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␈↓ α∧␈↓␈↓ u1


␈↓ α∧␈↓α␈↓ β@A CARTESIAN PRODUCT THEORY OF COUNTERFACTUALS

␈↓ α∧␈↓α␈↓ ∧wby John McCarthy, Stanford University

␈↓ α∧␈↓␈↓ αTThis␈α∀note␈α∀proposes␈α∀a␈α∀theory␈α∀of␈α∀certain␈α∀counterfactual␈α∀conditional␈α∀sentences␈α∀based␈α∪on
␈↓ α∧␈↓informal␈αapproximate␈αcartesian␈αproduct␈α
theories␈αof␈αaspects␈αof␈α
the␈αworld.␈α Don't␈αconfuse␈αmy␈α
theory
␈↓ α∧␈↓of counterfactuals with the theory that I claim a counterfactual refers to.

␈↓ α∧␈↓␈↓ αTCounterfactual␈α
conditional␈αsentences␈α
are␈αimportant␈α
in␈αmaking␈α
computer␈αprograms␈α
that␈αcan
␈↓ α∧␈↓do␈α
common␈α
sense␈α
reasoning.␈α
 For␈α
example,␈α∞an␈α
action␈α
is␈α
considered␈α
intentional␈α
only␈α
if␈α∞the␈α
action
␈↓ α∧␈↓would have been different had the intentions been suitably different.

␈↓ α∧␈↓␈↓ αTOur theory has several aspects.

␈↓ α∧␈↓␈↓ αT1.␈α∀Within␈α∃a␈α∀theory␈α∀that␈α∃refers␈α∀to␈α∀a␈α∃cartesian␈α∀product␈α∀structure␈α∃for␈α∀a␈α∀state␈α∃space,␈α∀a
␈↓ α∧␈↓counterfactual␈α∩conditional␈α⊃sentence␈α∩that␈α⊃involves␈α∩an␈α⊃alternate␈α∩value␈α⊃of␈α∩one␈α⊃component␈α∩has␈α⊃a
␈↓ α∧␈↓definite␈α∞meaning,␈α∞because␈α∞a␈α∂definite␈α∞state␈α∞of␈α∞the␈α∂system␈α∞is␈α∞referred␈α∞to.␈α∂ Thus␈α∞if␈α∞a␈α∞state␈α∂has␈α∞the
␈↓ α∧␈↓components (3,4,7) called ␈↓↓x,␈↓ ␈↓↓y,␈↓ and ␈↓↓z,␈↓ then "if ␈↓↓x␈↓ were 5" refers to the state (5,4,7).

␈↓ α∧␈↓␈↓ αT2.␈α⊃We␈α∩propose␈α⊃to␈α∩refer␈α⊃other␈α∩counterfactuals␈α⊃to␈α⊃this␈α∩easy␈α⊃case␈α∩by␈α⊃using␈α∩such␈α⊃cartesian
␈↓ α∧␈↓product theories as "approximations" to more general situations.

␈↓ α∧␈↓␈↓ αT3.␈α∞At␈α∂first␈α∞sight␈α∂such␈α∞approximations␈α∞may␈α∂seem␈α∞arbitrary,␈α∂but␈α∞such␈α∂approximate␈α∞theories
␈↓ α∧␈↓often enjoy an objective preferred status.

␈↓ α∧␈↓␈↓ αTWe base our considerations on an example.

