Epistemic vs Metaphysical modalities

With the triangle in jeopardy, and the Kantian link between conceivability and possibility broken, one could choose to preserve either one of the other sides: only Frege or only Carnap. Kripke is certainly not neutral in that debate: his instincts about language and about meaning push him towards one option (to keep the metaphysical/Carnap side rather than the cognitive/Frege side).

One could also be of the view that those sides offer independent, valuable contributions to our understanding of meaning. They may be incompatible in a traditional, "one-dimensional" view of semantics, but some, most notably Chalmers, Stalnaker and Kaplan, have developed a two-dimensional semantics where a "Carnap concept" and a "Frege concept" of meaning can cohabit, and where what we call meaning is actually the pair of them. Before we delve further into that option, we will examine the parallelism and discrepancies between these two sides of the triangles, these two possible routes for a 1D semantics.

The parallelism is already evident from some of the notations we’ve introduced. We can now fully cash this out by defining a new concept of epistemic modality as follows:

• \(\square \phi\) : \(\phi\) is metaphysically necessary.

• \(\Diamond \phi\) : \(\phi\) is metaphysically possible.

• \(\blacksquare \phi\) : \(\phi\) is epistemically necessary (\(\hat{\leftrightarrow}\) a priori).

• \(\blacklozenge \phi\) : \(\phi\) is epistemically possible (\(\hat{\leftrightarrow}\) conceivable).

Indicative vs Subjunctive conditional

It is crucial to get a good grasp of this distinction, lest the whole framework, which is about orthogonal dimensions of meaning, falls apart. We can get a clearer view by moving from this theoretical definition of modalities to the way we express them in natural language. The semantic distinction at play is reflected in our daily expressions through the use of indicative (or material) conditional, or of subjunctive (or counterfactual) conditional.

Consider this pair of sentences:

• It is not raining and it may actually be raining (for all I know).

• It is not raining and it could have been raining (possibly).

The first sentence, an indicative, is wrong: once I have ascertained that it’s not raining, it may not be actually raining; that would be a contradiction. The second sentence, which involves a subjunctive mode, is obviously right: in normal conditions on Earth, it could be raining, or it could be dry; the fact that it is actually not raining is irrelevant.

The connection with modality is quite clear. Provided that it is not raining (which I’ve observed), it is inconceivable that it may be raining: that is a statement about epistemic possibility, where the epistemic basis canonically includes an empirical observation of the weather. Separately, it is metaphysically possible that it could be raining; whether it rains or not is contingent, and that doesn’t hinge on an epistemic basis at all, just on external reality.

This further example is courtesy of David Lewis:

• JFK was assassinated; if Oswald did not kill JFK, then someone else did.

• JFK was assassinated; if Oswald had not killed JFK, then someone else would have.

The indicative (b1) is obviously right. With an epistemic basis that includes knowledge of JFK’s assassination, it is epistemically necessary that someone killed him. So if Oswald didn’t do it, then someone else must be guilty. Counterfactually however, as per (b2), it is not impossible that JFK would have lived, of course: there is no metaphysical fatality to his fate.

I’ve invoked the concept of an epistemic basis here. In the case of a human, we could think of it as a knowledge base. But the concept of a priori that we shall deploy is more general, as explained above, and assumes an ideally rational cognizer who also ignores all empirical data.

• Superman has revealed his secret identity to Lois. For all that Lois knows, Jimmy Olsen may not be Superman (on that basis, it is epistemically necessary that Clark Kent is Superman)

• For all that Lois knows a priori, Jimmy Olsen may well be Superman (Lois may have been told by Clark who he really is, but that’s a posteriori knowledge).

• Jimmy Olsen could not have been Superman (because Clark and Superman rigidly designate the same being, and Jimmy doesn’t).

All of those are true (in the relevant fictions): (c1) is an indicative with a partly empirical knowledge base, (c1’) is another indicative, whose epistemic modality is aligned with apriority, and (c2) is a subjunctive statement.

