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?

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.

Incompleteness

For years in my logic classes I’ve been giving a rough but fairly accessible sketch of the fact that there are unprovable arithmetical truths (a special case of Tarski’s indefinability of truth), using an explicit Goedel sentence using concatenation of strings of symbols rather than Goedel encoding and the diagonal lemma.

I’ve finally revised the sketch to give the full First Incompleteness theorem, using Rosser’s trick. Here is a draft.

First-order naturalism

In a lovely paper, Leon Porter shows that semantic naturalism is false. One way to put the argument is as follows:

1. If semantic naturalism is true, truth is a natural property.

2. All natural properties are first order.

3. Truth is not a first order property.

4. So, truth is not a natural property.

5. So, semantic naturalism is not true.

One can show (3) by using the liar paradox or just take it as the outcome of Tarski’s Indefinability of Truth Theorem.

Of course, naturalism entails semantic naturalism, so the argument refutes naturalism.

But it occurred to me today, in conversation with Bryan Reece, that perhaps one could have a weaker version of naturalism, which one might call first-order naturalism that holds that all first order truths are natural truths.

First-order naturalism escapes Porter’s argument. It’s a pretty limited naturalism, but it has some force. It implies, for instance, that Zeus does not exist. For if Zeus exists, then that Zeus exists is a first-order truth that is not natural.

First-order naturalism is an interestingly modest naturalist thesis. It is interesting to think about its limits. One that comes to mind is that it does not appear to include naturalism about minds, since it does not appear possible to characterize minds in first-order language (minds represent the world, etc., and talk of representation is at least prima facie not first-order).

Representation and truth

For a while, I’ve been thinking of a teleological/normative account of representation. The basic idea is that:

1. State S represents reality being such that r if and only if one’s teleology specifies that one should be in state S only if r.

But I’ve also been worried that this makes representation much too common in the world. If a bacterium’s nature says that some behavior that should only be triggered under some circumstances, then on this account, the bacterium’s behavior represents the occurrence of these circumstances.

I am kind of willing to bite that bullet. But perhaps I don’t need to.

For a long time I’ve been sensitive to the difference between a proposition p and the second-order proposition that p is true, but this sensitivity has largely been a matter of nitpicking. But today I realized that this distinction may help save the teleological account of normativity with a very small tweak:

2. State S represents a proposition p if and only if one’s teleology specifies that one should be in state S only if p is true.

It is plausible that only higher organisms have a teleology that makes reference to truth as such.

Remark 1: If we want, we can have both (1) and (2) by distinguishing between “simple representation” and “alethic representation”. Alethic representation is then related to simple representation as follows:

3. State S alethically represents reality being such that r if and only if S simply represents reality being such that it is true that r.

Remark 2: Given Leon Porter’s argument that truth is not a physical property, it is interesting to note that on the alethic version, representation requires a being that has normative properties that make reference to something nonphysical. In particular, this kind of normativity cannot be grounded in evolution.

More steps in the open future and probability dialectics

I’ve often defended a probabilistic objection to open future views on which either future-tensed contingents are all false or are neither true nor false. If T(q) is the proposition that q is true, then:

1. P(T(q)) = P(q).

But on the open future views, the left-hand-side is zero, since it’s certain that q is not true. So the right-hand-side is zero. But then both q and its negation have zero probability, and we can’t make any predictions about the future.

An open futurist might push the following response. First, deny (1). Then insist that P(q) for a future contingent q is the objective tendency or chance towards q turning true. Thus, P(coin will be heads) is 1/2 for a fair indeterministic coin, since the there is an objective tendence of magnitude 1/2 for the coin to end up heads.

In this post I want to discuss my next step in the dialectics. I think there may be a problem with combining the objective tendency response with epistemic probabilities. Suppose that yesterday a fair coin was flipped. If the coin was heads, then tomorrow two fair indeterministic coins will be flipped, and if the coin is tails, then tomorrow one fair indeterministic coin will be flipped. Let H be the proposition that tomorrow at least one coin will be heads. If yesterday we had heads, then the objective tendency of H is 3/4. If yesterday we had tails, then the objective tendency of H is 1/2. But we need to be able to say:

2. P(H) = (1/2)(3/4) + (1/2)(1/2) = 5/8.

Now note that we are quite certain that 5/8 is not the objective tendency of H. The objective tendency of H is either 1/2 or 3/4.

So the open futurist needs a more sophisticated story. Here seems the right one. We say that P(q) is the average of the objective tendencies towards q weighted by the subjective probabilities of these tendencies. This is basically causal probability. The story requires that there be a present fact about all the objective tendencies.

