Game Theory (Part 2) - The Formal Ingredients

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This is part two of my series on game theory. For an index, see here. The series follows the lectures of Ben Polak, which are available on the Open Yale Courses website.

Last time out, we had a gentle introduction to the kind of thinking required in game theory. Game theory can be a lot of fun since it frequently involves playing games. But there is also some serious mathematics underpinning the whole field. Game theorists use the formal language of mathematics to build models of real world situations. In this post we’ll introduce some of this formal language.


1. The Ingredients of a Game
A game is any interaction between two or more rational (or purposive) agents. In any such interaction, a rational decision-maker will have to take into consideration what the other agent is likely to do. In other words, he will have to anticipate the actions and choices of others. This differentiates a game from a decision problem, which involves no anticipation.

Whenever you set about modelling a strategic interaction, you must ask yourself four key questions:

• (1) Who are the players?
• (2) What are their strategies or actions?
• (3) What are their payoffs?
• (4) What kinds of information do the players have at their disposal?

A word or two must be said about these questions.

The first question is relatively straightforward and simply forces us to consider who the decision-making entities are in the game. One thing to be wary of is that the player in a game is not necessarily isomorphic with an individual human being or organism. Players could be entire organisations who, for the purposes of building the model, can be treated as one agent.

The second question raises the distinction between an action and a strategy. An action is simply a choice that an agent can take at a particular round or node in a game. A strategy is a set of actions covering every round or node in a game. In games which have only one round, actions and strategies are equivalent. In games that have two or more rounds, they are distinct.

The third question raises the thorny issue of what exactly is a payoff. For the purpose of this series we won’t get into this issue. A payoff is taken as a measure of the satisfaction or desirability of an outcome for a player. One distinction to keep in mind is that between ordinal and cardinal payoffs. An ordinal payoff is a ranking; a cardinal payoff is supposed to be some kind of real number value. If one is dealing with uncertainties or probabilities, cardinal payoffs are required.

The fourth question is about the knowledge that the players have of the formal structure of the game. For the first part of this series, we will assume that the players have perfect information about the games. This means that they know who the other players are, what actions and strategies are available to them, and what payoffs they associate with particular outcomes.

Formally, we use the notation in the following diagram to represent each of these elements (minus information).

2. Notation in Action
We can now employ this notation in the analysis of a game. We can also use it to define some of the concepts, such as strict domination, that we introduced in part 1. Consider the game in the following diagram. There is no particular story or scenario motivating the model; it is purely an abstract mathematical entity.

Now ask yourself, are there any strictly dominated strategies in this game? Consider it first from the perspective of player 1. Hold player 2’s strategies fixed and determine which of 1’s strategies is best response to each of player 2’s. It is pretty clear when you perform this exercise that none of player 1’s strategies are strictly dominated. Each is a best response under different assumptions about player 2.

Now look at the game from the perspective of player 2. Again, hold player 1’s strategies fixed and work out which of player 2’s strategies are best responses. If we perform this exercise it becomes clear that although no single one of 2’s strategies dominates all the others, the strategy “centre” does strictly dominate “right”. This is because centre always yields a higher payoff than right, irrespective of what player 1 does.

This leads us to the following definition of strict dominance. It is stated in the diagram above but since it is so important it deserves to be repeated here:

Strict Dominance: Player i’s strategy Si* is strictly dominated by player i’s strategy Si if: 

• Ui (Si, S-i) > Ui (Si*, S-i) for all S-i

If we were to continue our analysis of the game given in the previous diagram, we would delete player 2’s strictly dominated strategy (Right) and continue to solve it as a 2 x 2 game, instead of as a 3 x 2. This leads us to the concept of iterated deletion of strictly dominated strategies, which we’ll be discussing in future entries.

Course on Game Theory

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I think I’ve mentioned, on occasion, the role that game theory plays in my own thinking. Despite my fascination with this analytical tool, I have never actually taken a formal course in game theory. Instead, I dabble with various elements of it as they seem relevant to my own research work.

To make up for my lack of training in this area, I have gone through several of the online video lecture series on game theory. In particular, the following three:

Ben Polak, Open Yale Courses - Game Theory

John Fountain, University of Canterbury, New Zealand

Kathleen Bawn, UCLA - Politics, Strategy and Game Theory

Of these three courses, I found Ben Polak’s to be, by far and away, the best. He is an engaging lecturer and, on the whole, his classes are fun to watch. That said, the other courses cover some areas that he doesn’t and so can be beneficial as supplements. For example, Bawn’s course focuses on political examples, which I happen to like.

Anyway, in order to consolidate what I have learned from these courses, and to share it with those who would like to learn more about this topic, I have decided to write up my own introductory series on game theory. This series will follow the format and sequence of Polak’s Yale lectures, but will hopefully be self-contained (i.e. you won’t need to watch the lectures to fully understand it). It will not shy away from the formal and mathematical concepts employed in game theory, but it will try to explain those concepts in a fairly elementary way.

This post will serve as an index to the entire series.

