The Mere Addition Paradox

In this post, I formulate the Mere Addition Paradox, a paradox in so-called "population ethics". I would like to know what others think of the paradox.

Consider the following three possible populations:
Population A: A population of 10,000 people each of whose level of well-being is +100,000
Population A+: A population of 20,000 people, 10,000 of whom have a level of well-being of
+100,000 and the other 10,000 of whom have a level of well-being of +90,000
Population B: A population of 20,000 people each of whose level of well-being is +95,000
(I should explain what I mean by "well-being" here. The well-being of an individual is simply how well things go for that individual. The well-being of one individual is greater than the well-being of another individual just in case things go better for the first individual than they do for the second individual.)

There are then two versions of the Mere Addition Paradox:

Mere Addition Paradox (Version 1)
1. The existence of Population A+ is better than the existence of Population A.
2. The existence of Population B is better than the existence of Population A+.
3. If (1) and (2), then the existence of Population B is better than the existence of Population A.
4. Therefore, the existence of Population B is better than the existence of Population A.

Mere Addition Paradox (Version 2)
1*. The existence of Population A+ is just as good as the existence of Population A.
2. The existence of Population B is better than the existence of Population A+.
3*. If (1) and (2), then the existence of Population B is better than the existence of Population A.
4. Therefore, the existence of Population B is better than the existence of Population A.

Premise (3) is justified by appeal to the transitivity of being better than: For all x, y, and z, if x is better than y and y is better than z, then x is better than z. Premise (3*) is justified by appeal to the slightly different principle that for all x, y, and z, if x is just as good as y and z is better than y, then z is better than x. On the assumption that both of these principles are true, the two versions of the Mere Addition Paradox show that whether we accept premise (1) or premise (1*), so long as we accept premise (2) we get the conclusion that the existence of Population B is better than the existence of Population A.

I have already stated why one might accept premises (3) and (3*). Why might one accept premises (1), (1*), and (2)?

Let's begin with (2). The reasoning for two goes as follows: The existence of Population B is better than the existence of Population A+ because the average level of well-being does not differ between the two populations and Population B is more equitable than Population A+. Assuming that equality is a good-making feature of a population, then, the existence of Population B is better than the existence of Population A+.

What about premises (1) and (1*)? Few have attempted to argue for one of these over the other, preferring to argue instead for disjunction. But arguing for their disjunction is good enough, since which of the disjuncts is true doesn't matter for the success of the argument. If (1) and (2) are true, then (given (3)), the conclusion follows and if (1*) and (2) are true, then (given (3*)), the conclusion follows. So, why believe the disjunction of premises (1) and (1*)? Well, it seems that merely adding some people with high well-being to a population each of whose members has even higher well-being can't make the existence of the resultant population worse than the existence of the original. After all, presumably the existence of people with high well-being is a good thing and so cannot detract from an already good thing. Thus, either the existence of Population A+ is better than the existence of Population A or the existence of Population A+ is just as good as the existence of Population A.

Let me end by saying why the Mere Addition Paradox is supposed to be paradoxical. Consider Population B+, a population of 40,000, 20,000 of whom have a level of well-being +95,000 and 20,000 of whom have a level of well-being of +85,000. By reasoning similar to the reasoning in favor of the disjunction of premises (1) and (1*), the existence of Population B+ is either better than or just as good as the existence of Population B. But then, by reasoning similar to the reasoning in favor of premise (2), the existence of Population C (a population of 40,000 people each of whose level of well-being is +90,000) is better than the existence of Population B+. So, by reasoning similar to the reasoning in favor of premises (3) and (3*), the existence of Population C is better than the existence of Population B and hence, since the existence of Population is better than the existence of Population A, the existence of Population C is better than the existence of Population A. By repeated applications of the same sort of argument, then, we reach the conclusion (sometimes called "The Repugnant Conclusion") that the existence of a large population each of whose members has the same very low positive level of well-being is better than the existence of Population A.

So, what do you all think of this paradox? What should one say to the arguments? Should one accept that the existence of a very large population each of whose members has a very low positive level of well-being (described by some as having a life barely worth living) is better than the existence of a smaller population each of whose members has a very high level of well-being? If not, what should one say against the reasoning in favor of that conclusion?

[I should note before closing that the premises and conclusion of the arguments should probably include ceteris paribus clauses. That is, we are wondering about whether the existence of Population B is better than the existence of Population A assuming that the only relevant difference between the two populations is how many people they contain and the level of well-being each person has. I should also note that if you want to find out more about the Mere Addition Paradox and potential solutions to it, you should look at the SEP's entry entitled "The Repugnant Conclusion": http://plato.stanford.edu/entries/repugnant-conclusion/]

Is looking the same as transitive?