On the technical side, this works. But here is a philosophical worry. If P(H) = 5/8 neither represents the objective tendency of H (which is either 1/2 or 3/4) nor one’s credence that H is true (which is zero on open-futurism), why is it that we should be making our decisions about the future in the light of P(H)?

Pursuing a thing for its own sake

Suppose you pursue truth for its own sake. As we learn from Aristotle, it does not follow that you don’t pursue truth for the sake of something else. For the most valuable things are both intrinsically and instrumentally valuable, and so they are typically pursued both for their own sake and for the sake of something else.

What if you pursue something, but not for the sake of something else. Does it follow that you pursue the thing for its own sake? Maybe, but it’s not as clear as it might seem. Imagine that you eat fiber for the sake of preventing colon cancer. Then you hear a study that says that fiber doesn’t prevent colon cancer. But you continue to eat fiber, out of a kind of volitional inertia, without any reason to do so. Then you are pursuing the consumption of fiber not for the sake of anything else. But merely losing the instrumental reason for eating fiber doesn’t give you a non-instrumentally reason. Rather, you are now eating fiber irrationally, for no reason.

Perhaps it is impossible to do something for no reason. But even if it is impossible to do something for no reason, it is incorrect to define pursuing something for its own sake as pursuing it not for the sake of something else. For that you pursue something for its own sake states something positive about your pursuit, while that you don’t pursue it for the sake of anything else states something negative about your pursuit. There is a kind of valuing of the thing for its own sake that is needed to pursue the thing for its own sake.

It is tempting to say that you pursue a thing for its own sake provided that you pursue it because of the intrinsic value you take it to have. But that, too, is incorrect. For suppose that a rich benefactor tells you that they will give you a ton of money if you gain something of intrinsic value today. You know that truth is valuable for its own sake, so you find out something. In doing so, you find out the truth because the truth is intrinsically valuable. But your pursuit of that truth is entirely instrumental, despite your reason being the intrinsic value.

Hence, to pursue a thing for its own sake is not the same as to pursue it because it has intrinsic value. Nor is it to pursue it not for the sake of something else.

I suspect that pursuing a thing for its own sake is a primitive concept.

Truth-directedness and propriety of scoring rules does not imply strict propriety

A scoring rule assigns a score to a credence assignment (which can but need not satisfy the axioms of probability), where a score is a random variable measuring how close the credence assignment is to the truth.

A scoring rule is strictly truth-directed provided that if c′ is a credence assignment that is closer to the truth than c is at ω, then c′ gets a better a score at ω. A scoring rule is proper provided that for all probabilities p, the p-expected value of the score of a probability p is at least as good as the p-expected value of the score of any other credence, and is strictly proper.

Propriety for a scoring rule is a pretty plausible condition, but it’s a bit harder to argue philosophically for strict propriety. But scoring-rule based philosophical arguments for probabilism—the doctrine that credences ought to be probabilities—require strict propriety.

In a clever move, Campbell-Moore and Levinstein showed that propriety plus strict truth-directedness and additivity (the idea that the score can be decomposed into a sum of single-event scores) implies strict propriety.

Here’s an interesting fact I will show: propriety plus strict truth-directedness do not imply strict propriety in the absence of additivity. Further, my counterexample will be bounded, infinitely differentiable and strictly proper on the probabilities. Personally don’t find additivity all that plausible, so I conclude the Campbell-Moore and Levinstein move does not move the discussion of strict propriety and probabilism ahead much.

Let Ω = {0, 1}. Given a credence function c (with values in [0,1]) on the powerset of Ω, define the credence function c^* which has the same value as c on the empty set and on Ω, but where c^*({0}) is the number z in [0,1] that minimizes (c({0})−z)² + (c({1})−(1−z))², and where c^*({1}) = 1 − c^*({0}). In other words, c^* is the credence function closest to c in the Euclidean metric such that c^*({0}) + c^*({1}) = 1.

Now let b^*(c) = b(c^*). Then b^* agrees with b score on the probabilities, and hence is strictly proper on them. Further, every value of b^* is a Brier score of some credence, and hence b^* is proper.

We now check that it is strictly truth-directed. Brier scores are strictly truth-directed. Thus, replacing a credence function with one that is closer to the truth on Ω or on the empty set will improve the b^* score. Moreover, it is easy to check that c^*({0}) = (1+c({0})−c({1}))/2. It’s easy to check that if we tweak c({0}) to move us closer to the truth at some fixed ω ∈ {0, 1}, then c^* will be closer to the truth at ω as well, and similarly if we tweak c({1}) to be closer to the truth at ω, and in both cases we will improve the score by the strict truth-directedness of Brier scores.

