Possible futures

Given a time t and a world w, possible or not, say that w is t-possible if and only if there is a possible world w_t that matches w in all atemporal respects as well as with respect to all that happens up to and including time t. For instance, a world just like ours but where in 2027 a square circle appears is 2026-possible but not 2028-possible.

Here is an interesting and initially plausible metaphysical thesis:

1. The world w is possible iff it is t-possible for every finite time t.

But (1) seems false. For imagine this:

2. On the first day of creation God creates you and promises you that on some future
   day a butterfly will be created ex nihilo. God never makes any other promises. God never makes butterflies. And nothing else relevant happens.

I assume God’s promises are unbreakable. The world described by (2) seems to be t-possible for every finite time t. For the fact that no butterfly has come into existence by time t does not falsify God’s promise that one day a butterfly will be created. But of course the world described by (2) is impossible.

(It’s interesting that I can’t think of a non-theistic counterexample to (1).)

So what? Well, here is one applicaiton. Amy Seymour in a nice paper responding to an argument of mine writes about the following proposition about situation where there are infinitely many coin tosses in heaven, one per day:

3. After every heads result, there is another heads result.

She says: “The open futurist can affirm that this propositional content has a nearly certain general probability because almost every possible future is one in which this occurs.” But in doing so, Seymour is helping herself to the idea of a “possible future”, and that is a problematic idea for an open futurist. Intuitively:

4. A possible future is one such that it is possible that it is true that it obtains.

But the open futurist cannot say that, since in the case of contingent futures, there can be no truth about its obtaining. The next attempt at accounting for a possible future may be to say:

5. A future is possible provided it will be true that it is possible that it obtains.

But that doesn’t work, either, since any future with infinitely many coin tosses (spaced out one per day) is such that at any time in the future, it is still not true that it is possible that it obtains, since its obtaining still depends on the then-still-future coin tosses. The last option I can think of is:

6. A future is possible provided that for every future time t it is t-possible.

But that fails for exactly the same reason that the t-possibility of worlds story fails.

Here is one way out: Deny classical theism, say that God is in time, and insist that God has to act at t in order to create something ex nihilo at t. But God, being perfect, can’t make a promise unless he has a way of ensuring the promise to come true. But how can God make sure that he will one day create the butterfly? After all, on any future day, God is free not to create it then. Now, if God promised to create a butterfly by some specific date, then God could be sure that he would follow through, since if he hadn’t done so prior to the specified date, he would be morally obligated to do so on that day, and being perfect he would do so. So since God can’t ensure the promise will come true, he can’t make the promise. (Couldn’t God resolve to create the butterfly on some specific day? On non-classical theism, maybe yes, but the act of resolving violates the clause “nothing else relevant happens” in (2).)

This way out doesn’t work for classical theism, where God is timeless and simple. For given timelessness, God can timelessly issue the promise and “simultaneously” timelessly make a butterfly appear on (say) day 18, without God being intrinsically any different for it. So I think the classical theist has reason to deny (1), and hence has no account of “possible futures” that is compatible with open futurism, and thus probably has to deny open futurism. Which is unsurprising—most classical theists do deny open futurism.

Unrestricted quantification and Tarskian truth

It is well-known—a feature and not a bug—that Tarski’s definition of truth needs to be given in a metalanguage rather than the object language. Here I want to note a feature of this that I haven’t seen before.

Let’s start by considering how Tarski’s definition of truth would work for set theory.

We can define satisfaction as a relation between finite gappy sequences of objects (i.e., sets) and formulas where the variables are x₁, .... We do this by induction on formulas.

How does this work? Following the usual way to formally create an inductive definition, we will do something like this:

1. A satisfaction-like relation is a relation between finite sequences of sets and formulas such that:

   1. the relation gets right the base cases, namely, a sequence s satisfies x_n ∈ x_m if and only if the nth entry of s is a member of the mth entry of s) and satisfies x_n = x_m if and only if the nth entry of s is identical to the mth entry

   2. the relation gets right the inductive cases (e.g., s satisfies ∃x_nϕ if and only if for every sequence s′ that includes an nth place and agrees with s on all the places other than the nth place we have s′ satisfying ϕ, etc.)

