Axiology

Little by little, though not, I hope, painstakingly, we have reached the last step on this ladder, which I call the axiology section. Axiology is the study of value, but we have some distance to travel before reaching the shores of that concept. For now, consider this simple scenario:

• Peter is hiking and sees a large rock overhead that’s about to fall on the trail.

• Peter sees that a child is standing right there.

• Peter jumps and pushes the child out of harm’s way.

In a way, this chain of events is perfectly at home within the type of models we’ve described so far, both from the physical and metaphysical perspectives. The rock’s fall and Peter’s leap are similar events in that they describe the motion of solid bodies that can be approximately described in classical mechanics, and certainly must comply with an ultimate theory \(\mathfrak{ToP}\). With that lens, Peter’s adventure is not that different from two billiard balls on a collision course.

But of course, that is not the lens that any onlooker would take. Here are some statements one could make, and the cluster of associated concepts.

Action:

Peter decided to jump.

Teleology:

Peter’s aim was to save the child.

Reasons:

Peter had reasons to jump.

Rationality:

Peter’s decision to jump was rational.

Justification:

Peter was justified in trying to save the child.

Epistemic norm:

Peter saw the rock, so he knew that it would fall.

Deontic norm:

Peter was morally obligated to save the child.

Evaluative norm:

Given the context, Peter’s action was right.

Value:

Peter’s action was absolutely good.

All of those are part of our regular human discourse, yet they seem conceptually distinct from everything we’ve discussed so far. In a modern physical theory, events are explained by causes, not justified by aims. It feels natural to say that Peter’s jump was caused by some tension in his muscles which itself can be explained by electrical activity in his neurons, etc. But it simultaneously feels right to say his jump was justified by his aim to save another human being. The rock, though, has no aims and does not "behave" according to any norm.

At least, that is the prima facie view promoted by common sense. Is there something to all this, some part of our ontology that we have heretofore ignored, or is it another illusory language game? That is what we’re about to explore

Reasons: what is the fundamental unit of reason?

Position: Reasons fundamentalism

Action

That is the first concept I’ve identified as departing from a factual or descriptive account of reality. In its daily usage, the term denotes some intentional manifestation of an agent’s will. Colloquially, most speakers would agree that Peter’s jump is an action, whilst the rock falling is just an event.

One could easily add that an action, unlike purely physical events, also involves a mental ground. Ludwig Wittgenstein, always crafting useful images, explained action as "what is left over if I subtract the fact that my arm goes up from the fact that I raise my arm". Action can then be seen as a sort of transmission belt between a mental state and a physical event. The convention I will adopt here is to see action as a mental state in its own right.

The sequence would then look something like this: \[\text{[Desire to drink]$_{\mathbf{2}} \rightarrow$ [Action: to grab a glass of water]$_{\mathbf{2}} \rightarrow$ [Arm moving]$_{\mathbf{1}}$}\]

This concept of action makes sense in two main ways. First, it helps capture a sense in which action may exist even when the corresponding physical fact never obtains. Wishing to do something does not always entail that one takes the corresponding action. But also, at least in my own way of using this terminology, taking action doesn’t entail that its physical manifestation be realized. Peter may take action to jump, but then realize that having been sitting so long, his legs are numb, so nothing happens. Yet there definitely was a mental discrepancy between the simple desire to jump and the clear intention to instil bodily movement.

The second point is that action is not correlated with activity in the active/passive sense. Let’s say that Peter now encounters a bear on his hike. He needs to balance a strong impulse to run with the thought it is safer to stand still. By doing the latter, Peter is definitely taking an action by deciding to not move. It is difficult to define physical events in relative terms versus a baseline - either everything is an event or nothing is. It is then easier to define action as a mental state, rather than some physical/mental pairing.

Teleology

That term denotes the type of explanation that refers to purposes, ends or aims rather than causes. As already mentioned above, the natural sciences, like the style of fact-based metaphysics we’ve developed, have largely eschewed teleology. Although ends and aims are still haunting scientific discourse ("snakes have evolved to produce venom in order to paralyze their prey"), the reducibility of all natural sciences to physics should entail that any rigorous exposé be entirely causal.

However, one key instance of teleology that has not been waived so successfully is that of subjective minds, in particular human minds. I, like all human beings, seem to have aims in the forms of mental states: my wish to drink some coffee, my intention to go to the gym tomorrow, my fear of roller coasters. Since action has been identified as another mental state, we can also recognize that an action, usually the prelude to some physical event, is geared towards that event as its end. Agency thus seems inseparable from teleology.

The term of the art for the group of mental states that exhibit aims is that of pro-attitude (one could say that fear is an anti-attitude, or rather a tendency to be pro-not doing what one fears). None of this is too surprising, since we’ve introduced intentional mental states earlier in this book. Remember that those have an object they point to, a content, and a mode (belief, wish, fear, etc.).

Content, we’ve gone on and on about in our semantics travels. Now it seems we are to focus a bit more on the modes, which we had quickly breezed over. It seems clear that the form of directedness that’s characteristic of intentionality must be linked with the teleology of subjective mentality. Yet, all intentional states are not teleological. My seeing a chair is a perceptual state that’s directed at an object, but there is no aim involved. All in all, it appears that aims are the wont of a subset of intentional mental states, and that action is one of those mental states that can exhibit aims.

Since we’ll be manipulating mental states quite a bit, let’s equip ourselves with some symbols:

• \(\textsf{Bel}[x\phi]\): \(x\) believes that \(\phi\).

• \(\textsf{Des}[x\phi]\): \(x\) wishes that \(\phi\).

• \(\textsf{Act}[x\phi]\): \(x\) brings about \(\phi\) (by acting).

\(x\) is an agent — formally, an ordinary object that we associate with mental subjectivity; for instance a human being. To say that "\(x\) believes that \(\phi\)" is equivalent to an existence statement about the mental state "\(x\)’s belief that \(\phi\)".

Rationality

It is not obvious how rationality can be connected to this conceptual cluster. To elaborate, we will first need to agree on what that is... starting perhaps with what it’s not!

