Informative Aboutness

ABSTRACT

Pretheoretically, (B) ‘all believers are immortal’ is about all believers, but (1) B is not about any unbeliever. Similarly, (M) ‘all mortals are unbelievers’ is not about any immortal, but (2) M is about all mortals. But B and M are logically equivalent universal generalizations, so arguably they are about exactly the same objects; by (2), they are about those mortals who are unbelievers, contradicting (1). If one responds by giving up (1), is there still a sense in which B treats unbelievers differently from believers? I argue that there is. B is uninformative about unbelievers but informative about believers, in the following sense: for any object o, the information that B provides only about o—namely, ‘o is a believer only if o is immortal’—is entailed (and thus rendered redundant) by ‘o is an unbeliever’ but not by ‘o is a believer’.

KEYWORDS:

• aboutness
• information
• conditional informativeness
• contraposition
• logical equivalence

Notes

¹ By an argument parallel to the one I gave in the text, one can reach the conclusion that every universal generalization is about all objects: for example, ‘all swans are white’ is logically equivalent to ‘every object is white or not a swan’, which is about all objects (Lambert and van Fraassen [1972: 88]; Armstrong [1983: 42]). I find this conclusion acceptable, but Hart [1981: 5–6] objects in effect that, because every proposition is logically equivalent to some universal generalization or other, the conclusion that every universal generalization is about all objects has the unacceptable consequence that every proposition is about all objects. I reply that a proposition that is logically equivalent to a universal generalization need not be about exactly the same objects as the universal generalization. For example, the proposition that the Eiffel Tower is metallic (which is not about the Parthenon) is logically equivalent to the universal generalization (which is, inter alia, about the Parthenon) that everything non-metallic is distinct from the Eiffel Tower.

² To object to this assumption, it is not enough to argue that ‘all swans are white or Australian’ is about the class of swans (Goodman [1961: 7 n.1]; cf. Putnam [1958]), or about the concept ‘swan’ [Frege 1884: 60], or about the property of being a swan (cf. Dretske [1977: 252–3]; Sober [1985: 17]): something may be about that class or concept or property and also be about all swans (i.e. about every individual swan: cf. Lamarque [2014: 262]). Goodman argues that ‘every x is P’ is not about all objects: “‘about’ behaves somewhat as ‘choose’ does.… Choosing something involves not choosing something else.… Likewise, saying so and so about an object involves not saying so and so about some other” [1961: 5]. Goodman provides no reason, however, to accept this as a good analogy. Moreover, even if ‘all swans are white or Australian’ is shown by Goodman's argument not to be about all objects, it is not shown not to be about all swans, since it does say something about every swan (namely that it is white or Australian) that it does not say about any non-swan. (On Goodman's views on ‘about’, see Ullian [1962], Rescher [1963], Patton [1965], Putnam and Ullian [1965], Tichý [1975: 88–90], and Hart [1981: 18–42].)

³ Some authors find this assumption hard to contest: ‘That logically equivalent statements should thus be about just the same things would seem a minimal condition of adequacy that any acceptable definition of aboutness must satisfy’ (Goodman [1961: 12]; cf. Putnam [1958: 125], Tichý [1975: 88], and Hart [1981: 4, 8–9]). Other authors, however, contest the assumption (Yourgrau [1987: 135–6]; Demolombe and Jones [1999: 115–16]; Yablo [2014]; cf. Sober [1985: 15–16]), and it might argued that by making the assumption one deviates from a pretheoretic concept of aboutness. In reply, I can grant this: the concept of aboutness that I consider in this paper corresponds to tutored (instead of raw) intuitions. One might object to the assumption as follows: ‘all unmarried men are men’, which is about men, is logically equivalent to ‘all even numbers are numbers’, which is not about men. In the present context, however, this objection is question-begging: one might reply that ‘all even numbers are numbers’ is about men, since it says, about each man, that he is an even number only if he is a number. Alternatively, one might object to the assumption by appealing to a fine-grained theory of propositions—e.g. a theory of structured propositions [Russell 1903; Salmon 1986; Soames 1987; King 2014]—which holds that logically equivalent propositions may be distinct. I reply that the view that logically equivalent propositions may be distinct is compatible with the assumption that logically equivalent universal generalizations are about exactly the same objects (cf. Hoffmann [manuscript]): distinct propositions may be about exactly the same objects.