Finally, however, note that b^* is not strictly proper and does not have a domination theorem of the sort used in arguments for probabilism, since the b^*-score of any credence c that fails to be a probability due to its being the case c({0}) + c({1}) ≠ 1 but that gets the right values on the empty set and Ω (zero and one, respectively) is equal to the b^*-score of c^*, and c^* will be a probability in that case.

Note that in the example above we don't have quasi-strict propriety either.

When truth makes you do less well

One might think that being closer to the truth is guaranteed to get one to make better decisions. Not so. Say that a probability assignment p₂ is at least as true as a probability assignment p₁ at a world or situation ω provided that for every event E holding at ω we have p₂(E)≥p₁(E) and for every event E not holding at ω we have p₂(E)≤p₁(E). And say that p₂ is truer than p₁ provided that strict inequality holds in at least one case.

Suppose that a secret integer has been picked among 1, 2 and 3, and p₁ assigns the respective probabilities 0.5, 0.3, 0.2 to the three possibilities while p₂ assigns them 0.7, 0.1, 0.2. Then if the true situation is 1, it is easy to check that p₂ is truer than p₁. But now suppose that you are offered a choice between the following games:

• W₁: on 1 win $2, on 2 win $1100, and on 3 win $1000.

• W₂: on 1 win $1, on 2 win $1000, and on 3 win $1100

If you are going by p₁, you will choose W₁ and if you are going by p₂, you will choose W₂. But if the true number is 1, you would be better off picking W₁ (getting $2 instead of $1), so the truer probabilities will lead to a worse payoff. C’est la vie.

Say that a scoring rule for probabilities is truth-directed if it never assigns a poorer score for a truer set of probabilities. The above example shows that a proper scoring rule need not be truth-directed. For let s(p)(n) be the payoff you will get if the secret number is n and you make your decision between W₁ and W₂ rationally on the basis of probability assignment p (with ties broken in favor of W₁, say). Then s is a proper (accuracy) scoring rule but the above considerations show that s(p₂)(1)<s(p₁)(1), even though p₂ is truer at 1. In fact, we can get a strictly proper scoring rule that isn’t truth-directed if we want: just add a tiny multiple of a Brier accuracy score to s.

Intuitively we would want our scoring rules to be both proper and truth-directed. But given that sometimes we are pragmatically better off for having less true probabilities, it is not clear that scoring rules should be truth-directed. I find myself of divided mind in this regard.

How common is this phenomenon? Roughly it happens whenever the truer and less-true probabilities disagree on ratios of probabilities of non-actual events.

Proposition: Suppose two probability assignments are such that there are events E₁ and E₂ with probabilities strictly between 0 and 1, with ω₁ in neither event, and such that the ratio p₁(E₁)/p₁(E₂) is different from the ratio p₂(E₁)/p₂(E₂). Then there are wagers W₁ and W₂ such that p₁ prefers W₁ and p₂ prefers W₂, but W₁ pays better than W₂ at ω₁.

Truth directed scoring rules on an infinite space

A credence assignment c on a space Ω of situations is a function from the powerset of Ω to [0, 1], with c(E) representing one’s degree of belief in E ⊆ Ω.

An accuracy scoring rule s assigns to a credence assignment c on a space Ω and situation ω the epistemic utility s(c)(ω) of having credence assignment c when in truth we are in ω. Epistemic utilities are extended real numbers.

The scoring rule is strictly truth directed provided that if credence assignment c₂ is strictly truer than c₁ at ω, then s(c₂)(ω)>s(c₁)(ω). We say that c₂ is strictly truer than c₁ if and only if for every event E that happens at ω, c₂(E)≥c₁(E) and for every event E that does not happen at ω, c₂(E)≤c₁(E), and in at least one case there is strict inequality.

A credence assignment c is extreme provided that c(E) is 0 or 1 for every E.

Proposition. If the probability space Ω is infinite, then there is no strictly truth directed scoring rule defined for all credences, or even for all extreme credences.

In fact, there is not even a scoring rule that strictly truth directed when restricted to extreme credences, where an extreme credence is one that assigns 0 or 1 to every event.

This proposition uses the following result that my colleague Daniel Herden essentially gave me a proof of:

Lemma. If PX is the power set of X, then there is no function f : PX → X such that f(A)≠f(B) whenever A ⊂ B.

Now, we prove the Proposition. Fix ω ∈ Ω. Let s be a strictly truth directed scoring rule defined for all extreme credences. For any subset A of PΩ, define c_A to be the extreme credence function that is correct at ω at all and only the events in A, i.e., c_A(E)=1 if and only if ω ∈ E and E ∈ A or ω ∉ E and E ∉ A, and otherwise c_A(E)=0. Note that c_B is strictly truer than c_A if and only if A ⊂ B. For any subset A of PΩ, let f(A)=s(c_A)(ω).

