Kripke's paper contains some notation and technical terms that are probably unfamiliar to most of you. I didn't mention these earlier since they largely occur in section 2, which should just be skimmed or skipped since he's not really addressing the main issue there (i.e., whether the considerations in Donnellan's paper refute Russell's theory of descriptions). But I thought I'd provide some comments on notation and technical terms here, in case you're interested. (We'll be talking at greater length about some of this later, but it's not very crucial right now.)
First, a familiar one:
Backwards 'E' should be read as 'there exists a' or 'there is a' or 'some'. It is called 'the existential quantifier'.
I also mentioned that sometimes '&' is used for conjunction, and sometimes '^' is used for conjunction. Kripke uses the latter.
Similarly, sometimes '->' is used for a conditional, and sometimes a sideways horseshoe is used. Kripke uses the latter.
Upside down 'A' does not occur in Kripke's paper. It's read as 'for all' or 'all' or 'every'. It is called 'the universal quantifier'. It is sometimes expressed using upside down 'A', and sometimes it is understood without being explicitly written. That's (unfortunately) how Kripke is using it in the first example of notation in his paper.
So look for instance at page 384 under 'Preliminary Considerations'. Go to the parenthetical remark '(I.e., (Ex)(phi!(x)^ psi(x)), where "phi!(x)" abbreviates "phi(x) ^ (y)(phi(y) -> y = x")'. First, a note on the notation: the occurrence of '(y)' to the immediate right of the '^' should be read as 'for all y'. So here the universal quantifier is being understood without being explicitly written. So 'phi!(x)' basically says that there is a unique phi, since what it abbreviates should be read as 'x is phi and for all y, if y is phi then y is identical to x'.
'De dicto' is a Latin expression that is often used in philosophy of language. It means roughly what 'of the proposition' means. 'De re' is Latin too. It means roughly what 'of the object' means. The distinction can be gleaned from Kripke's remarks and we'll talk more about it later so I won't belabor it too much now. But the difference is one of scope, which we talked about in connection with Russell's theory. We pointed out that on Russell's view, descriptions can take different scopes with respect to negation, for example. So contrast (i) and (ii):
(i) It's not the case that the present king of France is bald.
(ii) The present king of France is a non-bald thing.
In (i), the negation has wide (sometimes called 'large') scope over the description, so a proper representation of its logical form should reflect that. We can mark the same distinction by saying that the description has narrow (sometimes called 'small') scope with respect to the negation. In (ii), the description has wide (large) scope over the negation. So in (ii) the negation has small (narrow) scope with respect to the description.
On Russell's view, these scope distinctions make a difference in the truth conditions of the sentences. (i) is true on Russell's view, since it's false that anything is both the unique king of France and bald. (ii) is false on Russell's view since for it to be true there would have to be a unique king of France that belongs to the category of non-bald things.
The notion of scope does not only apply to descriptions and negations, however. We can apply it to other expressions as well. Kripke's doing this when he talks in section 2a about the behaviour of descriptions in modal or intensional contexts. Metaphysical modality has to do with what's possible and what's necessary. Modal expressions include 'necessarily', 'possibly', 'it's possibly the case that', 'it's necessarily the case that', etc. Kripke points out that 'The number of planets is necessarily odd' is subject to a scope distinction: the description can take wide scope with respect to the modal expression, or vice-versa. See Kripke's explanation of the difference.
Similarly, propositional attitude verbs like 'believes' are sometimes said to create intensional contexts. Sentences that contain descriptions along with attitude verbs are likewise subject to scope ambiguities. Consider Kripke's example: 'Jones believes that the richest debutante in Dubuque will marry him'. There are two ways this may be true: Jones may have a belief that he would report in English by saying 'The richest debutante in Dubuque will marry me'. That is, he bears the relation expressed by 'believes' to a proposition that has the property of being the richest debutante in Dubuque as a constituent. This is the de dicto reading; the attitude verb takes wide scope over the description. But the sentence may also be true because he believes of the richest debutante in Dubuque, whether he thinks of her that way or not, whether he knows she is the richest debutante in Dubuque or not, that she will marry him. This is the de re reading; the description takes wide scope over the attitude verb.
If this is confusing, don't worry. We'll talk more about it later. It's not super crucial to our main interests in the Russell-Donnellan-Kripke dispute.
Back to modality: Kripke says that the claim that the actual number of planets (nine) has the property of necessary oddness is true according to "essentialists like him". First, ignore the flap over Pluto. For what it's worth, there are already claims to have discovered a ninth planet under the new definition of 'planet'. But let's just ignore all that. The more important thing is what Kripke means by 'essentialist'. To be an essentialist is to hold that objects have non-trivial necessary properties. So here's a trivial necessary property: being such that if dogs bark, then dogs bark. We all have that property. Everything does. And we couldn't have failed to have it since it is necessarily true that if dogs bark, then dogs bark. The idea here is that 'if dogs bark, then dogs bark' is a logical truth. (It's symbolized in propositonal logic as 'P -> P'. Using truth tables for '->' that we discussed at the beginning of the term you can verify that any sentence of this form must be true.) So trivial necessary properties of things are properties that all things have and that all things must have, as a matter of logic. Non-trivial necessary properties can be defined in terms of this: to be a non-trivial necessary property is to be a necessary property (a property a thing must have) that is not had trivially, as a matter of logic. So consider yourself. You could have been different from the way you are. You could have worn all purple today, for instance. But are there things about you that could not have been different, and not just as a matter of logic? Could you have failed to be human, for instance? Or an organism of some sort? If you think you could not have failed to be an organism, then you are an essentialist: being an organism is, according to you, a non-trivial necessary feature of you. So that's essentialism. (More on that later too.)
On pp. 388 Kripke introduces some semi-formalized sentences to illustrate his point about scope: the sentences labeled '(2a)', (2b)', and '(2c)' on the bottom of the page. Those sentences contain some boxes and diamonds. The box abbreviates 'necessarily' or 'it is necessarily the case that'. The diamond abbreviates 'possibly' or 'it is possibly the case that'.
Annoyingly, Kripke introduces another symbolization for definite descriptions on pp. 389 in the first sentence of section 2b. The upside down iota abbreviates 'the'. So the symbolism in that sentence should be read as 'the x such that x is phi' or 'the phi'. It is common to use upside down iota to formalize definite descriptions. I mentioned this briefly in class.
Next Kripke introduces the notion of a rigid definite description. An expression is rigid if and only if it refers to the same thing in every counterfactual situation in which it refers to anything. So 'the even prime number' is rigid since it denotes the number 2 in any possible world state. Contrast this with 'the tallest student in FPL'. Suppose this actually denotes Travis. Would it still denote Travis if things had worked out differently? Presumably not; it depends on which counterfactual circumstances we consdier. If Shaq had enroled in FPL, it would denote Shaq and not Travis. Or if Travis had dropped, it would not have denoted him. So 'the tallest student in FPL' is not rigid since in some counterfactual situations it would not denote Travis. We'll talk more about this later on as well.
As I said, none of this is crucial to our main interests in Kripke's paper. So don't let this stuff freak you out. No need to master it in order to get the main point. My purpose here was just to introduce some of the notation and technical terms Kripke uses in the event that you're interested in decoding the difficult section 2 of Kripke's paper.