Reductive analyses of desert

This blog post explores the possibility of giving a reductive analysis of the concept of desert, as in, “getting what you deserve”. I briefly talk about the current state of research on desert, which seems to fail to give any reductive analysis; then I give my preferred reductive analysis, as well as some utilitarian reductive analyses which I developed for my friend Tetraspace, so that she would not need to use a trivialized view.

Here is a natural and appealing idea: it is a good thing when people get what they deserve. If I am distributing raises, and only Jones deserves a raise, then it would be better for me to give the raise to Jones than to anyone else. —Feldman and Skow, SEP entry on “Desert”

• SEP on desert
• Deontological reduction of desert
• Utilitarian reductions of desert
• Trivialized desert
• Desert as time-bound desert-from a possibly elided distributor
• Desert as improvement in felicific power
• FDT-style desert

SEP on desert

Recently, as an outgrowth of a curiosity about contemporary value theory, I was reading the SEP article on the concept of desert. The following infographic depicts the state of philosophical research on the topic, as portrayed by the SEP article.

Infographic of SEP on desert, drawn by Nano Banana. Please click for full image.

What struck me about the SEP article was that all the given theories took the concept of desert as a primitive concept. This concept can be analysed in terms of its relata: all desert claims involve someone who deserves it (the deserver), something deserved (the desert), and something about the deserver in virtue of which the desert is deserved (the desert base), and optionally a fourth element, which is someone from whom the deserver deserves the desert (the distributor). It can also be distinguished from entitlement: desert holds in virtue of intrinsic features of the deserver, while entitlement holds in virtue of social institutions. But none of the theorists cited in the article seemed to be offering a reductive analysis of desert in terms of other concepts.

Deontological reduction of desert

Personally, I really don’t want to take desert as a primitive concept. I already have a moral theory, so everything looks like a nail. Using my moral theory to define desert, I think my own view of desert is this:

x deserves y (from p) ≝ x has a moral right to receive y (from p), in at least the broad sense of morality. If x has this moral right in the narrow sense of morality, we might say x strongly deserves y (from p).

My moral theory is such that this analysis works perfectly, as far as I can tell. 

Utilitarian reductions of desert

My friend Tetraspace is some sort of utilitarian, I think. When I was reading the SEP article, I was reminded of her views on desert.

Trivialized desert

For we each of us deserve everything, every luxury that was ever piled in the tombs of the dead kings, and we each of us deserve nothing, not a mouthful of bread in hunger. —Ursula K. Le Guin, The Dispossessed (quote)

In some old posts to X (formerly Twitter), now deleted, Tetraspace made comments to the effect of espousing a trivialized view of desert: every sentient being, regardless of personal traits, deserves whatever they would get in an ideal world, which includes all of their wildest dreams and infinite eternal happiness, since that’s what everyone receives in the ideal world. I call this view trivialized because everyone deserves the same thing, regardless of who or what they are. (Or rather, maybe different people deserve different things to the extent that they enjoy different things, but still, the linguistic difference is very small.)

She appealed to her consequentialist sympathies to support this view, since she was unable to think of how to arrive at a nontrivial view of desert within her consequentialist framework. Given the state of research in the SEP article (where desert was always taken as a primitive), I really can’t blame her, but I still thought she could do better, even if we insist on a reductive analysis of desert within utilitarianism.

Desert as time-bound desert-from a possibly elided distributor

In a reply to Tetraspace on X (formerly Twitter), in May 2025, I had once offhandedly suggested that she should use this analysis:

x deserves y ≝ ∃ p ∈ person, ∃ t ∈ time, EV(⌜p doesn’t give y to x within t⌝) < 0.

Of course, we can get the version with a distributor by simply freeing the variable:

x deserves y from p ≝ ∃ t ∈ time, EV(⌜p doesn’t give y to x within t⌝) < 0.

Come to think of it now, while this analysis can recover many of the commonsense linguistic phenomena, it has some grave flaws. Namely, it implies that a terrorist kidnapper deserves to receive a ransom, as long as his threat that he will kill his hostages is credible. So much, then, for the time-bound desert-from view.

Note (2026-02-16): I had been silently using the little ⌜corners⌝ for quasiquotation, and this apparently confused Tetraspace at first, so I am noting here that this is what they are.

Desert as improvement in felicific power

After thinking about it now, my best attempt for Tetraspace is that Tetraspace should have a concept of someone’s “power to make things better”, and then she should use this analysis:

x deserves y (from p) ≝ at least one of these holds:

1. y is not rivalrous, and receiving y (from p) will increase x’s power to make things better
2. y is rivalrous, and receiving y (from p) will increase x’s power to make things better more than if y is given (by p) to anyone else

I think this analysis produces most of the commonsense theorems:

• random abuse is undeserved, because random suffering usually makes you worse at making things better,
• random gifts are usually undeserved but sometimes deserved,
• rewards for good actions are deserved, because of the incentive that they provide, and
• punishment is deserved if, and only if, (and precisely to the extent that,) it’s rehabilitative.

I have no idea whether Tetraspace would take exception to a powers ontology. I think, to account for science, she needs the concept of powers anyway, regardless whether it is primitive; so she might as well use it for the desert theory. Under this analysis, desert is reduced to a metaphysical component (the concept of power) and an evaluative component (an order of world-states from better to worse), where the latter component can be taken wholly from her own value theory. My own moral theory proves beyond a shadow of a doubt that her value theory is wrong, but there is no reason why it should lead her to say weird-sounding things about what people deserve.

FDT-style desert

Upon reading the above, Tetraspace herself suggested adding a counterfactual conditional to the naïve time-bound reduction, which would allow avoiding the result where terrorists deserve ransom. An FDT-style approach that I believe is likely to capture her preferred reasoning is given below.

x deserves y (from p) ≝ The expected utility of the world where the decision algorithm A (which p runs) is logically fixed to output “give y to x” is higher than the world where A is fixed to output “don’t give y.” That is,

Desert(x, y) ≝ EV(⌜A □→ Give⌝) > EV(⌜A □→ ¬Give⌝)

Where the evaluation includes all subjunctive dependencies (past, present, and future) linked to algorithm A.

The box-arrow notation is, of course, David Lewis’s box-arrow for counterfactuals.

This formulation is somewhat opaque to me, but it seems to give the following commonsense results:

• People deserve to be compensated for completed work or good deeds, because the optimal decision algorithm must reliably reward past cooperation to ensure future cooperation.
• Bad actors do not deserve to have their demands met (e.g., ransoms), because an algorithm that rewards threats logically guarantees that more threats will be made.
• Criminals deserve punishment even if it will not rehabilitate them, because fulfilling the algorithm's prior commitment to punish is necessary to deter others from committing the same crime.
• Offenders deserve penalties that fit the crime, because an optimal incentive structure balances the necessary deterrent effect against the societal friction and cost of extreme, draconian penalties.
• Victims do not deserve random bad luck, natural disasters, or random abuse, because unpredictable suffering serves no logical or functional role in shaping positive behavior.

It also seems to yield the result that you deserve your lottery win because you intentionally acquired a ticket, but not a different kind of windfall that really had nothing to do with your decision policy.

I am glad that Tetraspace was able to use her own resources to get closer to commonsense views on desert; the possibility of getting threatened by terrorists is obviously a common decision theory example, but for whatever reason I did not think of this, despite having looked into similar stuff before. I suppose I have to hand this one to the folks at LessWrong. Counterfactuals are still somewhat unclear and metaphysical nowadays, but there is more research going into clarifying them than into powers ontology (or at least I think so), so this seems like an improvement upon the previous proposals.

Funny “loading screen” art made with Nano Banana, also posted here.

Chesterton’s fence is easy to get over

Many people who I’ve talked to seem to think that Chesterton’s fence consists in a very strong presumption against changing institutions. Some praise it as the very symbol of their own strong attitude against changing institutions, and refer to any change in institutions as “tearing down Chesterton’s fences”; others condemn Chesterton’s fence as being an excessively strong constraint, arguing we should be readier to change institutions, and hence should abandon Chesterton’s fence.

