Meaning, truth and reference

Semantics in context

Linguistic items

: singular/general terms (names, indexicals, descriptions, etc.), quantifiers, connectives, sentences

Semantics is the study of meaning: it seeks to connect our linguistic expressions to things in the world.

Matters of human language may seem quite remote from an endeavour like the bestiary of abstract entities we’re building in this section. Why introduce this topic now? As I will try to illuminate, semantics finds itself at the nexus of metaphysics, logic and epistemology. Does this entail that new abstract entities are needed? As always, there are three main options at hand: eliminativists would say there’s no such thing as semantic entities (only the sounds we make or the symbols we write) and that philosophy has nothing to say about meaning; alternately, we might feel able to build up objects relevant to semantics using existing abstract objects, like properties or sets; lastly, one may have it that sui generis semantic entities are required.

To examine these cases, we will need to go on a brief historical tour. Let’s first rewind towards linguistics and its various branches, to clarify what we’ll be dealing with. For instance, even the colloquial use of word is ambiguous with regards to types and tokens, a concept we’ve introduced above. Sentences like "a five-minute speech should have less than 700 words" clearly refer to word-tokens — in this case utterances spoken by a human being. In contrast: "he uses the word ’like’ a lot" refers to a word-type, and not to one specific utterance. "Low-level" branches of linguistics deal with tokens, be they uttered (phonology), written (graphology), or thought (psycholinguistics). Morphology, on the other hand, deals with word-types and their internal structure.

Beyond morphology, syntax studies the structure of well-formed clauses and sentences. The basic units in syntactic analysis, called constituents, may be composed of one or several word-types: "Paris" and "New York City" play the exact same syntactic role despite the difference in word count.

Natural language formalized

The joint development, over centuries, of philosophical logic and of mathematical notations gave form to ideas such as Leibniz’s characteristica universalis: an ideal language representing the alphabet of human thought, to be used not only for international communication but also to express mathematical, scientific and metaphysical concepts. The idea was revived in a rigorous way by Gottlob Frege with his Begriffsschrift ("concept-script"), which was an inspiration to Russell, Wittgenstein, Church and Turing. That throughline towards informatics and computer science made the notion of formal language almost as natural in our modern world as that of speech or script.

Whilst phonology and morphology are integral to the study of natural languages, a formal language can be solely defined by its syntax, together with a set of symbols. A syntax can be defined directly as the set of well-formed phrases, or via a grammar (a set of formation rules). But in this section, we’ll be looking at the semantics of natural languages or of a hypothetical mental language — why go into a discussion of formal syntaxes?

First, let’s highlight the mundane fact that in daily use, our languages are chock-full of ambiguities, be they lexical ("I am renting this flat", "British left waffles on Falklands"), or grammatical ("the chicken is ready to eat", "they are flying planes"). The atoms of natural language semantics are neither speech acts nor sentence-types but uniquely parseable syntactic forms. Semantics is thus working over a controlled form of natural language whose syntax can be formalized. The meaning of the sentence "a boy climbed every tree" can hardly be examined until we’ve decided which syntactic form we’re talking about, i.e. whether there is one persistent boy who’s climbed every tree, or whether every tree has been climbed by a boy once.

This formalization is equally implicit in the field of logic. Today, a formal logic is defined as the combination of a formal syntax with a set of axioms and rules of inference. Equipped with this schema, we can prove theorems, the conclusions of inferential chains whose sole premises are axioms. Since ancient times, logical inferences have been applied to statements expressed in natural language — this can only work for unambiguous, uniquely parseable ones.

The objects of semantics

With the playing field hopefully a little clearer, we can move to our current interest: semantics. It is an inseparable part of our natural languages: surely, our prehistoric ancestors used language to convey meaning, not to deploy interesting syntactic schemata. Artificially constructed languages themselves are reliant on semantic interpretation, except in metalinguistic contexts, where symbols may just be symbols.

Semantics is commonly defined as the theory of meaning, but we shall distinguish between reference, truth, and meaning proper.

Reference

Read this English sentence: "The razoction is saftomaliac". It seems morphologically and syntactically acceptable, yet it is meaningless in the most obvious way: half of its words don’t mean anything. We cannot point to something and say "that’s a razoction", nor can we offer any definition.

It appears, then, that prior to any meaning comes the notion of reference, the process by which we interpret expressions in a language as things in the world. Then we can say those expressions denote, signify or refer to something.

Everyone would probably agree that the name "David Chalmers" refers to the man named David Chalmers. Anything beyond that is up for debate, though. What can be outlined is the basis of our modern formulation, laid out by (once again!) Gottlob Frege in his aforementioned Begriffsschrift. Frege operates a key distinction between singular terms and functional expressions:

• Singular terms are paradigmatic referring expressions: they denote one specific thing in the world. They include:

  • Proper names, like "Bjorn Borg" or "Paris".

  • Definite descriptions, like "the 1976 Wimbledon champion" or "the capital of France".

• Functional expressions do not denote any specifc object: they map some objects onto others. Frege calls such expressions unsaturated. For instance "the capital of" maps countries onto their capital cities (and objects that are not countries onto nothing). This unsaturated expression actually refers to a function which we could represent in familiar mathematical notation as \(C(\cdot)\), or \(x \mapsto C(x)\), or in lambda calculus as \(\lambda x(Cx)\).

Frege also introduces the concept of extension. For a singular term, it is simply its reference. For a function, it is the set of argument/result pairs. The extension of "capital of" would then be \(\{(\textrm{France}, \textrm{Paris}), (\textrm{U.S.A.}, \textrm{Washington D.C.}),(\textrm{Italy}, \textrm{Rome})... \}\).

We’ve explained above that semantic theory operates over a formalized, controlled language that is rid of any lexical ambiguity. Therefore, the reference process is similary unambiguous. In principle, the word-type "London" could designate a city in the U.K. or one in Canada (among others). In semantics, we assume that determination is already made, one way or the other. The converse, however, is false: distinct singular terms can share extensions, for instance "London", "the capital of the U.K.", "Charlie Chaplin’s birthplace" denote the same thing. We can say those terms co-refer.

A demonstrative like "this city" also co-refers with "London" under the circumstances these lines were written in (I am in London). Demonstratives, as well as pronouns or words like "here" and "now" are indexicals. They can be seen as functional expressions from contexts of utterance to objects. In a context when I am speaking, the reference of "I" is then Jonathan Teyssandier.

Truth-value and propositions

This basic account of reference seems robust enough, but inquisitive readers surely find it lacking. We have talked a good deal about singular terms and functional expressions, but not much about adjectives, verbs or whole sentences. What would be the referents of "blue", "to run" or "Harry loves Sally"?

The answer, in Frege’s model, is found at the confluence between his analysis of language and his study of philosophical logic. First, we can identify adjectives and verbs with predicates, properties that can be affirmed or denied of a given object. As we showed in our own discussion of properties, the fundamental distinction between subject and predicate has been common currency since at least Aristotle.

Frege’s incredibly seminal contribution was in replacing Aristotle’s traditional propositional logic with a predicate logic. This allows to analyze language by re-phrasing sentences into semantically parseable forms. In a sentence like "navy is darker than turquoise", the colors play the syntactic role of a noun phrase, and thus "look like" singular terms. Indeed, since the sentence superficially follows the same template as "Jack is darker than Bob", which is really about singular terms, we are naturally inclined to consider "navy" and "turquoise" as such. However we can re-phrase the sentence as "for every pair of objects \(x\) and \(y\), IF [\(x\) is navy and \(y\) is turquoise] THEN [\(x\) is darker than \(y\)]" (to be clear, it is acceptable to alter the syntax, while keeping the semantic value intact). By quantifying over all objects, we’ve brought to light the predicative nature of those words.

To Frege, predicates belong to the class of unsaturated, functional expressions: once proper analysis has been performed, they can only appear alongside a saturated term they apply to. However the unsaturated predicate \([\cdot \textrm{ is blue}]\) does not return a new object: its role in logic is to tell us if a proposition like [\(x \textrm{ is blue}\)] is true or false.

Now, truth is often considered colloquially as a property itself. We might say a sentence or belief is true or false, the same way a ball is blue or red. Frege’s revolutionary move is to consider "the True" and "the False" as objects; he called them truth-values. This was somewhat anticipated in the logical works of George Boole and of C.S. Peirce, but Frege first established the link with philosophy and semantics.

