Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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Rule-follow<strong>in</strong>g as Coord<strong>in</strong>ation: A Game-<strong>the</strong>oretic Approach<br />
Giacomo Sillari, Philadelphia, Pennsylvania, USA<br />
Make <strong>the</strong> follow<strong>in</strong>g experiment: say “It’s cold here”<br />
<strong>and</strong> mean “It’s warm here”.<br />
Can you do it?<br />
Ludwig Wittgenste<strong>in</strong>, Philosophical Investigations, §510.<br />
I can’t say “it’s cold here” <strong>and</strong> mean “it’s warm here”—<br />
at least, not without a little help from my friends.<br />
David Lewis, Convention.<br />
1. Rule-follow<strong>in</strong>g, coord<strong>in</strong>ation <strong>and</strong><br />
normativity<br />
The slogan that “mean<strong>in</strong>g is normative” is best understood<br />
<strong>in</strong> <strong>the</strong> context of strategic <strong>in</strong>teraction <strong>in</strong> a community of<br />
<strong>in</strong>dividuals. Famously, Kripke has argued <strong>in</strong> (Kripke 1982)<br />
that <strong>the</strong> central portion of <strong>the</strong> Philosophical Investigations<br />
describes both a skeptical paradox <strong>and</strong> its skeptical solution.<br />
Solv<strong>in</strong>g <strong>the</strong> paradox <strong>in</strong>volves <strong>the</strong> element of <strong>the</strong><br />
community, which determ<strong>in</strong>es conditions of assertability <strong>in</strong><br />
<strong>the</strong> language. A battery of argument is used to show that<br />
mean<strong>in</strong>g (or, <strong>in</strong> general, rule-follow<strong>in</strong>g) cannot be expla<strong>in</strong>ed<br />
by resort<strong>in</strong>g to an <strong>in</strong>dividual’s mental states, or her<br />
past use, or her dispositions. By exclusion, this <strong>in</strong>dicates<br />
that no descriptive fact is constitutive of mean<strong>in</strong>g, <strong>and</strong><br />
hence that “mean<strong>in</strong>g is normative.” Arguably, <strong>the</strong> normativity<br />
of mean<strong>in</strong>g stems from <strong>the</strong> assertability conditions hold<strong>in</strong>g<br />
<strong>in</strong> <strong>the</strong> society (<strong>in</strong>deed, membership <strong>in</strong> <strong>the</strong> community<br />
depends on one’s record of compliance.) But how exactly<br />
is <strong>the</strong> existence of such conditions susta<strong>in</strong>ed <strong>in</strong> <strong>the</strong> community?<br />
And is it accurate to say that <strong>the</strong>re is no fact to <strong>the</strong><br />
matter of rule-follow<strong>in</strong>g?<br />
I need an important caveat here: To answer <strong>the</strong>se<br />
questions, I momentarily step back from analyz<strong>in</strong>g<br />
mean<strong>in</strong>g <strong>and</strong> elaborate on <strong>the</strong> more general notion of rulefollow<strong>in</strong>g<br />
<strong>in</strong>stead. Kripke himself uses <strong>the</strong> terms mean<strong>in</strong>g<br />
<strong>and</strong> rule-follow<strong>in</strong>g ra<strong>the</strong>r <strong>in</strong>terchangeably <strong>in</strong> (Kripke 1982).<br />
I will conform to <strong>the</strong> ambiguous usage for ease of<br />
exposition, <strong>and</strong> mention my justification for it <strong>in</strong> <strong>the</strong> last<br />
section of this contribution.<br />
Wittgenste<strong>in</strong> states (§§198, 199) that a rule is<br />
followed <strong>in</strong>sofar as <strong>the</strong>re exists a custom, a convention. I<br />
argue that this <strong>and</strong> similar remarks <strong>in</strong> <strong>the</strong> Philosophical<br />
Investigations are illum<strong>in</strong>ated when looked at through <strong>the</strong><br />
lens of David Lewis’s <strong>the</strong>ory of convention. Lewis argues <strong>in</strong><br />
