Modal details in Unger's argument against his existence

Unger famously argues that he doesn’t exist, by claiming a contradiction between three claims (I am quoting (1) and (2) verbatim, but simplifying (3)):

1. I exist.

2. If I exist, then I consist of many cells, but a finite number.

3. If I exist and I consist of many but a finite number of cells, then removal of the least important cell does not affect whether I exist.

Unger then says:

these three propositions form an inconsistent set. They have it that I am still here with no cells at all, even while my existence depends on cells. … One cell, more or less, will not make any difference between my being there and not. So, take one away, and I am still there. Take another away: again, no problem. But after a while there are no cells at all.

But taken literally this is logically invalid. Premise (2) says that I consist of many but a finite number of cells. But to continue applying premise (3), Unger needs that premise (2) would still be true no matter how many cells were taken away. But premise (2) does not say anything about hypothetical situations. It says that either I don’t exist, or I consist of a large but finite number of cells. In particular, there are no modal operators in (2).

Now, no doubt this is an uncharitable objection. Presumably (2) is not just supposed to be true in the actual situation, but in the hypothetical situations that come from repeated cell-removals. At the same time, we don’t want (2) to be ad hoc designed for this argument. So, probably, what is going on is that there is an implied necessity operator in (2), so that we have:

4. Necessarily, if I exist, then I consist of many cells, but a finite number.

The same issue applies to (3), since (3) needs to be applied over and over in hypothetical situations. Another issue with (3) is that to apply it over and over, we need to be told that removal of the cell is possible. So now we should say:

5. Necessarily, if I exist and I consist of many but a finite number of cells, then removal of the least important cell is possible and does not affect whether I exist.

Now, I guess, we can have a valid argument in S4.

Is this a merely technical issue here? I am not sure. I think that once we’ve inserted “Necessarily” into (4) and (5), our intuitions may start to shift. While (2) is very plausible if we grant the implied materialism, (4) makes us wonder whether there couldn’t be weird situations where I exist but don’t consist of many but a finite number of cells. First, it’s not obviously metaphysically impossible for me to grow an infinitely long tail? That, however, is a red herring. The argument can be retooled only to suppose that I necessarily have many cells and I actually have a finite number. But, second, and more seriously, is it really true that there is no possible world where I exist with only a few cells? In fact, perhaps, I once did exist with only a few cells in this world!

Similarly about (5). It’s clear that right now I can survive the loss of my least important cell. But it is far from clear that this is a necessary truth. It could well be metaphysically possible that I be reduced to some state of non-redundancy where every cell is necessary for my existence, where removal of any cell severs an organic pathway essential to life. I would be in a very different state in such a case than I am right now. But it’s far from clear that this is impossible.

Perhaps, though, the modality here isn’t metaphysical modality, but something like nomic modality. Maybe it’s nomically impossible for me to be in a state where every cell is non-redundant. Maybe, but even that’s not clear. And it’s also harder to say that the removal of the least important cell has to (in the nomic necessity sense) be nomically possible. Couldn’t it be that nomically the only way the least important cell could be removed would be by cutting into me in ways that would kill me?

Furthermore, once we’ve made our modal complications to the argument, it becomes clear that of the three contradictory premises (1), (4) and (5), premise (1) is by far the most probable. Premise (1) is a claim about my own existence, which seems pretty evident to me, and is only a claim about how things actually are now. Premises (4) and (5) depend on difficult modal details, on how things are in other worlds, and on metaphysical intuitions that are surely more fraught than those in the cogito.

(One of the things I’ve discovered by teaching metaphysics to undergraduates, with a focus on formulating logically valid arguments, is that sometimes numbered arguments in published work by smart people are actually quite some distance from validity, and it’s hard to see exactly how to make them valid without modal logic.)

Sexual symmetry and asymmetry

I want to think a bit about conservative Christian views of sex and gender, but before that I want to offer two stories to motivate a crucial distinction.