2. A sequence s satisfies a formula ϕ provided that every satisfaction-like relation holds between s and ϕ.

The problem is that in (2) we quantify over satisfaction-like relations. A satisfaction-like relation is not a set in ZF, since any satisfaction-like relation includes ((a),ϕ₌) for every set a, where (a) is the sequence whose only entry is a at the first location and ϕ₌ is x₁ = x₁. Thus, a satisfaction-like relation needs to be a proper class, and we are quantifying over these, which suggests ontological commitment to these proper classes. But ZF set theory does not have proper classes. It only has virtual classes, where we identify a class with the formula defining it. And if we do that, then (2) comes down to:

3. A sequence s satisfies ϕ if for every satisfaction-like formula F the sentence F(s,ϕ) is true.

And that presupposes the concept of truth. (Besides which, I don’t know if we can define a satisfaction-like formula.) So that’s a non-starter. We need genuine and not merely virtual classes to give a Tarski-style definition of truth for set theory. In other words, it looks like the meta-language in which we give the Tarski-style definition of truth for set theory not only needs a vocabulary that goes beyond the object-language’s vocabulary, but it needs a domain of quantification that goes beyond the object-language’s domain.

Now, suppose that we try to give such a Tarskian definition of truth for a language with unrestricted quantification, namely quantification over literally everything. This is very problematic. For now the satisfaction-like relation includes the pair ((a),ϕ₌) for literally every object a. This relation, then, can neither be a set, nor a class, nor a proper superclass, nor a supersuperclass, etc.

I wonder if there is a way of getting around this difficulty by having some kind of a primitive “inductive definition” operator instead of quantifying over satisfaction-like relations.

Another option would be to be a realist about sets but a non-realist about classes, and have some non-realist story about quantification over classes.

I bet people have written on this stuff, as it’s a well-explored area. Anybody here know?

Immortality of the soul and the soul's proper operation

This is an attempt to make an argument for the natural immortality of the soul from the premise that the soul has a proper operation that is independent of the body. The argument is going to be rather odd, because it depends on my rather eccentric four-dimensionalist version of Aristotelian metaphysics.

Start with the thought of how substances typically grow in space. They do this by causing themselves to have accidents in new locations, and they come to exist where these new accidents are. Thus, if I eat and my stomach becomes distended, I now have an accident of stomachness in a location where previously I didn’t, and normally I come to be partly located where my accidents are.

It is plausible (at least to a four-dimensionalist) that spatiotemporal substances grow in time like they grow in space. Thus, they produce accidents in a new temporal location, a future one, and typically come to be located where the accidents are—maybe they come to be there by being active in and through the accidents. (There are exceptions: in transsubstantiation, the bread and wine don’t follow their accidents. But I am focusing on what naturally happens, not on miracles.)

Suppose now that the soul has a proper operation that is independent of the body. Given the fact that my intellectual function is temporal in nature, it is plausible that in this proper operation, my soul is producing a future accident of mine—say, a future accident of grasping some abstract fact—and does so regardless of how sorry and near-to-death a state my body has. But a substance normally stretches both spatially and temporally to become partly located where its accidents are. So by producing a future accident of mine the soul normally ensures that I will be there in that future to be active in and through that accident. Thus the soul, in exercising that future-directed proper activity, makes me exist in the future.

Now that I’ve written this down, I see a gap. The fact that the soul has a proper operation independent of the body does not imply that the soul always engages in that operation. If it does not always engage in that operation, then there is the danger that if my body should perish at a time when the operation is not engaged in, the soul would fail to extend my existence futureward, and I would perish entirely.

On this version of the proper function argument, we thus need a proper operation that the soul normally or naturally always engages in. We might worry, however, that the intellectual operations all cease when we are in dreamless sleep. However, we might suppose that the soul by its nature always carries forward in time some aspect of the understandings or abstractions that it has gained, and this carrying forward in time is indeed a proper operation that occurs even in dreamless sleep, since we do not lose our intellectual gains when we are asleep. (We should distinguish this carrying forward of an aspect of the intellectual gains from the aspects of memory that are mediated by the brain. The need to do this is a weakness of the argument.)

The above depends on my idiosyncratic picture of persistence over time: substances cause their future existence. Divine sustenance is divine cooperation with this causation. The argument has holes. But I feel I may be on to something.