Purely descriptive vs purely causal reference

Talking about reference above, I’ve outlined two models. Under the causal model, expressions point to a specific ordinary object. Expressions defined in that way (archetypally, proper names) refer to the same ordinary object (under a certain relative identity relation) in each possible world. Kripke calls those expressions metaphysically rigid. Going back to our earlier example: it is (metaphysically) necessary that Aurore Dupin and George Sand refer to the same human being: \[\Box ( \mathbf{D}\texttt{"AD"}[a] \wedge \mathbf{D}\texttt{"GS"}[a])\] Yet it is an a posteriori necessity: it is conceivable that Aurore Dupin and George Sand may actually refer to separate humans (in some scenario, for all we know).

As it turns out, we can show that expressions that fully espouse the descriptive model of reference are, in a nice parallelism, epistemically rigid. What could be a better archetype for a descriptively referring expression but a description? That’s generally right, but with a caveat. Just like names can be in certain contexts disguised descriptions, descriptions can be disguised names. The paradigmatic example is the Holy Roman Empire, which was neither holy, nor Roman, nor an Empire. More subtly even, the way a descriptive expression refers is dependent on the reading of the sentence it appears in. Take these sentences:

• The French president was born in Amiens because Jean-Michel et Françoise Macron lived there in 1977.

• The French president was born in Amiens because Emmanuel Macron won the 2022 election.

  The first sentence works with a de re reading: it is "about the thing". Indeed it is true of the actual current president, but would be wrong of nearly anyone else. If we replaced the description with "Emmanuel Macron", there would be a difference in terms of pragmatics but not of semantics. De re expressions can then be seen as disguised names. The second sentence seems to be only consistent with a de dicto reading: it is "about what is said". Indeed, we have a president who’s from Amiens because Macron, an Amiens native, happens to have won the election. Unlike the first sentence, replacing the description with the reference’s name doesn’t work at all: to say that Macron was born in Amiens (in 1977) in virtue of his winning the election in 2022 is patently absurd.

  For an even starker example, I propose this sentence: "Even if Macron had never been president, the president of France would still be Jean-Michel Macron’s son". On a de re reading, it is true: Macron would still be his father’s son if he had never won the election (provided that lineage or DNA are involved in the way we define trans-world identity for human beings). On a de dicto reading it is obviously false: if Macron had never been president, someone else would have, and that man is most likely not Macron’s sibling.

  Coming back to the topic at hand: what we are saying is that descriptive reference goes and points to an ordinary object, but instead of it being fixed (and endowed with trans-world identity), it is simply defined by way of the description operator. "The president of France" designates the ordinary human being who actually exemplifies that property. Formally: \[\mathbf{D}\texttt{"The PoF"}(\imath_{w_\mathfrak{a}} [x] (P[x]))\] The \(w_\mathfrak{a}\) index designates the actual world (’our’ world), and shows that the description operator must be fine-grained instead of trans-world: the expression designates someone who fulfills a description actually — the actual president, not all the people who could possibly have been president.

  We can see that this reference relation is not metaphysically rigid. Let’s call \(w_\mathfrak{AJ}\) a possible world where Alain Juppé won the election. In that world, Emmanuel Macron (the actual president from my standpoint) does not exemplify the property of being president: \[w_\mathfrak{AJ} \Vdash \neg (\mathbf{D}\texttt{"The PoF"}(\imath_{w_\mathfrak{a}} [x] (P[x])))\]

  However, we know that the designation relation holds in that world in relation to the president from that world: \[w_\mathfrak{AJ} \Vdash \mathbf{D}\texttt{"The PoF"}(\imath_{w_\mathfrak{AJ}} [x] (P[x]))\] We need to shift the scope of the description operator. That shift, as opposed to the previous one, is reflective of epistemic modality: I’m not trying to think whether something would be true of that possible world, as seen from my perspective. Instead I’m evaluating whether it would be true in that world, if I was part of it and saw it as actual.

  It sounds like a truism: all we’re saying is that when someone thinks or talk about "the president", they refer to someone who’s actually the president from their standpoint. And indeed we can say that fact is a priori. Remember that the ideally rational cognizer is deemed to know all facts that are not empirical as part of their epistemic basis. A causal-historical designation that originates in an initial baptism is definitely empirical. The knowledge certain properties are exemplified by certain beings (e.g. that Macron is the president) is too. But the understanding that an expression like "the president of France" is related to the specific property \(P\) that we seek to capture with those terms is purely analytic: it is part of an understanding of language that our ideal cognizer has (notwithstanding the brute reality that language needs to be learnt empirically, semantics needs to work within a fixed epistemic basis that must include some conventionally delineated analytic facts, lest we face a predicament where everything is empirical or nothing is).