1. The First Four Lessons
2. The Formal Ingredients
3. Weak Dominance, Iterated Deletion and Common Knowledge
4. The Median Voter Theorem

Game Theory (Part 1) - First Four Lessons

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As announced, I will be writing a series of posts outlining the basic concepts in game theory. The series will follow Ben Polak’s lectures, which are available here. This first post covers the first of Polak’s lectures. Although officially Polak’s lecture offers five lessons, I have reduced this to four since the fifth is just a joke about Yale students.

We begin, as we should, with a game. And since the source material comes from a college course, what better way to get the students enthused than with a game about their grades.


1. Grading Game
Two players (A and B) must play a game in order to determine which grade they will get in a course. They do so by choosing to put either α or β in a box. Their choice will then be paired with the choice of the other player. As a result, there are four possible outcomes in this game:

• (a) If A puts α in the box and B puts β in the box, then A gets an A-grade while B gets a C-grade.
• (b) If both A and B put α in the box they will both get B-minus grades.
• (c) If A puts β in the box and B puts α in the box, then A gets a C-grade while B gets an A-grade.
• (d) If both A and B put β in the box then they will both get B-plus grades.

This information can be a little difficult to follow when expressed in these terms. We can make things easier by arranging it in a box (called either a payoff or outcome matrix). In this box, A is the row-player and B is the column-player. The grades that are written in the four different boxes represent the “payoffs” to the respective players, with player A’s grades being written first.

If you’ve never come across any game theory before, you might like to stop reading at this point and think about what you would do if you were playing this game.


2. Solving the Game
So, what did you choose to do? The prediction (and, indeed, the recommendation) from game theory is that you would choose α. Furthermore, the prediction is that your opponent will also choose α. This means, of course, that you will both end up with B-minus grades. This is disappointing since if you both played β you would have received a higher grade.

Why do we assume that both will choose α? The answer is that α strictly dominates β. A more formal definition of strict domination will be given in a subsequent post. For now, an informal definition will suffice:

• Strict Domination = a strategy (α) strictly dominates a strategy (β) if the payoff from α is greater than the payoff from β, regardless of what the other player chooses to do.

It should be apparent from the payoff matrix given above that α strictly dominates β because it always yields a higher grade than β, irrespective of what the other player does. This brings us to our first lesson:

Lesson 1: You should not play a strictly dominated strategy.

The grading game as described has the structure of a Prisoners’ Dilemma. This means that there is a “temptation payoff” (the A-grade in this case) and a “sucker’s payoff” (the B-grade) that brings about the strict domination. It also yields a sub-optimal outcome for both players. This gives us our second lesson:

Lesson 2: Rational choice can lead to outcomes that are sub-optimal (Pareto Inefficient)

This is a significant lesson since PDs arise in many socially important situations.


3. Evil Gits and Indignant Angels
The grading game outlined above can be depicted using numbers instead of grades as in the following matrix. The numbers are supposed to represent the utilities that players attach to the respective outcomes.

Note that this has been called the “evil gits”-version of the grading game. Why is this? Well, it is because the solution to a game is determined by the payoffs that the players attach to the respective outcomes. In this case, we assume that the players want to secure as high as possible a grade for themselves, even if this comes at the expense of others. This assumption could be wrong.

It could be that the players, far from being evil gits, are indignant angels. This would mean that they get upset (experience a disutility) if they profit at someone else’s expense, and, similarly, they get indignant if someone else profits at their expense. In the context of the grading game, this results in the following change in the game:

This dramatically alters our proposed solution to the game. It is no longer the case that one strategy strictly dominates another. To jump the gun a little bit, α would be your best response if your opponent played α, whereas β would be your best response if your opponent played β. What you end up choosing depends entirely on your assumption about your opponent. This gives us our third lesson:

Lesson 3: Payoffs matter - changing the payoffs changes the strategy

We can now imagine a final variant of the game. This one involves an evil git (A) playing against an indignant angel (B). It has the following structure:

The analysis of this game is slightly more complicated than previous two.

First, look at it from the perspective of the evil git. It’s pretty clear that from his/her perspective α strictly dominates β.

The indignant angel might reason his/her way through the game in the following manner:

• (1) If A plays α, my best response would be to play α.
• (2) If A plays β, my best response would be to play β.
• (3) A is almost certainly going to play α, since that is a strictly dominant strategy for him/her.
• (4) Therefore, I should also play α.

Again, this leads to a sub-optimal result and provides us with our fourth and final lesson:

Lesson 4: When analysing a game, put yourself in the other player’s shoes and try to figure out what they would do.

Okay, that’s it for this first post on game theory. In the next entry, we will look in more detail at the formal ingredients of a game.

God and Moral Autonomy - James Rachels

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In an earlier post, I went through Scott Aikin’s discussion of the problem of worship. Those who read that post will know that I had a number of criticisms of Aikin’s presentation of the problem. In the end, I think the problems stemmed from the fact that Aikin tried to do too much with his argument. He tried: (i) show that God is not worthy of worship; (ii) show that no rational moral agent can engage in acts of worship and (iii) show that God does not exist because no being is worthy of worship. As I pointed out at the time, some of those aims are conflict with one another.