Enough is enough. This blog post argues that Chesterton’s fence is actually a very weak constraint. I argue that this is how it was meant by Chesterton, and that it is not useful to interpret it in any other sense. This is partly a point in favor of the fence: if a reformer can’t even meet Chesterton’s very weak constraint, which is very easy to meet, then he’s a very poor reformer who hasn’t even done his very easy homework, and certainly shouldn’t be admitted to the rule of our society. But it also means that getting over Chesterton’s fence is not the be-all-end-all of whether you’ll allow someone to change your institutions; you need more defenses than Chesterton’s waist-high garden fence.

• G.K. Chesterton’s original text
• Chesterton only required a debunking origin story, and this is very easy to provide
• Examples of debunking origin stories
• Isn’t that the genetic fallacy?
• Against interpreting Chesterton’s fence in a stronger sense
• Conclusion

G.K. Chesterton’s original text

Chesterton’s fence is a parable that was given entirely in two paragraphs in his book “The Thing: Why I Am A Catholic”, in the chapter “The Drift from Domesticity”. In full, they run as follows:

In the matter of reforming things, as distinct from deforming them, there is one plain and simple principle; a principle which will probably be called a paradox. There exists in such a case a certain institution or law; let us say for the sake of simplicity, a fence or gate erected across a road. The more modern type of reformer goes gaily up to it and says, “I don’t see the use of this; let us clear it away.” To which the more intelligent type of reformer will do well to answer: “If you don’t see the use of it, I certainly won’t let you clear it away. Go away and think. Then, when you can come back and tell me that you do see the use of it, I may allow you to destroy it.”

This paradox rests on the most elementary common sense. The gate or fence did not grow there. It was not set up by somnambulists who built it in their sleep. It is highly improbable that it was put there by escaped lunatics who were for some reason loose in the street. Some person had some reason for thinking it would be a good thing for somebody. And until we know what the reason was, we really cannot judge whether the reason was reasonable. It is extremely probable that we have overlooked some whole aspect of the question, if something set up by human beings like ourselves seems to be entirely meaningless and mysterious. There are reformers who get over this difficulty by assuming that all their fathers were fools; but if that be so, we can only say that folly appears to be a hereditary disease. But the truth is that nobody has any business to destroy a social institution until he has really seen it as an historical institution. If he knows how it arose, and what purposes it was supposed to serve, he may really be able to say that they were bad purposes, or that they have since become bad purposes, or that they are purposes which are no longer served. But if he simply stares at the thing as a senseless monstrosity that has somehow sprung up in his path, it is he and not the traditionalist who is suffering from an illusion. We might even say that he is seeing things in a nightmare. This principle applies to a thousand things, to trifles as well as true institutions, to convention as well as to conviction. It was exactly the sort of person, like Joan of Arc, who did know why women wore skirts, who was most justified in not wearing one; it was exactly the sort of person, like St. Francis, who did sympathise with the feast and the fireside, who was most entitled to become a beggar on the open road. And when, in the general emancipation of modern society, the Duchess says she does not see why she shouldn’t play leapfrog, or the Dean declares that he sees no valid canonical reason why he should not stand on his head, we may say to these persons with patient benevolence: “Defer, therefore, the operation you contemplate until you have realised by ripe reflection what principle or prejudice you are violating. Then play leapfrog and stand on your head and the Lord be with you.”

Chesterton moves on from this to say that “among the traditions that are being thus attacked, not intelligently but most unintelligently, is the fundamental human creation called the Household or the Home”. Chesterton was a social conservative, and the fence was only introduced as a device to defend the family against leftist critiques. He had no broader, general theory of the fence beyond these two paragraphs, and he did not come back to it.

Chesterton’s fence, when stated as a “principle”, is sometimes summarized as, “Do not remove a fence until you know why it was put up in the first place.” As you can see from the full text above, this does capture Chesterton’s advice. What I intend to emphasize in this blog post is that it fully captures all of his advice.

Chesterton only required a debunking origin story, and this is very easy to provide

Chesterton’s fence, when stated like this, is a really easy constraint to get over. To get over Chesterton’s fence, the reformer who would like to abolish an institution F need only provide, and support, two claims:

1. The institution F was originally instituted with the intention of fulfilling the purpose P.
2. The purpose P is not a purpose that we currently need, or want, or should want, to fulfill anymore. (Or alternatively, P is already equally well or better fulfiflled by a different institution G, which has made F redundant; or, F does not fulfill P anymore.)

That’s it. I call those two claims together a “debunking origin story”: the claim 1 gives an origin story for the institution F, and the claim 2 uses that origin story to debunk F with respect to our current society.

Quick thinkers will already notice that it’s still possible that the institution F is a good institution even though it is possible for a critic of F to get over Chesterton’s fence when criticizing it, and that is my point: Chesterton’s fence is only a weak constraint. For slower thinkers, however, I will give a list of debunking origin stories for some presently-standing institutions, which will emphasize my point.

Examples of debunking origin stories

Here are 20 examples of still-existing institutions/policies that people sometimes condemn by pointing to “bad origins” (racism, eugenics, colonial violence, etc.). Although I did not put any care into collecting the list (and hence the examples ended up somewhat U.S.-centric), I personally endorse all of these arguments:

1. Minimum-wage laws — Critics point out that some Progressive Era advocates explicitly used eugenics / “unfit worker” arguments in support of wage floors.
2. Planned Parenthood / the early U.S. birth-control movement — Often condemned because Margaret Sanger embraced eugenics rhetoric (which Planned Parenthood itself acknowledges, while arguing the history is complicated).
3. Modern American policing — Frequently criticized as having roots in slave patrols and racialized social control (with some historians/commentators disputing how direct that lineage is).
4. The prison system + prison labor — Condemned because the 13th Amendment’s “punishment” exception and post–Civil War practices like convict leasing are seen as reconstituting coerced labor.
5. The U.S. Electoral College — Criticized as a constitutional compromise entangled with slavery-era power and representation (including incentives tied to enslaved populations).
6. The U.S. Senate filibuster — Often condemned for its long association with blocking civil-rights legislation (even when writers disagree about whether it originated that way).
7. Single-family zoning — Condemned for exclusionary roots tied to racial segregation and keeping certain groups out of neighborhoods.
8. Homeowners associations (HOAs) / deed restrictions (historical) — Criticized because many neighborhood governance structures grew alongside racially restrictive covenants and enforced segregation norms (even though those covenants are now unenforceable).
9. U.S. housing finance & underwriting institutions — Condemned due to the legacy of redlining and government-backed mortgage systems that excluded Black neighborhoods (even though explicit redlining was later outlawed).
10. Standardized testing (IQ tests, SAT/aptitude testing) — Criticized for eugenicist intellectual roots and early claims about racial hierarchy used to justify exclusion.
11. The “War on Drugs” policy apparatus — Condemned as structurally racialized, with critics citing both historical intent claims and racially disparate impacts in enforcement.
12. The U.S. Border Patrol / border enforcement institutions — Criticized as emerging from restrictionist eras and racialized immigration politics; critics describe “racist origins” in early 20th-century enforcement formation.
13. The modern U.S. visa/quota border-control framework (legacy of 1924) — Condemned because the Immigration Act of 1924 was strongly shaped by racial hierarchy and eugenics-influenced restrictionism, and it helped institutionalize modern border controls.
14. Vagrancy / loitering policing and related ordinances — Criticized as inheriting logic from Black Codes that criminalized idleness and constrained freedom after emancipation.
15. Major museums with colonial-era collections — Condemned because many collections were built through colonial conquest/looting, prompting restitution and repatriation disputes.
16. Elite universities with slavery ties — Condemned because many prominent schools benefited from enslavement, slave-trade wealth, or related institutions, and some now publish formal reports acknowledging it.
17. Land-grant universities (Morrill Act system) — Criticized as having been financed through expropriated Indigenous land, with ongoing debates about restitution and revenue.
18. Child welfare systems in Indigenous contexts — Condemned due to the legacy of state-sponsored family separation (e.g., federal Indian boarding schools) and continuing debates about removals and sovereignty.
19. Psychiatry’s diagnostic institutions (e.g., DSM tradition) — Criticized because the field historically pathologized homosexuality and used institutional power in ways now seen as abusive; advocates cite this to challenge current authority.
20. Columbus Day (and similar “heritage” civic holidays) — Condemned for celebrating a figure linked to colonization and Indigenous dispossession, which is why many jurisdictions have shifted to Indigenous Peoples’ Day while the federal observance still exists.

