Does Any Noncontingent Proposition Entail a Contingent Proposition?

This post continues the discussion in the comment thread of an earlier post.  

Propositions divide into the contingent and the noncontingent.  The noncontingent divide into the necessary and the impossible.  A proposition is contingent iff it is true in some, but not all, broadly logical possible worlds, 'worlds' for short.   A proposition is necessary iff it is true in all worlds, and impossible iff it true in none.  A proposition p entails a proposition q iff there is no world in which p is true and q false.

The title question divides into two:  Does any impossible proposition entail a contingent proposition?  Does any necessary proposition entail a contingent proposition?

As regards the first question, yes.  A proposition A of the form p & ~p is impossible.  If B is a contingent proposition, then there is no possible world in which  A is true and B false.  So every impossible proposition entails every contingent proposition.  This may strike the reader as paradoxical, but only if he fails to realize that 'entails' has all and only the meaning imputed to it in the above definition.

As for the second question, I say 'No' while Peter Lupu says 'Yes.'  His argument is this:
1. *Bill = Bill* is necessary.
2. *Bill = Bill* entails *(Ex)(x = Bill)*
3. *(Ex)(x = Bill)* is contingent.
Ergo
4. There are necessary propositions that entail contingent propositions.

Note first that for (2) to be true, 'Bill' must have a referent and indeed an existing referent.  'Bill' cannot be a vacuous (empty) name, nor can it have a nonexisting 'Meinongian' referent.  Now (3) is surely true given that 'Bill' is being used to name a particular human being, and given the obvious fact that human beings are contingent beings.  So the soundness of the argument rides on whether (1) is true.

I grant that Bill is essentially self-identical: self-identical in every world in which he exists.  But this is not to say that Bill is necessarily self-identical: self-identical in every world.  And this for the simple reason that Bill does not exist in every world.  So I deny (1).  It is not the case that Bill = Bill in every world.  He has properties, including the 'property' of self-identity, only in those worlds in which he exists.

My next post will go into these matters in more detail.

Addendum 28 May 2011.  Seldom Seen Slim weighs in on Peter's argument as follows:

I believe your reply to Peter is correct. It follows from how we should define constants in 1st order predicate logic. A domain or possible world is constituted by the objects it contains. Constants name those objects. If a domain has three objects, D = {a,b,c}, then the familiar expansion for identity holds in that domain, i.e., (x) (x = x) is equivalent to a = a and b = b and c = c. But notice that this is conditional and the antecedent asserts the existence in D of (the objects named by) a, b, and c. Thus premise 2 of Peter's argument is actually a conditional: IF a exists in some domain D, then a = a in D. The conclusion (3) must also be a conditional: if a exists in D , then something  in D is self-indentical. That of course does not assert the existential Peter wants from (x)(x = x). Put simply, a = a presumes [presupposes] rather than entails that a exists.

Comments

[Account Deleted]

Bill,

Consider the following argument:

(i) ((x)(x=x));

(ii) a=a, for any arbitrarily chosen object a; (from (i))

(iii) (Ex)(x=a); (from (ii) by existential generalization);

Now, (i) is necessary, but (iii) is contingent. Yet (i) entails (iii) via (ii), which is also necessary. So I simply do not see how the principle (1*) which you and Jan seem to accept applies in modal logics that include quantification plus identity.

You say: "I grant that Bill is essentially self-identical: self-identical in every world in which he exists. But this is not to say that Bill is necessarily self-identical: self-identical in every world. And this for the simple reason that Bill does not exist in every world."

But this can't be right. Are you saying that the proposition 'it is possible that Bill is not self-identical' is true in some world? Which world would that be?

It cannot be in any world in which Bill exists. I believe we agree with that. So the only remaining possibility is that it is true in worlds in which Bill fails to exist.

But in worlds in which Bill fails to exist 'Bill is not self-identical' cannot have a truth-value at all and, therefore, cannot be true. For if we assume that it is true, then it means that the property of 'being not identical to itself' is true of some object in this world; namely, Bill. But, no object can have such a property. And in particular, Bill cannot have this property, since ex-hypothesis, Bill does not exist in this world.

[Bill Vallicella]

Peter,

I'd say that Jan has already refuted your view. If you insist that *Bill = Bill* is necessary, then you are committed to saying that *Bill exists* is also necessary.