Electrons and Positrons

Electrons and positrons (a positron is a positively charged anti-particle to the electron) are very different in one way but not so much in another. If you take some system of electrons and positrons, and swap in a positron for an electron, the system will behave very differently—it will be attracted to the things that the electron was repelled by and vice versa. But if you replace all the electrons by positrons and all the positrons by electrons, it won’t make a significant difference (technically, there may be some difference due to the weak force, but that’s dominated by electromagnetic interaction). Similarly, a cloud of electrons behaves pretty much like a cloud of positrons, but a mixed cloud of electrons and positrons will behave very differently (electrons and positrons will collide releasing energy).

Electrons and positrons are significantly pairwise non-interchangeable, but globally approximately interchangeable.

We might conclude: electrons and positrons significantly differ relationally but do not differ much intrinsically.

On the other hand, if you have a system made of photons and electrons, and you swap out a photon and replace it by an electron, it will make a significant difference, but likewise typically if you swap out all the photons and electrons, it will also make a significant difference (unless the system was in a rare symmetric configuration). Thus, photons and electrons are significantly pairwise and globally interchangeable, and hence significantly differ both relationally and intrinsically.

Heterothallic Isogamous Organisms

Isogamous sexually-reproducing organisms have equally sized gametes among their sexes, and hence cannot be labeled as “female” and “male” (biologists define “female” and “male” in terms of larger and smaller gametes, respectively). Instead these sexes get arbitrarily labeled as plus and minus (I will assume there are only two mating types for simplicity). In heterothallic organisms, the sexes are located in different individuals, so two are needed for reproduction. Humans are heterothallic but not isogamous. But there are many species (mostly unicellular, I believe) that are heterothallic and isogamous.

We can now suppose a heterothallic and isogamous species with pretty symmetric mating roles. In such a species, again, we have significant individual non-interchangeability in a system. If Alice is a plus and Bob is a minus, they can reproduce, but if you swap out Bob for a plus, you get a non-reproductive pair. But if the mating roles are sufficiently similar, you can have global approximate interchangeability: if in some system you put pluses for the minuses and minuses for the pluses, things could go on much as before. A group of pluses may behave very much like a group of minuses (namely, over time the population will decrease to zero), but a mixed group of pluses and minuses is apt to behave very differently. We thus have pairwise non-interchangeability but approximate global interchangeability.

We might similarly say: pluses and minuses in our heterothallic and isogamous species significantly differ relationally but do not differ much intrinsically. On the other hand, cats and dogs significantly differ both relationally and intrinsically.

The Distinction

We thus have a distinction between two kinds of differences, which we can label as relational and intrinsic. I am not happy with the labels, but when I use them, please think of my two examples: particles and isogamous organisms. These two kinds of differences can be thought of as denying different symmetries: intrinsic differences are opposed to global interchange of the types of all individuals; relational differences are opposed to pairwise interchange of the types of a pair of individuals.

Conservative Christian Views of Sex and Gender

Conservative Christians tend to think that there are significant differences between men and women. In addition to cultural traits, there are two main theological reasons for thinking this:

1. Marriage asymmetry: Men and women can marry, but men cannot marry men and women cannot marry women.

2. Liturgical asymmetry: Only men can serve in certain liturgical “clerical” roles.

Of these, the marriage asymmetry is probably a bit more widely accepted than the liturgical asymmetry. (Some also think there is an authority asymmetry in the family where husbands have a special authority over wives. This is even more controversial among conservative Christians than the liturgical asymmetry, so I won’t say more about it.)

We could suppose an arbitrary divine rule behind both asymmetries. But this is theologically problematic: a really plausible way of reading the difference between the Law of Moses and the Law of the Gospel is at that in the Law of the Gospel, we no longer have arbitrary rules whose primary benefit is obedience, such as the prohibition on eating pork.