Then f(A)<f(B) whenever A ⊂ B. Hence f is a strictly monotonic function from PPΩ to the reals. Now, if Ω is infinite, then the reals can be embedded in PΩ (by the axiom of countable choice, Ω contains a countably infinite subset, and hence PΩ has cardinality at least that of the continuum). Hence we have a function like the one the Lemma denies the existence of, a contradiction.

Note: This suggests that if we want strict truth directedness of a scoring rule, the scoring rule had better take values in a set whose cardinality is greater than that of the continuum, e.g., the hyperreals.

Proof of Lemma (essentially due to Daniel Herden): Suppose we have f as in the statement of the Lemma. Let ON be the class of ordinals. Define a function F : ON → A by transfinite induction:

• F(0)=f(⌀)

• F(α)=f({F(β):β < α}) whenever α is a successor or limit ordinal.

I claim that this function is one-to-one.

Let H_α = {F(δ):δ < α}.

Suppose F is one-to-one on β for all β < α. If α is a limit ordinal, then it follows that F is one-to-one on α. Suppose instead that α is a successor of β. I claim that F is one-to-one on α, too. The only possible failure of injectivity on α could be if F(β)=F(γ) for some γ < β. Now, F(β)=f(H_β) and F(γ)=f(H_γ). Note that H_γ ⊂ H_β since F is one-to-one on β. Hence f(H_β)≠f(H_γ) by the assumption of the Lemma. So, F is one-to-one on ON by transfinite induction.

But of course we can’t embed ON in a set (Burali-Forti).

Reasons from the value of true belief

Two soccer teams are facing off, with a billion fans watching on TV. Brazil has a score of 2 and Belgium has a score of 0, and there are 15 minutes remaining. The fans nearly unanimously think Brazil will win. Suddenly, there is a giant lightning strike, and all electrical devices near the stadium fail, taking the game off the air. Coincidentally, during the glitch, Brazil’s two best players get red cards, and now Belgium has a very real chance to win if they try hard.

But the captain of the Brazilian team yells out this argument to the Belgians: “If you win, you will make a billion fans have a false belief. A false belief is bad, and when you multiply the badness by billion, the result is very bad. So, don’t win!”
Great hilarity ensues among the Belgians and they proceed to trounce the Brazilians.

The Belgians are right to laugh: the consideration that the belief of a billion fans will be falsified by their effort carries little to no moral weight.

Why? Is it that false belief carries little to no disvalue? No. For suppose that now the game is over. At this point, the broadcast teams have a pretty strong moral reason to try to get back on the air in order to inform the billion fans that they were mistaken about the result of the game.

In other words, we have a much stronger reason to shift people’s beliefs to match reality than to shift reality to match people’s beliefs. Yet in both cases the relevant effect on the good and bad in the world can be the same: there is less of the bad of false beliefs and more of the good of true beliefs. An immediate consequence of this is that consequentialism about moral reasons is false: the weight of moral reasons depends on more than the value of the consequences.

It is often said that belief has a mind-to-world direction of fit. It is interesting that this not only has repercussions for the agent’s own epistemic life, but for the moral life of other parties. We have much more reason to help others to true belief by affecting their beliefs than by affecting the truth and falsity of the content of the beliefs.

Do the Belgians have any moral reason to lose, in light of the fact that losing will make the fans have correct belief? I am inclined to think so: producing a better state of affairs is always worthwhile. But the force of the reason is exceedingly small. (Nor do the numbers matter: the reason’s force would remain exceedingly small even if there are trillions of fans because Earth soccer was famous through the galaxy.)

There is a connection between the good and the right, but it is quite complex indeed.

Epistemology and the presumption of (im)permissibility

Normally, our overt behavior has the presumption of moral permissibility: an action is morally permissible unless there is some specific reason why it would be morally impermissible.

Oddly, this is not so in epistemology. Our doxastic behavior seems to come along with a presumption of epistemic impermissibility. A belief or inference is only justified when there is a specific reason for that justification.

In ethics, there are two main ways of losing the presumption of moral permissibility in an area of activity.

The first is that actions falling in that area are prima facie bad, and hence a special justification is needed for them. Violence is an example: a violent action is by default impermissible, unless we have a special reason that makes it permissible. The second family of cases is areas of action that are dangerous. When we go into a nuclear power facility or a functioning temple, we are surrounded by danger—physical or religious—and we should refrain from actions unless we have special reason to think they are safe.

Belief isn’t prima facie bad. But maybe it is prima facie dangerous? But the presumption of impermissibility is not limited to some special areas. There indeed are dangerous areas of our doxastic lives: having the wrong religious beliefs can seriously damage us psychologically and spiritually while having the wrong beliefs about nutrition and medicine can kill us. But there seem to be safe areas of our doxastic lives: whatever I believe about the last digit in the number of hairs on my head or about the generalized continuum hypothesis seems quite safe. Yet, having the unevidenced belief that the last digit in the number of hairs on my head is three is just as impermissible as having the unevidenced belief that milk cures cancer.

