Mereology, plural quantification and free lunches

It is sometimes claimed that arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a new way of talking without any deep philosophical (or at least metaphysical) commitments.

I think this is false.

Consider this Axiom of Choice schema for mereology:

1. If for every x and y such that ϕ(x) and ϕ(y), either x = y or x and y don’t overlap, and if every x such that ϕ(x) has a part y such that ψ(y), then there is a z such that for every x such that ϕ(x), there is common part y of x and z such that ψ(y).

Or this Axiom of Choice schema for pluralities:

2. If for all xx and yy such that ϕ(xx) and ϕ(yy) either xx and yy are the same or have nothing in common, then there are zz that have exactly one thing in common with every xx such that ϕ(xx).

If arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a handy way of talking, then whether (1) or (2) is correct is just a verbal question.

But (1) and (2) respectively imply mereological and plural Banach-Tarski paradoxes:

3. If z is a solid ball made of points, then it has five pairwise non-overlapping parts, of which the first two can be rigidly moved to be pairwise non-overlapping and compose another ball of the same size as z, and the last three can likewise be so moved.

4. If the xx are the points of a solid ball, then there are aa, bb, cc, dd and ee which have nothing pairwise in common and such that together they make up xx, and there are rigid motions that allow one to move aa and bb into pluralities that have nothing in common but make up a solid ball of the same size as xx and to move cc, dd and ee into pluralities that have nothing in common and make up another solid ball of the same size.

Conversely, assuming ZF set theory is consistent, there is no way to prove (3) and (4) if we do not have some extension to the standard axioms of mereology or plurals like the Axiom of Choice. The reason is that we can model pluralities and mereological objects with sets of points in three-dimensional space, and either (3) or (4) in that setting will imply the Banach-Tarski paradox for sets, while the Banach-Tarski paradox for sets is known not to be provable from ZF set theory without Choice.

But whether (3) or (4) is true is not a purely verbal question.

One reason it’s not a purely verbal question is intuitive. Banach-Tarski is too paradoxical for it or its negation to be a purely verbal thing.

Another is a reason that I gave in a previous post with a similar argument. Whether the Banach-Tarski paradox holds for sets is not a purely verbal question. But assuming that the Axiom of Separation can take formulas involving mereological terminology or plural quantification, each of (3) and (4) implies the Banach-Tarski paradox for sets.

Some stuff about models of PA+~Con(PA)

Assume Peano Arithmetic (PA) is consistent. Then it can’t prove its own consistency. Thus, there is a model M of PA according to which PA is inconsistent, and hence, according M, there is a proof of a contradiction from a finite set of axioms of PA. This sounds very weird.

But it becomes less weird when we realize what these claims do and do not mean in M.

The model M will indeed contain an M-natural number a that according to M encodes a finite sequence of axioms of PA, and it will also contain an M-natural number p that according to M encodes a proof of a contradiction using the axioms encoded in A.

However, here are some crucial qualifications. Distinguish between the M-natural numbers that are standard, i.e., correspond to an actually natural number, one that from the point of view of the “actual” natural numbers is finite, and those that are not. The latter are infinite from the point of view of the actual natural numbers.

First, the M-natural number a is non-standard. For a standard natural number will only encode a finite number of axioms, and for any finite subtheory of PA, PA can prove its consistency (this is the “reflexivity of PA”, proved by Mostowski in the middle of the last century). Thus, if a were a standard natural number, according to M there would be no contradiction from the axioms in a.

Second, while every item encoded in a is according to M an axiom of PA, this is not actually true. This is because any M-finite sequence of M-natural numbers will either be a standardly finite length sequence of standard natural numbers, or will contain a non-standard number. For let n be the largest element in the sequence. If this is standard, then we have a standardly finite length sequence of standard natural numbers. If not, then the sequence contains a non-standard number. Thus, a contains something that is not axiom of PA.

In other words, according to our model M, there is a contradictory collection of axioms of PA, but when we query M as to what that collection is, we find out that some of the things that M included in the collection are not actually axioms of PA. (In fact, they won’t even be well-formed formulas, since they will be infinitely long.) So a crucial part of the reason why M disagrees with the “true” model of the naturals about the consistency of PA is because M disagrees with it about what PA actually says!

