tag:blogger.com,1999:blog-3891434218564545511Tue, 12 Nov 2024 02:28:15 +0000Godprobabilitytimeinfinitylanguagecausationmoralityfree willmindloveevilknowledgenaturalismpresentismtruthBayesianismdecision theoryconsciousnessmodalityexplanationmathematicssciencehumorparadoxintentionlyingethicsactionPrinciple of Double EffectbeliefdeterminismepistemologyreasonslogicChristianityPrinciple of Sufficient Reasonlaws of naturesexAristotelianismevolutionphysicalismquantum mechanicsdualismproblem of evilvirtueAristotleontologyassertionprogrammingfunctionalismcausal finitismspacevaguenessPlatonismNatural LawkillingmereologyrationalityvalueSt. Thomas AquinaspainsubstanceA-theorymaterialismdeathmultiversenormativitypromisespunishmentresponsibilityqualiacompatibilismcosmological argumentchangebeautyexistencegroundingpersonal identityphysicsscoring rulesabortioncreationgooddesireontological argumentset theoryteleologyeternalismliar paradoxmurderLeibnizfreedomDeep ThoughtsbookspropositionsAxiom of 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fallacybatterybeginningsbelievebigotrybinarybindingbirthdaysblindnessbloggingblurtingbranching actualismbranching spacetimebriberybringing aboutbrutenessbutterfliescanon lawcapitalismcarecaringcarpentrycastigationcategoriescausa suicausal decision theorycausal loopscausalitycauselessnesscavescellscentral limit theorychain of beingchairscheatingcivic friendshipcivilityclassclassical physicsclassificationclergyclericalismcloningcloudsclumpscoffeecognitive sophisticationcoherentismcoincidencecoinstantiationcold warcommon descentcommunication boardscommunion of the saintscompactnesscompanionshipcomparisoncompatibilitycompetitioncompositionalitycompulsionconciliationismconcretenesscondemnationconditionsconflict of interestconsciousconsconsciousnessconsecrationconsequence argumentconstrualconstructioncontestcontextualismcontinental philosophycontractarianismcontractscontrastcontrolconvenienceconventionalismconvexityconvincingcopyrightcoralscorporationscorrespondencecounselingcountable additivitycounterpartscouragecourtcreativitydark nebuladark nightdarknessdatadata consolidationde Finettide dictodegrees of freedomdenialdenotationdependancydeposit of faithdepressiondepthderelictionderivative valuederivativesdeterrencedetractiondeviant logicdevilsdevotiondevotionsdiagonal lemmadialoguedictionariesdifferential equationsdifficultydignitary harmdisbeliefdisclosurediscoverydisgustdishonestydislikingdisquotationdissentditheringdivine permissiondivinitydoctrinedog whistlesdoxastic goodsdoxinsdraftdramadrivingdrunkennessduck-rabbitduct tapeduellingdurationduresseartheconomicsectopic pregnancyefficient causationegalitarianismembeddingembryologyemergentismemotivismemphasisencryptionendenforcementengagementenhancementenlightenmentenmityens rationisentanglemententitiesepistemic conversionepistemic gapepistemic harmepistemic injusticeepistemic possibilityeroseroteticsessenceessentialismevagelicalismevangelicalismevangelizationexceptionsexclude middleexduranceexerciseexistentialismexobiologyexperiment philosophyexperimental philosophyexpressingfactory farmingfailurefaithfulnessfallibilismfalsemakersfamiliarityfashionfelix culpafertilityfictional charactersfigurative speechfine-tunefinkingfishfive-dimensionalismfleetingnessfoliationfollowershipfootballforced choiceforesightforgeryfour causesfunctionfunctionsfundamentalismfuture selvesgalaxiesgardeninggeneralizationsgenetic fallacygenetic manipulationgenocidegenregerrymanderinggivingglorificationgodsgraspgreedgriefgripinggritgroup rightsgroup theoryguisesgunshaeecceitieshaeeceitieshalf-lifeharmonyhatredheirloomsheroismheuristicshierarchyhindsighthistory of scienceholismholodeckhomonymhomophobiahomosexual activityhousehold hintshtmlhumor?hungerhypnosishypochondriaiconsillucutionary forceimmoralityimmortalityimpanationimpartialityimperativesimpositionimpossible attemptsimpressivenessin virtue ofincompatibilityinconstencyindependence axiomindexindicative 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particularismmoral philosophymoral realismmoral responsibilitymoral standingmoralsmultiplicationmultitaskingmysticismnarrativenationalismnaturaismnatural theologynaturalistnecessitismneednegandnoisesnon-cognitivismnon-deductive reasoningnon-realismnonlocalitynormative statusnorms natural lawnothingnessnuclear weaponsnumbernuminousnuminousnessobjectivismobservable universeoff topicoffersoligonismomnipresence.operationalismoppositionoptimalismoptimizationoral contraceptionorderorder of experienceordinary languageorektinsorgan salesorganicismorganizationsorgansorientationoriginal valueoriginalsimoriginsorphansorthodoxyoscilloscopeoutsideoxygenpairspanentheismpanexperientialismpanomnipsychismpanteleologypapal infallibilityparableparadiseparallel processingparameciaparticularspatiencepeopleper se causal sequencesperceptualperfect beingperjurypermissivismpersistent vegetative statepersonpersonal qualitative identitypersonality identitypersuasionperverse rewardsperversionpessimismpetsphenomenaphilosophical theologypiecesplayplayspocket oraclepoetic justicepointingpollutionpolytheismpornographypositionpositive psychologypossessionspotato chipspowerspracticespragmatic contradictionpragmatic encroachmentpragmaticspreemptionspreestablished harmonypresenitsmpresentismlpresocraticspresuppositionprevision.probabilityprimary qualitiesprintingpro tanto reasonsprobabilitiesprodigal sonprofessionprofessional philosophyprofessionalsprofitprojectibilityprojectionpromulgationpronounspronunciationpropensityproperitespropositionprotestprotocolproverbsproxiesproxypseudonymitypseudosciencepublic domainpublic goodpublic squarepunspurity of heartpurposepursuitqualitative differencequantifierquantitiesquasi-substancequestionraceradical translationrainrandom walkrather thanrational choicerationalismrattlesnakesreceivingreciprocityrecombinationredreductio ad absurdumredundancyreference classreference framereflectionrefutationregionsrelationshipreligious disagreementremarriageremote causationrepairsrepetitionrepresentationalismreptilesreputationrequestresistancerestitutionrestricted compositionretinal imagesreverse enngineeringrhetoricriddlesrightnessriotsrisk compensationrocketsrunningsacerdotalitysafetysame sex marriagesame-sex relationssame-sex sexual activitysayingscandalscattered objectsscepticsscientismscrupulositysculpturesecond order desiresecularismseductionseemingselection effectself-consciousnessself-defeatself-interestself-knowledgeself-organizationselvessentence typesserialseriousnesssermonssexismsexual ethissexualityshallownessshameshavingshoessicknessside-effectssilencingsimilaritysimonyskillskillssmall governmentsnowflakesocial choicesocial contractsocial conventionsocial interactionssocietysolid objectssophistrysoulmatessoundsovereigntyspecies relativityspecificationspeedsperm donationspiessplittingspookinessstarsstatesstringsstrugglestubbornnessstupiditysubjunctivessublimesubsists insubtractionsuggestionsupernormalcysuperpositionsurprise examsurrealsswampmanswarmssymbolismsympathysynecdochesynonymtamingtaskstastetaxatechnologytelekinesistelevisiontendencytensismtenuretextualismtheatertheistic Platonismtheistic determinismtheological virtuestheoremstheoretical reasontheoriatheorytheory choicetheosisthinkingthoughthrowing a matchtiebreakerstop-down causationtracestradeoffstragedytragedy of the commonstransfer problemtransitiontransworld identitytreatmenttruth paradoxturthmakingtwinningtypologyunificationunitarityunitsuniverseupbringingusefulnessusingusuryutility monstersutopiavampiresvarietyvectorsvegetaranismvicious circlevirtual partsvirtue epistemologyvisibilityvon Balthasarvoyeurismvulnerabilitywaitingwalkingwallswave-particle dualityways of beingweak transitivityweirdnesswell-foundednesswhite lieswholeswickednesswinningwisdomwishful thinkingwordworkworksxiangqizebrasAlexander Pruss's Bloghttp://alexanderpruss.blogspot.com/noreply@blogger.com (Alexander R Pruss)Blogger4319125tag:blogger.com,1999:blog-3891434218564545511.post-4447042100104950465Mon, 11 Nov 2024 21:29:00 +00002024-11-11T20:27:42.209-06:00countingGoodmanmereologyQuinesecond order quantificationtransitivityGoodman and Quine and transitive closure<p>In the <a
href="https://alexanderpruss.blogspot.com/2024/11/goodman-and-quine-and-shared-bits.html">previous
post</a>, I showed that <a
href="https://sites.pitt.edu/~rbrandom/Courses/2023%20Sellars/Sellars%20texts/Goodman-StepsTowardConstructive-1947.pdf">Goodman
and Quine’s</a> counting method fails for objects that have too much
overlap. I think (though the technical parts here are more difficult)
that the same is true for their definition of the ancestral or
transitive closure of a relation.</p>
<p>GQ showed how to define ancestors in terms of offspring. We can try
to extend this definition to the transitive closure of any relation
<em>R</em> over any kind of
entities:</p>
<ol type="1">
<li><em>x</em> stands in the transitive
closure of <em>R</em> to <span
class="math inline"><em>y</em></span> iff for every object <span
class="math inline"><em>u</em></span> that has <span
class="math inline"><em>y</em></span> as a part and that has as a part
anything that stands in <em>R</em> to a
part of <em>u</em>, there is a <span
class="math inline"><em>z</em></span> such that <span
class="math inline"><em>R</em><em>x</em><em>z</em></span> and both <span
class="math inline"><em>x</em></span> and <span
class="math inline"><em>z</em></span> are parts of <span
class="math inline"><em>R</em></span>.</li>
</ol>
<p>This works fine if no relatum of <span
class="math inline"><em>R</em></span> overlaps any other relatum of
<em>R</em>. But if there is overlap, it
can fail. For instance, suppose we have three atoms <span
class="math inline"><em>a</em></span>, <span
class="math inline"><em>b</em></span> and <span
class="math inline"><em>c</em></span>, and a relation <span
class="math inline"><em>R</em></span> that holds between <span
class="math inline"><em>a</em> + <em>b</em></span> and <span
class="math inline"><em>a</em> + <em>b</em> + <em>c</em></span> and
between <em>a</em> and <span
class="math inline"><em>a</em> + <em>b</em></span>. Then any object
<em>u</em> that has <span
class="math inline"><em>a</em> + <em>b</em> + <em>c</em></span> as a
part has <em>c</em> as a part, and so
(1) would imply that <em>c</em> stands
in the transitive closure of <em>R</em>
to <span
class="math inline"><em>a</em> + <em>b</em> + <em>c</em></span>, which
is false.</p>
<p>Can we find some other definition of transitive closure using the
same theoretical resources (namely, mereology) that works for
overlapping objects? No. Nor even if we add the “bigger than” predicate
of GQ’s attempt to define “more”. We can say that <span
class="math inline"><em>x</em></span> and <span
class="math inline"><em>y</em></span> are equinumerous provided that
neither is bigger than the other.</p>
<p>Let’s work in models made of an infinite number of mereological
atoms. Write <em>u</em> ∧ <em>v</em>
for the fusion of the common parts of both <span
class="math inline"><em>u</em></span> and <span
class="math inline"><em>v</em></span> (assuming <span
class="math inline"><em>u</em></span> and <span
class="math inline"><em>v</em></span> overlap), <span
class="math inline"><em>u</em> ∨ <em>v</em></span> for the fusion of
objects that are parts of one or the other, and <span
class="math inline"><em>u</em> − <em>v</em></span> for the fusion of all
the parts of <em>u</em> that do not
overlap <em>v</em> (assuming <span
class="math inline"><em>u</em></span> is not a part of <span
class="math inline"><em>v</em></span>). Write <span
class="math inline">|<em>x</em>|</span> for the number of atomic parts
of <em>x</em> when <span
class="math inline"><em>x</em></span> is finite. Now make these
definitions:</p>
<ol start="2" type="1">
<li><p><em>x</em> is finite iff an atom
is related to <em>x</em> by the
transitive closure (with respect to the kind <em>object</em>) of the
relation that relates an object to that object plus one atom.</p></li>
<li><p><span
class="math inline"><em>A</em><em>x</em><em>y</em><em>w</em></span> iff
<em>x</em> and <span
class="math inline"><em>y</em></span> are finite and whenever <span
class="math inline"><em>x</em>′</span> is equinumerous with <span
class="math inline"><em>x</em></span> and does not overlap <span
class="math inline"><em>y</em></span>, then <span
class="math inline"><em>x</em>′ ∨ <em>y</em></span> is equinumerous with
<em>w</em>. (This says <span
class="math inline">|<em>x</em>| + |<em>y</em>| = |<em>w</em>|</span>.)</p></li>
<li><p>Say that <span
class="math inline"><em>D</em><sub><em>y</em></sub><em>u</em><em>v</em></span>
iff <span
class="math inline"><em>A</em>(<em>u</em>−<em>y</em>,<em>u</em>−<em>y</em>,<em>v</em>−<em>y</em>)</span>
(i.e., <span
class="math inline">|<em>v</em>−<em>y</em>| = 2|<em>u</em>−<em>y</em>|</span>)
and either <em>v</em> does not overlap
<em>y</em> or and <span
class="math inline"><em>u</em> ∧ <em>y</em></span> is an atom or <span
class="math inline"><em>v</em></span> and <span
class="math inline"><em>y</em></span> overlap and <span
class="math inline"><em>u</em> ∧ <em>y</em></span> consists of <span
class="math inline"><em>v</em> ∧ <em>y</em></span> plus one atom. (This
treats <em>u</em> and <span
class="math inline"><em>v</em></span> as basically ordered pairs <span
class="math inline">(<em>u</em>−<em>y</em>,<em>u</em>∧<em>y</em>)</span>
and <span
class="math inline">(<em>v</em>−<em>y</em>,<em>v</em>∧<em>y</em>)</span>,
and it makes sure that from the first pair to the second, the first
component is doubled in size and the second component is decreased by
one.)</p></li>
<li><p>Say that <span
class="math inline"><em>Q</em><sup>0</sup><em>y</em><em>x</em></span>
iff <em>y</em> is finite and for some
atom <em>z</em> not overlapping <span
class="math inline"><em>y</em></span> we have <span
class="math inline"><em>y</em> ∧ <em>z</em></span> related to something
not overlapping <em>x</em> by the
transitive closure of <span
class="math inline"><em>D</em><sub><em>y</em></sub></span>. (This takes
the pair (<em>z</em>,<em>y</em>), and
applies the double first component and decrease second component
relation described in (4) until the second component goes to zero. Thus,
it is guaranteed that <span
class="math inline">|<em>x</em>| = 2<sup>|<em>y</em>|</sup></span>.)</p></li>
<li><p>Say that <span
class="math inline"><em>Q</em><em>y</em><em>x</em></span> iff <span
class="math inline"><em>y</em></span> is finite and <span
class="math inline"><em>Q</em><sup>0</sup><em>y</em><em>x</em>′</span>
for some non-overlapping <em>x</em>′
that does not overlap <em>y</em> and
that is equinumerous with <span
class="math inline"><em>x</em></span>.</p></li>
</ol>
<p>If I got all the details right, then <span
class="math inline"><em>Q</em><em>y</em><em>x</em></span> basically says
that <span
class="math inline">|<em>x</em>| = 2<sup>|<em>y</em>|</sup></span>.</p>
<p>Thus, we can define use transitive closure to define binary powers of
finite cardinalities. But <a
href="https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/interleaving-logic-and-counting/E8C36FFD28218F157BE9D150D7B55AA6">the
results</a> about the expressive power of monadic second-order logic
with cardinality comparison say that we can only define semi-linear
relations between finite cardinalities, which doesn’t allow defining
binary powers.</p>
<p><strong>Remark:</strong> We don’t need equinumerosity to be defined
in terms of a primitive “bigger”. We can define equinumerosity for
non-overlapping finite sets by using transitive closure (and we only
need it for finite sets). First let <span
class="math inline"><em>T</em><sub><em>y</em></sub><em>u</em><em>v</em></span>
iff <em>v</em> − <em>y</em> exists and
consists of <em>u</em> − <em>y</em>
minus one atom and <span
class="math inline"><em>v</em> ∧ <em>y</em></span> exists and consists
of <em>v</em> ∧ <em>y</em> minus one
atom. Then finite <em>x</em> and <span
class="math inline"><em>y</em></span> are equinumerous<span
class="math inline"><sub>0</sub></span> iff they are non-overlapping and
<em>x</em> ∨ <em>y</em> has exactly two
atoms or is related to an object with exactly two atoms by the
transitive closure of <span
class="math inline"><em>T</em><sub><em>y</em></sub><em>u</em><em>v</em></span>.