Perhaps it is simply that moral and epistemic normativity are not as analogous as they have seemed to some.

But there is another option. Perhaps, despite what I said, our doxastic lives are always dangerous. Here is one way to suggest this. Perhaps truth is sacred, and so dealing with truth is dangerous just as it is dangerous to be in a temple. We need reason to think that the rituals we perform are right when we are in a temple—we should not proceed by whim or by trial and error in religion—and perhaps similarly we need reasons to think that our beliefs are true, precisely because our doxastic lives always, no matter how “secular” the content, concern the sacred. Our beliefs may be practically safe, but the category of the sacred always implicates a danger, and hence a presumption of impermissibility.

I can think of two ways our doxastic lives could always concern the sacred:

1. God is truth.

2. All truth is about God: every truth is contingent or necessary; contingent truths tell us about what God did or permitted; necessary truths are all grounded in the nature of God.

All this also fits with an area of our moral lives where there is a presumption of impermissibility: assertion. One should only make assertions when one has reason to think they are true. Otherwise, one is lying or engaging in BS. Yet assertion is not always dangerous in any practical sense of “dangerous”: making unwarranted assertions about the number of hairs one one’s head or the general continuum hypothesis is pretty safe practically speaking. But perhaps assertion also concerns the truth, which is something sacred, and where we are dealing with the sacred, there we have spiritual danger and a presumption of impermissibility.

The Math Tea argument

The Math Tea argument is an argument that there are real numbers that can’t be defined. The idea is this: there are only countably many definitions of real numbers (e.g., π^e or "The middle root of the polynomial x³ − 5x² + 2x + 4"), and uncountably many real numbers, so there are real numbers that have no definitions.

Elegant as this argument is, it has crucial set-theoretic flaws. For instance, there is no guarantee that there is a set of all the definable real numbers. The axioms of set theory tell us that for any predicate F in the language of set theory there is a set of all the numbers that satisfy F. But the predicate "is definable" is in English, not in set theory.

We can, however, argue for the following weaker claim. Assume set theory is true. Then either:

1. There is a real number that cannot be defined in the language of set theory, or

2. "A real number is missing": there is an English language formula F(n) whose only semantic predicate is set-theoretic satisfaction such that there is no real number x whose nth digit after the decimal point is 1 if F(n) and is 0 if not F(n).

Here is the argument. A formula of set-theory defines a real number if it has exactly one free variable and is satisfied by precisely one real number. Say that F(n) if and only if the nth formula of set theory (in lexicographic ordering) defining a real number defines a real number that does not have a 1 in the nth place after the decimal point. The only semantic predicate in F(n) is set-theoretic satisfaction. Suppose (2) is false. Then there is a real number x whose nth digit after the decimal point is 1 if F(n) and is 0 if not F(n). If x can be defined in the language of set theory by a formula ϕ, then suppose ϕ is the nth real-number-defining formula. Then F(n) if and only if x does not have a 1 in the nth place. But x has a 1 in the nth place if and only if F(n). Contradiction! So, x cannot be defined, and hence (1) is true.

Logically speaking, if ZF is consistent, ZFC is consistent both with (1) (this follows by letting the digits of x be defined by the set of all set-theoretic truths and noting that if ZF is consistent, we can consistently suppose there is a set of all set-theoretic truths, but that set of course cannot be defined) and with the denial of (1).

But philosophically speaking, we might reasonably say that (2) would imply that "there aren’t enough real numbers", which sounds wrong, so it seems more reasonable to accept (1) instead.

Is there a set of all set-theoretic truths?

Is there a set of all set-theoretic truths? This would be the set of sentences (in some encoding scheme, such as Goedel numbers) in the language of set theory that are true.

There is a serious epistemic possibility of a negative answer. If ZF is consistent, then there is a model M of ZFC such that every object in M is definable, i.e., for every object a of M, there is a defining formula ϕ(x) that is satisfied by a and by a alone in M (and if there is a transitive model of ZF, then M can be taken to be transitive). In such a model, it follows from Tarski’s Indefinability of Truth that there is no set of all set-theoretic truths. For if there were such a set, then that set would be definable, and we could use the definition of that set to define truth. So, if ZF is consistent, there is a model M of ZFC that does not contain a set of all the truths in M.

