Collective Inconsistency and Plural Predication

We often say things like

1. The propositions p, q, r are inconsistent.

Suppose, to keep things simple, that each of the three propositions is self-consistent.  It will then be false that each proposition is self-inconsistent. (1), then, is a plural predication that cannot be given a distributive paraphrase.  What (1) says is that the three propositions are collectively inconsistent.  This suggests to many of us  that there must be some one single entity that is the bearer of the inconsistency.  For if the inconsistency does not attach distributively to each of p, q, and r, then it attaches to something distinct from them of which they are members.  But what could that be?

If you say that it is the set {p, q, r} that is inconsistent, then the response will be that a set is not the sort of entity that can be either consistent or inconsistent.  Note that it is not helpful to say

A set is consistent (inconsistent) iff its members are consistent (inconsistent).

For that leaves us with the problem of the proper parsing of the right-hand side, which is the problem with which we started.

And the same goes for the mereological sum (p + q + r).  A sum or fusion is not the sort of entity that can be either consistent or inconsistent.

What about the conjunction p & q & r?  A conjunction of propositions is itself a proposition.  (A set of propositions is not itself a proposition.) This seems to do the trick. We can parse (1) as

2. The conjunctive proposition p & q & r is (self)-inconsistent.

In this way we avoid construing (1) as an irreducibly plural predication.  For we now have a single entity that can serve as the logical subject of the predicate ' . . . is/are inconsistent.'  We can avoid saying, at least in this case, something that strikes me as only marginally intelligible, namely, that there are irreducible monadic non-distributive predicates.  My problem with irreducibly plural predication is that I don't know what it means to say of some things that they are F if that doesn't mean one of the following: (i) each of the things is F; (ii) there is a single 'collective entity' that is F; or (iii) the predicate 'is F'  is really relational. 

One could conceivably object that in the move from (1) to (2) I have 'changed the subject.'  (1) predicates inconsistency of some propositions, while (2) predicates (self)-inconsistency of a single conjunctive proposition.  Does this amount to a changing of thr subject?  Does (2) say something different about something different?

Comments

[Phoenix Kyle MacGregor]

I understood from my introduction to logic, which I endured at the insistence of Peter Lupu, that it is precisely sets of propositions which are consistent or inconsistent. Isn't that so?

[Quinn]

It seems to me that consistency/inconsistency is a relation, and that

1. The propositions p, q, r are inconsistent.

is an elliptical sentence which fully spelled out should read

1*. The propositions p, q, r are inconsistent with one another.

Then the notion of (self)-inconsistency would be sort of parasitic on that primary notion.

2. The conjunctive proposition p & q & r is (self)-inconsistent.

can be read as

2*. The conjunctive proposition p & q & r is inconsistent with itself.

Or is there a problem with a proposition having a relation with itself?