Conceivable scenarios vs possible worlds

Where there is talk of modality, possible worlds can’t be too far. We’re familiar with metaphysically possible worlds: what is metaphysically possible is deemed to obtain in at least one possible world. Can we paint epistemic modality with a similar brush? It certainly seems feasible to define conceivable situations, and individuate them when they are not a priori the same. Following Chalmers, we will call them scenarios to suggest a contrast from possible worlds. Pursuing the analogy between these forms of modality, we can assume that something is epistemically possible (conceivable) when it is true in at least some scenarios, and epistemically necessary (a priori) when it is true in every scenario.

What exactly are those scenarios? So far, they appear to describe situations, corresponding to ways that things may be, which could turn out to be different from the way things actually are. In that sense they seem very close to possible worlds as a concept. One added twist is that scenarios incorporate an indexical basis. That is because epistemic modality is sensitive to indexicality, unlike metaphysical modality. I am French (that’s actually the case), but I could have been Italian (counterfactually), in a possible world. That is made true in worlds where Jonathan is Italian (since there is no counterfactual scenario where I would not be Jonathan). However we can devise a scenario centred on somebody else, say Carlo Rovelli. In that scenario the statement “I am Italian” is made true when the state of affairs that Carlo Rovelli is Italian obtains. For instance, it obtains in our world. This dependence on indexicality has led to consider that scenarios may simply be centered worlds: pairs of possible worlds and indexical bases.

The result of that approach would be to advocate for modal monism: metaphysical and epistemic modality would then really explore the same modal domain, albeit with perspectives different enough that a 2D approach is not merely redundant. Antonella Mallozzi’s remarkable papers have shown in an enlightening way how modal monism is incompatible with the combination of

1. accepting Kripkean views about names.

2. including descriptive content in our understanding of meaning.

We must then accept modal dualism, and the fact that the class of scenarios is ontologically distinct from that of possible worlds, regardless of indexicality.

Let’s try to understand this practically by coming back to our earlier example of Aurore Dupin and her nom de plume, George Sand. As we’ve seen, these two names rigidly designate the same human being, and their referents are thus rigidly equal. There is no possible world where Aurore Dupin and George Sand are not the same person. Either the corresponding human being exists and both names refer to her, or she doesn’t exist and neither name refers. The extent of what is metaphysically possible is then influenced by the way things are: David Armstrong used the slogan: “the possible swims in the actual”. But, as we’ve pondered before, it is entirely conceivable that George Sand may not be Aurore Dupin. That is because to know they are the same is a posteriori knowledge. There are lots of scenarios that can accommodate that idea. For instance, there is a scenario where Aurore Dupin is the human we associate to that name (Maurice Dupin’s daughter), and where an entirely different human named George Sand writes novels that include “La Mare au Diable”, “François le Champi”, etc. The corresponding situation is not incompatible with the laws of metaphysics, or even of physics for that matter. So why again is it metaphysically impossible? Our names, which are fixed by causal-historical inheritance through baptism, cannot be used in that way: it would contradict metaphysical definitions that we’re operating under. There is definitely a possible world where Aurore Dupin doesn’t write books and someone the locals call “George Sand” wrote all the George Sand’s novels. But the name that possible inhabitants of a possible world utter when talking about a hypothetical writer is of no importance in semantics. From our perspective, that other person is either an identifiable human that also exists in our world, or a new one we ought to give a new name (George Schmand?). So, the conceivable scenario where George Sand is not Aurore Dupin is different from the possible world I’ve just described in terms of its semantic assumptions, even tough it looks superficially similar.