The argument does not establish that we necessarily are immortal. We are only naturally immortal, in that normally we do not perish. It is possible, as far as the argument goes, that the proper operation should fail to succeed in extending us into the future, if only because God might choose to stop cooperating in the way that constitutes sustenance (but I trust he won’t).

Aquinas' argument for the immortality of the soul

Aquinas argues that because the human soul has a proper operation—abstract thought—that does not depend on the body, the soul would survive the destruction of the body.

I’ve never quite understood this argument. It seems to show that there could be a point to the soul surviving the destruction of the body, but that doesn’t show that it will.

It seems that by the same token one could say that because my fingers have an operation independent of my toes, my fingers would survive the destruction of the toes. But that need not be true. I could simultaneously have my toes and fingers crushed, and the fingers’ having an operation independent of the toes would do nothing to save them. In fact, in most cases, fingers perish at the same time as toes do. For in most, though not all, human lives, fingers perish when a person dies, and the toes do so as well. So the argument can’t be that strong.

Still, on reflection, there may be something we can learn from the fingers and toes analogy. We shouldn’t expect the fingers to perish simply as a metaphysical consequence of my toes perishing. By analogy, then, we shouldn’t expect the soul to perish simply as a metaphysical consequence of the body perishing. That’s not the immortality of the soul, but it’s some progress in that direction. After all, the main reason for thinking the soul to perish at death is precisely because one thinks this is a metaphysical consequence of the body perishing.

And I am not denying that there are good arguments for the immortality of the human soul. I think there may be an argument from proper operation that makes even more progress towards immortality, but I’ll leave that for another occasion. Moreover, I think the immortality of the human soul follows from the existence of God and the structure of human flourishing.

Snakes and finitude

For years I have thought the finite to be mysterious, and needs something metaphysical like divine illumination or causal finitism to pick it out. Now I am not sure. I think snakes and exact duplicates can help. And if that’s right, then the argument in my other post from today can be fixed.

Here are some definitions, where the first one is supposed to work for snakes that may be in the same or in different worlds:

• Snake a is vertebrally equal to snake b provided that there is a possible world with exact duplicates of a and b such that in that world it would be possible to line up the two snakes vertebra by vertebra, stretching or compressing as necessary but neither destroying nor introducing vertebra.

• Snake a is the vertebral successor of snake b provided there is a possble world with exact duplicates of a and b such that in that world it is possible to line up the two snakes vertebra by vertebra with exactly one vertebra of a outside the lineup, again stretching or compressing as necessary but neither destroying nor introducing vertebra.

• A world w is abundant in snakes provided that w has a snake with no vertebrae (say, an embryonic snake) and every snake in w has a vertebral successor in w.

• A snake a is vertebrally finite provided that in every world in which snakes are abundant there is a snake vertebrally equal to a.

• A plurality is finite provided that it is possible to put it in one-to-one correspondence with the vertebrae of a vertebrally finite snake.

These definitions require, of course, that one take metaphysical possibility seriously.

A dialectically failing argument for truth-value realism about arithmetic

Truth-value realism about (first-order) arithmetic is the thesis that for any first-order logic sentence in the language of arithmetic (i.e., using the successor, addition and multiplication functions along with the name “0”), there is a definite truth value, either true or false.

Now, consider the following argument for truth-value realism about arithmetic.

Assume eternalism.

Imagine a world with an infinite space and infinite future that contains an ever-growing list of mathematical equations.

At the beginning the equation “S0 = 1” is written down.

Then a machine begins an endless cycle of alternation between three operations:

1. Scan the equations already written down, and find the smallest numeral n that occurs in the list but does not occur in an equation that starts with “Sn=”. Then add to the bottom of the list the equation “Sn = m” where m is the numeral coming after n.

2. Scan the equations already written down, and find the smallest pair of numerals n, m (ordered lexicographically) such that n + m= does not occur in the list of equations, and write at the bottom of the list n + m = r where r is the numeral representing the sum of the numbers represented by n and m.

3. Scan the equations already written down, and find the smallest pair of numerals n, m (ordered lexicographically) such that n ⋅ m= does not occur in the list of equations, and write at the bottom of the list n ⋅ m = r where r is the numeral representing the product of the numbers represented by n and m.

No other numerals are ever written down in that world, and no equations disappear from the list. We assume that all tokens of a given numeral count as “alike” and no tokens of different numerals count as “alike”. The procedure of producing numerals representing sums and products of numbers represented by numerals can be given entirely mechanically.