Notice that all of these criticisms, if true, successfully get over Chesterton’s fence: they claim that the institution F arose to fulfill a purpose P, and that the purpose P is not a purpose that we currently need, or want, or should want, to fulfill anymore. (Of course, they do need to be true, which is debatable, but this is beyond the scope of this blog post.)

I hope using a slew of claims that are very controversial in current politics helped you notice how weak of a constraint Chesterton’s fence is.

Isn’t that the genetic fallacy?

The genetic fallacy is when you criticize an argument because of the motivations of the person who argued it. So the criticisms above technically don’t commit the genetic fallacy in the usual sense. However, in the broader sense where having bad origins doesn’t mean something is itself bad, the criticisms in the list do commit the genetic fallacy if used by themselves, as the first and only criticism of the institutions.

But if a defender of each of these institutions had decided to appeal to Chesterton’s fence as a defense of the institutions, then the debunking origin story provided by the criticisms fully fulfills everything Chesterton asked for. Nothing more is needed to get past Chesterton’s challenge. Chesterton’s fence has been successfully gotten over by these critics, because they provided a debunking origin story, and Chesterton’s fence, as such, does not ask for anything more, or anything else.

Against interpreting Chesterton’s fence in a stronger sense

If someone says Chesterton did not mean what I said he meant, then such a person is certainly wrong and irrational, and not worth listening to, and should certainly be excluded from society; I will not say they should be killed, but certainly no one would miss them. But let us suppose a more reasonable opponent of my blog post. Such a reasonable opponent says:

Sure, I concede Chesterton meant what you said he meant, since I am reasonable, unlike an irrational and unlovable person who does not concede this. But I think you are making Chesterton’s fence into something useless and irrelevant, and hence I think we should use the name “Chesterton’s fence” to refer to the stronger principle that, if we wish to tear down an institution, we should find out what purpose it currently serves. After all, Chesterton’s fence is only relevant to the extent that the original purpose of an institution sheds some light on its current purpose, and that’s obviously what we should care about, so it is not even worth having a name for the other thing.

It is the nature of a reasonable opponent that, although he is reasonable, I oppose him. My reason is as follows: defenders of an institution can always make up an infinity of purposes that they think (or claim to think) the institution currently serves for them. It is not clear which of these deserve a hearing. Maybe you should just do your best to answer the best arguments against your view that you can think of, but then Chesterton’s fence is not distinct from the “principle of charity”, or “steel manning”; it is certainly not worth having a third name for an idea that already has two.

More to the point, Chesterton’s fence has Chesterton’s name on it, and Chesterton is a famous traditionalist conservative. It makes sense that Chesterton’s fence should refer to something that talks about deferring to tradition, and our forefathers, etc. Chesterton’s fence clearly ties into Burkean conservative defenses of tradition, such as the view of tradition that Ed Feser attributes to F.A. Hayek. An interpretation of Chesterton’s fence should honor its conservative and traditionalist roots.

Chesterton’s idea, as I reconstruct it, has to do with an intuition that when we actively or passively support the continued existence of an institution, we don’t only (or even necessarily) do so because we have our own reasoning to think it’s a good idea, but also (or sometimes even exclusively) because we trust the tradition of our society that has been maintaining the institution up to now. Our trust in our ancestors, in turn, is only as good as their trust in their ancestors, and so on, and the line of people trusting each other has to end at the people who first instituted the institution. Chesterton is saying that, even before you get to the merits of the discussion of an institution, you should be able to at least overcome our commonsense trust in our ancestors. And I add: if you can’t even do that, how good of a reformer can you be, exactly?

Conclusion

In summary, to recapitulate:

1. Chesterton’s fence only requires that you provide a debunking origin story of an institution that you seek to abolish: describe the original purpose of the institution, and show that that purpose is no longer a good reason for the institution.
2. This is what Chesterton meant by it.
3. This is what is interesting to mean by it, because (1) interpreting it as being about purposes currently served by an institution makes it too similar to charitableness/steelmanning, and (2) the debunking-origin-story interpretation also best captures Chesterton’s conservative and traditionalist worldview, and (3) a weak constraint is more interesting as a challenge to a reformer, because if a reformer can’t even meet a weak constraint, then he is certainly a bad reformer.

Illustration for this blog post, drawn by Nano Banana.

Simplified modal ontological argument

This blog post defends S5 modal logic by arguing that atheists should not reject its axioms merely to avoid the theistic conclusion of Plantinga-style modal ontological arguments. It does so by emphasizing that a modal ontological argument is, after all, an ontological argument: it moves from a certain substantive definitional claim about God to an affirmative existential claim about God. The real work in the argument is done by the substantive definitional claim, and the logic, regardless how strong of a modal logic it is, is just scaffolding. Or so I claim, and I defend this by recasting the modal ontological argument in a weaker modal logic with stronger premises, showing that the argument is still valid and can still be equally well motivated by informal conceptions of theism in this way, and then only afterwards comparing the argument with something that you’d need S5 to show. S5 is then seen to be a perfectly innocent, and delightfully accurate, model of how alethic modalities work in natural language, so that non-theists who would like to reject modal ontological arguments would be best served by either sticking to an agnostic or sceptical claim about theism or by adopting some theory that denies the very possibility of the truth of theism, rather than trying to make such strange, hard to interpret claims as that “God’s existence is contingent since it isn’t actual, but it could have been actual, in which case it would be necessary”.

I first came up with the central argument of this blog post on 2025-08-03, but the blog post came largely from a discussion on 2026-02-02 in which I used the argument to defend S5, much the same as in this blog post (in fact, a few words in this blog post were reused from the discussion).

• 1. Introduction
• 2. The argument restated
• 2.1. Premise 1: God’s existence is not contingent
• 2.2. Premise 2 (God’s existence is possible) and the derivation
• 3. Options for non-theists, and why rejecting S5 isn’t one
• 3.1. How S5 enters into the argument, and why rejecting it is a bad idea
• 3.2. The real live options for non-theists
• 3.2.1. Agnosticism or scepticism
• 3.2.2. Believe that God’s existence is impossible, and hence reject Premise 2
• 3.2.3. Dispute the theist’s definition

1. Introduction

Alvin Plantinga famously considered his own modal ontological argument unconvincing. Maybe his pessimism about natural theology was warranted in some way, but it wasn’t helped by his needlessly overcomplicated presentation of the argument.

Plantinga’s argument, like all ontological arguments, moves from a substantive, definitional claim about God to an affirmative existential claim. But Plantinga took it for granted that he would be able to use the strongest modal logic, S5, and hence, like a logician seeking to use only the minimal necessary axioms, pared down his substantive understanding of God to a minimal claim that allowed S5 to do as much of the work of the proof as possible. This is perfectly fine if the argument is to remain academic, but naturally, since it is a proof-of-God, it made its way to apologists, who merely repeated Plantinga’s argument without deeper reflection on what was being said in it. This was the first time many atheists heard of modal logic, and many of them reflexively denied the S5 axioms instead of the premises, which is what the argument’s phrasing was meant to pressure them to do. But while modal logicians can laugh at atheists being made to look foolish by restricting themselves to a uselessly weak version of alethic modal logic, laypersons don’t really see what’s so funny about this. And I think it’s a dismal state of affairs, because I don’t want to be unable to use the nice, perfect, beautiful, S5 modal logic when modelling alethic modalities in my conversations with atheists, just because they’re afraid of Plantinga. Enough is enough.

