A meagre existence

I would like to consider the Vulcan example once more, explicating the argument that 'Vulcan does not exist' in Frege-Russell terms.  We start, I think, with some background propositions on Newtonian celestial mechanics and observations of the orbit of Mercury.  We don't have to make these explicit, suffice it to say that the orbit of Mercury is anomalous and hence it is hypothesised that there is exists a planet within the orbit of Mercury whose gravitation is responsible for the anomalies.  I think it's useful to note that this does not say that one of the planets lies within the orbit of Mercury. This is in flat contradiction to the fact that the planets are Mercury, Venus, Earth, Mars,..., which is an extensional claim.  Instead the claim is that the concept 'intra-Mercurial planet responsible...', which we could call LeVerrier(), is instantiated.  In FR

∃x. LeVerrier(x),                                                           (H)

that is, there is an object that satisfies the LeVerrier() predicate.  Under this hypothesis the rules of deduction within FR allow us to introduce a new name, 'Vulcan', say, for this object, and hence we have

LeVerrier(Vulcan).

At this point it's worth considering the import of the statement 'Vulcan exists'.   In my view it neither extends what we have already nor contradicts it.  It simply conveys no new information at all, and need not be said.  Hence, perhaps, there is no FR notation to express it.  All the logical work is done by the name 'Vulcan', legitimate use of which itself rests on the hypothesis H.

The argument then continues along these lines.  From the experimental fact that no such planet has been observed we infer that the diameter of Vulcan is less than a certain length d₁, say.  And from Newtonian mechanics, and a reasonable assumption about the typical density of planetary matter we infer that in order to affect the orbit of Mercury in the prescribed way the diameter of Vulcan is greater than a second length d₂, say.  When we do the calculations we find d₁<d₂.  We thus reach a contradiction: by the transitivity and asymmetry of <, the diameter of Vulcan is both less than and not less than d₂, say.

So now we have a second anomaly.  The hypothesis put forward to dissolve the anomaly in the orbit of Mercury is itself anomalous.   In this situation we look for an escape route.  Perhaps we can explain why a larger planet might still be unobserved.  Or why a smaller one could account for Mercury's peculiarities.  Perhaps Newtonian celestial mechanics isn't quite right.  But if we cling to Newton it seems we have no choice but to abandon hypothesis H.  And this is what we mean by saying that Vulcan does not exist:  The existential claim that legitimates use of the name 'Vulcan' is denied.

Now this is a very specific example.  What about the general run of singular existential statements, like 'Socrates exists'?  My claim is that we are in an analogous position.  At some point in our past some authoritative figure or author has told us 'In fifth century BC Athens there was a philosopher called Socrates', or something similar.  This general existential claim introduces a new individual to us and gives us a name to use to refer to him in talk with ourselves and others.  Later we may learn more about this ancient figure.  Unless there is a suggestion that there may have been more than one Athenian called Socrates that it might be worth us knowing about, everything we are told about the man called Socrates we ascribe to just the one individual.  To be told 'Socrates really existed' is to invite the reply 'I never doubted it'.   But it seems that everything we know about the historical Socrates comes down to us in the work of Plato, Xenophon, and 'fragmentary remains of some other authors (eg, Aeschines)', according to the Oxford Companion to Philosophy.  So it doesn't require too great a stretch of the imagination to suppose that Plato may have invented Socrates as a convenient vehicle for expressing certain, perhaps unpopular, philosophical ideas.  And Xenophon may have followed suit.  If the evidence we have had been much less, and this could easily have been the case, then the discovery of evidence that Plato had indeed made up Socrates would be overwhelming.  In these circumstances we would be forced to revoke the assertion that there was an ancient Athenian philosopher called Socrates who did and said just what Plato tells us.  And then we would say 'Socrates didn't exist'.  But maybe there was another Socrates whom Plato was unaware of who did do all those things.  So Socrates didn't exist but another Socrates did.

To summarise:  there is an important asymmetry between singular existential assertions and denials.  If the name 'Vulcan' has been properly introduced by a general existential assertion then 'Vulcan exists' tells us nothing new.  In contrast,  'Vulcan doesn't exist' amounts to a denial of the general existential statement by which the name was introduced to us.   On the other hand,  if the name 'Vulcan' has not been properly introduced then 'Vulcan exists' is meaningless to us.  All we can do is shrug our shoulders.  Likewise, in similar circumstances, 'Vulcan does not exist' invites a blank stare.  Both statements are 'conversationally ill-formed'.  This account seems to give precedence to general existential statements over singular ones, at least from the point of view of logic and language, of saying and inferring what there is.   So I remain a Fressellian.   But no doubt the individuals of whom we talk are prior to language---or at least the non-fictional ones are.