Semantics of syntactically incorrect language

As anyone who has talked with a language-learner knows, syntactically incorrect sentences often succeed in expressing a proposition. This is true even in the case of formal languages.

Formal semantics, say of the Tarski sort, has difficulties with syntactically incorrect sentences. One approach to saving the formal semantics is as follows: Given a syntactically incorrect sentence, we find a contextually appropriate syntactically correct sentence in the vicinity (and what counts as vicinity depends on the pattern of errors made by the language user), and apply the formal semantics to that. For instance, if someone says “The sky are blue”, we replace it with “The sky is blue” in typical contexts and “The skies are blue” in some atypical contexts (e.g., discussion of multiple planets), and then apply formal semantics to that.

Sometimes this is what we actually do when communicating with someone who makes grammatical errors. But typically we don’t bother to translate to a correct sentence: we can just tell what is meant. In fact, in some cases, we might not even ourselves know how to translate to a correct sentence, because the proposition being expressed is such that it is very difficult even for a native speaker to get the grammar right.

There can even be cases where there is no grammatically correct sentence that expresses the exact idea. For instance, English has a simple present and a present continuous, while many other languages have just one present tense. In those languages, we sometimes cannot produce an exact grammatically correct translation of an English sentence. One can use some explicit markers to compensate for the lack of, say, a present continuous, but the semantic value of a sentence using these markers is unlikely to correspond exactly to the meaning of the present continuous (the markers may have a more determinate semantics than the present continuous). But we can imagine a speaker of such a language who imitates the English present continuous by a literal word-by-word translation of “I am” followed by the other language’s closest equivalent to a gerund, even when such translation is grammatically incorrect. In such a case, assuming the listener knows English, the meaning may be grasped, but nobody is capable of expressing the exact meaning in a syntactically correct way. (One might object that one can just express the meaning in English. But that need not be true. The verb in question may be one that does not have a precise equivalent in English.)

Thus we cannot account for the semantics of syntactically incorrect sentences by applying semantics to a syntactically corrected version. We need a semantics that works directly for syntactically incorrect sentences. This suggests that formal semantics are necessarily mere approximate models.

Similar issues, of course, arise with poetry.

Truth-value realisms about arithmetic

Arithmetical truth-value realists hold that any proposition in the language of arithmetic has a fully determined truth value. Arithmetical truth-value necessists add that this truth value is necessary rather than merely contingent. Although we know from the incompleteness theorems that there are alternate non-standard natural number structures, with different truth values (e.g., there is a non-standard natural number structure according to which the Peano Axioms are inconsistent), the realist and necessist hold that when we engage in arithmetical language, we aren’t talking about these structures. (I am assuming either first-order arithmetic or second-order with Henkin semantics.)

Start by assuming arithmetical truth-value necessitism.

There is an interesting decision point for truth-value necessitism about arithmetic: Are these necessary truths twin-earthable? I.e., could there be a world whose denizens who talk arithmetically like we do, and function physically like we do, but whose arithmetical sentences express different propositions, with different and necessary truth values? This would be akin to a world where instead of water there is XYZ, a world whose denizens would be saying something false if they said “Water has hydrogen in it”.

Here is a theory on which we have twin-earthability. Suppose that the correct semantics of natural number talk works as follows. Our universe has an infinite future sequence of days, and the truth-values of arithmetical language are fixed by requiring the Peano Axioms (or just the Robinson Axioms) together with the thesis that the natural number ordering is order-isomorphic to our universe’s infinite future sequence of days, and then are rigidified by rigid reference to the actual world’s sequence of future days. But in another world—and perhaps even in another universe in our multiverse if we live in a multiverse—the infinite future sequence of days is different (presumably longer!), and hence the denizens of that world end up rigidifying a different future sequence of days to define the truth values of their arithmetical language. Their propositions expressed by arithmetical sentences sometimes have different truth values from ours, but that’s because they are different propositions—and they’re still as necessary as ours. (This kind of a theory will violate causal finitism.)

One may think of a twin-earthable necessitism about arithmetic as a kind of cheaper version of necessitism.

Should a necessitist go cheap and allow for such twin-earthing?