As a result, predicates find their place quite naturally in Frege’s system, as a special type of functional expression: one whose result is a truth-value. The predicate \([\cdot \textrm{ is prime}]\) then associates true to prime numbers and false to anything else. As per our definition, its extension would look like this: \[\{(1, \textrm{False}), (2, \textrm{True}), (3, \textrm{True}), (…4,\textrm{False}),… (\textrm{Bjorn Borg}, \textrm{False}), (\textrm{London},\textrm{False})...\}\] Isomorphically, the extension of a predicate is defined to be the set of objects that satisfy it. In this instance: \(\{2,3,5,7,11...\}\).

What applies to predicates can easily be expanded to n-place relations by noting those are just predicates applying to ordered n-uples. "Paul introduced George to John" entails that \((\textrm{Paul},\textrm{George}, \textrm{John})\) satisfies the predicate \([\cdot \textrm{ introduces } \cdot \textrm{ to } \cdot]\).

With predicates in tow, we can now consider the reference of simple sentences. Those are just the combination of a subject and a predicate, for instance: “Paul is a musician”. Frege keeps following his model, and fits sentences of this type into his classification. Is the sentence saturated or unsaturated? It is clearly not lacking any extra term to be complete; we can then confidently claim it is a saturated expression. We also know it is the linguistic expression of a function (the predicate \([\cdot \textrm{ is a musician}]\)) being applied to an object (\(\textrm{Paul}\)). Its reference should then be the result of the function, in this instance its truth-value: \(\textrm{True}\).

That is quite an extreme conclusion: that all possible declarative sentences, if they do have a reference, denote one of two truth-values: \(\textrm{True}\) or \(\textrm{False}\). Just like different proper names or descriptions can refer to the same person, all the true sentences are co-referring names for \(\textrm{True}\), and all the false sentences for \(\textrm{False}\).

That’s a bit hard to swallow... We may want sentences using completely different terms and different syntactic forms to have a different reference. Instead we’re collapsing all possible sentences’ references over only two logical values. To be fair, Frege made this daring choice in a very pragmatic way: it helped link up semantics with his logical investigations. Another way to frame this is by highlighting the importance of substitutivity. In classical logic, the principle of the indiscernibility of identicals is often invoked, that in effect, if \(a=b\), then replacing \(a\) with \(b\) in any expression will not change it. Adapting it to Frege’s semantics, we may say that if \(a\) and \(b\) co-refer, then substituting \(a\) with \(b\) in an expression will not change its reference.

The line of reasoning that is then harnessed to justify sentences referring to truth-values, and truth-values only, is called the slingshot argument. It consists in showing that a series of sentences share the same reference. Here is an example:

• J.R.R. Tolkien wrote The Lord of the Rings.

• J.R.R. Tolkien wrote 3 volumes of The Lord of the Rings.

• 3 volumes of The Lord of the Rings were written by J.R.R. Tolkien.

• 3 is the number of The Lord of the Rings volumes written by J.R.R. Tolkien.

• 3 is the number of countries in North America.

• There are 3 countries in North America.

The Lord of the Rings is a literary work that is made of three volumes, so (i) and (ii) co-refer by virtue of substituting co-referring terms. (ii), (iii) and (iv) are simple syntactic variations; I’m not saying they carry the same semantic value — as we shall see, they don’t have the same meaning indeed — but it seems that incremental re-phrasings should not yield a change of reference, lest the whole generality of semantics crumble (what would we then make of translation in different languages, for instance?). (iv) and (v) is another substitution, this time between terms that undeniably refer to the same mathematical object: the number 3. (v) and (vi) are syntactic variations of each other, the type of which we’ve encountered above. All in all, we find that (i) and (vi) co-refer, while they have absolutely nothing in common… except for their truth-value: they are both true.

So far we’ve established that (1) sentences refer to truth-values (if they refer at all) and (2) predicates are functions from objects to truth-values. Frege’s system allows us to go even further by admitting functions from truth-values to truth-values. This is another bridge from semantics to logic: logical connectives like the conjonction (“and”), disjunction (“or”) or negation (“not”) are defined as such functions. For instance, the negation, applied to \(\textrm{True}\), returns \(\textrm{False}\), and vice versa. Syntactically, compound sentences are independent clauses connected by coordination conjunctions. Transcribed in the realm of semantics, these are simple sentences together with logical connectives.

The introduction of truth-values as objects has then allowed to seamlessly expand Frege’s simple model and identify the reference of most linguistic expressions. It’s also established a firm bridge between semantics and logic, now presented as the science, not only of valid inference, but of truth value. Frege reprised the Platonic triangle of truth, beauty and good (following in the footsteps of his Neo-Kantian teacher Wilhelm Windelband), to set forth logic as the counterpart of aesthetics and ethics.

Meaning — the modal and structural theories

Reference and truth are indispensable to provide a bridge between language and the world. This is one thing we expected from semantics. But is it enough to close the book on our investigation? Clearly not, for reference is not nearly fine-grained enough to account for the semantic value of our thoughts and expressions.

Frege (once more!) developed this insight via a dual puzzle (which you’ve previewed in the section about mental content). Classical logic, as well as Frege’s predicate logic are extensional. This means that any two terms that co-refer can be substituted for one another without affecting the reference of compound expressions or the truth-value of sentences. For instance, the proper name "Isabelle Huppert" and the description "the 2001 Cannes Best Actress laureate" both designate the actress Isabelle Huppert. The two singular terms indeed seem interchangeable: the 2001 Cannes Best Actress laureate’s birthday is the same as Isabelle Huppert’s birthday. If it is true that today, Isabelle Huppert is in Paris, then it is a valid inference that the 2001 Cannes Best Actress laureate is in Paris today.

Frege’s first point is that the sentences below, although strictly equivalent based on substitution rules, are not the same. They differ starkly in cognitive significance.

• Isabelle Huppert is Isabelle Huppert.

• Isabelle Huppert is the 2001 Cannes Best Actress laureate.

His second, and related, point is that the two sentences below can be true:

• Pierre believes that Isabelle Huppert is Isabelle Huppert.

• Pierre does not believe that Isabelle Huppert is the 2001 Cannes Best Actress laureate.

The substitutivity of co-referring terms is then violated for belief ascriptions and similar contexts, which we’ll call propositional attitudes. It appears these contexts are not extensional but intensional.

Certainly, there is more to meaning than reference. This is not a novel idea, and it can be traced all the way back to ancient Greece by way of the scholastics, John Locke and J.S. Mill. Yet the type of treatment that Frege instituted and which would become typical of analytic philosophy established semantics as a general discipline, free of ties with language use and speech acts.

A purely referential theory of meaning is then insufficient. A more robust concept of meaning should accomplish at least the following:

• To explain expressions’ behavior in intensional contexts.

• To be plausibly known by any competent speaker.

• To determine reference.

We’ll consider three models; they are not incompatible, actually they represent a progression from coarse-grained to fine-grained.

Modal theory of meaning

The first theory of meaning we’ll examine involves other, possible worlds. That may seem an awfully abstract thing to explain the meaning of expressions, but it is not that outlandish.

What really differentiates co-referring expressions? If Huppert is indeed a 2001 Cannes laureate, it is by virtue of Huppert winning that award that year at that film festival. A bit more technically, we can say that the two terms co-refer by virtue of that state of affairs obtaining in our world (conventionally called: the actual world). If Naomi Watts or Nicole Kidman had won the award then one of their names would co-refer with "the 2001 Cannes Best Actress laureate".

Nothing too surprising there: we’ve just highlighted that the reference of certain expressions depends on their circumstances of evaluation (or in modal lingo, on which world we’re at). "Isabelle Huppert" just refers to Huppert, but "the 2001 Cannes Best Actress laureate" could possibly be Huppert, Kidman, Watts, among others. More so, it appears that the semantic values of these terms may be distinguished by their sensitivity to circumstance.

What I call a modal theory of meaning (more commonly "possible-worlds semantics") takes a pleasingly minimalist next step. If meanings are things that turn into a certain reference in certain possible worlds, then why not say that meanings are just functions from worlds to references? In that context:

• The meaning of a singular term is a function from worlds to objects.

• The meaning of a functional expression is a function from worlds to functions.

• The meaning of a sentence, called a proposition, is a function from worlds to truth-value — equivalently it can be seen as the set of worlds where the proposition is true.