Plato and teaching philosophy to the young

In the Republic, Plato says philosophy education shouldn’t start until age 30. I’ve long worried about Plato’s concern about providing young people with tools that, absent intellectual and moral maturity, can just as well be used for sophistry.

Exegetically, however, I think I was missing an important point: Plato is talking about his utopian society, where one can (supposedly) count on society raising the young person to practice the virtues and live by the truth (except for the noble lie). We do not live in such a society. It could well be the case that in our society, young people need the tools.

We might make a judgment like this. Absent the tools of a philosophical education, an intelligent young person set afloat on the currents of our society maybe is 50% likely to be led astray by these currents. The tools are unreliable especially in the hands of the young: perhaps the tools have a 65% chance of leading to the right and 35% of leading to ill. That’s still better than letting the young person navigate society without the tools. But if our society were better—as Plato thinks is the case in his Republic—then the unreliable tools might be worse than just letting society form one.

A puzzle about consistency

Let T₀ be ZFC. Let T_n be T_n − 1 plus the claim Con(T_n − 1) that T_n − 1 is consistent. Let T_ω be the union of all the T_n for finite n.

Here’s a fun puzzle. It seems that T_ω should be able to prove its own consistency by the following reasoning:

If T_ω is inconsistent, then for some finite n we have T_n inconsistent. But Con(T_n) is true for every finite n.

This sure sounds convincing! It took me a while to think through what’s wrong here. The problem is that although for every finite n, T_ω can prove Con(T_n), it does not follow that T_ω can prove that for every finite n we have Con(T_n).

To make this point perhaps more clear, assume T_n is consistent for all n. Then Con(T_n) cannot be proved from T_n. Thus any finite subset of T_ω is consistent with the claim that for some finite n the theory T_n is inconsistent. Hence by compactness there is a model of T_ω according to which for some finite n the theory T_n is inconsistent. This model will have a non-standard natural number sequence, and “finite” of course will be understood according to that sequence.

Here’s another way to make the point. The theory T_ω proves T_ω consistent if and only if T_ω is consistent according to every model M. But the sentence “T_ω is consistent according to M” is ambiguous between understanding “T_ω” internally and externally to M. If we understand it internally to M, we mean that the set that M thinks consists of the axioms of ZFC together with the ω-iteration of consistency claims is consistent. And this cannot be proved if T_ω is consistent. But if we understand “T_ω” externally to M, we mean that upon stipulating that S is the object in M’s universe whose members^M correspond naturally to the members^V of T_ω (where V is “our true set theory”), according to M, it will be provable that the set S is consistent. But there is a serious problems: there just may be no such object as S in the domain of M and the stipulation may fail. (E.g., in non-standard analysis, the set of finite naturals is never an internal set.)

(One may think a second option is possible: There is such an object as S in M’s universe, but it can’t be referred to in M, in the sense that there is no formula ϕ(x) such that ϕ is satisfied by S and only by S. This option is not actually possible, however, in this case.)

Or so it looks to me. But all this is immensely confusing to me.

Existential inertia and spacetime

According to the principle of existential inertia:

1. If x exists at t₁ and t₂ > t₁ and there is no cause of x’s not existing at t₂, then x exists at t₂.

This sounds weird, and one way to get at the weirdness for me is to put it in terms of relativity theory. Times are spacelike hypersurfaces. So, then:

2. If x exists somewhere on a spacelike hypersurface H₁ and H₂ is a later spacelike hypersurface and there is no cause of x’s not existing on H₂, then x exists on H₂.

This seems weird to me. Why should being in one specific area of spacetime metaphysically push one to exist in another specific area of spacetime? I can see how existing in one area of spacetime could physically push one to exist in another. But metaphysically? That seems odd.

Non-formal provability

A simplified version of Goedel’s first incompleteness theorem (it’s really just a special case of Tarski’s indefinability of truth) goes like this:

• Given a sound semidecidable system of proof that is sufficiently rich for arithmetic, there is a true sentence g that is not provable.

Here:

• sound: if s is provable, s is true

• semidecidable: there is an algorithm that given any provable sentence verifies in a finite number of steps that it is provable.