The first breeding ground of confusion seems to be the interface between rationality and logic. It is certainly encouraged by colloquial overlaps; to say that "X is acting rationally" or that "X is acting logically" is not very different in commonplace usage. However these two notions operate over different domains. Logic is about rules of inference between different propositions, and is prior to any metaphysical assumption. Rationality is broadly about the coherence between various mental states that make up a subjective mind. For instance, the "law of non-contradiction" (that some proposition and its negation cannot both be true) is a purely logical statement: \[\forall \phi (\neg( \phi \wedge \neg \phi))\] The rational equivalent would then be that a belief in \(\phi\) cannot rationally be combined with a belief in not-\(\phi\): \[\forall \phi \forall x (( \textsf{Bel}[x\phi] \wedge \textsf{Bel}[x\neg\phi]) \text{ is not rational})\]

It might be true, although at this stage it is still a pure guess, that all logical principles have an equivalent in terms of rationality. But not all rational principles have logical roots. Imagine that Peter doesn’t want to eat coconut, and that Peter believes (correctly) that ordering a coconut-based dish will imply his being served coconut. If Peter does order the coconut, we would conclude (absent more information) that the action was not rational. There’s nothing illogical here; what doesn’t cohere is the group of mental states involved. We could intuit the general principle that if \(x\) doesn’t want \(\phi\) but believes that \(\alpha\) will surely lead to \(\phi\), then to be rational, \(x\) shouldn’t bring about \(\alpha\): \[(\textsf{Des}[x \neg\phi] \wedge \textsf{Bel}[x(\alpha \rightarrow\phi)]] \wedge \textsf{Act}[x \alpha]) \text{ is not rational}\] The discrepancy between rationality and logic was only the first pitfall I had to mention. The next one is borne out of the contrast between rationalism and empiricism. These opposing numbers, which any student in Philosophy 101 is familiar with, unfortunately give the wrong idea about rationality. Rationalism is an epistemological thesis that regards a priori reasoning as the key source of knowledge, while empiricism teaches that sensory experience is. But the concept of rationality we’re dealing with in this section is much broader and absolutely compatible with both theses, regardless of apriority. To say that "Peter seeing a dog" entails that "Peter believes a dog to be nearby" is a rather obvious empirical principle, but that is also an example of rationally coherent mental states.

So, it appears that rationality is some general quality that is fulfilled by mental states considered together, or by a sequence thereof. Indeed, if intentional mental states like the ones from folk psychology do exist (beliefs, desires, feelings...), then they clearly operate within intentional systems. What we are not talking about here is the descriptive approach, that would seek to depict and explain causally why some mental state leads to another in a given mind. If such explanation is even possible, then it’s part of psychology, and at its most fundamental level of \(\mathfrak{ToPM}\), the "scientific theory" of the natural (i.e. physico-mental) world.

Our talk of rationality here is instead normative. We are saying that some assemblages of mental states are good, worthwhile, obligatory in a way that we call "rational", and that others are bad, undesirable, forbidden in a way we call "irrational". Much more will be said about norms and normativity in the next subsection.

For now, we want to keep the spotlight on rationality, and more particularly to get to the bottom of that concept by isolating its base unit.

Reasons

Distancing myself from many philosophical writings that erect Reason as a singular primitive ideal, I would argue this investigation is better served by dealing with the fundamental unit, the "atom of Reason". In most contemporary accounts, that is identified as... a reason.

First let’s rebrand "Reason" as a relation between:

• a fact \(\text{f}\): the reason

• an ordinary object \(x\): the agent

• a mental state \(\psi\): the response

• a mathematical object \(\delta\): the degree

Then, the following proposition: \[\mathcal{R}[\text{f}x\psi \delta]\] reads as "\(\text{f}\) is a reason for \(x\) to \(\psi\) with degree \(\delta\)".

What makes an ordinary object \(x\) an agent goes well beyond this book’s scope. Suffice it to say that we know human beings to be agents; which animals or other beings are is obviously less clear-cut, not to mention the infinite realms of actual or possible agents, unfathomable to us.

The degree is also an essential notion to formalize reasoning. I have reasons to go outside in the sun, but I also have reasons to stay in the office and write. It’s clear that any human action will ultimately result from a comparison between the magnitude of a huge array of reasons. The natural incline is to consider \(\delta\) as a real number: I may be 30% attracted by the sun outside but 70% deterred by my work deadline. Yet I won’t make any imposition, since we could obviously be way more creative, introducing multi-dimensional frameworks, etc. Degrees will still need to share some essential features of real numbers though, like having an order (if we can’t say if a reason is stronger than another, that seems to defeat the purpose) or some addition operation (similarly, we want to be able to tell if reasons 1 and 2 can together "out-rank" reason 3).

This lets us define the key concept of justification: \(x\) is justified to \(\psi\) when \(x\) has more reasons to \(\psi\) than to not \(\psi\). We could write, very roughly, that: \[\mathcal{J}x\psi \hat{\equiv} \int_{\mathcal{D}_\text{f}} \left( \imath \delta (\mathcal{R}[\text{f}x\psi \delta]) - \imath \delta'(\mathcal{R}[\text{f}x(\neg \psi) \delta']) \right) d\mu(\text{f}) > 0\] This is a mere sketch obviously. Developing a full topology of reasons could fill a whole book. In plain English, we’ve just expressed that, going over the whole universe of facts and counting reasons to \(\psi\) positively (pro rata to their degree), and reasons not to \(\psi\) negatively), a positive result means that \(\psi\) is justified.

A bit more technically, I may add that this sum is meant to be a Lebesgue-style integral over the domain of facts \(\mathcal{D}_\text{f}\), equipped with some measure \(\mu\). The expression \(\imath \delta (\mathcal{R}[\text{f}x\psi \delta])\) uses the definite description operator: it is the degree of \(\text{f}\) if \(\text{f}\) is a reason, with the convention that if \(\text{f}\) is not a reason, the outcome is \(0\) ("the zero" of whatever mathematical set was picked).

Of course this amiable setup hides devilish difficulties. The reason relation is not monotonous, meaning that if \(\text{f}\) is a reason, \(\text{f}\) and \(\text{g}\) together are not necessarily a reason. That doesn’t prevent double-counting and other infinity-prone features. The hope is that some adequate topology can be developed (or has been?) that could flesh out such a formalization.

Coming back to the reason relation, although it may not seem like it at first sight, to say that reasons are facts is quite momentous. As a reminder from previous sections, a fact is an obtaining state of affairs, so it exists if and only if the corresponding state of affairs obtains, which itself is equivalent to the corresponding proposition being true. So for instance, "John F. Kennedy died in 1997" is a perfectly valid proposition, that turns out to be false in the actual world. So there is no such fact "that JFK died in 1997". Something that doesn’t exist cannot be an argument of a relation, therefore (colloquially) "only things that are true" can be reasons.

That fits quite nicely with our linguistic intuitions. It would seem a bit strange to say "the reason for Ana to take an umbrella was that it was raining, but it wasn’t raining". We are then faced with a dilemma. We may simply say there was no reason for Ana to take the umbrella. Or we could say that the reason was her belief that it was raining. With a bit more sophistication, one could suggest the reason is, say, that she heard on the weather forecast that it was raining — but that is every bit as agent-relative as the previous suggestion, since there is no way to progress from hearing the forecast to taking the action without involving a belief about the weather. Why should that quite natural interpretation make us uncomfortable?

Say that Ed believes he’s a bird, and jumps from a tall building, to his demise. It seems equally just to say that:

• Ed believed that he’s a bird, that birds can fly, and that jumping from a tall building is safe for flying creatures. From his perspective, his was then a rational decision.