James Rachels has an argument that focuses on (iii), and I thought I would share it here as well. This argument has been floating around for some years now, and is easily available on the internet, so I’ll only offer the briefest of summaries (see here for another version).


1. What is worship?
Rachels begins with the observation that whatever else he is -- creator, necessary being, morally perfect, immaterial, eternal etc. -- God is supposed to be worthy of worship. On this, there should be widespread agreement among the religious. After all, they seem to spend a great deal of their time worshipping him. But this raises the question: what exactly is it to worship another being?

Rachels answers by way of analogy. He asks us to imagine a man named McBlank who is a devout pacifist. He goes to jail for dodging the draft, and he protests and writes pamphlets in an effort to prevent war. Rachels says there are three things to note about McBlank:

• (i) He recognises certain facts about warfare and the suffering it causes;

• (ii) These facts are taken to have some radical implications for how McBlank acts, i.e. they have moral significance for him; and

• (iii) The facts affect McBlank’s conception of himself and his place in the world.

Having pointed these things out, Rachels goes on to say that the religious worshipper is in a similar position to McBlank. After all, the following three things seem to be true of the worshipper:

• (i) They recognise certain facts, namely, that a being called God exists, who has created the universe and all that’s in it, and who has a divine purpose for his creation;

• (ii) These facts have normative significance for how they behave, they must behave in accordance with god’s plan;

• (iii) The facts affect the worshipper’s image of themselves and their relationship to the world, in particular in that they see themselves as a child of God.

Rachels suggests that these three things, particularly (iii), are features of the worshipful state of mind. Believing that there is a being to whom you must bow down and express your loyalty and obedience is central to worship.

[Rachels then discusses Wittgenstein’s non-cognitive theory of religious language. I’ll leave this out.]


2. What is ritual?
Having established the characteristics of the worshipful state of mind, Rachels turns his attention to the function of ritual.

He begins by looking at a political ritual: the investiture of a prince by the Queen of England. He argues that the ritual seems to serve two major purposes:

• (i) It creates a socio-political entity called a “prince”;

• (ii) The person who occupies this social role, the prince, assumes certain duties as a result of the ritual.

Rituals of this sort are common and they all make sense within a certain social context. For example, most holders of senior political office undergo some ritual whereby they take an oath of office or are presented with a medal or seal of office. In the process, these politicians take on a social role (president, prime minister) and assume duties. Whether they are any good at fulfilling those duties in another matter.

Religious rituals are similar in their function, but, obviously, the content is somewhat different:

• (i) They signify the status of the believer relative to God. The believer is inferior to God, is his “child”. Worship, after all, is not a symmetrical relationship or a relationship of mutual trust.

• (ii) The believer accepts that they have certain duties to God, e.g. to respect his commands, to pursue his purposes etc.

Rachels argues that these aspects of religious rituals take precedence over the more commonly mentioned ones. For example, it is sometimes said that the purpose of ritual is to glorify God. Rachels suggests that this is derivative from the purpose of signifying the worshipful relationship to God.

At this stage, Rachels makes one of his central claims. He maintains that it is not possible for a believer, as a child of God, to set their judgment against God’s.

To illustrate his point he highlights some of the Biblical stories concerning Abraham. In the classic tale of Abraham and Isaac, Abraham accepts God’s will, despite its prima facie immorality, and attempts to sacrifice his only son. In doing so, Abraham acts consistently with his role as God’s child. In another story, God threatens to destroy Sodom and Gomorrah and Abraham starts bargaining with him. This would not seem consistent with his role. In this case, he sets his judgment against God’s.

Rachels goes on to argue that it is simply not logically possible to defy God. His argument turns on the idea that God is a not a proper name (like “Dave” or “Paul”) but rather a title. The title in question is “the being who must be worshipped”. It would not be possible to simultaneously believe in such a being and reject its authority.


3. The Moral Autonomy Argument
This leads us, finally, to the main argument. It begins with the observation that we do seem to think of ourselves as morally autonomous agents. That is to say, we ultimately make our own determinations about what we ought to do (we may listen to the counsel of others in the process). We can either accept that as a truism or make some argument for it. An argument for moral autonomy might focus on how it necessarily flows from rationality or something like that.

But if it is true that we are morally autonomous agents, then it seems like God cannot exist. Why not? Because it would mean there is never going to be a being whose moral determinations should replace our own. Since that is what God is supposed to be, we must reject his existence.

Rachels presents this argument in the following formal terms:

• (1) If any being is God, he must be a fitting object of worship.
• (2) No being could ever be a fitting object of worship because this would require the abandonment of one’s moral autonomy.
• (3) Therefore, there is no God.

I like the simplicity of this argument. It avoids trying to explain why none of God’s properties make him worthy of worship, which was the trap Aikin’s article seemed to fall into. If being morally autonomous means that one could never worship another entity, then such an explanation is unnecessary.


4. Objections and Replies
Rachels finishes off by considering the more obvious objections and replies to his argument. Some of these target premise (1), some target premise (2).