In this blog post, I restate the modal ontological argument by using a weaker modal logic combined with a stronger substantive definitional claim about God, so as to make the background understanding of God explicit, and show that nothing is unusual about it. With this, I aim to show that the substantive definitional claim about God is the real core of the ontological argument, not anything about S5. I review the options for non-theists very thoroughly so as to make it clear that they really don’t need to deny S5, and there’s really no reason to do so. The main goal of this blog post, in summary, is to emphasize the advantages and innocence of S5, so as to show that S5, by itself, says nothing more than natural language intuitively says about alethic modalities, so that the core of any modal ontological argument is something else.

2. The argument restated

The argument uses classical propositional modal logic with axiom T, and uses g as a constant that means “God exists”. (Axiom T is the same as what is called Axiom M by the SEP, which seems to be an outlier on this.) The exposition uses the definition ◇p ≝ ¬□¬p freely, and also freely speaks of “theists” as believers in g and “atheists” as believers in ¬g, with “agnostics” being those who suspend judgment, and “sceptics” being those who deny that it is possible to know whether g is the case (more on this in section 3); “non-theists” are whichever persons aren’t theists, of course.

The argument is stated as follows:

1. □g ∨ □¬g (premise)
2. ¬(□¬g) (premise)
3. □g (1,2 DS)
4. g (3 via axiom T)

This all went by very fast, so I am going to explain it line-by-line.

2.1. Premise 1: God’s existence is not contingent

Premise 1 states that God’s existence is either necessary or impossible. This is to deny that God is a contingent being, in the sense that God might either have been or not have been. The negation of Premise 1 is ◇g ∧ ◇¬g, i.e., possibly God exists AND possibly God doesn’t exist. Theists who accept Premise 1 deny that it’s possible that God doesn’t exist, and atheists who accept Premise 1 deny that it’s possible that God exists. Hence, neither theism nor atheism are forced by Premise 1 by itself.

The motivation of Premise 1 for theists is that they accept one of the disjuncts. The motivation of Premise 1 for atheists is that, if they affirm ◇g ∧ ◇¬g, then it’s unclear that their statement ¬g is really denying the same thing that theists affirm. After all, theists conceive of God as a necessary being, not as a contingent being. If the atheist denies Premise 1, the theist is free to say that the atheist hasn’t really denied what he affirms.

The very use of “contingent” as standard modal terminology hints at this way of conceiving of necessary truths, since it comes from Leibnizian metaphysics. In the Leibnizian metaphysics, whenever you said something was “contingent”, you also said it was “contingent on” something else: the “things that might have been and might not have been” were coextensive with the “things that depend on something else for their existence”. Leibniz took this simply as a linguistic datum, and it went unquestioned for a long time (I’m still not sure who exactly questions it), because it just works with natural language so perfectly.

But affirming Premise 1 does not require a commitment to this thesis of Leibnizian metaphysics. Using “fundamental being” for beings that aren’t ontologically dependent, then we might say that, regardless whether Leibniz is right in saying that all-and-only fundamental beings are necessary beings, it remains that theists conceive of God as a necessary being, quite apart from whether theists conceive of God as a fundamental being (which I think they also do, but which the argument does not claim). If the atheist conceives of God as a contingent being, which might either be or not be, then the atheist is not meaning the same thing by “God” that the theist means by it.

I say that the theist conceives of God as a necessary being. There is, of course, no universal agreement among theists about anything, so which theist do I mean? Well, Alvin Plantinga, for one, certainly conceives of God as a necessary being, and hence accepts Premise 1. Leibniz, of course, also does. More generally, I can’t think of any theist in the history of philosophy or of religion who ever explicitly denied Premise 1, at least about the greatest god, although not all of them always explicitly affirmed it. The atheist who denies Premise 1 is being very weird, historically, and hence I think I am saying something very plausible in saying that he is not denying something that any theists affirm. So if the atheist wants to deny what theists affirm, he must accept Premise 1, and hence (if he wants to remain an atheist in the present sense, rather than an agnostic or sceptic) claim that God’s existence is impossible, not just “possible but not actual”. More on options for atheists in section 3.

2.2. Premise 2 (God’s existence is possible) and the derivation

Premise 2 states that God’s existence is not impossible. Together with Premise 1, this easily implies theism, of course. Plantinga also uses Premise 2, so this argument isn’t an improvement on Plantinga in this respect. The atheist is free to deny Premise 2 if he wants; this is discussed in section 3.

To be fully explicit in case someone can’t read the notation: Since Premise 1 said God’s existence is either necessary or impossible, and Premise 2 denied that it’s impossible, then by Disjunctive Syllogism you get that God’s existence is necessary, which is the proposition 3. Axiom T says that necessary propositions are actually true, hence from proposition 3, via axiom T, you get proposition 4. I don’t expect any of this derivation to be controversial, it’s the premises that are controversial.

3. Options for non-theists, and why rejecting S5 isn’t one

The argument did not use the full strength of the modal logic system S5; it relied more or less entirely on the fact that Premise 1 is required to capture the theist’s informal understanding of God.

Granted, it also used Axiom T. But Axiom T, unlike the more bespoke S5 axioms that involve iterated modalities, is certainly obviously true in natural language. When would you ever say that something is “necessarily true but not actually true”? This seems simply ungrammatical. Hence, Axiom T is an analytic truth.

3.1. How S5 enters into the argument, and why rejecting it is a bad idea

How does S5 even come into it, then? The answer may be surprising: S5 only comes into it if the atheist wants to accept Premise 2 (i.e., deny that God’s existence is impossible; together with atheism and Axiom T, this already implies the negation of Premise 1, but nevermind this for now) and furthermore affirm that, “if God had existed, then God would exist necessarily”, in the subjunctive mood, interpreted under a standard modal reading of subjunctives. S5 is then required to show that such an atheist is inconsistent, such that the subjunctive statement combined with Premise 2 would force theism. I will explain.

If the atheist merely wants to say, in the indicative mood, that “if God exists, then God exists necessarily”, then this can be interpreted as a mere material conditional, g → □g. Even for the atheist, this can coexist in a classical S5 model with ◇g (God’s existence is not impossible), because, since the atheist accepts ¬g, then accepting g would lead to inconsistency. In classical logic, “from a contradiction, anything follows”, and in particular, you have g → □g.

But if the atheist wants to say, in the subjunctive mood, “if God had existed, then God would exist necessarily”, then under a standard modal reading of subjunctive conditionals, S5 can show that this atheist cannot, as an atheist, grant that it is possible that God exists. This standard interpretation, seen for instance here and here, interprets a subjunctive statement “if A had been the case, then B would have been the case” as materially implying, at least, that ◇A → ◇B. So for the atheist who says, “if God had existed, then God would exist necessarily”, this amounts to ◇g → ◇□g. Combined with his commitment to ◇g from his acceptance of Premise 2, the atheist becomes committed to ◇□g. This is where the characteristic axiom of S5, which is called either Axiom 5 or Axiom E depending on the source, comes into play, which says that ◇□p → □p. Instantiating the scheme with g, you have ◇□g → □g. By modus ponens, you have □g, and then Axiom T yields the theistic conclusion, as before. So such an atheist would be inconsistent, committed to theism and atheism, under S5.

However, such an atheist is certainly inconsistent in a more ordinary way, not just given S5. If he claims that God’s existence is possible (given his acceptance of Premise 2) but not actual (given his acceptance of atheism), then he claims that God’s existence is contingent. (That is, atheism and Premise 2, together with Axiom T, commit him to the negation of Premise 1.) So it is simply very strange, in natural language, that he claims, about God’s existence, that “if this proposition were to be true, it would be a necessary truth”. If you were to ever say the just-quoted sentence about any other proposition in your life, certainly you would be saying it about something which is either a tautology or a contradiction, such as a mathematical conjecture, not about something contingent. It is very weird, in natural language, to say that a contingent proposition would be necessary if it were true. The modal translation, and the S5 axiom, are only capturing this reality about how language is used, not introducing anything out of the ordinary. The dispute is about the premises, not about S5.