Here is a reason not to. On such a twin-earthable necessitism, there are possible universes for whose denizens the sentence “The Peano Axioms are consistent” expresses a necessary falsehood and there are possible universes for whose denizens the sentence expresses a necessary truth. Now, in fact, pretty much everybody with great confidence thinks that the sentence “The Peano Axioms are consistent” expresses a truth. But it is difficult to hold on to this confidence on twin-earthable necessitism. Why should we think that the universes the non-standard future sequences of days are less likely?

Here is the only way I can think of answering this question. The standard naturals embed into the non-standard naturals. There is a sense in which they are the simplest possible natural number structure. Simplicity is a guide to truth, and so the universes with simpler future sequences of days are more likely.

But this answer does not lead to a stable view. For if we grant that what I just said makes sense—that the simplest future sequences of days are the ones that correspond to the standard naturals—then we have a non-twin-earthable way of fixing the meaning of arithmetical language: assuming S5, we fix it by the shortest possible future sequence of days that can be made to satisfy the requisite axioms by adding appropriate addition and multiplication operations. And this seems a superior way to fix the meaning of arithmetical language, because it better fits with common intuitions about the “absoluteness” of arithmetical language. Thus it it provides a better theory than twin-earthable necessitism did.

I think the skepticism-based argument against twin-earthable necessitism about arithmetic also applies to non-necessitist truth-value realism about arithmetic. On non-necessitist truth-value realism, why should we think we are so lucky as to live in a world where the Peano Axioms are consistent?

Putting the above together, I think we get an argument like this:

1. Twin-earthable truth-value necessitism about arithmetic leads to skepticism about the consistency of arithmetic or is unstable.

2. Non-necessitist truth-value realism about arithmetic leads to skepticism about the consistency of arithmetic.

3. Thus, probably, if truth-value realism about arithmetic is true, non-twin-earthable truth-value necessitism about arithmetic is true.

The resulting realist view holds arithmetical truth to be fixed along both dimensions of Chalmers’ two-dimensional semantics.

(In the argument I assumed that there is no tenable way to be a truth-value realist only about Σ₁⁰ claims like “Peano Arithmetic is consistent” while resisting realism about higher levels of the hierarchy. If I am wrong about that, then in the above argument and conclusions “truth-value” should be replaced by “Σ₁⁰-truth-value”.)

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.

Causal Robinson Arithmetic

Say that a structure N that has a distinguished element 0, a unary function S, and binary operations + and ⋅ is a causal Robinson Arithmetic (CRA) structure iff:

1. The structure N satisfies the axioms of Robinson Arithmetic, and

2. For any x in N, x is a partial cause of the object Sx.

The Fundamental Metaphysical Axiom of CRA is:

• For every sentence ϕ in the language of arithmetic, ϕ is either true in every metaphysically possible CRA structure or false in every metaphysically possible CRA structure.

Causal Finitism—the doctrine that nothing can have infinitely many things causally prior to it—implies that any CRA is order isomorphic to the standard natural numbers (for any element in the CRA structure other than zero, the sequence of predecessors will be causally prior to it, and so by Causal Finitism must be finite, and hence the number can be mapped to a standard natural number), and hence implies the Fundamental Metaphysical Axiom of CRA.

Given the Fundamental Metaphysical Axiom of CRA, we have a causal-structuralist foundation for arithmetic, and hence for meta-mathematics: We say that a sentence ϕ of arithmetic is true if and only if it is true in all metaphysically possible CRA structures.

Sensory-based hacking and consent

Suppose human beings are deterministic systems.

Then quite likely there are many cases where the complex play of associations combined with a specific sensory input deterministically results in a behavior in a way where the connection to the input doesn’t make rational sense. Perhaps I offer you a business deal, and you are determined to accept the deal when I wear a specific color of shirt because that shirt unconsciously reminds you of an excellent and now deceased business partner you once had, while you have found the deal dubious if I wore any other color. Or, worse, I am determined to reject a deal offered by some person under some circumstances where the difference-maker is that the person is a member of a group I have an implicit and irrational bias against. Or perhaps I accept the deal precisely because I am well fed.

If this is true, then we are subject to sensory-based hacking: by manipulating our sensory inputs, we can be determined to engage in specific behaviors that we wouldn’t have engaged in were those sensory inputs somewhat different in a way that has no rational connection with the justification of the behavior.

Question: Suppose a person consents to something (e.g., a contract or a medical procedure) due to deliberate deterministic sensory-based hacking, but otherwise all the conditions for valid consent are satisfied. Is that consent valid?