What connects them? Well, it’s important to realize that epistemic modality flips Armstrong’s saying: it considers that “the actual swims in the possible”. So, where possible worlds are counterfactual situations from the perspective of our actual world, conceivable scenarios are situations considered as actual. By assuming that Aurore and George are different persons, we’re already adopting (unknowingly) the semantics of a different world, where Aurore Dupin and George Schmand live. We can then evaluate possibilities from the standpoint of that world, considered as actual. From there, ours is only a possible, counterfactual world among many. From there, Aurore Dupin and George Schmand are as different in our world than anywhere else since that’s a metaphysically rigid fact. Denizens of that world, if they could observe ours, would simply notice that George Schmand doesn’t seem to exist here, and that Aurore Dupin wrote all her books, but that wouldn’t jeopardise any metaphysically rigid proposition.

The difficulty with that approach is that the linkage of scenarios with worlds is far from clean. It was a massive shortcut to assume that I could present the proposition that Dupin and Sand are different persons, and that, if considered actual, it would correspond to a world where Schmand wrote Sand’s books. It could match a world instead where Dupin wrote all the books, but George Sand was the nom de plume of Mary Ann Evans instead, so Aurore could not use it. That’s where the Fregean mechanics come into play: saying that Dupin and Sand are different is not enough, we’re actually talking about the senses of these names. For the scenarios to be correctly identified and singled out, they must incorporate the corresponding sense, lest they remain hopelessly ambiguous. At a very high level, we can say that scenarios include the descriptive content we associate with names. The scenario relative to the Schmand-world will then correspond to a sense of “George Sand” that’s basically limited to “the author of …”, without much biographical detail. That said, it’s also important to remember that senses cannot be pure descriptions, even complex ones, as that would re-open the Pandora’s box of Kripke’s arguments. We will see shortly that we also don’t have to accept them as sui generis entities like Frege did.

Admittedly, lots of what precedes may seem counter-intuitive, but that is largely a result of confusing the names we use to designate things with those same things (in other words: the semantics/metaphysics interface is not properly represented in most of our discourse, with the possible exception of mathematics). Perhaps an example from a “hard science” will make that clearer. Helium as an element was discovered in 1868. Alpha decay, a form of radioactive decay involving the emission of alpha particles was discovered in 1899. In 1907, Rutherford proved that alpha particles were Helium nuclei. As a matter of fact, alpha particles always were Helium nuclei. One can simplistically define the sense of Helium as “the element with atomic number 2” and of alpha particles as “what’s emitted in the process of radioactive decay that emits nucleons”. We can conceive of many scenarios where alpha particles are not Helium, for instance a world where alpha decay involves the emission of Carbon nuclei. But that cannot correspond to a possible world from my standpoint. A Helium nucleus and a Carbon nucleus are clearly not the same nucleus, the same particle, or whatever other relative identity relation we want to use. And when Rutherford originally named alpha particles, he didn’t mean it in a descriptive way. He observed a phenomenon, (metaphorically) pointed to the particles, and baptised them. To allow plausibly causally-baptised names to not be trans-world equals invites mayhem, especially when you consider that descriptions can be fulfilled by literally anything. For all I know a priori, alpha particles may be tiny dragons or bullets from Liliputian rifles.

Taking another step back, this should be an invitation to reflect on the nature and objectivity of semantics. All of these examples are couched in natural language (English here), and about beings or things that you, reader, know about. But we also know that semantics is not about humans and the way they speak or think. It is part of a theory that would be valid even if no one ever uttered a word. Then if we got rid of words and expressions and languages, wouldn’t we just end up with a simpler formalism? Since the metalanguage we write that formalism in would involve fixed definitions (an ordinary object is represented by one symbol), we would end up with a stable, one-dimensional model of the world (the counterfactual/Kripkean one). That is entirely correct, but the other dimension, the Fregean one, is precisely there to emulate the cognitive value of meaning as we observe it in human minds and that is reflected in behaviour. By entertaining scenarios that would not be metaphysically possible worlds from our standpoint, and observing what happens to the object of reference when we vary those, we can formally represent another dimension of content, called narrow content (more on this shortly). The mathematical apparatus around that stays valid regardless of whether humans exist or not.