S5 is the best formal model of natural alethic modal language. It corresponds to frames where the accessibility relation is reflexive, symmetric, and transitive, or equivalently, reflexive and Euclidean. This is simply how natural-language alethic modalities work, and there is really nothing else to use, if you want a model of alethic modalities in formal logic. There is no reason to weaken it. I encourage you to look further into modal logic with an open mind, without thinking so much about theistic arguments, so that you will see what I am talking about. (Do not confuse alethic modalities with other modalities, of course, such as epistemic, deontic, or temporal modalities, which I do not address here.)

3.2. The real live options for non-theists

If you don’t want to accept theism, your options are the following:

3.2.1. Agnosticism or scepticism

This is to suspend judgment on whether God exists (agnosticism), or alternatively, to claim that is impossible to know whether God exists (scepticism about theism). These can be challenged, but I don’t care to do so here.

3.2.2. Believe that God’s existence is impossible, and hence reject Premise 2

Since the ontological argument merely proves that there is a necessary being, the most direct way to do this is to deny that there are any necessary beings at all, which is certainly something some people have believed, such as David Hume; if there are no necessary beings, then any purported necessary being is impossible.

If the atheist wants to make a more modest claim, he can use the typical arguments that some particular version of God’s purported attributes leads to inconsistency, such as arguments from evil, or from the omnipotence paradox. Those arguments are, of course, tied to a specific notion of what God’s attributes are, which may be disputed by the theist. Alternatively, the atheist can restrict necessary beings in a more local way, such as by claiming that the only necessary beings are (for instance) mathematical objects, and God isn’t (for instance) a mathematical object. You need to motivate the restriction on necessary beings somehow, and the more kinds of necessary beings you allow, of course, the more room there is for theists to argue that God fits into one of your allowed kinds.

3.2.3. Dispute the theist’s definition

I am including this option for logical completeness: technically, you may reject Premise 1 and try, somehow, to answer the theist’s challenge that this does not capture his conception of God. You may claim that the theist does not really believe that it is impossible for God to fail to exist, despite his protestations to the contrary; you may try to interpret the religious and theological tradition as somehow best explained by a contingent view of God. I do not think this makes any sense, but I can’t let my blog post be anything but exhaustive.

Illustration for this blog post, drawn by Nano Banana.

Mathematics is a model of language

This blog post outlines a philosophy of mathematics centered on representational structures, rejecting both the view that math is merely a tool for physics and the view that it is solely a game of analytic derivations in abstract logic. I draw on Robert Stalnaker’s pragmatics and the solution used by Stalnaker to address the “problem of equivalence”, which is the puzzle that, since all mathematical truths are necessary, they should logically all have the same content. The solution is that mathematics is not just about the truths themselves, but about how we represent them: it is the study of the specific connections between formulas, proof patterns, and axiomatic systems. Therefore, to learn mathematics is to master these symbolic manipulations, and to explore the capabilities of different representational systems.

Under this view, mathematics acts as a formal model of natural language. Mathematical concepts (like numbers or triangles) originate from ordinary language practices (counting or measuring) and are then “reified” into rigorous systems to allow for consistent operations. This perspective explains why mathematics is universally applicable (because these structural patterns are portable across domains) and why “auxiliary” concepts like hyperreals are useful (they serve as temporary representational bridges). It also explains why reducing mathematical objects to set theory often feels wrong (producing “junk theorems”): such reductions might work formally, but they sever the connection to the original representational roles that the concepts play in our natural language.

We cannot deny the axioms of mathematics; for they exhibit nothing more than a consistent use of words, and affirm of some idea that it is itself and not something else. —William Godwin, Political Justice, book 1, chapter 4

• Philosophies of mathematics
• What mathematics is about
• How mathematics is learned
• How mathematics is applied
• How mathematics is motivated
• How mathematics is interpreted
• Geometries as precisifications of spatial intuitions
• Reduction and junk theorems

Philosophies of mathematics

This blog post is about philosophy of mathematics. As I’ve said before, there are roughly three types of philosophy of mathematics:

1. an excuse to do more mathematics: here we have how intuitionism created intuitionist mathematics, or how there are foundation movements that try to recast all of mathematics in terms of type theory or category theory.
2. “math is only real if it’s applied”: here we have so-called “mathematical naturalism”, which gives us the famous arguments that mathematical objects must be real because, and only insofar as, they are indispensable to physics.
3. you can accept all of current math if you just accept this very unclear idea: the idea here is that you can give an illuminating philosophical interpretation to all of current mathematics—not just the parts that get applied practically—, and without even needing to reinterpret it in any restricted logic (as in intuitionism) or foundational theory (such as category theory or type theory). This is an ambitious goal, and the catch seems to be that every proposal that claims to achieve it seems to do its work by way of providing a very vague and unsatisfying interpretive scheme, such as by saying mathematics is a meaningless game of symbols, or that it consists of tautologies, etc., where it isn’t clear what makes it true that all-and-only mathematics fits the interpretive scheme, and how this fits the various appearances about mathematical practice.

I don’t think any idea about philosophy of mathematics fully escapes being one of these types, and this blog post belongs to type 3, but I hope I have mitigated the vagueness enough that it says something interesting.

The fact that I’m vaguely gesturing at something that isn’t very precise makes it all important to give the reader a feel for the kind of thing that was said in the literature that constitutes the broad kind of background that I’m coming from, without making the reader go read all of those books in full; which I hope explains why I’ve thrown all these quotations in there, including the famous Kant quotation that everyone has seen before (but who knows if someone hasn’t?).

What mathematics is about

Now if mathematical truths are all necessary, then on the possible worlds analysis there is no room for doubt about the truth of the propositions themselves. There are, it seems, only two mathematical propositions, the necessarily true one and the necessarily false one, and we all know that the first is true and the second false. But the functions that determine which of the two propositions is expressed by a given mathematical statement are just the kind that are sufficiently complex to give rise to reasonable doubt about which proposition is expressed by a statement. Hence it seems reasonable to take the objects of belief and doubt in mathematics to be propositions about the relation between statements and what they say. —Robert Stalnaker, Inquiry, p. 73

Stalnaker’s pragmatics, as explained in his books Inquiry (1984; especially chapter 4) and Context and Content (1999; especially chapter 12), begins with what he calls a “coarse-grained” semantic picture. Propositions, at the most basic level, are sets of possible worlds (or functions from worlds to truth-values). With that in hand, you can model assertion as eliminating worlds from the conversational context, presupposition as fixing a common-ground set, and inquiry as progressively narrowing what remains live. As spare as it is, that model earns its keep: Stalnaker can explain a lot of linguistic practice while keeping the semantic core simple and truth-conditional.

But the same simplicity creates what Stalnaker calls the problem of equivalence. If propositions are just sets of worlds, then necessarily equivalent statements—especially in mathematics—collapse to the same content. If every mathematical truth is necessary, it can look as if there are, informationally, only two mathematical propositions: the necessary truth and the necessary falsehood. And yet we plainly distinguish “17 is prime” from “the angles of a Euclidean triangle sum to 180°,” can believe one without believing the other, and can learn one without learning the other.

Stalnaker’s motivating response is to treat this not as a reason to abandon the possible-worlds model of informational content, but as a reason to notice something special about mathematical inquiry: much of what we learn in mathematics is about representational structures and their relations to that coarse-grained content. We do not just “latch onto” a necessary truth; we learn (and can be ignorant of) whether a given formula, proof pattern, or axiom-system representation determines that truth. Mathematics, on this view, is not merely a repository of necessary contents; it is, centrally, a disciplined study of structures of representation—structures that can be instantiated in different languages and notations, and manipulated by calculation and proof.

