In a model a name is a pointer to an individual, so two names for one individual make an identity statement true. Yet “Hesperus is Phosphorus” was a discovery. Evaluate it in a model, see what the model cannot show, and use Frege's distinction between sense and reference to say what is missing.
Core ideas
The last lesson let a sentence be evaluated in a model. This one asks what a name contributes to the value, and finds that it contributes less than its meaning.
The model says which individual each name picks out, and a sentence containing the name is evaluated using that individual and nothing else. Two names that pick out the same individual make the same contribution to every value.
“Hesperus is Hesperus” is known to anyone. “Hesperus is Phosphorus” had to be found out. In the model both are true, for the same reason, and the model cannot tell them apart.
Frege's answer is that a name has a sense as well as a referent: the evening star and the morning star are two ways of being given one planet. The model records the planet, so it records the reference and leaves the sense out.
The material
The model that makes the identity true, the model that makes it false, and what neither of them records.
The reference of a name is the individual it picks out. Its sense, on Frege's view, is the way that individual is presented, the information that goes with using the name. Two names can share a reference and differ in sense.
The old example is a pair of names. People saw a bright light in the evening sky and named it Hesperus. They saw a bright light in the morning sky and named it Phosphorus. It was later established that these were one planet, Venus. Let h be the name Hesperus and p the name Phosphorus.
A model for this has two individuals, a for the planet and b for something else, say another planet. The predicate Shines holds of a alone. The names are given referents: h picks out a, and p picks out a. Evaluate h = p. The sentence is true when its two terms pick out the same individual; they both pick out a; the lab prints True.
The model also makes ∀x (x = h → x = p) true: whatever is Hesperus is Phosphorus. And because the value of a sentence is built from the referents of the names in it, you can put p where h stood, anywhere, and the value cannot move. Shines(h) and Shines(p) are true together or false together.
Now give p the referent b. The identity is False, and the uniqueness sentence fails at a: the individual a is Hesperus but is not Phosphorus. This model is just as easy to write down as the first. It is a perfectly good description of how things might have been, and for the early astronomers it was a live candidate for how things were. Learning that h = p was true was learning which of the two models described the sky.
So the model contains the discovery in one sense and misses it in another. It contains it as the difference between the two referent assignments. It misses why an astronomer could sincerely say that Hesperus is visible in the evening and wonder whether Phosphorus is, which no sentence the lab evaluates can express once h and p share an individual. Frege's account is that the two names come with different senses, and that the astronomer's belief goes by sense. The lab has no senses to hold and no beliefs to report, and the lesson does not pretend it does.
Putnam's Twin Earth shows the same gap from the other side. Take the sentence “water is H2O”, written ∀x (Water(x) ↔ H2O(x)). In the first model the individual a is water and H2O, and b is a different liquid, XYZ. The sentence is True. In the second the word Water is applied to the stuff around the speakers there, which is XYZ, so its extension holds b and the sentence is False; the lab names a as the first place it fails, since a is H2O and not water there. The sentence is the same string. What the speakers have in their heads can be the same on both planets, and Putnam's claim is that the meaning of the word is not in the head alone. The lab shows only that extension is fixed by the model, not by the sentence.
The corrective to take away is narrow. Two names with the same reference give the same value in every sentence the lab can evaluate. That does not make them the same name, or give them the same meaning in the sense of the word that explains why “Hesperus is Phosphorus” was news.
Interactive
A first-order sentence evaluated over a domain you can edit
Method
Five steps for any identity statement that looks like news.
Say in the model which individual each name picks out. An identity statement is true exactly when its two terms pick out one.
Read the value. A True here means the names are co-referring; it does not say the names mean the same.
Replace one by the other in a sentence the model can evaluate. If the names share a referent, the value cannot change.
If a speaker could accept one sentence and doubt the other, the difference is not in the reference. It is in the way the referent is given.
Name the sense, or the ignorance of which model is actual, that the table has no place for.
One sentence in two models, and a second sentence in each.
In the first model the names are interchangeable everywhere the lab can look, and still “Hesperus is Phosphorus” is not the trivial sentence that “Hesperus is Hesperus” is. That is Frege's point: the extra is the sense, and the model does not hold it.
Given two names and a model, you should be able to say whether their identity is true, show that swapping them cannot change a value, and say in a sentence what an informative identity carries that the model does not.
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