Is the following thesis true?:

Transitivity of Looking the Same As: For all w, x, y, and z, if x looks the same as y to w and y looks the same as z to w, then x looks the same as z to w.

Many philosophers have denied this thesis. However, the following argument appears to be a compelling argument in its favor:
1. Let w, x, y, and z be arbitary individuals and suppose that x looks the same as y to w and y looks the same as z to w.
2. For all x, y, and z, x looks the same as y to z iff the way x looks to z is the same as the way y looks to z.
3. For all x, y, and z, the way x looks to z is the same as the way y looks to z iff the way x looks to z=the way y looks to z.
4. The way x looks to w=the way y looks to w and the way y looks to w=the way z looks to w. [From (1),( 2), and (3)]
5. For all x, y, and z, if x=y and y=z, then x=z.
6. The way x looks to w=the way z looks to w. [From (4) and (5)]
7. x looks the same as z to w. [From (2), (3), and (6)]
8. For all w, x, y, and z, if x looks the same as y to w and y looks the same as z to w, then x looks the same as z to w. [(1)-(7), conditional proof]

What, if anything, should one say to this argument in favor of the Transitivity of Looking the Same As? Should we accept its conclusion? Deny one of its premises? If deny one of its premises, which one?

[Delia Graff Fara seems to present such an argument in her "Phenomenal Continua and the Sorites", available at http://www.princeton.edu/~graff/papers/mindcontinua.pdf]

God and Ethics

Some theists seem to think that there is an important connection between God and ethics, although those who are not philosophically trained often have a difficult time articulating what they take the connection to be.

One suggestion is that these theists accept some form of divine command theory (DCT) according to which the property of being right just is the property of having been commanded by God. However, this form of DCT is problematic. There are, for instance, Euthyphro-type worries. And, in addition, there is the worry that the property of being right and the property of being commanded by God are not necessarily extensionally equivalent, and so not identical. Whether or not one is persuaded by these objections, it is interesting to explore other options concerning what the connection between God and ethics might be.

Another suggestion is that whereas theists can be motivated to do what is right, atheists cannot. Unfortunately, this suggestion fails empirically. There clearly are atheists who are motivated to do what is right, and thus can be motivated to do what is right.

This leads me to my suggestion concerning what such theists should say: Whereas theists are rational in being motivated to do what is right, atheists are not rational in being motivated to do what is right. (This suggestion presupposes that it makes sense to say that someone is rationally motivated to do something. Those who do not agree needn't read any further.) Let's flesh out the suggestion. Suppose that we believe that people are sometimes rational in being motivated to do something. We might adopt an analogue of foundationalism for rational motivation according to which some of our motivations, our basic motivations, are simply rational regardless of their relation to other motivations but that other motivations are rational because they are supported by our basic motivations. If we adopt this view, which we might call "motivational foundationalism", a theist might say the following: There is a basic motivation to avoid suffering, but there is not a basic motivation to do what is right. Theists may be rational in being motivated to do what is right because they are rationally motivated to avoid suffering and they believe (rationally?) that if they do what is right they will avoid suffering (as they will avoid Hell). However, atheists cannot be rational in being motivated to do what is right because there are no basic motivations and beliefs that they have that jointly make it rational for them to be motivated to do what is right. So, whereas theists are rational in being motivated to do what is right, atheists are not.

It seems to me that this is an interesting suggestion concerning what a theist might mean when he or she claims that there is an important connection between God and ethics. It does not suffer from the same worries that DCT and the claim that atheists cannot be motivated to do what is right suffer from. On the other hand, it is not very well-developed. To develop it a theist would have to give accounts of rational motivation and of basic motivation that supports his or her claim that an atheist cannot be rationally motivated to do what is right. Regardless, I take it that this would be an interesting undertaking for a theist who was convinced that there is an important connection between God and ethics. Not only that, but any motivational foundationalist, whether a theist or an atheist, should be interested in giving a theory of basic motivation. And if that motivational foundationalist is an atheist, then he or she should be interested in explaining how an atheist can be rationally motivated to do what is right.

Arguments and Evidence

The following principle seems plausible:

Arguments and Evidence (AE): Necessarily, for all x and y, if x is a valid argument and y has evidence for each of x's premises, then y has evidence for x's conclusion.