It is tempting to answer in the negative. But if one answers in the negative, then quite a lot of our consent is in question. For even if we are not victims of deliberate sensory-based hacking, we are likely often impacted by random environmental sensory-based hacking—people around us wear certain colors of shirts or have certain shades of skin. So the question of whether determinism is true impacts first-order questions about the validity of our consents.

Perhaps we should distinguish three kinds of cases of consent. First, we have cases where one gives consent in a way that is rational given the reasons available to one. Second, we have cases where one gives consent in a way that is not rational but not irrational. Third, we have cases of irrational consent.

In cases where the consent is rational, perhaps it doesn’t matter much that we were subject to sensory-based hacking.

In cases where the consent is neither rational nor irrational, however, it seems that the consent may be undermined by the hacking.

In cases where the consent is irrational, one might worry that the irrationality undercuts validity of consent anyway. But that’s not in general true. It may be irrational to want to have a very painful surgery that extends one’s life by a day, but the consent is not invalidated by the irrationality. And in cases where one irrationally gives consent it seems even more plausible that sensory-based hacking undercuts the consent.

I wonder how much difference determinism makes to the above. I think it makes at least some difference.

We have systematic overdetermination in our movements

The causal exclusion argument requires us to deny that there is systematic overdetermination between mental and physical causes.

But it is interesting to note that in the real world there is systematic overdetermination of physical movements. Suppose I raise my arm. My muscle contraction is caused by a bunch of electrons moving in the nerves between the brain and the muscle. Suppose there are N electrons involved in the electrical flow, for some large number N. But now note that except in extremely rare marginal cases, any N − 1 of the electrons are sufficient to produce the same muscle contraction. Thus, my muscle contraction is overdetermined by at least N groups of electrons. Each of these groups differs from the original N electron group by omitting one of the electrons. And each group is sufficient to produce the effect.

One might try to defend the no-systematic-overdetermination view by saying that what doesn’t happen is systematic overdetermination by non-overlapping causes. There are two problems with this approach. First, it is not empirically clear that there isn’t systematic overdetermination by non-overlapping causes. It could turn out that typically twice as many electrons are involved in nerve impulses as are needed, in which case there are two non-overlapping groups each of which is sufficient. Second, the anti-physicalist can just say that there is overlap between the mental cause and the physical cause—the mental cause is not entirely physical, but is partially so.

Alternately, one might say that there may be systematic overdetermination of physical events by physical events, but not of physical events by physical and mental events. This would need an argument.

More on God causing infinite regresses

In my previous two posts I focused on the difficulty of God creating an infinite causal regress of indeterministic causes as part of an argument from theism to causal finitism. In this post, I want to drop the indeterministic assumption.

Suppose God creates a backwards infinite causal regress of (say) chickens, where each chicken is caused by parent chickens, the parent chickens by grandparent chickens, and so on. Now, I take it that the classical theist tradition is right that no creaturely causation can function without divine cooperation. Thus, every case where a chicken is caused by parent chickens is a case of divine cooperation.

Could God’s creative role here be limited to divine cooperation? This is absurd. For then God would be creating chickens by cooperating with chickens!

So what else is there? One doubtless correct thing to say is this: God also sustains each chicken between its first moment of life and its time of death. But this sustenance doesn’t seem to solve the problem, because the sustenance is not productive of the chickens—it is what keeps each chicken in existence after it has come on the scene. So while there is sustenance, it isn’t enough. God cannot create chickens by cooperating with chickens and by sustaining them.

Thus God needs to have some special creative role in the production of at least some of the chickens, fulfilling a task over and beyond cooperation and sustenance. Furthermore, this special task must be done by God in the case of an infinite number of the chickens, since otherwise there would be a time before which that task was not fulfilled—and yet God created infinitely the chickens before that time, too, since we’re assuming an infinite regress of chickens.

What happens in these cases? One might say is that in these special cases, God doesn’t cooperate with the parent chickens. But since no creaturely causation happens without divine cooperation, in these cases the parent chickens don’t produce their offspring, which contradicts our assumption of the chickens forming a causal regress. So that won’t do.

So in these cases, we seem to have two things happening: divine cooperation with chicken reproduction and divine creation of the chicken. Since divine cooperation with chicken reproduction is sufficient to produce the offspring, and divine creation of the chicken is also sufficient, it follows that in these cases we have causal overdetermination.