The beauty of 2D semantics as I understand it at least, is that not only can Fregean content contribute meaningfully to meaning without jeopardizing Kripkean intuitions. It also provides a way out of the conundrum of senses’ nature as sui generis abstract objects. When I introduced scenarios in more detail, I used our knowledge of Fregean senses, abundantly discussed in these pages, to give a feel of how scenarios were individuates. But in the formalism, I would propose, scenarios would come first. They are not possible worlds, but I have high hopes that a hitherto unformalized part of the theory will allow to systematically construct scenarios from possible worlds via the adjunction of an indexical basis and some semantic re-shuffling. Once scenarios are set as a primitive data of semantics, senses can be defined in an analogous way to intensions, as functions from scenarios to reference. For instance the assertion that "George Sand is Aurore Dupin", in a scenario where the empirical knowledge of that fact has been acquired, refers to the truth-value "True" in all possible worlds. But in our world, that expression can refer to "True" or "False" depending on the scenario. That distribution is precisely what makes up the sense of that expression, its epistemic modal value. This also allows to naturally avoid the simplistic idea that we could associate one linguistic expression with one linguistic sense. We’ve seen that Frege tried to defend that correspondence for a moment but quickly abandoned the idea, starting the slide toward the cluster vision of Wittgenstein, later formalized by Searle. Indeed we have to recognize that our expressions are often radically opaque, which is not a comforting thought. This is starkly the case when varying scenarios. Consider this pair of sentences:

• If Caesar had been in the Korean War in 1951, he would have used atomic weapons.

• If Caesar had been in the Korean War in 1951, he would have used catapults and ballistae.

The scenarios that would plausibly make that true are clear, with some basic comprehension. In the former, Caesar was born in 1880 and ends up leading the U.S. forces instead of MacArthur: he’s got the same strategic inclinations and personality as ’our’ Caesar, but modern military knowledge. In the latter, the ancient Caesar is somehow transported to the Korean battlefield with his legions, and has to fight for his life using what he knows. So the ’if’ clause as an expression is very far from delineating a clear sub-set of "topologically close" scenarios.

Narrow vs wide mental content

Lastly, let’s not forget that even if semantics in a formalized ontology such as ours is mind-independent, i.e. it’s not grounded in mental activity and should be purely expressible in terms of abstracta, there is still a strong link with the philosophy of mind: the mental content or intentional content from the intentional component of mental states is bound to be connected to meaning. To make that link evident, it may be time to invoke some famous thought experiments.

Putnam’s Twin-Earth story illustrates the externalist view that, as he put it, "meanings just ain’t in the head". Twin-Earth is another planet (it can be in our world, not even a different possible world), where everything is exactly identical, except that everywhere there is water (H2O) on our Earth, one finds instead a chemical component called XYZ. Apart from its chemical composition, XYZ looks, tastes and feels the same.

In the thought experiment, we consider humans in the equivalent of the year 1770 on each planet: they don’t know about the chemical composition of most elements — and their sensory interactions with those prevalent liquids are exactly the same. Humans from Twin-Earth also speak English, and they happen to call this XYZ liquid "water" as well. Now since hydrogen had been discovered recently, consider an Earth human called Bill, and his Twin-Earth counterpart Twin-Bill who think "Water is made, in parts, of hydrogen". For Bill from Earth, the term "water" is associated through the causal reference model with H2O (even if Bill doesn’t know it’s the case). The expression is made true when H2O is made of hydrogen, which is the case. Meanwhile, to Twin-Bill, "water" doesn’t mean H2O but XYZ, through the causal chain of reference.The expression is made true when XYZ contains hydrogen, which it doesn’t. The truth-conditions between the two thoughts are different, but even their referents are different truth-values, so the associated mental content can’t be the same.

Yet, Bill and Twin-Bill are supposed to be physically identical beings: they have lived the same life on planets that, to their eyes, were exactly identical, and have undergone the same sensory experience, since we’ve hypothesized that H2O and XYZ behave the exact same way from the perspective of pre-modern-chemistry humans. Therefore, Bill and Twin-Bill’s brains must be identical, as should be their phenomenal state of consciousness. And yet, their mental contents are different. This is purported to show that when we consider meaning from a Kripkean perspective, considering only metaphysical modality, content is wide: it does not depend on the internal state of the person having the thought; instead it only depends on the external world.