We now say that <em>x</em> and <span
class="math inline"><em>y</em></span> are equinumerous provided that
they are finite and either <span
class="math inline"><em>x</em> = <em>y</em></span> (i.e., they have the
same atoms) or both <span
class="math inline"><em>x</em> − <em>y</em></span> and <span
class="math inline"><em>y</em> − <em>x</em></span> are defined and
equinumerous<sub>0</sub>.</p>
http://alexanderpruss.blogspot.com/2024/11/goodman-and-quine-and-transitive-closure.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-8875862007404059255Fri, 08 Nov 2024 18:12:00 +00002024-11-08T13:15:58.026-06:00countingGoodmanlogicQuinesecond order quantificationNo fix for Goodman and Quine's counting<p>In <a
href="https://alexanderpruss.blogspot.com/2024/11/goodman-and-quine-and-shared-bits.html">yesterday’s
post</a>, I noted that Goodman and Quine’s nominalist mereological <a
href="https://sites.pitt.edu/~rbrandom/Courses/2023%20Sellars/Sellars%20texts/Goodman-StepsTowardConstructive-1947.pdf">definition</a>
of what it is to say that there are more cats than dogs fails if there
are cats that are conjoint twins. This raises the question whether there
is some <em>other</em> way of using the same ontological resources to
generate a definition of “more” that works for overlapping objects as
well.</p>
<p>I think the answer is negative. First, note that GQ’s project is
explicitly meant to be compatible with there being a finite number of
individuals. In particular, thus, it needs to be compatible with the
existence of mereological atoms, individuals with no proper parts, which
every individual is a fusion of. (Otherwise, there would have to be no
individuals or infinitely many. For every individual has an atom as a
part, since otherwise it has an infinite regress of parts. Furthermore,
every individual must be a fusion of the atoms it has as parts,
otherwise the supplementation axiom will be violated.) Second, GQ’s
avail themselves of one non-mereological tool: size comparisons (which I
think must be something like volumes). And then it is surely a condition
of adequacy on their theory that it be compatible with the logical
possibility that there are finitely many individuals, every individual
is a fusion of its atoms <em>and</em> the atoms are all the same size. I
will call worlds like that “admissible”.</p>
<p>So, here are GQ’s theoretical resources for admissible worlds. There
are individuals, made of atoms, and there is a size comparison. The size
comparison between two individuals is equivalent to comparing the
cardinalities of the sets of atoms the individuals are made of, since
all the atoms are the same size. In terms of expressive power, their
theory, in the case of admissible worlds, is essentially that of <a
href="https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/interleaving-logic-and-counting/E8C36FFD28218F157BE9D150D7B55AA6">monadic
second order logic with counting</a>, MSO(#), restricted to finite
models. (I am grateful to Allan Hazen for putting me on to the
correspondence between GQ and MSO.) The atoms in GQ correspond to
objects in MSO(#) and the individuals correspond to (extensions of)
monadic predicates. The differences are that MSO(#) will have empty
predicates and will distinguish objects from monadic predicates that
have exactly one object in their extension, while in GQ the atoms are
just a special (and definable) kind of individual.</p>
<p>Suppose now that GQ have some way of using their resources to define
“more”, i.e., find a way of saying “There are more individuals
satisfying <em>F</em> than those
satisfying <em>G</em>.” This will be
equivalent to MSO(#) defining a <em>second-order</em> counting
predicate, one that essentially says “The set of sets of satisfiers of
<em>F</em> is bigger than the set of
sets of satisfiers of <em>G</em>”, for
second-order predicates <em>F</em> and
<em>G</em>.</p>
<p>But <a
href="https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/interleaving-logic-and-counting/E8C36FFD28218F157BE9D150D7B55AA6">it
is known</a> that the definitional power of MSO(#) over finite models is
precisely such as to define semi-linear sets of numbers. However, if we
had a second-order counting predicate in MSO(#), it would be easy to
define binary exponentiation. For the number of objects satisfying
predicate <em>F</em> is equal to two
raised to the power of the number of objects satisfying <span
class="math inline"><em>G</em></span> just in case the number of
singleton subsets of <em>F</em> is
equal to the number of subsets of <span
class="math inline"><em>G</em></span>. (Compare in the GQ context: the
number of atoms of type <em>F</em> is
equal to two the power of the number of atoms of type <span
class="math inline"><em>G</em></span> provided that the number of atoms
of type <em>F</em> is one plus the
number of individuals made of the atoms of type <span
class="math inline"><em>G</em></span>.) And of course equinumerosity can
be defined (over finite models) in terms of “more”, while the set of pairs
(<em>n</em>,2<sup><em>n</em></sup>) is
clearly not semi-linear.</p>
<p>One now wants to ask a more general question. Could GQ define
counting of individuals using <em>some other</em> predicates on
individuals besides size comparison? I don’t know. My guess would be no,
but my confidence level is not that high, because this deals in logic
stuff I know little about.</p>
http://alexanderpruss.blogspot.com/2024/11/no-fix-for-goodman-and-quines-counting.htmlnoreply@blogger.com (Alexander R Pruss)1tag:blogger.com,1999:blog-3891434218564545511.post-7319840669029780079Thu, 07 Nov 2024 19:06:00 +00002024-11-07T13:06:40.003-06:00countingGoodmannominalismQuineGoodman and Quine and shared bits<p><a
href="https://sites.pitt.edu/~rbrandom/Courses/2023%20Sellars/Sellars%20texts/Goodman-StepsTowardConstructive-1947.pdf">Goodman
and Quine</a> have a clever way of saying that there are more cats than
dogs without invoking sets, numbers or other abstracta. The trick is to
say that <em>x</em> is a bit of <span
class="math inline"><em>y</em></span> if <span
class="math inline"><em>x</em></span> is a part of <span
class="math inline"><em>y</em></span> and <span
class="math inline"><em>x</em></span> is the same size as the smallest
of the dogs and cats. Then you’re supposed to say:</p>
<ol type="1">
<li>Every object that has a bit of every cat is bigger than some object
that has a bit of every dog.</li>
</ol>
<p>This doesn’t work if there is overlap between cats. Imagine there are
three cats, one of them a tiny embryonic cat independent of the other
two cats, and the other two are full-grown twins sharing a chunk larger
than the embryonic cat, while there are two full-grown dogs that are not
conjoined. Then a bit is a part the size of the embryonic cat. But
(assuming mereological universalism along with Goodman and Quine) there
is an object that has a bit of every cat that is no bigger than any
object has a bit of every dog. For imagine an object that is made out of
the embryonic cat together with a bit that the other two cats have in
common. This object is no bigger than any object that has a bit of each
of the dogs.</p>
<p>It’s easy to fix this:</p>
<ol start="2" type="1">
<li>Every object that has an unshared bit of every cat is bigger than
some object that has an unshared bit of every dog,</li>
</ol>
<p>where an unshared bit is a bit <span
class="math inline"><em>x</em></span> not shared between distinct cats
or distinct dogs.</p>
<p>But this fix doesn’t work in general. Suppose the following atomistic
thesis is true: all material objects are made of equally-sized
individisible particles. And suppose I have two cubes on my desk, <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span>, with <span
class="math inline"><em>B</em></span> having double the number of
particles as <em>A</em>. Consider this
fact:</p>
<ol start="4" type="1">
<li>There are more pairs of particles in <span
class="math inline"><em>A</em></span> than particles in <span
class="math inline"><em>B</em></span>.</li>
</ol>
<p>(Again, Goodman and Quine have to allow for objects that are pairs of
particles by their mereological universalism.) But how do we make sense
of this? The trick behind (1) and (2) was to divide up our objects into
equally-sized pieces, and compare the sizes. But any object made of the
parts of all the particles in <span
class="math inline"><em>B</em></span> will be the same size as <span
class="math inline"><em>B</em></span>, since it will be made of the same
particles as <em>B</em>, and hence will
be bigger than any object made of parts of <span
class="math inline"><em>A</em></span>.</p>
http://alexanderpruss.blogspot.com/2024/11/goodman-and-quine-and-shared-bits.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-8472093260091751062Tue, 05 Nov 2024 21:35:00 +00002024-11-05T15:35:09.310-06:00changeessentiality of originspleasurequaliatropesTrope theory and merely numerical differences in pleasures<p>Suppose I eat a chocolate bar and this causes me to have a trope of
pleasure. Given assentiality of origins, if I had eaten a numerically
different chocolate bar that caused the same pleasure, I would have had
had a numerically different trope of pleasure.</p>
<p>Now, imagine that I eat a chocolate bar in my right hand and it
causes me to have a trope of pleasure <span
class="math inline"><em>R</em></span>, and immediately as I have
finished eating that one chocolate bar, I switch to eating the chocolate
bar in my left hand, which gives me an exactly similar trope of
pleasure, <em>L</em>, with no temporal
gap. Nonetheless, by essentiality of origins, trope <span
class="math inline"><em>L</em></span> is numerically distinct from trope
<em>R</em>.</p>
<p>To some (perhaps Armstrong) this will seem absurd. But I think it’s
exactly right. In fact, I think it may even an argument for trope
theory. For it seems pretty plausible that as I switch chocolate bars,
something changes in me: I go from one pleasure to another exactly like
it. But on heavy-weight Platonism, there is no change: I instantiated
pleasure and now I instantiate pleasure. On non-trope nominalism,
likewise there is no change. It’s trope theory that gives us the change
here.</p>
http://alexanderpruss.blogspot.com/2024/11/trope-theory-and-merely-numerical.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-1674296095396862069Sun, 03 Nov 2024 17:07:00 +00002024-11-03T11:09:11.385-06:00decision theoryexpected utilityprobabilityvotingDoes one's vote make a difference?<p>Suppose that there is a simple majority election, with two
candidates, and there is a large odd number of voters. Suppose polling
data makes the election too close to call. How likely is it that you can
decide which candidate wins?</p>
<p>I could look up this stuff, but it’s more fun to figure it out.</p>
<p>A quick and dirty model is this. We have <span
class="math inline"><em>N</em></span> people other than you voting, each
choosing between candidates <em>A</em>
and <em>B</em> with probabilities <span
class="math inline"><em>p</em></span> and <span
class="math inline">1 − <em>p</em></span> respectively. You don’t know
what <em>p</em> and <span
class="math inline">1 − <em>p</em></span> are, but polling data tells
you that <em>p</em> is between <span
class="math inline">1/2 − <em>a</em></span> and <span
class="math inline">1/2 + <em>b</em></span> for some positive numbers
<em>a</em> and <span
class="math inline"><em>b</em></span>. Your vote decides the election
provided that exactly <em>N</em>/2
people vote for candidate A. This requires that <span
class="math inline"><em>N</em></span> be even (if <span
class="math inline"><em>N</em></span> is odd, at best you can decide
between a candidate winning and the election being undecided, so you
can’t decide <em>which</em> candidate wins), which has probability <span
class="math inline">1/2</span>. Given that <span
class="math inline"><em>N</em> = 2<em>n</em></span> is even, the
probability that the other votes are exactly balanced is <span
class="math inline">(<em>a</em>+<em>b</em>)<sup>−1</sup> <em>C</em>(2<em>n</em>,<em>n</em>)∫<sub>1/2−<em>a</em></sub><sup>1/2+<em>b</em></sup><em>p</em><sup><em>n</em></sup>(1−<em>p</em>)<sup><em>n</em> − 1</sup><em>d</em><em>p</em></span>,
where <em>C</em>(<em>m</em>,<em>n</em>)
is the binomial coefficient. Assuming <span
class="math inline"><em>n</em></span> is large as compared to <span
class="math inline"><em>a</em></span> and <span
class="math inline"><em>b</em></span>, the integral can be approximated
by replacing its bounds by 0 and <span
class="math inline">1</span> respectively, and some work with
Mathematica shows that for large <span
class="math inline"><em>n</em></span> the probability is approximately
<span
class="math inline">1/(<em>N</em>(<em>a</em>+<em>b</em>))</span>.</p>
<p>So what? Well, suppose you think that candidate <span
class="math inline"><em>A</em></span> will on average make a person in
the jurisdiction be <em>u</em> units of
flourishing better off than candidate <span
class="math inline"><em>B</em></span> will, and there are <span
class="math inline"><em>K</em></span> persons, where <span
class="math inline"><em>K</em> ≥ <em>N</em> + 1</span> (there are at
least as many persons as candidates). So, the expected amount of
difference that your voting for <span
class="math inline"><em>A</em></span> will make is at least <span
class="math inline"><em>K</em><em>u</em>/(2<em>N</em>(<em>a</em>+<em>b</em>))</span>.