If we are to avoid supposing an arbitrary divine rule, we need to suppose differences between men and women to explain the theologically grounded asymmetries. And this is apt to lead conservative Christians to philosophical and theological theorizing about normative differences such as women being called more to “receptivity” and men more to “givingness”, or searching through sociological, psychological and biological data for relevant differences between the behavior and abilities of men and women. The empirical differences tend to lie on continua with wide areas of overlap between the sexes, however, and the normative differences are either implausible or likewise involve continua with wide areas of overlap (men, too, are called to receptivity).

But I think we are now in a position to see that there is a logical shortcoming behind the focus of this search. For differences between men and women can be relational or intrinsic, and the search has tended to focus on the intrinsic.

However, I submit, purely relational differences are sufficient to explain both the marriage and liturgical asymmetries. One way to see this is to pretend that we are a heterothallic isogamous species (rather than heterothallic anisogamous species that we actually are), consisting of pluses and minuses rather than females and males.

Then, if marriage has an ordering to procreation, that would neatly explain why pluses and minuses can marry each other, but pluses can’t marry pluses and minuses can’t marry minuses. No intrinsic difference between pluses and minuses is needed to explain this. Thus, as soon as we accept that marriage has an ordering to procreation, we have a way to explain the marriage asymmetry without any supposition of intrinsic differences.

Likewise, if there is going to be an incarnation, and only one, and the incarnate God is going to be incarnate as a typical organism of our species, then this incarnation must happen as a plus or a minus. And if married love is a deep and passionate love that is a wonderful symbol for the love between God and God’s people, then if the incarnation is as an individual of one of the sexes, God’s Church would then symbolically have the opposite sex. And then those whose liturgical role it is to stand in for the incarnate God in the marriage-like relationship to the Church would most fittingly have the sex opposite to that of the Church. Thus, if the incarnate God is incarnate as a plus, the Church would be figured as a minus, one can explain why it is fitting that the clergy in the relevant liturgical roles would be pluses; if the incarnate God is incarnate as a minus, we have an explanation of why the clergy in these roles would be minuses as well. (Interestingly, on this story, it’s not that the clergy are directly supposed to be like the incarnate God in respect of sex, but that their sex is supposed to be the opposite to that of the Church, and given that in the species there are only two sexes, this forces them to have the same sex as the incarnate God: the clergy need to have a sex opposed to the sex opposed to that of the incarnate God.)

Now, we are not isogamous, and we have female and male, not plus and minus. But we can still give exactly the same explanations. Even though in an anisogamous species there are significant intrinsic biological differences between the sexes, we need not advert to any of them to explain either the marriage or the liturgical asymmetry. The marriage asymmetry is tied to the pairwise non-interchangeability of the sexes and explained by the procreative role of marriage. The liturgical asymemtry is tied to the marriage asymmetry together with the symmetry-breaking event of God becoming incarnate in one of the sexes.

As far as this story goes, there need not be any morally significant intrinsic differences between male and female to explain the marital and liturgical asymmetries. The relational difference, that you need male and female for a mating pair, is morally significant on this story, but in a way that is entirely symmetric between male and female. And then we have one symmetry-breaking event: God becomes incarnate as a male. We need not think that there is any special reason why God becomes incarnate as a male or a female—it could equally well have been as a female. The decision whether to become incarnate as a male or a female could be as arbitrary as the decision about the exact eye color of the incarnate God (though, of course, eye color does not ground either significant intrinsic or significant relational differences). But if it were an incarnation as a female, other changes would be fitting: the clergy who symbolize the nuptial role of the incarnate God would fitting be female, in the exodus story it would fitting be female lambs and goats that would be sacrificed, and it would be fitting that Sarah be asked to sacrifice her first-born daughter.

I am not saying that there are no morally significant intrinsic differences between male and female. There may be. We are, after all, not only heterothallic but also anisogamous, and so there could turn out to be such intrinsic differences. But we need not suppose any such to explain the two asymmetries, and it is safer to be agnostic on the existence of these intrinsic differences.