Now, if ϕ is an arithmetical sentence, then we say that ϕ is true provided that ϕ would be true in a world such as above under the following interpretation of its basic terms:

4. The domain consists of the first occuring token numerals in the giant list of equations (i.e., a token numeral in the list of equations is in the domain if and only if no token alike to it occurs earlier in the list).

5. 0 refers to the zero token in the first equation.

6. The value of Sn for a token numeral n is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a capital S token followed by a token alike to n.

7. The value of n + m for token numerals n and m is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a token alike to n follow by a plus sign followed by a token alike to m.

8. The value of n ⋅ m for token numerals n and m is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a token alike to n follow by a multiplication sign followed by a token alike to m.

It seems we now have well-defined truth-value assignments to all arithmetical sentences. Moreover, it is plausible that these assignments would be correct and hence truth-value realism about arithmetic is correct.

But there is one serious hole in this argument. What if there are two worlds w₁ and w₂ with lists of equations both of which satisfy my description above, but ϕ gets different truth values in them? This is difficult to wrap one’s mind around initially, but we can make the worry concrete as follows: What if the two worlds have different lengths of “infinite future”, so that if we were to line up the lists of equations of the two worlds, with equal heights of lines, one of the two lists would have an equation that comes after all of the equations of the other list?

This may seem an absurd worry. But it’s not. What I’ve just said in the worry can be coherently mathematically described (just take a non-standard model of arithmetic and imagine the equations in one of the lists to have the order-type of that model).

We need a way to rule out such a hypothesis. To do that, what we need is a privileged notion of the finite, so that we can specify that for each equation in the list there is only a finite number of equations before it, or (equivalently) that for each operation of the list-making machine, there are only finitely many operations.

I think there are two options here: a notion of the finite based on the arrangement of stuff in our universe and a metaphysically privileged notion of the finite.

There are multiple ways to try to realize the first option. For instance, we might say that a finite sequence is one that would fit in the future of our universe with each item in the sequence being realized on a different day and there being a day that comes after the whole sequence. (Or, less attractively, we can try to use space.) One may worry about having to make an empirical presupposition that the universe’s future is infinite, but perhaps this isn’t so bad (and we have some scientific reason for it). Or, more directly in the context of the above argument, we can suppose that the list-making machine functions in a universe whose future is like our world’s future.

But I think this option only yields what one might call “realism lite”. For all we’ve said, there is a possible world whose future days have the order structure of a non-standard model of arithmetic, and the analogue to the mathematicians of our world who employed the same approach as we just did to fix the notion of the finite end up with a different, “more expansive”, notion of the finite, and a different arithmetic. Thus while we can rigidify our universe’s “finite” and or the length of our universe’s future and use that to fix arithmetic, there is nothing privileged about this, except in relation to the actual world. We have simply rigidified the contingent, and the necessity of arithmetical truths is just like the necessity of “Water is H₂O”—the denial is metaphysically impossible but conceivable in the two-dimensionalist sematics sense. And I feel that better than this is needed for arithmetic.

So, I think we need a metaphysically privileged notion of the finite to make the above argument go. Various finitism provide such a notion. For instance, finitism simpliciter (necessarily, there are only finitely many things), finitism about the past (necessarily, there are always only finitely many past items), causal finitism (necessarily, each item has only finitely many causal antecedents), and compositional finitism (necessarily, each item has at most finitely many parts). Finitism simpliciter, while giving a notion of the finite, doesn’t work with my argument, since my argument requires eternalism, an infinite future and an ever-growing list. Finitism about the past is an option, though it has the disadvantage that it requires time to be discrete.

I think causal finitism is the best option for what to plug into the argument, but even if it’s the best option, it’s not a dialectically good option, because it’s more controversial than the truth-value realism about arithmetic that is the conclusion of the argument.

Alas.

Causation and counterfactuals

Suppose that an extremely reliable cannon is loaded with a rock, and pointed at a window, and the extremely reliable timer on the cannon is set for two minutes. Two minutes later, the cannon shoots out the rock causing the window to break.

