# Stephen Yablo's Aboutness and logical subtraction

I was finishing reading Aboutness by Yablo, but there is an intuitive definition that I do not get:

He says on page 148 that:

What is this relation of adding falsity, or being additionally false, or being false not just because B is false? I want to say that X adds falsity to B 5 when B&X is false for a reason that does not trade on B being false, as is shown by its being instantiable even when B is true. This is the same as B→¬X being true for a reason that can obtain even when B is true. Reasons like this, that do not trade on B’s falsity, are the kind we above called B-compatible or B-friendly.

Why should it make intuitive appeal that we define the fact that A adds falsity to its consequence B when there is a reason for A to be false even when B is true? I do not get intuitively why this should make sense. Can anyone help me on this?

Suppose that B is "All five original Take That members are American", and X is "Robbie is American". There's an intuitive sense in which X does not add falsity to B, in that the content that X has that is false is part of the content of B.

But if X was, say "Robbie and Mick Jagger are both American", then X would be adding falsity to B, in that, sure, the part about Robbie being American overlaps with what is said by B, but the Mick Jagger bit is an altogether new false bit.

One way to make this intuition slightly more precise, Yablo argues, is by using the notion of B-compatible reasons: there is a reason for X to be false that is compatible with the (counterfactual) truth of B; namely, that Mick Jagger is not American.

This is, simply on the strength of logical manipulation and the semantics of the material conditional, the same as saying that B → ¬X can be true even in a case in which B is true: if B is true, the only way that the conditional can be true is if X is false. That is to say, the fact that this conditional is possibly true means that there is a way for X to be false that does not depend on B being false.

(You might feel uneasy about this way of using semantic content as if it was a substance that can be taken from here to there, can be shared and so on; but this is precisely the kind of talk that Yablo is trying to make precise in his book.)

• Thanks for the answer. I still do not get what justifies Yablo's reasoning behind the definition. Why should we choose that X&B be false not just because B is false to talk about adding falsity? Mar 17 at 21:42
• Just to see how best I can help: do you find it somewhat plausible that "Paris is in Italy" adds some extra falsity to "London is in France" in the sentence "Londin is in France and Paris is in Italy"? Mar 18 at 10:20
• Yes, it is plausible to me. Apart from my comment, reading again the page and your answer, I came up to the conclusion that, intuitively, A adds falsity to B iff there is a reason why A is false that it is not already a reason why B is false - if there is a reason for which B is false that is already a reason for which A is false, there would be no introduction of falsemaker, and so A would not add falsity to B. Mar 18 at 11:35
• This still does not explain to me why having a reason for which ¬A is true that does not force the falsity of B is equivalent to having a reason for the truth of B->¬A that does not force the falsity of B. Mar 18 at 11:36
• This is just logical manipulation, and the meaning of the conditional. Let me know if the edit makes this clearer. Mar 18 at 11:54

B→¬X being true for a reason that can obtain even when B is true

One might interpret this as follows. Say that X adds falsity to B, when there exists a set of true premises from which we can derive B→¬X, where the premises are not inconsistent with B.

Edit: This set of premises ought to be further constrained, e.g. to be chosen as a subset of a fixed, larger set of premises. Otherwise we could simply let the premises be "B→¬X" itself, trivially satisfying the condition. It seems Yablo does this by taking a world w and allowing the set of premises to be the existence predicates for objects in w.

The passage you quote appears in the context of a discussion of what it means to speak of the subtraction of two propositions "A-B" when A implies B. A-B should have the property that when combined with B it yields A. One proposal is that A-B is the material implication B → A. Yablo rejects this because it gives the wrong result. It would make A-B true whenever B is false, and would require B to be true if A-B is false. This runs against our expectation that in general the truth of A-B is independent of the truth of B.

Intuitively, A-B should be true when something about the semantic relation between A and B makes it true, not just trivially because B happens to be false. This 'something' he characterises as a relation whereby A-B is false if A "adds falsity" to B, meaning that A&B is false because of some semantic content that A brings and not simply because B is independently false. This can be expressed counterfactually by saying that B → ¬A could remain true even when B is true.

It might help to understand this in the context of accounts of conditionals. Material implication is the simplest conditional, and the only one that is a truth function. This is both a strength and a weakness. It is a strength because it makes its evaluation very simple using a truth table and because it is readily interdefinable with classical conjunction, disjunction and negation. It is a weakness because it is unable to express the idea that the consequent 'follows from' the antecedent in some way.

A material implication A → B is true if A is false, whereas typically with conditionals the truth or acceptability of "if A then B" is independent of the truth of A. We would not choose to say, "if A then B" just because we believed A to be false. We expect "if A then B" to be robust with respect to the truth of A and to hold even if (counterfactually) A is true. This means that for real conditionals, we expect there to be some connection between A and B, whether logical, semantic or pragmatic.

This is analogous to what Yablo says about A-B. We do not hold it to be trivially true just because B is false. We require that some semantic connection ensures that B → ¬A is robust with respect to the truth of B.

• Thanks a lot. So, form what I get, for instance, A adds falsity to B in a world w iff were B true in w, then A would be false. There is a point that I do not think I understand properly: how is this equivalent to B→¬A being true for a reason compatibile with the truth of B? Mar 17 at 8:09
• The idea is that there is some reason or ground that makes B → ¬A true by virtue of a connection between B and A, and hence we do not accept B → ¬A trivially just because we believe B to be false. Whatever that reason is, it continues to hold were B to be true. If you understand counterfactuals in terms of possible worlds, then it means that ¬A is true in the relevant possible worlds in which B holds. Mar 17 at 10:01
• Thanks. There is still a point that by intuition I do not get. Why should we arbitrarily say that for A to add falsity to B, their conjunction should be false for reasons that can hold even when B is true? Mar 17 at 16:58

I just put here an add-on on the previous question, in order to specify a point.

" I want to say that X adds falsity to B 5 when B&X is false for a reason that does not trade on B being false, as is shown by its being instantiable even when B is true. "

This is a point that I can't make sense of: why should we intuitively regard X as adding falsity to B if and only if X&B is false for a B compatibile reason?