[Account Deleted]

Bill,

"I'd say that Jan has already refuted your view."

If you think so, then you must also think that the following examples are all contingent:

(a) Bill is either a male or it is not the case that Bill is a male;

(b) Bill is either seven feet tall or it is not the case that Bill is seven feet tall.

(c) Bill is not taller than Himself.

(d) (If Bill is taller than Mike and Mike is taller than Peter, then Bill is taller than Peter)

etc.

Reason: Since Bill fails to exist in some possible worlds, it follows that by yours and Jan's reasoning (a),(b),(c), (d) and countless other apparently necessary truths are in fact contingent.

I think that the problem we face is this. Interpretation precedes modal evaluation: i.e., strictly speaking, one cannot evaluate the truth status or the modal status of propositions unless one first assigns an interpretation to the terms in a given domain (or multiple domains when it comes to possible world semantics). Hence, in order to evaluate the modal status of forms such as 'a=a' one must have already assigned an interpretation to the term 'a' in every domain of every possible world (for the quantified modal logic case), o/w 'a=a' is not a well-formed-formula and, therefore, cannot be given a truth value.

I presuppose that such an interpretation has been already given before an assessment of the modal status of propositions is possible. I think you and Jan somehow overlook this fact.

"If you insist that *Bill = Bill* is necessary, then you are committed to saying that *Bill exists* is also necessary."

This is not quite right. You are reading the consequence of 'Bill=Bill' in a de-dicto fashion, whereas the proper consequence should be a de-re reading. Strictly speaking the relevant inference is the following:

(*) Nec(Bill=Bill)

Therefore,

(**)(Ex)Nec(x=Bill).

But due to the combination of modal operators and quantifiers, (**) should receive a de-re reading: i.e., there exists at least one particular object x; i.e., Bill, such that it is true of it that in every possible world it is identical to Bill. So read, (**) is false in worlds in which the particular object in question fails to exist.

[Jan]

Peter,

I have come to appreciate your argument that the whole thing depends on what particular logic we employ.

Let me try to summarise (and perhaps speculate about / develop) your views:

1) Statements invoking an object o might induce a proposition in a world w only if o exists in w. For example, a statement 'Bill is a man' induces the proposition *Bill is a man* in every world in which Bill exists.

2) In worlds in which Bill does not exist, 'Bill is a man' has meaning, but induces something less than a proposition. Let's call it a proto-proposition. In particular, a proto-proposition *Bill is a man* does not have a truth value in those worlds at all (it cannot be a proposition by this fact alone).

3) A modal proposition is something that reduces to propositions or proto-propositions in particular possible worlds. For example, a m-proposition *Bill is a man* reduces to a proposition in every world where Bill exists (and thus has a truth value), and to a proto-proposition in worlds where Bill does not exist (and thus does not have a truth value).

4) A m-proposition p implies a m-proposition q iff in every world in which p reduces to a true proposition, q reduces to a true proposition also. In particular, there's no requirement on q in worlds in which p reduces to a proto-proposition.

5) A m-proposition p is called necessary iff in every world in which p reduces to a proposition, it reduces to a true proposition.

Given all that is written above, I concede that *Bill=Bill* implies *(Ex) x = Bill* and that the antecedent is necessary and the consequent contingent.

However, it is not true that in this model (1*) fails for identity statements only. First, note that for any essential property P of Bill m-proposition *Bill is P* is necessary. Indeed, by definition of P *Bill is P* is true in every world in which Bill exists. Now, *(Ex) x = Bill* is true in every world in which *Bill is P* is, which means that *Bill is P* entails *(Ex) x = Bill*. Such is the less than desirable cost of making propositions of the type *Bill is not taller than himself* necessary.

[Jan]

I've just noticed that in this model every possible being is also a necessary being. Is there a way of construing Peter's words so as not arrive at this conclusion? I cannot see it at the moment.

[Jan]

Correction: in this model the notions 'o exists in all possible worlds' and 'o is such that *o exists* is a necessary proposition' diverge. Only the latter is true for all objects. If necessary existence is defined as existence in all possible worlds, the problem that I wrote about in the previous comment does not occur.

[Account Deleted]

Jan,

I think that you got what I was trying to say throughout several of my posts. Thanks for the clarifications and input.