␈↓ α∧␈↓␈↓ αTSuppose␈α
two␈α
ski␈α
instructors␈α
observe␈α
their␈α
student␈α
fall.␈α
 The␈α
first␈α
ski␈α
instructor␈α
says,␈α
"If␈α
he
␈↓ α∧␈↓had␈α
bent␈α∞his␈α
knees␈α∞he␈α
wouldn't␈α∞have␈α
fallen".␈α∞ The␈α
second␈α∞instructor␈α
replies,␈α∞"No,␈α
he␈α∞would␈α
still
␈↓ α∧␈↓have␈α∞fallen,␈α
but␈α∞if␈α
he␈α∞had␈α
put␈α∞his␈α
weight␈α∞on␈α
his␈α∞downhill␈α
ski,␈α∞he␈α
wouldn't␈α∞have␈α∞fallen".␈α
 Then
␈↓ α∧␈↓they␈α∂look␈α∂at␈α∂a␈α∂videotape␈α∂of␈α∂the␈α∂event,␈α∂and␈α∂the␈α∂second␈α∂instructor␈α∂agrees␈α∂that␈α∂the␈α∂first␈α⊂one␈α∂was
␈↓ α∧␈↓correct.␈α
 If␈αhe␈α
had␈α
bent␈αhis␈α
knees,␈α
he␈αwouldn't␈α
have␈α
fallen,␈αand␈α
putting␈α
his␈αweight␈α
on␈αhis␈α
downhill
␈↓ α∧␈↓ski␈α⊃wouldn't␈α⊃have␈α⊃helped.␈α⊃ How␈α⊃can␈α⊃we␈α⊃explain␈α⊃their␈α⊃ability␈α⊃to␈α⊃agree␈α⊃on␈α⊃the␈α⊃truth␈α∩of␈α⊃these
␈↓ α∧␈↓counterfactual conditional sentences?

␈↓ α∧␈↓␈↓ αTWe␈αbelieve␈αthat␈αthe␈αinstructors␈αshare␈αa␈αtheory␈αof␈αskiing,␈αand␈αwhen␈αthis␈αtheory␈αis␈αapplied␈αto
␈↓ α∧␈↓the␈α∂information␈α∂obtained␈α⊂from␈α∂the␈α∂videotape,␈α⊂it␈α∂gives␈α∂a␈α⊂definite␈α∂answer.␈α∂ A␈α⊂suitable␈α∂computer
␈↓ α∧␈↓program␈αequipped␈αwith␈αthe␈αtheory␈αand␈αthe␈αinformation␈αon␈αthe␈αvideotape␈αwould␈αcome␈αto␈αthe␈α
same
␈↓ α∧␈↓conclusion.

␈↓ α∧␈↓␈↓ αTThe␈αtheory␈αmodels␈αthe␈αskier␈αas␈αa␈αstick␈αfigure␈αwith␈αjoints␈αthat␈αhe␈αoperates.␈α The␈αlimbs␈αhave
␈↓ α∧␈↓masses␈αand␈αmoments␈αof␈αinertia.␈α The␈αhill␈αhas␈αa␈αshape.␈α The␈αoutcome␈αis␈αa␈αfunction␈αof␈αthe␈αmotions
␈↓ α∧␈↓of␈α∞the␈α∞skiers␈α∞joints␈α∞and␈α∞the␈α∞slope␈α∞of␈α∞the␈α
hill.␈α∞ Built␈α∞on␈α∞this␈α∞theory␈α∞there␈α∞is␈α∞a␈α∞theory␈α∞of␈α
discrete
␈↓ α∧␈↓actions:␈α
bending␈αthe␈α
knees␈α
or␈αnot,␈α
putting␈α
the␈αweight␈α
on␈α
one␈αski␈α
or␈α
the␈αother␈α
or␈α
both,␈αswinging␈α
the
␈↓ α∧␈↓hips, and all the other actions described in the books written by ski instructors.

␈↓ α∧␈↓␈↓ αTNotice␈α∃that␈α∃this␈α∃theory␈α∃does␈α∃not␈α∃say␈α∃what␈α∃the␈α∃student␈α∃will␈α∃do;␈α∃it␈α∃merely␈α∃gives␈α∃the
␈↓ α∧␈↓consequences␈α
of␈α
the␈α
sequences␈α
of␈α
actions.␈α
 In␈α
the␈α
language␈α
of␈α
automata␈α
theory,␈α
the␈α
automaton␈αis
␈↓ α∧␈↓not␈α
autonomous␈α
but␈αhas␈α
inputs␈α
from␈αthe␈α
outside.␈α
 If␈α
whether␈αthe␈α
student␈α
bends␈αhis␈α
knees␈α
is␈αone␈α
of
␈↓ α∧␈↓␈↓ u2


␈↓ α∧␈↓the␈αinputs␈αto␈αthe␈αsystem,␈αthen␈αthe␈αtheory␈αdetermines␈αwhat␈αwill␈αhappen␈αif␈αthis␈αinput␈αis␈αaltered␈αand
␈↓ α∧␈↓the others are left unchanged.