This is at least <span
class="math inline"><em>u</em>/(<em>a</em>+<em>b</em>)</span>. Thus, if
the polling data gives you a range between <span
class="math inline">0.48</span> and <span
class="math inline">0.52</span> for the probability of a person’s
preferring candidate <em>A</em>, and
half of the people in the jurisdiction vote, the expected amount of
difference that your vote makes is <span
class="math inline">25<em>u</em></span>. This is quite a lot if you
think that which candidate wins makes a significant difference <span
class="math inline"><em>u</em></span> per governed person.</p>
<p>Interestingly, some numerical work with Mathematica also shows that
as number of people increases, then the expected amount of difference
your vote makes also increases asymptotically, up to the limit of <span
class="math inline"><em>K</em><em>u</em>/(2<em>N</em>(<em>a</em>+<em>b</em>))</span>.
So for larger jurisdictions, even though the probability of your vote
making a difference is smaller, the expected difference from your vote
is a bit bigger.</p>
<p>My quick and dirty model is not quite right. Of course, people don’t
come to the polls and randomly choose whom to vote for. A more likely
source of randomness has to do with who actually makes it to the polls
(who gets sick, who has something come up, who decides it’s pointless to
vote, etc.). A better model might be this. We have <span
class="math inline"><em>M</em></span> people eligible to vote, of whom
<em>p</em><em>M</em> want to vote for A
and (1−<em>p</em>)<em>M</em> want to
vote for B. Some random subset of the <span
class="math inline"><em>M</em></span> people then votes. My probabilist
intuitions say that this is not that different from my model if the
number of actual voters is, say, half of the eligible voters. If I had
an election that I was eligible to vote in coming, I might try to figure
our the more complex model, but I don’t.</p>
http://alexanderpruss.blogspot.com/2024/11/does-ones-vote-make-difference.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-6212999303467879459Mon, 28 Oct 2024 15:13:00 +00002024-10-28T12:15:12.888-05:00BibleexegesisGodinspirationintentiontheologyTorahTheology and source critical analysis<p>There is reason to think that a number of biblical
texts—paradigmatically, the Pentateuch—were redacted from multiple
sources that scholars have worked to tease apart and separately analyze.
This is very interesting from a scholarly point of view. But I do not
know that it is <em>that</em> interesting from the theological point of
view.</p>
<p>Vatican II, in <a
href="https://www.vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_const_19651118_dei-verbum_en.html">Dei
Verbum</a>, famously teaches:</p>
<blockquote>
<p>since everything asserted by the inspired authors or sacred writers must be
held to be asserted by the Holy Spirit, it follows that the books of
Scripture must be acknowledged as teaching solidly, faithfully and
without error that truth which God wanted put into sacred writings for
the sake of salvation. … However, since God speaks in Sacred Scripture
through men in human fashion, the interpreter of Sacred Scripture, in
order to see clearly what God wanted to communicate to us, should
carefully investigate what meaning the sacred writers really intended,
and what God wanted to manifest by means of their words.</p>
</blockquote>
<p>Presumably many other Christian groups hold something similar.</p>
<p>Now, in the case of a text put together from multiple sources, the
question is who the “sacred writers” are. I want to suggest that in the
case of such a text, the relevant “sacred writers” are the editors who
put the texts together, and especially the ones responsible for a final
(though this is a somewhat difficult to apply concept) version, and the
intentions relevant to figuring out “What God wanted to communicate to
us” are the intentions of the final layer of editing. The books in
question, such as Genesis, are not anthologies. In an anthology, an
editor has some purposes in mind for the anthologized texts, but the
texts belong, often in a more or less acknowledged fashion, to the
individual authors. The editorial work in putting the Biblical works
together from source material is much more creative—it is genuine form
of authorship—which is obvious from how much back-and-forth movement
there is. Like in an anthology, we should not take the editor’s
intentions to align with the intentions of the source material authors,
but unlike in an anthology, the final work comes with the editor’s
authority, and counts as the assertion of the editor, with the editor’s
intentions being the ones that determine the meaning of the work.</p>
<p>If this is right, then I think we can only be fully confident of
dealing with inspired teaching in the case of what the editors intend to
assert through the final works. Writers typically draw on a multiplicity
of sources, and need not be asserting what these sources meant in their
original context—think of the ways in which a writer often repurposes a
quote from another. Think here of how Homer draws upon a rich variety of
fictional and nonfictional source material, but when he adapts them for
inclusion in his work, the intentions relevant to “What the
<em>Iliad</em> and <em>Odyssey</em> say” are Homer’s intentions.</p>
<p>If what we want to be sure of is “what God wanted to communicate to
us”, then we should focus on the redactors’ intentions. In particular,
when there is a tension in text between two pieces of source material,
exegetically we should focus on what the editor meant to communicate to
us by the choice to include material from both sources. (In a text
without divine inspiration, we might in the end attribute a tension to
editorial carelessness, but in fact scholars rarely make use of
“carelessness” as an explanation for phenomena in great works of secular
literature.) I think we should be open even to the logical possibility
that the editor misunderstood what the source material meant to
communicate, but it is the editor’s understanding that is normative for
the interpretation of what the text as a whole is saying.</p>
<p>From a scholarly point of view, earlier layers in the composition
process are more interesting. But I think that from a theological point
of view, it is what the editor wanted to communicate that matters.</p>
<p>I don’t want to be too dogmatic about this, for three reasons. First,
it is <em>possible</em> that the source material is an inspired text in
its own right. But, I think, we typically don’t know that it is (though
in a Christian context, an obvious exception is where the New Testament
quotes Jesus’ inspired teaching). Second, it is possible for a writer or
editor who has a deep respect for a piece of source material to include
the text with the intention that the text be understood in the sense in
which the original authors intended it to be understood, in which case
the intentions of the authors of the source material may well be
relevant. Third, this is not my field—I could be really badly
confused.</p>
http://alexanderpruss.blogspot.com/2024/10/theology-and-source-critical-analysis.htmlnoreply@blogger.com (Alexander R Pruss)2tag:blogger.com,1999:blog-3891434218564545511.post-3691683257577665500Thu, 24 Oct 2024 21:36:00 +00002024-10-24T16:36:08.481-05:00David Lewisimpartialityinductionmodal realismself-locating beliefAn impartiality premise<p>In <a
href="http://alexanderpruss.com/papers/ActualAndPossible.html">an
argument</a> that David Lewis’s account of possible worlds leads to
inductive skepticism, I used this premise:</p>
<ol type="a">
<li>If knowing that <em>x</em> is <span
class="math inline"><em>F</em></span> (where <span
class="math inline"><em>F</em></span> is purely non-indexical and <span
class="math inline"><em>x</em></span> is a definite description or
proper name) does not epistemically justify inferring that <span
class="math inline"><em>x</em></span> is <span
class="math inline"><em>G</em></span> (where <span
class="math inline"><em>G</em></span> is purely non-indexical), then
neither does knowing <em>x</em> is
<em>F</em> and that <span
class="math inline"><em>x</em></span> is I (now, here, etc.: any pure
indexical will do) justify inferring that <span
class="math inline"><em>x</em></span> is <span
class="math inline"><em>G</em></span>.</li>
</ol>
<p>This is less clear to me now than it was then. Self-locating evidence
<em>might</em> be a counterexample to this principle. I know that the
tallest person in the world is the tallest person in the world. But
suppose I now learn that I am the tallest person in the world. It
doesn’t seem entirely implausible to think that at this point it becomes
reasonable (or at least more reasonable) to infer that the number of
people in the world is small. For on the hypothesis that the number of
people is small, it seems more likely that I am the tallest than on the
hypothesis that the number of people is large. (Compare: That I won some
competition is evidence that the number of competitors was small.)</p>
<p>But I think I can fix my argument by using this premise:</p>
<ol start="2" type="a">
<li>If knowing that <em>x</em> is <span
class="math inline"><em>F</em></span> (where <span
class="math inline"><em>F</em></span> is purely non-indexical and <span
class="math inline"><em>x</em></span> is a definite description or
proper name) and that a uniformly randomly chosen person (or other
occupied location) is <em>x</em> would
not epistemically justify inferring that <span
class="math inline"><em>x</em></span> is <span
class="math inline"><em>G</em></span> (where <span
class="math inline"><em>G</em></span> is purely non-indexical), then
neither does knowing <em>x</em> is
<em>F</em> and that <span
class="math inline"><em>x</em></span> is I (now, here, etc.: any pure
indexical will do) justify inferring that <span
class="math inline"><em>x</em></span> is <span
class="math inline"><em>G</em></span>.</li>
</ol>
<p>There are multiple versions of (b) depending on how the random choice
works, e.g., whether it is a random choice from among actual persons or
from among possible persons (cf. self-sampling vs. self-indication).</p>
<p>It takes a bit of work to convince oneself that the rest of the
argument still works.</p>
http://alexanderpruss.blogspot.com/2024/10/an-impartiality-premise.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-7392803699168548799Thu, 24 Oct 2024 03:57:00 +00002024-10-23T23:08:04.395-05:00geometryprogrammingvisionA new kind of projectI did something new and fun this fall: I wrote <a href="https://arxiv.org/abs/2410.17997">a computer science paper</a>. It's an analysis of the conditions under which a device equipped with a camera and an accelerometer can identify its position relative to two observed landmarks with known positions. Except for a measure zero set of singular cases with infinitely many solutions, there are always at most two solutions for device positions (this was previously known), and I found necessary and sufficient conditions for there to be a single solution. In particular, if the two landmarks are at the same altitude, there is always a single solution, unless the device is at the same altitude as the landmarks.<div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRbfOE5LZCA4iyOjDRKz1skwIP3AQLwXHhPhyphenhyphendI259oKKPhSPP62lT5z2ahs8-1EWt78nlVxY7h3k_z_SqnBiTlrr9UYB7rG0sM60RO6EIqTDSrirdaecjpHKw2J8_NTnur9_bYKhb_lx-RXR716teFS3WefLfuvIfLAYU7V5CvuWq4YAhqA-rSuC3F0Y/s2102/screenshot-cropped.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="2102" data-original-width="1173" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRbfOE5LZCA4iyOjDRKz1skwIP3AQLwXHhPhyphenhyphendI259oKKPhSPP62lT5z2ahs8-1EWt78nlVxY7h3k_z_SqnBiTlrr9UYB7rG0sM60RO6EIqTDSrirdaecjpHKw2J8_NTnur9_bYKhb_lx-RXR716teFS3WefLfuvIfLAYU7V5CvuWq4YAhqA-rSuC3F0Y/s320/screenshot-cropped.png" width="179" /></a></div>I implemented the algorithm on a phone (code <a href="https://github.com/arpruss/p2pexperiment">here</a>). In the screenshot, the markers 1 and 2 are landmarks, identified and outlined in green with OpenCV library code, and then the phone uses their positions and the accelerometer data to predict where the control markers 3 and 4 are on the screen, outlining them in red.</div><div><br /></div><div>For someone like me who does some philosophy of science, it was an interesting experience to actually do a real experiment and collect data from it.<br /><div><br /></div>I am planning at some point to try to implement the algorithm using infrared LEDs under a TV and the accelerometer and infrared camera inside a right Nintendo Switch joycon. To that end, over the last couple of days I've <a href="https://github.com/CTCaer/jc_toolkit/issues/99">reverse-engineered</a> two of the joycon infrared camera blob identification modes.<br /><div><br /></div></div>http://alexanderpruss.blogspot.com/2024/10/a-new-kind-of-project.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-2454198770469709295Wed, 23 Oct 2024 16:19:00 +00002024-10-23T11:19:01.768-05:00AristotelianismscientiaAristotelian sciences<p>There is an Aristotelian picture of knowledge on which all knowable
things are divided exhaustively and exclusively into sciences by subject
matter. This picture appears wrong. Suppose, after all, that <span
class="math inline"><em>p</em></span> is a fact from one science—say,
the natural science fact that water is wet—and <span
class="math inline"><em>q</em></span> is a fact from another
science—say, the anthropological fact that people pursue pleasure. Then
the conjunction <em>p</em> and <span
class="math inline"><em>q</em></span> does not belong to either of these
science, or any other science.</p>
<p>One might cavil that a conjunction isn’t another fact over and beyond
the conjuncts, that to say <em>p</em>
and <em>q</em> is to say <span
class="math inline"><em>p</em></span> and to say <span
class="math inline"><em>q</em></span>. I am sceptical, but it’s easy to
fix. Just replace my counterexample with something that isn’t a
conjunction but is logically equivalent to it, say the claim that it’s
not the case that either <em>p</em> or
<em>q</em> is false.</p>
http://alexanderpruss.blogspot.com/2024/10/aristotelian-sciences.htmlnoreply@blogger.com (Alexander R Pruss)2tag:blogger.com,1999:blog-3891434218564545511.post-9114672124366741162Tue, 22 Oct 2024 01:14:00 +00002024-10-21T20:14:28.731-05:00utilitarianismActual result utilitarianism implies a version of total depravity<p>Assume actual result utilitarianism on which there are facts of the
matter about what would transpire given any possible action of mine, and
an action is right just in case it has the best consequences.</p>
<p>Here is an interesting conclusion. Do something specific, anything.