• Ed’s action was grounded on his belief that he was a bird. He wasn’t a bird, therefore he didn’t have a good reason to jump.

We can see that a good old analysis is warranted here, as our language is misleading us. Despite the superficial similarity, two distinct conceptions emerge:

• the former example, which John Broome would call structural rationality, is a judgment over the coherence of an agent’s intentional mental states. It is a narrow, internalist concept, which doesn’t depend upon the external world beyond Ed’s subjective mind.

• the latter, which Broome calls substantial rationality is wide and externalist: something can be a reason or not based on facts whose existence the agent has no accurate knowledge of.

Can we still salvage some unity and link the two concepts? I believe we can. Let’s rephrase the structural, "narrow" assessment of rationality as: "It is the case that if Ed was a bird, he would be justified to jump. Ed believed that, and in addition, believed that he was a bird. Lastly, Ed did jump. So the action was rational". It is then possible to build up rationality based on that model: \[\mathcal{Rational}[x\psi] \hat{\equiv} \begin{cases} \phi \rightarrow \mathcal{J}x\psi \\ \mathsf{Bel}[x(\phi \rightarrow \mathcal{J}x\psi)]\\ \mathsf{Bel}[x\phi] \\ \psi \end{cases}\]

In this subsection, I aimed to demonstrate the singularity of rational judgements vs the factual-descriptive mode we had become accustomed to, introduce a form of rational atomism with reasons, and provide a convincing sketch for a mathematical formalization. With that done, let’s plunge into a topic that was just grazed here: normativity.

Norms: do normative facts exist?

Position: Normative pluralism

The normativity of reasons

It has been common to operate a distinction between motivating reasons and normative reasons. The former are traditionally described as the reasons someone has to be in a certain state; the latter cover the reasons that “count in favour” of being in a certain state. In a nutshell, normativity is about the "ought": it is preoccupied with how things ought to be, rather than with how they are.

However, I am inclined to reject that distinction altogether, and to accept that reasons are essentially normative. First, looking at the (supposed) motivating vs normative difference, it seems to be re-litigating the rationality/reasons distinction from the last subsection. That Ed believed he was a bird qualifies (by most accounts) as a motivating “reason” for jumping off a building. But there is no normative reason for him to do so (nothing objectively “counts in favor” of that). Ultimately, I would argue that what some authors call motivating reasons simply evidence instances of an agent being rational. And what fits the definition of reasons that I have proposed is really a normative reason.

This conclusion can easily seem counter-intuitive, in virtue of the habit we have of reflecting on morality as the pure archetype of a normative system. Let’s come back to Peter, who jumped to save a child from a falling rock. There is no debate that one reason to jump was normative: Peter was morally compelled to save a child in danger. Now, imagine that someone else spotted the rock as it was about to fall, and quickly offered Peter one million dollars to save the child. It is a fact the offer was made, and we may want to say that counted for at least something in Peter’s quick thinking, but it seems awkward to put that reason under the normative category. Seeking retribution for a good action should indeed not be the bedrock of anyone’s moral code, but the point here is that normativity is much broader. The rule that, all else being equal, the promise of a prize increases, ever so slightly, someone’s bundle of reasons to make the correct move, seems like the kind of common-sense norm that guides human behavior, for better or worse.

To accept that is not to make a value judgement, though, like accepting some moral norms would be. To take another example, being hungry is a reason for me to initiate the action “grabbing my sandwich”. Again, the implicit norm that being hungry is good reason to grab any food around is common-sense, and not borne out of some intricate moral stance. If anything, it is grounded in biology, but can be counter-acted by a plethora of social norms. I regard it as a norm nonetheless, notwithstanding a difference of nature and degree with something like “Thou shalt not kill”.

A typology of norms

So, normative reasons are multifarious, but we can still try and provide some structure to their domain. I will outline three categories:

• Deontic reasons are reasons to act. They are connected to notions of obligation: to say I am overwhelmingly justified in acting a certain way (and not at all justified to do nothing) is equivalent to saying that I must act that way. Moral or ethical norms are a natural fit for this category.

• Evaluative reasons are reasons to feel or to wish. Aesthetic norms are built upon evaluation: to find something beautiful is to have reasons to feel pleasure, emotion, appreciation. Moral obligation can also be framed as an evaluation of situations: I may not be the agent but feel an action is good if I have reasons to wish someone would take it.

• Epistemic reasons are reasons to believe. All subjective minds spin a web of beliefs. But those don’t exist in a vacuum: they connect to each other, but also to facts that justify them: what the agent has seen, has been told...

I’ve been using the term norm without a proper definition. Now I will state that any proposition that involves the reason relation \(\mathcal(R)\) is a norm. Further to that, I will single out a sub-set of those norms which I’ll call principles. Colloquially, it is quite clear what the difference is: principles, for instance moral principles, are “laws”, or “rules” with limitless purview, while regular norms can pertain to very specific situations. Let’s formalise things a little bit.

We can start again from our very simple example. The fact that Camille (a child) is in danger of being crushed by a rock (\(Dc\)) is a reason for Peter to take the action: try and save Camille (\(Spc\)). \[\mathcal{R}[[\lambda Dc] p (\textsf{Act}[Spc]\delta])\] This is:

• agent-relative: about the agent’s judgment of his own (purported) actions.

• special: about a given, specific agent (Peter).

• particular: involving a specific object (Camille).

To make a norm agent-neutral instead of agent-relative, one needs to have the reason and the response be independent of the agent’s identity. For instance: the fact that Camille is in danger is good reason for Peter to wish that someone would save Camille. In other words, whomever may save Camille, Peter would garner normative support towards that scenario. \[\forall \, h \exists \delta>\delta_0 \ \left( \mathcal{R}[[\lambda Dc]p (\textsf{Des}[Shc]) \delta] \right)\]

To make a norm general instead of special, it needs to be true of any agent. For instance: the fact that Camille is in danger is good reason for anyone to wish that Peter saves her. \[\forall h \, \exists \delta>\delta_0 \left( \mathcal{R}[[\lambda Dc]h (\textsf{Des}[Spcl]) \delta] \right)\]

To make a norm universal instead of particular, the reason and response need to be rid of any proper name (i.e. any direct-reference, Kripkean acquaintance that is attached to a specific ordinary object) other than the agent. For instance: the fact that a human is in danger is good reason for Peter to try and save them: \[\forall h \, \exists \delta>\delta_0 \left( \mathcal{R}[[\lambda Dh]d (\textsf{Act}[Sph]) \delta] \right)\]

We shall define a principle as a norm that is agent-neutral, general and universal. For instance: the fact that a human is in danger is good reason for anyone to try and save them: \[\forall h \, \forall j \, \exists \delta>\delta_0 \left( \mathcal{R}[[\lambda Dh]j (\textsf{Act}[Sjh]) \delta] \right)\]

From a given principle, or, for that matter, from universal, general or agent-neutral norms, more specific norms can stem at all time. That’s one of the basics of rationality: to take one’s principles and apply them to actual situations. Which reasons, or types of reasons, can be derived from principles is naturally a key question for normative metaphysics. It determines the structure of the entire normative system.