Interestingly, however, there is also a serious epistemic possibility of a positive answer. If ZF is consistent, then there is a model M of ZFC that does contain a set of all the truths in M. Here is a proof. If ZF is consistent, so is ZFC. Let ZFC^T be a theory whose language is the language of set theory with an extra constant T, and whose axioms are the axioms of ZFC with the schemas of Separation and Replacement restricted to formulas of ZFC (i.e., formulas not using T), plus the axiom:

1. ∀x(x ∈ T → S(x))

where S(x) is a sentence saying that x is the code for a sentence (this is a syntactic matter, so it can be specified explicitly), and the axiom schema that has for every sentence ϕ with code n:

2. ϕ ↔ n ∈ T.

Any finite collection of the axioms of ZFC^T is consistent. For let M be a model of ZFC (if ZF is consistent, so is ZFC, so it has a model). Then all the axioms of ZFC will be satisfied in M. Furthermore, for any finite subset of the additional axioms of ZFC^T, there is an interpretation of the constant T under which those axioms are true. To see this, suppose that our finite subset contains (1) (no harm throwing that in if it’s not there) and the instances ϕ_i ↔ n_i ∈ T of (2) for i = 1, ..., m. It is provable from ZF and hence true in M that there is a set t such that x ∈ t if and only if x = n₁ and ϕ₁, or x = n₂ and ϕ₂, …, or x = n_m and ϕ_m.

Moreover, any such set can be proved in ZF to satisfy:

3. ∀x(x ∈ t → S(t)).

Interpreting T to be that set t in M will make the finite subset of the additional axioms true.

So, by compactness, ZFC^T has an interpretation I in some model M. In M there will be an object t such that t = I(T). That object t will be a set of all the truths in M that do not contain the constant T. Now consider the interpretation I⁻ of ZFC in M, which is I without any assignment of a value to the constant T (since T is not a constant of ZFC). Then ZFC will be true in M under I⁻. Moreover, the object t in M will be a set of all the truths in M.

So, if ZF is consistent, then there is a model of ZFC with a set of all set-theoretic truths and a model of ZFC without a set of all set-theoretic truths.

The latter claim may seem to violate the Tarski Indefinability of Truth. But it doesn’t. For that set of all truths will not itself be definable. It will exist, but there won’t be a formula of set theory that picks it out. There is nothing mathematically new in what I said above, but it is an interesting illustration of how one can come close to violating Indefinability of Truth without actually violating it.

Now, what if we take a Platonic view of the truths of set theory? Should we then say that there really is a set of all set-theoretic truths? Intuitively, I think so. Otherwise, our class of all sets is intuitively “missing” a subset of the set of all sentences. I am inclined to think that the Axioms of Separation and Replacement should be extended to include formulas of English (and other human languages), not just the formulas expressible in set-theoretic language. And the existence of the set of all set-theoretic truths follows from an application of Separation to the sentence “n is the code for a sentence of set theory that is true”.

Is it too risky to do philosophy if there is no God?

If we are created by a loving God, there is good reason to expect that what is good for us to believe—maybe even good for us as moral agents—and what is true tend to go together in the case of the most important beliefs. But if we’re not created by a loving God, then I wouldn’t expect the true and the beneficial to go together, except in the case of straightforward empirical beliefs about the external world, such as that apples are nutritious and that lions eat us. If there is no loving God, it would seem pretty likely to me that—as some non-theist philosophers indeed worry—it is good for us to have various philosophical illusions (say, that God exists).

This means that if one is sure there is no loving God, there is a pretty decent argument against doing philosophy. For either philosophy leads to truth or not. If it doesn’t lead to truth, there is little point to doing it: for then philosophy fails to promote the non-instrumental value of truth and we have no reason to think that it would be any more beneficial instrumentally than our pre-philosophical views. But even if it leads to truth, then unless we think there is a correlation between truth and utility, we are still risking endangering beliefs—such as in moral responsibility—that are crucial for human society’s functioning. Given how much is at stake here, it seems not to be worth the risk. One might hope, of course, that philosophy would lead to beliefs—true or false—that would let society function much better than it has done in the past, and that the hope of this benefit at least cancels out the fear of harm. But I think this is unrealistically optimistic: it seems far easier to undermine society than to build it up. (Think of the sweeping tragedies arising from Marxist and fascist philosophies in the 20th century.)

That said, one doesn’t need to be confident that there is a God to justify doing philosophy. One just needs a sufficiently high probability that once one takes into account the possibility that God exists and hence that truth and utility are correlated, the expected value of doing philosophy is positive.

And the above line of thought doesn’t apply to the kind of abstruse philosophy which is unlikely to connect with everyday life.

Vagueness and degrees of truth

Consider the non-bivalent logic solution to the problem of vagueness where we assign additional truth values between false and true. If the number of truth values is finite, then we immediately have a regress problem once we ask about the boundaries for the assignment of the finitely many truth values: for instance, if the truth values are False, 0.25, 0.50, 0.75 and True, then we will be able to ask where the boundary between “x is bald” having truth value 0.50 and having truth value 0.75 lies.

