr(e|k)

In previous entries we saw that one of Swinburne's key claims is that the intrinsic probability of theism, Pr(h|k), is high because God is the simplest possible being. And the reason for God's simplicity, according to Swinburne, is that he possesses certain properties in infinite quantities. The properties in question are freedom, knowledge and power.

We have already seen how Gwiazda challenges Swinburne's claim that infinite values are necessarily simpler, but for the purposes of this post we will accept it. Our focus instead will be on the problems that stem from Swinburne's seemingly contradictory assertions about God's properties.

1. Restricted Knowledge and Power?

Swinburne sees the properties of freedom, knowledge and power as being central to the person of God. Furthermore, in his most famous work, The Existence of God, Swinburne asserts (p. 7) that God has perfect freedom, omniscience and omnipotence. That is to say, he has the maximal* or infinite amount of these properties.

And yet, in The Coherence of Theism, Swinburne claims (p. 177) that it is impossible for a being to be perfectly free and omniscient; that God must be perfectly free; and, consequently, that God's knowledge must be curtailed or restricted.

As it happens, there is a reasonably compelling argument for this conclusion. Roughly:

• Perfect freedom entails the absence of sufficient causal predetermination of your actions. In other words, it entails that your decisions are exempt from ordinary causal laws.
• If your decisions are exempt from ordinary causal laws, it is impossible to predict or know what they would be in advance.
• If a being had perfect knowledge of the future, it would have to have knowledge of its future decisions.
• But this would mean that its future decisions were not perfectly free.
• Therefore, a being cannot be both perfectly free and omniscient.

The same reasoning could apply to God's power, which should not be able to affect future decisions.

This strikes me as being about right, but it would seem to contradict Swinburne's claim in The Existence of God, and, more importantly, his argument for God's simplicity. The latter depending entirely on the claim that infinite values of properties are the simplest.

Gwiazda identifies two possible lines of response for Swinburne.


2. The Relativity of Simplicity
First, Swinburne could argue that God has the maximal amount of knowledge and power that is consistent with his perfect freedom. In other words, God has as much knowledge and power as it is possible to have, relative to perfect freedom. And that this makes it as simple as it could be.

Gwiazda argues that this response can do nothing to rehabilitate Swinburne's argument for God's simplicity. Once you accept that power (or any other property) can be simple provided it is maximal relative to a constraint, you open up the possibility of simplicity being claimed for much lower levels of power.

To see this, imagine the following two cases. In the first, we have two supernatural beings: God and Schmod. God has perfect freedom and constrained power. Schmod has perfect power and constrained freedom. Swinburne would be claiming that God is simple even though his power is less than what it could be. But why could the same reasoning not be applied to Schmod? Is freedom somehow more important? We will return to this question at the end.

In the second case, we have God and a regular mortal human named Alex. We assume that Alex has as much power as it is possible for a human being with mental and physical limitations to have. In other words, that his power is maximal relative to other constraints. Can we now claim that Alex's power is simple? If not, why not?

If simplicity can be defined as maximality relative to a constraint, and if all levels of a property are maximal relative to some constraint or other, it is difficult to see how simplicity has any distinguishing features.


3. One Simple Property
The other response open to Swinburne is to pin the claim of simplicity on one of God's properties and not on the collection of freedom, knowledge and power.

Gwiazda detects two places in Swinburne's writings where this seems to be done. The first is in The Christian God where he claims that God possesses 'pure, limitless intentional power' and that this property is simple. The second is in an article entitled "Can There be More than One God?"** where he discusses God's almightiness and argues that it is simple.

Gwiazda thinks there are problems with these single-property arguments. First, it is difficult to see how intentional power and almightiness are not ultimately constituted by freedom, knowledge and power. Second, the actual argument to God's simplicity in The Existence of God only talks about freedom, knowledge and power, not these other single properties.

Thus, Swinburne would need to reformulate his argument for God's simplicity.