␈↓ α∧␈↓␈↓ αTIf␈α∞we␈α∂take␈α∞the␈α∞theory␈α∂as␈α∞given,␈α∞the␈α∂truth␈α∞of␈α∞the␈α∂counterfactual␈α∞conditional␈α∂is␈α∞determined.
␈↓ α∧␈↓Notice␈α
the␈αcartesian␈α
product␈αstructure␈α
of␈αthe␈α
space␈α
of␈αthe␈α
skier's␈αactions.␈α
 Since␈αthe␈α
action␈α
of␈αthe
␈↓ α∧␈↓student␈α∂is␈α⊂described␈α∂by␈α⊂a␈α∂collection␈α∂of␈α⊂independent␈α∂inputs␈α⊂to␈α∂the␈α∂system,␈α⊂␈↓↓within␈α∂the␈α⊂theory␈↓␈α∂the
␈↓ α∧␈↓notion of changing one input and leaving the others fixed is well defined.

␈↓ α∧␈↓␈↓ αTWhile␈α∃interpreting␈α∃counterfactuals␈α∃relative␈α∃to␈α∃a␈α∃cartesian␈α∃product␈α∃theory␈α∃is␈α∃relatively
␈↓ α∧␈↓unproblematic,␈α⊃the␈α⊃more␈α∩usual␈α⊃philosophical␈α⊃goal␈α⊃is␈α∩to␈α⊃assign␈α⊃truth␈α⊃values␈α∩to␈α⊃counterfactuals
␈↓ α∧␈↓regarded␈α∩as␈α⊃statements␈α∩about␈α∩the␈α⊃world.␈α∩ A␈α∩statement␈α⊃relative␈α∩to␈α∩a␈α⊃particular␈α∩theory␈α∩can␈α⊃be
␈↓ α∧␈↓regarded␈α∩as␈α∩a␈α⊃statement␈α∩about␈α∩the␈α⊃world␈α∩provided␈α∩it␈α⊃can␈α∩be␈α∩argued␈α⊃that␈α∩this␈α∩theory␈α∩has␈α⊃a
␈↓ α∧␈↓preferred␈α
position␈α
among␈α
theories␈α∞of␈α
the␈α
world.␈α
 Thus␈α
we␈α∞argue␈α
that␈α
the␈α
student␈α∞wouldn't␈α
have
␈↓ α∧␈↓fallen␈α
if␈α
he␈α
had␈α
bent␈α
his␈αknees␈α
by␈α
claiming␈α
that␈α
the␈α
theory␈αused␈α
by␈α
the␈α
two␈α
instructors␈α
best␈αfits␈α
the
␈↓ α∧␈↓whole␈αphenomenon␈αof␈αskiing.␈α
 We␈αcan␈αweaken␈αthis␈αcondition␈α
slightly␈αby␈αrequiring␈αmerely␈αthat␈α
the
␈↓ α∧␈↓statement be true in all good theories of skiing.

␈↓ α∧␈↓␈↓ αTWhat␈α⊂makes␈α⊂a␈α⊂good␈α⊃theory␈α⊂of␈α⊂skiing.␈α⊂ Like␈α⊂a␈α⊃formal␈α⊂scientific␈α⊂theory,␈α⊂a␈α⊃common␈α⊂sense
␈↓ α∧␈↓theory␈α∩becomes␈α∩preferred␈α∩by␈α∩accounting␈α⊃for␈α∩a␈α∩wide␈α∩range␈α∩of␈α⊃phenomena␈α∩-␈α∩in␈α∩this␈α∩case␈α⊃the
␈↓ α∧␈↓experience␈α∞of␈α
ski␈α∞instructors.␈α∞ Since␈α
the␈α∞theory␈α
of␈α∞skiing␈α∞used␈α
by␈α∞ski␈α
instructors␈α∞depends␈α∞on␈α
the
␈↓ α∧␈↓vast␈α⊃experience␈α⊃of␈α⊃generations␈α∩of␈α⊃ski␈α⊃instructors␈α⊃and␈α⊃writers␈α∩about␈α⊃skiing,␈α⊃we␈α⊃should␈α∩not␈α⊃be
␈↓ α∧␈↓surprised␈α
that␈α
it␈αis␈α
difficult␈α
to␈αgive␈α
a␈α
definition␈αof␈α
"if␈α
the␈αstudent␈α
had␈α
bent␈αhis␈α
knees"␈α
that␈αdoes
␈↓ α∧␈↓not␈α⊂depend␈α∂on␈α⊂a␈α∂theory.␈α⊂ Indeed␈α∂I␈α⊂would␈α∂argue␈α⊂that␈α∂many␈α⊂common␈α∂sense␈α⊂statements␈α∂resemble
␈↓ α∧␈↓statements␈α∂in␈α∞the␈α∂natural␈α∂sciences␈α∞in␈α∂being␈α∞meaningful␈α∂only␈α∂within␈α∞elaborate␈α∂theories␈α∂based␈α∞on
␈↓ α∧␈↓large fragments of experience.

