Thursday, September 6, 2012

Expunging Plato: Necessity and Naming.

In the philosophical world, the most commonly accepted theory of naming seems to be Kripke's rigid designators (as first laid out in Naming and Necessity). I think Kripke is wrong. Further I think Kripke's views, in his own hands, and others, have lead to numerous fallacies. Reproduced below is an adaptation of an email I sent my Phil of Language classmates+prof introducing my objections to rigid designation. 

Suppose that “Aristotle is the teacher of Alexander” is necessarily true so long as “the teacher of Alexander” is an element of the descriptive definition of 'Aristotle'; and suppose, following convention, we may formalize “necessarily 'Aristotle is the teacher of Alexander'” as: □(∀x)((∀y)(Fy ≡ x=y) ⊃ Fx) where F() predicates "is the teacher of Alexander". That is: "necessarily: for all things x, if x is the unique teacher of Alexander, then x is the teacher of Alexander"; note the bracketing carefully. As it happens, Aristotle is in fact the unique teacher of Alexander. Informally: necessarily, whichever thing is the teacher of Alexander, is the teacher of Alexander.
Similarly we may symbolize “Aristotle is necessarily the teacher of Alexander” as: (∀x)((∀y)(Fy ≡ x=y) ⊃ □Fx). That is: "for all things x, if x is the unique teacher of Alexander, then it is necessary that x is the teacher of Alexander"
Where I break with (seeming) philosophical consensus (and my last semester's Phil of Language class), is that I think that not only is the former formalization a true proposition, but so is the latter. Consider what □Fx means. By definition a proposition is necessary iff it is true in every model of the language. The first sentence in question encodes a logical tautology: “Fx ⊃ Fx”, “A is A”. So there is no consistent model under which it evaluates as false, and thus we accept that it is a necessary truth.
The second sentence I concede intuitively appears to not be a necessary truth. It seems like Aristotle, the x which is infact Alexander's teacher, could have been not-the-teacher-of-Alexander. But I hold that this intuition is mistaken, and that the mistake results from a subtle equivocation, in the plain English form: an equivocation on the name “Aristotle”, in the formal version: an equivocation on the variable x!
We agree that the first sentence is necessary exactly because the descriptive cash-out of “Aristotle” includes “is the teacher of Alexander”. But when we feel that it is not the case that Aristotle is necessarily the teacher of Alexander, the “Aristotle” we are using no longer includes “is the teacher of Alexander” in it's cash-out. Instead we hope to point to the guy who we're usually talking about when we say “Aristotle”, and we hope that this reference can be held across possible worlds. This is fine, we can simply define “Aristotle” by all of it's usual descriptive terms, less “is the teacher of Alexander”. Then, as intuition suggests, it is not true that “Aristotle is necessarily the teacher of Alexander”. But that is only because we are using a different definition of “Aristotle”, strictly speaking: we are using—in an argument—the same token for two different words, the very definition of equivocation.
Formally: let F() be a predicate meaning “() is the teacher of Alexander”. Then (∀x)(~□Fx); for this to hold, we need only to accept that two distinct entities each could possibly have been the one teacher of Alexander. And so it appears that ~[(∀x)((∀y)(Fy ≡ x=y) ⊃ □Fx)], since (∀x)(~□Fx). What does it mean for a universally quantified implication to be false? Simply that there is some case where the antecedent is true while the consequent is false. Suppose ~[(∀x)((∀y)(Fy ≡ x=y) ⊃ □Fx)], then (by negation of a universal, equivalent reformulation of the implication, de Morgan, and a double negation elimination): (∃x)((∀y)(Fy ≡ x=y) ∧ (~□Fx)). Then by the logic of necessity there is some model on which ~Fx, since ~□Fx. But x is not a free variable, x is bound by a quantified conjunction as being such that Fx, and there is no consistent model, of any modal logic, on which Fx∧~Fx, thus □Fx for this particular way of binding x! This is a contradiction, hence it is not the case that ~[(∀x)((∀y)(Fy ≡ x=y) ⊃ □Fx)], and it is the case that (∀x)((∀y)(Fy ≡ x=y) ⊃ □Fx), Q.E.D.
I have similar arguments to give to the effects that: “Hesperus is Phosphorus” is not necessary, if it is not a priori;
 “I am here now” is necessary, if it is analytic; and “All cats are animals” is necessary and a priori, if it is analytic!

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