The idea is that we start with a precisely defined ‘formal’ notion of proof that yields semidecidability of provably, and show that this concept of proof is incomplete—there are truths that can’t be proved.

But I am thinking there is another way of thinking about this stuff. Suppose that instead of working with a precisely defined concept of proof, we have something more like a non-formal or intuitive notion of proof, which itself is governed by some plausible axioms—if you can prove this, you can prove that, etc. That’s kind of how intuitionists think, but we don’t need to be intuitionists to find this approach attractive.

Note that I am not explicitly distinguishing axioms.

The idea is going to be this. The predicate P is not formally defined, but it still satisfies some formal constraints or axioms. These can be formulated in a formal language (Brouwer wouldn’t like this) that has a way of talking about strings of symbols and their concatenation and allows one to define a quotation function that given a string of symbols returns a string of symbols that refers to the first string.

One way to do this is to have a symbol ′α′ for any symbol α in the original language which refers to α, and a concatenation operator +, so one can then quote αβγ as ′α′ + ′β′ + ′γ′. I assume the language is rich enough to define a quotation function Q such that Q(x) is the quotation of a string x.

To formulate my axioms, I will employ some sloppy quotation mark shorthand, partly to compensate for the difficulty of dealing with corner quotes on the web. Thus, ′αβγ′ is shorthand for ′α′ + ′β′ + ′γ′, and as needed I will allow substitution inside the quotation marks. If there are nested quotation marks, the inner substitutions are resolved first.

1. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ϕ′), then P(′∼ϕ′).

2. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ψ→ϕ′), then P(′ϕ′).

3. For all sentences ϕ, we have P(′P(′ϕ′)→ϕ′).

4. If ϕ has a formal intuitionistic proof from sufficiently rich axioms of concatenation theory, then P(′ϕ′).

Here, (1) and (2) embody a little bit of facts about proof, both of which facts are intuitionistically and classically acceptable. Assumption (3) is the philosophically heaviest one, but it follows from its being axiom that if ϕ is provable, then ϕ, together with the fact that all axioms count as provable. That a formal intuitionistic proof is sufficient for provability is uncontroversial.

Using similar methods to those used to prove Goedel’s first incompleteness theorem, I think we should now be able to construct a sentence g and the prove, in a formal intuitionistic proof in a sufficiently rich concatenation theory, that:

5. g ↔︎  ∼ P(′g′).

But these facts imply a contradiction. Since 5 can be proved in our formal way, we have:

6. P(′g↔︎∼P(′g′)′). By 4.

7. P(′P(′g′)→g′). By 3.

8. P(′g′). By 6, 7 and 2.

9. P(′∼g′). By 6, 8 and 1.

Hence the system P is inconsistent in the sense that it makes both g and  ∼ g are provable.

This seems to me to be quite a paradox. I gave four very plausible assumptions about a provability property, and got the unacceptable conclusion that the provability property allows contradictions to be proved.

I expect the problem lies with 3: it lets one ‘cross levels’.

The lesson, I think, is that just as truth is itself something where we have to be very careful with the meta- and object-language distinction, the same is true of proof if we have something other than a formal notion.

Our Baylor alligator

A year or two ago, some juvenile alligators moved into our area or were moved in (we are right on the edge of the alligator zone in Waco). I've been hoping to see one of them. Finally, today, I did, right by Baylor's marina, as you can see from the reflection of the Baylor logo.

A new argument for causal finitism

I will give an argument for causal finitism from a premise I don’t accept:

1. Necessary Arithmetical Alethic Incompleteness (NAAI): Necessarily, there is an arithmetical sentence that is neither true nor false.

While I don’t accept NAAI, some thinkers (e.g., likely all intuitionists) accept it.

Here’s the argument:

2. If infinite causal histories are possible, supertasks are possible.

3. If supertasks are possible, for every arithmetical sentence, there is a possible world where someone knows whether the sentence is true or false by means of a supertask.

4. If for every arithmetical sentence there is a possible world where someone knows whether the sentence is true or false by means of a supertask, there is a possible world where for every arithmetical sentence someone knows whether it is true or false.