␈↓ α∧␈↓␈↓ αTWe proceed formally as follows:

␈↓ α∧␈↓␈↓ αT1.␈α⊂Relative␈α⊂to␈α⊂a␈α⊂space␈α⊂A␈α⊂described␈α⊂as␈α⊂a␈α⊂cartesian␈α⊂product,␈α⊂we␈α⊂can␈α⊂define␈α⊃changing␈α⊂one
␈↓ α∧␈↓component of an element of the space and leaving the others fixed.

␈↓ α∧␈↓␈↓ αT2.␈αWhen␈αa␈α
component␈αof␈αA␈α
changes,␈αthe␈αvalue␈αof␈α
a␈αfunction␈αf␈α
defined␈αon␈αthe␈αspace␈α
changes
␈↓ α∧␈↓in a definite way.

␈↓ α∧␈↓␈↓ αT3.␈αTherefore,␈α
the␈αtruth␈α
of␈αa␈α
counterfactual␈αconditional␈αof␈α
the␈αlike␈α
"If␈αthe␈α
␈↓↓i␈↓th␈αcomponent␈αof␈α
x
␈↓ α∧␈↓were 3, the value of f(x) would be 7" relative to some initial point x0.

␈↓ α∧␈↓␈↓ αT4.␈α⊃When␈α⊃a␈α∩description␈α⊃of␈α⊃the␈α⊃state␈α∩of␈α⊃an␈α⊃aspect␈α⊃of␈α∩the␈α⊃world␈α⊃has␈α⊃a␈α∩cartesian␈α⊃product
␈↓ α∧␈↓structure␈α∂in␈α∂a␈α∞theory␈α∂T,␈α∂then␈α∞we␈α∂can␈α∂interpret␈α∞counterfactual␈α∂conditional␈α∂sentences␈α∂involving␈α∞a
␈↓ α∧␈↓changed␈α∞value␈α∂of␈α∞a␈α∂function␈α∞of␈α∂the␈α∞on␈α∂the␈α∞aspect␈α∂caused␈α∞by␈α∂a␈α∞changed␈α∂value␈α∞of␈α∂a␈α∞component.
␈↓ α∧␈↓This interpretation is relative to the theory ␈↓↓T.␈↓

␈↓ α∧␈↓␈↓ αT5.␈αIf␈αwe␈αhave␈αgood␈αscientific␈αor␈αcommon␈α
sense␈αreasons␈αfor␈αpreferring␈αthe␈αtheory␈α␈↓↓T,␈↓␈αthen␈α
we
␈↓ α∧␈↓can regard such counterfactuals as objectively true or false.

␈↓ α∧␈↓␈↓ αTThe␈α∀motivation␈α∃for␈α∀proposing␈α∃this␈α∀theory␈α∃of␈α∀counterfactuals␈α∃comes␈α∀from␈α∃research␈α∀in
␈↓ α∧␈↓artificial␈α∞intelligence.␈α∞ We␈α∞would␈α∞like␈α∞robots␈α∞and␈α∞computer␈α∞programs␈α∞to␈α∞use␈α∂counterfactuals␈α∞and
␈↓ α∧␈↓concepts␈α∞based␈α∞on␈α∞them␈α∞in␈α∞the␈α∂same␈α∞circumstances␈α∞as␈α∞do␈α∞people.␈α∞ The␈α∞cartesian␈α∂product␈α∞theory
␈↓ α∧␈↓seems␈αsufficiently␈αcomputational␈αthat␈αwe␈αcan␈αimagine␈αprogramming␈αthe␈αrobot␈αto␈αuse␈αit␈αto␈αgenerate
␈↓ α∧␈↓appropriate counteractuals.
␈↓ α∧␈↓␈↓ u3


