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...