• The meaning of a predicate is a function from worlds to sets of objects.

We can thus explain the difference between the meaning of sentences (1) and (2) above. (1) is true in all worlds (necessarily true), whilst (2) is true only in a set of worlds representing the proposition it expresses. (3) and (4) also don’t seem inconsistent, as competent speakers can presumably recognize the difference in content between the two beliefs ascribed to Pierre.

A difference between co-extensive predicates can also be explained. For instance, it so happens that in our world, all creatures with a heart also have kidney, and vice versa. The predicates "having a heart" and "having kidneys" thus co-refer, since their extension is the same set of creatures. They differ in meaning though: there could be worlds where they don’t designate the same objects.

Since the formalization of this theory in the 1950s and 60s, meanings in the modal interpretation are often called intensions. As serviceable as it is, this model is however not sufficient to explain the granularity of meaning. That is evident from a simple fact: if intensions are functions of worlds, then propositions that are true at all worlds (necessary ones), or false at all worlds (impossible ones), would all have the same meaning. For instance, (5), (6) and (7) below are all necessary:

• 2+2=4

• 83 is a prime number.

• Bachelors are not married.

Yet we expect our theory of meaning to distinguish between those, even though they may share intensions.

Structural theory of meaning

There is an obvious, layman answer to that issue. (5), (6) and (7) express different meanings, because they involve different objects. Only (5) involves the number \(2\), only (6) involves the concept of "being prime", and only (7) involves bachelors. This pre-theoretic remark is the basis of what I’d call structural theories of meaning.

These theories apply to semantic expressions the same type of analysis as for syntax. For instance, proposition (5) from above, \(2+2=4\), puts into play four distinct objects: \(2\) (twice), \(4\), and two binary relations (addition and equality). Proposition (6) involves two objects: the number \(83\) and the property “being prime”. Proposition (7) will undergo the same type of linguistic analysis as we’ve encountered from Frege: \[\text{for all $x$, IF $x$ is bachelor THEN $x$ is unmarried }\] which involves logical connectives (\(\rightarrow\)) and a universal quantifier (\(\textrm{for all}\)).

In Frege’s theory of reference, sentences were interpreted as functions of their components’ references to truth-values. In a seemingly natural move, structural theories of meaning interpret propositions (the meanings of sentences) as functions of their components’ meanings.

Structure can also help further distinguish between proper names and descriptions. Bertrand Russell introduced the definite description operator \(\iota\). “The 2001 Cannes Best Actress winner” thus becomes \[\iota x \textrm{2001CBA}(x)\] This reads as "the one object \(x\) that verifies the property of being the 2001 Cannes Best Actress awardee: \(\textrm{2001CBA}\)". Since the meaning of this term involves the definite description operator as well as a property, it is automatically differentiated from the proper name “Isabelle Huppert”, without the need to use intensions.

Speaking of Russell, I will be very careful around the adjective “Russellian”, which is common currency in semantics, but highly misleading: Russell lived a long life, was highly prolific, and changed his mind often. Using the “Russellian” moniker with abandon, at least outside of a solidly established context, will render any work incomprehensible.

So, very carefully, I will say that the Russellian theory of meaning is a structural theory that accepts universals as the meanings of atomic functional expressions (i.e. those that analysis cannot break down further). Both the reference and meaning of the proper name “Isabelle Huppert” are the person thus named. The meaning of the property “being an actress” is an abstract object associated with this property. The meaning of the sentence “Isabelle Huppert is an actress” is a composite function of the woman and the abstract object. Russell doesn’t need intensions from modal semantics at any point (and rightly so, since modal intensions were formalized decades after he dabbled in semantics), yet structural theories of meaning in general are not incompatible with modal ones at all.

We can devise a structural theory of meaning which also relies on the resources and benefits from the modal theory of meaning described in the previous section. It accepts that the meaning of atomic expressions, be they names or properties, are functions from possible worlds, as stipulated in the modal framework. However it resolves subsequent issues by saying the meaning of sentences and other composite expressions is not these expressions’ intensions, but a structured intension whose components are the atomic components’ intensions. The meaning of “Isabelle Huppert is an actress” is then not composed of a woman and a property, but of two intensions (one of a singular term, one of a predicate).

Meaning — names, senses & descriptions

Structure did help us gain a more fine-grained view of semantics: one that distinguishes between two necessary propositions, for instance. However there still is an important problem to deal with. Consider this new version of Frege’s puzzles:

• Romain Gary is Romain Gary.

• Émile Ajar is Romain Gary.

and

• Alex believes that Romain Gary fought in World War II.

• Alex does not believe that Émile Ajar fought in World War II.

"Romain Gary" and "Émile Ajar" refer to the same person (they’re both noms de plume of Roman Kacew). Yet the first pair of sentences seems to differ in cognitive significance. As to the second pair, competent speakers could conceivably hold both of Alex’s beliefs at the same time. Since both terms are proper names, our first instinct is to assume that none of the previous models can help us: (1) the two names not only co-refer, they also share intensions and (2) they are a priori devoid of an underlying structure.

How can we deal with these two puzzles — the first about identity, the second about propositional attitudes (of which belief ascriptions are but one example)? Here are the three options we’ll consider:

• Fregeans about names answer that there is more to names’ semantic value than reference: names also have a sense.

• Russellians about names answer that there is more about most names’ semantic value than reference: that’s because they are not really names but disguised descriptions.

• Kripkeans about names maintain that names’ whole semantic value is just their reference: they deny that Frege’s puzzles are in any way puzzling.

Frege’s senses

Frege’s views on names were in reaction to those expounded by John Stuart Mill: the Millian take is that names are meaningless tags (marks on paper, sounds in the air...) that denote objects but have no connotation. In other words, there’s nothing more to names’ meaning than reference.

Frege saw his two puzzles about names as an indictment of Millian assumptions. Let’s invoke a familiar instance of the identity puzzle.

• Clark Kent is Clark Kent.

• Clark Kent is Superman.

To Frege, the difference in cognitive significance entails that the semantic value of these sentences must differ. Therefore, from compositionality, the semantic value of the two names must differ. Since their reference is identical, Frege posits the existence of a sense: the meaning or mode of presentation of that name.

Once we admit that the Superman-sense and the Clark-sense are distinct, both the identity puzzle and the belief puzzle are dispelled: we don’t expect sentences with different meanings to have the same cognitive significance, nor do we expect people to have identical beliefs about different propositions. This is why the present ideas belong to the group of theories of mediated reference: sense acts as a screen between language and the world.

Let’s note here that Frege’s linkage of meaning with belief and knowledge is not innocent nor obvious. Senses’ identity conditions are grounded in cognition: two names have different senses (mean different things) if there is a competent speaker whose belief can be reverted by substituting one name for the other.

What is the relationship between names, senses and reference? First, different names can express the same sense: synonyms within a common language, or translations, share the same meaning, by definition. Conversely, one name can also express different senses. On that particular point, Frege has been fairly shifty, exclaiming at one point: "As if it were permissible to have different [senses] with the same wording! This contradicts the rule of unambiguousness, the most important rule that logic must impose on language.". Should we infer that the relation between name and sense is one-to-one? Well, Frege later adopts another stance: "In the case of an actual proper name such as ’Aristotle’ opinions as to the sense may differ. It might, for instance, be taken to be the following: the pupil of Plato and teacher of Alexander the Great. [...] So long as the reference remains the same, such variations of sense may be tolerated, although they are to be avoided in the theoretical structure of a demonstrative science." It appears that the one-to-one matching of names and sense is to be reserved to an ideal, uniquely parseable language.

Furthermore, having multiple senses waiting to be grasped provides semantic riches that can account for the granularity of our thoughts. Wittgenstein himself took a similar position, with this example in tow: "If one says ‘Moses did not exist’, this may mean various things. It may mean: the Israelites did not have a single leader when they withdrew from Egypt — or: their leader was not called Moses — or: there cannot have been anyone who accomplished all that the Bible relates of Moses... But when I make a statement about Moses, [...] has the name ‘Moses’ got a fixed and unequivocal use for me in all possible cases?"

There is then a many-to-many relationship between linguistic names and senses. However, an important Fregean tenet is that sense determines reference. Any given sense, if it refers, has one and only one reference. This dual role of senses as conveyors of meaning and determinants of reference is fundamental to this worldview, but also the source of some problems, as we shall see when discussing Kripke’s criticism.