␈↓ α∧␈↓␈↓ αTContrast␈αthis␈αexplanation␈αof␈αcounterfactuals␈αwith␈αDavid␈αLewis's␈α(1973).␈α He␈αassigns␈αtruth␈αto
␈↓ α∧␈↓a␈αcounterfactual␈αprovided␈αthe␈αconsequent␈αis␈αtrue␈αin␈αthe␈αclosest␈αpossible␈αworld␈αto␈αthe␈αpresent␈αworld
␈↓ α∧␈↓in␈αwhich␈αthe␈αantecedent␈αis␈αtrue.␈α Since␈αthe␈αworld␈αis␈αsubstantially␈αdeterministic,␈αwe␈α
have␈αproblems
␈↓ α∧␈↓with␈α
what␈αis␈α
the␈α
closest␈αworld␈α
in␈αwhich␈α
the␈α
student␈αbent␈α
his␈αknees.␈α
 Perhaps␈α
his␈αknees␈α
are␈αtoo␈α
stiff
␈↓ α∧␈↓to␈αbend␈αbecause␈αof␈αa␈αchildhood␈αaccident␈αcaused␈αby␈αa␈αmosquito␈αannoying␈αhis␈αfather␈αwhile␈αdriving.
␈↓ α∧␈↓However,␈αif␈αthe␈αaccident␈αhadn't␈αoccurred␈αhe␈αwould␈αhave␈αworried␈αless␈αand␈αeaten␈αmore␈α
and␈αwould
␈↓ α∧␈↓have␈αfallen␈αanyway␈αbecause␈α
of␈αbeing␈αtoo␈αfat.␈α
 There␈αdoesn't␈αseem␈αto␈α
be␈αany␈αway␈αto␈α
program␈αour
␈↓ α∧␈↓robot to generate appropriate counterfactuals using Lewis's theory.

␈↓ α∧␈↓␈↓ αTThe␈αcartesian␈αmodel␈αof␈αcounterfactuals␈αavoids␈α
such␈αproblems␈αand␈αsticks␈αto␈αthe␈α
theory␈αused
␈↓ α∧␈↓by␈αthe␈αski␈αinstructors,␈αwho␈αcan␈αonly␈αbe␈αpersuaded␈αto␈αthink␈αabout␈αthe␈αchildhood␈αaccident␈αand␈αthe
␈↓ α∧␈↓mosquito after they have had several beers.

␈↓ α∧␈↓␈↓ αTConsider␈αLewis's␈αexample␈αof␈αthe␈αparty.␈α If␈αOtto␈αhad␈αcome␈αit␈αwould␈αhave␈αbeen␈αa␈αgood␈αparty
␈↓ α∧␈↓but␈α∞not␈α∞if␈α∞Anna␈α∞had␈α∞also␈α∞come.␈α∞ This␈α∞is␈α∞quite␈α∞understandable␈α∞in␈α∞terms␈α∞of␈α∞a␈α∞suitable␈α∞theory␈α∞of
␈↓ α∧␈↓what makes a good party.

␈↓ α∧␈↓␈↓ αTCartesian␈α⊂counterfactuals␈α⊂also␈α⊂seem␈α⊂to␈α⊂agree␈α⊂with␈α⊂intuition␈α⊂in␈α⊂cases␈α⊂where␈α⊃the␈α⊂sentences
␈↓ α∧␈↓don't␈α
seem␈αto␈α
have␈αdefinite␈α
truth␈αvalues.␈α
 For␈αexample,␈α
"If␈αwishes␈α
were␈αhorses␈α
beggars␈αwould␈α
ride"
␈↓ α∧␈↓is␈α
not␈α
associated␈α∞with␈α
any␈α
approximate␈α∞theory.␈α
 Sentences␈α
beginning␈α
"If␈α∞the␈α
South␈α
had␈α∞won␈α
the
␈↓ α∧␈↓civil␈αwar␈α.␈α.␈α."␈αseem␈αto␈αbe␈αmeaningful␈αor␈αnot␈αaccording␈αto␈αhow␈αmuch␈αtheory␈αof␈αhistory␈αwe␈αimagine
␈↓ α∧␈↓them imbedded in.