The Lewisian counterfactual account of causation accounts for the causation by the counterfactual:

1. Were the cannon not to have fired the rock, the window wouldn’t have broken.

But imagine that a risk-taking undersupervised kid was walking by towards the end of the the two minutes, and on a whim considered swapping the rock in the cannon for their steel water bottle. The decision whether to do the swap was an extremely conflicted one, and a single neuron’s made the difference, and resulted in the swap not happening.

We can set up the story in such a way that on Lewis’s way of measuring the closeness of worlds, a world where the kid swapped the rock for the water bottle is closer than any worlds where the timer wasn’t set or where the cannon misfired or where the cannon wasn’t loaded or anything like that. In that case on a Lewisian analysis of counterfactuals:

2. Were the cannon not to have fired the rock, the window would still have broken.

But surely whether the kid walks by or not, the cannon’s firing the rock caused the window to break.

Temporal purism

Say that a fact is temporally pure about an instantaneous time t provided that it holds solely in virtue of how things are at t. (The term is due to Richard Gale, but I am not sure he would have wanted the “instantaneous” restriction.) Thus, that Alice is swallowed the fatal poison at noon is not temporally pure because a part of why it holds is that she died after noon. The concept of a temporally pure fact is intuitively related to the Ockhamist notion of a hard fact: any fact that’s temporally pure about the past or present is a hard fact.

I will allow two ways of filling out “instantaneous time t” in the definition of temporal purity: the time t can be a B-theoretic time like “12:08 GMT on January 2, 2084 AD” and it can be an A-theoretic time like “exactly three hours ago” or “now”.

We can now define a theory:

• Temporal purism: Necessarily, all temporal facts are grounded in the temporally pure facts and/or facts about the existence (including past and future existence) of instantaneous times and of their temporal relationships.

Presentists, open futurists and eternalists can all embrace temporal purism.

Probably the best way to deny temporal purism is to hold that there are fundamental truths about temporal reality that irreducibly hold over an interval of times—this is the temporal equivalent of holding that there are fundamental distributional properties.

I think there are reasons to deny temporal purism. First, it is plausible that (a) some states of our consciousness are fundamental features of reality and (b) they irreducibly occur over an interval of time of some positive length. Claim (a) is pretty standard among dualists. Claim (b) seems to follow from the plausibility that no state of consciousness shorter than, say, a nanosecond can be felt by us, but of course there are no unfelt states of consciousness.

Second, temporal purism pushes one pretty hard to an at-at analysis of change, and many people don’t like that.

Eternalists can deny temporal purism. This is pretty clear: eternalists have no difficulty with temporally distributional properties.

I think it is difficult for open futurists to deny temporal purism. For suppose that some fundamental feature F of our temporal reality occurs over an interval from t₁ to t₂, and cannot occur over a much shorter interval. Then at some time very shortly after t₁, but well before t₂, the feature is already present. But its being present seems to depend on a future that is open. So open futurism plus temporal impurism pushes one to a view on which the present and even the near past is open, because it depends on what will happen in the future.

Closed-future presentists can deny temporal purism. However, this feels uncomfortable to me. There is something odd on presentism about the idea that a present reality depends on the near past and/or the near future. At the same time, many odd things are actually true.

I think the denial of temporal purism pushes one somewhat towards eternalism.

Laziness is the mother of invention

For about 20 years, we've been using a set of Logitech Z-2200 speakers as our TV speakers. But they have an inconvenient volume control that requires one to get up from the sofa to turn the knob. Most of the time, it's enough to adjust the volume with the TV's internal control, but sometimes the speakers' knob needs to be tweaked. I had a gear motor lying around that never got used for a project, and so I set it with a blue pill microcontroller board, an IR receiver scavenged from a broken toy, a drv8833 driver, and some 3D printed parts, some lasercut plywood to keep things in place, a skate bearing, and a laser cut case, so now I can turn the volume control knob by pressing some unused buttons on our Blu-ray player remote. There is a satisfying whirring noise when it turns.

Arduino source code is here.

Presentism, multiverses and discrete time

Suppose time is in fact continuous and modeled by the real numbers.

It seems odd indeed to me that the real numbers should be the only possible way for time to run. The real numbers are a very specific mathematical system. There are other systems, such as the hyperreals or the rationals or even the integers, that seem to be plausible alternatives. I know of no argument that the time sequence has to be numbered by the real numbers.