4. A Question of Priority
The final question mark hanging over Swinburne's understanding of God's properties is his prioritisation of freedom over knowledge and power. For some reason, Swinburne thinks that this prioritisation gives rise to the simplest possible being. Gwiazda is not convinced. He thinks there is another, even simpler being.

To see what this might be, let's first represent God's three properties -- freedom, knowledge and power, as an ordered triple. "R" stands for a restricted property and "∞" stands for an infinite property. The simplest being would be (∞, ∞, ∞), but since perfect freedom is not compatible with omniscience and omnipotence, Swinburne's god is (∞, R, R).

Now, we saw in the previous part  how Swinburne thinks that zero is probably as simple as infinity and definitely more simple than any finite value. But wouldn't this imply that a being with no freedom, but infinite power and knowledge (0, ∞, ∞) would be simpler than Swinburne's god?

It is hard then to see why priority is given to freedom.





* "Maximal" is understood here to cover infinite values, but leaves the door open to the possible of some logical or absolutely necessary limits.
** (1988) 5 Faith and Philosophy 225

Gwiazda on Swinburne (Index)

Tweet

This is an index for my recent series on Jeremy Gwiazda's criticisms of Richard Swinburne's argument for the existence of God.


1. Introduction

2. Complexity Quotients and God's Simplicity

3. Alternatives to God

4. The Principle P

5. God's Properties

Gwiazda on Swinburne: The Principle P

Tweet

This post is part of my series on Jeremy Gwiazda's criticisms of Richard Swinburne's argument for the existence of God. For the essential introduction and background to Swinburne's argument, see here.

To briefly recap, Swinburne claims that the probability of God (h) given certain empirical evidence (e) and background tautological evidence (k) is high. Or, at least, higher than any of the alternatives. This argument makes use of Bayes' theorem:

• Pr(h|e&k) = Pr(e|h&k) Pr(h|k) / Pr(e|k)

We have seen in previous entries that one of the things Swinburne relies upon to support his claim that Pr(h|e&k) is high is the argument that intrinsic probability of theism, or Pr(h|k), is high due to God's simplicity.

To support the argument for God's simplicity, Swinburne appeals to something he calls the Principle P:

Hypotheses attributing infinite values to properties of objects are simpler than ones attributing large finite values.

God's three key properties are his knowledge, freedom and power, and according to orthodox theism these properties are infinite in their values. On principle P, this would make God the simplest possible being.

Principle P does a lot of heavy lifting for Swinburne. It allows him to rule out, as profoundly unlikely, any explanation that appeals to a finite or limited god. This undercuts some of Hume's classic objections to theistic explanations. Given the heavy lifting that P is doing, it is essential to investigate whether or not it is justified. That is what Gwiazda does and that is what we will do in this post.


Four Arguments for P
Gwiazda identifies four justifications of P, sprinkled throughout Swinburne's oeuvre. We will respond to each in sequence.


i. Mathematical Simplicity
The first justification comes from the mathematical understanding of simplicity. Swinburne argues that "a  law is mathematically simpler than another in so far as the latter uses terms defined by the terms used in the former but not vice versa". So, for example, addition is said to be simpler than multiplication because the latter operation depends on the concept of addition.

Applying this to infinity, Swinburne argues that we can understand what the idea of an infinite quantity (or quantity without limit) is without needing to understand the specific large numbers, such as a googolplex, that are part of that infinite quantity.

This, as Gwiazda points out, is a rather silly argument. For we can also understand what a googolplex is without understanding what infinity is. Neither is simpler than the other, even if we accept Swinburne's definition of mathematical simplicity.


ii. Scientific Practice
The second justification of the principle P comes from the history of scientific practice. Swinburne uses one example over-and-over again. It is the example of the speed of light.

Swinburne argues that scientists -- or natural philosophers as they were then known -- originally preferred to posit an infinite velocity for the speed of light. It was only later, when other considerations came into play, that the finite value was arrived at. This, he thinks, is suggestive of the simplicity of an infinite value.