First, he notes that some religious believers might argue that God’s command to us is to be morally autonomous. This is the kind of the view put forward by Thomists. They think God gave us our rational powers for a reason: to figure out what is morally right on our own steam. Rachels responds by saying that most religious believers think God has commanded a good deal more than this. And in any event, it means our autonomy rests on the contingency of God’s command: if God retracted the command we would lose our autonomy.

Second, there’s the response that since God is perfectly good, we could never do anything morally wrong by following his command. That would, again, be consistent with moral autonomy. Rachels argues that we can’t know whether a being is perfectly good until we have evidence of his moral character. This is presented to us in his commands. But a command to unconditionally obey and worship seems indicative of bad moral character, not good.

Third, it could be argued, along the lines of St. Augustine, that people are sinful and fallible and so cannot trust their own moral consciences. Rachels argues that this is flat out illogical. To know that our own judgment is corrupt we would have to (a) know that certain actions are morally required that (b) are not being revealed as morally required by our consciences. But (a) is the kind of thing we can’t know if our consciences are sinful and fallible. Furthermore, even if one did accept this kind of view, it would seem like one is first making an autonomous judgment about the rightness of God’s commands (and the wrongness of one’s own reasoning) and then submitting to God. Either way, you can't seem to escape autonomous judgment.

Fourth, some have argued that our own moral judgment may simply be the voice of God in our minds. Rachels suggests that this would reveal a troubling character on the part of God since he tricks us into thinking our judgments are really our own. Additionally, Rachels thinks a kind of Euthyphro dilemma could be employed here.

Fifth, and finally, some believers have argued that worthiness of worship is to be kept distinct from duties to worship. A being can be worthy of worship without us having a corresponding duty to worship them. An analogy would be the case of Mrs Brown who, although we acknowledge that she is worthy of being married (must one be worthy of this?), we do not propose to because she is already married.

Rachels points out that in the case of Mrs. Brown there is a contingent set of facts that blocks the duty (I also doubt you would ever have a duty to marry someone, but anyway...). If those facts changed, the duty might apply. The same cannot be said of God because, if Rachels argument is correct, it is never permissible to worship any being, under any circumstances.

This concludes my discussion of Rachels’s argument.

The Problem of Worship - Aikin

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I must say, I’m looking forward to Scott Aikin and Robert Talisse’s new book Reasonable Atheism. From what I gather, the book will deal with the status of belief in God in a liberal democratic state. This is a topic I’m interested in and have covered on the blog before.

I’m currently preparing an article on this topic (as a spinoff from my own PhD research), and I’m interested to see whether Aikin and Talisse’s book has usurped me by making the argument I want to make. Since the book has only just been released, and will probably take several weeks to reach me here in Ireland, I’ve decided to go through the following article by one of the book’s co-authors:

• “The Problem of Worship” Summer 2010, Think, 101-113

The article makes the argument that God cannot exist because there can be no being who is a proper object of worship (which is what God is supposed to be). Along the way to reaching that conclusion, Aikin makes some interesting comments about the nature of moral agency and the nature of worship.

In this post, I try to summarise and reconstruct the parts of Aikin’s article that focus on worship and the obligations of moral agency. I do not really focus on the claim that God does not exist. This is for an important reason: Much and all as I enjoyed Aikin’s article, I had some serious problems when I tried to reconstruct the argument that he offered. I doubt readers will be interested in these problems, but if they are, they might like to read this exegetical note. Anyway, as a result of my problems, I have deemed it appropriate to reconstruct large parts of what he had to say.


1. God and Worship
We begin with some propositions about the nature of God and worship. The first proposition is:

• (1) If God exists, then he is a proper object of worship.

This statement seems relatively uncontroversial. As far as I am aware, most theists think that God is worthy of worship. Indeed, many structure their lives around the worship of God. Later, I will suggest that there are two types of worship and that ultimately God is only an object of one of them.

For now, we move on to the question: what does it mean for something to be a “proper object of worship”? Aikin suggests the following answer:

(2) For any object X, if X is the proper object of worship, then all rational moral agents are either:

• Obliged to worship X;

• If they worship, they are obliged to worship only X.

Aikin reckons that 2(a) is more typical of the religious attitude toward God, but 2(b) allows a chink of light for those with a more liberal attitude to worship. 2(a) seems more typical because it is often claimed (by some Christians, at any rate) that those who do not worship are somehow guilty of a moral crime.


2. Rational Moral Agency and Worship
We turn now to consider what the act of worship actually entails and whether it is compatible with the demands of rational moral agency. This is where my first major deviation from Aikin’s presentation occurs. To make sense of his argument, I find it necessary to distinguish between two types of worship: (i) conditional-worship or c-worship; and (ii) unconditional-worship or u-worship.

The act of c-worship can be said to look something like this:

(3) For any rational moral agent (A), if A c-worships X, A’s c-worship consists of three elements: 

• (a) Conditional obedience to X and the demands that X’s existence and properties place on A. 

• (b) Belief that X is worthy of worship; 

• (c) Performance of communicative and ritual acts expressing (a) and (b).

The act of u-worship differs from this in one respect only: the obedience is unconditional (we’ll give the proposition stating u-worship the number 4). As will be seen, part of the argument here is that the worship of God could only come in an unconditional form. This is bad because it violates the demands of rational moral agency.