5. Necessarily, if someone knows whether p is true or false, then p is true or false.

6. So, if infinite causal histories are possibly, possibly all arithmetical sentences are true or false. (2-5)

7. So, infinite causal histories are impossible. (1, 6)

The thought behind (3) is that if for every n it is possible to check the truth value of ϕ(n) by a finite task or supertask, then by an iterated supertask it is possible to check the truth values of ∀xϕ(x) (and equivalently ∃xϕ(x)). Since every arithmetical sequence can be written in the form Q₁x₁...Q_kx_kϕ(x₁,...,x_k), where the truth value of ϕ(n₁,...,n_k) is finitely checkable, it follows that every arithmetical sequence can have its truth value checked by a supertask.

The thought behind (4) is that one can imagine an infinite world (say, a multiverse) where for every arithmetical sentence ϕ the relevant supertask is run and hence the truth value of the sentence is known.

Provability and truth

The most common argument that mathematical truth is not provability uses Tarski’s indefinability of truth theorem or Goedel’s first incompleteness theorem. But while this is a powerful argument, it won’t convince an intuitionist who rejects the law of excluded middle. Plus it’s interesting to see if a different argument can be constructed.

Here is one. It’s much less conclusive than the Tarski-Goedel approach. But it does seem to have at least a little bit of force. Sometimes we have experimental evidence (at least of the computer-based kind) for a mathematical claim. For instance, perhaps, you have defined some probabilistic setup, and you wonder what the expected value of some quantity Q is. You now set up an apparatus that implements the probabilistic setup, and you calculate the average value of your observations of Q. After a billion runs, the average value is 3.141597. It’s very reasonable to conclude that the last digit is a random deviation, and that the mathematically expected value of Q is actually π.

But is it reasonable to conclude that it’s likely provable that the expected value of Q is π? I don’t see why it would be. Or, at least, we should be much less confident that it’s provable than that the expected value is π. Hence, provability is not truth.

Reducing promises to assertions

To promise something, I need to communicate something to you. What is that thing that I need to communicate to you? To a first approximation, what I need to communicate to you is that I am promising. But that’s circular: it says that promising is communicating that I am promising. This circularity is vicious, because it doesn’t distinguish promising from asking: asking is communicating that I am asking.

But now imagine I have a voice-controlled robot named Robby, and I have programmed him in such a way that I command him by asserting that Robby will do something because I have said he will do it. Thus, to get him to vacuum the living room, I assert “Robby will immediately vacuum the living room because I say so.” As long as what I say is within the range of Robby’s abilities, any statement I make in Robby’s vicinity about what he will do because I say he will do it is automatically true. This is all easily imaginable.

Now, back to promises. Perhaps it works like this. I have a limited power to control the normative sphere. This normative power generates an effect in normative space precisely when I communicate that I am generating that effect. Thus, I can promise to buy you lunch by asserting “I will be obligated to you to buy you lunch.” And I permit you to perform heart surgery by asserting “You will cease to have a duty of respect for my autonomy not to perform heart surgery on me.” As long as what I say is within my normative capabilities, by communicating that I am making it true by communicating it, I make it be true, just as Robby will do what I assert he will do because of my say-so, as long as it is within his physical capabilities.

This solves the circularity problem for promising because what I am communicating is not that I am promising, but the normative effect of the promising:

1. x promises to ϕ to y if and only if x successfully exercises a communicative normative power to gain an obligation-to-y by ϕing

2. a communicative normative power for a normative effect F is a normative power whose object is F and whose successful exercise requires the circumstance that one express that one is producing F by communicating that one is so doing.

There are probably some further tweaks to be made.

Of course, in practice, we communicate the normative effect not by describing it explicitly, but by using set phrases, contextual cues, etc.

This technique allows us to reduce promising, consenting, requesting, commanding and other illocutionary forces to normative power and communicating, which is basically a generalized version of assertion. But we cannot account for communicating or asserting in this way—if we try to do that, we do get vicious circularity.

A curious poker variant

In some games like Mafia, uttering falsehoods is a part of the game mechanic. These falsehoods are no more lies than falsehoods uttered by an actor in a performance are lies.

