/LMAR=0/XLINE=3/FONT#0=BAXL30/FONT#1=BAXM30/FONT#2=BASB30/FONT#3=SUB/FONT#4=SUP/FONT#5=BASL35/FONT#6=NGR25/FONT#7=MATH30/FONT#8=FIX25/FONT#9=GRKB30
␈↓ α∧␈↓␈↓ u1


␈↓ α∧␈↓α␈↓ β∂EXPLAINING COUNTERFACTUALS WITH CARTESIAN PRODUCTS

␈↓ α∧␈↓␈↓ αTSome␈α␈↓↓counterfactual␈αconditional␈αsentences␈↓␈α(we␈αwill␈αcall␈αthem␈α␈↓↓counterfactual␈↓␈αfor␈αshort)␈αcan␈αbe
␈↓ α∧␈↓explained␈α⊂as␈α⊂follows.␈α⊂ Consider␈α⊂a␈α⊂function␈α⊂␈↓↓f(x,y,z)␈↓␈α⊂and␈α⊂suppose␈α⊂that␈α⊂␈↓↓x = 1,␈α⊂y = 2␈↓,␈α⊂and␈α∂␈↓↓z = 3␈↓.
␈↓ α∧␈↓Now␈α
consider␈αthe␈α
assertion␈α␈↓↓"If␈α
y␈αwere␈α
5,␈α
then␈αf␈α
would␈αbe␈α
7"␈↓.␈α The␈α
assertion␈αis␈α
considered␈α
true␈αif
␈↓ α∧␈↓and only if ␈↓↓f(1,5,3) = 7␈↓.

␈↓ α∧␈↓␈↓ αTWe␈α∂first␈α⊂remark␈α∂that␈α⊂the␈α∂truth␈α⊂of␈α∂the␈α∂counterfactual␈α⊂is␈α∂dependent␈α⊂on␈α∂the␈α⊂particular␈α∂co-
␈↓ α∧␈↓ordinate␈αsystem.␈α If␈αwe␈αchange␈αto␈αa␈αco-ordinate␈αsystem␈α␈↓↓(x,y,z')␈↓␈αwhere␈α␈↓↓x␈↓␈αand␈α␈↓↓y␈↓␈αare␈αas␈αbefore,␈αbut
␈↓ α∧␈↓␈↓↓z' = z + y␈↓,␈αthen␈αthe␈αtruth␈αof␈αthe␈αabove␈αsentence␈αdepends␈αon␈αwhether␈α␈↓↓f(1,5,0) = 7␈↓.␈α Therefore,␈αwe
␈↓ α∧␈↓will␈α⊂consider␈α⊂that␈α⊂this␈α⊂kind␈α⊂of␈α⊂explanation␈α∂of␈α⊂a␈α⊂counterfactual␈α⊂depends␈α⊂on␈α⊂the␈α⊂existence␈α⊂of␈α∂a
␈↓ α∧␈↓preferred co-ordinate system.

␈↓ α∧␈↓␈↓ αTSuch␈αpurely␈αmathematical␈αexamples␈αof␈αcounterfactuals␈αoffer␈αno␈αproblem␈αand␈αtherefore␈αlittle
␈↓ α∧␈↓interest.␈α⊂ Let␈α∂us␈α⊂now␈α∂consider␈α⊂a␈α∂counterfactual␈α⊂conditional␈α∂sentence␈α⊂about␈α∂the␈α⊂real␈α∂world␈α⊂-␈α∂an
␈↓ α∧␈↓example discussed by David Lewis.

␈↓ α∧␈↓1)␈↓ αt␈α␈↓↓"If␈αOtto␈αhad␈αcome␈αto␈αthe␈αparty,␈αit␈αwould␈αhave␈αbeen␈αa␈αgood␈αparty␈α-␈αunless␈αAnna␈αhad␈αcome
␈↓ α∧␈↓↓too,␈αin␈αwhich␈αcase␈αit␈αwould␈αhave␈αbeen␈αa␈αbad␈αparty.␈α However,␈αif␈αMax␈αhad␈αcome␈αin␈αaddition␈αto␈αOtto
␈↓ α∧␈↓↓and Anna, it would have been a good party"␈↓.

␈↓ α∧␈↓␈↓ αTIf␈αwe␈αdenote␈αthe␈αpresence␈αof␈αOtto,␈αAnna␈αand␈αMax␈αat␈αthe␈αparty␈αby␈αBoolean␈αvariables␈α␈↓↓o,␈↓␈α␈↓↓a,␈↓
␈↓ α∧␈↓and␈α
␈↓↓m␈↓␈α
respectively,␈α
we␈α
can␈α
make␈α
an␈α
abstract␈α
model␈α
characterized␈α
by␈α
a␈α
function␈α␈↓↓goodness(o,a,m)␈↓␈α
in
␈↓ α∧␈↓which (1) becomes the formal assertion

␈↓ α∧␈↓2)␈↓ αt ␈↓↓goodness(1,0,0) = good ∧ goodness(1,1,0) = bad ∧ goodness(1,1,1) = good␈↓.