How does this all gel with what we’ve presented before, namely the modal and structural views of semantics? Very well, it turns out. Modal theories assume that the meanings of expressions are functions from possible worlds to objects; it is entirely possible to equate the sense of expressions with those intensions. Of course senses are also more granular — we’ve seen it in the case of names, but it can be true of descriptions or predicates: it’s now often said that Frege’s theory can explain hyperintensionality (the difference between expressions that are co-intensional). A similar comment can be made about propositional structures, only Frege would say that the sense of a proposition is not a compound of objects or of intensions, but of senses. In both cases, senses are more granular, and can use all resources from the more general models.

This granularity is the first major criticism that Fregean semantics typically face: the cognitive identity conditions for senses can give the impression that any given word has as many senses as there are speakers, or perhaps as many senses as tokens of that word. Frege had pre-emptively countered that argument by insisting that senses are not mental items and are not grounded in words being uttered or thought. But that in itself gives rise to the second challenge: Frege concludes that senses are part of a Third Realm (we would say: of the abstract world), but cannot explain them in terms of other objects. His senses are truly sui generis. The way in which minds can grasp sense is similarly left vague, the concept given as a posit.

Russell’s disguised descriptions

Bertrand Russell, a reader of Frege’s work, proposed his own framework for mediated reference. Instead of ascribing abstract senses to names, he argued that most were in fact disguised descriptions. The name Romain Gary could then be shorthand for "the author of La Promesse de l’aube" and Émile Ajar for "the author of La Vie devant soi". Clark Kent could be "the meek, bespectacled journalist from the Daily Planet" and Superman "the flying hero from Krypton". Regardless of reference, substituting one member of each pair for the other in a sentence automatically changes the semantic value, thus dispelling both the identity and belief puzzles.

Of course, not all names can be equated with descriptions. If I’ve just been introduced to a man named Bob, I can express thoughts about him without being able to associate any description with that individual. In that case, my Bob-thoughts have no deeper meaning than a direct reference to Bob. But Russell is quite stingy with cases when that is admissible: he imposes a principle of acquaintance, following which our knowledge about names is always through description unless we have direct awareness of the referred object. Anyone talking today about George Washington may mean something like "the first U.S. President" or "the commander of the Continental Army", but to people who knew him personally he might just have been "George".

Like the Fregean model, this one is in principle compatible with modal and structural theories. The linguistic analysis is just pushed to another level by inserting descriptions where names were, at which point those theories can apply. But what is the relationship between Russellian descriptivism and Fregean senses? Intuitively, it must be a strong one: in his own writings, Frege himself often used descriptions to make explicit his concept of sense. If one decreed that senses are just descriptions or combinations thereof, we could still construct the resources established by Frege: descriptions can indeed provide meaning, and they can also determine reference. This similarity has encouraged talk of Frege-Russell descripivism. But despite Frege’s apparent openness to using descriptions, it’s important to keep the distinction: Fregean senses are much more fine-grained as they can establish differences in cognitive significance that are not expressible in terms of descriptions.

One way we might take exception with descriptivism is by questioning the uniqueness of descriptions. Somewhat of a straw man argument for sure: it is doubtful Russell would have explicitly defended that uniqueness, but he was quite vague on the topic. Searle applied a "cluster" approach to the problem. To him, a sentence about a name does not assert that a single description, or a group of descriptions, are satisfied, but it presupposes the reference satisfies "enough of them": a sufficient but unspecified (weighted) amount. Searle embraced the vagueness as an asset. Names are not abbreviated descriptions but "pegs on which to hang descriptions". This brings us back to Wittgenstein’s Moses example from above: "Moses didn’t exist" doesn’t have a clear sense. Does this mean that no Hebrew prophet was really speaking to God? Or that Hebrew slaves did not have a single leader during their exodus from Egypt? Or that there was never a Hebrew boy who was found in a basket and raised at Pharaoh’s court? The cluster of descriptions associated with this ancient name is quite spread out, and the semantic value of sentences about Moses will heavily depend on what context pegs on this name.

All in all, Russellian descriptivism has all the makings of a robust theory, and accordingly was a dominant view for decades. Its weaknesses were pointed through the revival of direct reference theory, with no critique as tranchant as Saul Kripke’s.

Kripke’s direct reference

Remember that direct reference as expounded by J.S. Mill had been shaken to its core by Frege’s puzzles, leading to the development of descriptivism. In a famous series of 1970s lectures, Saul Kripke patly claimed that his forebears’ views were plain wrong, and proceeded to undermine the mere concept of disguised description. To that end, he developed three arguments.

The first argument is called modal as it revisits in more details the interface between modal reasoning (counterfactual scenarios in possible worlds) and descriptivism. To start with, Kripke called attention to the concept of rigid designation, building on the work of Ruth Barcan Marcus. A name is said to designates rigidly if it denotes the same object in every possible world — in other words, if it necessarily refers to that object. Kripke argues that all of our ordinary proper names are actually rigid, on that basis: somebody else than the current U.S. President might have been the current U.S. President (therefore, that description is not rigid); but nobody else than Joe Biden might possibly have been Joe Biden (so that name is rigid). Kripke would justify the latter statement thus: someone points to an image of Joe Biden, saying "this guy won the election, but he could have lost!"; then someone replies "if you mean to designate "the winner of the last presidential election", then it is not true that he might have lost, since the winner necessarily won". Kripke remarks that the former reflects our regular use of language, whilst the latter could only be uttered by an academic... If we follow this idea along, proper names cannot be abbreviated descriptions, nor clusters of descriptions. In general, a description is not rigid: the U.S. president in 1971 was Richard Nixon, but it could have been Hubert Humphrey or Ronald Reagan (note that I said "in general": descriptions such as "the sum of 1 and 2"designate their reference as a matter of necessary truth, in all logically possible worlds; but it’s not this type of description we’re considering here). If proper names, by contrast, are rigid, then they can still satisfy a description, but cannot be equated with it.

Kripke’s second argument is called epistemic, as it hinges on the concept of a priori knowledge. Since Kant, a priori is how we qualify knowledge that can be acquired independently from empirical experience (via purely rational means, for instance mathematical or logical inference). Kripke notes another case of failed substitution. Let’s assume that the name Rutherford Birchard Hayes simply abbreviates the description: "the 19th U.S. President". It is knowable a priori that the 19th U.S. President was President (to any competent speaker of the language, that is a clear logical inference). But it is not knowable a priori that Rutherford B. Hayes was President: if you had not read it or been told so, you couldn’t have deduced it using reason only; it is a posteriori knowledge. Kripke concludes, once again, that descriptivism cannot be correct, for we can’t substitute descriptions for names.

A third argument is called semantic. Essentially, Kripke denies the Russellian idea that all our knowledge about objects we’re not acquainted with is descriptive. In most cases, he argues, we’re incapable of associating definite descriptions to names. There are people who would accurately, and uniquely, describe Feynman as "the physicist who pioneered the path integral formulation of quantum mechanics". But to most people who have heard that name, he is "some physicist". Some of these people may have also heard of Steven Weinberg, whom they would also say is "some physicist". Yet they are able to speak competently of these two persons, despite their incapacity to explain the difference via descriptions. How can names possibly refer, if it’s not via acquaintance or description? That can be explained via a causal-historical model: names would refer in virtue of a chain, going all the way back to an initial "baptism". The person who has just heard of "some physicist" named Feynman is then the latest rung in a chain going back to someone who actually knew and was acquainted with him.