Now, we have some problems. First, does this overdetermination happen in all cases of chicken reproduction or only in some? It doesn’t need to happen in all of them, since it is overdetermination after all. But if it happens only in some, then it is puzzling to ask how God chooses which cases he overdetermines and which he does not.

Second, when there is overdetermination, the overdetermination is not needed for the effect. So it seems that if God’s additional role is that of overdetermining the outcome, that role is an unnecessary role, and the chickens could be produced by mere divine cooperation, which we saw is absurd. This isn’t perhaps the strongest of arguments. One might say that while in each particular case the overdetermining divine creative action is not needed, it is needed that it occur in some (indeed, infinitely many) cases.

Third, just as it is obviously absurd if God creates chickens merely by cooperating with chickens, it seems problematic, and perhaps absurd, that God creates chickens merely by cooperating with chickens and overdetermining that cooperation.

Famously, Aquinas thinks that God could have created an infinite regress of fathers and sons, and hence presumably of chickens as well. At this point, I can think of only one plausible way of getting Aquinas out of the above arguments, and it’s not a very attractive way. Instead of saying that God cooperates with the production of offspring, we can say that occasionalism holds in every case of substantial causation, that all causation of one substance’s existence by another is a case of direct divine non-cooperative causation, with the creaturely causation perhaps only limited to the transmission of accidents. Like all occasionalism, an occasionalism about substance causation is unappealing philosophically and theologically.

God and chancy infinite causal regresses

Suppose that a dod is a critter that chancily, with probability 1/2, causes one offspring during its life. The lifespan of a dod is one year. Further, imagine that like Sith, there are only ever one or two dods at a time, because each dod dies not long after reproducing, and if there were two or more mature dods at once, they’d fight to the death.

Now, imagine we have an infinite regress of dods, because each dod comes from an earlier dod. This would be hard to believe! After all, at any time at which we have a dod, we should be extremely (infinitely?) surprised that the dods haven’t died out yet. After all the probability that, given a dod at some time that there would be a dod in n years exponentially decreases with n.

Assuming causal finitism is false, it seems God could intentionally create an infinite regress of dods. But what would that look like? Here’s one story. God overrides the chances and directly and intentionally creates a backwards-infinite (and maybe even forwards-infinite, if he so chooses) sequence of dods. In that case, within that sequence the 1/2 chance of dod reproduction plays no explanatory role whatsoever. It seems we have occasionalism or a miracle or both. In any case, it does not appear that we actually have an infinite causal regress of dods in this case—the causation between dods, with its 1/2 chance, seems not to have any explanatory role. So the “overriding” story doesn’t work.

The other option is the Thomistic story. God doesn’t override chances. Instead, through primary causation, God concurs in creaturely causation and makes the finite cause produce its effect in such a way that the finite cause is fully acting as an indeterministic cause (this goes along with a view on which God can make us freely and indeterministically choose things). But this is very strange. For what explanatory role does the 1/2 in the chancy causation play? Assuming God wanted there to be an infinite sequence of dods, he could do exactly the same thing if the chance were 1/10 or 9/10 or even 1. It seems that the dod reproduces if and only if God intends the dod to reproduce, and whether God intends the dod to reproduce seems to have nothing to do with the “1/2” in the dod’s reproductive probabilities—it’s not plausible that God has probability 1/2 of intending each given dod to reproduce. And if God had probability 1/2 of intending each given dod to reproduce, how could he intentionally ensure that there ever are any dods, since the probability that God has infinitely many of these individual-dod-reproduction intentions is zero.

So we have problems. This gives further evidence that theism implies causal finitism.

From theism to causal finitism

Causal Finitism—the thesis that nothing can have an infinite causal history—implies that there is a first cause, and our best hypothesis for what a first cause would be is God. Thus:

1. If Causal Finitism is true, God exists.

But I think one can also argue in the other direction:

2. If God exists, Causal Finitism is true.

Aquinas wouldn’t like this since he thought that God could create a per accidens ordered backwards-infinite causal series.

In this post, I want to sketch an argument for (2). The form of the argument is this.