But we’ve eschewed much more momentous questions. This axiology section started by introducing reasons as a common ground to a cluster of seemingly disparate concerns. Then the connection between reasons and normativity has been reinforced. But that’s still just talk, discourse, pixels, inkblots or soundwaves... Indeed, we may have talked a great deal about reasons and norms, but we have done so light-heartedly, as we might in daily conversation. It’s time to get a bit philosophical and to follow our usual line of questioning:

1. The semantic question: do normative sentences express truth-apt propositions?

2. The ontological question: do normative facts exist?

3. The metaphysical question: what grounds normative properties?

4. The epistemological question: how can we come to know normative facts?

I won’t answer these questions here unequivocally — there are many flavors of normative sentences that may fall in one category or the other. This subsection is then placed under the sign of normative pluralism. The next one, centered around value-based norms, will be more trenchant.

The semantics of norms

It is time to confront the foremost bogeyman in philosophy. Are we fooled by language? Remember that we started talking about action, rationality and normativity after acknowledging their importance as a domain of discourse. There are normative sentences, that’s an unassailable fact.

But we’ve also started to use the reason relation \(\mathcal{R}\) in a carefree way, writing propositions, implicitly admitting its existence in the world. It’s time to question these priors. We’re only spending time speaking about normativity because it is strange, unlike anything else we’ve built so far. Perhaps it’s just an illusion, then?

Let’s summon some examples of normative sentences that will be our guinea pigs:

• Deontic: It is forbidden to swim in the river.

• Evaluative: ’Penny Lane’ is a beautiful song.

• Epistemic: You must believe what you see.

We know that these sentences exist, because they have actually been uttered. It is a natural impulse to automatically assume they express a proposition. Indeed, their syntactic structure is similar to non-normative sentences:

• It is cold to swim in the river.

• ’Penny Lane’ is a pop song.

• Alice did believe what she had seen.

Those sentences, when uttered, are reflective of a cognitive state. In short, to say that \(\phi\) is to express a belief in \(\phi\), that is, to signal that I am in the mental state \(\textsf{Bel}[\phi]\). Since \(\phi\) is a proposition, it is truth-apt: it can be either true or false. Is it the case for normative sentences?

To accept that is to espouse a position called normative cognitivism. If that is the branch we pick, we need to face more questions, at the very least the ontological one, which we’ll discuss in a moment. If we don’t accept normative cognitivism, that makes us (drum roll...) normative non-cognitivists. Sometimes theses opposite positions are called descriptivism and non-descriptivism.

Now is as good a time as any to say this: the answer to the semantic question (and to the ones that follow) need not be the same for all kinds of normative sentences! One may well be a moral non-cognitivist but an aesthetic cognitivist (for the sake of argument).

If non-cognitivists believe that normative sentences are not really about normative propositions, then what do they make of them? Well, the standard stance is to say they are disguised expressions of conative states: mental states that include desire, fear, appreciation... It is common to group them under the more neutral guise of "a pro- (or anti-) attitude" towards something.

Indeed, lots of our non-sentential utterances express conative states. Shouting "Boooooo, Novak!" in a stadium reflects a wish that the so-named sportsman lose. "Yes, he lost!" arguably doesn’t reflect a belief that he did lose (which is likely obvious in context) but mostly expresses the pleasure the speaker experiences.

Similarly, non-cognitivists include the following main groups:

• emotivists would say that the impermissibility of swimming in the river is just an alternate way of "booing" that action, really expressing a strong dislike for it.

• prescriptivists would rather say that sentence expresses a very strong wish or a direction, best rendered as an imperative: "don’t swim in the river!"

• expressivists cover a more varied, and dare we say sophisticated, range of positions which boil down to a generalization of the above towards more generic pro- and anti- conative attitudes.

The ontology of norms

Do normative facts exist? For non-cognitivists, the question is moot: a fact is an obtaining proposition, so if there’s no normative proposition, there can be no normative fact. What of cognitivists? They accept that there are truth-apt propositions about normativity. Those can be true or false. If a normative proposition is true, the corresponding state of affairs obtains, and a corresponding fact exists. Accepting that makes us proponents of normative realism.

The complementary stance is normative anti-realism. Of course, non-cognitivism is a flavor of anti-realism. But is it possible to be a cognitivist and an anti-realist? It is, if one believes that all normative propositions are false. That is why the resulting position is often called error theory: it argues that all of our beliefs in normative propositions are misguided, as there is no fact of the matter.

That may seem like a peculiar position: what’s the use of an entire class of propositions that are uniformly false? Like other anti-realists, error theorists are struck by the queerness of (purported) normative properties; refusing the existence of normative facts avoids dealing with that. Unlike non-cognitivists however, they don’t deny that there’s genuine cognitive content behind our normative sentences that can play a similar explanatory role as "morals" do in folk psychology. The general conceit of error theory is actually not new in this work. It is very close to mathematical fictionalism or to our treatment of fictional entities proper, to such an extent that we could develop a sub-set of error theory we’d call normative fictionalism.

About fictional entities, we had established, in summary, that:

• "Rastignac is ambitious" is false.

• "Rastignac is a fictional character" is true.

• "Rastignac is ambitious" is true in Balzac’s literary world.

• "Rastignac" is an abstract object that encodes "being ambitious"

Normative discourse is not so much about objects than about properties, so it might be more useful to flip the perspective, as in this example:

• "Romulus was a demigod" is false.

• "’Demigod’ is a fictional kind" is true.

• "Romulus was a demigod" is true in Roman myths.

• "Being a demigod in Roman myths" is an abstract object that encodes "being exemplified by Romulus"

If we believe in mathematical fictionalism, we can similarly say that:

• "\(2\) is even" is false.

• "\(2\) is a mathematical object" is true.

• "’Being even’ is a mathematical property" is true.

• "\(2\) is even" is true in Peano number theory.

• "\(2\)" is an abstract object that encodes "being even".

• "Being even" is an abstract object that encodes "being exemplified by \(2\)".

Moving on to a normative sentence, we get:

• "Stealing is wrong" is false.

• "’Being wrong’ is a moral property" is true.

• "Stealing is wrong" is true in the moral code \(\mathfrak{M}\).

• "Being wrong in \(\mathfrak{M}\)" is an abstract object that encodes "being exemplified by stealing".

Let’s use a last example to shed a different light on this idea:

• "Moving the pawn backwards is illegal" is false.

• "’Being illegal’ is a deontic property" is true.