␈↓ α∧␈↓␈↓ αTI␈αdon't␈αknow␈αwhether␈αall␈αcounterfactuals␈αfit␈αthe␈αcartesian␈αmodel.␈α Proposed␈α
counterexamples
␈↓ α∧␈↓will be welcomed.

␈↓ α∧␈↓Challenge␈α
for␈α
the␈α
reader:␈αConstruct␈α
a␈α
theory␈α
in␈αwhich␈α
if␈α
someone␈α
had␈αsaid␈α
to␈α
Fermat,␈α
"If␈α2␈↓∧32␈↓+1
␈↓ α∧␈↓were␈αa␈αprime,␈αtwice␈αit␈αwould␈αbe␈αa␈αprime",␈αFermat␈αwould␈αhave␈αcorrectly␈αreplied,␈α"False".␈α Note␈αthat
␈↓ α∧␈↓2␈↓∧32␈↓+1 is divisible by 641, but Fermat didn't know it.


␈↓ α∧␈↓αREFERENCES

␈↓ α∧␈↓␈↓αLewis, David␈↓ (1973), ␈↓↓Counterfactuals␈↓, Harvard University Press.
␈↓ α∧␈↓␈↓ u4


␈↓ α∧␈↓We here propose a new theory based on the approximate theories of the some aspect of the world

␈↓ α∧␈↓␈↓ αTOur object is to propose a new theory of counterfactual conditional sentences.

␈↓ α∧␈↓␈↓ αTSuppose␈α
our␈α
space␈α
X␈α∞of␈α
possibilities␈α
is␈α
a␈α∞cartesian␈α
product.␈α
 X␈α
=␈α
X1␈α∞x␈α
.␈α
.␈α
.␈α∞Xn.␈α
 Suppose
␈↓ α∧␈↓that f : X → A is a function whose value interests us, and the point

␈↓ α∧␈↓␈↓ αTx↑0 = (x1↑0, . . . ,xn↑0)

␈↓ α∧␈↓is␈α
the␈α
current␈α
value␈α
of␈α
x.␈α
 Consider␈α
the␈α
counterfactual␈α
sentence:␈α
␈↓↓If␈α
xi␈α
were␈α
b,␈α
then␈α
f␈α
would␈αhave␈α
the
␈↓ α∧␈↓↓value c␈↓.  We consider it synonymous with

␈↓ α∧␈↓␈↓ αTf(x1↑0, . . . x[i-1]↑0, b, x[i+1]↑0, . . . xn↑0) = c.

␈↓ α∧␈↓Notice␈α
that␈αthe␈α
meaning␈αof␈α
this␈αkind␈α
of␈α
counterfactual␈αdepends␈α
on␈αthe␈α
particular␈αrepresentation␈α
of
␈↓ α∧␈↓X␈α∞as␈α∞a␈α∞cartesian␈α∞product.␈α∞ Our␈α∞idea␈α∞is␈α∞that␈α∞in␈α∞many␈α∞(perhaps␈α∞most)␈α∞cases␈α∞where␈α
counterfactual
␈↓ α∧␈↓conditional␈α∩sentences␈α∪are␈α∩intuitively␈α∪meaningful,␈α∩there␈α∪is␈α∩a␈α∪distinguished␈α∩representation␈α∪of␈α∩a
␈↓ α∧␈↓certain relevant space as a cartesian product, so that the above definition is meaningful.

␈↓ α∧␈↓␈↓ αTThe␈α
problem␈α
is␈α
made␈αmore␈α
complex␈α
by␈α
the␈αfact␈α
that␈α
the␈α
space␈αin␈α
question␈α
is␈α
not␈α"the␈α
world"
␈↓ α∧␈↓but␈α∞a␈α∞space␈α∞that␈α∞approximates␈α∞the␈α∞relevant␈α∂aspect␈α∞of␈α∞the␈α∞world.␈α∞ The␈α∞world␈α∞does␈α∞not␈α∂have␈α∞the
␈↓ α∧␈↓distinguished␈α∞cartesian␈α∞product␈α∞structure,␈α∞but␈α∞a␈α∞certain␈α∞distinguished␈α∞approximating␈α∞space␈α∞does.
␈↓ α∧␈↓The counterfactual is then only approximately meaningful.