Friday, June 01, 2012

Circularity and descent to singulars

Here’s a different argument for the circularity of the ‘thin conception’ of existence. First, some preliminaries. In scholastic logic (and in modern logic, for that matter) there is an explication of the truth conditions of general statements in terms of singular statements. The universal proposition ‘every man is running’ is explicated as a conjunction of singular statements ‘Socrates is running and Plato is running and .. etc’, where every man who exists is the referent of exactly one proposition in the conjunction. Likewise, the existential proposition ‘some man is running’ is explicated as a disjunction of singular statements thus ‘Socrates is running or Plato is running or .. etc’. This is called ‘descending to singulars’.

With that in mind, let’s consider the thin definition of the term ‘exists’:

(def) ‘Some American philosopher exists’ = ‘Some philosopher is American’

It’s evident that we can descend to singulars in either of two ways. We can take the defininiens statement on the right, and descend to all the singulars falling under ‘philosopher’. For example, suppose the only three living philosophers are Plato, Socrates and Quine. Then the statement on the right explicates to ‘Plato is American or Socrates is American or Quine is American’. Obviously the first two are Greek, so they won’t do, but Quine is American so the disjunction is true.

However, the statement on the left, the statement to be defined, the definiendum is more difficult. We have to take all the singulars falling under ‘American philosopher’, and form a disjunction of the form ‘a exists or …’. Since the only American philosopher in our domain is Quine, that gives ‘Quine exists’. But we can’t define ‘Quine exists’ in terms of any non-existential statement. The problem is that the thin definition of ‘exists’ only works for general existential statements. The defining propositions must have the form ‘some F is G’, which is general. There is no equivalent explication for singular existentials. Therefore the ‘descent to singulars’ proves that we cannot eliminate the term ‘exists’ from our discourse, and the thin conception is therefore circular.

Will that do? Well no, but more tomorrow.


Anthony said...

"But we can’t define ‘Quine exists’ in terms of any non-existential statement."

In context, you certainly can. By "Some American philosopher exists" you seem to mean that some American philosopher lives (since you were restricting your consideration only to living philosophers).

Edward Ockham said...

OK let's remove the word 'living' from that sentence. Let's suppose the only philosophers are etc.

Anthony said...

>> OK let's remove the word 'living' from that sentence.

Drat. :)

This seems similar to the first in this series from Mav. He dismissed the definition of "Quine exists" as "Quine is Quine". But IIRC, he used the notion of possible worlds to do so.