Thus, given our initial supposition, it should be possible to have time sequences corresponding to ordered sequences numbered by the integers or the hyperreals. Here, then, is a further intuition. It is possible to have a multiverse with radically different spacetime structures in each universe of it. If so, then we would expect the possibility of a multiverse where different universes in a multiverse have time sequences based on very different ordered sets.

Suppose presentism is necessarily true. Then even in such a multiverse, there would be an absolute present running across all of these timelines in the different universes. And that would be rather odd. Imagine that in one universe the time-line is corresponds to the integers and in the other it corresponds to the reals, and both are found in one multiverse. What happens in the universe whose time-line is based on the integers when the line of the present moves continuously across the uncountable infinity of times numbered by the real numbers? Does it stay for infinitely moments at the same integer? But then at infinitely many moments of time it would be at one moment, which is a contradiction. Or does the universe with the integer time-line pop out of existence when the present doesn’t meet up with these integers? Maybe that’s the best view, but it’s a weird view.

Perhaps the presentist’s best bet is to say that there is a privileged mathematical structure that models what a time-line could be like. If so, my intuition says that the only candidate for that privileged structure would be a discrete structure like the integers. For there are arguments in the history of philosophy for time having to be discrete (arguments from Zeno through myself), but none for time having to be modeled by specifically the real numbers, or the rational numbers, or some specific hyperreal field.

Darwin and Einstein against the shared-form interpretation of Aristotle

Assume an Aristotelian account of substantial form on which forms are found in the informed things. A classic question is whether substantial forms are shared between members of the same kind or whether each individual has their own numerically (but maybe not qualitatively) distinct form.

Here’s a fun argument against the shared-form view. For evolution to work with substantial forms, sometimes organisms of one metaphysical kind must produce organisms of another kind. For instance, supposing that wolves are a different metaphysical kind from dogs, and dogs evolved from wolves, it must have happened that two wolves reproduced and made a dog. (I suspect wolves and dogs are metaphysically the same kind, but let’s suppose they aren’t for the sake of the argument.) If we are to avoid occasionalism about this, we have to suppose that the two wolves had a causal power to produce a dog-form under those circumstances.

Plausibly dogs evolved from wolves in Siberia, but there was also a Pleistocene wolf population in Japan, and imagine that the causal power to produce a dog was found in both wolf populations. Suppose, counterfactually, that a short period of time after a pair of wolves produced a dog in Siberia, a pair of Japanese wolves also produced a dog. On a shared-form view, when the Siberian wolves produced a dog, they did two things: they produced a dog-form and they made a dog composed of the dog-form and matter. But when the Japanese wolves produced a dog, the dog-form already existed, so they only thing they could do is make a dog composed of matter and that dog-form.

The first oddity here is this. Our (perhaps imaginary) Japanese wolves didn’t know that there was already a dog in Siberia, so when they produced a dog, they exercised exactly the same causal powers that their Siberian cousins did. But their exercise of these causal powers had a different effect, because it did not produce a new form, since the form already existed, and instead it made the form get exemplified in some matter in Japan. It is odd that the exercise of the same causal power worked differently in the same local circumstances.

Second, there is an odd action-at-a-distance here. The dog-form was available in Siberia, and somehow the Japanese wolves in the story made matter get affected by it thousands of kilometers away.

In fact, to make things worse, we can suppose the Japanese wolves only lagged a two or three milliseconds after their Siberian cousins. In that case, the Siberian wolves caused the existence of the dog-form, which then affected the coming-into-existence of a dog in Japan in a faster-than-light way. Indeed, in some reference frames, the Japanese dog came into existence shortly before the dog-form came into existence in Siberia. In those reference frames we have backwards causation: the Siberian wolves make a dog-form and that dog form organizes matter in Japan earlier.

If, on the other hand, every dog has a numerically distinct form, there is no difficulty: the Japanese wolves’ activity can be entirely causally independent of the Siberian ones’.

Inferentialism and the fictitious isolated hydrogen atom

This is another attempt at an argument against inferentialism about logical constants.

Given a world w, let w^* be a world just like w except that it has added to it an extra spatiotemporally disconnected island universe containing exactly one hydrogen atom with a precisely specified wavefunction ψ₀. Suppose that in in the actual world there is no such isolated hydrogen atom. Now, given a nice first-order language L describing our world, let L^* be a language whose constants are the same as the constants of L with an asterisk added to every logical constant, name and predicate. Given a sentence ϕ of L, let ϕ^* be the corresponding sentence of L^*—i.e., the sentence with all of L’s logical constants asterisked.