How mathematics is learned

Suppose there were a community of English speakers that grew up doing its arithmetic in a base eight notation. The words “eight” and “nine” don’t exist in its dialect; the words “ten” and “eleven,” like the numerals “10” and “11” denote the numbers eight and nine. Now suppose that some child in this community has a belief that he would express by saying “twenty-six times one hundred equals twenty-six hundred.” Would it be correct to say that this child believes that twenty-two times sixty-four equals fourteen hundred eight? This does not seem to capture accurately his cognitive state. His belief, like our simple arithmetical beliefs, is not really a belief about the numbers themselves, independently of how they are represented. —Robert Stalnaker, Context and Content, p. 237

The representational-structure picture fits the phenomenology of learning mathematics unusually well. Mathematical competence is inseparable from activities like: transforming inscriptions, applying rules of inference, calculating, constructing proofs, translating between equivalent forms.

Those are not optional “presentational” extras. They are how mathematical information is acquired and used. If learning that 689×43=29627 were merely ruling out worlds where a necessary truth fails, it would be mysterious how computation helps—since computation doesn’t teach us which worlds are actual. But if what you are learning is (roughly) that this representational procedure connects these numerals and operations to this result, then calculation is exactly the right kind of epistemic act: it is an exploration of the representational system’s structure.

So the “object” of mathematical belief is often best seen as involving claims like: a representation with this structure has this content; a derivation of this form preserves truth relative to these axioms; this symbolic construction yields a representation equivalent to that one. Those are propositions we can genuinely be ignorant of and come to know by manipulating the system.

How mathematics is applied

An initial hint that numbers are magnitudes comes from their algebra. The natural numbers, the positive real numbers and the ordinal numbers each have an associative operation of addition which defines a linear order. I call a system with that precise algebraic structure a positive semigroup. We find positive semigroups cropping up not only in mathematics but in the physical sciences as well. The fundamental physical magnitudes have this same algebraic structure: for example, mass is a positive semigroup because addition of masses is associative, and the addition defines a linear order. The same is true for length, area, volume, angle size, time and electric charge: these and other physical magnitudes all have the structure of a positive semigroup. —Keith Hossack, Knowledge and the Philosophy of Number, p. 1

Our picture also helps with a classic puzzle: mathematics is generally applicable—it shows up everywhere—yet it also seems to be about very particular objects (the number 2, the empty set, π, triangles, and so on).

If mathematics is fundamentally about representational structures, its generality is no surprise. Representational structures are general: they are patterns of form and transformation that can be implemented in many domains. The same algebraic structure can organize bookkeeping, physics, logic, probability, and geometry because those practices can be represented in ways that share the relevant structure.

At the same time, it feels like mathematics is about particular objects because our linguistic practice encourages reification. We talk as if “2” names an object; we quantify over numbers; we introduce constants (“0”, “∅”); we build theories that treat these as a stable domain. That practice is extremely useful: it streamlines inference and stabilizes coordination. But on the representational view, the “particularity” of mathematical objects is tightly connected to how our public language fixes and reuses representational roles.

How mathematics is motivated

The trouble with this objection is that it completely ignores history: the theory of real numbers, and the theory of differentiation etc. of functions of real numbers, was developed precisely in order to deal with physical space and physical time and various theories in which space and/or time play an important role, such as Newtonian mechanics. Indeed, the reason that the real number system and the associated theory of differentiation etc. is so important mathematically is precisely that so many of the problems to which we want to apply mathematics involve space and/or time. It is hardly surprising that mathematical theories developed in order to apply to space and time should postulate mathematical structures with some strong structural similarities to the physical structures of space and time. It is a clear case of putting the cart before the horse to conclude from this that what I’ve called the physical structure of space and time is really mathematical structure in disguise. —Hartry Field, Science Without Numbers

Mathematics developed, initially, from ordinary language, and then grew more complex to account for more technical, scientific language. The natural numbers are a model of the counting numbers we deploy in ordinary language: “two apples,” “three steps,” “four chairs.” We naturally used them for measurement of continuous quantities as well, since measurement tools (such as rulers) require counting (of markings on the rulers, for instance) to be used.

Extensions beyond the naturals—negative numbers, rationals, reals, complex numbers—are not forced on us by everyday counting talk, but they can be similarly motivated by ordinary counting contexts themselves, insofar as there is a practical need to perform operations (subtraction, division, solving equations) in a way that preserves and extends successful inferential patterns. We introduce new entities so that the representational system remains closed under operations we already treat as coherent.

On this view, the payoff of “new numbers” is that they yield correct, systematic consequences about the original practice. We extend the representational framework, do the work there, and recover truths that constrain ordinary counting-number claims.

Nonstandard analysis is a good illustration, since it introduces the hyperreal numbers, but only does so in order to formulate theorems of calculus, whose interest lies in what they say about real numbers. Hyperreal numbers function as an auxiliary representational domain: you begin with real-valued problems, temporarily move into the hyperreals to make certain reasoning patterns (infinitesimals, transfer principles) tractable, and then return with a theorem stated purely in terms of the reals. The hyperreals are not forced on us as “the real subject matter”; they are a representational device that systematizes certain operations and delivers results about the original target domain.

More exotic mathematical theories, which are more removed from ordinary practices, must be seen to correspond to more unusual, specialized, or tightly constrained contexts of representation—ways of carving up possibilities that aren’t our everyday default, but that could in principle become useful for unusual particular tasks.

How mathematics is interpreted

Give a philosopher the concept of a triangle, and let him try to find out in his way how the sum of its angles might be related to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, and in it the concept of equally many angles. Now he may reflect on this concept as long as he wants, yet he will never produce anything new. He can analyze and make distinct the concept of a straight line, or of an angle, or of the number three, but he will not come upon any other properties that do not already lie in these concepts. But now let the geometer take up this question. He begins at once to construct a triangle. Since he knows that two right angles together are exactly equal to all of the adjacent angles that can be drawn at one point on a straight line, he extends one side of his triangle, and obtains two adjacent angles that together are equal to two right ones. Now he divides the external one of these angles by drawing a line parallel to the opposite side of the triangle, and sees that here there arises an external adjacent angle which is equal to an internal one, etc. In such a way, through a chain of inferences that is always guided by intuition, he arrives at a fully illuminating and at the same time general solution of the question. —Immanuel Kant, Critique of Pure Reason (Cambridge ed.), A716/B744

Kant famously treated mathematics as synthetic a priori: not merely unpacking meanings, but extending knowledge while remaining non-empirical. Others—especially in the Fregean and later formalist traditions—press the opposite thought: mathematics is analytic, derivable by logic and definitions.

The representational-structure view suggests a reconciliation: Within an axiom system, many results are “analytic” in a formal sense: they follow by rule-governed transformations from the axioms. But choosing the axiom system is not itself analytic. It is a modeling decision about which representational structure we are using to regiment some practice (counting, measuring, spatial reasoning, etc.). So mathematics is “analytic” only conditional on the axioms and rules—on the representational framework you adopt. As a model of natural-language practices (counting, describing space, tracking quantities), it is not analytic in any simple way, because natural-language meanings are not fixed enough to determine a unique axiom system. We can adopt different axioms and get different theorems—precisely because we are building different precise models of an imprecise practice.

