In addition to being interesting in its own right, this thesis will, if correct, make plural quantification available as an innocent but extremely powerful tool in metaphysics, philosophy of mathematics, and philosophical logic.
For instance, George Boolos has used plural quantification to interpret monadic second-order logic and has argued on this basis that monadic second-order logic qualifies as “pure logic”.
It is therefore both natural and useful to consider a slightly richer language: arguments?
Lots of English predicates work this way, for instance “… So if our primary interest was to analyze natural language, we would probably have to allow such predicates.
\] (That is, for any things, there is something that is one of them.) Let be the theory based on the language \(L_\) which arises in an analogous way, but which in addition has the following axiom schema of extensionality: \[ \tag \Forall\Forall [\Forall(u \prec xx \leftrightarrow u \prec yy) \rightarrow(\phi(xx) \leftrightarrow \phi(yy))] \] (That is, for any things\(_1\) and any things\(_2\) (if something is one of them\(_1\) if and only it is one of them\(_2\), then they\(_1\) are \(\phi\) if and only if they\(_2\) are \(\phi)\).) This axiom schema ensures that all coextensive pluralities are indiscernible. For ease of communication we will use the word “plurality” without taking a stand on whether there really exist such entities as pluralities.
Statements involving the word “plurality” can always be rewritten more longwindedly without use of that word.
In two important articles from the 1980s George Boolos challenges this traditional view (Boolos 19a).
He argues that it is simply a prejudice to insist that the plural locutions of natural language be paraphrased away.
A predicate \(P\) that isn’t distributive is said to be For instance, the predicate “form a circle” is non-distributive, since it is not analytic that whenever some things \(xx\) form a circle, each of \(xx\) forms a circle.
Another example of non-distributive plural predication is the second argument-place of the logical predicate \(\prec\): for it is not true (let alone analytic) that whenever \(u\) is one of \(xx, u\) is one of each of \(xx\).