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01-08-2014

The Logic of Knowledge and the Flow of Information

Author: Simon D’Alfonso

Published in: Minds and Machines | Issue 3/2014

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Abstract

In this paper I look at Fred Dretske’s account of information and knowledge as developed in Knowledge and The Flow of Information. In particular, I translate Dretske’s probabilistic definition of information to a modal logical framework and subsequently use this to explicate the conception of information and its flow which is central to his account, including the notions of channel conditions and relevant alternatives. Some key products of this task are an analysis of the issue of information closure and an investigation into some of the logical properties of Dretske’s account of information flow.

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Although the k variable which occurs in Dretske’s definition relativises information to what the receiver already knows concerning the possibilities at the source, this relativisation is only meant to accommodate the way information is thought about, that the information one can get from a signal depends on what they already know; it does not undermine the essential objectivity of information.

This idea of channel conditions/relevant alternatives will be explored more in section "Information Closure". There is no determinate method to decide what counts and what does not count as a relevant alternative. In general the selection will depend upon the knowing agent and their environment and will also be a pragmatic decision. Particularly due to this lack of clear determination the notion of relevant alternatives has been a point of philosophical contention and from one point of view its application is seen to be somewhat ad hoc. Nonetheless, it is a valuable idea that can serve as a foundation for accounts that afford a flexible way to realistically talk about information (and knowledge).

A precis of Knowledge and the Flow of Information plus some commentaries can be found in Dretske (1983).

Technically there are some exceptions involving formulas with a probability of zero and hence undefined calculations. With monotonicity, Pr(p|p) = 1 [$p \sqsupset p$] but $\hbox{Pr}(p | p \wedge \neg p)$ is undefined [not $(p \wedge \neg p) \sqsupset p$]. With Reverse Disjunction, $\hbox{Pr}(p | p \vee (q \wedge \neg q)) = 1$ [$(p \vee (q \wedge \neg q)) \sqsupset p$] but $\hbox{Pr}(p | q \wedge \neg q)$ is undefined [not $(q \wedge \neg q) \sqsupset p$].

The account given by C&M is itself quite problematic, with one reason being that properties of a counterfactual conditional discord too much with properties commonly associated with information flow. If an account of information is to abandon the universality of these properties, it could at least offer some accommodation by delineating the restricted applicability of these properties and account for why they are linked to an ordinary conception of information flow. Demir (2011) recently pointed a problematic consequence of the counterfactual theory of information that arguably makes it untenable. He shows that given the standard possible worlds account of counterfactuals, according to C&M’s definition “‘A carries information that B’ necessarily implies ‘A carries information that B and C’ for any C such that the closest not-C world is more remote than the closest not-B world".

The reflexivity property also gives us other desirable validities which would not hold without it. For example, if A carries the information that B carries the information that C, then A and B together carry the information that $C: \,\square(A \supset \square(B \supset C)) \vdash \square((A \wedge B) \supset C)$

Other clearly problematic results follow from further additions. For example, adding the zoological truth $\square_{1}(Pa \supset \neg Fa)$ leads to the problematic $\square_{2}(Pa \supset \neg Ba). $

Perhaps expressed more clearly with the equivalent: $\diamondsuit_{2}Pa \supset \neg \square_{2}(Ba \supset Fa)$

Regarding $\square_{1}, \,\square_{1}(Ba \supset Fa)$ is illegitimate and leads to the incorrect $\square_{1}(Pa \supset \neg Pa). \,\square_{1}(Ba \wedge Pa \supset \neg Fa)$ is derivable in the system so the addition of $\square_{1}(Ba \wedge Pa \supset Fa)$ would lead to the unacceptable $\neg \diamondsuit_{1}(Ba \wedge Pa) \vee \diamondsuit_{1}(Fa \wedge \neg Fa). $ Hence $\square_{1}(Ba \wedge Pa \supset Fa)$ is also illegitimate.

Perhaps in less practical discourse, a catch-all clause C could act as a stand-in of sorts, such that $\square(A \supset B) \supset \square \neg C. $

See Luper (2010) for an overview.

$\rightarrow$ is being used here as a general representation of an implication/conditional

This is sometimes talked about in terms of nearby possible worlds

This is further evidenced by a subsequent case that Shackel makes, according to which, Dretske is committed to closure for a modified version of closure, a general closure principle for signalled information.

On a related note, another part of Dretske’s account which would need to be revised is information nesting as mentioned in section “Dretske on Information and Knowledge”:

$\ldots$ if a signal carries the information that s is F, it also carries all the information nested in s’s being $F.\, \ldots$
$$ \hbox{The information that }t\hbox{ is }G\hbox{ is nested in }s'\hbox{s being }F = s'\hbox{s being }F\hbox{ carries the information that }t\hbox{ is }G.\hbox{ ( Dretske 1981, p. 71)} $$

He also distinguishes between information that is analytically nested and information that is nomically nested. As the zebra-mule example shows, caution must be exercised with nesting and its applicability regulated, for although ‘not-mule’ is necessarily implied by ‘zebra’, the information that ‘not-mule’ is not necessarily analytically nested in the information that ‘zebra’.

Heller (1999) offers a good explication of Dretske’s type of relevant alternatives account that answers opponent contextualists such as Stine.

As suggested by an anonymous referee, perhaps a better name for Dretske’s view would be ‘situationism’, since it relies only on objective features of the epistemic agent’s situation and not on contexts of discourse and such.

Kripke presented this and some other examples at an APA session in the early 1980s. An extensive published critique of Nozick’s account, including discussion of the red barn example, can be found in Kripke (2011).

Dretske (2003, p. 117) briefly discusses this example and offers the case for why it does not affect his account, even if it affects Nozick’s. Adams (2005, p. 8) offers the case for why Kripke’s example does not undermine Nozick’s or Dretske’s account either.

In its full form this statement is an analytic truth. For our purposes this is a simplified version, since if something is an equid it could also be a horse or donkey. Adding these would not affect the argument though, since all the options are mutually exclusive.

See Chellas (1980) for a comprehensive look at such systems of non-normal modal logic.

Adams, F. (2005). Tracking theories of knowledge. Veritas 50, 1–35, draft available online at http://revistaseletronicas.pucrs.br/ojs/index.php/veritas/article/viewFile/1813/1343.

Baumann, P. (2006). Information, closure, and knowledge: On jager’s objection to dretske. Erkenntnis, 64, 403–408.CrossRefMATH

Black, T. (2006). Contextualism in epistemology. In J. Fieser & B. Dowden (Eds.), Internet encyclopedia of philosophy, july 15, 2006 edn, URL: http://www.iep.utm.edu/contextu/.

Chellas, B. F. (1980). Modal logic: An introduction. Cambridge: Cambridge University Press.CrossRefMATH

Cohen, J., & Meskin, A. (2006). An objective counterfactual theory of information. Australasian Journal of Philosophy, 84(3), 333–352.CrossRef

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Demir, H. (2011). The counterfactual theory of information revisited. Australasian Journal of Philosophy, 1–3 (online first).

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Title: The Logic of Knowledge and the Flow of Information
Author: Simon D’Alfonso
Publication date: 01-08-2014
Publisher: Springer Netherlands
Published in: Minds and Machines / Issue 3/2014
Print ISSN: 0924-6495
Electronic ISSN: 1572-8641
DOI: https://doi.org/10.1007/s11023-013-9310-x

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