Nothing in this post is meant as an argument for either the marriage asymmetry and the liturgical asymmetry. I have argued for the marriage asymmetry elsewhere, but here I am just saying that it could be explained if we grant the procreative ordering of marriage. And my arguments for the liturgical asymmetry are based on fittingness. But fittingness considerations do not constrain God. While we can explain why the clergy are of the same sex as the incarnate God by the nuptial imagery story that I gave above, God could instead have chosen to make the clergy be of the opposite sex as the incarnate God, in order to nuptially signify the people with the clergy, or God could chosen to make the clergy be of both sexes, to emphasize the fact that salvation is tied to the humanity (see St. Athanasius on this) and not the sex of the incarnate God. But when many things are fitting, God can choose one, and we can then cite its fittingness as a non-deterministic explanation.

Though, I suppose, I have at least refuted this argument:

3. The only way to explain the marriage and liturgical asymmetries is by supposing morally significant intrinsic differences between female and male.

4. There are no such intrinsic differences.

5. So, probably, the asymmetries don’t exist either.

I have refuted it by showing that (3) is false.

Against group intentional action

Alice, Bob and Carl are triumvirate that unanimously votes for some legislation, for the following reasons:

1. Alice thinks that hard work and religion are intrinsically bad while entertainment is intrinsically good, and believes the legislation will decrease the prevalence of hard work and religion and increase that of entertainment.

2. Bob thinks that hard work and entertainment are intrinsically bad while religion is intrinsically good, and believes the legislation will decrease the prevalence of hard work and entertainment and increase that of religion.

3. Carl thinks that religion and entertainment are intrinsically bad while hard work is intrinsically good, and believes the legislation will decrease the prevalence of religion and entertainment and increase that of hard work.

If groups engage in intentional actions, it seems that passing legislation is a paradigm of such intentional action. But what is the intention behind the action here?

When I first thought about cases like this, I thought they were a strong argument against group intentional action. But then I became less sure. For we can imagine an intrapersonal version. Suppose Debbie the dictator was given a card by a trustworthy expert that she was informed contains a truth, with the expert departing at that point. Before she could read it, however, she accidentally dropped the card in a garbage can. Reaching into the garbage can, she found three cards in the expert’s handwriting, two of them being mere handwriting exercises and one being the advice card:

1. Hard work and religion are intrinsically bad while entertainment is intrinsically good, and the legislation will decrease the prevalence of hard work and religion and increase that of entertainment.
2. Hard work and entertainment are intrinsically bad while religion is intrinsically good, and the legislation will decrease the prevalence of hard work and entertainment and increase that of religion.
3. Religion and entertainment are intrinsically bad while hard work is intrinsically good, and the legislation will decrease the prevalence of religion and entertainment and increase that of hard work.

Oddly, Debbie’s own prior views are so undecided that she just sets her credence to 1/3 for each of these propositions, and enacts the legislation. What is her intention?

But now I think there is a plausible answer: Debbie’s intention is to increase whichever one of the trio of entertainment, religion and hard work is good and decrease whichever two of them are bad.

Could we thus say that that is what the triumvirate intends? I am not sure. Nobody on the triumvirate has such an abstract intention.

So perhaps we still have an argument against group intentional action, of the form:

1. If there is group intentional action, the triumvirate acts intentionally.

2. Something only acts intentionally if it has an intention.

3. The triumvirate has no intention.

4. So, there is no group intentional action.

Reasons of identity

In paradigm instances of parental action, my reason for action is the objective fact that I am a parent, not because of the subjective fact that I think I'm a parent or identify with being a parent. There are times when it makes sense to act on the subjective fact. If I'm asked by someone (say, a counselor) whether I identify with being a parent, my answer needs to be based on the subjective fact that I so identify. But those are atypical cases. 

I suspect this is generally true: cases when one acts on what one is are primary and cases when one acts on what one identifies as are secondary. It is, thus, problematic to define any feature that is significantly rationally relevant to ordinary action in terms of what one identifies with. 

Goodman and Quine and transitive closure

In the previous post, I showed that Goodman and Quine’s counting method fails for objects that have too much overlap. I think (though the technical parts here are more difficult) that the same is true for their definition of the ancestral or transitive closure of a relation.