Now consider a variant of poker where a player is permitted to utter falsehoods when and only when they have a Joker in hand. In this case when the player utters a falsehood with Joker in hand, there is no lie. The basic communicative effect of uttering s is equivalent to asserting “s or I have a Joker in hand (or both)”, though there may be additional information conveyed by bodily expression, tone of voice, or context.

If this analysis of poker variant is correct, then the following seems to follow by analogy. Suppose, as many people think, that it is morally permissible to utter falsehoods in “assertoric contexts” to save innocent lives. (An assertoric context is roughly one where the speaker is appropriately taken to be asserting.) Given that we are always playing the “morality game”, by analogy this would mean that in paradigm instances when we utter a declarative sentence s, we are actually communicating something like “s or I am speaking to save innocent lives.” If this is right, then it is impossible to lie to save innocent lives, just as in my poker variant it is impossible to lie when one knows one has the Joker in hand (unless maybe one is really bad at logic).

The above argument supports this premise:

1. If it is morally permissible to utter falsehoods in assertoric contexts to save innocent lives, it is not possible to lie to save innocent lives.

But:

2. It is possible to lie to save innocent lives.

I conclude:

3. It is not morally permissible to utter falsehoods in assertoric contexts to save innocent lives.

In short: lying is wrong, even to save innocent lives.

Evolution of my views on mathematics

I have for a long time inclined towards ifthenism in mathematics: the idea that mathematics discovers truths of the form "If these axioms are true, then this thing is true as well."

Two things have weakened my inclination to ifthenism.

The first is that there really seems to be a privileged natural number structure. For any consistent sufficiently rich recursive axiomatization A of the natural numbers, by Goedel’s Second Incompleteness Theorem (plus Completeness) there is a natural number structure satisfying A accordingto which A is inconsistent and there is a natural number structure satisfying A according to which A is consistent. These two structures can’t be on par—one of them needs to be privileged.

The second is an insight I got from Linnebo’s philosophy of mathematics book: humans did mathematics before they did axiomatic mathematics. Babylonian apparently non-axiomatic but sophisticated mathematics came before Greek axiomatic geometry. It is awkward to think that the Babylonians were discovering ifthenist truths, given that they didn’t have a clear idea of the antecedents of the ifthenist conditionals.

I am now toying with the idea that there is a metaphysically privileged natural number structure but we have ifthenism for everything else in mathematics.

How is the natural number structure privileged? I think as follows: the order structure of the natural numbers is a possible order structure for a causal sequence. Causal finitism, by requiring all initial segments under the causal relation to be finite, requires the order type of the natural numbers to be ω. But once we have fixed the order type to be ω, we have fixed the natural number structure to be standard.

Definitions

In the previous post, I offered a criticism of defining logical consequence by means of proofs. A more precise way to put my criticism would be:

1. Logical consequence is equally well defined by (i) tree-proofs or by (ii) Fitch-proofs.

2. If (1), then logical consequence is either correctly defined by (i) and correctly defined by (ii) or it is not correctly defined by either.

3. If logical consequence is correctly defined by one of (i) and (ii), it is not correctly defined by the other.

4. Logical consequence is not both correctly defined by (i) and and correctly defined by (ii). (By 3)

5. Logical consequence is neither correctly defined by (i) nor by (ii). (By 1, 2, and 4)

When writing the post I had a disquiet about the argument, which I think amounts to a worry that there are parallel arguments that are bad. Consider the parallel argument against the standard definition of a bachelor:

6. A bachelor is equally well defined as (iii) an unmarried individual that is a man or as (iv) a man that is unmarried.

7. If (6), then a bachelor is either correctly defined by (iii) and correctly defined by (iv) or it is not correctly defined by either.

8. If logical consequence is correctly defined by one of (iii) and (iv), it is not correctly defined by the other.

9. A bachelor is not both correctly defined by (iii) and correctly defined by (iv). (By 9)

10. A bachelor is neither correctly defined by (iii) nor by (iv). (By 6, 7, and 10)

Whatever the problems of the standard definition of a bachelor (is a pope or a widower a bachelor?), this argument is not a problem. Premise (9) is false: there is no problem with saying that both (iii) and (iv) are good definitions, given that they are equivalent as definitions.