After all, we are concerned with valid arguments because we are concerned with evaluating evidence for and against different claims. Using valid arguments helps us to determine whether we have evidence for their conclusions, and the reason for this seems to be that valid arguments are evidence-preserving: If we have evidence for the premises of a valid argument, then we have evidence for its conclusion.

However, (AE) also seems to have implausible consequences. For instance, suppose that, at time t1, Alan's total evidence supports each of the premises of a modus ponens argument:
1. P
2. If P, then Q.
3. Therefore, Q.
Then, at t2, Alan gains some additional evidence without losing any of his previous evidence and that the evidence he has gained makes it the case that, at t2, his total evidence, while still supporting that if P, then Q, supports the negation of P. In addition, the evidence Alan gains at t2 does not support the negation of Q. Surely such a case is possible. But if (AE) is true, then it follows that, at t2, Alan has evidence in favor of Q, since he still has evidence in favor of P and if P, then Q, since he has not lost any of the evidence he previously had. Making certain other plausible assumptions, it also follows that Alan has just as much evidence in favor of Q at t2 as he had at t1. And this seems implausible. Are we then to reject (AE)?

One suggestion would be that we should reject (AE) and replace it with:

Arguments and Total Evidence (ATE): Necessarily, for all x and y, if x is a valid argument and y's total evidence supports each of x's premises, then y has evidence for x's conclusion.

However, (ATE) cannot do all of the explanatory work that (AE) did. For instance, consider the following argument:
1. The fact that the physical constants allow for the existence of life is more probable given theism than given atheism.
2. If (1), then theism is true.
3. Therefore, theism is true.
I take it that I have some evidence in favor of each of the premises of this argument and that because of this, I have some reason to believe its conclusion. However, I also believe that my total evidence supports the negation of premise (2). So, (ATE) does not help to explain why I have evidence in favor of the conclusion of this argument, whereas (AE) does. Thus, (ATE) cannot do all of the explanatory work (AE) did.

Let me end with some questions: Is (AE) false, as I have suggested? Are the reasons I have given to think that (AE) is false convincing? If (AE) is false, can we replace it with a principle that does not have its implausible consequences and yet still explains my situation with respect to the fine-tuning argument mentioned above and other such situations? More generally, what is the relationship between valid arguments and evidence?

Sextus Empiricus', I mean Hume's, Problem

Okay so I apologize for the epistemology, but here comes some. Recall Hume's problem of induction: Either principles of induction are justified inductively or deductively. Principles of induction (like x% of observed As are Bs so x% of unobserved As are Bs) are not necessary truths, so they are not justified deductively. But justifying them inductively is viciously circular; so they are not so justified.

One can extend this argument as follows: If inductive principles are not justified, then nothing is justified inductively. But deduction is justified either inductively or deductively. And it's not justified deductively since that would be circular. So deduction is not justified.

Rich considers the former, but not the latter, argument in his book. His take on it is that inductive principles like the above are not justified inductively or deductively. But there are necessary a priori truths like 'knowing that x% of observed As are Bs gives one a reason to believe that x% of unobserved As are Bs' that justify inductive inferences. One can then block the extended argument by denying that nothing is justified inductively.

I am attracted to the claim that there are a priori truths that justify inductive inferences. What worries me is that it is not very plausible to suppose that deduction is justified inductively. So it seems that, ultimately, pressure must be put on the 'that's circular' part of the argument. But circular reasoning is bad, right?

It strikes me that there are a couple of options here: distinguish good and bad circularity and say why the justification of deduction involves the former, hold that any justified deductive principle stands in an infinitely long chain of reasons, or hold that there are justified basic beliefs that justify deductive principles (maybe those principles themselves are such beliefs). But these are exactly the options at play in the familiar infinite regress argument concerning the structure of justified beliefs--the only difference is that the familiar arguments focus on empirical beliefs.

I think foundationalism is just as plausible in the a priori case as in any other. But I won't plump for that here. I am more interested in whether the following are correct:

(A) Coherentism is an attempt to distinguish good and bad circularity for empirical beliefs. But coherentism seems especially implausible when brought to bear on a priori beliefs about principles of induction and deduction. This is because conditions on coherence typically include things like a requirement that the relevant propositions "jointly probabilify" each other and that they are logically consistent.

More interestingly:

(B) At bottom, Hume's problem of induction does not raise any problem not solved by a sufficiently adequate response to the infinite regress problem. It's really just a special case of that problem--not a fundamentally different one, as has been traditionally supposed.

Word?