• "Moving the pawn backwards is illegal" is true in the rules of chess

• "Being illegal in the rules of chess" is an abstract object that encodes "being exemplified by ’moving the pawn backwards’".

We can see the close relation between fictionalism and formalism. Balzac’s books and Roman myths are like guidebooks to their universes: they make a large number of statements, from which we can ourselves extrapolate, using some form of metaphysical inference. Number theory and chess rules are an even clearer fit since they are explicitly axiomatic systems from which an infinity of theorems can be inferred. In this worldview, normative codes such as morals or aesthetics can follow either an informal, pragmatic, individual "code" made up of countless micro-rules, or a codified set of "tables of the Law".

The formalization itself is similar to what we’ve done for the aforementioned topics of mathematics and fiction. We can start by writing that stealing is wrong in a certain moral code: \[\mathfrak{M} \vDash W s\] Then the concept of being wrong (in the code) is just the abstract object that encodes all the things that being wrong exemplifies in the code: \[W_{\mathfrak{M}}=\imath y ( \kappa y \wedge \forall \mathcal{F} ( y\mathcal{F} \equiv \mathfrak{M} \vDash \mathcal{F} W )\] The code itself can be defined as the abstract object that encodes all propositional properties that can be inferred from its axioms: \[\mathfrak{M} = \imath x ( \mathbf{A}x \wedge \forall P ( xP \equiv \exists \phi ( P=[\lambda y \phi] \wedge \alpha_1,...\alpha_n \vdash_M \phi_T ) ))\]

We should now find it easier to relate to error theory. It is fairly consistent with our daily experiences to say that something is permissible based on a certain set of rules, or is beautiful based on the tenets of some aesthetic movement. The obvious drawback is that relativism is part of the package, and can quickly veer into nihilism. If moral or epistemic codes are inferred from arbitrary axiom schemas, then we are seemingly disarmed against people arguing that the Earth is flat, or that hurting children is a good thing...

The metaphysics of norms

This one only matters to realists. It’s about the place of normative things in our ontology. Anti-realists don’t believe there are normative facts, and a fortiori, don’t believe in normative things outside of the fictionalist theories. Realists however don’t have that luxury. Our ontology so far has made use of two procedures to generate properties and abstracta. The first one, the "natural route", starts from fundamental natural properties (originating in \(\mathfrak{ToPM}\)); even though it can make heavy use of abstraction via the comprehension principle, and even combine the outcome with mathematical properties, the end result is recognizable as an entity whose existence depends on the physico-mental world.

"Having an electric charge", a physical property, "feeling pain", a mental one, obviously fall into this category. More complex properties like "being a bicameral constitution" or "being a sonnet" are too, since they can be exemplified by objects that are themselves typified by ordinary things. The other route is purely abstract and is represented by the procedure we’ve just reminded ourselves of, that can lead to the sprouting of mathematical theories, fictional worlds or legal rulebooks.

Adopting the latter solution brings us back to anti-realism. The choice is then simply between the natural route, and an additional solution, one that is always at hand: the profligate route that simply posits new, independent entities.

The natural route leads to an attitude called normative naturalism. A minima, its central tenet is the idea that normative facts are grounded in natural facts. As we’ve seen when exploring inter-level metaphysics, there are many sub-cases. We’ll distinguish two here.

• Reductive normative naturalism has it that normative properties are not just grounded in natural properties, they are natural properties, in the sense that the right bridge laws can reduce the former group to the latter. A very simple version would be a direct equivalence like "to be a good thing is to elicit joy" or "to be beautiful is to be symmetrical". Of course, a serious attempt in this direction would result in an ultra-Leibnizian stance, whereby goodness or beauty would be defined as fearsomely complex algebraic combinations of physico-mental primitives. In other words, we would need to argue that there are a “formula for good” and a "formula for beauty" with which an omniscient being could evaluate every normative proposition.

• Synthetic normative naturalism says that normative properties are grounded in, but not reducible to natural ones. The relationship between the two would then occupy the space that’s broadly referred to as emergence. That is not a great fit for our own model, at least at first sight: the principle that properties, relations and abstract objects are built algebraically enshrines reduction as the primary type of grounding between properties and tends to conflate emergence with independence, which is the next position we’ll examine.

The profligate route eschews the idea of emergence indeed, to simply conclude that normative properties form a metaphysically primitive category, another category of being altogether. Because of the existence and independence of normative properties in that worldview, it is often called normative Platonism. That can be quite easy to accept in a religious or mystical context. Taking the example of monotheism, normative properties are commonly related to God (e.g. what is obligatory is what God wants), and since God and God’s "cognitive states" are already considered to be beyond the natural world, normative concepts have a cozy home to themselves already. It is also possible to accept this outside of religious settings: that would entail the addition of new "tables of the law" to the existing "tables of natural law", namely a normative complement to those causal and descriptive principles that govern our universe.

The epistemology of norms

This is a basic philosophical question, but also a profund one that has been at the core of belief systems of all eternity: how do we come to know what’s good or what’s bad? There’s not enough space here to delve into the complexities of this topic; suffice to say that the answers to previous questions will come to determine the realm of epistemic possibility.

Moral expressivists and formalists could readily consider normative epistemology to be moot since they defined normative judgments on the basis of actual mental states (conative ones for the expressivist, knowledge of a formal system for the fictionalist).

Reducive naturalists would rely in our times on evolutionary psychology: natural properties we come to see as synonyms for "good" must be related causally to some traits that’s been selected as advantageous for our species. Knowledge of "good behavior" is thus pre-programmed in subjective agents through their genetic heritage. Pure naturalists would perhaps suggest that a belief in universal values is a hallucination that self-conscious beings make up when confronted with the evidence of that behavioral bias.

As to Platonists themselves, they must contend with the same issue as realists of every obedience: if some norms are neither mental nor physical, then how can we physico-mental beings know of them?

Value: is there such a thing as intrinsic Good?

Position: Axiological pluralism

The concepts of norms and reasons as philosophical staples is fairly recent. In a more distant past, discussions of a meta-ethical or meta-aesthectic nature were more readily couched in the language of value.

Re-framing norms as value judgments

Values are certainly part of everyday speech and as such, have become fair game for philosophical theorization early on. The "classical triad", which morphed during the medieval era into the "transcendentals" is based on Plato’s trilogy of beauty, goodness and truth. More complex distinctions have emegerged, among which we’ll quote a single example, out of Frankena’s Ethics. Frankena sorts various purported values into six groups:

1. Beauty, harmony, proportion, aesthetic experience

2. Moral disposition, virtue, pleasure, happiness, contentment

3. Truth, knowledge, understanding, wisdom, honour, esteem

4. Life, health, strength, peace, security

5. Love, affection, friendship, cooperation

6. Power, achievement, freedom, adventure, novelty

The term’s usage has become a bit muddled, running the gamut from Platonic universals, to individual traits (someone’s "personal values"). To re-center the topic, I should start by pointing out once more the equivalence between statements about reasons, normativity, and values.