GQ showed how to define ancestors in terms of offspring. We can try to extend this definition to the transitive closure of any relation R over any kind of entities:

1. x stands in the transitive closure of R to y iff for every object u that has y as a part and that has as a part anything that stands in R to a part of u, there is a z such that Rxz and both x and z are parts of R.

This works fine if no relatum of R overlaps any other relatum of R. But if there is overlap, it can fail. For instance, suppose we have three atoms a, b and c, and a relation R that holds between a + b and a + b + c and between a and a + b. Then any object u that has a + b + c as a part has c as a part, and so (1) would imply that c stands in the transitive closure of R to a + b + c, which is false.

Can we find some other definition of transitive closure using the same theoretical resources (namely, mereology) that works for overlapping objects? No. Nor even if we add the “bigger than” predicate of GQ’s attempt to define “more”. We can say that x and y are equinumerous provided that neither is bigger than the other.

Let’s work in models made of an infinite number of mereological atoms. Write u ∧ v for the fusion of the common parts of both u and v (assuming u and v overlap), u ∨ v for the fusion of objects that are parts of one or the other, and u − v for the fusion of all the parts of u that do not overlap v (assuming u is not a part of v). Write |x| for the number of atomic parts of x when x is finite. Now make these definitions:

2. x is finite iff an atom is related to x by the transitive closure (with respect to the kind object) of the relation that relates an object to that object plus one atom.

3. Axyw iff x and y are finite and whenever x′ is equinumerous with x and does not overlap y, then x′ ∨ y is equinumerous with w. (This says |x| + |y| = |w|.)

4. Say that D_yuv iff A(u−y,u−y,v−y) (i.e., |v−y| = 2|u−y|) and either v does not overlap y or and u ∧ y is an atom or v and y overlap and u ∧ y consists of v ∧ y plus one atom. (This treats u and v as basically ordered pairs (u−y,u∧y) and (v−y,v∧y), and it makes sure that from the first pair to the second, the first component is doubled in size and the second component is decreased by one.)

5. Say that Q⁰yx iff y is finite and for some atom z not overlapping y we have y ∧ z related to something not overlapping x by the transitive closure of D_y. (This takes the pair (z,y), and applies the double first component and decrease second component relation described in (4) until the second component goes to zero. Thus, it is guaranteed that |x| = 2^|y|.)

6. Say that Qyx iff y is finite and Q⁰yx′ for some non-overlapping x′ that does not overlap y and that is equinumerous with x.

If I got all the details right, then Qyx basically says that |x| = 2^|y|.

Thus, we can define use transitive closure to define binary powers of finite cardinalities. But the results about the expressive power of monadic second-order logic with cardinality comparison say that we can only define semi-linear relations between finite cardinalities, which doesn’t allow defining binary powers.

Remark: We don’t need equinumerosity to be defined in terms of a primitive “bigger”. We can define equinumerosity for non-overlapping finite sets by using transitive closure (and we only need it for finite sets). First let T_yuv iff v − y exists and consists of u − y minus one atom and v ∧ y exists and consists of v ∧ y minus one atom. Then finite x and y are equinumerous₀ iff they are non-overlapping and x ∨ y has exactly two atoms or is related to an object with exactly two atoms by the transitive closure of T_yuv. We now say that x and y are equinumerous provided that they are finite and either x = y (i.e., they have the same atoms) or both x − y and y − x are defined and equinumerous₀.

No fix for Goodman and Quine's counting

In yesterday’s post, I noted that Goodman and Quine’s nominalist mereological definition of what it is to say that there are more cats than dogs fails if there are cats that are conjoint twins. This raises the question whether there is some other way of using the same ontological resources to generate a definition of “more” that works for overlapping objects as well.