(Lewis 1969) that coord<strong>in</strong>ation games (situations of<br />
strategic <strong>in</strong>teraction <strong>in</strong> which <strong>the</strong> <strong>in</strong>terest of <strong>the</strong> players<br />
roughly co<strong>in</strong>cide) underlie every <strong>in</strong>stance of convention, <strong>in</strong><br />
that a convention is a regularity <strong>in</strong> <strong>the</strong> solution (equilibrium)<br />
of recurrent coord<strong>in</strong>ation games. The agents participat<strong>in</strong>g<br />
<strong>in</strong> <strong>the</strong> convention conform to <strong>the</strong> regularity because <strong>the</strong>y<br />
prefer conformity over non-conformity, conditional on o<strong>the</strong>r<br />
agents’ conform<strong>in</strong>g. They form <strong>the</strong> belief about o<strong>the</strong>r<br />
agents’ conformity through some coord<strong>in</strong>ation device:<br />
explicitly—through agreement—or tacitly—because a<br />
certa<strong>in</strong> action st<strong>and</strong>s out as <strong>the</strong> one that most likely<br />
(almost) everyone will pick. Such an action is salient to <strong>the</strong><br />
parties. In <strong>the</strong> case of a recurrent coord<strong>in</strong>ation problem, a<br />
special k<strong>in</strong>d of salience—precedent—is at play.<br />
Conventionality <strong>in</strong> <strong>the</strong> sense of Lewis is sufficient for<br />
some degree of normativity to arise. Indeed, <strong>in</strong> a<br />
community <strong>in</strong> which a certa<strong>in</strong> custom is <strong>in</strong> place—say, <strong>the</strong><br />
custom of go<strong>in</strong>g by sign-posts—<strong>the</strong>re is an equilibrium <strong>in</strong><br />
<strong>the</strong> actions <strong>and</strong> beliefs of <strong>the</strong> agents <strong>in</strong>volved such that <strong>the</strong><br />
agents prefer conformity to <strong>the</strong> custom, provided that all<br />
o<strong>the</strong>r members <strong>in</strong> <strong>the</strong> community act accord<strong>in</strong>g to <strong>the</strong><br />
convention. If I do not go by sign-posts, or I go by <strong>the</strong>m <strong>in</strong><br />
a funny, abnormal way (for <strong>in</strong>stance, go<strong>in</strong>g <strong>in</strong> <strong>the</strong> direction<br />
opposite to <strong>the</strong> one <strong>in</strong>dicated) I act contrary to both my<br />
preferences—because I will not get where I <strong>in</strong>tend to go—<br />
<strong>and</strong> <strong>the</strong> preferences of o<strong>the</strong>r members of <strong>the</strong> community—<br />
because, say, I will end up be<strong>in</strong>g late, or not show<strong>in</strong>g up at<br />
all. My reputation will suffer. This <strong>in</strong>dicates that, <strong>in</strong> general,<br />
parties to a convention feel, to a larger or smaller extent,<br />
<strong>the</strong> pressure to conform. As Lewis puts it, conventions are<br />
a k<strong>in</strong>d of social norm. But are we entitled to cast <strong>the</strong> rulefollow<strong>in</strong>g<br />
phenomenon <strong>in</strong> a game-<strong>the</strong>oretic account of<br />
convention?<br />
In its most general terms, <strong>the</strong> communitarian view<br />
ma<strong>in</strong>ta<strong>in</strong>s that, while many <strong>in</strong>terpretations of a given rule<br />
may arise, <strong>the</strong>re is (roughly speak<strong>in</strong>g) only one correct way<br />
to abide by <strong>the</strong> rule, as determ<strong>in</strong>ed by <strong>the</strong> community. In<br />
particular, <strong>the</strong> customary action is <strong>the</strong> action that<br />
accurately corresponds to <strong>the</strong> rule. The problem with<br />
arguments of this general form is that <strong>the</strong> same skeptical<br />