Now, once again, descriptivism developed as a way to solve Frege’s puzzles. Aren’t they a hurdle to Kripke’s direct reference model? Let’s look at the identity puzzle again through the prism of (7) and (8). Both Romain Gary and Émile Ajar are proper names, and therefore rigid designators of their reference (the human being whose legal name was Roman Kacew). Because of direct reference, there is nothing more to these terms’ meanings than that reference. The two sentences under consideration, "Romain Gary is Romain Gary" (7) and "Émile Ajar is Romain Gary" (8) then have the exact same meaning. Even more, since both names designate an object rigidly (that is, in every possible world where that object exists), then those two sentences are also true in every possible world — they are necessary. The consequences of following Kripke’s path are wide-ranging. What precedes could be seen as a damning contradiction. From Kant to Frege by way of the positivists, what is necessary (true in any possible world) has been equated with the a priori (what can be known using reason only). Conversely, the contingent (true only in some words) was assumed to be the same as a posteriori knowledge. Kripke simply rejects the premise. "Émile Ajar is Romain Gary" or "Clark Kent is Superman" are examples of necessary a posteriori statements. Even more, Kripke also shows that common terms like indexicals are an example of contingent a priori: it is a priori for any competent speaker that the sentence "I am here" is always true, whoever utters it. However the fact that I am here (i.e. that Jonathan is sitting in a living room in London) is clearly contingent: in other worlds, I could be elsewhere. So, the identity puzzle can be safely ignored, as direct reference entails the separation of meaning and modality from knowledge (and therefore, cognitive significance). A similar treatment can be applied to the belief puzzle. What Kripke does to address it is to change the setting: instead of using a pair of names like Gary/Ajar or Kent/Superman, he takes the example of Paderewski, who was a Polish Prime Minister and a famous musician. It is conceivable that, hearing about this man in different contexts, one would come to simultaneously believe that "Paderewski has musical talent" (thinking of the musician) and "Paderewski has no musical talent" (thinking of the politician). Kripke’s intent with such examples is to show that the truth-makers of belief ascriptions and other propositional attitudes lie in extra-semantic contexts anyway, so it is not the adoption of direct reference that is cause for belief "puzzles" this type: from the perspective of semantics, there is no puzzle.

Direct reference is probably a dominant view at the time of writing, but I’m not without qualms towards it. The concept of rigidity is a strong one, and in my view not nearly as "natural" as Kripke would like to make it appear, jumping as he does from speech acts and usage back to semantics. In a possible world where Roman Kacew writes all the books by Romain Gary and it is Jean d’Ormesson who calls himself Émile Ajar and writes all the books we know from Ajar, rigidity still imposes that the counterpart from "our" Ajar is "their" Kacew, and "Romain Gary is Émile Ajar" stays true in all possible worlds, including that one. Puzzle lovers can also have a field day by imagining limit cases, such as a possible world where Kacew writes Gary’s books, and has an identical twin who writes Ajar’s books...

Signs and semiosis

Signs can at first glance be seen as a generalization of semantic meaning. Using terms introduced by Ferdinand de Saussure in his study of semiology, we can say that a signifier is a type (exemplified by replicas) associated with a signified. The sign "[cat]", for instance, associates the signifier "cat" (a word-type) to a signified: the idea of a cat. This tool is nimble and was swiftly expanded to visual signs (a stop sign), bio-signs (that elicit behaviour in living beings), and much more. Yet, this approach is still linguistic above all else. Saussure sees signs primarily as nodes in the vast network of language; he assigns to them a relational or structural role rather than freestanding existence. Furthermore, signifier and signified are unequivocally psychological in nature to Saussure.

An alternate view of semiotics by C.S. Peirce took the theory of signs from a linguistic endeavour to a holistic model of metaphysical import, mirroring in pragmatic philosophy what Frege was developing in proto-analytic philosophy. Peirce’s key insight was to describe signs as a triadic entity involving a representamen, an interpretant, and an object. (As an aside, I need to note that I’ll mostly make use of Peirce’s 1903 terminology here — adapting it for my own purposes a needs be. Peirce developed his model over decades, and not only are there several markedly different accounts, his terminological creativity knew no bound, arguably leading to an overabundance of neologisms. Readers should see no harshness here: I empathize with Peirce’s quest for consistent and evocative neologisms).

A representamen would be a type (the utterance of a word, a physical object, a thought) that stands for something else and is in effect the sign’s vehicle. The object is what the representamen stands for. Peirce takes an expansive view of objects, explicitly stating that the object of a sign does not need to exist to fulfil that role; Platonists or Meinongians would find no fault with that idea. Lastly, the interpretant is all that the sign can in itself signify, mean or convey. Peirce steered clear of the term "meaning", perhaps lest his subtle notions be conflated with how others used that term; he also rejected "signification" as too narrow, since too him an interpretant could be an emotion or an action, rather than a verbal expression.

So far we are in familiar territory. Representamen and interpretant seem close enough to, respectively, signifier and signified. Interpretant and object remind us of Frege’s sense and reference. The resulting "semiotic triangle" does not by any means originate with Peirce’s work: it was used in some way, shape or form by Stoics and by medieval scholastics. After Peirce, it became a hallmark with Richards and Ogden’s historically informed book on meaning. But for all the explanatory value of that trichotomy, one may still wonder how signs could make a claim for their own ontological category: can’t they simply be considered as a molecular complex of simpler entities, be they physical, mental or abstract? In the case of the sign for "cat", the triad associates a vehicle, the word "cat", which is determines by an object, the kind "cat", and which determines an interpretant: the meaning of "cat". Can’t the sign simply be an ontological compound made from those semantic atoms? That is where Peirce makes his mark: to him, the relation between a sign’s components is genuinely triadic. That statement rests on his own work in algebraic logic, later referred to as the reduction thesis. Under Peirce’s chosen logical framework, it is possible to prove that the general triadic relation cannot be decomposed into dyadic and monadic ones. At the same time, tetradic, pentadic, and all larger polyadic relations can be decomposed. Therefore monads, dyads and triads appear as the only fundamental relations.

With that idea, we’re reaching the core of what makes this semiotic model special. Peirce doesn’t see triadic relations as a convenient representation of signs only, he identifies the two: any triadic relation is a sort of sign, and is intertwined with the ideas of mediation and of meaning. With that, we’re already traveling beyond the bounds of language interpretation and human communication. Unlike Saussure, Peirce does not see the semiotic process as necessarily involving sentient utterers and interpreters. He attempts to explain his stance through the notion of quasi-mind. That is consistent with his pragmatic approach, which sees human thought as governed by "laws of mind" or habits. The purpose that distinguishes sentient behavior from pure physical consequence is already apparent, to Peirce, in the quasi-purpose of biological evolution, itself a byproduct of physical evolution. Reverting those principles allows to consider structures "expressed" by quasi-utterers (like a crystal, or a beehive) as the sign-vehicles denoting their object (a law or organizing principle of some type). In addition, Peirce vehemently rejects the idea of a single sign existing in a vacuum — the interpretant of a sign can itself be a sign (in some versions of his taxonomy, Peirce calls it the "interpretant sign"), leading to a process of infinite semiosis, which is characteristic of thinking, but obviously broader when applied to quasi-minds. Such a strand of thought can lead to postulating pansemiotic theses, and see the whole universe as a semiotic machine, as some have.

This foundational nature of triadicity/semioticity led Peirce to erect it as a full-blown ontological category he called Thirdness (not his most inspired neologism, but I have nothing better to propose at this time...). As can be expected from our considerations about irreducible relations, his system (which he called cenopythagorean) appeals to three categories.

• Firstness is monadic. Its characterization is as a quality or feeling. Modally it indicates possibility. Other terms Peirce associated with it include "complete in itself, freedom, measureless variety, freshness, multiplicity, manifold of sense, peculiar, idiosyncratic"

• Secondness is dyadic. Its characterization is as a reaction or resistance. Modally it indicates actuality. Other terms Peirce associated with it include "otherness, comparison, dichotomies, mutual action, will, volition, involuntary attention, shock, sense of change"

• Thirdness is triadic. Its characterization is as a representation or habit. Modally it indicates necessity. Other terms Peirce associated with it include "composition, continuity, moderation, comparative, reason, sympathy, intelligence, structure, regularities, representation".

Peirce did not conceive his ontology by appealing to familiar entities like concrete objects and properties. We can only surmise, simplistically, that his firsts (characterized by firstness) could be qualia, that seconds could be denizens of the physical world, and that thirds would cover abstract entities from our third realm, alongside perhaps mental states that are not purely qualitative.