Now, mental externalism is in no way a newfangled idea that emerged from Kripke’s semantics; Hegel’s "objective spirit" or some aspects of Sartre and Heidegger’s phenomenologies may well reach the same conclusion. Be that as it may, it still flies in the face of both a well-established Cartesian tradition as well as modern conclusions from scientific psychology. For Descartes, a subject can know the content of their own thoughts "from the armchair", that is, through rational introspection. Of course, that couldn’t be the case if content was at least partially grounded in the external world. Modern psychology has also got us used to consider that all behavior, which is prompted by mental causation, originates in the internal psychic state of subjects. Proponents of those views would say that Bill and Twin-Bill’s thoughts then must have the same content, and would typically explain by way of Bill and Twin-Bill’s common concept of "water" (H2O and XYZ both) as a transparent, drinkable liquid that fills the seas, lakes and rivers, falls when it rains, etc. In other words, mental internalists, who have it that mental content is completely supervenient on the internal state of subjects, believe in a descriptive form of content, or at least one that can be expressed via some Fregean sense. This is usually called narrow content and is the type of meaning we’ve arrived at by considering epistemic modality. Narrow content theorists have their own favorite thought experiment, which Descartes called "the evil demon", but that has been more recently reframed as "brain-in-the-vat" (or "the Matrix"): a brain in a vat (or whatever more expansive part of our bodies) that would receive the exact same stimuli that the nervous and hormonal systems send to our actual brains would find itself in the same mental content. Free-Jon and Jon-in-a-vat would then be in the same mental state while being in shockingly different physical states. In a reversal of the Twin-Earth story, this invalidates the basic tenets of externalism.

Two-dimensional semantics

With two-dimensional semantics, we consider that there are causal and descriptive components to the reference of individual expressions (i.e. both are part of semantics and not dependent on pragmatic trappings like contexts of utterance, tone or phrasing). We consider that metaphysical and epistemic modality, indicative and subjunctive modes contribute to meaning. Most importantly perhaps, we thereby cut the main Gordian knot of the internalism/externalism debate.

The only way for partisans of each camp to justify their position in the face of reasonable folk intuitions is to take an excessively picky view of what qualifies as content. With 2D semantics, every thought automatically has a wide component (its intension) and a narrow one (its sense), representing metaphysical and epistemic modality respectively. The framework was already introduced in our discussion of scenarios above. Assuming we can devise a consistent way to generate conceivable scenarios from possible worlds via a semantic reshuffling (re-attribution of causal-reference label) and the adjunction of indexical bases (and subsequent collapse of some epistemic dimensions via their analyticity, e.g. making "I am not here" inconceivable), we’ve defined a super-space of (world, scenario) pairs.

The intension ("metaphysical meaning") of an expression is a function from possible worlds (as seen from a specific scenario considered as actual) to reference. The sense ("epistemic meaning") of an expression is a function from conceivable scenarios (in a given world) to reference. Meaning as a whole is the function from (world, scenario) pairs to reference.

To close, let’s consider a worked example. Remember two facts about our world, the author known as George Sand (who wrotes novels such as La mare au diable) was born Aurore Dupin, and the author known as George Eliot was born Mary Ann Evans. We will consider three possible worlds:

• \(w_0\) (our world): George Sand is the nom de plume of Aurore Dupin.

• \(w_1\): Aurore Dupin did not write anything; George Sand is the birth name of a different person, who wrote the books.

• \(w_2\): Mary Ann Evans took "George Sand" as her nom de plume; Aurore Dupin published novels under her birth name.

Now we will define diagonal scenarios: \(s_0\), \(s_1\), \(s_2\). This means that everything in \(s_0\) considered as actual is true at \(w_0\), and so on.

Remember that meaning, in this framework, is function from world/scenario pairs to reference. The complete distribution is of course infinite, but for illustrative purposes we can just consider a three-by-three semantic table, which is why we’ve introduced three worlds and three scenarios. We’ll consider the semantic tables for some representative expressions.

Proper name:

George Sand

George Sand is a proper name, the archetype of a Kripkean expression whose reference is grounded in causal-historical acquaintance. As we have seen before, it is an a posteriori necessity that George Sand denotes the human being Aurore Dupin.