Let the rules of inference of L^* be the same as those of L with asterisks added as needed.

Let the semantics of L^* be as follows:

• Every predicate P^* in L^* means the same thing as P in L.

• Every name a^* in L^* means the same thing as a in L.

• Any sentence ϕ^* in L^* without quantifiers means the same thing as ϕ in L.

• But if ϕ^* has a quantifier, then ϕ^* means that ϕ would be true if there were an extra spatiotemporally disconnected island universe containing exactly one hydrogen atom with wavefunction ψ₀.

Thus, ϕ^* is true in world w if and only if ϕ is true in w^*.

Observe that because L contains only names for things that exist in the actual world, and hence not for the extra hydrogen atom or its components, an atomic sentence P(a₁,...,a_n) in L is true if and only if the corresponding sentence P^*(a₁^*,...,a_n^*) is true in L^*.

Logical inferentialism tells us that the logical constants of L^* mean the same thing as those of L, modulo asterisks. After all, modulo asterisks, we have the same inferences, the same meanings of names, and the same meanings of predicates. But this is false: for if ∃^* in L^* were an existential quantifier, then it would be true that there exists an isolated hydrogen atom with wavefunction ψ₀. But there is none such.

Probabilities of regresses of chickens

Suppose we have a backwards-infinite sequence of asexually reproducing chickens, ..., c₋₃, c₋₂, c₋₁, c₀ with c_n having a chance p_n of producing a new chicken c_n + 1 (chicken c₀ may or may not have succeeded; the earlier ones have succeeded). Suppose that the p_n are all strictly between 0 and 1, and that the infinite product p₋₁p₋₂p₋₃... equals some number p strictly between 0 and 1.

Intuitively, we should be surprised that chicken c₀ exists if p is low and not surprised if p is high. If we have observed c₀ and are considering theories as to what the chances p_n are, other things being equal, we should prefer the theories on which the product p is high to ones on which it’s low.

But what exactly does p measure? It seems to be some kind of a chance of us getting c₀. But it doesn’t measure the unconditional probability of getting an infinite sequence of chickens leading up to c₀. For that is very tiny indeed, since it is extremely unlikely that the world would contain chickens at all. It seems to be a kind of conditional probability. Let q_n be the proposition that chicken c_n exists. Then P(q₀∣q_n) = p₀p₋₁p₋₂...p_n, and so p is the limit of the conditional probabilities P(q₀∣q_n). It is plausible thus to think of p as a conditional probability of q₀ on q_−∞, which is the infinite disjunction of all the q_n.

But q_−∞ is a rather odd proposition. It is grounded in q_n for every finite n, assuming that a disjunction, even an infinite one, is grounded in its true disjuncts. Thus every one of the q_n is explanatorily prior to q_−∞. But this means that P(q₀∣q_−∞) is actually a conditional probability of q₀ on something that isn’t explanatorily prior to q₀—indeed, that is explanatorily posterior to q₀. This challenges the interpretation of p as a chance of getting chicken c₀.

I am not quite sure what conclusion to draw from the above argument. Maybe it offers some support for causal finitism, by suggesting that things are weird when you have a backwards infinite causal sequence?

Inferentialism and the completeness of geometry

The Quinean criterion for existential commitment is that we incur existential commitment precisely by affirming existentially quantified sentences. But what’s an existential quantifier?

The inferentialist answer is that an existential quantifier is anything that behaves logically like an existential quantifier by obeying the rules of inference associated with quantifiers in classical logic.

Here is a fun little problem with the pairing of the above views. Tarski proved that, with an appropriate axiomatization, Euclidean geometry is complete and consistent, i.e., for every geometric sentence ϕ, exactly one of ϕ and its negation is provable from the axioms. Now let us stipulate a philosophically curious language L^*. Syntactically, the symbols of L^* are the symbols of L but with asterisks added after every logical connective, and the sentences are of L^* are the sentences of L with an asterisk added after every connective and predicate. The semantics of L^* are as follows: the sentence ϕ of L^* means that the sentence of L formed by dropping the asterisks from ϕ is provable from the axioms of Euclidean geometry.

Inferentially, the asterisked connectives of L^* behave exactly like the corresponding non-asterisked connectives of L.