Maybe wiggle your right thumb a certain way. There are many—perhaps even
infinitely many—other things you could have done (e.g., you could have
wiggled the thumb slightly differently) instead of that action whose
known consequences are no different from the known consequences of what
you did. We live in a chaotic world where the butterfly principle very
likely holds: even minor events have significant consequences down the
road. It is very unlikely that of all the minor variants of what you
did, all of which have the same known consequences, the variant you
chose has the best overall consequences down the road. Quite likely, the
variant action you chose is middle of the road among the variants.</p>
<p>So, typically, whatever we do, we do wrong on actual result
utilitarianism.</p>
http://alexanderpruss.blogspot.com/2024/10/actual-result-utilitarianism-implies.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-5005364120077757534Sat, 19 Oct 2024 16:57:00 +00002024-10-21T14:26:44.285-05:00comparative probabilityconditional probabilityThere is no canonical way to define a regular comparative probability in terms of a full conditional probability<p>I claim that there is no general, straightforward and satisfactory
way to define a total comparative probability with the standard axioms
using full conditional probabilities. By a “straightforward” way, I mean
something like:</p>
<ol type="1">
<li><em>A</em> ≲ <em>B</em> iff <span
class="math inline"><em>P</em>(<em>A</em>−<em>B</em>|<em>A</em><em>Δ</em><em>B</em>) ≤ <em>P</em>(<em>B</em>−<em>A</em>|<em>A</em><em>Δ</em><em>B</em>)</span></li>
</ol>
<p>or:</p>
<ol start="2" type="1">
<li><em>A</em> ≲ <em>B</em> iff <span
class="math inline"><em>P</em>(<em>A</em>|<em>A</em>∪<em>B</em>) ≤ <em>P</em>(<em>A</em>|<em>A</em>∪<em>B</em>)</span>
(Pruss).</li>
</ol>
<p>The standard axioms of comparative probability are:</p>
<ol start="3" type="1">
<li><p>Transitivity, reflexivity and totality.</p></li>
<li><p>Non-negativity: ⌀ ≤ <em>A</em>
for all <em>A</em></p></li>
<li><p>Additivity: If <span
class="math inline"><em>A</em> ∪ <em>B</em></span> is disjoint from
<em>C</em>, then <span
class="math inline"><em>A</em> ≲ <em>B</em></span> iff <span
class="math inline"><em>A</em> ∪ <em>C</em> ≲ <em>B</em> ∪ <em>C</em></span>.</p></li>
</ol>
<p>A “straightforward” definition is one where the right-hand-side is
some expression involving conditional probabilities of events definable
in a boolean way in terms of <em>A</em>
and <em>B</em>.</p>
<p>To be “satisfactory”, I mean that it satisfies some plausible
assumptions, and the one that I will specifically want is:</p>
<ol start="6" type="1">
<li>If <span
class="math inline"><em>P</em>(<em>A</em>|<em>C</em>) < <em>P</em>(<em>B</em>|<em>C</em>)</span>
where <span
class="math inline"><em>A</em> ∪ <em>B</em> ⊆ <em>C</em></span>, then
<em>A</em> < <em>B</em>.</li>
</ol>
<p>Definitions (1) and (2) are straightforward and satisfactory in the
above-defined senses, but (1) does not satisfy transitivity while (2)
does not satisfy the right-to-left direction of additivity.</p>
<p>Here is a proof of my claim. If the definition is straightforward,
then if <em>A</em> ≲ <em>B</em>, and
<em>A</em>′ and <span
class="math inline"><em>B</em>′</span> are events such that there is a
boolean algebra isomorphism <em>ψ</em>
from the algebra of events generated by <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span> to the algebra of events generated
by <em>A</em>′ and <span
class="math inline"><em>B</em>′</span> such that <span
class="math inline"><em>ψ</em>(<em>A</em>) = <em>A</em>′</span>, <span
class="math inline"><em>ψ</em>(<em>B</em>) = <em>B</em>′</span> and
<span
class="math inline"><em>P</em>(<em>C</em>|<em>D</em>) = <em>P</em>(<em>ψ</em>(<em>C</em>)|<em>ψ</em>(<em>D</em>))</span>
for all <em>C</em> and <span
class="math inline"><em>D</em></span> in the algebra generated by <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span>, then <span
class="math inline"><em>A</em>′ ≲ <em>B</em>′</span>.</p>
<p>Now consider a full conditional probability <span
class="math inline"><em>P</em></span> on the interval <span
class="math inline">[0,1]</span> such that <span
class="math inline"><em>P</em>(<em>A</em>|[0,1])</span> is equal to the
Lebesgue measure of <em>A</em> when
<em>A</em> is an interval. Let <span
class="math inline"><em>A</em> = (0,1/4)</span> and suppose <span
class="math inline"><em>B</em></span> is either <span
class="math inline">(1/4,1/2)</span> or <span
class="math inline">(1/4, 1/2]</span>. Then there is an isomorphism
<em>ψ</em> from the algebra generated
by <em>A</em> and <span
class="math inline"><em>B</em></span> to the same algebra that swaps
<em>A</em> and <span
class="math inline"><em>B</em></span> around and preserves all
conditional probabilities. For the algebra consists of the eight
possible unions of sets taken from among <span
class="math inline"><em>A</em></span>, <span
class="math inline"><em>B</em></span> and <span
class="math inline">[0,1] − (<em>A</em>∪<em>B</em>)</span>, and it is
easy to define a natural map between these eight sets that swaps <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span>, and this will preserve all
conditional probabilities. It follows from my definition of
straightforwardness that we have <span
class="math inline"><em>A</em> ≲ <em>B</em></span> if and only if we
have <em>B</em> ≲ <em>B</em>. Since the
totality axiom for comparative probabilities implies that either <span
class="math inline"><em>A</em> ≲ <em>B</em></span> or <span
class="math inline"><em>B</em> ≲ <em>A</em></span>, so we must have both
<em>A</em> ≲ <em>B</em> and <span
class="math inline"><em>B</em> ≲ <em>A</em></span>. Thus <span
class="math inline"><em>A</em> ∼ <em>B</em></span>. Since this is true
for both choices of <em>B</em>, we
have</p>
<ol start="7" type="1">
<li><span
class="math inline">(0,1/4) ∼ (1/4,1/2) ∼ (1/4, 1/2]</span>.</li>
</ol>
<p>But now note that ⌀ < {1/2} by
(3) (just let <em>A</em> = ⌀, <span
class="math inline"><em>B</em> = {1/2}</span> and <span
class="math inline"><em>C</em> = {1/2}</span>). The additivity axiom
then implies that <span
class="math inline">(1/4,1/2) < (1/4, 1/2]</span>, a
contradiction.</p>
<p>I think that if we want to define a probability comparison in terms
of conditional probabilities, what we need to do is to weaken the axioms
of comparative probabilities. My current best suggestion is to replace
Additivity with this pair of axioms:</p>
<ol start="8" type="1">
<li><p>One-Sided Additivity: If <span
class="math inline"><em>A</em> ∪ <em>B</em></span> is disjoint from
<em>C</em> and <span
class="math inline"><em>A</em> ≲ <em>B</em></span>, then <span
class="math inline"><em>A</em> ∪ <em>C</em> ≲ <em>B</em> ∪ <em>C</em></span>.</p></li>
<li><p>Weak Parthood Principle: If <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span> are disjoint, then <span
class="math inline"><em>A</em> < <em>A</em> ∪ <em>B</em></span> or
<span
class="math inline"><em>B</em> < <em>A</em> ∪ <em>B</em></span>.</p></li>
</ol>
<p>Definition (2) satisfies the axioms of comparable probabilities with
this replacement.</p>
<p>Here is something else going for this. In <a
href="https://arxiv.org/abs/2010.07366">this paper</a>, I studied the
possibility of defining non-classical probabilities (full conditional,
hyperreal or comparative) that are invariant under a group <span
class="math inline"><em>G</em></span> of transformations. Theorem 1 in
the paper characterizes when there are full conditional probabilities
that are strongly invariant. Interesting, we can now extend Theorem 1 to
include this additional clause:</p>
<ol start="6" type="i">
<li>There is a transitive, reflexive and total relation <span
class="math inline">≲</span> satisfying (4), (8) and (9) as well as the
regularity assumption that <span
class="math inline">⌀ < <em>A</em></span> whenever <span
class="math inline"><em>A</em></span> is non-empty and that is invariant
under <em>G</em> in the sense that
<em>g</em><em>A</em> ∼ <em>A</em>
whenever both <em>A</em> and <span
class="math inline"><em>g</em><em>A</em></span> are subsets of <span
class="math inline"><em>Ω</em></span>.</li>
</ol>
<p>To see this, note that if there is are strongly invariant full
conditional probabilities, then (2) will define <span
class="math inline">≲</span> in a way that satisfies (vi). For the
converse, suppose (vi) is true. We show that condition (ii) of the
original theorem is true, namely that there is no nonempty paradoxical
subset. For to obtain a contradiction suppose there is a non-empty
paradoxical subset <em>E</em>. Then
<em>E</em> can be written as the
disjoint union of <span
class="math inline"><em>A</em><sub>1</sub>, ..., <em>A</em><sub><em>n</em></sub></span>,
and there are <span
class="math inline"><em>g</em><sub>1</sub>, ..., <em>g</em><sub><em>n</em></sub></span>
in <em>G</em> and <span
class="math inline">1 ≤ <em>m</em> < <em>n</em></span> such that
<span
class="math inline"><em>g</em><sub>1</sub><em>A</em><sub>1</sub>, ..., <em>g</em><sub><em>m</em></sub><em>A</em><sub><em>m</em></sub></span>
and <span
class="math inline"><em>g</em><sub><em>m</em> + 1</sub><em>A</em><sub><em>m</em> + 1</sub>, ..., <em>g</em><sub><em>n</em></sub><em>A</em><sub><em>n</em></sub></span>
are each a partition of <em>E</em>.</p>
<p>A standard result for additive comparative probabilities in Krantz et
al.’s measurement book is that if <span
class="math inline"><em>B</em><sub>1</sub>, ..., <em>B</em><sub><em>n</em></sub></span>
are disjoint, and <span
class="math inline"><em>C</em><sub>1</sub>, ..., <em>C</em><sub><em>n</em></sub></span>
are disjoint, with <span
class="math inline"><em>B</em><sub><em>i</em></sub> ≲ <em>C</em><sub><em>i</em></sub></span>
for all <em>i</em>, then <span
class="math inline"><em>B</em><sub>1</sub> ∪ ... ∪ <em>B</em><sub><em>n</em></sub> ≲ <em>C</em><sub>1</sub> ∪ ... ∪ <em>C</em><sub><em>n</em></sub></span>.
One can check that the proof only uses One-Sided Additivity, so it holds
in our case. It follows from <span
class="math inline"><em>G</em></span>-invariance that <span
class="math inline"><em>A</em><sub>1</sub> ∪ ... ∪ <em>A</em><sub><em>m</em></sub> ∼ <em>E</em> ∼ <em>A</em><sub><em>m</em> + 1</sub> ∪ ... ∪ <em>A</em><sub><em>n</em></sub></span>.