3. God cannot create a sequence of beings ..., A₋₃, A₋₂, A₋₁, A₀ where each being causes the next one.

4. If God cannot create such a sequence, such a sequence is impossible.

5. The best explanation of the impossibility of such a sequence is Causal Finitism.

Claim (4) comes from omnipotence. Claim (5) is I think the weakest part of the argument. Causal Finitism follows logically from the conjunction of two theses, one ruling out backwards-infinite causal chains and the other ruling out infinite causal cooperation (a precise statement and a proof is given in Chapter 2 of my Infinity book). But I am now coming to think that there is a not crazy view where one accepts the anti-chain part of Causal Finitism but not the anti-cooperation part. However, (a) the main cost of Causal Finitism come from the anti-chain part (the anti-chain part is what forces either a discrete time or a discrete causal reinterpretation of physics), (b) there are significant anti-paradox benefits to maintaining the anti-cooperation part, and (c) the theory may seem more unified in having both parts.

Now let’s move on to (3). Here is an argument. Say that an instance of causation is chancy provided that the outcome has a probability less than one.

6. If God can create a backwards-infinite causal sequence of beings, he can create a backwards-infinite chancy causal sequence of beings as the only thing in creation.

7. Necessarily, if God creates a backwards-infinite chancy causal sequence of beings as the only thing in creation, then there is no creature x such that God determines x to exist.

8. Necessarily, if God creates, he acts in a way that determines that something other than God exists.

9. Necessarily, if God determines that something other than God exists then there is a creature x that God determines x to exist.

10. Necessarily, if God creates a backwards-infinite chancy causal sequence of beings, then there is a creature x such that God determines x to exist. (8,9)

11. Hence, God cannot create a backwards-infinite chancy causal sequence of beings. (7,10)

12. Hence, God cannot create a backwards-infinite causal sequence of beings. (6,11)

The thought behind (6) is an intuition about modal uniformity. I think (6) is probably the most vulnerable part of the argument, but I don’t think it’s the one Aquinas would attack. What I think Aquinas would attack would most likely be (7). I will get to that shortly.

But first a few words about (8). In theory, it is possible to determine that something exists without determining any particular thing to exist. One can imagine a being with a chancy causal power such that if it waves a wand necessarily either a bunny or a pigeon is caused to exist, with the probability of the bunny being 1/2 and the probability of the pigeon being 1/2. But God is not like that. God’s will is essentially efficacious and not chancy. God can play dice with the universe, but only by creating dice. Thus, if God wanted to ensure there is a bunny or a pigeon without ensuring which specific one exists, he would have to create a random system that has chancy propensities for a bunny and for a pigeon and that must exercise one of the two propensities.

In fact, I think divine simplicity may imply this. For by divine simplicity, any two possible worlds that differ must differ in something outside God. Now consider a world w₁ where God determines a bunny to exist, and a world w₂ where God merely determines that a bunny or a pigeon exists and in fact a bunny is what comes about. There seems to be no difference outside God between these two worlds (one might wonder about the relation of being-created: could there be an relation of being-created-chancily and being-created-non-chancily? this seems fishy to me, and suggests a regress—how are the two relations differently related to God? and do we want to multiply such relations, saying there is such a thing as being-created-chancily-with-probability-0.7?). If both worlds are possible, by divine simplicity they must be the same, which is absurd. So at least one must be impsosible. And w₂ is a better candidate for that than w₁.

That still doesn’t establish (8). For I admitted that God can play dice if he creates dice. Thus, it seems that God could determine that something exists without determining where it’s A or B or C (say) by determining there to be dice that decide whether A or B or C are produced. But on this story, God still determines there to be dice, so there is an x—a die—that God determines to exist. I think a bit more could be said here, but as I said, I don’t think this is the main thing Aquinas would object to.

Back to (7). Why can’t God create a chancy backwards-infinite causal sequence while determining some item A_n in it to exist? Well, the sequence is chancy, so the probability that A_n − 1 causes A_n given that A_n − 1 exists is some p < 1. But, necessarily, if one creature causes another, it does so with divine cooperation (Aquinas will not disagree), and conversely if God cooperates with one creature to cause another, the one creature does cause the other. That the probability that God cooperates with A_n − 1 to cause A_n is equal to the probability that A_n − 1 causes A_n, because necessarily one thing happens if and only if the other does. Thus, the probability that God cooperates with A_n − 1 to cause A_n, given that A_n − 1 exists, is p. But p < 1, so it sure doesn’t look like a case of God determining A_n to exist!

