An objective counterfactual theory of information

Abstract

We offer a novel theory of information that differs from traditional accounts in two respects: (i) it explains information in terms of counterfactuals rather than conditional probabilities, and (ii) it does not make essential reference to doxastic states of subjects, and consequently allows for the sort of objective, reductive explanations of various notions in epistemology and philosophy of mind that many have wanted from an account of information.

Notes

¹This work is fully collaborative; the authors are listed in alphabetical order.

²Here and in what follows, x's being F and y's being G are construed as actual events; it follows that (on (W)), one event carries information about a second only if the second actually occurs.

³Our interest in (W), and why it goes wrong, is partly autobiographical; see Meskin and Cohen forthcoming.

⁴The same result follows from Dretske's slightly different setup (cf. §II). If there is no possible alternative outcome for p, then the conditional probability of p given k alone is unity; hence, on this definition, too, nothing can carry the information that p; see Dretske 1981: 12].

⁵This formulation leaves it open that events can carry information about themselves. One could avoid this outcome (if desired) by stipulating that informational relations only obtain between distinct events. We take no stand as to the advisability of this stipulation.

⁶Like Dretske's and Shannon's understandings of information (cf. §I.A), (S*) entails that no information can be carried about necessary events. On the other hand, there are two ways in which this assumption might be relaxed without giving up on the spirit of the present proposal. The first option would be to simply stick with (S) as opposed to (S*). A drawback of this option is that (S) entails that all signals carry information about every necessary event; but presumably anyone interested in making room for the possibility of information-carrying about necessary events would find this entailment intolerable. A second alternative would be to use a non-standard account of the semantics of counterfactuals according to which not all counterlegals are vacuously true. Thanks to Fred Dretske for encouraging us to think about this issue.

⁷The remarks that follow are applicable to other accounts of information (both semantic and quantitative) that are grounded in conditional probabilities. Most saliently, consider the setup of Shannon 1948: let {s₁, … ,s_n} be discrete alternative states of a source s with probabilities {P(s₁), … ,P(s_n)} respectively, and let {r₁, … ,r_k} be discrete alternative states of a receiver r with probabilities {P(r₁), … ,P(r_k)} respectively; assume that P(s_i) > 0 for all i, that P(r_j) > 0 for all j, and that . Shannon defines the mutual information between s and r as follows:. So defined, mutual information makes ineliminable reference to the same sorts of inverse conditional probabilities as Dretske's theory, and so is vulnerable to the concerns we raise about the interpretation of those probabilities. Considerations of convenience and specificity suggest that we should restrict attention to a single probabilistic account in what follows; given that it is already familiar to so many philosophers, it seems reasonable to choose Dretske's account as our focus.

⁸The classic statement of the problem that propensity theories have with ‘inverse’ probabilities (often called ‘Humphreys's paradox’) occurs in Humphreys 1985. For reviews of the literature on Humphreys's paradox, see Gillies 2000 and Humphreys 2004.