Gwiazda points out that Swinburne is being selective in his treatment of scientific history. Although it is true that figures like Aristotle and Descartes preferred the infinite value for the speed of light, it is equally true that Empedocles, Francis Bacon and Galileo preferred a finite value. Furthermore, it is surely not an insignificant point that those preferring the infinite value turned out to be wrong.

There are a few more points to be made here. First, Swinburne would need a much broader survey of scientific history to justify his claim -- one example won't cut it. Second, Gwiazda thinks that there might be psychological reasons why we prefer infinite values over finite values. For instance, it might be due to an unwillingness to admit to imprecision in our measurements; or it might be due to a primacy-recency effect in which the limits or endpoints of a sequence are remembered with greater felicity than the intermediate points.

I wouldn't overemphasis those psychological observations myself.


iii. Finite Limitations Need Explanation
On occasion, Swinburne appeals to the notion that a finite limitation is begging for explanation, in a way that limitlessness does not. Gwiazda thinks that this would need to be developed into an actual argument if it is to succeed. There is no further detail given in Swinburne's work (according to Gwiazda anyway).


iv. Zero and Infinity are "Neat"
On other occasions, Swinburne appeals to the notion that zero, one and infinity are mathematically neater and simpler quantities than all other finite numbers. The idea here is that 0 and 1 must be understood jointly, because one can't have the concept of zero-ness without also having the concept of one-ness but one could have the concept of one-ness without having the concept of two-ness. Likewise, as discussed earlier, Swinburne thinks that infinity can be understood without needing to understand other finite quantities.

Again, there is little support for this intuition and it could even be developed into an objection to Swinburne's understanding of God. We will do this in a moment.

On the whole, Swinburne's support for P is pretty thin. Next, Gwiazda develops two arguments for the falsity of P.


2. Why P might be false
The first argument against P focuses on the fact that the postulation of infinite values is often a sign of complexity and error in scientific explanations. This is a direct refutation of Swinburne's second argument for P.

Gwiazda uses the example of singularities in cosmology to make his case. A singularity is a point at which some measurement takes on an infinite value. The best-known examples are the points of infinite density found in black holes and at the origin of the universe (Big Bang). Scientists are not particularly happy with these infinite values. Indeed, they usually take them as indications of errors in their present theories.

The second argument against P takes onboard Swinburne's claim about the simplicity or neatness of zero, one and infinity. If Swinburne is right when he makes this claim, it would seem to follow that a God with no freedom (or knowledge or power) would be just as simple as one with infinite freedom. It would also seem to follow that a God with one unit of freedom (or knowledge or power) would be just as simple as an infinite God.

This is surely not something Swinburne would wish to entertain. But if that's the case, he will have to do a lot more work ironing out his definition of simplicity and his arguments in favour of an infinite God.

That's it for this post. In the next entry we will consider Swinburne's understanding of the relationship between God's three key properties (freedom, knowledge and power) and how this understanding further damages his argument for the infinite and simple God. 

Gwiazda on Swinburne: Alternatives to God

Tweet

This post is part of a series on Jeremy Gwiazda's criticisms of Richard Swinburne's argument for the existence of God. For an introduction to the series, and to Swinburne's argument, see here.

To quickly recap, Swinburne argues that the probability of God's existence (h), given certain empirical evidence (e) and background tautological evidence (k) is high. This is stated in Bayesian form as:

• Pr(h|e&k) = Pr(e|h&k) Pr(h|k) / Pr(e|k)

Obviously, this equation is meaningless without actual numerical values. We saw in part two that problems arise when we combine some of Swinburne's proposed values and with his assumptions about probability. In particular, we saw that if he wants h to confer a high probability on e, he may have to abandon his commitment to God's simplicity.

In this part we will look at another mathematical difficulty. One that arises when Swinburne considers alternatives to theism.