To be a rational moral agent is, by definition, to exercise autonomy over one’s moral life, i.e. to not be ultimately under anyone else’s authority. It is, however, rational to be conditionally obedient or deferent to another being. For example, I am conditionally obedient to the officers of the law in my home state. But I reserve the right to retract this obedience.

I could not be unconditionally obedient to another being. This would be tantamount to a denial of my rational moral agency. It it would involve relinquishing my moral autonomy. So:

• (5) Unconditional submission, obedience or deference to another being contravenes the requirements of rational moral agency.

Now we are ready to make some arguments.



3. Argument: It is not reasonable to believe that God is a proper object of c-worship
The first argument claims that God cannot reasonably be believed to be a proper object of c-worship. We begin by stating the conditions under which c-worship would be reasonable:

• (6) If A is to reasonably believe that X is a proper object of c-worship, then A must reasonably believe that X has some property (W) that makes X worthy of c-worship.

This principle is an internalist and evidentialist one. It states that it is never reasonable to accept a conclusion about X’s worthiness of worship without having some evidence or internal knowledge of a W-property to support that conclusion. Non-evidentialists or externalists might be inclined to dispute this principle, but it seems unimpeachable to me. U-worship would not depend on this evidentialist/internalist premise.

We follow this principle with a claim:

• (7) God has no properties that would make him worthy of c-worship.

And this leads to the obvious conclusion:

• (8) A cannot reasonably believe that God is a proper object of c-worship.

The argument is sound, but in need of considerable support. Premise (7) is the key. What might encourage us to think that God has no properties that make him worthy of c-worship? Well, if we consider all the properties that are traditionally ascribed to God (separately and jointly) and it turns out that none of them would warrant conditional obedience to him, then the claim might be justified.

So let’s go through all of these properties.

We can take the properties of necessity, immateriality and personality as a group. It is pretty clear that none of these properties would warrant even conditional worship. For example, I am not obedient to other human beings simply because they are persons, they have to earn obedience in other ways. Similarly, immateriality and necessity seem insufficient by themselves to warrant worship or conditional obedience.

Maybe God deserves c-worship for creating us and the world in which we live? The analogy often given here is that of giving thanks to someone who gives you a gift. Two things need to be said about this. First, giving thanks is not equivalent to worship (of either variety). Second, creation seems to be a pretty mixed-bag: some parts of it are good, but others a genuinely horrid. It would be difficult to say that a being deserved worship for providing us with such a mixed-blessing.

How about omnipotence? As has been noted by others, submission to an all-powerful being simply in virtue of the fact that it is all powerful seems to be akin to accepting a fascistic, or totalitarian authority over your life. This would not be compatible with the strictures of moral agency.

Maybe omniscience is a better candidate? Deference to somebody who knows everything seems rational, right? Not so, says Aikin. He makes an interesting argument here. He suggests that any being who deserves our conditional worship would have to respect our right to privacy. After all, the right to think one’s own thoughts is a core component of moral agency. But a being who knows everything would not respect your right to privacy and so would not deserve c-worship.

Then there’s omnipresence. This is even worse than omniscience. A being who was always with you, who never let you alone, would be a harasser. It would not respect your right to be let alone. As such, it could never deserve c-worship.

Perhaps the best candidate property is omnibenevolence. After all, obedience to a being that is perfectly good would seem compatible with our moral agency. Well, perhaps, but then we get into a problem about worshipping a specific being or worshipping an ideal of moral goodness. Worshipping the ideal would not bring with it obedience or the ritual acts of worship. And if we are worshipping a specific being then there is the problem of knowing whether the being truly is truly omnibenevolent. Evidence from scripture and from the problem of evil should give us pause for thought on this front. These problems are often resolved by appeals to faith or to the limitations of human knowledge, but if we were to accept these solutions we would be heading into u-worship territory, i.e. worship without an evidential basis.

According to the Bible, God commands us to worship him and him alone. It’s right there in the first commandment. Should we therefore worship him because he commands us to do so? Obviously not. You would not worship me if I commanded you to do so, and you would be right not to do so.

This raises another possibility though. Perhaps a combination of omnibenevolence and commandment would suffice for c-worship? If God is all-good, then God is a reliable source of moral instruction. And if one of his instructions is to worship him, why not follow it? The response is that a being who commands us to worship him would seem, by that very act, to lose its claim to moral perfection. To command us to worship would be a petulant, self-aggrandising action, not one we would expect of a morally perfect being.

Finally, maybe God deserves worship because he is the redeemer, saver and the keeper of the keys to eternal life? This won’t do either, these properties might make it prudent to worship God, but they do nothing to say that God is worthy of worship.

In conclusion, none of the properties traditionally ascribed to God make him worthy of c-worship. This allows us to construct the following argument in support of (7), above:

• (9) God is traditionally said to have the following set of properties: (i) necessity; (ii) immateriality; (iii) personality; (iv) creator; (v) omnipotence; (vi) omniscience; (vii) omnipresence; (viii) omnibenevolence; (ix) commander; (x) redeemer.