That is because I hold the belief that proper name denotes Aurore Dupin in every counterfactual world where she exists (thus considering \(s_0\) as actual). But that belief comes from empirical experience. If I considered instead \(s_2\) as actual, then I would use the name George Sand to refer to Mary Ann Evans. In every counterfactual world, it would do so. It would just so happen that in \(w_2\) that belief was true.

Definite description:

"The author of La Mare au Diable"

"The author of La Mare au Diable" is a definite description. When used by denizens of a given world who are competent speakers, it is analytical that this will pick out the human who wrote La Mare au Diable in their world. Descriptions are indeed epistemically rigid designators.

Simple sentence (exemplification):

"George Sand wrote La Mare au Diable"

This semantic table is a combination of the previous two. Here’s how to read it:

• At (\(s_0\),\(w_2\)), the sentence is true.
  If we consider as actual the scenario \(s_0\) where the name George Sand denotes the human Aurore Dupin, then the sentence is true in \(w_2\) because it is a fact that the human Aurore Dupin wrote that book in that world.

• At (\(s_2\),\(w_2\)), the sentence is false.
  If we consider as actual the scenario \(s_2\) where the name George Sand denotes the human Mary Ann Evans, then the sentence is false in \(w_2\) because it is a fact that the human Mary Ann Evans did not write that book in that world.

We can see that the expression is neither metaphysically nor epistemically rigid. It is both contingent and a posteriori.

Simple sentence (encoding):

Sherlock Holmes is a detective (\(\mathfrak{sh}D\))

Unlike the previous sentence, this one should be formalized using the encoding copula. In the fictional world of Conan Doyle, the human Sherlock Holmes exemplifies the property of being a detective. But in our world, the abstract object \(\mathfrak{sh}\) encodes that property.

In fact that is the case both in our world but also in every other counterfactual world. That is because encoding is a metaphysically rigid satisfaction relation, which is axiomatic (all the way back in the mathesis formalis ("formal ontology") section).

Besides, it is also a priori that \(\mathfrak{sh}\) encodes being a detective. It might seem like a strange statement. I certainly wasn’t born with the knowledge that "Sherlock Holmes is a detective"; I needed to learn that empirically. The case I’m making here is rather that the statement is, in a certain way, analytic. If the abstract object \(\mathfrak{sh}\) is defined as the encoding cluster of "being a detective", "living on Baker Street" and "being Watson’s partner", then to know that "Sherlock Holmes is a detective" can be seen as similar for competent, idealized cognizers, as to know that "bachelors are unmarried"/

Since the sentence is both necessarily and a priori true, we will call it wholly rigid. This is completely consistent with our ontological axiom which states that abstract objects are essences, entirely characterised by their definition. The human being Aurore Dupin is characterised by her DNA, her parents, or any other characteristic we use for the kind “human being”. Besides that, all bets are off. It is both possible and conceivable that she was a circus artist instead of a writer, or that she called herself Robert Silver instead of George Sand. Sherlock Holmes, however, despite the superficial linguistic similarity, is not a human being. Referring to “him” as a detective actually picks out an abstract object, which could not possible nor conceivably encode any property beyond its essential characteristics.

Crucially, however, abstract objects not only encode properties, they can exemplify! When we consider sentences where abstract objects satisfy exemplification relations, we find that those are not at all super-rigid, and fall back on the same cases we’ve studied before.

We chose this set of worlds:

• \(w_0\) (our world): Sherlock Holmes is the name of a fictional detective imagined by Arthur Conan Doyle.

• \(w_3\): Sherlock Holmes is the name of a fictional baker imagined by Arthur Conan Doyle.

• \(w_4\): Sherlock Holmes is the name of a fictional detective imagined by Agatha Christie.

\(s_0\), \(s_3\), \(s_4\) are the corresponding diagonal scenarios.

Proper name:

Sherlock Holmes

Definite description:

“The most famous fictional detective imagined by Conan Doyle”

Simple sentence (exemplification):

“Sherlock Holmes is the most famous fictional detective imagined by Conan Doyle”