Peirce uses this model to propose his much-discussed classifications of signs. You may note the plural. He was constantly revising his system throughout his life, yielding countless arrangements. I’ll refer to his 10-sign system from 1903 here (in following years, he would develop, albeit not as fully, a 28- and a 66-sign system). The method is based on three trichotomies (Peirce would pleasantly admit he suffered from "triadomania"). The representamen itself could be a first (for instance, an emotion), a second (an uttered word) or a third (the corresponding word-type). The respective names for those are qualisign, sinsign and legisign. The next trichotomy examines the way the object of the sign determines its representamen. If that determination is a first, the representamen is akin to the object in terms of some quality — the relation is said to be iconic (for instance, a stick figure for a person). If it is a second, then the representamen is in an actual connection with the object — an indexical relation (meow sounds indicating a cat is around). If it is a third, the representamen is associated with the object by virtue of a law or habit — that is a symbolic relation (a national flag). The third and last trichotomy is concerned with the way the representamen determines the interpretant. Should that relation be a first, it is called rhematic (for instance, the single name "Joe", the unsaturated predicate "is a man", or a logotype are rhemes: they refer to their object in a symbolic way, but are interpreted as evoking a unitary concept). When that relation is a second, it is called dicent (a proposition or a saturated predicates qualify, since they are interpreted as dyadic relations, or compounds thereof, and associated with an idea of actuality, given their definite truth-value). Lastly, if that relation is a third, the sign is called an argument, and as the name suggests it is exemplified by logical sequences such as a syllogism. Most thoughts are considered by Peirce as arguments.

Careful readers are surely wondering why a system based on a triple trichotomy does not yield 27 kinds of signs, instead of 10. That is because all combinations are not admissible. Peirce showed that the valence of a representamen cannot be exceeded by that of the R-O relationship. Similarly, the valence of an R-O relationship cannot be exceeded by that of the R-I relationship. The 10 resulting types of signs are (1) (rhematic iconic) qualisigns (2) (rhematic) iconic sinsigns (3) rhematic indexical sinsigns (4) dicent (indexical) sinsigns (5) (rhematic) iconic legisigns (6) rhematic indexical legisigns (7) dicent indexical legisigns (8) rhematic symbol(ic legisign) (9) dicent symbol(ic) legisign (10) argument(ative symbolic legisign).

Taking stock of this brief account of Peirce’s semiotics, was I justified in introducing signs as a possibly distinct category alongside meanings, properties, etc.? It seems like a halfway measure: either one adheres to Peirce’s notions of Thirdness and pansemiosis, and then signs are perhaps the only type of abstract entity. Or one doesn’t and then signs can be decomposed and do not constitute a sui generis class of abstracta. But again, what I’m building here is an almost cladistic account of entities that could be the basic bricks of my ontology. I have no qualms over-populating it at the moment.

Semantics

Reference: how do linguistic terms come to denote things?

Position: Reference dualism

As we’ve seen a few chapters above, semantics is about the reference (or denotation) of expressions in our language, about the truth value of our statements, and about the meaning (or connotation) of expressions. It is crucial to distinguish these general linguistic matters from the semantics of the formal language used in our ontology. I’ve tried to dispel misunderstandings by calling the latter "logical semantics". The formal language and its semantic interpretation in terms of ontology is prior to notions of ordinary objects and concepts that we will encounter here...

But let’s come back to our linguistic semantics... The matters of reference, truth and meaning are intermingled to a degree that theories of reference often seem to be talking at cross-purpose. I’ll try to simplify by considering three questions.

What is reference?

Conceptually, the answer is simple: reference is a relation between an expression-type and an object. It realizes the familiar mechanism whereby the name-type "Everest" denotes the object that is Mount Everest. Formally we will write: \[\mathbf{D}\texttt{"Everest"}\big[e\big]\] That may seem out of place. Previous paragraphs were about metaphysics or modality, and didn’t seem to assume the existence of language or of humans. Here I’m introducing expressions from a natural language (English) out of the gate. But the point of formal semantics is that the workings of our natural language point us to a deeper truth that’s part of the fabric of our world, whether a single word was ever spoken or not. Despite the use of concrete natural-language examples, the formalism should make that evident.

In the above, \(\mathbf{D}\) represents the denotation relation. \(\texttt{"Everest"}\) stands for the name-type (as opposed to the token: every time I have written that word in this section I’ve minted a new token; but it’s always the same type). \(\big[e\big]\) is the ordinary object, a mountain located in the Himalaya. The only constraint on objects of reference is that they exist in an ontological sense. But that still leaves us with a broad playing field that includes concretes objects as well as all abstract objects.

Colloquially it may seem that Everest is referring to something as concrete as it gets, but it’s important to remember that in our framework, ordinary objects are equivalence classes. I cannot direct my thoughts towards a specific heap of atoms that would mentally represent Everest (and no other heap would). Even if I could, atoms are not fundamental and discourse about them is ordinary discourse already: it’s "turtles all the way down" towards whatever fundamental substance there is, which we do not know yet, and thus cannot think or speak about.

Therefore, better to consider that all statements about the concrete world will be made by referring to ordinary objects. Since ordinary objects correspond to a specific kind, the denotation relation involves a choice of a kind, through the ordinary object it picks out. It was implicit, but easy to guess that \(\big[e\big]\) is of the kind "mountain". It is nonetheless instrumental, since some things can be "the same mountain" but not "the same rock" as a concrete Mount Everest. That is how natural language works as well: except in especially ambiguous situations, it should be clear what kind of thing an expression denotes to competent speakers.

What grounds reference?

Or to put in another way: in virtue of what do expressions refer? I will eschew the distinction between "direct" and "mediated" reference. Ultimately, reference is grounded in something, which we can call the mediator between language and object. As we’ve seen in various sections, the ground is two-fold.

• On the descriptivist model, expressions refer by virtue of a certain descriptive content that identifies an object as the referent. For instance: "the King of England".

• On the causal model, they refer in virtue of a causal chain, or rather of a "web of usage" that harks back to an original baptism.

The descriptivist way is generic: any sort of semantic expression can be denoted in that way (by talking of "semantic expression", I aim to exclude logical or formal connectives like "and" or "the"). The causal model is rather restrained to singular objects; on the linguistic side, proper names, kind names and demonstratives are archetypes. It is conceivable that I hear of "William IV, some king" and never associate any descriptive content to him beyond being a monarch, which does not single out an individual. Yet I can ascribe to specific beliefs about \(\texttt{"William IV"}\) , which can be evaluated based on which properties or relations the human being \(\big[ W_4\big]\) did satisfy. The chain of use on which this is grounded goes through my teachers, the authors of books I’ve read, back to people who knew William, and ultimately to his parents.

This could seem a bit of a specious distinction: after all, an ordinary object like \(\big[ W_4\big]\) is defined via certain properties: it is a set that includes all concrete objects that (1) satisfy the property of being human and (2) "are the same human" as a distinguished instance \(w_4\). The latter relation is also defined by properties, for instance "having as parents \(\big[ f\big]\) and \(\big[ m \big]\)" or "having DNA code that includes ’...’". This all amounts to a canonical definition or \(\big[ W_4\big]\), and isn’t that descriptive content?

Well, the key point is that canonical descriptions are typically not known by the speaker. To use a common example, Saul Kripke insists that natural kinds like "water" refer by virtue of a causal chain. Nowadays, we have good candidates for a canonical definition such as "a liquid that is mostly \(\textrm{H}_\textrm{2}\textrm{O}\)". But by pointing to the transparent liquid they saw in river or falling from the sky and calling it water, people initiated a causal-historical chain that for the most part has always denoted the same ordinary liquid as it does today, even as, for most of human history, the concept \(\textrm{H}_\textrm{2}\textrm{O}\) was not known, thought about or spoken about. In contrast, to refer by virtue of descriptive content requires mastery of that content. If someone speaks of "the longest-serving Taoiseach", it is hard to see how they could use that description to pick out a specific human whilst not knowing what a Taoiseach is.

Should we even talk about reference?

It’s always good to revise our priors. There have been many who’ve expressed doubts about the concept of reference itself. Does it amount to anything useful or is it hopelessly simplistic? Quine famously called reference "inscrutable". His point was that by considering language on the one hand, and the way the world is on the other, we can make educated guesses about what exactly words refer to, but there will never be a way to adjudicate between various hypotheses. Quine does have a point: there is multiple realizability when looking at a specific world and a specific context of utterance. And reference alone is indeed inscrutable.

That does not mean that reference as a concept should be discarded, only that it’s inseparable from a theory of meaning. Our theory of meaning will not be an overlay on the theory of reference but rather its foundation. Anticipating a bit: meanings will be construed as functions from worlds (our actual world or other possible worlds) and scenarios (representing epistemic states, for instance but not limited to the cognitive state of an ideal language user) to referents. The exact referent could then vary depending on the mode (belief, fear, hope...) or the context of utterance, since those would be reflected in the scenario of evaluation. Lastly, it’s always good to remind ourselves that lots of fine distinctions we may be tempted to make (for instance between "blue", "blueness", "being blue") do not require different referents since the possible difference in truth valuation will depend on pragmatics or on the mechanics of specific natural languages.