• (10) None of the properties (i) - (x) make (or should make one think) God a proper object of c-worship.

• (7) Therefore, God has no (or should not be thought to have) properties that make him worthy of c-worship.

This conclusion might seem a little strong. It could, after all, be the case that God has other properties, unknown to us, that make him worthy of worship. But to worship him on the basis of unknown properties would amount to u-worship, not c-worship. If the believer wants to appeal to such properties in support of c-worship, then they need to produce the goods and tell us what they are.



4. Argument: We are obliged not to worship God
The next argument follows on from the previous one. It could proceed in one of two ways. I’m going to outline a shorter version of it. This is quite different from the version presented by Aikin.

The argument comes in two stages. The first looks like this:

• (11) If it is not reasonable to believe X worthy of c-worship, then it would only be possible to u-worship X.

It is not reasonable to believe God worthy of c-worship.

• (12) It is not reasonable to believe God worthy of c-worship.

• (13) Therefore, it would only be possible to u-worship God.

We then add to this argument a premise drawn from our earlier claims about what is and is not compatible with the demands of moral agency:

• (14) Rational agents are obliged not to u-worship any being (from 5).

The idea here is, obviously, that unconditionally surrendering to another being would be contrary to one’s obligations as a moral agent. This is a strong claim and it has interesting implications:

• (15) All rational moral agents are obliged not to u-worship God (from 13 and 14).

And since we have already (2) said that worship involves communicative and ritual acts, it would follow that:

• (16) All rational moral agents are obliged not to engage in communicative or ritual acts of u-worship.

This is a conclusion that could have some interesting political and legal consequences, to say the least.

5. Argument: God (probably) does not exist
The final conclusion of Aikin’s article is that God does not exist. It is difficult for me to reach a similar conclusion given my reformulation of what Aikin had to say. Particularly, my effort to draw a distinction between two varieties of worship would weaken the argument significantly to one about the inadequacy of current conceptions of God.

Rather than give my own argument, I will offer a simplified version of the one Aikin offers:

• (1) If God exists then he is a proper object of worship.
• (2) If X is a proper object of worship, then A would be obliged to worship X.
• (3) There is no being whom A is obliged to worship (from preceding argumentation).
• (4) Therefore, there is no being who is a proper object of worship.
• (5) Therefore, there is no God.

I’ll look at James Rachels version of this same argument in another post. That’s it for now.

Exegetical Note: Aikin's "The Problem of Worship"

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I like arguments. I particularly like arguments with a formal structure made up of a numbered propositions and inferential relations. As is apparent from other posts on this blog, I also like constructing maps that represent the inferential relationships between the numbered propositions.

Given all these likes, one would think that Scott Aikin’s article “The Problem of Worship” would be right up my alley. After all, it consists of 15 numbered propositions, linked together by a series of inferential relationships. Alas, the article is not entirely satisfactory because the formal argument that Aikin presents suffers from a number of deficiencies.

I think there are three main reasons for these deficiencies (i) typographical errors result in conclusions being drawn from the wrong premises; (ii) inconsistent use of terminology leads to conclusions that are unrelated to the premises; and (iii) two major conclusions are reached which, although related, are distinct.

I’ll give examples of each of these problems below. On the whole, I don’t think any of these deficiencies substantially detract from the persuasiveness of Aikin’s article. I wouldn’t discuss it if I thought it without merit. But they do, I think, justify my attempt to reconstruct his argument rather than directly summarise it.


Problem One: Typographical Errors
It is common in philosophical argumentation to state from which premises one is drawing a particular conclusion. Aikin does this at several points in his article (conclusions 9, 10, 11, 13, 14 and 15). The problem is that, on two of these occasions, the premises from which the conclusions are drawn are misidentified. The first example is (10) which states the following (p. 110):

• (10) Therefore, A is not obliged to hold any object to be worthy of worship (from 6 and 9).

The problem is that premises (6) and (9) state the following, respectively:

• (6) If A has no reason to see any property W of X as making X worthy of worship then A is obliged not to worship X.

• (9) Therefore, there are no W-properties any A may reasonably see as making any x worthy of worship.

The difficulty this poses should be apparent. Although (9) is fine, (6) is a principle stating that A is under a positive obligation (duty) not to worship X in certain circumstances. But the conclusion reached is that A is not obliged to hold any object to be worthy of worship (i.e. not worshipping is morally permissible). This is not a conclusion about A’s positive obligations so could not be drawn from premise (6).

The problem is easily corrected. (10) can be drawn from a combination of (5) and (9) because (5) says the following:

• (5) If A is obliged to view X as worthy of worship, then A must have reason to see some property W of X as making X worthy of worship.

Given the facile nature of this correction, I presume the problem is the result of typographical error.

The same basic problem occurs again in relation to (11), which states:

• (11) Therefore, A is obliged not to hold any object to be worthy of worship (from 7 and 9).

This can’t be right because (7) is principle stating when it is reasonable to believe that an entity is worthy of worship, not a principle stating what our positive obligations are with respect to worship. Once again this problem is easily corrected. Premise (6) does state a principle relating to positive obligations, so (11) can be drawn from a combination of (6) and (9).

