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.