Truth: what type of things can be true; what makes them so?

Position: Logical subatomism

Truth is a central subject in philosophy. Any theory of truth should answer at least two fundamental questions

1. What kind of thing can be true or false? (the nature of truth-bearers)

2. What makes something true? (the nature of truth-makers)

Although its consequences range far beyond that field, a theory of truth belongs, at its core, to semantics, since truth-bearers tend to be semantic objects (like utterances, sentence-types, propositions…) or at least semantics-adjacent ones (beliefs, facts…). For instance, we have already seen that in Frege’s theory, the True and the False serve as the reference of propositions.

Theories of truth come in many flavours, but the metaphysically realist worldview I’ve been advocating is usually associated with one strand of thought. Painting with a broad stroke, we can describe it as the correspondence theory of truth. It can already be found, in a vague form, in the teachings of the Ancient Greeks. Aristotle proposed that “to say ‘it is’ about what is, or ‘it is not’ about what is not, is true”. To put it more clearly: what makes a statement true or false is the way that things are (or are not): every truth corresponds to an objective fact, or a state of affairs. In modern terms: a statement is true when the corresponding fact obtains. So if it is true that Sean is a physicist, it is by virtue of the corresponding fact (that Sean is a physicist).

The correspondence theory thus feeds into a broadly “objective” vision of truth. The reliance on facts (presumably abstract objects) encourages us to commit to their existence. And the bridge between semantics and metaphysic allows to import the logical laws which often are the underpinning of the latter. With classical propositional logic, this leads to a bivalent, non-contradictory truth.

This can lead astute readers to harbor concerns about the sheer adequacy of the concept. It is not clear at all that such commitments can help us build a coherent view of day-to-day assertions like "It is true that ’Penny Lane’ is a good song". This type of statement is called normative, and will be presented in the next section.

The most influential version of the correspondence theory is Tarski’s. The Polish logician dispensed with facts and states of affairs to ground truth in reference, satisfaction and logic. The theory follows from two simple principles.

The truth definition states that a sentence is true if and only if the corresponding formula is satisfied. This merits some development. The way this is often formalized is: \[\Vdash \phi \leftrightarrow \phi\] That can be misleading: Tarski insisted on the difference between the right-hand-side \(\phi\), which belongs to the object language and tells us something about the world, and the left-hand-side \(\phi\), which is a statement in the metalanguage. So even as we’re using the same symbol, or the same string of characters, on either side, we must remember they represent two distinct objects that just happen to be in correspondence.

A concrete example of the truth definition is that “snow is white” is true when snow is white. Due to the way we express this in a natural language (in this instance, English), this sounds like a tautology, but it’s not. A less natural but more precise version would be: the metalinguistic sentence “snow is white” is true when the object ‘snow’ satisfies the property ‘being white’. Every formula from the object language can be expressed as a sentence in the metalanguage, but the reverse is not true. That is actually a key point to consider. If the two languages were just mirror of each other, then the theory could be accused of redundancy. That position was defended by Frege. But what Tarski established rigorously is that any language that includes its own concept of truth as a predicate will inevitably face paradoxes. Confining truth to the metalanguage allows to dispel them.

Beyond the truth definition, the second element of Tarski’s model is the truth schema. It basically helps find the truth value of complex metalinguistic expressions by importing logical connectives and quantifiers from the object language’s syntax. For instance: \[”A \textrm{ and } B \textrm{ is true}” \leftrightarrow A \wedge B\], and so on. By using this schema recursively, the valuation of any metalinguistic sentence can in principle be reduced to multiple applications of the truth definition.

Marian David has called this approach “logical subatomism” as a way to highlight the filiation with Russell and Wittgenstein’s logical atomism. They designed a form of truth schema that allowed to walk back from complex expressions to atomic ones but they still promoted correspondence with facts and states of affairs. By relating the truth of atomics sentences to the satisfaction relation between two objects, Tarski could be said to have “split the atom”.

The system we have been building is very well-suited to Tarski’s theory of truth. It has been shown in Zalta’s work that \([\lambda \phi] \leftrightarrow \phi\) is a theorem. We can identify \([\lambda \phi]\) to truth-bearing propositions, that is to elements from the metalanguage, and \(\phi\) to the truth-making formula from the object language, landing exactly on Tarski’s general truth definition. The dual copula means that we have two kinds of atomic satisfaction relations: exemplification and encoding. As to the recursive truth schema, it can also be proved from the axioms.

Meaning: is linguistic meaning exhausted by reference?

Position: Semantic distributionalism

It is now time to reach the last step on the semantic ladder, one that is essential to many of the disciplines and concepts we shall consider. Essential, in fact, to any thought we think, any sentence we utter. That piece is our theory of meaning. Let’s first remind ourselves of the building blocks we’ve presented so far in the "abstract realm" section:

• Carnap’s modal semantics describes meaning in terms of intensions: functions from possible worlds to extensions. “Macron” and “the French president” have the same extension (or referent) but different intensions: “Macron” could only be the same human, in any possible world, whilst “the French president” could have been Alain Juppé. Hence these singular terms have different meanings.

• Russell’s structural semantics is more fine-grained. It helps explain our intuition that co-intensional terms like “\(28 + 32\)” and “\(60\)” appear to have different meanings nonetheless. It posits that complex expressions have meanings derived from their syntactic structures: their meaning is not the intension of the sentence, but a function of its elementary units’ intensions.

• The semantics of names hopes to explain why co-referential, co-intensional terms with no underlying structure may have different meanings, like “Superman” and “Clark Kent”.

  • Frege assumed that meanings are carried by senses, which determine reference. Although they point to the same referent in every world, “Superman” and “Clark Kent” would have different senses in virtue of the speaker or thinker’s being in a different cognitive state.

  • Russell thought that most names were disguised descriptions rather than elementary units in an expression. If Superman is a disguised way to say “the flying, Kryptonian protector of Metropolis”, it is naturally different in meaning from “Clark Kent”.

  • Kripke denied that a more fine-grained theory was possible: since "Superman" and "Clark Kent" co-refer in every world (to be precise, in every world where they refer at all), then they have the same meanings. The clear cognitive difference between "Superman is Clark Kent" and "Clark Kent is Clark Kent" must then find a ground outside of semantics — most likely in pragmatics or psychology.

Now, if one ponders these options for the first time, it is tempting to dismiss Kripke’s contribution. It appears to impose an arbitrary cut-off on the granularity of meaning. Intuitively it’s an attractive proposition that meaning, what we mean, is individuated by some descriptive or mental load that’s intrinsic to the language user’s intention. The notion that "Superman" and "Clark Kent" may have the exact same semantic value is enough to separate Kripke’s idea from layman notions of meaning as a sort of dictionary definition.

It is then important to re-state Kripke’s argument as clearly as possible, showing in simple terms why there’s no escaping it.

Meaning, modality, reason: the golden triangle

I’ll start with an image I’m borrowing from David Chalmers, which says well enough what has been lost: the golden triangle of semantics. Pre-Kripke, the semantic vulgate could accommodate a belief in both Carnap’s modal approach and some version of a Frege-Russell descriptivism (which equated Fregean senses with complex descriptions). Each of those forms one side of a triangle whose vertices are meaning, necessity, reason. Let’s consider its sides in turn.

1. Meaning and necessity is the title of Rudolf Carnap’s most prominent work on semantics, so it should come as no surprise that his ideas about intension constitute the link between these concepts. His core idea is that two expressions with the same meaning must have the same intension: they must share the same reference in every possible world where they exist. The reverse is not enforced in the same way: "Superman" and "Clark Kent" have the same intension, but perhaps different meanings. Here are equivalent ways to expound this idea:

   • Same meaning \(\implies\)Same intension

   • There is no difference in intension without a difference in meaning.