Since <em>E</em> is the disjoint union
of <span
class="math inline"><em>A</em><sub>1</sub> ∪ ... ∪ <em>A</em><sub><em>m</em></sub></span>
with <span
class="math inline"><em>A</em><sub><em>m</em> + 1</sub> ∪ ... ∪ <em>A</em><sub><em>n</em></sub></span>,
this violates the Weak Parthood Principle.</p>
http://alexanderpruss.blogspot.com/2024/10/there-is-no-canonical-way-to-define.htmlnoreply@blogger.com (Alexander R Pruss)1tag:blogger.com,1999:blog-3891434218564545511.post-1592114551613776494Fri, 18 Oct 2024 01:44:00 +00002024-10-17T21:01:43.101-05:00laws of naturemereological universalismrestricted compositionRestricted composition and laws of nature<p>Ted Sider famously argues for the universality of composition on the
grounds that:</p>
<ol type="1">
<li><p>If composition is not universal, then one can find a continuous
series of cases from a case of no composition to a case of
composition.</p></li>
<li><p>Given such a continuous series, there won’t be any abrupt cut-off
in composition.</p></li>
<li><p>But composition is never vague, so there would have to be an
abrupt cut-off.</p></li>
</ol>
<p>Consider this argument that every velocity is an escape velocity:</p>
<ol start="4" type="1">
<li><p>If it’s not the case that every velocity is an escape velocity
from a spherically symmetric body of some fixed size and mass, then one
can find a continuous series of cases from a case of insufficiency to
escape to a case of sufficiency to escape.</p></li>
<li><p>Given such a continuous series, there won’t be any abrupt cut-off
in escape velocity.</p></li>
<li><p>But escape velocity is never vague, so there would have to be an
abrupt cut-off.</p></li>
</ol>
<p>It’s obvious that we should deny (5). There <em>is</em> an abrupt
cut-off in escape velocity, and there is a precise formula for what it
is: <span
class="math inline">(2<em>G</em><em>M</em>/<em>r</em>)<sup>1/2</sup></span>
where <em>G</em> is the gravitational
constant, <em>M</em> is the mass of the
spherical body, and <em>r</em> is its
radius. As the velocity of a projectile gets closer and closer to the
<span
class="math inline">(2<em>G</em><em>M</em>/<em>r</em>)<sup>1/2</sup></span>,
the projectile goes further and further before turning back. When the
velocity reaches <span
class="math inline">(2<em>G</em><em>M</em>/<em>r</em>)<sup>1/2</sup></span>,
the projectile goes out forever. There is no paradox here.</p>
<p>Why think that composition is different from escape velocity? Why not
think that just as the laws of nature precisely specify when the
projectile can escape gravity, they also precisely specify when a bunch
of objects compose a whole?</p>
<p>My suspicion is that the reason for thinking the two are different is
thinking that composition is something like a “logical” or maybe
“metaphysical” matter, while escape is a “causal” matter. Now,
universalists like David Lewis do tend to think that the whole is a free
lunch, nothing but the “sum of the parts”, in which case it makes sense
to think that composition is not something for the laws of nature to
specify. But if we are not universalists, then it seems to me that it is
very natural to think of composition in a <em>causal</em> way: when a
proper plurality of <em>x</em>s are
arranged a certain way, they <em>cause</em> the existence of a new
entity <em>y</em> that stands in a
composed-by relation to the <span
class="math inline"><em>x</em></span>s, just as when a projectile has a
certain velocity, that causes the projectile to escape to infinity.</p>
<p>Some may be bothered by the fact that laws of nature are often taken
to be contingent, and so there would be a world with the same parts as
ours but different wholes. That would bother one if one thinks that
wholes are a free lunch. But if we take wholes seriously, it should no
more bother us than a world where particles behave the same way up to
time <em>t</em><sub>1</sub>, and then
behave differently after <span
class="math inline"><em>t</em><sub>1</sub></span> because the laws are
different.</p>
<p>Humeans have good reason to reject the above view, though. If the
laws of composition are to match our intuitions about composition, they
are likely to be extremely complex, and perhaps too complex to be part
of the best system defining the laws on a Humean account of laws. But if
we are not Humeans about laws, and think the simplicity of laws is
merely an epistemic virtue, the explanatory power of laws of composition
might make it reasonable to accept very complex such laws.</p>
<p>That said, we all have reject the simple causal version of the above
view, where a proper plurality composing a whole causes the whole’s
existence. For instance, I am composed by a plurality of parts that
includes my hair, but my hair is not a cause of my existence: I would
have just as much existed had I never developed hair. So a more complex
version of the causal view is needed: initial parts (maybe the DNA in
the zygote that I started as) causally contribute to the existence of
the whole, but the causal relation runs in a different direction with
respect to later parts, like teeth: perhaps I and my teeth together
cause the teeth to be parts of me.</p>
<p>(I don’t endorse the more complex causal view either. I prefer, but
still do not endorse, an Aristotelian alternative: when <span
class="math inline"><em>y</em></span> is in a certain condition, it
causes the existence of all of the parts. This is much neater because
the causation always runs in the same direction.)</p>
http://alexanderpruss.blogspot.com/2024/10/restricted-composition-and-laws-of.htmlnoreply@blogger.com (Alexander R Pruss)1tag:blogger.com,1999:blog-3891434218564545511.post-6398656196302332741Wed, 16 Oct 2024 00:56:00 +00002024-10-21T14:27:57.206-05:00comparative probabilityconditional probabilityinvariancesymmetryMore on full conditional probabilities and comparative probabilities<p>I claim that there is no general, straightforward and satisfactory
way to define a total comparative probability with the standard axioms
using full conditional probabilities. By a “straightforward” way, I mean
something like:</p>
<ol type="1">
<li><em>A</em> ≲ <em>B</em> iff <span
class="math inline"><em>P</em>(<em>A</em>−<em>B</em>|<em>A</em><em>Δ</em><em>B</em>) ≤ <em>P</em>(<em>B</em>−<em>A</em>|<em>A</em><em>Δ</em><em>B</em>)</span></li>
</ol>
<p>or:</p>
<ol start="2" type="1">
<li><em>A</em> ≲ <em>B</em> iff <span
class="math inline"><em>P</em>(<em>A</em>|<em>A</em>∪<em>B</em>) ≤ <em>P</em>(<em>B</em>|<em>A</em>∪<em>B</em>)</span>.</li>
</ol>
<p>The standard axioms of comparative probability are:</p>
<ol start="3" type="1">
<li><p>Transitivity, reflexivity and totality.</p></li>
<li><p>Non-negativity: ⌀ ≤ <em>A</em>
for all <em>A</em></p></li>
<li><p>Additivity: If <span
class="math inline"><em>A</em> ∪ <em>B</em></span> is disjoint from
<em>C</em>, then <span
class="math inline"><em>A</em> ≲ <em>B</em></span> iff <span
class="math inline"><em>A</em> ∪ <em>C</em> ≲ <em>B</em> ∪ <em>C</em></span>.</p></li>
</ol>
<p>A “straightforward” definition is one where the right-hand-side is
some expression involving conditional probabilities of events definable
in a boolean way in terms of <em>A</em>
and <em>B</em>.</p>
<p>To be “satisfactory”, I mean that it satisfies some plausible
assumptions, and the one that I will specifically want is:</p>
<ol start="6" type="1">
<li>If <span
class="math inline"><em>P</em>(<em>A</em>|<em>C</em>) < <em>P</em>(<em>B</em>|<em>C</em>)</span>
where <span
class="math inline"><em>A</em> ∪ <em>B</em> ⊆ <em>C</em></span>, then
<em>A</em> < <em>B</em>.</li>
</ol>
<p>Definitions (1) and (2) are straightforward and satisfactory in the
above-defined senses, but (1) does not satisfy transitivity while (2)
does not satisfy the right-to-left direction of additivity.</p>
<p>Here is a proof of my claim. If the definition is straightforward,
then if <em>A</em> ≲ <em>B</em>, and
<em>A</em>′ and <span
class="math inline"><em>B</em>′</span> are events such that there is a
boolean algebra isomorphism <em>ψ</em>
from the algebra of events generated by <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span> to the algebra of events generated
by <em>A</em>′ and <span
class="math inline"><em>B</em>′</span> such that <span
class="math inline"><em>ψ</em>(<em>A</em>) = <em>A</em>′</span>, <span
class="math inline"><em>ψ</em>(<em>B</em>) = <em>B</em>′</span> and
<span
class="math inline"><em>P</em>(<em>C</em>|<em>D</em>) = <em>P</em>(<em>ψ</em>(<em>C</em>)|<em>ψ</em>(<em>D</em>))</span>
for all <em>C</em> and <span
class="math inline"><em>D</em></span> in the algebra generated by <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span>, then <span
class="math inline"><em>A</em>′ ≲ <em>B</em>′</span>.</p>
<p>Now consider a full conditional probability <span
class="math inline"><em>P</em></span> on the interval <span
class="math inline">[0,1]</span> such that <span
class="math inline"><em>P</em>(<em>A</em>|[0,1])</span> is equal to the
Lebesgue measure of <em>A</em> when
<em>A</em> is an interval. Let <span
class="math inline"><em>A</em> = (0,1/4)</span> and suppose <span
class="math inline"><em>B</em></span> is either <span
class="math inline">(1/4,1/2)</span> or <span
class="math inline">(1/4, 1/2]</span>. Then there is an isomorphism
<em>ψ</em> from the algebra generated
by <em>A</em> and <span
class="math inline"><em>B</em></span> to the same algebra that swaps
<em>A</em> and <span
class="math inline"><em>B</em></span> around and preserves all
conditional probabilities. For the algebra consists of the eight
possible unions of sets taken from among <span
class="math inline"><em>A</em></span>, <span
class="math inline"><em>B</em></span> and <span
class="math inline">[0,1] − (<em>A</em>∪<em>B</em>)</span>, and it is
easy to define a natural map between these eight sets that swaps <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span>, and this will preserve all
conditional probabilities. It follows from my definition of
straightforwardness that we have <span
class="math inline"><em>A</em> ≲ <em>B</em></span> if and only if we
have <em>B</em> ≲ <em>B</em>. Since the
totality axiom for comparative probabilities implies that either <span
class="math inline"><em>A</em> ≲ <em>B</em></span> or <span
class="math inline"><em>B</em> ≲ <em>A</em></span>, so we must have both
<em>A</em> ≲ <em>B</em> and <span
class="math inline"><em>B</em> ≲ <em>A</em></span>. Thus <span
class="math inline"><em>A</em> ∼ <em>B</em></span>. Since this is true
for both choices of <em>B</em>, we
have</p>
<ol start="7" type="1">
<li><span
class="math inline">(0,1/4) ∼ (1/4,1/2) ∼ (1/4, 1/2]</span>.</li>
</ol>
<p>But now note that ⌀ < {1/2} by
(3) (just let <em>A</em> = ⌀, <span
class="math inline"><em>B</em> = {1/2}</span> and <span
class="math inline"><em>C</em> = {1/2}</span>). The additivity axiom
then implies that <span
class="math inline">(1/4,1/2) < (1/4, 1/2]</span>, a
contradiction.</p>
<p>I think that if we want to define a probability comparison in terms
of conditional probabilities, what we need to do is to weaken the axioms
of comparative probabilities. My current best suggestion is to replace
Additivity with this pair of axioms:</p>
<ol start="8" type="1">
<li><p>One-Sided Additivity: If <span
class="math inline"><em>A</em> ∪ <em>B</em></span> is disjoint from
<em>C</em> and <span
class="math inline"><em>A</em> ≲ <em>B</em></span>, then <span
class="math inline"><em>A</em> ∪ <em>C</em> ≲ <em>B</em> ∪ <em>C</em></span>.</p></li>
<li><p>Weak Parthood Principle: If <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span> are disjoint, then <span
class="math inline"><em>A</em> < <em>A</em> ∪ <em>B</em></span> or
<span
class="math inline"><em>B</em> < <em>A</em> ∪ <em>B</em></span>.</p></li>
</ol>
<p>Definition (2) satisfies the axioms of comparable probabilities with
this replacement.</p>
<p>Here is something else going for this. In <a
href="https://arxiv.org/abs/2010.07366">this paper</a>, I studied the
possibility of defining non-classical probabilities (full conditional,
hyperreal or comparative) that are invariant under a group <span
class="math inline"><em>G</em></span> of transformations. Theorem 1 in
the paper characterizes when there are full conditional probabilities
that are strongly invariant. Interesting, we can now extend Theorem 1 to
include this additional clause:</p>
<ol start="6" type="i">
<li>There is a transitive, reflexive and total relation <span
class="math inline">≲</span> satisfying (4), (8) and (9) as well as the
regularity assumption that <span
class="math inline">⌀ < <em>A</em></span> whenever <span
class="math inline"><em>A</em></span> is non-empty and that is invariant
under <em>G</em> in the sense that
<em>g</em><em>A</em> ∼ <em>A</em>
whenever both <em>A</em> and <span
class="math inline"><em>g</em><em>A</em></span> are subsets of <span
class="math inline"><em>Ω</em></span>.</li>
</ol>
<p>To see this, note that if there is are strongly invariant full
conditional probabilities, then (2) will define <span
class="math inline">≲</span> in a way that satisfies (vi). For the
converse, suppose (vi) is true. We show that condition (ii) of the
original theorem is true, namely that there is no nonempty paradoxical
subset. For to obtain a contradiction suppose there is a non-empty
paradoxical subset <em>E</em>. Then
<em>E</em> can be written as the
disjoint union of <span
class="math inline"><em>A</em><sub>1</sub>, ..., <em>A</em><sub><em>n</em></sub></span>,
and there are <span
class="math inline"><em>g</em><sub>1</sub>, ..., <em>g</em><sub><em>n</em></sub></span>
in <em>G</em> and <span
class="math inline">1 ≤ <em>m</em> < <em>n</em></span> such that
<span
class="math inline"><em>g</em><sub>1</sub><em>A</em><sub>1</sub>, ..., <em>g</em><sub><em>m</em></sub><em>A</em><sub><em>m</em></sub></span>
and <span
class="math inline"><em>g</em><sub><em>m</em> + 1</sub><em>A</em><sub><em>m</em> + 1</sub>, ..., <em>g</em><sub><em>n</em></sub><em>A</em><sub><em>n</em></sub></span>
are each a partition of <em>E</em>.</p>
<p>A standard result for additive comparative probabilities in Krantz et
al.’s measurement book is that if <span
class="math inline"><em>B</em><sub>1</sub>, ..., <em>B</em><sub><em>n</em></sub></span>
are disjoint, and <span
class="math inline"><em>C</em><sub>1</sub>, ..., <em>C</em><sub><em>n</em></sub></span>
are disjoint, with <span
class="math inline"><em>B</em><sub><em>i</em></sub> ≲ <em>C</em><sub><em>i</em></sub></span>
for all <em>i</em>, then <span
class="math inline"><em>B</em><sub>1</sub> ∪ ... ∪ <em>B</em><sub><em>n</em></sub> ≲ <em>C</em><sub>1</sub> ∪ ... ∪ <em>C</em><sub><em>n</em></sub></span>.