1. What are the Alternatives?
Swinburne is trying to offer an explanation for the evidence we observe in this universe. He thinks theism is the most probable explanation. But he identifies three alternatives:

• h1 = "There are many Gods responsible for the creation of the universe"
• h2 = "There is an eternal physical universe (or multiverse)"
• h3 = "There is no explanation; the universe is just a brute fact"

So we end up with four possible explanations for the observed universe:

2. Putting Figures on the Alternatives

For the purposes of analysing the argument, we can assume that these four possibilities are mutually exclusive and jointly exhaustive. In other words, they amount to a complete carving-up of the available probability space. And since all probabilities have a value between 0 and 1, we must accept the following equation:

• Pr(h|k) + Pr(h1|k) + Pr(h2|k) + Pr(h3|k) = 1

Now we need some estimate of the different probabilities. Interestingly, as Gwiazda points out, Swinburne thinks that the Pr(h|k) could be very small indeed. His reason being that the existence of something rather than nothing is highly improbable. This leads Gwiaza to put a figure of 0.001 on it.

You may think this figure is slightly suspicious (I did) but I think Gwiazda is only using it to make a point about the combination of Swinburne's reluctance to put precise figures into his equation with his philosophical assumptions about what makes something probable or not.

Anyway, if we accept the figure of 0.001 for Pr(h|k) we must also accept that:

• Pr(h1|k) + Pr(h2|k) + Pr(h3|k) = 0.999

3. Muddled Reasoning

Now let's go through Swinburne's thoughts on each of these alternative explanations. He starts off by saying that the probability of there being many gods (h1) must be much lower than the probability of a single God. This would seem to make sense.

He goes on to say that the probability of no explanation (h3) must be "infinitesimally low". I'm not sure how defensible that is. I guess it depends on how we understand probability. If we are working from a subjective or epistemic understanding, then I guess the thought that the universe had no explanation would be pretty surprising. This would then lead to an infinitesimally low figure for Pr(h3|k). Still, I have my doubts.

Nonetheless, if we accept Swinburne's reasoning we must ask: Where does that leave the probability of the eternal physical universe (h2)? In pretty good shape actually. For if Swinburne is correct, h1 and h3 must be less intrinsically probable than h. This means that they must be less than 0.001.

That can mean only one thing: h2 must have a high intrinsic probability. Indeed, it could have a probability of up to 0.998. This would make h2 the most attractive explanation by a mile.

Obviously, Swinburne does not agree. He thinks the physical universe is highly complex and so highly improbable. Now we saw how this assumption was problematic in the previous post, but if its going to work here Swinburne will need to raise his estimate for the intrinsic probability of theism, or consider more alternative explanations.

Either way, he has work to do.

4. Summing up and Moving On

The first two entries in this series have dealt with some technical problems that arise from Swinburne's use of Bayes' theorem. Gwiazda's main argument has been that Swinburne's philosophical assumptions about probability have placed him somewhere between a rock and a hard place when it comes to proving his central thesis that God is a better explanation of complex universe than anything else.

In future entries in this series we will move on from these technical problems to consider Swinburne's understanding of God's simplicity.

Gwiazda on Swinburne: Complexity Quotients and God's Simplicity

Tweet

This post is the second in a series on Jeremy Gwiazda's criticisms of Richard Swinburne's argument for the existence of God. For an introduction to Swinburne's argument see here.

We saw in the first part that Swinburne's basic argument is Bayesian in form. He claims that the probability of theism (h) given certain empirical evidence (e) and background tautological evidence (k) is high. We derive this probability using Bayes theorem, as follows:

• Pr(h|e&k) = Pr(e|h&k) Pr(h|k) / Pr(e|k)

Swinburne estimates that Pr(e|h&k) is 0.5. He then argues that the intrinsic probability of h, or Pr(h|k), is much higher than the intrinsic probability of e, or Pr(e|k). He does so on the grounds that h is much simpler than e and that the simpler hypothesis must have the higher probability.

