2O. Again, he admits that the identification of heat with molecular motion and water with H
2O are both contingent a posteriori discoveries. However, this has no effect on their necessary identities. Indeed there's another contingent fact about these necessary identities. Kripke says that of course we can "imagine heat without molecular motion" and a mental state "without any corresponding brain state". None of this affects the necessary identities. (Note: Kripke believes that the identity of heat with molecular motion is necessary; though he doesn't think the same about the identity of a particular mental state with a particular brain state.)
The same is true of H
2O and water. There may be other examples of stuff that has the macro-qualities of water; though it would not thereby be water. In Kripke’s case, the macro-properties of water aren't the standard by which we determine or define water. That standard falls within the ambit of water’s micro-properties – that is, H
2O molecules. It is these micro-properties that make water a natural kind, not water’s macro—properties; which may, after all, be shared by other substances. Water is also H
2O whether or not we discover this to be the case.
Why doesn’t all this apply to mental states and brain states? Because mental states are defined exclusively in terms of their phenomenal qualities; unlike water. That is, if we come across phenomenal qualities that don’t coincide with particular brain states, then such mental qualities aren't necessary identical to such brain states. There is, however, a contingent identity between mental states and brain states. There is no distinction between macro- and micro-properties when it comes to mental states.
needn't go to another possible world to meet an alternative inventor of bifocals. In this world someone else might have invented bifocals. So "the inventor of bifocals" is a non-rigid (or flaccid) designator. It's non-rigid because the description refers to different persons at different worlds and might have referred to a different person even in our own world.
Why is “the square root of 25” the same as, say, “Tony Blair”? Again, the former appears to be in some sense descriptive; though the name “Tony Blair” doesn’t seem to be descriptive, at least not at a prima facie level.
Kripke gives his own example of necessary existents: mathematical entities. If we get back to rigid designations of non-necessary beings, such as Benjamin Franklin, then the name “Benjamin Franklin’ must designate Benjamin Franklin “in any possible world where the object in question does exist, in any situation where the object would exist”. Of course, if Benjamin Franklin didn't exist, the name would have no designation. It would have no referent.
Kripke is in essence emphasising the importance of objects rather than names. More than that, he's emphasising the essences of objects which make it possible for Tony Blair to exist at different possible worlds - even in those at which he has a totally different name. He'll still be the same object. Moreover, he'll still have the same essence.
Firstly he asks: "What do we mean by calling a statement necessary?" His answer is: Firstly, the statement is true. Secondly, "it could not have been otherwise". Contingent truth, on the other hand, is a matter of a statement being true; though it could have been the case that it isn't true. Kripke says that these are metaphysical issues. He then discusses a priori truth and says that such a thing "can be known independently of all experience". Because of the concern with our knowledge of these statements, they're assigned to the realm of epistemology. Questions of a priori truth are epistemological because they're concerned "with the way we can know certain things to be in fact true". As Kripke was well aware, traditionally it was thought that all necessarily true statements could be known a priori. Of course Kripke questions this assumption. In fact he offers his own alternative. Some things or statements may be necessarily true; though only knowable a posteriori (that is, our knowledge depends on experience). Kripke offers his own example: the Goldbach conjecture. This conjecture claims that every even number is the sum of two primes. Because this is a mathematical statement, it must be necessarily true (if it is true). However, the Goldbach conjecture isn't known a priori. Here Kripke qualifies the notion of the a priori. It's not simply a question of what is known independently of experience; but also what "can be known independently of experience".
Another addition to the a priori argument, in relation to Goldbach's conjecture, is that part of its - possible - truth would be our ability to prove it true if it were true. Kripke denies this too. It's been known since Gödel, Kripke argues, that within certain mathematical systems there's at least one theorem that's not provable within that system. So there can be no absolute and total guarantor of truth within any mathematical system. This means, again, that not all mathematical truths are provable. Therefore they certainly aren't known to be true a priori. (Gödel’s stance on mathematical systems may be applicable to systems of various descriptions outside of pure mathematics.)
P ⊃ P
To get back to the example that opened the paper, Kripke says that “one could not possibly have a situation in which…Hesperus would not have been Phosphorus”. So if both names rigidly designate the same object, say, Venus, then both names are necessarily identical.
Hesperus is Phosphorus.
“'Hesperus’ and ‘Phosphorus’ are names of the same heavenly body.”
Kripke gives an excellent example of what’s at issue here. Take the statement “2+2=4”. If we're talking names exclusively, this statement wouldn't be necessarily true, or perhaps even true at all. If we're talking about the accepted designations of these inscriptions, then the statement is necessarily true. Kripke elaborates. He says, “’2’ and ‘4’ might have been used to refer to two different numbers” (to the ones they do now refer). If the inscription ‘2’ referred to the mathematical object 3, then the statement “2+2=4” would be necessarily false. In this instance, “2+2=4” should be “2+2=6” because, again, the inscription ‘2’ refers to the object 3.
Take ‘heat’ and ‘the motion of molecules’. Both terms could be seen to refer to the same thing. That heat is the motion of molecules is a scientific fact. It is an a posteriori judgment. The motion of molecules isn't “contained in the concept” - as Kant would have put it - of [heat]. As Kripke put its, “scientific investigation might have turned out otherwise”. However, the discovery was indeed contingent or a posteriori; though the connection between heat and the motion of molecules is necessary. (Note: not between the names ‘heat’ and the desctiption ‘the motion of molecules’). Regardless of our knowledge, our words, etc., there's a necessary connection or identity between heat and the motion of molecules.