But now can’t the inferentialist say the same thing about premise (3) of my original argument?

No. Here’s why. That ψ has a tree-proof from ϕ is a different fact from the fact that ψ has a Fitch-proof from ϕ. It’s a different fact because it depends on the existence of a different entity—a tree-proof versus a Fitch-proof. We can put the point here in terms of grounding or truth-making: the grounds of one involve one entity and the grounds of the other involve a different entity. On the other hand, that Bob is an unmarried individual who is a bachelor and that Bob is a bachelor who is unmarried are the same fact, and have the same grounds: Bob’s being unmarried and Bob’s being a man.

Suppose one polytheist believes in two necessarily existing and essentially omniscient gods, A and B, and defines truth as what A believes, while her coreligionist defines truth as what B believes. The two thinkers genuinely disagree as to what truth is, since for the first thinker the grounds of a proposition’s being true are beliefs by A while for the second the grounds are beliefs by B. That necessarily each definition picks out the same truth facts does not save the definition. A good definition has to be hyperintensionally correct.

Logical consequence

There are two main accounts of ψ being a logical consequence of ϕ:

• Inferentialist: there is a proof from ϕ to ψ

• Model theoretic: every model of ϕ is a model of ψ.

Both suffer from a related problem.

On inferentialism, the problem is that there are many different concepts of proof all of which yield an equivalent relation of between ϕ and ψ. First, we have a distinction as to how the structure of a proof is indicated: is a tree, a sequence of statements set off by subproof indentation, or something else. Second, we have a distinction as to the choice of primitive rules. Do we, for instance, have only pure rules like disjunction-introduction or do we allow mixed rules like De Morgan? Do we allow conveniences like ternary conjunction-elimination, or idempotent? Which truth-functional symbols do we take as undefined primitives and which ones do we take as abbreviations for others (e.g., maybe we just have a Sheffer stroke)?

It is tempting to say that it doesn’t matter: any reasonable answers to these questions make exactly the same ψ be logical consequence of the same ϕ.

Yes, of course! But that’s the point. All of these proof systems have something in common which makes them "reasonable"; other proof systems, like ones including the rule of arbitrary statement introduction, are not reasonable. What makes them reasonable is that the proofs they yield capture logical consequence: they have a proof from ϕ to ψ precisely when ψ logically follows from ϕ. The concept of logical consequence is thus something that goes beyond them.

None of these are the definition of proof. This is just like the point we learn from Benacerraf that none of the set-theoretic “constructions of the natural numbers” like 3 = {0, 1, 2} or 3 = {{{0}}} gives the definition of the natural numbers. The set theoretic constructions give a model of the natural numbers, but our interest is in the structure they all have in common. Likewise with proof.

The problem becomes even worse if we take a nominalist approach to proof like Goodman and Quine do, where proofs are concrete inscriptions. For then what counts as a proof depends on our latitude with regard to the choice of font!

The model theoretic approach has a similar issue. A model, on the modern understanding, is a triple (M,R,I) where M is a set of objects, R is a set of relations and I is an interpretation. We immediately have the Benacerraf problem that there are many set-theoretic ways to define triples, relations and interpretations. And, besides that, why should sets be the only allowed models?

One alternative is to take logical consequence to be primitive.

Another is not to worry, but to take the important and fundamental relation to be metaphysical consequence, and be happy with logical consequence being relative to a particular logical system rather than something absolute. We can still insist that not everything goes for logical consequence: some logical systems are good and some are bad. The good ones are the ones with the property that if ψ follows from ϕ in the system, then it is metaphysically necessary that if ϕ then ψ.

A praise-blame asymmetry

There is a certain kind of symmetry between praise and blame. We praise someone who incurs a cost to themselves by going above and beyond obligation and thereby benefitting another. We blame someone who benefits themselves by failing to fulfill an obligation and thereby harming another.

But here is a fun asymmetry to note. We praise the benefactor in proportion to the cost to the benefactor. But we do not blame the malefactor in proportion to the benefit to the malefactor. On the contrary, when the benefit to the malefactor is really small, we think the malefactor is more to be blamed.