So, the number of truth values had better be infinite. But it seems to be worse than that. It seems there cannot be a set of truth values. Here is why. If x has any less hair than y, but neither is definitely bald or non-bald, then “x is bald” is more true than “y is bald”. But how much hair one has is quantified in our world with real numbers, say real numbers measuring something like a ratio between the volume of hair and the surface area of the scalp (the actual details will be horribly messy). But there will presumably be possible worlds with finer-grained distances than we have—distances measured using various hyperreals. Supposing that Alice is vaguely bald, there will be possible people y who are infinitesimally more or less bald than Alice. And as there is no set of all possible infinitesimals (because there is no set of all systems of hyperreal), there won’t be a set of all truth values.

Moreover, there will be vagueness as to comparisons between truth values. One way to be less bald is to have more hairs. Another way is to have longer hairs. And another is to have thicker hairs. And another is to have a more wrinkly scalp. Unless one adopts epistemicism, there are going to be many cases where it will be vague whether “x is bald” is more or less or equally or incommensurably true as “y is bald”.

We started with a simple problem: it is vague what is and isn’t bald. And the non-bivalent solution led us to a vast multiplication of such problems, and a vast system of truth values that cannot be contained in a set. This doesn’t seem like the best way to go.

Do all positive truths have truthmakers?

Consider this thesis:

1. Every positive true proposition has a truthmaker.

This seems plausible. But I think it is only reasonable to accept (1) if one accepts:

2. Any plurality of objects has a mereological sum or fusion which essentially has the members of the plurality as parts.

To see this, consider some plurality, the xs of existing things. Then, surely:

3. The proposition, E!xx, that the xs exist is positive.

But what object is suited to be the truthmaker of E!xx? The truthmaker of E!xx will have to be some object o with the property that, necessarily, if o exists, so do all the xs. Our best candidate for that object is some object that has all the xs as essential parts. But we also don’t want to include irrelevancies in the truthmaker, so we shouldn’t include in o anything that overlaps none of the xs. In other words, o will very plausibly be the mereological sum of the xs.

Since I don’t believe in fusions, I have to deny (1). But at least I may be able to accept:

4. Every positive true proposition has a plural truthmaker,

where a plural truthmaker of p is a plurality of objects that collectively make p true. Note that pluralities need not in general be objects themselves, so we do not have the same problem as above.

Uncontroversial examples of truthmaking?

I used to think that the following would be an uncontroversial example of truthmaking:

1. Any elephant is a truthmaker for the proposition that there are elephants.

But that’s only true if every elephant is essentially an elephant, i.e., couldn’t exist without being an elephant. For if x is a truthmaker for p, then x’s existence has to entail p. If Jumbo were accidentally an elephant, then Jumbo’s existence wouldn’t entail that there are elephants.

Given that essentialism is controversial, it seems that if we are to give uncontroversial examples of truthmaking, they have to be something like;

2. Jumbo is a truthmaker for the proposition that Jumbo exists.

3. Alice is a truthmaker for the proposition that at least one of Alice, Bob and Carl exists.

Truth and probabilistic consistency

Suppose Alice has an inconsistent probabilistic assignment P^A. Then, famously, there is a series of bets on single propositions (call these binary bets) that is a Dutch Book against Alice: i.e., Alice by her lights will accept each bet, and is guaranteed to lose money.

But now suppose Bob has a probabilistic assignment P^B—perhaps a consistent one—that is strictly further from the truth than Alice’s inconsistent one in the sense that

1. for any p, if p is false, then P^B(p)≥P^A(p),

2. for any p, if p is true, then P^B(p)≤P^A(p), and

3. at least one of the inequalities is strict.

Then Alice will do at least as well as Bob on every portfolio of offers of binary bets, and on some portfolios she will do strictly better than Bob. In particular, even if Bob’s probabilistic assignment is consistent, and there is a binary bet Dutch Book against Alice, Bob will fare no better than Alice with respect to that book.

Thus, if we start with a consistent assignment and then by some process move towards truth, we will do better (against binary bet portfolios) even if we lose consistency.

So why is Alice’s probabilistic assignment supposed to be rationally bad in a way that Bob’s isn’t? Well, the difference is this. A bookie can fleece Alice simply on the basis of knowing Alice’s probability assignment. But simply knowing Bob’s probability assignment won’t be enough to know which portfolio will fleece him.