One can check that the proof only uses One-Sided Additivity, so it holds
in our case. It follows from <span
class="math inline"><em>G</em></span>-invariance that <span
class="math inline"><em>A</em><sub>1</sub> ∪ ... ∪ <em>A</em><sub><em>m</em></sub> ∼ <em>E</em> ∼ <em>A</em><sub><em>m</em> + 1</sub> ∪ ... ∪ <em>A</em><sub><em>n</em></sub></span>.
Since <em>E</em> is the disjoint union
of <span
class="math inline"><em>A</em><sub>1</sub> ∪ ... ∪ <em>A</em><sub><em>m</em></sub></span>
with <span
class="math inline"> <em>A</em><sub><em>m</em> + 1</sub> ∪ ... ∪ <em>A</em><sub><em>n</em></sub></span>,
this violates the Weak Parthood Principle.</p>
http://alexanderpruss.blogspot.com/2024/10/more-on-full-conditional-probabilities.htmlnoreply@blogger.com (Alexander R Pruss)1tag:blogger.com,1999:blog-3891434218564545511.post-4662325303562789519Mon, 14 Oct 2024 20:10:00 +00002024-10-14T15:10:38.778-05:00beautylaws of naturesimplicityThe epistemic force of beauty in laws of nature does not reduce to simplicity<p>Some people think that simplicity of laws of nature is a guide to
truth, and some think beauty of laws of nature is. One might ask: Is the
beauty of laws of nature a guide that goes <em>beyond</em> simplicity?
Are there times when one could make epistemic decisions about the laws
of nature on the basis of beauty where simplicity wouldn’t do the
job?</p>
<p>I think so. Here is one case. Suppose we live in a Newtonian
universe, and we are discovering fundamental forces. The first one has
an inverse cube law. The second has an inverse cube law. These two laws
account for most phenomena, but a few don’t fit. Scientists think there
is a third fundamental force. For the third force law, we have two
proposals that fit the data: an inverse square law and a slightly more
complicated inverse cube law. It is, I think, quite reasonable to go for
an inverse cube law by induction over the laws.</p>
<p>There is something indeed beautiful about the idea that the same
power law applies to all the forces of nature. But if we just go with
simplicity, we will go for an inverse square law. However, going for the
inverse cube law seems clearly reasonable, and it is what beauty
suggests—but not simplicity.</p>
<p>Here is another thought. Sometimes a fundamental law has some
particularly lovely mathematical implications. For instance, a <a
href="https://en.wikipedia.org/wiki/Conservative_force">conservative
force law</a> is connected in a lovely way with a potential. But it need
not be the case that a conservative force law is <em>simpler</em> than a
non-conservative alternative. (It is true that a conservative force is
the gradient of a potential. If the potential can be particularly simply
expressed, this makes it easier to express the conservative force law.
But we can have a case where the potential is harder to express than the
force itself.)</p>
http://alexanderpruss.blogspot.com/2024/10/the-epistemic-force-of-beauty-in-laws.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-4664370351578844266Mon, 14 Oct 2024 19:08:00 +00002024-10-21T14:30:43.842-05:00comparative probabilityconditional probabilityde Finettiqualitative probabilityDefining comparative probabilities in terms of conditional probabilities<p>Suppose we have a full conditional probability <span
class="math inline"><em>P</em>(<em>A</em>∣<em>B</em>)</span> defined for
all pairs of events (stipulating that <span
class="math inline"><em>P</em>(<em>A</em>∣⌀) = 1</span> if we wish). I've proposed two
methods for defining a probability comparison using
conditional probabilities:</p>
<ol type="1">
<li><p><em>A</em> ⪅ <em>B</em>
iff <span
class="math inline"><em>P</em>(<em>A</em>∣<em>A</em>∪<em>B</em>) ≤ <em>P</em>(<em>B</em>∣<em>A</em>∪<em>B</em>)</span>.</p></li>
<li><p><span
class="math inline"><em>A</em> ⪅ <em>B</em></span> iff <span
class="math inline"><em>P</em>(<em>A</em>−<em>B</em>∣<em>A</em><em>Δ</em><em>B</em>) ≤ <em>P</em>(<em>B</em>−<em>A</em>∣<em>A</em><em>Δ</em><em>B</em>)</span>,
where <span
class="math inline"><em>A</em><em>Δ</em><em>B</em> = (<em>A</em>−<em>B</em>) ∪ (<em>B</em>−<em>A</em>)</span>
is the symmetric difference.</p></li>
</ol>
<p>In a footnote in a <a
href="http://alexanderpruss.com/papers/nonmeasurableOrNonregular.pdf">paper</a>,
I wrote about the second ordering, which I incorrectly attributed to de Finetti: “This ordering has the advantage
that if <em>A</em> is a proper subset
of <em>B</em>, then <span
class="math inline"><em>A</em> < <em>B</em></span>, but it is
somewhat harder to prove transitivity”.</p>
<p>Well, that was an understatement! It’s not just harder to prove
transitivity: it’s impossible.</p>
<p>Define:</p>
<ul>
<li><p><em>Ω</em>: all
integers</p></li>
<li><p><em>E</em><sub>0</sub>:
non-negative even integers</p></li>
<li><p><em>E</em>: positive even
integers</p></li>
<li><p><em>D</em>: positive odd
integers.</p></li>
</ul>
<p>Let <em>P</em> be a full conditional
probability such that:</p>
<ol start="3" type="1">
<li><p><span
class="math inline"><em>P</em>(<em>E</em><sub>0</sub>∣<em>E</em><sub>0</sub>∪<em>D</em>) = 1/2 = <em>P</em>(<em>D</em>∣<em>E</em><sub>0</sub>∪<em>D</em>)</span></p></li>
<li><p><span
class="math inline"><em>P</em>(<em>D</em>∣<em>D</em>∪<em>E</em>) = 1/2</span>
= <i>P</i>(<i>E</i>|<em>D</em>∪<em>E</em>).</p></li>
</ol>
<p>It is easy to see from (3) and (4) that because <span
class="math inline"><em>E</em><sub>0</sub></span> and <span
class="math inline"><em>D</em></span> are disjoint, and so are <span
class="math inline"><em>D</em></span> and <span
class="math inline"><em>E</em></span>, then by definition
(2) we have <span
class="math inline"><em>E</em><sub>0</sub> ⪅ <em>D</em></span> and <span
class="math inline"><em>D</em> ⪅ <em>E</em></span>. (For disjoint <span
class="math inline"><em>A</em></span> and <span
class="math inline"><em>B</em></span>, the definition (2) of
<em>A</em> ⪅ <em>B</em> is equivalent
to thedefinition (1).) However, <span
class="math inline"><em>E</em><sub>0</sub> − <em>E</em> = {0}</span>,
<span
class="math inline"><em>E</em> − <em>E</em><sub>0</sub> = ⌀</span>, and
<span
class="math inline"><em>E</em><sub>0</sub><em>Δ</em><em>E</em> = {0}</span>,
so <span
class="math inline"><em>P</em>(<em>E</em>−<em>E</em><sub>0</sub>∣<em>E</em><sub>0</sub><em>Δ</em><em>E</em>) = 0</span>
while <span
class="math inline"><em>P</em>(<em>E</em><sub>0</sub>−<em>E</em>∣<em>E</em><sub>0</sub><em>Δ</em><em>E</em>) = 1</span>,
and thus we cannot have <span
class="math inline"><em>E</em><sub>0</sub> ⪅ <em>E</em></span>.</p>
<p>The only question is whether there actually is a full conditional
probability satisfying (3) and (4). If there is, then (2) is not transitive in our case.</p>
<p>There <em>is</em> such a full conditional probability. Let <span
class="math inline"><em>Q</em><sub><em>n</em></sub>(<em>A</em>) = ∣ <em>A</em> ∩ [−<em>n</em>,<em>n</em>] ∣ /(2<em>n</em>+1)</span>,
where $B$ is the cardinality of a set <span
class="math inline"><em>B</em></span>. Then <span
class="math inline"><em>Q</em><sub><em>n</em></sub></span> is a
probability. Let <em>Q</em> be a limit
of the <em>Q</em><sub><em>n</em></sub>
along an ultrafilter. This is a finitely additive hyperreal probability
which is non-zero for all non-empty sets. Define <span
class="math inline"><em>P</em>(<em>A</em>∣<em>B</em>)</span> as the
standard part of <span
class="math inline"><em>Q</em>(<em>A</em>∩<em>B</em>)/<em>Q</em>(<em>B</em>)</span>
for <em>B</em> non-empty. This is a
full conditional probability. Moreover, <span
class="math inline"><em>P</em>(<em>A</em>∣<em>B</em>) = lim <em>Q</em><sub><em>n</em></sub>(<em>A</em>∩<em>B</em>)/<em>Q</em><sub><em>n</em></sub>(<em>B</em>)</span>
whenever the latter limit is defined. That limit is defined in the cases
of the events involved in (3) and (4), and it is easy to evaluate the
limits and see that (3) and (4) are true.</p>
<p>I don’t know what I was thinking when I wrote that footnote. My guess
is that I had in my mind a proof sketch that doesn’t work (I have some
idea what that might have been).</p>
<p>Whew! I noticed this afternoon that Theorem 1 of <a
href="https://arxiv.org/pdf/2010.07366">this paper</a> of mine was
incompatible with the transitivity of comparison (2).
This made me really worried that my Theorem 1 was false. But since the
comparison (2) isn’t transitive, I can relax.</p>
<p>This raises an interesting potential research problem. The Pruss
definition of ⪅ does not satisfy the
additivity axiom for comparative probabilities, namely that if <span
class="math inline"><em>C</em></span> is disjoint from <span
class="math inline"><em>A</em> ∪ <em>B</em></span>, then <span
class="math inline"><em>A</em> ≲ <em>B</em></span> if and only if <span
class="math inline"><em>A</em> ∪ <em>C</em> ≲ <em>B</em> ∪ <em>C</em></span>
(it only preserves the left-to-right implication). Definition (2) does satisfy the additivity axiom, which is what I liked
about it.</p>
<p>I suspect there is no good definition of comparative probabilities in
terms of full conditional probabilities that satisfies the additivity
axiom. (One reason for this intuition has to do with the fact that in
Figure 1 <a href="https://arxiv.org/pdf/2010.07366">here</a> there are
entries with a Yes in column 3 and a No in column 5.)</p>
<p>So, I now wonder: Is there some good combination of a definition of
comparative probabilities in terms of full conditional probabilities
with some weakened version of the additivity axiom?</p>
http://alexanderpruss.blogspot.com/2024/10/the-de-finetti-definition-of.htmlnoreply@blogger.com (Alexander R Pruss)2tag:blogger.com,1999:blog-3891434218564545511.post-8358523563079428440Thu, 10 Oct 2024 15:56:00 +00002024-10-10T10:56:01.955-05:00compatibilismconditionals of free willdilemmasfree willMolinismA really bad moral dilemma<p>Here would be a really bad kind of moral dilemma:</p>
<ul>
<li>It is certain that unless you murder one innocent person now, you
will freely become a mass murderer, but if you do murder that innocent
person, you will freely repent of it later and live an exemplary
life.</li>
</ul>
<p>If compatibilism is true, such dilemmas are possible—the world could
be so set up that these unfortunate free choices are inevitable. If
compatibilism is false, such dilemmas are impossible, absent Molinism.</p>
<p>We might have a strong intuition that such dilemmas are impossible.