⁴ As I explain later (section 5), this is not to say that ‘all non-Australian swans are white’ is uninformative about the class of Australian swans.

⁵ 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. I am not claiming that this is a full explanation, but discussing other (partial) explanations lies beyond the scope of this paper.

⁶ Since I am talking about propositions rather than sentences, I see no problem with infinite conjunctions. Other authors, by contrast, take sentences rather than propositions to be about objects (cf. Ryle [1933: 10]; Carnap [1937: 284–92]; Hodges [1971: 5]).

⁷ One might object that this consequence is intuitively unappealing because Q₁ also provides, for example, the information that Proust is a writer or a philosopher, which is only about Proust and is distinct from Q₁. I reply that in the text I am talking about the full (or strongest) information that Q₁ provides only about Proust; the proposition that Proust is a writer or a philosopher is only partial information that Q₁ provides only about Proust.

⁸ This example might suggest that a proposition Q is only about an object o if and only if o is the only object that is a constituent of Q. Arguably, however, both parts of this suggestion fail. Against the ‘only if’ part, one might argue that, if the tallest spy is François, then the proposition that the tallest spy is French is only about François but does not have François as a constituent [Fitch and Nelson 1997]. Against the ‘if’ part, one might argue that, although the Sphinx is the only object that is a constituent of the proposition that the Sphinx is made out only of limestone, that proposition is also about every proper part of the Sphinx. I am not taking a stand on these arguments.

⁹ One can similarly see that the information that the negation of the proposition that Proust is a writer provides only about Proust amounts to that negation itself (since that negation is also only about Proust).

¹⁰ One might object that a necessary proposition is not about any object (cf. Goodman [1961: 4]; contrast Lewis [1988: 140–1]), so, no necessary proposition is only about Proust. If so, I reply, then modify Definition 2 by specifying that, if no proposition is both only about o and entailed by Q, then the information that Q provides only about o is (for example) the necessary proposition that o exists if o exists.

¹¹ Proof of (a). Since Q entails every proposition that is both only about o and entailed by Q, Q entails the conjunction of all these propositions—namely, Inf_o(Q). Moreover, if Q is only about o, then Q is a proposition that is both only about o and entailed by Q, so Q is entailed by the conjunction of all these propositions—namely, by Inf_o(Q). Proof of (b). If Q entails R, then every proposition that is both only about o and entailed by R is also a proposition that is both only about o and entailed by Q, so the conjunction of all the latter propositions—namely, Inf_o(Q)—entails the conjunction of all the former propositions—namely, Inf_o(R). Proof of (c). Suppose that R is only about o. By (b), Inf_o(Q & R) entails Inf_o(R) & Inf_o(Q), and thus entails R & Inf_o(Q)—since, by (a), R is logically equivalent to Inf_o(R). Conversely, to prove that R & Inf_o(Q) entails Inf_o(Q & R), consider any proposition T that is both only about o and entailed by Q & R, and prove that T is also entailed by R & Inf_o(Q)—i.e. prove that the following proposition (call it Y) is necessary: if R & Inf_o(Q) is true, then T is true. Let Z be the disjunction of Q with the negation of Inf_o(Q). Then Z & Inf_o(Q) is logically equivalent to Q & Inf_o(Q), and thus, by (a), to Q. Since Q & R entails T, (Z & Inf_o(Q)) & R entails T, so Z entails Y. Since Y is only about o (because R, Inf_o(Q), and T are only about o), to prove that Y is necessary it is enough to prove that any proposition that is both only about o and entailed by Z is necessary. To prove this, let X be such a proposition. Since X is entailed by Z, X is entailed by Q, and X is also entailed by the negation of Inf_o(Q). By contraposition, ∼X (i.e. the negation of X) entails Inf_o(Q), so ∼X entails every proposition that is both only about o and entailed by Q. But X is such a proposition; so ∼X entails X, and thus X is necessary. (By the way, this proof also shows that, for any proposition Q, there is a proposition Z such that both Inf_o(Z) is necessary and Q is logically equivalent to Z & Inf_o(Q).)