But perhaps there is something like overdetermination, but between determination and chanciness (so not exactly over-determination). Perhaps God both determines A_n to exist and chancily cooperates with A_n − 1 to produce A_n. One problem with this hypothesis is with divine simplicity: it does not seem that there is any difference outside God between a world where God does both and God merely cooperates or concurs. But Aquinas may respond: “Yes, exactly. Necessarily, when one creature chancily causes another, God’s primary causation determines which specific outcome results. Thus there is no world where God merely cooperates.” So now the view is that whenever we have chancy causation, necessarily God determines the outcome. But suppose I chancily toss a coin, and it has chance 1/2 of heads and chance 1/2 of tails. Then on this view, I get heads if and only if God determines that I get heads. Hence the chance that God determines I get heads is 1/2. But it seems plausible that God’s determinations are not measured by numerical probabilities, and in any case that they are not measured by numerical probabilities coming from our world’s physics!

Megethology as mathematics and a regress of structuralisms

In his famous “Mathematics is Megethology”, Lewis gives a brilliant reduction of set theory to mereology and plural quantification. A central ingredient of the reduction is a singleton function which assigns to each individual a singleton of which the individual is the only member. Lewis shows that assuming some assumptions on the size of reality (namely, that it’s very big) there exists a singleton function, and that different singleton functions will yield the same set theoretic truths. The result is that the theory is supposed to be structuralist: it doesn’t matter which singleton function one chooses, just as on structuralist theories of natural numbers it doesn’t matter if one uses von Neumann ordinals or Zermelo ordinals or anything else with the same structure. The structuralism counters the obvious objection to Lewis that if you pick out a singleton function, it is implausible that mathematics is the study of that one singleton function, given that any singleton function yields the same structure.

It occurs to me that there is one hole in the structuralism. In order to say “there exists a singleton function”, Lewis needs to quantify over functions. He does this in a brilliant way using recently developed technical tools where ordered pairs of atoms are first defined in terms of unordered pairs and an ordering is defined by a plurality of fusions, relations on atoms are defined next, and so on, until finally we get functions. However, this part can also be done in a multiplicity of ways, and it is not plausible that mathematics is the study of singleton functions in that one sense of function, given that there are many sense of function that yield the same structure.

Now, of course, one might try to give a formal account of what it is for a construction to have the structure of functions, what it is to quantify not over functions but over function-notions, one might say. But I expect a formal account of quantification over function-notions will presumably suffer from exactly the same issue: no one function-notion-notion will appear privileged, and a structuralist will need to find a way to quantify over function-notion-notions.

I suspect this is a general feature with structuralist accounts. Structuralist accounts study things with a common structure, but there are going to be many accounts of common structure that by exactly the same considerations that motivate structuralism require moving to structuralism about structure, and so on. One needs to stop somewhere. Perhaps with an informal and vague notion of structure? But that is not very satisfying for mathematics, the Queen of Rigor.

Metaphysical universism

Here’s a metaphysical view I haven’t seen: the fundamental obejcts (priority version) or the only objects (existence version) are universes, but there can be more than one of these. Call this metaphysical universism (as distinguished from Quisling’s philosophy).

If in fact there is only one universe, metaphysical universism extensionally coincides with monism. But even in that case, metaphysical universism is a different theory, because it has different modal implications. And if we live in a multiverse, metaphysical universism is extensionally different from monism, since monism says that the one fundamental (priority) or one and only (existence) entity is the multiverse as a whole, not the universes.

I can think of two main advantages of metaphysical universism over monism.

First, suppose there is only one universe. It is plausible that there could be another in addition to this one. Metaphysical universism embraces this possibility. Monism only says that The One could have been bigger so as to comprise two spatiotemporally disconnected regions.

Second, there is an old intuition that being and unity are connected. In a multiverse, monism violates this intuition, for in a multiverse it is the universes that have unity, not the multiverse. Indeed quantum entanglement arguments for monism in the context of a non-Everettian multiverse seem to me to point more towards metaphysical universism than monism.

On the other hand, monism has a significant advantage over metaphysical universism insofar as monism solves the problem of truthmakers of negative and universal claims by making The One be the truthmaker of all of them.

Of course, both theories are false.

The ethics of plant care

If someone devotes a significant part of their life to affectionately caring for plastic flowers, there is something wrong with them. Not so in the case of real, living plants. This points to me to the idea that life as such, and not just conscious life or the life of animals, is a valuable thing.

