Why Abstract Propositions are of Utility, by way of Jabberwocky.

To illustrate the utility of abstract propositions, consider that Jabberwocky, in talking of ‘slithy toves’, references two fictional worlds, fw, at which a ‘slithy tove’ is a salamander, and fw’, at which a ‘slithy tove’ is an element absent from the inventory of the world of the text, iw.  In both cases, ‘slithy toves did gyre and gimble’ is true for fw at iw.  To delineate fw and fw’ let elements and the terms that reference them range across worlds separately. The domain of the term ‘slithy tove’ includes both fw and fw’, but at fw picks out a salamander, and at fw’ picks out the element unknown at iw. The term-element relation holds between ‘slithy tove’ and salamander at fw.  However, in order that this has meaning at iw, ‘salamander’ has been allowed to range across worlds as a term and an element together, contrary to the programme of separating them out.  This is a partisan approach, as if to say that the inhabitants of fw are misguided: a ‘slithy tove’ is properly called a ‘salamander’.

In order to be non-partisan about worlds it is proper to treat iw, fw and fw’ as worlds for which the terms ‘salamander’ and ‘slithy tove’ variously pick out two elements in an overlapping way. The  term ‘slithy tove’ has import at fw and fw’, but the two worlds are delineated in that the element referred to as a slithy tove at fw is also part of the inventory of iw. The relation between a term and an element is unique for each of the three worlds, so that they are separated out.  At this point, though, to name the elements we have to hand means falling back into undemocratic ways and again letting a term-and-element range across worlds. An inhabitant at iw might pick out fw as the world at which ‘slithy tove’ refers to the element referred to at iw as a ‘salamander’ and fw’ as ‘the world at which ‘slithy tove’ refers to an element not in the inventory of iw.  This is alright, catering to two worlds. But this sort of reference is cumbersome, and more so if more worlds, terms and objects are under consideration.  For alethic space entire the list that begins ‘the element referred to at iw as ‘salamander’, and at fw as ‘slithy tove’..’ is infinitely long.

Abstract propositions generalise this list and give a democratic account of terms and elements. Call p the element that at iw is termed ‘salamander’. Allow p to range across worlds, independently of any term. Let terms be elements of worlds that name other elements of those worlds. Abstract propositions are as good for elements we cannot describe as for those we can. The proposition q represents the unknown element to which the term ‘slithy tove’ refers at fw’. The propositions r and s represent the terms ‘salamander’ and slithy tove’. So iw = {p, r} and r names p, fw ={p, s} and s names p, and fw’ ={q, r} and r names q.

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