Have we achieved anything at all here? Or did we only show value discourse to be one re-phrasing away from the normative or the rational discourse we’ve been tackling so far? That is indeed the case. However the shift to a value perspective will help see through the mists of language and answer important questions about the true nature of normativity.

Intrinsic value

There’s a traditional distinction between intrinsic and instrumental value. An archetypical example is money. Most agents would say it has no intrinsic value, but only instrumental value (it is not good in itself, but good because of what one can get with it). For most reasons, the corresponding value is then instrumental, but as we hark back through the rational arguments that make up the links of our reasons’ chain, we supposedly could reach some "ultimate", primitive reasons that are not grounded in other reasons, and would reflect intrinsic value.

It is clear there are many forms of intrinsic value, hence the endorsement in this subsection of axiological pluralism. As it turns out, the taxonomy that we’ve developed will not lead us to choose one option, but instead will help classify reasons and acknowledge their diverse ontological status.

1. For dessert, Alice may get an apple and Elly some berries. The ultimate ground for many reasons to choose one thing over the other seems to be that "there’s no accounting for taste". Subjective minds develop their thoughts based on rationality, but when you pull back that thread, it happens you end up on some irreducible, purely individual preference. That looks a lot like expressivism. Alice’s reason to eat an apple might then be identified with a deeply rooted desire for apples.

   We also need to acknowledge that expressivism doesn’t have to be the last word here. It seems sensible to assume that these deeply rooted preferences supervene on the physico-mental features associated with subjective minds. Such reasons may then be simply grounded in natural properties.

2. There is a more overt kind of normative naturalism that’s directly predicated on the biology of subjective beings. Being hungry is a reason to seek some sustenance. That simple statement is plausibly an ultimate reason, irreducible to simpler ones, and is clearly grounded in the biological or evolutionary features of our bodies. Ghrelin and other hormones elicit physical reactions and specific mental states that drive food-seeking behavior. So, that normative principle (or even a more universal one like "all animals want to eat enough to stay alive") can theoretically be shown as reflective of a causal chain that’s firmly embedded in the natural world. Reductive normative naturalism is thus the relevant thesis for large swathes of our reasons.

3. We’ve already evoked the case of artifical rules. When we play chess, follow etiquette or replicate a dance routine, our reasons are grounded in a number of axioms forming an internalized rulebook. The corresponding value — what makes a legal chess move, a correct social behavior, or a good dance step — does not stem directly from natural properties but from a formal, fictional normative system that becomes an abstract add-on to our rationality posits.

All of those approaches have something in common: they can accommodate some form of normative relativism. Whatever we may think should be an absolute moral or aesthetic principle, true in all scenarios and in all possible worlds, seems bound to be jeopardized in all these models.

Expressivism is about the attitudes of speakers, which undermines the mere concept of principle. If only one agent has a genuine pro-attitude towards the idea of murdering someone (in itself, independent from any motive), and if moral cognitivism was wrong, then we couldn’t erect any commandment-style principle putting murder firmly in the "bad column". Humanity has learnt the hard way that any depraved moral or aesthetic attitude we could conjure has likely been hailed as good by someone, somewhere...

Formalism fares no better, as it suggests that murder is bad by virtue of a formal moral code we’ve internalized. But there’s really nothing to respond to individuals, or societies, that hold opposite moral tenets. To say that some civilizations were wrong to encourage human sacrifice would then be nothing more than a formal "move" from players in the "game" of Western morality. (I’m asking for some leniency here, being conscious I may present a caricature of some views; I’m lacking space and time to do them justice. That ever-present issue somehow seems more fraught in the context of moral discourse).

As to naturalism, it seems more likely to accommodate absolute principles at first sight, since natural properties are grounded into an evolutionary process that is independent from individualities and prior to social organization. Yet, we should not forget about the infinite riches of (perhaps) our own universe, and of (surely) the modal realm of possible worlds. Whatever natural property \(N\) we reduce "good" to, there might be a planet and a species somewhere for which murdering their offspring is \(N\). Is it murder then possibly good?

Meta-epistemology as a remedy to relativism

Relativism is certainly, to me, hard to countenance. But even partial acceptance of normative Platonism can seem equally uncomfortable at first. To better understand my view of it, I suggest a detour via epistemology. It is most definitely normative, yet distinct from, and prior to, deontic and evaluative studies.

Epistemology is traditionally presented as the study of knowledge (or other forms of cognitive success, but to simplify here we’ll only talk of knowledge). The now-canonical definition of knowledge is as justified true belief (JTB), Epistemology thus finds itself at the nexus of normativity (justification), semantics (truth) and mind (belief). Let’s consider those in turn.

• Belief: to know something is at the very least to believe it. What I know is a distinguished sub-set of my beliefs.

• Truth might be slightly less obvious. Colloquially, someone may know something that is false; the word then serves to emphasize that they really, really, believe it. But in philosophical usage, truth is always a condition precedent to knowledge. So I cannot know that Mitt Romney was President (in the actual world), because that is not true. Aristotle thought that he knew heavy bodies fell faster than lighter bodies; but he didn’t, for that was false.

• Justification, as we’ve seen, is a quality possessed by what we should, or what we’re entitled to believe. It was shown to be a necessary condition for knowledge from Plato’s Meno. Plato insists that not everything we believe is believed knowledgeably, for it may lack aitias logismos (reasoned explanation). Dharmottara in medieval India, or Peter of Mantua in Renaissance Italy, would offer examples of the distinction.

  If I walk in the desert and see a vast blue oasis, I may form the belief that there’s water there. Once arrived, I may realize that it was a mirage, but still find out that behind some rocks there was a small stream. My belief that there would be water turned out to be true, but clearly wasn’t justified. Technically I didn’t know there was water

  On another topic, some people firmly believe, and others firmly disbelieve, there is another technically advanced species in our universe. One of of those groups believes a truth, yet it is doubtful that its members have proper reasons for their belief...

Of course justification, and thus the relevant laws of rationality, are at the center of all epistemological questions. Justification as it was traditionally understood, or even the mere acceptance of its historical prominence, have been hotly debated. But I would argue that critics are arguing to reform justification, making it more sophisticated, rather than abolishing it as a concept. If everyone agrees that there is more to knowledge than just true belief, then justification is just what fills the gap.

The goal here is not to develop an epistemology, though, it is to flesh out a meta-epistemology, which hopefully might then be a guiding light towards a general view of meta-normativity. I would sum up the JTB view as such:

1. Epistemic principles + Rational inference \(\rightarrow\) Justification

2. Justification + Belief \(\rightarrow\) Justified belief

3. Justified belief + Truth \(\rightarrow\) Knowledge

Then I would generalize it in this way:

1. Normative principles + Rational inference \(\rightarrow\) Justification

2. Justification + Mental state \(\rightarrow\) Rationality

3. Rationality + Absolue Value \(\rightarrow\) Virtue

The first line is straightforwardly derived from our own definition of justification: reasons to believe, act or feel are generated via rational inference, but their ultimate ground resides in irreducible principles.