¹² Although, as I said, I am talking about logical necessity and entailment throughout this paper, it is worth noting that different kinds of necessity and entailment correspond to different kinds of informativeness. For example, the proposition that Bucephalus is not a philosopher entails both logically and metaphysically the proposition that Bucephalus is a philosopher only if Bucephalus is a writer. So, say that the proposition Q₃ that all philosophers are writers is both logically and metaphysically uninformative about Bucephalus given that Bucephalus is not a philosopher. By contrast, if it is metaphysically but not logically necessary that no horse is a philosopher, then the proposition that Bucephalus is a horse entails metaphysically but arguably not logically the proposition that Bucephalus is a philosopher only if Bucephalus is a writer. If so, say that Q₃ is metaphysically uninformative but logically informative about Bucephalus given that Bucephalus is a horse.

¹³ One might object that Q_B does provide only about Camus the information that Camus exemplifies the property of admiring Zola: Q_B does not provide this information about Zola or about anyone else. I reply that the information that Camus exemplifies the property of admiring Zola is about both Camus and Zola, and thus is not only about Camus. My claim that Q_B provides no information only about Camus is not the claim that there is no property that Q_B attributes only to Camus; it is instead the claim that no non-necessary proposition entailed by Q_B is only about Camus. Compare: the proposition that Camus and Zola are both French does provide information (which is) only about Camus—namely, the proposition that Camus is French—although it does not attribute the property of being French only to Camus.

¹⁴ I take both the proposition that o is an Australian swan and the proposition that o exemplifies the property of being an Australian swan to be the singular proposition with respect to o that it is an Australian swan (see Cartwright [1997: 73–6]).

¹⁵ One can prove that this definition of unconditional informativeness has the desirable consequence that Q is unconditionally uninformative about o exactly if Q is (conditionally) uninformative about o given some (equivalently: any) necessary proposition R—equivalently, given any proposition R that is only about o. By contrast, if Q is unconditionally uninformative about o, Q may still be (conditionally) informative about o given a non-necessary proposition R that is not only about o. To see this, go back to the last example I gave in section 3: the proposition that Sartre is French is unconditionally uninformative about Proust, but is (conditionally) informative about Proust given the proposition that Sartre is French only if Proust is a writer. One might object in two ways to my definition of unconditional informativeness. (1) One might argue that the proposition that π is a transcendental number is unconditionally informative about π although the information that it provides only about π is necessary. I reply that this proposition is metaphysically (and maybe also conceptually) but not logically necessary; as I said, I am talking about logical necessity throughout this paper. (2) One might argue that the proposition that everything distinct from Socrates is material is unconditionally informative about Socrates—since it raises the probability that Socrates is also material—although the information that it provides only about Socrates (namely, the proposition that Socrates is distinct from Socrates only if Socrates is material) is necessary (given that, necessarily, Socrates is not distinct from Socrates). I reply that in this paper I consider only deductive (not inductive) informativeness.

¹⁶ I am grateful to John Bengson, Alan Hájek, John Mackay, Michael Titelbaum, Russ Shafer-Landau, Alan Sidelle, some anonymous reviewers, and especially Elliott Sober for comments, and to my mother for typing the bulk of the paper. Special thanks are due to Aviv Hoffmann for extremely helpful comments.