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.
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
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
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.