The second line is also a re-statement of our previous musings: the laws of rationality may be many, but it is clear that to \(X\) when one is justified to \(X\) is a hallmark of rational behavior.

The last line is more speculative and takes us right into our present topic, but it also does lots of the conceptual work we need by re-framing epistemic considerations in the value space... Truth is an absolute value in my framework, since I subscribe to the correspondence theory: a true proposition reflects a fact that exists in the actual world. Once we analyze language, eliminate the pragmatics and ambiguous terms, nothing can be true to me and false to somebody else.

However, it may well be that justification is relative. Even if the principles of rational inference are absolute (a point that has not been proved here), the ultimate normative principles may be relative, should we adopt a non-Platonist perspective. There are countless positions among epistemologists themselves that are in stark contrast to each other about what science can know, for instance. Put ten of them in a room, and you may find that knowledge is relative, for they may interpret the same true belief as either justified or unjustified according to their models, all perfectly rational perhaps, but differing in the deeply held principles they’re grounded upon.

The point here that saves us from "pessimistic" relativism, is that absolute truth can serve as the ultimate judge. Belief aims at truth, and justification should be an efficient mechanism that leads us to truth. A naive way to sort out "good" from "bad" epistemologies would be to take them on a test run. An epistemology that is based exclusively on horoscopes and hits the bull’s-eye of truth only half the time might reasonably be called deficient. Some epistemologies may correctly justify lots of true belief, but also incorrectly justify false beliefs. Conversely, other may avoid false beliefs, but be too skimpy with their justification, leading us to become radical skeptics.

Whatever methodology we choose, this should start to explain the vocabulary that I’ll be using. Knowledge is a virtue because it’s a way not only of believing the truth, but of believing well.

Virtue normativity

This helps re-frame normativity as a search for virtue, where that elusive notion is defined as doing something well: doing the right thing, for the right reasons.

That approach can help illuminate the reducibly normative "values" (from formalism or naturalism) as well as the irreducibly normative ones. In a game of cards, the only right thing, the ultimate value, is a win. But one can win by playing well or not. If a player wins by only making moves that are justified by the player’s aim to win (taking into account the laws of rationality and the rules of the game), it is sensible to say that she played well — that’s akin to what we call virtue. But a player who wins by sheer luck, after making ill-thought moves, did not play well. A player who misunderstood the rules and lost might still be called rational, provided they would have won, had the rules been such as she imagined.

We can spin the same thread for those normative principles that reduce to natural facts. Reducible normative principles can then be re-branded as the rules in a "game of life", and "acting well" as the behavioral strategy most conducive to natural success, the telos that emerges from the complexities of living systems (notwithstanding their purely causal roots).

But this parallel with epistemology was originally introduced to help cast out the specter of radical relativism. What if there were absolute values, analogues of truth; what if virtuous deeds and feelings were those that not only were worthy of those values, but also rationally justified ways to those ends?

Let’s recapitulate the concepts from epistemology we’ve evoked here as well as in previous sections:

1. Some propositions are true.

2. Knowledge is an epistemically justified belief in a true proposition.

3. A scenario is a set of beliefs, for instance those held by a given agent.

4. A priori propositions are true in every scenario. Every ideal epistemic agent believes them.

5. Conceivable propositions are true in at least some scenarios.

6. A posteriori propositions are false in at least some scenarios.

7. Inconceivable propositions appear false in every scenario. No ideal epistemic agent believes them.

Let’s define the analogues for the deontic value we call morality:

1. Some actions are right.

2. A righteous act is a morally justified right action.

3. A storyline is a set of actions, for instance those taken by a given agent.

4. Obligatory actions are righteous in every storyline. Every ideal moral agent takes them.

5. Permissible actions are righteous in at least some storylines.

6. Omissible is what we may call actions that, in at least some storylines, it is righteous to abstain from taking.

7. Forbidden actions aren’t righteous in any storyline. No ideal moral agent takes them.

And let’s proceed with the evaluative value of beauty:

1. Some things are beautiful.

2. Appreciation is an aesthetically justified feeling towards a beautiful thing.

3. Taste is a set of feelings, for instance those felt by a given agent.

4. Sublime things ought to be appreciated, regardless of taste. Every ideal aesthetic agent feels that way.

5. Admirable things can be appreciated, according to some tastes.

6. Imperfect things can be unappreciated, according to some tastes.

7. Awful things shouldn’t be appreciated, regardless of taste. No ideal aesthetic agent would.

The ontology of absolute values

These constructions invite some key questions. Are some things really absolutely right or absolutely beautiful, and if so, what would clear the bar? Are those just examples or could there be more values? Lastly, what ontological categories do these primitive concepts fit in?

To the first question, I would answer in the affirmative. After all, the value framework has been introduced precisely to keep relativism at bay. At the same time, it is important to not mistake this proposal for some neo-religious normative frenzy. The idea is not codify human action under a moral straitjacket, or to legislate on aesthetics with absolute edicts. Either attitude would be unwarranted, notwithstanding their continuing popularity.

Let’s go back, once more, to the question of knowledge. Some things are true and "always knowable": they are a priori truths (paradigmatically: mathematical facts or analytical ones). Nobody does in fact know them all, but in principle, an ideal cognizer doesn’t need epistemic justification (perception, conversations,...) to know them. Then, there are things that are in fact true, but a posteriori. To unearth as many of those as possible, cognizers develop their own epistemic principles, from which they deem a number of beliefs to be justified, building up a scenario of what the world is like.

Similarly, a number of actions are absolutely right. Moral agents are behaving after a set of moral principles to justify the acts they bring about. Some actions are "automatically" right and that judgment doesn’t require moral justification: ideal moral agents would always consider them as obligatory.

Now, couched in this colloquial language of morality, the last few sentences might seem slightly unhinged. We’re using that language for convenience, because it’s non-technical and easy to grasp, but one shouldn’t take it too literally and see "right" or morality as something that’s prone to open Heaven’s gate. Moral agents are not theologians or philosophers, they’re subjective beings with a set of attitudes they wouldn’t necessarily call a moral system.

A deranged person who takes pleasure in hurting others, even if he calls himself "evil" would still be aiming for right, because "right" in this context is just a way to label what individuals consider a measure of practical success. However, the absolutist point we’re making is that hurting others is in fact not right in itself: sadistic principles miss the mark as much as their epistemic principles fail flat-earthers in their quest for truth.