I don’t want to say that it’s always bad to affectionately care for artifacts. When the artifacts have an intimate and significant connection with human beings, as in the case of a chair that grandma made or a work of art, such affectionate care can make sense. But having an affectionate care for plants makes sense even in the absence of a connection to human beings.

What about microscopic forms of life? Can it make sense to fondly feed a bacterium? I think so, but I agree that the case is less clear.

Grim Toe-Cutters

Imagine that Fred has all ten toes at 10 am, and there are infinitely many grim reapers. When a grim reaper wakes up, it looks at Fred, and if he has all his ten toes, it cuts one off and destroys it; otherwise, it does nothing. There are no other toe-cutters around.

Suppose, further, that grim reaper wake-up times can be set by you to any times between 10 and noon, endpoints not included. If you set the activation times to be such that there is a first activation time after 10 am (e.g., the nth reaper wakes up 60/n minutes before noon), there is no paradox of any sort. But if you set the times such that they are all after 10 am, but before every activation time there is another activation time, then… well, then logic guarantees that Fred will get a toe cut off infinitely many times and will regrow a toe infinitely many times! For without toe-regrowing, we get a paradox.

This is, of course, logically and metaphysically possible. Toes can regrow, and it is metaphysically though perhaps not physically possible for them to do so quickly. But what is amazing is that just by setting wake-up times for grim toe-cutters, we can make this miracle happen.

Grim Reapers and logical impossibility

The main objection to the Grim Reaper paradox as an argument against infinite causal sequences is the Unsatisfiable Pair (UP) objection that notes that paradox sets up an impossible situation—and that’s why it’s impossible!

I’m exploring a response that distinguishes metaphysical and (narrowly) logical unsatisfiability. The Grim Reaper situation is not logically unsatisfiable. The UP objection (well, really, Unsatisfiable Quadruple) notes that the following cannot all be true:

1. For all n > 0, the nth reaper wakes up at 60/n minutes after 10 am and kills Fred if and only if Fred is alive.

2. Fred is alive at 10 am.

3. There are no possible causes of Fred’s death other than those described in (1).

4. There are no possible causes of Fred’s resurrection.

But all that’s needed to have these four claims hold is for each reaper to kill Fred and then have Fred causelessly come back to life before the next one kills him. And while I think causeless resurrections are metaphysically impossible, they are (narrowly) logically coherent.

In other words, for the UP objection to work, the unsatisfiability must be metaphysical, not merely narrowly logical. But this, I think, negatively affects the force of the UP objection. For instance, in my Infinity book I consider Grim Reapers with adjustable wake-up times, and I note that for some wake-up time settings (say, the nth reaper wakes up 60/n minutes before noon) there is no paradox, and I ask what metaphysical force prevents the wake-up time settings from being the paradoxical ones. Daniel Rubio in a review of the book responds (in the context of a parody) that “no metaphysical thesis is required to explain this impossibility; the fact that it would lead to a contradiction is enough.” But in fact a metaphysical thesis is required to explain the impossibility, since there is no contradiction (in the narrowly logical sense) in (1)–(4).

Perhaps this is not a big deal. After all the metaphysical thesis here, that causeless events are impossible, is one that I do accept. But nonetheless it is a metaphysical thesis, as such on par with causal finitism, and hence when we consider the explanation of the impossibility of the Grim Reaper story and the impossibility of various other of the causal paradoxes that I discuss, there is something appealing about seeing the case as nonetheless offering support for causal finitism, which explains all of them, while the thesis about causeless events being impossible does not.

Unreliable Grim Reapers

As usual, Fred is alive at 10 am, and there is an infinite sequence of Grim Reapers, where the nth has an alarm set for 60/n minutes after 10 am, and if the alarm goes off, it checks if Fred is dead, and swings its scythe at Fred if and only if Fred is alive. But here’s the twist. These Grim Reapers are unreliable killers. The probability that the nth Reaper’s swing would succeed in killing Fred is 1/n^p, where p is some positive real number, the same for each Reaper, and independently of all other relevant events.

Here’s the fun thing. It seems possible for Fred to survive the whole ordeal. All it takes is for every Grim Reaper to fail at killing Fred. Nothing absurd happens then. Moreover, it seems this isn’t the only way for absurdity to be avoided in this case. We could also suppose that the nth Reaper kills Fred, while Reapers n + 1, n + 2, … all fail.

