It is orthodoxy amongst truthmaker theorists that the existence of a truthmaker necessitates the truth of that which it makes true: if a thing A is a truthmaker for the proposition that p, then there is no possible world in which A exists and p is false. Is there any good argument for this claim?

The only argument I know of for truthmaker necessitarianism is that given by David Armstrong. Armstrong says:

If it is said that the truthmaker for a truth could have failed to make the truth true, then we will surely think that the alleged truthmaker was insufficient by itself and requires to be supplemented in some way. A contingently sufficient truthmaker will be true only in circumstances that obtain in this world. But then these circumstances, whatever they are, must be added to give the full truthmaker.

His thought, I take it, is this. Suppose that A makes p true but doesn’t necessitate the truth of p. In that case there are some possible worlds in which A exists and p is false: some set of circumstances in which the existence of A does not suffice for the truth of p. But then isn’t it overwhelmingly intuitive that the truthmaker for p is not simply A but rather A together with whatever makes it true that those circumstances do *not* in fact obtain?

Let’s spell this argument out in a little more detail, making explicit Armstrong’s implicit assumptions. A first shot is:

1) A makes p true. (Assumption)

2) The existence of A does not necessitate the truth of p. (Assumption for reductio)

3) There is a possible world in which A exists and p is false. (From 2)

4) There is a (probably highly disjunctive) proposition q which is true in exactly those worlds in which A exists and p is false. (q is the proposition that describes the circumstances in which the existence of A does not suffice for the truth of p.) (This assumption follows from the general assumption that for any set of possible worlds, there is a proposition that is true at exactly those worlds.)

5) What makes p true is A together with what makes not-q true. I.e. if not-q is made true by B, then p is made true by the sum of A and B. (Assumption)

6) p is not made true by both A and the sum of A and B. (Assumption)

7) p is not made true by A. (From 5 and 6)

8) Contradiction. (From 1 and 7)

But what right does Armstrong have for premise (6)? A proposition can have more than one truthmaker – you and I both make it true that there are humans – so there is no contradiction in holding that p is made true both by A and by the sum of A and B. I guess Armstrong’s thought is that the sum of A and B jointly necessitate the truth of p, and that *if* there is a necessitating truthmaker for a proposition, nothing which did not necessitate the truth of that proposition deserves to be called its truthmaker. That is, it would only be reasonable to think that p had a non-necessitating truthmaker if p had no necessitating truthmaker. (That is, I think, what is behind his comment that “the alleged truthmaker was insufficient by itself”.) That is a fairly plausible thought, so let us amend Armstrong’s argument as follows:

1) A makes p true. (Assumption)

2) The existence of A does not necessitate the truth of p. (Assumption for reductio)

3) There is a possible world in which A exists and p is false. (From 2)

4) There is a (probably highly disjunctive) proposition q which is true in exactly those worlds in which A exists and p is false. (q is the proposition that describes the circumstances in which the existence of A does not suffice for the truth of p.) (This assumption follows from the general assumption that for any set of possible worlds, there is a proposition that is true at exactly those worlds.)

5) The truth of p is necessitated by A together with what makes not-q true (call it B). A and B together are a necessitating truthmaker for p. (Assumption)

6) There is no non-necessitating truthmaker for p. (From 5)

7) Contradiction. (From 1, 2 and 6)

I used to think the best point of resistance for the denier of necessitarianism was the assumption, presupposed by premise (5), that every truth has a truthmaker. If we deny that negative facts such as not-q require a truthmaker then the argument doesn’t go through. But, I thought, one who upholds truthmaker maximalism is committed, by this argument, to truthmaker necessitarianism. Now I think that is wrong. No one, maximalist or non-maximalist, should be persuaded by this argument to accept truthmaker necessitarianism.

Ask yourself first: what right does Armstrong have to premise (5)? Why should we think that A together with whatever makes it true that not-q is a necessitating truthmaker for p? The only reason I can see is as follows. q is a complete specification of the possible circumstances in which A exists and p is false. So if not-q is made true then those circumstances do not obtain. So if not-q is made true then we are in a world in which the existence of A suffices for the truth of p. So, necessarily, if both the truthmaker for not-q and A exist, then p is true.

Unfortunately, this last step is no good. It is true that every world in which not-q is true is a world in which the existence of A suffices for the truth of p. There is no possible world in which not-q is true and A exists and p is false. But it only follows that the truthmaker for not-q cannot exist in a world in which A exists and p is false if we add the assumption that the truthmaker for not-q cannot exist in a world in which not-q is false (i.e. a world in which q is true). But of course we can’t assume that, for that would simply beg the question. Our only reason for thinking that the truthmaker for not-q couldn’t exist in a world in which q is true is if truthmakers must necessitate the truth of the propositions they are truthmakers of, and that is exactly what the argument is trying to establish.

If Armstrong is to avoid begging the question he must leave it open that B, the truthmaker for not-q, can exist in a world in which q is true. That is a world in which the existence of A does not suffice for the truth of p, so we have been given no reason to rule out the possibility of worlds in which A and B both exist and p is false. So we have no reason to think that A and B together are a necessitating truthmaker for p, and hence no reason to think that A is not an adequate truthmaker for p.

To focus discussion, let us consider a particular account of truthmakers according to which a truthmaker for p need not necessitate the truth of p: Josh Parsons’ theory that the truthmaker for p is that which is intrinsically such that p. Since an object needn’t have its intrinsic properties essentially, Parsons denies necessitarianism. A can make p true and exist in worlds in which p is false provided that, in each of those worlds, A differs intrinsically from how it actually is.

How should Parsons respond to Armstrong’s argument? Well, what should Parsons think the truthmaker for not-q is? q is the proposition that describes exactly the circumstances in which A exists and p is false. What are those circumstances on Parsons’ theory? They are the circumstances in which A differs intrinsically from how it actually exists. So not-q is the proposition that is true exactly when A is as it actually is intrinsically. So what makes this true? That which is intrinsically such that A has the intrinsic properties it actually has – namely, A.

So for Parsons, A is the truthmaker for not-q. What makes it true that we are not in the circumstances in which A can exist and p be false is just the truthmaker for p. A can exist in worlds in which not-q is false, of course – the existence of A does not necessitate that A is intrinsically as it actually is. But that is no objection; it is part of the theory that the existence of the truthmaker need not necessitate the truth of that which it makes true. All that is required is that in the worlds in which A exists and not-q is false, A differs intrinsically from how it actually is. And that is obviously guaranteed: A cannot be as it actually is intrinsically and q be true, since q *just is* the proposition that A differs intrinsically from how it actually is.

It is clear then that Armstrong’s argument has no pull against someone who accepts a theory like Parsons’. The argument only works if the truthmaker for not-q must be a necessitating truthmaker, but to assume this would beg the question.