There is another typographical error in the conclusion when Aikin refers to premises (8) and (9) in relation to evidentialism. I presume he meant to refer to premises (5) and (6).


Problem Two: Inconsistent Use of Terminology
The correction of (11) does highlight another problem with Aikin’s article. As can be seen above, (11) is a conclusion about A’s obligations to hold an entity worthy of worship, whereas (6) is a premise about obligations to worship in general. So even with my correction, (11) cannot properly be drawn from (6). A more general conclusion can reached, viz. “A is obliged not to worship any x”.

There is a way to solve this problem. It requires that we go back to premise (3) which defines what is involved in the act of worship:

(3) For any rational moral agent A, if A worships x, A’s worship is the joint performance of three acts:  

• (a) A is unconditionally obedient to X and to the demands that X’s existence and properties place on A;  

• (b) A views x as absolutely worthy of worship;   

• (c) A performs rituals or communicative acts expressing 3(a) and 3(b)

As can be seen, believing x to be worthy of worship is part and parcel of the act of worshipping, so it would be implied by reaching the more general form of (11). Of course, we could repair this in another way by going back to (6) and changing the principle so that it is restricted to “obliged not to hold x worthy of worship”. The problem is that this approach seems unnecessary.

Allowing for the more general form of (11) does create some downstream problems. In particular it means that the additional argumentation that Aikin offers for (13) is redundant since (13) just states that rational moral agents are obliged not to worship any entities - a conclusion that will have already been reached by (11). I think Aikin is aware of this himself since just after stating (11) in the narrow form, he states (p. 110) “given that we shouldn’t worship God (on 11)”. This is clearly a general claim about worship not a claim about worthiness of worship.

There is another problem arising from terminology. This one relates to the use of the word “unconditional” in premise (3), given above. The inclusion of the word suggests that Aikin thinks the act of worship necessarily involves unconditional obedience. Aikin follows this up by stating (well, to be fair, he does offer an argument) that:

• (4) Unconditional submission to any authority contravenes the requirements of moral agency.

If we accept this premise (and I think we probably should), then it automatically follows that we should not worship any being. Why? Because Aikin has already defined worship as including unconditional obedience (I am assuming obedience and submission are interchangeable).

This is weird because Aikin goes on to stress that conditional obedience is allowable. He then considers whether God possesses any properties that would make him worthy of conditional obedience (he actually refers to worship, not obedience, further muddying the waters). This argumentation seems unnecessary if he thinks worship involves unconditional obedience. Only if he thinks some kind of conditional-obedience-based worship were permissible would it be necessary to consider those arguments.

Thus I would be inclined to correct premise (3) by removing the term unconditional from the definition of worship. This would leave premise (4) intact.



Problem 3: What is the Argument?
A final problem I have with Aikin’s argument is that it seems to mix-up related but distinct arguments and avoid making necessary arguments.

One of these arguments, probably the main one, leads to the conclusion that God does not exist. This conclusion is backed up by (roughly) the following chain of reasoning: there is no being that we are obliged to worship; God is supposed to be worthy of obligatory worship; so there must not be a God.

[Note: This chain of argumentation is problematic given the repairs undertaken above. In particular, if we drop the “unconditional obedience” clause from our definition of worship, then it is possible that there might be some entities who are worthy of worship. Thus we will have to offer an argument more like the following: God is supposed to be worthy of obligatory worship; None of God’s properties make him worthy of obligatory worship; so there must not be a God (at least going by present descriptions of his properties).]

There is also another significant argument about the practical or social implications of the fact that no rational moral agent is allowed to unconditionally surrender to another being. The argument is that since surrender to God would be unconditional, and would involve communicative and ritual acts of worship, it must be the case that communicative and ritual acts of worship are impermissible. Aikin makes this point in premise (12).

I think this is really interesting, but I think it is distinct from the main argument that Aikin is making and so is worth separating out from that argument. Maybe this desire to keep these points separate is just a personal quirk of my own. I am, after all, primarily interested in the socio-political aspects of what Aikin has to say.

A final problem arises from my confusion over whether Aikin is making an argument that is specifically about God or about all possible beings X. The conclusions that he reaches are certainly phrased in terms of “all possible beings X” (see his 9, 10, 11, 13 and 14), but the argumentation he offers focuses on God, not all possible beings.

This problem comes to the fore when Aikin infers (9) from (7) and (8). (9) was stated above and is clearly about the properties possessed by all possible beings. But (7) and (8), and the preceding paragraphs, are clearly about God’s properties. It seems then like an additional argument is needed which would run along the following lines:

• (a) If any being is worthy of (unconditional) worship, then it is God.
• (b) None of God’s properties make him worthy of (unconditional) worship.
• (c) Therefore, it is unlikely that any being is worthy of (unconditional) worship.

This conclusion (c) would then effectively be equivalent to Aikin’s (8).