Gwiazda thinks that this line of reasoning gets Swinburne into probability theory's version of hot water. We are about to see why.

1. Complexity Quotients

To appreciate the flaws we need to introduce the concept of a complexity quotient. This is a figure that arises from the division of one intrinsic probability by another. It gives some indication of how complex one theory is when compared to another.

In the case at hand, the relevant quotient is derived when we divide the intrinsic probability of theism by the intrinsic probability of the evidence. Before we consider that case, let's consider an instructive example.

Let w be the hypothesis "exactly one wooden block exists". W must have some intrinsic probability based on the background tautological evidence. But when you think about, w is ambiguous: wooden blocks come in a variety of sizes and shapes.

For illustrative purposes we can divide w into four more specific hypotheses. Since we are going purely on background tautological evidence, we can assume that these more specific hypotheses are equiprobable. The hypotheses are as follows:

• w1 = "one small light wooden block exists"
• w2 = "one large light wooden block exists"
• w3 = "one small dark wooden block exists"
• w4 = "one large dark wooden block exists"

Because these are more specific versions of w, and because they are equiprobable, it follows that:

• Pr (w|k) = 4*Pr(w1|k)

Which implies the following complexity quotient:

• Pr (w|k) / Pr(w1|k) = 4

Which means that w is four times more probable than w1. This makes sense since w is the more general, non-specific hypothesis. 

So good, so far.

2. Swinburne's Error

Swinburne's error becomes apparent once we bring the concept of the complexity quotient to bear on his original equation.

• Pr (h|e&k) = Pr(e|h&k) Pr(h|k) / Pr (e|k)

Recall that all probabilities must be greater than 0 but less than 1. This means that the maximum value for each side of this equation is 1. Recall also that Swinburne put a figure of 1/2 or 0.5 on Pr(e|h&k). This would give us:

• 1 ≥ 1/2 * Pr(h|k) / Pr(e|k)

Which we can multiply through by 2 to give us:

• Pr(h|k) / Pr(e|k) ≤ 2

This is an odd result. Whereas the intrinsic probability of one wooden block existing was 4 times greater than the probability of one small light wooden block, we are now forced to conclude that the intrinsic probability of God is only 2 times greater than the intrinsic probability of all the evidence that Swinburne wants to explain.

In other words, despite his claims that the evidence was incredibly complex and so in need of an explanation, Swinburne's own figure for Pr(e|h&k) would force him to accept that the evidence is not that complex when compared with God.

3. Where did he go wrong?

Gwiazda suggests that there is a rational explanation for Swinburne's error. You see, Swinburne wants the posterior probability of God's existence -- that is, Pr(h|e&k) -- to be reasonably high (> 0.5). He wants this because he wants his argument to have some persuasive force. After all, if you were told that the probability that God explains the observable universe was, say, 0.1 you would be relatively unimpressed.

Because he wants the figure to be relatively high, he needs h to confer a relatively high probability on the evidence. This is why he comes up with the figure of 1/2 for Pr(e|h&k). 

But at the same time he is committed to: (i) the idea that simple things are more probable than complex things; and (ii) the idea that God is simple whereas the observable evidence is complex. This commitment implies that a simple thing can never confer a high probability on a complex thing.

Consequently, he would be better off abandoning his claim that God is simple (or that the evidence is complex).

That's it for now. In the next part we will consider an additional mathematical problem that arises when Swinburne addresses alternative explanations for e.

Gwiazda on Swinburne: Introduction

Tweet

Richard Swinburne is a highly respected contemporary Christian philosopher. In his impressive back-catalogue you will find painstaking evidentialist arguments supporting everything from the existence of God, the truth of the resurrection, to the validity of core Christian doctrines such as the trinity and the incarnation.