If so, maybe that gives us another reason to reject compatibilism and Molinism.</p>
http://alexanderpruss.blogspot.com/2024/10/a-really-bad-moral-dilemma.htmlnoreply@blogger.com (Alexander R Pruss)3tag:blogger.com,1999:blog-3891434218564545511.post-2033665646122008309Wed, 09 Oct 2024 16:16:00 +00002024-10-09T11:16:17.300-05:00deterrenceintentionkillingpolicyPrinciple of Double EffectproportionalitywarProportionality and deterrence<p>There are many contexts where a necessary condition of the
permissibility of a course of action is a kind of proportionality
between the goods and bads resulting from the course of action. (If
utilitarianism is true, then given a utilitarian understanding of the
proportionality, it’s not only necessary but sufficient for
permissibility.) Two examples:</p>
<ul>
<li><p>The Principle of Double Effect says it is permissible to do
things that are foreseen to have a basic evil as an effect, if that evil
is not intended, and <em>if</em> proportionality between the evil effect
and the good effects holds.</p></li>
<li><p>The conditions for entry into a just war typically include both a
justice condition <em>and</em> a proportionality condition (sometimes
split into two conditions, one about likely consequences of the war and
the other about the probability of victory).</p></li>
</ul>
<p>But here is an interesting and difficult kind of scenario. Before
giving a general formulation, consider the example that made me think
about this. Country <em>A</em> has a
bellicose neighbor <em>B</em>. However,
<em>B</em>’s regime while bellicose is
not sufficiently evil that on a straightforward reading of
proportionality it would be worthwhile for <span
class="math inline"><em>A</em></span> to fight back if invaded. Sure,
one would lose sovereignty by not fighting back, but <span
class="math inline"><em>B</em></span>’s track record suggests that the
individual citizens of <em>A</em> would
maintain the freedoms that matter most (maybe this is what it would be
like to be taken over by Alexander the Great or Napoleon—I don’t know
enough of history to know), while a war would obviously be very bloody.
However, suppose that a policy of not fighting back would likely result
in an instant invasion, while a policy of fighting back would have a
high probability of resulting in peace for the foreseeable future. We
can then imagine that the benefits of likely avoiding even a non-violent
takeover by <em>B</em> outweigh the
small risk that despite <em>A</em>’s
having a policy of armed resistance <span
class="math inline"><em>B</em></span> would still invade.</p>
<p>The general case is this: We have a policy that is likely to prevent
an unhappy situation, but following through on the policy violates a
straightforward reading of proportionality if the unhappy situation
eventuates.</p>
<p>One solution is to take into account the value of follwing through on
the policy with respect to one’s credibility in the future. But in some
cases this will be a doubtful justification. Consider a policy of
fighting back against an invader—at least initially—even if there is no
chance of victory. There are surely many cases of bellicose countries
that could successfully take over a neighbor, but judge that the costs
of doing so are too high given the expected resistance. But if the
neighbor has such a policy, then in case the invasion nonetheless
eventuates, whatever is done, sovereignty will be lost, and the policy
will be irrelevant in the future. (One might have some speculation about
the benefits for other countries of following through on the policy, but
that’s very speculative.)</p>
<p>One line of thought on these kinds of cases is that we need to forego
such policies, despite their benefits. One can’t permissibly act on
them, so one can’t have them, and that’s that. This is unsatisfying, but
I think there is a serious chance that this is right.</p>
<p>One might think that the best of both worlds is to make it seem like
one has the policy, but not in fact have it. A problem with this is that
it might involve lying, and I think lying is wrong. But even aside from
that, in some cases this may not be practicable. Imagine training an
army to defend one’s country, and then having a secret plan, known only
to a very small number of top commanders, that one will surrender at the
first moment of an invasion. Can one really count on that surrender? The
deterrent policy is more effective the fiercer and more patriotic the
army, but those factors are precisely likely to make them fight despite
the surrender at the top.</p>
<p>Another move is this. Perhaps proportionality itself takes into
account not just the straightforward computation of costs and benefits,
but also the value of remaining steadfast in reasonably adopted
policies. I find this somewhat attractive, but this approach has to have
limits, and I don’t know where to draw them. Suppose one has invented a
weapon which will kill every human being in enemy territory. Use of this
weapon, with a Double Effect style intention of killing only the enemy
soldiers, is clearly unjustified no matter what policies one might have,
but a policy to use this weapon might be a nearly perfect protection
against invasion. (Obviously this connects with the question of nuclear
deterrence.) I suppose what one needs to say is that the importance of
steadfastness in policies affects how proportionality evaluation go, but
should not be decisive.</p>
<p>I find myself pulled to the strict view that policies we should not
have policies acting on which would violate a straightforward reading of
proportionality, and the view that we should abandon the straightforward
reading of proportionality and take into account—to a degree that is
difficult to weigh—the value of following policies.</p>
http://alexanderpruss.blogspot.com/2024/10/proportionality-and-deterrence.htmlnoreply@blogger.com (Alexander R Pruss)0tag:blogger.com,1999:blog-3891434218564545511.post-4722737977564343784Mon, 07 Oct 2024 16:59:00 +00002024-10-07T13:12:02.577-05:00adulterydivorcemarriageremarriageAnother argument on the interpretation of Matthew 5:32 and 19:9<p>Mark (10:11-12) and Luke (16:18) have rather simple and
straightforward statements on divorce and remarriage: if you divorce and
remarry, you’re in adultery. A standard interpretation is the Strict
View:</p>
<ul>
<li>(SV) Divorce does not actually remove the marriage, and so if you
remarry, you’re still married to the previous party, and hence are
committing adultery.</li>
</ul>
<p>It’s usual in the Christian tradition to restrict this to consummated
Christian marriage, and I will take that for granted.</p>
<p>However, Matthew has a more complex set of prohibitions:</p>
<ul>
<li><p>Matthew 5:32: Anyone who divorces his wife, except on account of
<em>porneia</em>, makes her commit adultery, and anyone who marries a
divorced woman commits adultery.</p></li>
<li><p>Matthew 19:9: Anyone who divorces his wife, except due to
<em>porneia</em>, and marries another commits adultery.</p></li>
</ul>
<p>There are several puzzles here. First, unlike in Mark and Luke, we
have exceptions for <em>porneia</em>, a generic term for sexual
immorality. There are two main interpretations of this exception:</p>
<ol type="1">
<li><p>Except when the wife has committed sexual immorality (most
commonly, adultery).</p></li>
<li><p>Except when the “marriage” constitutes sexual
immorality.</p></li>
</ol>
<p>Reading (1) supports the Less Strict View:</p>
<ul>
<li>(LSV) Except when your spouse has committed adultery, divorce does
not actually remove the marriage, and so if you remarry, you’re still
married to the previous party, and hence are committing adultery.</li>
</ul>
<p>Reading (2) is based on the observation that not every legal marriage
is genuinely a marriage: the Romans, for instance, might have allowed a
couple to marry despite their being too closely related from the
Christian point of view. In such a case, their “marriage” is not a real
marriage but incest, a form of sexual immorality, and divorce is not
only permissible, but a very good idea. Note that on reading (2), we can
but need not suppose that Jesus verbally included the exception—the
inspired author might have added it for clarification because the issue
came up for converts, much as we put things in square brackets within a
quote to clarify the author’s meaning (there were no brackets in Greek,
of course).</p>
<p>Reading (2) has the advantage that it explains how all three Gospels
can be inspired, even though Mark and Luke have unqualified statements
of SV, since on reading (2) it <em>is</em> true that divorcing one’s
<em>wife</em> and remarrying is never permitted, but it is permissible,
of course, to divorce one’s partner in an immoral sexual relationship
that non-Christian society may call “marriage”. Note that the Greek for
“his wife” can literally just mean “his woman”, which makes the
disambiguation especially appropriate.</p>
<p>But I want to turn towards a different and more complex argument for
SV. Notice that in Matthew 5:32, instead of us being told that the man
who divorces his wife (or woman) commits adultery, we are oddly told
that he makes <em>her</em> commit adultery. But being a betrayed spouse
does not constitute adultery! What’s going on? Well, the good
interpretations that I’ve seen note that the social context is a society
where it is very difficult to be a woman without a husband. There will
thus be significant social pressure to marry or become a concubine,
either of which would constitute adultery against the first husband. The
realities of the day were such that very likely she <em>would</em>
succumb to the pressure, and the first husband would have caused her to
commit adultery, and thereby he would have earned himself something
worse than a millstone about the neck (Matthew 18:6). This reading also
nicely explains why Matthew 5:32, unlike the three other texts, does not
mention the man marrying another. For the woman is going to be exposed
to the social pressure to join herself to another man whether or not her
(first) husband marries another.</p>
<p>Note that this reading of “makes her commit adultery” <em>prima
facie</em> works on both readings of the <em>porneia</em> exception. On
the reading where the <em>porneia</em> is the wife’s adultery against
her husband, obviously if she is already committing adultery, by
divorcing her he isn’t making her commit adultery. On the reading where
the <em>porneia</em> is constituted by the immorality of the first
“marriage”, because the woman wasn’t really married to the man, if she
goes and marries another, she isn’t committing adultery.</p>
<p>Nonetheless, there is a serious problem for this reading of “makes
her commit adultery” on the Less Strict View and reading (1). While
Matthew 5:32 does not talk of the man marrying another, often the man
will marry another. So now imagine this story. There is a valid marriage
between Alice and Bob with no adultery, but Bob divorces Alice, and
marries Charlene. At this point, Bob <em>is</em> committing adultery
against Alice on both SV and LSV. Thus, if LSV is correct, then Alice is
entitled to divorce Bob and marry another, say Dave. But if she avails
herself of this, she isn’t committing adultery. In other words, if LSV
is correct, in many cases the first wife will be able to avoid
committing adultery without going against social pressures: she need
only wait for her first husband to marry, and then the “except on
account of <em>porneia</em>” clause on interpretation (1) frees her (and
since he’s already legally divorced her, she doesn’t need to do any
legal paperwork). (Of course, there will still be less common cases
where she is stuck, namely when the man fails to remarry. But such a
case wouldn’t be the rule, and Matthew 5:32 implies that leading the
woman to adultery is the rule rather than an exception.)</p>
<p>On SV, the problem for the reading of “makes her commit adultery”
entirely disappears. Whether or not the man remarries, there is social
pressure for the divorced wife to marry, and in doing so, she would be
committing adultery against the man.</p>
<p>Interestingly, there <em>is</em> a historically represented view that
avoids the Strict View, allows our interpretation of “makes her commit
adultery” and avoids the above interpretative problem, namely the quite
awful Asymmetric View:</p>
<ul>
<li>(AV) A woman is not permitted to remarry after a divorce, whether or
not the first husband committed adultery against her, but a man is
permitted to remarry after a divorce if, and only if, the first wife
committed adultery against him.</li>
</ul>
<p>Additionally, AV also explains why neither of the texts in Matthew
has an exception for <em>porneia</em> in the “anyone who marries a
divorced woman” clause, a minor weak point for LSV. (On SV and reading
(2) of <em>porneia</em>, we just note that one need not repeat a
parenthetical clarification every time.)</p>
<p>In fact, while there was controversy in the early centuries of
Christianity over remarriage and divorce following adultery, I
understand that it was mainly a controversy between advocates of SV and
AV, not between advocates of SV and LSV. However, AV was rightly
lambasted by St. Jerome for being sexist, and I assume almost nobody
wants to defend it now.</p>
<p>Thus to sum up my argument for SV:</p>
<ol type="i">
<li><p>One of SV, AV and LSV is true, as they are the historically
plausible Christian views on marriage.</p></li>
<li><p>The right interpretation of “makes her commit adultery” is the
social pressure interpretation.</p></li>
<li><p>This interpretation is incompatible with LSV.</p></li>
<li><p>AV is false.</p></li>
<li><p>Therefore, SV is true.</p></li>
</ol>
http://alexanderpruss.blogspot.com/2024/10/another-argument-on-interpretation-of.htmlnoreply@blogger.com (Alexander R Pruss)2tag:blogger.com,1999:blog-3891434218564545511.post-4704217261191992055Wed, 02 Oct 2024 19:24:00 +00002024-10-02T14:24:47.247-05:00A-theorychangeenduranceeventsMcTaggartsubstancetimeEvents and the unreality of time<p>When I think about McTaggart’s famous argument against the A-theory
of time—the theory that it is an objective fact about the universe what
time it is—I sometimes feel like it’s just a confusion but sometimes I
feel like I am on the very edge of getting it, and that there is
something to the argument. When I try to capture the latter feeling in
an argument that actually has a chance of being sound, I find it
slipping away from me.</p>
<p>So for the <em>n</em>th time in my
life, let me try again to make something of McTaggart style arguments.
