Iterative Conception of Propositions

(Cross Posted at joshuaspencer.net)

The iterative conception of propositions works like this. First, we start out with some propositions at the bottom level. None of these propositions contain as a constituent the property of being true. These propositions are combined in a Boolean kind of way to get conjunctions, disjunctions, conditionals etc. Then, for each proposition at the base level and for each proposition constructed in a Boolean kind of way, there is the proposition that that proposition is true. That is, for any P such that P is a proposition at the base level or P is a proposition formed by Boolean operations on propositions at the base level, there is the proposition that P is true. This is the second level of propositions. Now, we combine these propositions and the ones at the base level and the ones formed by Boolean operations in Boolean kind of way. We get even more propositions and we do the trick over again ad infinitum. There are no propositions other than the ones formed by this series of operations.

This iterative conception of propositions will look a lot like typed propositions but will not involve a hierarchy of truth predicates. Moreover, I believe this conception of propositions will avoid certain kinds of paradoxes. There will be no liar proposition. Moreover, for any proposition P, the proposition that P is true is not a conjunct of P. Hence, there will be no Russell proposition that has itself as a conjunct iff it does not. This iterative conception of propositions is rather attractive given that it avoids these paradoxes.

We might also add to our iterative conception of propositions a kind of anti-deflationist principle. That is, for any proposition P, the proposition that P is true is not identical to P. This anti-deflationist principle will be attractive to all those who believe that there really is a property of truth.

Here is a principle that seems to follow from the iterative conception of propositions combined with the anti-deflationist principle:

Particularized Principle of Sufficient Reason (PPSR):

For any true proposition, P, there is a particular sufficient reason, S, such that (i) S is identical to the proposition that P is true, (ii) S is true, (ii) necessarily: S only if P, (iii) S is not identical to P or to any contingent conjunct of P.

However, in An Essay on Free Will, Peter van Inwagen presented a strong argument against the principle of sufficient reason (which is implied by the particularized principle above). Hence, there is a strong argument against PPSR. Very briefly, I will set out the assumptions that underlie van Inwagen’s argument and present the argument against PPSR. My formulation will follow Hudson’s formulation of van Inwagen’s argument (“Brute Facts” AJP March 1997).

Here are the five assumptions that underlie the argument:

A1. There are contingently true propositions.

A2. Any conjunction of contingently true propositions is itself contingently true.

A3. Any true proposition is either contingently true or necessarily true.

A4. For any P and Q, if both Necessarily: P and Necessarily: P only if Q, then Necessarily: Q

A5. If there are contingently true propositions, then there is a conjunction of all contingently true ptopositions.

The argument against PPSR is rather straightforward.

1) P is a conjunction of all contingently true propositions (A1, A5)

2) Hence, P is contingently true (1, A2)

3) Hence, there is a sufficient reason for P, S, such that S is true and S is identical to the proposition that P is true. (2, PPSRi, PPSRii)

4) Hence S is either contingently true or necessarily true. (3, A3)

5) S is not necessarily true (proof below)

6) Hence S is contingently true. (4, 5)

7) But, S is not contingently true. (proof below)

The proof of 5 is as follows:

a) S is necessarily true (reductio assumption)

b) Hence, necessarily: S only if P (3, PPSRii)

c) Hence, necessarily: P (b, A4)

d) It is not the case that P is necessary (2)

e) Hence S is not necessarily true (from a-d)

The proof of 7 is as follows:

f) S is contingently true (reductio assumption)

g) Hence, S is a contingent conjunct of P (e, 1)

h) S is not a contingent conjunct of P (2, 3, PPSRiii)

i) Hence, S is not contingently true (from f-g)

Since PPSR followed from the iterative conception of propositions and the anti-deflationary principle, we can conclude that A1-A5, the iterative conception of propositions and the anti-deflationary principle are jointly inconsistent. I take it that A1-A4 are all very strong. So, that leaves us with a choice. We must decide on whether we are going to give up on the iterative conception of propositions, the anti-deflationary principle, or the assumption that if there are contingent truths then there is a conjunction of all such truths. Those who are attracted to the iterative conception of propositions because it sidesteps paradox must decide between being a deflationist about truth (that is accepting that for any proposition P, the proposition that P is identical to the proposition that P is true) or denying a conjunction of all contingent truths.

Let’s consider the second option. It seems that the second option is inconsistent with the iterative conception of propositions. After all, at any level of proposition construction, we are supposed to perform Boolean operations on all those propositions constructed up to that point. We should get a conjunction of all contingent propositions at the end of our (infinitely long) construction procedure. Remember that the construction procedures should not be thought of as an actual process. Rather, we should think of the iterative conception as a thesis that involves a base clause (about the existence of certain propositions) and a recursive clause (that tells us what propositions exist given the existence of those in our base clause). The recursive clause will have a quantifier over propositions. Hence, if we are to hold on to the iterative conception without accepting a conjunction of all contingently true propositions, we must say that the quantifier in the recursive clause is indefinitely extensible. If this is all correct, then the only way a person who believes in the iterative conception of propositions can avoid a conjunction of all contingently true propositions is by endorsing a rather radical thesis about quantification (namely that the quantifiers that range over propositions are indefinitely extensible).

Hence, it looks like the defender of the iterative conception of propositions is faced with a choice. Become a deflationist about truth. Say that for every proposition P, the proposition that P is true is identical to P. Or, alternatively, claim that quantifiers that range over propositions are indefinitely extensible. Neither option seems too attractive.