Kripke's Naming and Necessity

[Wyman]

I will break Kripke's main ideas down one at a time and see if anyone is interested in discussing them.

First, the modal problem. Most philosophers would agree that x=x is a necessary truth. The question is, given that y=x, does it follow that y necessarily equals x? It does seem to follow directly from the first identity by the rule of substitution. Let 'x necessarily equals x' be written #(x=x). Since we assume that x=y, we just substitute y for x and get #(x=y). So it would seem necessary that, if x=y, then #(x=y).

However, there seems to be a problem here. Some identity statements seem to be contingent and not necessary at all. Example - Venus can be seen in the morning in one location and then at night in another. Before modern times, it was thought that these two bright objects in the sky were two different stars. The one was called the morning star and the other the evening star. The Greeks (or some ancient peoples) called the former Phosphorus and the latter Hesperus.

Now we know that Hesperus=Phosphorus (and both are just Venus) is a true statement. Given what we said above about the necessity of identity statements, it must then be necessary that Hesperus=Phosphorus (or #(Hesperus=Phosphorus)). However, as Quine and others pointed out, such identity statements are contingent and not necessary. For it was only after the empirical discovery that both were in fact Venus that we came to know the identity statement was true.

Kripke goes on to attempt a solution to this problem which, after a lengthy and groundbreaking analysis, eventually leads to an interesting take on the mind/body problem.

[odysseus]

A little short on Kripke here. Anyway, if I remember how this goes, the problem lies with what is meant by "equals". If one term can serve as an unqualified substitute in all conceivable cases for another, then logical identity is established. But in the world differences extend beyond mere reference. Venus may be both the morning and evening star, but the morning star is not the evening star given simply that morning is not evening. You could argue that these are incidental differences, but then, you would have to disambiguate what "equals" means. This is what happens when you use think about the concept of things being equal in the strict logical sense of identity, then you make a move to the world where names are tossed around carelessly. The number 2: certainly 2=2=4, but 2 fingers of mine, which are designated by the general sense of '2' with no accommodation for the actual singularity, are certainly not equal to that of another person. It's just the numerical designation that is equal, the pure logical form of '2' as Kant would say.

[attofishpi]

Lucifer

[Speakpigeon]

However, there seems to be a problem here. Some identity statements seem to be contingent and not necessary at all. Example - Venus can be seen in the morning in one location and then at night in another. Before modern times, it was thought that these two bright objects in the sky were two different stars. The one was called the morning star and the other the evening star. The Greeks (or some ancient peoples) called the former Phosphorus and the latter Hesperus.

Now we know that Hesperus=Phosphorus (and both are just Venus) is a true statement. Given what we said above about the necessity of identity statements, it must then be necessary that Hesperus=Phosphorus (or #(Hesperus=Phosphorus)). However, as Quine and others pointed out, such identity statements are contingent and not necessary. For it was only after the empirical discovery that both were in fact Venus that we came to know the identity statement was true.

Simple... As the Greeks had it, Hesperus and Phosphorus didn't refer to the same thing and therefore didn't refer to Venus. So, it's definitely not a case of x = y.

And as to "contingent" and "necessary", in a deterministic universe everything is necessary. Contingently necessary. In a quantum world, everything is contingent. Necessarily contingent... Not exactly operational as a distinction.
EB

[Speakpigeon]

A little short on Kripke here. Anyway, if I remember how this goes, the problem lies with what is meant by "equals". If one term can serve as an unqualified substitute in all conceivable cases for another, then logical identity is established. But in the world differences extend beyond mere reference. Venus may be both the morning and evening star, but the morning star is not the evening star given simply that morning is not evening. You could argue that these are incidental differences, but then, you would have to disambiguate what "equals" means. This is what happens when you use think about the concept of things being equal in the strict logical sense of identity, then you make a move to the world where names are tossed around carelessly. The number 2: certainly 2=2=4, but 2 fingers of mine, which are designated by the general sense of '2' with no accommodation for the actual singularity, are certainly not equal to that of another person. It's just the numerical designation that is equal, the pure logical form of '2' as Kant would say.

Equality is defined between numbers and numbers are like words, conventional scribbling on a piece of paper or some physical support. What matters is that we should be able to make sense of the scribbling.
EB

[Logik]

First, the modal problem. Most philosophers would agree that x=x is a necessary truth.

It is not a necessary truth. It is an axiom that has stood for so long that nobody has bothered to challenge it, when it is in fact such a trivial and meaningless statement that it's not even necessary for a logic-system to function.

As demonstrated [1] here A = A is false, while B = B is true: https://repl.it/@LogikLogicus/Identity

The question is, given that y=x, does it follow that y necessarily equals x? It does seem to follow directly from the first identity by the rule of substitution. Let 'x necessarily equals x' be written #(x=x). Since we assume that x=y, we just substitute y for x and get #(x=y). So it would seem necessary that, if x=y, then #(x=y).

The problem is in the ambiguous semantics of the "=" symbol.

When you compare a thing to itself you are comparing identity which is poorly defined notion. Identity should really means "uniqueness". X is itself and there is no other thing like it. This conflation of meaning leads to the following oxymoron in English: Two things are identical.

When you compare two things to each other you are comparing their value, not their identities. Naturally - to speak of two things means that you have already individuated them from each other.

The most trivial way to navigate around this error is like this:
Identity: x = x (A thing is the same as itself)
Equality: x ⇔ y (two things are materially equivalent)
Falsity: x != y (two things cannot have the same identity)


### Notes
[1] There's another point of contention here. Some would say that the law of identity is mandatory in a consistent logical system, and that what I have contrived is a para-consistent logic so I am "cheating". I think this boils down to definition again. What is a "consistent system" exactly? One that adheres to the classic laws of logic?!?! That's a circular argument. A consistent system is one which doesn't trip over the principle of explosion. If I can introduce "A = A => False" then in locality that may be frowned upon, but on the whole the logic-system doesn't blow up - then it's consistent. Q.E.D Python.