␈↓ α∧␈↓␈↓ αTHowever,␈α∪the␈α∩English␈α∪language␈α∩assertions␈α∪about␈α∪Otto,␈α∩Anna␈α∪and␈α∩Max␈α∪are␈α∪offered␈α∩as
␈↓ α∧␈↓assertions␈αabout␈αthe␈αreal␈αworld,␈αand␈αimmediately␈αjumping␈αto␈αthe␈αabove␈αabstract␈αmodel␈αis␈αbegging
␈↓ α∧␈↓the␈α∂question.␈α∂ However,␈α∂if␈α∂both␈α∂the␈α∂speaker␈α∂and␈α∂the␈α∂hearer␈α∂were␈α∂adherents␈α∂of␈α∂a␈α∂mathematical
␈↓ α∧␈↓theory␈αof␈αparties,␈αthen␈αevaluating␈α
the␈αtruth␈αof␈α(2)␈αmight␈αbe␈α
exactly␈αthe␈αway␈αthey␈αwould␈α
settle␈αthe
␈↓ α∧␈↓truth of (1)

␈↓ α∧␈↓␈↓ αTOur␈α⊂contention␈α⊂is␈α⊂that␈α⊃this␈α⊂is␈α⊂a␈α⊂common␈α⊃situation␈α⊂when␈α⊂counterfactuals␈α⊂are␈α⊃used.␈α⊂ The
␈↓ α∧␈↓speaker␈α
and␈α
hearer␈α
share␈αan␈α
approximate␈α
theory␈α
of␈α
the␈αphenomenon␈α
being␈α
discussed,␈α
there␈α
is␈αa
␈↓ α∧␈↓function␈α∞dependent␈α∞on␈α∞a␈α∞state,␈α∞and␈α∞this␈α
state␈α∞has␈α∞a␈α∞preferred␈α∞Cartesian␈α∞product␈α∞structure.␈α∞ It␈α
is
␈↓ α∧␈↓then␈α
meaningful␈α∞to␈α
ask␈α∞the␈α
value␈α∞of␈α
the␈α
function␈α∞in␈α
a␈α∞state␈α
in␈α∞which␈α
specified␈α∞co-ordinates␈α
are
␈↓ α∧␈↓given␈α
specified␈α∞new␈α
values,␈α
and␈α∞the␈α
rest␈α
retain␈α∞the␈α
values␈α
of␈α∞the␈α
"base␈α
state".␈α∞ The␈α
truth␈α∞of␈α
the
␈↓ α∧␈↓counterfactual␈α
then␈αposes␈α
two␈α
requirements.␈α First,␈α
the␈α
statement␈αmust␈α
be␈α
true␈αin␈α
the␈α
theory,␈αand
␈↓ α∧␈↓second,␈α∂the␈α∂theory,␈α∂including␈α∂the␈α∂Cartesian␈α⊂product␈α∂structure␈α∂it␈α∂gives␈α∂to␈α∂the␈α∂situation,␈α⊂must␈α∂be
␈↓ α∧␈↓acceptable.

␈↓ α∧␈↓␈↓ αTA␈α
converse␈αis␈α
also␈α
possible;␈αnamely␈α
the␈αcounterfactual␈α
may␈α
be␈αa␈α
way␈αof␈α
expressing␈α
part␈αof
␈↓ α∧␈↓the␈αtheory.␈α Returning␈αto␈α
the␈αexample,␈αthe␈αspeaker␈α
and␈αthe␈αhearer␈αevaluate␈α
the␈αtruth␈αof␈α(1)␈α
within
␈↓ α∧␈↓a␈αtheory␈αof␈αthe␈αgoodness␈αof␈αparties␈αand␈αits␈αdependence␈αon␈αthe␈αpersonalities␈αof␈αthe␈αpeople␈αpresent.
␈↓ α∧␈↓Within␈α∩that␈α∩theory,␈α∩the␈α∩people␈α∩are␈α∪structures␈α∩of␈α∩objects␈α∩with␈α∩properties␈α∩and␈α∪relations.␈α∩ The
␈↓ α∧␈↓elementary␈α∞terms␈α∞of␈α∞that␈α∞theory␈α∞are␈α∂components␈α∞of␈α∞personality␈α∞and␈α∞are␈α∞not␈α∞atoms,␈α∂photons␈α∞and
␈↓ α∧␈↓molecules.␈α∩ In␈α∩fact,␈α∩an␈α∩attempt␈α∩to␈α∩evaluate␈α∩the␈α∩truth␈α∩of␈α∩(1)␈α∩in␈α∩terms␈α∩of␈α∩atoms,␈α∩photons␈α⊃and
␈↓ α∧␈↓molecules fails and generates only confusion.
␈↓ α∧␈↓␈↓ u2


␈↓ α∧␈↓␈↓ αTWhat␈α∂are␈α∞these␈α∂approximate␈α∞theories?␈α∂ How␈α∂do␈α∞they␈α∂approximate␈α∞reality?␈α∂ What␈α∂are␈α∞the
␈↓ α∧␈↓criteria for their acceptance.

␈↓ α∧␈↓John McCarthy
␈↓ α∧␈↓Artificial Intelligence Laboratory
␈↓ α∧␈↓Computer Science Department
␈↓ α∧␈↓Stanford University
␈↓ α∧␈↓Stanford, California 94305

␈↓ α∧␈↓ARPANET: MCCARTHY@SU-AI

␈↓ α∧␈↓␈↓εThis draft of COUNTE[E77,JMC] PUBbed at 17:36 on November 11, 1980.␈↓