INFORMATIVE ABOUTNESS Peter B. M. Vranas vranas@wisc.edu University of Wisconsin-Madison 11th International Conference in Philosophy, May 2016

INTRODUCTION I used to tell my students: If you propose a universal generalization and someone produces a counterexample to it, a standard strategy is to retreat to a restricted generalization that avoids the counterexample. For example, if you propose the universal generalization “all swans are white” and someone notes that there are black swans in Australia, you can retreat to the restricted generalization “all non-Australian swans are white”. The restricted generalization avoids the counterexample because (a) it is not about all swans: (b) it is only about non-Australian swans, (c) not about Australian ones. I now believe that (a), (b), and (c) are all false. Every time I have taught an introductory philosophy course, I have told my students something like the following: If you propose a universal generalization and someone produces a counterexample to it, a standard strategy is to retreat to a restricted generalization that avoids the counterexample. For example, if you propose the universal generalization “all swans are white” and someone notes that there are black swans in Australia, you can retreat to the restricted generalization “all non-Australian swans are white”. The restricted generalization avoids the counterexample because (a) it is not about all swans: (b) it is only about non-Australian swans, (c) not about Australian ones. The last sentence of the above passage used to sound platitudinous to me, but I now believe that claims (a), (b), and (c) are all false. There is a plausible argument for the conclusion that the generalization “all non-Australian swans are white” is about all swans—and thus is also about Australian swans, not only about non-Australian ones.

A SIMPLE ARGUMENT (1) “All swans are white or Australian” [like “All swans are white or pink”] is about all swans. (2) “All swans are white or Australian” is logically equivalent to “All non-Australian swans are white”. (3) Logically equivalent universal generalizations are about exactly the same objects. So: (4) “All non-Australian swans are white” is about all swans. The argument is simple. It starts with the commonsensical assumption that “all swans are white or Australian”—like “all swans are white or pink”—is about all swans (no matter what else, if anything, it may also be about). But “all swans are white or Australian” is logically equivalent to “all non-Australian swans are white”. And logically equivalent universal generalizations are about exactly the same objects. So “all non-Australian swans are white” is also about all swans—and thus is also about Australian swans, not only about non-Australian ones. Some people might question the premises of this argument, but my main goal here is not to defend its conclusion. I am more interested instead in how to move on if its conclusion is accepted. If “all non-Australian swans are white” is both about non-Australian swans and about Australian ones, then what can I tell my students instead of what I have been telling them?

MY PROPOSAL The information that “All non-Australian swans are white” provides about an object is that it is a non-Australian swan only if it is white; equiva-lently, it is Australian or white or not a swan. This information is entailed (hence rendered redundant) by the information that the object is an Australian swan. So “All non-Australian swans are white” is uninformative about Australian swans. Here is my proposal. The information that “all non-Australian swans are white” provides about an object is that it is a non-Australian swan only if it is white; equivalently (as one can show by elementary logic), it is Australian or white or not a swan. But this information is entailed, and in this sense rendered redundant, by the information that the object is an Australian swan. So my proposal is to tell my students that “all non-Australian swans are white” is uninformative about Australian swansand this is why it does not conflict with the information that some Australian swans are blackbut is informative about non-Australian swans. My proposal provides an explanation of the mistaken intuition that “all non-Australian swans are white” is not about Australian swans: it seems not to be about Australian swans because it is uninformative about Australian swans.

TWO DEFINITIONS Definition 1: Conditional informativeness. A proposition Q is uninformative about an object o given a proposition R iff the information that the conjunction of Q with R provides only about o is the information that R provides only about o. Definition 2: Information only about an object. The information that a proposition Q provides only about an object o is the conjunction of all propositions that are both only about o and entailed by Q. But how to make precise the distinction between uninformative and informative aboutness? This is the main topic of my talk. Start with the following definition: a proposition Q is uninformative about an object o given a proposition R exactly if the information that the conjunction of Q with R provides only about o is logically equivalent to the information that R provides only about o (otherwise, Q is informative about o given R). Intuitively, a proposition is uninformative about an object given R exactly if the information that the conjunction of the proposition with R provides only about the object is already provided by R. But what exactly is the information that a proposition provides only about an object? The following definition clarifies this concept: the information that a proposition Q provides only about an object o is the conjunction of all propositions that are both only about o and entailed by Q.

FOUR EXAMPLES (1) InfProust(<Proust is a writer>) = <Proust is a writer>. (If Q is only about o, then Info(Q) = Q.) (2) InfProust(<Proust is a writer or Sartre is a philosopher>) = the conjunction of all necessary propositions that are only about Proust. (3) InfProust(<Proust is a writer and Sartre is a philosopher>) = <Proust is a writer>. (If Q is only about o, then Info(Q & R) = Q & Info(R).) (4) InfProust(<All philosophers are writers>) = <Proust is a philosopher only if he is a writer>. To illustrate my definition of the information that a proposition provides only about an object, I will provide four examples. First example: consider the proposition that Proust is a writer. The information that this proposition provides only about Proust is this proposition itself. This can be shown by using my definition, but I omit the proof. More generally, it can be shown that, if a proposition Q is only about an object o (just like the proposition that Proust is a writer is only about Proust), then the information that Q provides only about o is just Q. Second example: consider the proposition that either Proust is a writer or Sartre is a philosopher. Intuitively, this proposition does not provide any information only about Proust. It can be shown that, according to my definition, the information that this proposition provides only about Proust is a necessary proposition. I propose to accept it as true by convention that providing no information only about an object amounts to providing necessary information. Third example: consider the proposition that Proust is a writer and Sartre is a philosopher. Intuitively, the information that this proposition provides only about Proust is the proposition that Proust is a writer, and it can be shown that my definition yields this result. More generally, it can be shown that, if a proposition Q is only about an object o, then the information that the conjunction of Q with a proposition R provides only about o is the conjunction of Q with the information that R provides only about o, which in this example is necessary, since the proposition R that Sartre is a philosopher provides no information only about Proust. Finally, the fourth example: consider the universal generalization that all philosophers are writers. This is the conjunction of the proposition that Proust is a philosopher only if Proust is a writer with the proposition that any philosopher distinct from Proust is a writer. So we have a variant of the third example, and it can be shown that, according to my definition, the information that this conjunction (or universal generalization) provides only about Proust is the proposition that Proust is a philosopher only if Proust is a writer. This example will be most relevant in what follows.