However, the more I think about this, the more I lose the intuition that all this shows there is something particularly rationally problematic about Alice’s assignments just because they are inconsistent. Why should game-theoretic performance against a competitor who knows one’s credences be particularly indicative of rationality or the lack thereof? When nature offers us betting portfolios (to pursue this trail or that trail after a wounded deer in the woods, say), these portfolios are normally independent of our credences. Of course, in business and war, we have to worry about mind-reading competitors. But much of our life, we don’t.

Suppose I find myself with inconsistent credences. What should I do? Should I force them to be consistent? If I am dealing with mind-reading competitors who have no more information about the external world than I do, then I should go for consistency. But going for consistency will force me to modify some of my probabilities, and for all I know, these probabilities may get modified away from truth. And that might be more harmful.

There may be interesting trade-offs. Maybe some intellectual strategies work better against mind-reading competitors and others work better with the portfolios set by nature. We should not take doing well with respect to one selection of portfolio to be particularly informative about the nature of rationality.

If classical theism rules out open theism, then classical theism rules out presentism

If presentism and most, if not all, other versions of the A-theory are true, then propositions change in truth value. For instance, on presentism, in the time of the dinosaurs it was not true that horses exist, but now it is true; on growing block, ten years ago the year 2019 wasn't at the leading edge of reality, but now it is. The following argument seems to show that such views are incompatible with classical theism.

1. God never comes to know anything.

2. If at t₁, x doesn’t know a proposition p but at t₂ > t₁, x knows p, then x comes to know p.

3. If propositions change in truth value, then there are times t₁ < t₂ and a proposition p such that p is not true at t₁ and p is true at t₂.

4. It is always the case that God knows every true proposition.

5. It is never the case that anyone knows any proposition that isn’t true.

6. So, if propositions change in truth value, then there are times t₁ < t₂ and a proposition p such that God doesn’t know p at t₁ but God does know p at t₂. (by 3-5)

7. So, if propositions change in truth value, God comes to know something. (by 2 and 6)

8. So, propositions do not change in truth value. (by 1 and 7)

I think the only controversial proposition is (1). Of course, some non-classical theists—say, open theists—will deny (1). But non-classical theists aren’t the target of the argument.

However, there is a way for classical theists to try to get out of (1) as well. They could say that the content of God’s knowledge changes, even though God and God’s act of knowing are unchanging. The move would be like this. We classical theists accept divine simplicity, and hence hold that God would not have been intrinsically any different had he created otherwise than he did. But had God created otherwise than he did, the content of his knowledge would have been different (since God knows what he creates). So the content of God’s knowledge needs to be partially constituted by created reality. (This could be a radical semantic externalism, say.) Thus, had God created otherwise than he did, God (and his act of knowledge which is identical to God) would have been merely extrinsically different.

But exactly the same move allows one to reconcile the denial of (1) with immutability. The content of God’s knowledge is partially constituted by created reality, and hence as created reality changes, the content of God’s knowledge changes, but the change in God is merely extrinsic, like a mother’s change from being taller than her daughter to being shorter than her daughter solely due to her daughter’s growth.

I agree that denying (1) is compatible with God’s being intrinsically unchanging. For a long time I thought that this observation destroyed the argument (1)-(8). But I now think not. For I am now thinking that even if (1) is compatible with immutability, (1) is a part of classical theism. For it is a part of classical theism that God doesn’t learn in any way, and coming to know is a kind of learning.

Here is one way to see that (1) is a part of classical theism. Classical theists want to reject any open theist views. But here is one open theist view, probably the best one. The future is open and propositions reporting what people will freely do tomorrow are now either false or neither-true-nor-false, but tomorrow they come to be true. An omniscient being knows all true propositions, but it is no shortcoming of omniscience to fail to know propositions that aren’t true. Then, our open theist says, God learns these propositions as soon as they become true. This is all that omniscience calls for.

Now, classical theists will want to reject this open theist view on the grounds of its violating immutability. But they cannot do so if they themselves reject (1). For the presentist (say) classical theist can reject (1) without violating immutability, so can our open theist. Indeed, our open theists can say exactly the same thing I suggested earlier: God changes extrinsically as time progresses, and the content of God’s knowledge changes, but God remains intrinsically the same.

So, what do I think the classical theist should say to our open theist? I think this: that God doesn’t come to know is not just a consequence of the doctrine of immutability, but is itself a part of the doctrine of immutability. A God who learns is mutable in an objectionable way even if this learning is not an intrinsic change in God. But if we say this, then of course we are committed to (1), and we cannot be presentists or accept any other of the theories of time on which propositions change in truth value.

I think the best response on the part of the classical theist who is an entrenched presentist would be to deny (1) and concede that classical theism does not rule out open theism. Instead, open theism is ruled out by divine revelation, and revelation here adds to classical theism. But it seems very strange to say that classical theism does not rule out open theism.