Suppose we adopt what seems the best alternative to Causal Finitism, namely the Inconsistent Pair response to the original Grim Reaper paradox, which says that the reason the original paradox is impossible is simply because it embodies an Inconsistent set of propositions—some Reaper has to kill Fred and none can. If that’s what’s wrong with the original Grim Reaper paradox, then it seems we have to accept my Unreliable Reaper story as possible.

But things are a little bit more complicated. The only way to avoid paradox in the Unreliable Reaper story is if there is some n ≥ 0 such that all the Reapers starting with Reaper n + 1 fail. But now suppose that 0 < p ≤ 1. Then the event that all the Reapers starting with Reaper n + 1 fail is less than or equal to (1−1/(n+1)^p)(1−1/(n+2)^p)(1−1/(n+3)^p)... = 0 (this is because Σ_k 1/k^p = ∞ if p ≤ 1). Thus the probability that we have avoided paradox is 0. Hence, if we have to avoid paradox, a specific zero probability event—namely, the event of paradox-avoidance—has to happen (the probability of a countable disjunction of zero probability events is zero). But if it has to happen, it can’t be probability zero, but must be probability one!

Perhaps here we bring back the Inconsistent Pair response. We say that my Unreliable Reaper story is impossible if p ≤ 1, because if p ≤ 1, then a zero probability event has probability one, which is inconsistent. No such problem occurs if p > 1. Thus, on this version of the Inconsistent Pair response, my Unreliable Reaper story is impossible if the success probability of the nth Reaper is 1/n^p for p ≤ 1 but possible if p > 1. And that’s pretty counterintuitive.

On finitistic addition

By a finite alphabet encoding of a set X, such as the real numbers, I mean a one-to-one function ψ from X to countably infinite sequences s₀s₁... taken from some finite alphabet. For instance, standard decimal encoding, with a decision whether to have infinite sequences of trailing nines or not, is a finite alphabet encoding of the reals, with the alphabet consisting of ten digits, a decimal point and a sign. Write ψ_k(x) for the kth symbol in the encoding ψ(x) of x.

A function f from Rⁿ to R is finitistic with respect to a finite alphabet encoding ψ provided that there is a function h from the natural numbers to the natural numbers such that the value of ψ_k(f(x₁,...,x_n)) depends only on the first h(k) symbols in each of ψ(x₁), ..., ψ(x_n).

This concept is related to concepts in “real computation”, but I am not requiring that the finite dependences be all implemented by the same Turing machine.

Theorem: Let X be any infinite divisible commutative group. Then addition on X is not finitistic with respect to any finite alphabet encoding.

A divisible group X is one where for every x ∈ X there is a y such that ny = x. The real numbers under addition are divisible. So are the rationals. So is the set of all rotations in the plane.

This has a somewhat unhappy consequence for Information Processing Finitism. If reality encodes real numbers in a discrete way consistent with IPF, we should not expect each real number to have a uniquely specified encoding.

Proof of Theorem: Suppose addition is finitistic with respect to ψ. Let F be the algebra on X generated by the sets of the form {x : ψ_k(x) = α}. If addition is finitistic, then for any A ∈ F, there is a finite sequence of pairs (A₁,B₁), ..., (A_N,B_N) of sets in F such that

1. {(x,y) : x + y ∈ A} = ⋃_i(A_i×B_i).

Therefore:

2. x + y ∈ A if and only if x ∈ ⋃{A_i : y ∈ B_i}.

Thus:

3.  − y + A =  ∪ {A_i : y ∈ B_i}.

Now as y varies over the members of X, there are at most 2^N different sets generated by the right hand side. Thus,  − y + A can take on only finitely many values. Hence, A has only finitely many translates.

But this is impossible. Let Z be the set of x such that x + A = A. This is an additive subgroup of X. Note that x + Z = y + Z iff x − y ∈ Z iff (x−y) + A = A iff x + A = y + A. Thus, if there are only finitely many x + A, there are finitely many x + Z. Hence X/Z is a finite group. Let n be its order. Then n[x] = 0 for every coset [x] = x + Z in R/Z. For any x ∈ X choose y such that ny = x. Then n[y] = 0, and so [x] = 0, thus Z = X. It follows that A is invariant under every translation, so it must be either ⌀ or X. Hence |F| ≤ 2, which is absurd since F is infinite as X is infinite and ψ is one-to-one.

(I got the main idea for this proof from the answer here.)