A THEOREM Definition 1 (reformulated): Q is uninformative about o given R iff Info(Q & R) = Info(R). Theorem: (a) Q is uninformative about o given R exactly if R entails Info(Q & R). (b) If R is only about o, then Q is uninformative about o given R exactly if R entails Info(Q). Example: <All philosophers are writers> is uninformative about Proust given that Proust is not a philosopher, but is informative about Proust given that Proust is a philosopher. Having clarified the concept of the information that a proposition provides only about an object, I return next to the concept of conditional informativeness. Recall that, according to Definition 1, a proposition Q is uninformative about an object o given a proposition R exactly if Info(Q & R) is logically equivalent to Info(R). I have proven a theorem which shows that this definition can be considerably simplified: First, Q is uninformative about o given R exactly if R entails Info(Q & R). Second, if R is only about o, then Q is uninformative about o given R exactly if R entails Info(Q). To illustrate the second part of the theorem, consider again the universal generalization that all philosophers are writers. This universal generalization is uninformative about Proust given that Proust is not a philosopher: the proposition that Proust is not a philosopher, which is only about Proust, entails the information that the universal generalization provides only about Proust, namely the proposition that Proust is a philosopher only if Proust is a writer. By contrast, the universal generalization is informative about Proust given that Proust is a philosopher: the proposition that Proust is a philosopher does not entail that Proust is a philosopher only if Proust is a writer.

CONDITIONAL INFORMATIVE ABOUTNESS Definition 3: Q is informative about objects given that they exemplify P iff for any object o, Q is informative about o given that o exemplifies P. V = <All non-Australian swans are white>. Info(V) = <o is a non-Australian swan only if o is white>. R = <o is an Australian swan>. R is only about o. For any o, R entails Info(V), so V is uninformative about Australian swans. R = <o is a non-Australian swan>. For any o, R does not entail Info(V), so V is informative about non-Australian swans. I am now in a position to make the main point of this talk, namely to make precise the distinction between uninformative and informative aboutness. Here is my definition: a proposition Q is informative about objects given that they exemplify a property P exactly if, for any object o, Q is informative about o given that o exemplifies P. For example, let V be the proposition that all non-Australian swans are white. On the one hand, V is uninformative about objects given that they exemplify the property of being an Australian swan (or, more succinctly, V is uninformative about Australian swans). This is because, for any object o, the proposition that o is a non-Australian swan only if it is white (which is the information that V provides only about o) is entailed by the proposition R that o is an Australian swan. On the other hand, V is informative about objects given that they exemplify the property of being a non-Australian swan (or, more succinctly, V is informative about non-Australian swans). This is because, for any object o, the proposition that o is a non-Australian swan only if it is white is not entailed by the proposition R that o is a non-Australian swan. One can similarly see that V is uninformative about non-swans and uninformative about white objects, but is informative about swans and informative about non-white objects.

THREE REMARKS “Informative about swans” ≠ “informative about all swans”: V is informative about swans but uninformative about Australian swans. “Informative about swans” ≠ “informative about the class of swans”: <All swans are white> is informative about swans and uninformative about white objects, but the class of swans may be the same as the class of white objects. “Informative about swans” = “informative about objects given that they are swans”. Let me now make three remarks on informative aboutness. First, to be informative about swans is not to be informative about all swans: as we saw, the proposition V that all non-Australian swans are white is informative about swans but is not informative about Australian swans. Second, to be informative about swans is not to be informative about the class (or the set) of swans: the proposition that all swans are white is informative about swans but not informative about white objects, even if (let us assume) the class of swans is the same as the class of white objects. Third, to be informative about swans is to be informative about objects given that they are swans. It’s important to keep in mind that the informative aboutness I am talking about is conditional; this is why its behavior differs from what one might initially expect.

AN OBJECTION The objection: V is informative about elephants, and this is counterintuitive. Reply: The intuition that V is uninformative about elephants relies on the background information that no elephants are swans. Relative to this background information B, V is indeed uninformative about elephants: B & <o is an elephant> entails that o is not a swan, which entails Info(V) = <o is a non-Australian swan only if o is white>. I conclude by addressing an objection to my definition of informative aboutness. According to my definition, the proposition V that all non-Australian swans are white is informative about elephants. This is because the proposition that an object is an elephant does not entail that the object is a non-Australian swan only if it is white. But this result is counterintuitive. In reply, note that the proposition that an object is an elephant, together with the proposition that no elephants are swans, entails that the object is not a swan, and thus does entail that the object is a non-Australian swan only if it is white. So I propose to say that, relative to the background information that no elephants are swans, the proposition that all non-Australian swans are white is indeed uninformative about elephants, and this is what explains the intuition that the proposition is uninformative about elephants.