paradox meant to show <strong>the</strong> impossibility of solipsistic rulefollow<strong>in</strong>g<br />
applies to <strong>the</strong> community. Which is <strong>the</strong> customary<br />
action? And why? Past use is no sufficient grounds to<br />
answer such questions for <strong>the</strong> community, as it is not<br />
sufficient grounds <strong>in</strong> <strong>the</strong> solipsistic case, s<strong>in</strong>ce <strong>the</strong><br />
community can come up with a variety of <strong>in</strong>terpretations of<br />
<strong>the</strong> rule, just as well as <strong>the</strong> <strong>in</strong>dividual can. However, if we<br />
<strong>in</strong>troduce a strategic element <strong>in</strong> <strong>the</strong> behavior of community<br />
members, <strong>the</strong>n <strong>the</strong> skeptical paradox disappears (or, as<br />
we shall see <strong>in</strong> <strong>the</strong> next section, gets “pushed towards<br />
bedrock.”) If we <strong>in</strong>terpret rule-follow<strong>in</strong>g as coord<strong>in</strong>ation<br />
equilibrium <strong>in</strong> a coord<strong>in</strong>ation problem, <strong>the</strong>n <strong>the</strong>re is a clear<br />
<strong>and</strong> compell<strong>in</strong>g fact to <strong>the</strong> matter of what “go<strong>in</strong>g by <strong>the</strong><br />
rule” consists of. In particular, <strong>in</strong>dividuals (<strong>and</strong> <strong>the</strong><br />
population <strong>the</strong>y <strong>in</strong>teract with) who go by <strong>the</strong> rule net a<br />
higher payoff than do <strong>in</strong>dividuals who transgress <strong>the</strong> rule.<br />
Moreover, transgress<strong>in</strong>g <strong>the</strong> rule comes at a price, both for<br />
<strong>the</strong> transgressor <strong>and</strong> for <strong>the</strong> agents <strong>in</strong>teract<strong>in</strong>g with him.<br />
Non-conformative behavior will end up be<strong>in</strong>g sanctioned<br />
(eventually with expulsion from <strong>the</strong> community), while<br />
conformative behavior will perpetuate itself, be<strong>in</strong>g based<br />
on <strong>the</strong> agreement to act accord<strong>in</strong>g to given rules. In this<br />
sense, agreement is <strong>the</strong> agreement <strong>in</strong> preferences <strong>and</strong><br />
beliefs that support a specific equilibrium <strong>in</strong> <strong>the</strong> recurrent<br />
coord<strong>in</strong>ation game.<br />
Thus, <strong>the</strong> “little help” needed by Lewis from his<br />
friends <strong>in</strong> <strong>the</strong> answer to <strong>the</strong> challenge of §510 reported <strong>in</strong><br />
<strong>the</strong> epigraph consists <strong>the</strong>n <strong>in</strong> <strong>the</strong>ir agree<strong>in</strong>g to change <strong>the</strong>ir<br />
preferences <strong>and</strong> beliefs, switch<strong>in</strong>g <strong>in</strong> so do<strong>in</strong>g from one<br />
solution of a recurrent coord<strong>in</strong>ation game to ano<strong>the</strong>r.<br />
Consider §224:<br />
The word “agreement” <strong>and</strong> <strong>the</strong> word “rule” are related<br />
to one ano<strong>the</strong>r, <strong>the</strong>y are cous<strong>in</strong>s. If I teach<br />
anyone <strong>the</strong> use of <strong>the</strong> one word, he learns <strong>the</strong> use<br />
of <strong>the</strong> o<strong>the</strong>r with it.<br />
I believe that <strong>the</strong> view expressed <strong>in</strong> this section captures<br />
<strong>the</strong> sense <strong>in</strong> which “agreement” <strong>and</strong> “rule” are related: A<br />
custom—<strong>and</strong> hence a rule—does not hold without an<br />
agreement <strong>in</strong> preferences <strong>and</strong> beliefs—<strong>and</strong> hence <strong>in</strong> co-<br />
325