Conclusion
This exegetical note has been long and, perhaps, uninteresting. I want to be very clear that despite my problems, I enjoyed Aikin’s article and found much of value in it. I’m also aware that I have limited experience and credentials in the great game of philosophy. It may be that Aikin’s argument is coherent and that the problems I have identified are products of my own misunderstanding.

If that is the case, I am willing to be corrected.

The Possibility of Historical Evidence for Miracles (Part 3)

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This post is part of my series on miracles. For an index, see here.

I am currently working my way through the following article:

• Luck, M “Against the Possibility of Historical Evidence for Miracles” (2005) 44 Sophia 7

In this article, Morgan Luck assesses Richard Swinburne’s argument claiming that there can be strong historical evidence for miracles. Luck notes that Swinburne’s argument relies on three key assertions. We have evaluated the first two of these assertions already. Now we turn our attention to the third:

• Assertion 3: There are conditions under which it is reasonable to assume a violation of a law of nature was caused by God

For the purposes of this evaluation, we are going to assume that we actually do have strong evidence for the occurrence of some kind of miracle. The question we now face is whether we can rightly infer that God is the cause of that miracle.

Swinburne concedes that we would never see God directly causing some miraculous effect. So we would have to make some kind of plausible inference from the effect to the existence of God. He thinks there are two methods for doing this.


1. Argument from Analogy
The first method is to make some kind of analogy between human intentional action and divine intentional action. In other words, we could say that a given miraculous effect seems to be the kind of thing that an agent would bring about (because it is similar to other things that agents like to bring about), but since it involves a violation of a law of nature it cannot be attributed to a human agent; therefore, it must be attributable to a supernatural agent.

To give an example, Swinburne asks us to imagine that 50-foot high fiery lettering appears in the sky reading “I caused these words to appear and I am a rational agent.” He then argues that we would be warranted in inferring that a rational agent caused the lettering to be.

That may be true, but does it warrant us in inferring that God caused it to be? Luck thinks not. He argues that the occurrence of such a violation of a law of nature, that is nonetheless typical of rational agency, is consistent with two other hypotheses: (i) a natural rational agent causing the event by natural means and (ii) a natural rational agent causing the event by means of an intermediary non-natural event.

The first of these hypotheses may elicit a raised eyebrow or two from the reader, but it is perfectly consistent given the definition of a law of nature with which Swinburne is working. As you no doubt recall from part one, for Swinburne a law of nature is a generalised axiom that achieves the best combination of simplicity and strength when applied to the available data.

Under that definition, it is perfectly possible for a natural rational agent to cause a violation of a law of nature by natural means. Presumably, this would just involve the rational agent making use of some as-yet-unknown natural process. Arthur C. Clarke’s famous comment that future technology would be indistinguishable from magic seems apposite here.

So it’s back to the drawing board for Swinburne. On his return, he may stipulate that when making inference to the existence of God, he is assuming that we have evidence that the miraculous event was caused by non-natural means.

One wonders what this evidence would look like, but let’s grant Swinburne the possibility. Even then, Luck argues, he would not be warranted in inferring God. It is, once again, perfectly possible for a natural rational agent to cause the event through a non-natural intermediary process. The possibility being invoked here is logical, not physical or metaphysical, but it is still enough to defeat Swinburne’s argument.


2. The Argument from Prior Belief
Swinburne can bounce back from this defeat. He can argue that we have prior evidence for the existence of God, and that this prior evidence gives us some indication of the kinds of things God would like to bring about. Combining this evidence with the prima facie occurrence of a miracle, might allow us to infer the existence of God.

This is certainly true, but it weakens the apologetic importance of the argument from miracles. No longer are we using the occurrence of a miracle to support the existence of God. Instead, we are using the existence of God to shape our interpretation of a prima facie miracle.

Even if we are willing to tolerate this approach to miracles, problems remain for Swinburne. Two are of particular importance.

First, it is still possible for the event to be attributable to a natural rational agent. Why so? Well, it all comes from the problem of deception. A natural agent, knowing the character of God, could try to mimic him by causing some apparently miraculous effect. (I covered this kind of objection previously in relation to a non-natural event).

Second, go back to part two and look at the discussion of the principle of causal closure (PCC). As noted there, even if we have good prima facie evidence for the occurrence of a miracle, the violation of the PCC makes it virtually impossible to know which particular miracle occurred. This makes it very difficult to know what kind of character is being revealed by a miraculous event.


3. Conclusion
That completes Luck’s evaluation of Swinburne’s case for historical evidence of miracles. We can summarise as follows:

• Assertion One - "A violation of a law of nature is possible": Luck agrees with this assertion. Adopting Lewis’s Systematic theory, a law of nature is a generalised axiom, from which conclusions about natural events can be deduced, with the best combination of simplicity and strength. It is possible for there to be non-repeatable counter-instances to such laws.
• Assertion Two - "The existence of particular present events could be indicative of the prior occurrence of particular violations of the laws of nature": Luck objected to this on the grounds that once we allow for the violation of the PCC, evidence can be consistent with any number of miracles.
• Assertion Three - "There are conditions under which it is reasonable to assume that a violation of a law of nature was caused by God": Luck objects to this on the grounds that it is possible for miracles to be caused by natural rational agents.