Jeremy Gwiazda is a recent CUNY PhD whose scholarly career is only in its spring. And yet, in a series of short papers (and in his PhD thesis), he has had the gall* to take on some of Swinburne's most important arguments.

It has all the makings of a modern day David and Goliath. Will the young upstart come out on top yet again? We'll see.

In this series I will look at Gwiazda's criticisms, using the following three papers as my guide:

• "Richard Swinburne, The Existence of God, and Exact Numerical Values" (2010) 38 Philosophia 357-363
• "Richard Swinburne's Argument to the Simplicity of God via the Infinite" (2009) 45 Religious Studies 487
• "Richard Swinburne, The Existence of God and Principle P" (2009) 48 SOPHIA 393-398

In this introductory post, I will provide some background on Swinburne's argument. Gwiazda focuses on the same basic deficiency in the all three of the articles, so this material will be essential reference as we go through them.

1. Bayes' Theorem

Ours is a Bayesian age. There are a number of excellent guides to Bayes' theorem online. I cannot hope to improve upon them here so I will provide a Spartan summary of the basic details.

Bayes' theorem is a formal result from probability theory. However it has some interesting philosophical implications for our understanding of rationality and induction. How so? Well, Bayes' theorem appears to give a mathematical formula for updating our degree of confidence in a hypothesis or belief based on our encounter with some evidence.

The theorem, along with a description of its basic terms, is given in the diagram below (click to enlarge).

In case you don't want to click on that and read it, the most important terms for the purposes of analysing Swinburne's argument are the likelihood of H and the prior probability of H.

The former represents the probability that a hypothesis confers on a piece of evidence. This tells us nothing about the probability of the hypothesis. Suppose you hear a knocking sound in your attic late at night. You propose that it is a group of gremlins getting rowdy. This hypothesis may confer a high probability on the knocking sound. After all, rowdy gremlins might be inclined to make noise. But this shouldn't encourage us to think that the gremlin hypothesis is itself highly probable. Before we can think that we need to know the values of the other quantities.

The prior probability, unsurprisingly, is the probability of the hypothesis without the evidence. Sometimes this figure is based on prior empirical data. However, in the case of Swinburne's argument there is no prior empirical data. So he is concerned with the intrinsic probability of the hypothesis. That is, the probability of the hypothesis given only certain logical/tautological truths.


2. Swinburne's Argument
Swinburne's central argument for the existence of God -- in the imaginatively titled The Existence of God -- is Bayesian in form. He looks at eleven pieces of evidences and feeds them through Bayes' theorem both individually and collectively. These pieces of evidence include things like purported religious experiences, apparent design, consciousness and so on.

Gwiazda only looks at the scenario in which Swinburne considers the conjunction of all eleven pieces of evidence. In the Bayesian calculus, this conjunction is labelled with an "e". Swinburne considers the probability of the hypothesis "God exists" (labelled h) given e and given a set of background "tautological" evidence (k).

This basic form is illustrated below. Obviously, Swinburne's goal is to show that Pr(h|e&k) is high (or at least higher than any alternative explanation of the evidence).

3. God's Simplicity
Giving the form is all well and good, but to make it meaningful some numerical values are needed. This, according to Gwiazda, is where the argument falls down. Before looking to Gwiazda's criticisms, we'll review what Swinburne actually says.

First, as noted in the diagram, Swinburne estimates that the likelihood of God's existence is about 1/2 or 0.5. This is the probability that God (and the tautological evidence) confers upon the evidence.

Second, Swinburne thinks that Pr(h|k) or the intrinsic probability of God's existence is high, whereas the Pr(e|k) is very low. The reason for this is that a simpler hypothesis is more probable than a complex one. And God is simple, whereas the evidence is complex.

Q.E.D, surely?

In the next part we'll see why Swinburne's approach is mistaken and why he needs to provide more precise figures to make his argument work.

*It has oft been noted that the internet needs a sarcasm-button. I add my voice to the swelling chorus supporting that idea.