Geometries as precisifications of spatial intuitions

When a man reasons, he does nothing else but conceive a sum total, from addition of parcels; or conceive a remainder, from subtraction of one sum from another: which (if it be done by words,) is conceiving of the consequence from the names of all the parts, to the name of the whole; or from the names of the whole and one part, to the name of the other part. And though in some things, (as in numbers,) besides adding and subtracting, men name other operations, as multiplying and dividing; yet they are the same; for multiplication, is but adding together of things equal; and division, but subtracting of one thing, as often as we can. These operations are not incident to numbers only, but to all manner of things that can be added together, and taken one out of another. For as arithmeticians teach to add and subtract in numbers; so the geometricians teach the same in lines, figures (solid and superficial,) angles, proportions, times, degrees of swiftness, force, power, and the like; the logicians teach the same in consequences of words; adding together two names, to make an affirmation; and two affirmations, to make a syllogism; and many syllogisms to make a demonstration; and from the sum, or conclusion of a syllogism, they subtract one proposition, to find the other. Writers of politics, add together pactions [contracts] to find men’s duties; and lawyers, laws, and facts, to find what is right and wrong in the actions of private men. In sum, in what matter soever there is place for addition and subtraction, there also is place for reason; and where these have no place, there reason has nothing at all to do. —Thomas Hobbes, Leviathan

In Kant’s time, only Euclidean geometry existed. Kant claimed geometry provided synthetic a priori knowledge about all possible sense experience, because it rested, according to him, on the spatial structure which is inherent to our sensory perception. The later development of non-Euclidean geometries has been thought to undermine Kant’s privileging of Euclidean geometry in this way; but as I see it, the fact that it’s even possible to see Euclidean geometry and alternative geometries as somehow in conflict, rather than simply talking about entirely different things, is a powerful illustration of the point I just made.

In both Euclidean and hyperbolic geometry, despite their different axioms and theorems, we really mean the same thing by the word “triangle”: a triangle is a figure determined by three non-collinear points joined pairwise by straight lines (geodesics). What changes across Euclidean and hyperbolic geometry is not that we suddenly mean something different by “triangle,” but that we add different background modeling assumptions about the ambient space—assumptions that determine which geometrical inferences are licensed. Once those modeling assumptions are in place, predictions diverge: Euclidean triangles have angle-sum 180°, while hyperbolic triangles have angle-sum less than 180°.

The key thought is that these are two precise mathematical models of the flexible, somewhat indeterminate natural-language apparatus of “straight,” “triangle,” and “space.” Different contexts of use—surveying small regions vs. reasoning about curved or non-Euclidean spaces—pull us toward different regimentations.

Reduction and junk theorems

I suggest that both of the above difficulties with the set-theoretical foundational consensus arise from the same source – namely, its strongly reductionistic tendency. Most mathematical objects, as they originally present themselves to us, are not sets. A natural number is not a transitive set linearly ordered by the membership relation. An ordered pair is not a doubleton of a singleton and a doubleton. A function is not a set of ordered pairs. A real number is not an equivalence class of Cauchy sequences of rational numbers. Points in space are not ordered triples of real numbers, and lines and planes are not sets of ordered triples of real numbers. Probability is not a normalized countably additive set function. A sentence is not a natural number. A proof is not a sequence of finite strings of symbols formed in accordance with the rules of some formal system. Each reader can supply his own examples of cases in which mathematical objects have been replaced in our thought and in our teaching by other, purely conceptual, objects. These conceptual objects may form structures which are isomorphic to relevant aspects of the structures formed by the objects we were originally interested in. They are, however, distinct from the objects we were originally interested in. Moreover, they do not fit smoothly into the larger structures to which the original mathematical objects belong. —Nicolas D. Goodman, The Knowing Mathematician

The representational view sheds light on how reductions of one mathematical theory to another always produce “junk theorems”, like “2 ∈ 3” or “1 is the powerset of 0.” (J.D. Hamkins, in his Lectures on the Philosophy of Mathematics, adds the example of John Conway’s account of numbers as games, which produces all familiar theorems about numbers, but where one may ask, “who wins 17?”) Inside a particular coding scheme (say, identifying numbers with particular sets), such statements can come out true. But we recoil because these are artifacts of the encoding, not stable generalizations that track our ordinary linguistic practice with numerals and membership talk.

In other words: reductions can preserve formal structure while scrambling representational roles that matter for interpretation. The resulting theorems may be harmless as bookkeeping inside the reduction, yet they fail as a model of how we actually use “2,” “element of,” or “powerset” in natural language. Our discomfort is pragmatic and semantic at once: the reduction has shifted us into a representational regime where the sentences no longer line up with the inferential and explanatory roles those expressions play in ordinary discourse.

Illustration for this blog post, drawn by Nano Banana.

Postscript: I never watch videos, but the day after I posted this, I watched this video about Euclid by Ben Syversen, which I think is related to this post somehow.

Distinguo Maiorem: On Predicate Relativization in Critical Inquiry

This blog post is geared around establishing two propositions, and deriving two conclusions from them. The two propositions concern the logical ubiquity and necessity of predicate relativization, defined as the act of subdividing a subject term to challenge universal claims. Drawing on the Scholastic tradition of distinguo maiorem (“I distinguish the major premise”), I argue that it is always linguistically possible to distinguish between types or senses of a term—for instance, conceding that “volant birds” fly while denying that “biological birds” (like penguins) do. Furthermore, I contend that such relativization is always permissible in critical inquiry; the alternative, “simpliciter-insistence” (demanding terms be accepted simply as stated), would block the way of inquiry in too many cases, and hence, the right to distinguish is essential for resolving conceptual disagreements and navigating edge cases where definitions break down.

From these premises, I derive two conclusions regarding the indeterminacy of rules and the interpretation of texts. First, I give an analysis of Kripke’s “plus/quus” paradox, which shows that rules cannot be defined tightly enough to prevent future distinctions; this analysis refutes Edward Feser’s argument for the immateriality of the human mind. Second, I argue that interpretive systems relying solely on fixed texts without a living authority—specifically, legal Originalism and theological Protestantism (Sola Scriptura)—are functionally impossible. Because a “dead” or unavailable author cannot settle distinctions regarding ambiguous predicates (e.g., “cruel” punishment or “killing”), these systems cannot yield definitive rules for a community, necessitating a living authority (such as a Supreme Court or Magisterium) to ratify interpretations and halt the infinite regress of relativization.

• (1) It is always possible to relativize a predicate. (This is an observation about language.)
• (2) It is always permissible, in critical discussion, to relativize any predicate. (This is a normative claim about critical inquiry.)
• (3) Conclusions and implications.
• (3.1) An analysis of the plus/quus problem.
• (3.2) The impossibility of Protestantism and Originalism.

(1) It is always possible to relativize a predicate. (This is an observation about language.)

It is a fundamental property of language and logic that it is always possible to relativize a predicate. In the course of any argument, when a speaker makes a universal claim—for example, “All Fs are Gs”—an interlocutor can always complicate the matter by subdividing the subject term F.

By “relativizing a predicate,” I mean the act of answering a universal claim by distinguishing between different senses or types of the subject. If the claim is “All Fs are G,” the relativizer counters by separating F into “A-type Fs” (meaning F understood in sense A) and “B-type Fs” (meaning F understood in sense B). The counter-claim then becomes: “All A-type Fs are Gs, but B-type Fs are not G.”

It is possible to relativize a predicate’s definition rather than the predicate. Even if we agree on the surface statement “All Fs are Gs,” we might disagree on what constitutes an F. Suppose we define F as “all-and-only the Hs that are J.” For example, let us define “bachelor” (F) as “an unmarried man” (H that is J). The relativizer might argue, “I grant that bachelors are unmarried men. However, I distinguish the term ‘man’ (H). All-and-only eligible, adult unmarried men (A-type Hs) are bachelors. But Pope Leo is an unmarried man (B-type H), yet we do not call him a bachelor.”

Predicate relativization has a rich history, most notably in the Scholastic tradition of medieval disputation. In these rigorous debates, the “proponent” would advance a syllogism, and the “opponent” was tasked with finding the flaw. If the proponent argued:

1. All birds fly. (major premise)
2. The penguin is a bird. (minor premise)
3. Therefore, the penguin flies. (conclusion)

The opponent could employ the distinguo maiorem (“I distinguish the major premise”). They might argue: “That all volant-type birds (A-type) fly, I concede; but that all biological birds (B-type) fly, I deny.” By relativizing the predicate “bird,” the opponent dismantles the universality of the major premise and blocks the conclusion.

Analysis confirms that this move is always logically available. No matter how precise a term appears, language is sufficiently fluid that a distinction can always be introduced to carve the concept into accepting and rejecting sub-types. Hence, (1) is true.