Consider the sentence ϕ of L^* that is written ∃^*x(x=^*x). This sentence, by stipulation, means that ∃x(x=x) is provable from the axioms of Euclidean geometry. According to the Quinean criterion plus inferentialism, it incurs existential commitment, because ∃^*x, since it behaves inferentially just like an existential quantifier, is an existential quantifier. Now, it is intuitively correct that ∃^*x(x=^*x) does incur existential commitment: it claims that there is a proof of ∃x(x=x), so it incurs existential commitment to the existence of a proof. So in this case, the inferentialist Quinean gets right that there is existential commitment. But rather clearly only coincidentally so! For now consider the sentence ψ that is written ∀^*x(x=^*x). Since ∀^*x behaves inferentially just like ∀x, by inferentialist Quineanism it incurs no existential commitment. But ψ means that there is a proof of ∀x(x=x), and hence incurs exactly the same kind of existential commitment as ϕ did, which said that there was a proof of ∃x(x=x).

What can the inferentialist Quinean respond? Perhaps this: The language L^* is syntactically and inferentially compositional, but not semantically so. The meaning of p∨^*q, namely that the unasterisked version of p∨^*q has a proof, is not composed from the meanings of p and of q, which respectively mean that p has a proof and that q has a proof. But that’s not quite right. For meaning-composition is just a function from meanings to meanings, and there is a function from the meanings of p and of q to the meaning of p∨^*q—it’s just a messy function, rather than the nice function we normally associate with disjunction.

Perhaps what the inferentialist Quinean should do is to insist on the intuitive non-inferentialist semantic compositional meanings for the truth-functional connectives, but not for the quantifiers. This feels ad hoc.

Even apart from Quineanism, I think the above constitutes an argument against inferentialism about logical connectives. For the asterisked connectives of L^* do not mean the same thing as their unasterisked variants in L.

Some issues concerning eliminative structuralism for second-order arithmetic

Eliminative structuralist philosophers of mathematics insist that what mathematicians study is structures rather than specific realizations of these structures, like a privileged natural number system would be. One example of such an approach would be to take the axioms of second-order Peano Arithmetic PA2, and say an arithmetical sentence ϕ is true if and only if it is true in every standard model of PA2. Since all such models are well-known to be isomorphic, it follows that for every arithmetical sentence ϕ, either ϕ or  ∼ ϕ is true, which is delightful.

The hitch here is the insistence on standard (rather than Henkin) models, since the concept of a standard depends on something very much like a background set theory—a standard model is a second-order model where every subset of Dⁿ is available as a possible value for the second-order n-ary variables, where D is the first-order domain. Thus, such an eliminative structuralism in order to guarantee that every arithmetical sentence has a truth value seems to have to suppose a privileged selection of subsets, and that’s just not structural.

One way out of this hitch is to make use of a lovely internal categoricity result which implies that if we have any second-order model, standard or not, that contains two structures satisfying PA2, then we can prove that any arithmetical sentence true in one of the two structures is true in the other.

But that still doesn’t get us entirely off the hook. One issue is modal. The point of eliminative structuralism is to escape from dependence on “mathematical objects”. The systems realizing the mathematical structures on eliminative structuralism don’t need to be systems of abstract objects: they can just as well be systems of concrete things like pebbles or points in space or times. But then what systems there are is a contingent matter, while arithmetic is (very plausibly) necessary. If we knew that all possible systems satisfying PA2 would yield the same truth values for arithmetical sentences, life would be great for the PA2-based eliminative structuralist. But the internal categoricity results don’t establish that, unless we have some way of uniting PA2-satisfying systems in different possible worlds in a single model. But such uniting would require there to be relations between objects in different worlds, and that seems quite problematic.

Another issue is the well-known issue that assuming full second-order logic is “too close” to just assuming a background set-theory (and one that spans worlds, if we are to take into account the modal issue). If we could make-do with just monadic second-order logic (i.e., the second-order quantifiers range only over unary entities) in our theory, things would be more satisfying, because monadic second-order logic has the same expressiveness as plural quantification, and we might even be able to make-do with just first-order quantification over fusions of simples. But then we don’t get the internal categoricity result (I am pretty sure it is provable that we don’t get it), and we are stuck with assuming a privileged selection of subsets.