However, the goal of moral platonism is not to evaluate all actions — far from it! To understand, we need a brief excursion into act individuation: we’ve been talking of actions in a freewheeling way, but it’s far from obvious how granular we can be. Alvin Goldman argues for a fine-grained account of coincident action. The act of pulling a trigger and the act of killing are different in that account. We could argue that context and generality are also arbitrary when discussing acts: the act of punching someone, the act of punching an attacker, the act of punching a thirteen-year-old attacker may be coincident but might call for different moral evaluations. In short, I don’t believe that moral realism can help to adjudicate concrete moral dilemmas.

To reprise a famous one: Sartre evokes the case of a student, brother of a slained résistant, who wants to join the Free French forces and fight the Germans, but also knows he ought to stay near his mother, being her last comfort in tragic times. Moral properties apply to acts exactly as they may be described, in isolation. The act of fighting a violent enemy who murders innocents indiscriminately might be an absolutely good thing, and comforting one’s mother might be another. Our modest and understated moral Platonism doesn’t provide a solution to the dilemma, and there probably isn’t one. But it acknowledges that in some situations, moral teleology is real — the famed "moral compass". Someone who’s adopted as part of their "code" the tenet that resisting an evil force and comforting one’s parent are both good, then uses rational means to weigh them, and come up with a justification to take one action or another, is making a decision we can call righteous. But someone who’s adopted as a code that none of those things matter and acts accordingly is misguided, regardless of individuality or circumstances.

One last point would be to evoke the distinction between merely good acts and obligatory ones. That is a similar notion to what a priori is in epistemology: an action that is self-evidently inevitable to a moral agent. Such an action would need to be extremely generic, because any real-life act, as "obviously good" as most people would consider it, could still be left unevaluated in storylines (moral scenarios) without contradiction. Obligatory actions should only reflect a sort of "moral analyticity". Those could embody meta-normative ideas such as "to acquire knowledge is obligatory". Coming to know something is then not only good, which is consistent with a normative view of epistemic matters, but good in every "storyline", every moral scenario adopted by agents.

That may seem to run counter to common sense: actual agents could deem it preferrable in many situations to not know some specific thing, or to never have known it. That is why action individuation is key. To know something, in isolation, is better than to have false beliefs, or lucky true beliefs, or no belief at all. But that is only the case in isolation. To know what Jane wrote in her diary is likely bad. The action of acquiring knowledge of Jane’s diary (bad) is coincident with the action of acquiring knowledge in general (good). Reasons to not do the former would likely win out in this simple case; it is left to agents to weigh the scales.

Some moral theory could also welcome more concrete obligatory actions. Hedonism may have it that actions leading to pleasure are obligatory — that is, that they are analytically good in every storyline, and that conscious disagreement by agents is just down to semantic obfuscation (so when someone consciously chooses "pain", it is just a label for what is actually pleasure to them, since they picked it, out of all things). Again, this needs to be understood in isolation, all else being equal. Hedonism doesn’t entail that the pleasure of agent supersedes the pain of others.

All these precautions apply in the same way to meta-aesthetics. Realism shouldn’t be interpreted as establishing an overarching classicist codification. Some "things" are absolutely beautiful, but again that is meant in isolation, and after going through the grinder of metaphysical individuation. Is "Guernica" beautiful in itself as an ordinary object? Is it beautiful given the historical context that inspired it? Is it beautiful given its place in the evolution of visual arts? The two latter cases would entail that the "beauty-bearing" things ought to be complex abstracta that encompass not only ordinary objects or types but their relations with a much broader sets of events. The attribution of objective aesthetic properties is not just about dividing all things into a "pretty" and an "ugly" columns.

I do strongly believe in genre eclectism. To a scholar of hip hop music, deeply rooted in the conventions of that genre and sensitive to its specific pleasure, a Mozart sonata might not seem worthy of appreciation. The reverse scenario is also easy to imagine. Aesthetic agents have tastes, sometimes passive, sometimes carefully constructed based on explicit critical tenets. If, as I believe, all genres of music can give rise to beautiful things (melodies, records,...), then a more open and eclectic taste system can be objectively better since it leads to more success in the search of beauty, just like good epistemic principles correctly aim for truth. But an excessively liberal taste can also overshoot and become unsuccessful by being too accepting.

Many questions remain. What type of things exactly are beauty-bearing? Can relative degrees of appreciation also be objective (e.g. "Penny Lane" being better than Schubert’s "Erlkönig")? Should there be negative degrees of appreciation (anything that’s not true is false, but everything that’s not beautiful is not necessarily ugly)? These open fields call for a fuller aesthetic theory that cannot be developed in these pages beyond this brief sketch.

Lastly, sublime things, in the same vein as obligatory actions, represent a sort of aesthetic analyticity, that can distinguish some meta-normative aspects, or very general principles. For the sake of argument, "symmetry is beautiful" might be one, which doesn’t at all entail that symmetrical things are all beautiful, or that they cannot be ugly. But like pleasure to the hedonist, it could be a principal component that shapes the distribution of a complex, multivariate function.

Can there be other irreducible values beyond the true, right or beautiful? This too warrants a much deeper study. This classical triad has stood the test of time, and I tend to believe there is a good case to be made for other values to be derivatives thereof; I will hold to that view in the present work.

This leaves us to discuss the place that value-statements occupy in our ontology. It’s important to distinguish value-statements from normative facts. The distinction is the same as that between truths and epistemic facts. The former are objective features of the actual world (a truth is equivalent to the mere existence of a fact), while the latter are about what cognizers know (and thus are grounded in justification, rationality, etc.).

So, value-statements about what’s right or beautiful are conceptually equivalent to truths. What is true depends on which facts exist, which in turn is guided by a hypothetical "ultimate" theory of the natural (physico-mental) world \(\mathfrak{ToPM}\). Out of all the possible well-formed facts allowed by the ontological formalism, a sub-set exists, but that space is not an amorphous blob: it is shaped by the aforementioned theory. Similarly, out of the vast expanse of possible actions and things (which, via individuation and context, is infinitely more vast than a simple set of discrete events or ordinary objects), a thin constellation is right, or beautiful.

One major difference however, is that value-statements cannot be considered a mere copula like truths. If they were, the consequences would be to establish a whole parallel, axiological semantics with its whole Tarskian T-schema, and so on. Perhaps that warrants further exploration, but that isn’t a hypothesis I would lightly deem viable. Instead, values will simply be distinguished properties. Just like the actual world is not chaotic because it is nomically possible, value-statements need to follow a (hitherto-unknown) theory of value \(\mathfrak{ToV}\) that constrains the shape that this outer garment takes, once draped over the natural world. In metaphysically possible worlds, "anything goes", but nomically possible worlds need to fit the laws of physics, of mentality, and of value (in that order, due to the corresponding inter-level dependence: mental intentionality supervenes on the physical-phenomenal interaction, and value in turn supervenes on the physico-mental facts). The model that truly governs \(\aleph\) then ought to be a theory of physics, mind and value \(\mathfrak{ToPMV}\).