⁹Another reason for thinking that the account presented here deserves to be thought of as a theory of information comes from consideration of the conditions (derived from the quantitative framework developed in Dretske 1981) that, according to Dretske, an adequate theory of information must respect [Dretkse 1981: 63 – 6]. While we lack the space to rehearse Dretske's quantitative framework and motivate his conditions here, we wish to point out that the counterfactual account satisfies two of his three conditions, and provides principled reasons for rejecting the third. (That the counterfactual account should follow Dretske's account reasonably closely on these matters is not so surprising, in view of the intimate relationship between probabilities and counterfactuals.) The counterfactual account respects condition B (i.e., that a signal carries the information that y is G only if y is G) [Dretske 1981: 64] because we have restricted informational relations to holding between actual events (n. 2); consequently, we may infer that if x's being F carries information about y's being G (as understood by (S*)), then y is G. The counterfactual account also respects condition C (i.e., that a signal carries the information that y is G only if the quantity of information the signal carries about y is (or includes) that quantity generated by y's being G) [Dretske 1981: 64]. For the satisfaction of the counterfactual conditional in (S*) guarantees that, no matter what other information is carried by x's being F, that state of affairs includes the information associated with y's being G—even if x's being F is correlated in the actual world with various actual states of affairs, we know that the one correlation that will persist as we move between worlds is that between x's being F and y's being G. However, the counterfactual account does not secure condition A (i.e., that a signal carries the information that y is G only if it carries as much information about y as would be generated by y's being G) [Dretske 1981:63]. This is because the non-vacuous truth of the counterfactual ┌if x were not F, then y would not have been G┐ allows that the signal (viz., x's being F) might have a non-zero equivocation (i.e., it leaves it open that P(y's being G|x and x's being F) < 1). We shall argue in §III.B that this result is unobjectionable—indeed, advantageous—for a theory of information (cf. Scarantino 2005). Consequently, pace Dretske, we neither regard condition A as a necessary condition for any theory of information worthy of the name, nor do we regard the failure of the counterfactual account to secure this condition as reason for thinking that it should not count as a theory of information.

¹⁰‘The transmission of information requires, not simply a set of de facto correlations, but a network of nomic dependencies between the condition at the source and the properties of the signal’ [Dretske 1981: 76 – 7]. Cf. also the quotation in n. 25.

¹¹We have modified the proposal in Loewer 1983: 76], inserting the insistence that the conditional's consequent be contingent, as a way of blocking objections about vacuous satisfaction—objections that are parallel to those that motivated the transition from (S) to (S*) in §I.

¹²For ease of exposition, here we use single letters as variables for events, although these should officially be understood as shorthand for predicational structures of the form ┌x is F┐.

¹³Thanks to Stephen Meskin for help thinking about the conditional probabilities here.

¹⁴As a referee points out, Dretske claims another benefit from the idea that the information relation requires a conditional probability of p on r (and k) of 1; namely, since he understands knowledge in terms of information (see §IV.A), this idea provides for him a response to the lottery paradox [Dretske 1981: 99 – 102]. In not making this requirement, then, we are also giving up Dretske's response to the paradox. We're not inclined to worry too much about this (even if, as per Dretske, knowledge is construed in terms of information); for there are a number of alternative responses to the paradox on offer (cf. Vogel 1992; Hawthorne 2003).

¹⁵At one point Dretske claims that the relativization at issue is supported by ordinary ways of thinking: ‘I propose to follow Dennett in relativizing the information contained in a signal [to the knowledge of the recipients] because I think this accurately reflects our ordinary ways of thinking about such matters’ [Dretske 1981: 79]. In response, we suggest: (1) it is not at all clear to us that ordinary ways of thinking do support relativization, and (2) even if relativization did reflect ordinary ways of thinking, we are unconvinced that the technical notion of information should be beholden to such thinking.

¹⁶Dretske himself raises this sort of issue with Kenneth Sayre's information theoretic account of recognition: ‘One cannot use the concept of information in the analysis of such epistemic concepts as knowledge, recognition, and memory if one turns around and uses epistemic concepts (such as learn, identify, or correctly interpreting) in the analysis of information. This makes the entire program circular’ [Dretske 1981: 244].

¹⁷Here and below we assume (for ease of exposition) that there is some coherent way of making sense of the conditional probabilities.

¹⁸Raymond Martin draws similar conclusions [Martin 1975; Martin 1983, and then uses them to argue against the counterfactual accounts of knowledge of Dretske 1971 and Nozick 1981. We agree with Martin about the context-sensitivity of our evaluations of counterfactuals; however, this is compatible with holding that there are non-doxastic facts of the matter about which counterfactuals are true (despite the context-sensitivity of our assessments). If information is construed as depending on which counterfactuals are true rather than which counterfactuals we take to be true, Martin's point is not damaging to the account of information we are exploring in the way Martin alleges it is damaging to counterfactual theories of knowledge. (We thank Dretske for bringing the Martin papers to our attention.)