   • \(\texttt{"A"}\) means \(\texttt{"B"} \implies\) Necessarily, \([A]=[B]\)

   • Possibly, \([A]\neq[B] \implies \texttt{"A"}\) doesn’t mean \(\texttt{"B"}\)

   • Necessity supervenes on meaning

2. Gottlob Frege’s concept of meaning as sense provides the connection between meaning and reason. Frege is not explicit about what type of object senses are, although he’s adamant that they are neither physical nor mental entities, and that they belong to a third realm. However, sense is characterized and individuated via a concept of ideal cognition we could call reason. Frege explains that where there is a difference in sense between two terms, there must be a difference in the cognitive significance of expressions, when we substitute one term for the other. The reverse is not obviously true: cognitive significance is poorly defined, but it seems like it could be infinitely nuanced. And semantics is only one of its ground, alongside pragmatics, context, etc. So imposing that a difference in cognitive significance entails a difference in sense could lead to a dangerously fine-grained (and ultimately unusable) semantics.

   • Same cognitive significance \(\implies\)Same sense

   • There is no difference in sense without a difference in conceivability.

   • \(\texttt{"A"}\) doesn’t mean \(\texttt{"B"} \implies\) Conceivably, \([A]\neq[B]\)

   • A priori, \([A]=[B] \implies \texttt{"A"}\) means \(\texttt{"B"}\)

   • Meaning supervenes on reason

3. The link between reason and necessity is provided by Imanuel Kant’s ideas about the a priori. A statement is a priori if an idealized thinker could determine its truth using reason only, at the exclusion of any empirical observation. Kant thought that if something is a priori then it is also necessary (true in every world), and vice versa. Conversely, if something is conceivable (i.e. if it is not a priori that it is false), then it is also possible (true in at least one world), and vice versa.

   • A priori, \([A]=[B] \iff\) Necessarily, \([A]=[B]\)

   • Conceivably, \([A]\neq[B] \iff\) Possibly, \([A]\neq[B]\)

   • Apriority is equivalent to necessity

   • Conceivability is equivalent to possibility

At first sight, this triangle is a beautiful achievement. After all, Carnap didn’t seek to undermine Frege: on the contrary, he wanted to explain senses using modal concepts. Even at the time, it was apparent that the variety of modality he used was conceptually distinct from apriority, yet Kant provided a perfect bridge between the two notions.

Unfortunately, the triangle had within it the seeds of its own downfall. It turns out that Kant’s imprudent equivalences jeopardized the whole edifice. Remember that there is some asymmetry in the way we’ve stated Carnap and Frege’s positions: necessity supervenes on meaning, but they are not equivalent; meaning supervenes on reason, but they are not equivalent. These two propositions together lead, transitively, to the idea that necessity supervenes on reason, which is consistent with the Kantian credo. But Kant imposes equivalence on these concepts. That belief entails that reason must supervene on necessity. So all implications in the above statement do end up being revertible, and become equivalences themselves! Rather than a triangle we then have a conjunction between meaning, reason and necessity: semantic value, cognitive value and modal value are then one and the same at heart.

This warrants a fair amount of discomfort, but Kripke cut through the knot by disproving Kant’s posits directly through examples. In what follows we will then operate a distinction between necessity and reason, or rather betwen necessity and apriority. The latter formulation makes it clearer that we’re dealing with idealized cognition: something could be inconceivable to Bob but quite conceivable to Alice (and thus, to an ideal being); Carl could find it hard to prove Fermat’s theorem using only reason, yet it is still true a priori.

The triangle, broken

Since we’ll be making that distinction, I’d like to introduce some notation to avoid ambiguities while still highlighting the parallels. We will say that:

• \(\square \phi\) indicates metaphysical necessity: \(\phi\) is true in every metaphysically possible world.

• \(\Diamond \phi\) indicates metaphysical possiblity: it is false that non-\(\phi\) is necessary (\(\Diamond \phi \equiv \neg \Box(\neg \phi)\) ).

• \(\blacksquare \phi\) indicates apriority: \(\phi\) is true to an ideal rational being, regardless of empirical experience.

• \(\blacklozenge \phi\) indicates conceivability: it is false that non-\(\phi\) is a priori (\(\blacklozenge \phi \equiv \neg \blacksquare(\neg \phi)\) ).

Let’s come back to Kripke’s arguments.

A posteriori necessity

Contra Kant, it shows that there are truths that can be both necessary and a posteriori. Consider these statements about the French author Aurore Dupin who’s known fame under the pen name George Sand.

• Could George Sand and Aurore Dupin possibly have been different humans? Mind you, this is fairly ambiguous, if asked in a common conversation. One could be tempted to answer: "sure, another person could have called herself George Sand, while Aurore Dupin picked another pen name"... But that’s missing the point, since our question wasn’t about word-types (sequences of symbols that we can utter, write or think), but about a singular term as a semantic unit.

  In this case, I believe that formalism is not a hindrance and can be a saving grace indeed. I formalized the satisfaction of a denotation relation as \[\mathbf{D}\texttt{"AD"}[a]\] \(\texttt{"AD"}\) is the name "Aurore Dupin" and \([a]\) is the ordinary object (kind: human being) representing the flesh-and-blood woman. So what we formally mean by "Aurore Dupin and George Sand are the same person" is: \[\mathbf{D}\texttt{"AD"}[a] \wedge \mathbf{D}\texttt{"GS"}[a]\]

  Could it have possibly been false? No, because ordinary objects are defined in such a way that they conserve trans-world identity, hence the denotation relation holds at all worlds. Kripke said that names designate rigidly. So, since "Aurore Dupin" denotes \([a]\) at all worlds and "George Sand" denotes \([a]\) at all worlds, they are necessarily the same.

• Could George Sand and Aurore Dupin conceivably be different humans? Of course! Now we’re talking about reason. If a perfectly rational being was never told or didn’t find out empirically that George Sand was Aurore Dupin’s pen name, they would have no way to conclude that it’s the case, and could hold a different belief. That both names denote the same human is then a posteriori.

It’s interesting to note that a posteriori necessity is throwing a wrench into Kant’s equivalence, but is quite consistent with the other two sides of the triangle. Combining Frege and Carnap, we find that apriority entails necessity. That isn’t inconsistent with necessity being a posteriori. Alas, this argument was not the end of Kripke’s undermining.

Contingent a priori

Some truths can be both contingent (they could have been false in different worlds) and knowable a priori. I will say straight away that to me, contingent a priori truths are always related, albeit perhaps in an implicit manner, to indexicality. Indexicals are terms whose reference is dependent on the context in which they are used, like "I", "here", "now", "this table", "my house", etc. It is often found that examples of contingent a priori sentences that are ostensibly non-indexical can host "hidden indexicals" in their midst. But for now we can consider a simple example like "I am here".

• I (Jonathan) just wrote that sentence in London. The way to analyze its reference would then be something like: \[\mathbf{D}\texttt{"I"}[jon] \wedge \mathbf{D}\texttt{"here"}[london] \wedge \mathbf{L}[jon][london]\] where \(\mathbf{L}\) is some ’ordinarized’ relation between an object and a location. Using our theory of truth, we can deduce that \[\mathbf{D}\texttt{"I am located here"}\top\]: the sentence expresses a true proposition. But the fact that \(\mathbf{L}\) is satisfied is obviously contingent: Jonathan could have been somewhere else. Hence the sentence "I am here" may always be actually true (that is, true in the context of utterance), but not necessarily so: it is a contingent truth.

• The sentence is also a priori: anyone who understands the rule of the language can know that it is true without any empirical input. If I woke up locked in a dark box, I wouldn’t know where I am, but I would still know that "I am here".

If we accept that some truths are contingent and a priori, that’s enough to break the link between Carnap’s and Frege’s views, at least in the way I’ve described them, since their combination entails that a priori truths must be necessary.

Now, these arguments can seem hard to grasp if they are new to you (they were to me). On top of that, the historiography of the topic is fairly confusing, so I’m insisting on some points which a cursory tour of the literature could lead anyone to misconstrue. In particular, this one: Kripke’s work did not invalidate Fregean senses as a concept. Kripke did successfully weaken the case for descriptivism in the sense of Russell’s disguised descriptions. But Frege’s theory of sense is much broader than descriptivism, which did have the advantage of defining the elusive senses, but in so doing lost its generality. Because Frege’s senses were so vague and essentailly sui generis, they could shoulder Kripke’s challenges.

What Kripke’s work implies is less strong but still of high import: that we can’t have Fregean senses and Carnap’s modal semantics at the same time: that’s the golden triangle broken.

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 "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”