I think the answer is negative. First, note that GQ’s project is explicitly meant to be compatible with there being a finite number of individuals. In particular, thus, it needs to be compatible with the existence of mereological atoms, individuals with no proper parts, which every individual is a fusion of. (Otherwise, there would have to be no individuals or infinitely many. For every individual has an atom as a part, since otherwise it has an infinite regress of parts. Furthermore, every individual must be a fusion of the atoms it has as parts, otherwise the supplementation axiom will be violated.) Second, GQ’s avail themselves of one non-mereological tool: size comparisons (which I think must be something like volumes). And then it is surely a condition of adequacy on their theory that it be compatible with the logical possibility that there are finitely many individuals, every individual is a fusion of its atoms and the atoms are all the same size. I will call worlds like that “admissible”.

So, here are GQ’s theoretical resources for admissible worlds. There are individuals, made of atoms, and there is a size comparison. The size comparison between two individuals is equivalent to comparing the cardinalities of the sets of atoms the individuals are made of, since all the atoms are the same size. In terms of expressive power, their theory, in the case of admissible worlds, is essentially that of monadic second order logic with counting, MSO(#), restricted to finite models. (I am grateful to Allan Hazen for putting me on to the correspondence between GQ and MSO.) The atoms in GQ correspond to objects in MSO(#) and the individuals correspond to (extensions of) monadic predicates. The differences are that MSO(#) will have empty predicates and will distinguish objects from monadic predicates that have exactly one object in their extension, while in GQ the atoms are just a special (and definable) kind of individual.

Suppose now that GQ have some way of using their resources to define “more”, i.e., find a way of saying “There are more individuals satisfying F than those satisfying G.” This will be equivalent to MSO(#) defining a second-order counting predicate, one that essentially says “The set of sets of satisfiers of F is bigger than the set of sets of satisfiers of G”, for second-order predicates F and G.

But it is known that the definitional power of MSO(#) over finite models is precisely such as to define semi-linear sets of numbers. However, if we had a second-order counting predicate in MSO(#), it would be easy to define binary exponentiation. For the number of objects satisfying predicate F is equal to two raised to the power of the number of objects satisfying G just in case the number of singleton subsets of F is equal to the number of subsets of G. (Compare in the GQ context: the number of atoms of type F is equal to two the power of the number of atoms of type G provided that the number of atoms of type F is one plus the number of individuals made of the atoms of type G.) And of course equinumerosity can be defined (over finite models) in terms of “more”, while the set of pairs (n,2ⁿ) is clearly not semi-linear.

One now wants to ask a more general question. Could GQ define counting of individuals using some other predicates on individuals besides size comparison? I don’t know. My guess would be no, but my confidence level is not that high, because this deals in logic stuff I know little about.

Goodman and Quine and shared bits

Goodman and Quine have a clever way of saying that there are more cats than dogs without invoking sets, numbers or other abstracta. The trick is to say that x is a bit of y if x is a part of y and x is the same size as the smallest of the dogs and cats. Then you’re supposed to say:

1. Every object that has a bit of every cat is bigger than some object that has a bit of every dog.

This doesn’t work if there is overlap between cats. Imagine there are three cats, one of them a tiny embryonic cat independent of the other two cats, and the other two are full-grown twins sharing a chunk larger than the embryonic cat, while there are two full-grown dogs that are not conjoined. Then a bit is a part the size of the embryonic cat. But (assuming mereological universalism along with Goodman and Quine) there is an object that has a bit of every cat that is no bigger than any object has a bit of every dog. For imagine an object that is made out of the embryonic cat together with a bit that the other two cats have in common. This object is no bigger than any object that has a bit of each of the dogs.

It’s easy to fix this:

2. Every object that has an unshared bit of every cat is bigger than some object that has an unshared bit of every dog,

where an unshared bit is a bit x not shared between distinct cats or distinct dogs.