¹⁹For reasons of brevity, we'll compare Dretske's doxastic theory only against an objective counterfactual theory of information in this section. As far as we can see, what we say about this comparison holds (mutatis mutandis) about the comparison between Dretske's view and an objective probabilistic theory of information as well.

²⁰It is plausible that, in such a case, the first investigator would have come to learn (and, in fact, know) that the peanut is under shell 4, even though the same does not appear plausible for the second investigator (even though both investigators received the information that the peanut was under shell 4). We take this to show that any non-doxastic information-based account of knowledge will have to locate the difference between the epistemic status of investigator one and investigator two in some further fact—consequently, on this sort of theory, the relationship between knowledge and information will be less straightforward than Dretske proposes 1981: 86]; cf. §IV.A. For suppose both investigators come to believe that the peanut is under shell 4 on the basis of the examination of shell 3. Dretske's (unmodified) account of knowledge would seem to entail that both investigators have come to know the location of the peanut—and this verdict seems implausible in the case of investigator 2. But the objective counterfactual account does not face insoluble problems, since there appear to be the resources to explain the different epistemic status of the two investigators: namely, it is plausible that the first investigator's knowledge of the location of the peanut depends (in part) on having received additional information from her prior examination of shells 1 and 2.

²¹We owe this objection to Craig Callender.

²²See Fine 1975 for one of the early objections to Lewis's semantics for counterfactuals, and Tooley 2003 and Schaffer 2004 for recent discussion of the problems with the standard account.

²³Lewis's counterfactual account of causation is especially relevant in the present context since that account, like the counterfactual account of information, rules out vacuous counterfactuals by fiat.

²⁴One standard answer to traditional preemption problems is appeal to the existence of chains of counterfactually related intermediary events (i.e., intervening causal processes or causal chains) between causes and effects that are ‘cut off’ in the case of preempted would-be-causes and effects [Lewis 1986a. While this move might be thought to handle some sorts of preemption, some authors have pointed to other sorts of preemption (e.g., so-called late and trumping preemption) that seem to resist such a solution [Lewis 1986b; Schaffer, 2000.

²⁵Cf. Dretske 1981; 245]: ‘In saying that the conditional probability (given r) of s's being F is 1, I mean to be saying that there is a nomic (lawful) regularity between these two event types, a regularity which precludes r's occurrence when s is not F’. In preemption cases, r's occurrence is not precluded when s is not F, since some other event might bring about r.

 On the other hand, (S*) allows that the thermometer in case 4 might carry the disjunctive information that either the room is 70°F or that an angel moved the thermometer to read 70°F. We'll ignore this complication.

²⁶For this reason, the proponent of the counterfactual account is likely to agree with Dretske's argument that the information relation is not a causal relation [Dretske 1981: 26 – 39].

²⁷For, on that view, of course, the presence of the informational relation is tantamount to the non-vacuous truth of the counterfactual ┌if y were not G, then x would not be F┐. Consequently, the counterfactual supposition that ¬G, conjoined with the counterfactual account, entails that ┌if y were not G, then x would not be F┐ is false—a result that is inconsistent with the critic's stipulation that ┌¬Fx┐ holds in all the closest ¬Gy-worlds.

²⁸For if the counterfactual supposition entails that the informational relation is present, then this, together with the counterfactual account, entails that ┌Fx┐ fails in all the closest ¬Gy-worlds.

²⁹As far as we can see, this response to Shope-style counterexamples is generalizable to other proposed counterfactual analyses of technical (i.e., non-folk) notions.

³⁰We are grateful to Craig Callender, Fred Dretske, Agustín Rayo, and Rob Rupert for helpful discussion of these matters and comments on early drafts. Two anonymous referees for this journal have also helped us improve the paper.