(2) It is always permissible, in critical discussion, to relativize any predicate. (This is a normative claim about critical inquiry.)

Not only is predicate relativization always possible, it is always allowed in critical discussion, or inquiry. By critical discussion, I mean a truth-seeking discussion along the lines of what Frans van Eemeren and Rob Grootendorst meant by it when they defined their normative pragma-dialectical model of critical discussion. Eemeren and Grootendorst correctly point out that certain norms are required by the practice of a dispute-resolution process designed to ensure rationality; a core tenet, the Freedom Rule, is that parties must be free to advance any standpoint or cast doubt on any standpoint. I am arguing, then, that blocking the ability to make distinctions stifles this process and prevents the resolution of the underlying conceptual disagreement.

To see the necessity of relativization, consider the alternative, which we might call “simpliciter-insistence.” This is the stance of a debater who refuses to accept distinctions, demanding that terms be taken simply (simpliciter) as stated. Such an opponent of relativization would argue something like the following:

I am not talking about A-type Fs and B-type Fs. I reject the relevance of your distinction. The question was about whether Fs, taken simpliciter and without any extra predicates, are Gs. You may grant that the object of our discussion is a B-type F, and deny that it is an A-type F; but in doing this, you change the subject. You must commit to an answer on whether the object of our discussion is an F, simply speaking, just an F, without any extra predicates.

For instance, in a normative discussion about freedom: “I reject your distinction between positive and negative freedom. I am talking about freedom, period. By distinguishing, you are changing the subject. You must answer whether the object of our discussion is in accordance with freedom, simply speaking.”

The fatal flaw of simpliciter-insistence is that it collapses when faced with “odd cases out-of-domain.” These are edge cases where a predicate is applied to a subject outside its usual context. For instance, suppose we know what “addition”, or summing, means for numbers. We may also have an intuition that “addition”, or summing, is possible with concepts, not only with numbers. But then there are conflicting, equally plausible answers as to what adding concepts means. For instance, what is the sum of the concepts “rational” and “animal”? If we demand that adding the concepts means adding their intensions, we know that the sum of the concepts is “rational and animal”, i.e., a human. But if we demand that adding the concepts means adding their extensions, we know then that the sum of the concepts is “rational or animal”, a concept that applies to humans, dogs, and, if there are any, rational non-animals such as angels. These result in opposite logical operators (AND vs. OR). If one insists on “addition simpliciter” in the domain of concepts, the question is unanswerable.

The defender of simpliciter-insistence might reply by saying that we can solve this by restricting a concept’s domain of application. As the philosopher Alvin Plantinga said, after all, “No prime minister is a prime number.” The opponent of relativization can argue, then, that we can simply restrict the domain to clear cases, e.g., addition is only defined for numbers, not for concepts. Very well. But here we have a regress, since the domain itself can be relativized: do only natural numbers count as numbers, or possibly complex numbers, and even more foreign algebraic systems? If we restrict the domain to algebraic systems with a well-defined addition, this is the same as leaving it to a disputant’s linguistic intuition whether some particular case is addition or not, which was the original problem. And in any purported particular application of addition, we may deny that this particular application constitutes a number in the sense required by arithmetic, and hence that 2+2=4 is relevant to an issue. This is always doable fully in good faith, in a truth-seeking way.

If the simpliciter-insistence defender restricts the domain, the relativizer can always relativize the domain restriction. Because critical discussants must be able to navigate these boundary cases to reach truth, the right to distinguish (distinguo maiorem) must be preserved, and hence, (2) is true.

(3) Conclusions and implications.

If we accept that predicates can always be relativized (1) and that we must allow this in discourse (2), two profound implications follow regarding rule-following and textual interpretation.

(3.1) An analysis of the plus/quus problem.

The inevitability of relativization sheds light on the famous “plus/quus” paradox presented by Saul Kripke in his reading of Wittgenstein.

Kripke challenges us to prove that when we used the “plus” function in the past, we didn’t actually mean “quus,” where x quus y equals x + y if x, y < 57, but equals 5 otherwise. If we have never performed a calculation with numbers larger than 57, all our past behavior is consistent with both “plus” and “quus.”

Since we ordinarily talk about very universal and general laws of arithmetic, such as the axiom of induction, it is not possible that, when we used “plus” in the past, we meant “quus” as specifically defined by Kripke. Nevertheless, although “plus” is indeed defined well outside the range of 57, it is not well-defined for every arbitrary argument. There are always borderline cases (e.g., adding transfinite numbers, or adding symbols) where the definition breaks down. The Kripkean “quus” proponent is essentially exploiting the infinite potential for predicate relativization. We cannot define a rule so tightly that it pre-empts every possible future distinction or out-of-domain application.

Edward Feser’s case for the immateriality of the human mind rested largely on the plus/quus problem: Feser says neural and behavioral features of humans are consistent with multiple interpretations of our words, which suggests our words should be indeterminate, but in fact when we use words, we actually mean an absolutely definite concept which is not indeterminate between all concepts that are consistent with our past behavioral and neural features, and hence, our minds must be immaterial, wholly independent from our behavioral and neural features. I argue that human concepts are not, after all, all that determinate, since in fact infinitely many questions can be raised about them which the concepts themselves do not determine, and hence, Feser has not shown that the human mind is immaterial.

(3.2) The impossibility of Protestantism and Originalism.

Finally, the universal possibility and permissibility of predicate relativization poses a devastating challenge to systems of interpretation that rely solely on a fixed written text without a living authority that can answer questions about it, such as, specifically, legal Originalism about the U.S. Constitution, and theological Protestantism, resting on the Sola Scriptura principle. The challenge is that it is impossible to use these systems consistently to determine rules of belief or practice for a collectivity, such as the community of believers or the community of U.S. citizens.

This is because, if we have a written text by an author who is dead or otherwise unavailable to answer questions (either in person or through a representative), then it is not always possible to apply the text to issues of theory or practice and get a definite result, which is acceptable to all reasonable parties to a discussion about the text. In particular, we cannot always apply the Bible to get definite answers on theology and on moral law; and we cannot always apply the U.S. Constitution to get definite answers on how to judge a federal issue. Some examples of relativization:

• Constitutional law: If the Constitution forbids “cruel and unusual punishment,” and we ask, “Is the death penalty cruel?”, we are asking for a judgment on a predicate. An opponent can always distinguish: “It is cruel in the modern sense (A-type), but not in the 18th-century sense (B-type).”

• Theology: If the Bible says “Thou shalt not kill,” does this apply to war? One can distinguish: “It forbids murder (A-type killing) but allows martial combat (B-type killing).”

Originalism and Protestantism both rely on a “dead author” in our current, Barthes-inspired sense, i.e., the Founding Fathers or the author(s) of the biblical texts, respectively. Protestants may argue that the true author of the Bible is God, who is indeed living rather than dead, but this does affect my point here, which is that the author is unavailable to answer clarifying questions. Some Protestants may here press onward, by arguing that God is available to the individual believer via the “inner witness of the Holy Ghost”; but this does not change the case I was arguing, which is that it is impossible for Sola Scriptura to get a definite result, which is acceptable to all reasonable parties to a discussion about the text, and hence to determine rules for the community of Christian believers; this is, of course, obvious now, in light of the manifold splinters of the Protestant community since Luther’s time.

If the author is available to answer questions, either in person or through a representative, then he can settle the matter: “I meant A-type, not B-type.” But if the author is unavailable to discussants, and we do not agree that anyone on earth is his representative, then we are left with the text “simpliciter.” Because the text cannot distinguish its own predicates when faced with new contexts or edge cases, and because the author cannot respond to a distinguo maiorem, the reader is forced to supply the distinction, leading to a fork in how the predicate is understood.

Thus, without a living authority (such as a Supreme Court or a Magisterium) to ratify which distinction is authoritative, the text alone cannot yield definite answers to theoretical or practical problems. It remains forever open to the infinite regress of relativization.

Illustration for this blog post, drawn by Nano Banana.