But this fix doesn’t work in general. Suppose the following atomistic thesis is true: all material objects are made of equally-sized individisible particles. And suppose I have two cubes on my desk, A and B, with B having double the number of particles as A. Consider this fact:

4. There are more pairs of particles in A than particles in B.

(Again, Goodman and Quine have to allow for objects that are pairs of particles by their mereological universalism.) But how do we make sense of this? The trick behind (1) and (2) was to divide up our objects into equally-sized pieces, and compare the sizes. But any object made of the parts of all the particles in B will be the same size as B, since it will be made of the same particles as B, and hence will be bigger than any object made of parts of A.

Trope theory and merely numerical differences in pleasures

Suppose I eat a chocolate bar and this causes me to have a trope of pleasure. Given assentiality of origins, if I had eaten a numerically different chocolate bar that caused the same pleasure, I would have had had a numerically different trope of pleasure.

Now, imagine that I eat a chocolate bar in my right hand and it causes me to have a trope of pleasure R, and immediately as I have finished eating that one chocolate bar, I switch to eating the chocolate bar in my left hand, which gives me an exactly similar trope of pleasure, L, with no temporal gap. Nonetheless, by essentiality of origins, trope L is numerically distinct from trope R.

To some (perhaps Armstrong) this will seem absurd. But I think it’s exactly right. In fact, I think it may even an argument for trope theory. For it seems pretty plausible that as I switch chocolate bars, something changes in me: I go from one pleasure to another exactly like it. But on heavy-weight Platonism, there is no change: I instantiated pleasure and now I instantiate pleasure. On non-trope nominalism, likewise there is no change. It’s trope theory that gives us the change here.

Does one's vote make a difference?

Suppose that there is a simple majority election, with two candidates, and there is a large odd number of voters. Suppose polling data makes the election too close to call. How likely is it that you can decide which candidate wins?

I could look up this stuff, but it’s more fun to figure it out.

A quick and dirty model is this. We have N people other than you voting, each choosing between candidates A and B with probabilities p and 1 − p respectively. You don’t know what p and 1 − p are, but polling data tells you that p is between 1/2 − a and 1/2 + b for some positive numbers a and b. Your vote decides the election provided that exactly N/2 people vote for candidate A. This requires that N be even (if N is odd, at best you can decide between a candidate winning and the election being undecided, so you can’t decide which candidate wins), which has probability 1/2. Given that N = 2n is even, the probability that the other votes are exactly balanced is (a+b)⁻¹ C(2n,n)∫_1/2−a^1/2+bpⁿ(1−p)^n − 1dp, where C(m,n) is the binomial coefficient. Assuming n is large as compared to a and b, the integral can be approximated by replacing its bounds by 0 and 1 respectively, and some work with Mathematica shows that for large n the probability is approximately 1/(N(a+b)).

So what? Well, suppose you think that candidate A will on average make a person in the jurisdiction be u units of flourishing better off than candidate B will, and there are K persons, where K ≥ N + 1 (there are at least as many persons as candidates). So, the expected amount of difference that your voting for A will make is at least Ku/(2N(a+b)). This is at least u/(a+b). Thus, if the polling data gives you a range between 0.48 and 0.52 for the probability of a person’s preferring candidate A, and half of the people in the jurisdiction vote, the expected amount of difference that your vote makes is 25u. This is quite a lot if you think that which candidate wins makes a significant difference u per governed person.

Interestingly, some numerical work with Mathematica also shows that as number of people increases, then the expected amount of difference your vote makes also increases asymptotically, up to the limit of Ku/(2N(a+b)). So for larger jurisdictions, even though the probability of your vote making a difference is smaller, the expected difference from your vote is a bit bigger.

My quick and dirty model is not quite right. Of course, people don’t come to the polls and randomly choose whom to vote for. A more likely source of randomness has to do with who actually makes it to the polls (who gets sick, who has something come up, who decides it’s pointless to vote, etc.). A better model might be this. We have M people eligible to vote, of whom pM want to vote for A and (1−p)M want to vote for B. Some random subset of the M people then votes. My probabilist intuitions say that this is not that different from my model if the number of actual voters is, say, half of the eligible voters. If I had an election that I was eligible to vote in coming, I might try to figure our the more complex model, but I don’t.