Last night I gave a talk at University of North Texas. When I gave the
talk, it was present, and afterwards it became past, and every second
that talk is receding another second into the past, becoming more and
more past, “older and older” we might say. There is something odd about
this, however, since the talk doesn’t exist now. Something that no
longer exists can’t change anymore. So how can the talk recede into the
further past, how can it become older and older?</p>
<p>Well, we do have a tool for making sense of this. Things that no
longer exist can’t <em>really</em> change, but they can have Cambridge
change, change relative to something else. Suppose a racehorse is
eventually forgotten after its death. The horse isn’t, of course,
<em>really</em> changing, but there is real change elsewhere.</p>
<p>More generally, we learn from McTaggart that events can’t
<em>really</em> change, but can only change relative to real change in
something other than events. The reasoning above shows that events can’t
<em>really</em> change in their A-determinations. And they can’t change
in their intrinsic non-temporal features, as McTaggart rightly insists:
it is eternally true that my talk was about God and mathematics; all the
flaws in the talk eternally obtain; etc. So if events can’t
<em>really</em> change, but only relatively to real change elsewhere,
and yet all of reality is just events, then there is no change.</p>
<p>But reality isn’t just events, and in addition to events changing
there is the possibility for enduring entities to change. Here’s perhaps
the simplest way to make the story go. The universe is an enduring
entity that continually gets older. My talk, then, recedes into the past
in virtue of the universe ever becoming older than it was when I gave
the talk. (If one is skeptical, as I am, that there is such an entity as
the universe, one can give a more complex story about a succession of
substances becoming older and older.)</p>
<p>Can one run any version of the McTaggart argument against a theory on
which fundamental change consists in a substance’s changing rather than
in the change of events? I am not sure, but at the moment I don’t see
how. If a person changes from young to old, we have two events: their
youth <em>A</em> and their old age
<em>B</em>. But we can now say that
neither <em>A</em> nor <span
class="math inline"><em>B</em></span> changes fundamentally: <span
class="math inline"><em>A</em></span> recedes into the past because of
the person’s (or the universe’s) growing old.</p>
<p>If this line of thought is right, then we do learn something from
McTaggart: an A-theorist should not locate fundamental change in events,
but in enduring objects.</p>
http://alexanderpruss.blogspot.com/2024/10/events-and-unreality-of-time.htmlnoreply@blogger.com (Alexander R Pruss)3tag:blogger.com,1999:blog-3891434218564545511.post-208396810184391010Mon, 30 Sep 2024 18:07:00 +00002024-09-30T13:07:50.225-05:00bioethicshistory of philosophyjobsphilosophy of scienceFour philosophy / adjacent jobs at Baylor<p>We have four jobs in philosophy or closely adjacent areas at Baylor, with most of the deadlines coming in mid-October:</p><p></p><ul style="text-align: left;"><li>Tenure-track open-area (with some preferences) <a href="https://apply.interfolio.com/149974">position in the Philosophy Department</a></li><li>Tenure-track position for a <u><a href="https://apply.interfolio.com/149589">philosopher in the Great Texts Department</a></u></li><li>Tenure-track position in <a href="https://apply.interfolio.com/152586">social psychology or history and philosophy of science in the Baylor Interdisciplinary Core</a></li><li>Senior position in <a href="https://apply.interfolio.com/149972">bioethics in the Religion Department</a>.</li></ul><p></p>http://alexanderpruss.blogspot.com/2024/09/four-philosophy-adjacent-jobs-at-baylor.htmlnoreply@blogger.com (Alexander R Pruss)1tag:blogger.com,1999:blog-3891434218564545511.post-6883416948480448889Fri, 27 Sep 2024 22:46:00 +00002024-09-27T17:46:11.239-05:00animalsdignityhumansvalueSpecial treatment of humans<p>Sometimes one talks of humans as having a higher value than other animals, and hence it being appropriate to treat them better. While humans do have a higher value, I don't think this is what justifies favoring them. For to treat something well is to bestow value on them. But it is far from clear why the fact that <i>x</i> has more value than <i>y</i> justifies bestowing additional value on <i>x</i> rather than on <i>y</i>. It seems at least as reasonable to spread value around, and preferentially treat <i>y</i>.</p><p>A confusing factor is that we do have reason to preferentially treat those who have more desert, and desert is a value. But the reason here is specific to desert, and does not in any obvious way generalize to other values.</p><p>I don't deny that we should treat humans preferentially over other animals, nor that humans are more valuable. But these two facts should not be confused. Perhaps we should treat humans preferentially over other animals because humans are persons and other animals are not--but this is a point about personhood rather than about value. I am inclined to think we shouldn't argue: humans are persons, personhood is very valuable, so we should treat humans preferentially. Rather, I suspect we should directly argue: humans are persons, so we should treat humans preferentially, skipping the value step. (To put it in Kantian terms, beings with dignity are valuable, but what makes them have dignity isn't just that they are valuable.)</p>http://alexanderpruss.blogspot.com/2024/09/special-treatment-of-humans.htmlnoreply@blogger.com (Alexander R Pruss)4tag:blogger.com,1999:blog-3891434218564545511.post-418621869313529794Thu, 26 Sep 2024 15:53:00 +00002024-09-26T10:55:03.849-05:00complexitylaws of natureLaws and mathematical complexity<p>Over the last couple of days I have realized that the laws of physics
are rather more complex than they seem. The lovely equations like <span
class="math inline"><em>G</em> = 8<em>π</em><em>T</em></span> and <span
class="math inline"><em>F</em> = <em>G</em><em>m</em><em>m</em>′/<em>r</em><sup>2</sup></span>
(with a different <em>G</em> in the two
equations) seem to be an iceberg most of which is submerged in the icy
waters of the foundations of mathematics where the foundational concepts
of real analysis and arithmetic are defined in terms of axioms.</p>
<p>This has a curious consequence. We might think that <span
class="math inline"><em>F</em> = <em>G</em><em>m</em><em>m</em>′/<em>r</em><sup>2</sup></span>
is <em>much</em> simpler than <span
class="math inline"><em>F</em> = <em>G</em><em>m</em><em>m</em>′/<em>r</em><sup>2</sup> + <em>H</em><em>m</em><em>m</em>′/<em>r</em><sup>3</sup></span>
(where <i>H</i> is presumably very, very small).
But if we fill out each proposal with the foundational mathematical
structure, the percentage difference in complexity will be slight, as
almost all of the contribution to complexity will be in such things as
the construction of real numbers (say, via Dedekind cuts).</p>
<p>Perhaps, though, the above line of thought is reason to think that
real analysis and arithmetic are actually fundamental?</p>
http://alexanderpruss.blogspot.com/2024/09/laws-and-mathematical-complexity.htmlnoreply@blogger.com (Alexander R Pruss)9tag:blogger.com,1999:blog-3891434218564545511.post-6689668012330988684Thu, 26 Sep 2024 15:44:00 +00002024-09-26T10:44:12.465-05:00conversionfree willHumepraiseresponsibilityMoral conversion and Hume on freedom<p>According to Hume, for one to be responsible for an action, the
action must flow from one’s character. But the actions that we praise
people for the most include cases where someone breaks free from a
corrupt character and changes for the good. These cases are not merely
cases of slight responsibility, but are central cases of
responsibility.</p>
<p>A Humean can, of course, say that there was some hidden determining
cause in the convert’s character that triggered the action—perhaps some
inconsistency in the corruption. But given determinism, why should we
think that this hidden determining cause was indeed in the agent’s
character, rather than being some cause outside of the character—some
glitch in the brain, say? That the hidden determining cause was in the
character is an empirical thesis for which we have very little evidence.
So on the Humean view, we ought to be quite skeptical that the person
who radically changes from bad to good is praiseworthy. We definitely
should not take such cases to be among paradigm cases of
praiseworthiness.</p>
http://alexanderpruss.blogspot.com/2024/09/moral-conversion-and-hume-on-freedom.htmlnoreply@blogger.com (Alexander R Pruss)1tag:blogger.com,1999:blog-3891434218564545511.post-5817485312901309334Wed, 25 Sep 2024 21:13:00 +00002024-09-26T10:32:44.900-05:00axiomsfundamentalityHumeanismlaws of natureHumeanism and knowledge of fundamental laws<p>On a "Humean" Best System Account (BSA) of laws of nature, the fundamental
laws are the axioms of the system of laws that best combines brevity and
informativeness.</p>
<p>An interesting consequence of this is that, very likely, no amount of
advances in physics will<br />
suffice to tell us what the fundamental laws are: significant advances
in mathematics will also be needed. For suppose that after a lot of
extra physics, propositions formulated in sentences <span
class="math inline"><em>p</em><sub>1</sub>, ..., <em>p</em><sub><em>n</em></sub></span>
are the physicist’s best proposal for the fundamental laws. They are
simple, informative and fit the empirical data really well.</p>
<p>But we would still need some very serious mathematics. For we would
need to know there isn’t a simpler collection of sentences <span
class="math inline">{<em>q</em><sub>1</sub>, ..., <em>q</em><sub><em>m</em></sub>}</span>
that is logically equivalent to <span
class="math inline">{<em>p</em><sub>1</sub>, ..., <em>p</em><sub><em>n</em></sub>}</span>
but simpler. To do that would require us to have a method for solving
the following type of mathematical problem:</p>
<ol type="1">
<li>Given a sentence <em>s</em> in some
formal language, find a simplest sentence <span
class="math inline"><em>s</em>′</span> that is logically equivalent to
<em>s</em>,</li>
</ol>
<p>in the case of significantly non-trivial sentences <span
class="math inline"><em>s</em></span>.</p>
<p>We might be able to solve (1) for some very simple sentences. Maybe
there is no simpler way of saying that there is only one thing in
existence than <span
class="math inline">∃<em>x</em>∀<em>y</em>(<em>x</em>=<em>y</em>)</span>.
But it is very plausible that any serious proposal for the laws of
physics will be <em>much</em> more complicated than that.</p>
<p>Here is one reason to think that any credible proposal for
fundamental laws is going to be pretty complicated. Past experience
gives us good reason to think the proposal will involve arithmetical
operations on real numbers. Thus, a full statement of the laws will
require including a definition of the arithmetical operations as well as
of the real numbers. To give a simplest formulation of such laws will,
thus, require us to solve the problem of finding a simplest
axiomatization of the portions of arithmetic and real analysis that are
needed for the laws. While we have multiple axiomatizations, I doubt we
are at all close to solving the problem of finding an optimal such
axiomatization.</p>
<p>Perhaps the Humean could more modestly hope that we will at least
know a part of the fundamental laws—namely the part that doesn’t include
the mathematical axiomatization. But I suspect that even this is going
to be very difficult, because different arithmetical formulations are
apt to need different portions of arithmetic and real analysis.</p>
http://alexanderpruss.blogspot.com/2024/09/humeanism-and-knowledge-of-fundamental.htmlnoreply@blogger.com (Alexander R Pruss)3tag:blogger.com,1999:blog-3891434218564545511.post-2388523288975114910Tue, 24 Sep 2024 18:54:00 +00002024-09-24T13:55:34.831-05:00chanceinfinitylotteriesparadoxprobabilityChanceability<p>Say that a function <span
class="math inline"><em>P</em> : <em>F</em> → [0,1]</span> where <span
class="math inline"><em>F</em></span> is a <span
class="math inline"><em>σ</em></span>-algebra of subsets of <span
class="math inline"><em>Ω</em></span> is <em>chanceable</em> provided
that it is metaphysically possible to have a concrete (physical or not) stochastic process with a
state space of the same cardinality as <span
class="math inline"><em>Ω</em></span> and such that <span
class="math inline"><em>P</em></span> coincides with the chances of that
process under some isomorphism between <span
class="math inline"><em>Ω</em></span> and the state space.</p>
<p>Here are some hypotheses ones might consider:</p>
<ol type="1">
<li><p>If <em>P</em> is chanceable,
<em>P</em> is a finitely additive
probability.</p></li>
<li><p>If <em>P</em> is chanceable,
<em>P</em> is a countably additive
probability.</p></li>
<li><p>If <em>P</em> is a finitely
additive probability, <em>P</em> is
chanceable.</p></li>
<li><p>If <em>P</em> is a countably
additive probability, <em>P</em> is
chanceable.</p></li>
<li><p>A product of chanceable countably additive probabilities is
chanceable.</p></li>
</ol>
<p>It would be nice if (2) and (4) were both true; or if (1) and (3)
were.</p>
<p>I am inclined to think (5) is true, since if the <span
class="math inline"><em>P</em><sub><em>i</em></sub></span> are
chanceable, they could be implemented as chances of stochastic processes
of causally isolated universes in a multiverse, and the result would
have chances isomorphic to the product of the <span
class="math inline"><em>P</em><sub><em>i</em></sub></span>.</p>
<p>I think (3) is true in the special case where <span
class="math inline"><em>Ω</em></span> is finite.</p>
<p>I am skeptical of (4) (and hence of (3)). My skepticism comes from
the following line of thought. Let <span
class="math inline"><em>Ω</em> = ℵ<sub>1</sub></span>. Let <span
class="math inline"><em>F</em></span> be the <span
class="math inline"><em>σ</em></span>-algebra of countable and
co-countable subsets (<em>A</em> is
co-countable provided that <span
class="math inline"><em>Ω</em> − <em>A</em></span> is countable). Define
<em>P</em>(<em>A</em>) = 1 for the
co-countable subsets and <span
class="math inline"><em>P</em>(<em>A</em>) = 0</span> for the countable
ones. This is a countably additive probability. Now let <span
class="math inline"><</span> be the ordinal ordering on <span
class="math inline">ℵ<sub>1</sub></span>. Then if <span
class="math inline"><em>P</em></span> is chanceable, it can be used to
yield paradoxes very similar to those of a countably infinite fair
lottery.</p>
<p>For instance, consider a two-person game (this will require the
product of <em>P</em> with itself to be
chanceable, not just <em>P</em>; but I
think (5) is true) where each player independently gets an ordinal
according to a chancy isomorph of <span
class="math inline"><em>P</em></span>, and the one who gets the larger
ordinal wins a dollar. Then each player will think the probability that
the other player has the bigger ordinal is <span
class="math inline">1</span>, and will pay an arbitrarily high fee to
swap ordinals with them!</p>
http://alexanderpruss.blogspot.com/2024/09/chanceability.htmlnoreply@blogger.com (Alexander R Pruss)4