Prolegomena 5 (2) 2006: 209–218 Making Sense of Questions in logic and Mathematics: Mill vs. Carnap ESTHER RAMHARTER University of Vienna – Institute of Philosophy Universitätsstrasse 7, 1010 Vienna, Austria esther.ramharter@univie.ac.at ORIGINAL SCIENTIFIC ARTICLE / RECEIVED: 06–09–06 ACCEPTED: 10–10–06 abstraCt:.Whether.mathematical.truths.are.syntactical.(as.Rudolf.Carnap.claimed). or.empirical.(as.Mill.actually.never.claimed,.though.Carnap.claimed.that.he.did). might.seem.merely.an.academic.topic ..However,.it.becomes.a.practical.concern. as.soon.as.we.consider.the.role.of.questions ..For.if.we.inquire.as.to.the.truth.of.a. mathematical.statement,.this.question.must.be.(in.a.certain.respect).meaningless. for.Carnap,.as.its.truth.or.falsity.is.certain.in.advance.due.to.its.purely.syntactical.(or.formal-semantical).nature ..In.contrast,.for.Mill.such.a.question.is.as.valid. as. any.other ..These.differing.views.have. their. consequences. for. contemporary. erotetic.logic . KeyworDs:.Empirical.propositions,.erotetic. logic,.mathematical. truth,.question,. scientific.inquiry . "I.thought:.are.we.now.back.with.John.Stuart.Mill?".(R ..Carnap) 1. Plan of Inquiry The.aim.of.this.paper.is,.first.of.all,.to.reveal.and.contrast.the.roles.played. by.questions.in.the.work.of.John.Stuart.Mill.and.Rudolf.Carnap.(sec ..2. and.3),.especially.in.mathematics.and.logic.(sec ..4);.and,.secondly,.to.argue.(sec ..5).that.deciding.between.the.theoretical.conceptions.of.Mill.and. Carnap.implies.a.decision.between.the. two.main.approaches. to.erotetic. logic ..Questions.in.mathematics.will.play.the.key.part.in.these.considerations,.for.mathematical.theorems.are.expected.to.be.true.a priori ..Thus. a.question.in.mathematics.makes.sense.only.with.respect.to.an.individual. searching.for.knowledge;.and.how.to.deal.with.the.knowledge-searching. individual.is.exactly.where.the.two.types.of.accounts.in.erotetic.logic.part. 210 Prolegomena 5 (2) 2006 ways:. those.which.use.an. intensional.operator. to.model. the. role.of. the. individual,.and.those.which.shift.the.matter.to.pragmatics.and.leave.it.out. of.syntax . 2. Questions in General Needless. to.say,.Mill's. theory.of. logic.differs.greatly. in. its.nature.from. that.of.Carnap ..Carnap's.theory.is.formal.in.the.modern.sense,.something. that.Mill.could.not.have.achieved ..Nevertheless,.their.theories.do.share.a. certain.feature:.they.are.concerned.not.only.with.ready-made.propositions,. but.also.with.the.questions.that.lead.to.those.propositions ..The.two.philosophers.are.also.connected.by.the.general.setting.in.which.they.examine. questions,.which.are.not.reduced.to.linguistic.or.logical.entities,.but.rather. placed.within.the.framework.of.scientific.inquiries . Of. the.various.questions.which.present. themselves. to.our. inquiring.faculties,.some.receive.an.answer.from.direct.consciousness,.others,.if.resolved. at.all,.can.only.be.resolved.by.evidence ..Logic.is.concerned.with.these.last .. But.before.inquiring.into.the.mode.of.questions,. it. is.necessary.to.inquire. what.are.those.which.offer.themselves;.what.questions.are.conceivable;.what. inquiries.are.there,.to.which.mankind.have.either.obtained,.or.been.able.to. imagine.it.possible.that.they.should.obtain,.an.answer ..This.point.is.best.ascertained.by.a.survey.and.analysis.of.Propositions ..(Mill,.2002:.12) Thus.the.topic.of.consideration.is.not.questions.per se,.isolated.and.studied.for.their.own.sake,.but.questions.that.have.de facto.been.of.interest .. Furthermore,.questions.are.not.only. located.within. the.framework,. they. also.determine.its.limits: With. the.original.data,.or.ultimate.premises.of.our.knowledge;.with. their. number.or.nature,.the.mode.in.which.they.are.obtained,.or.the.tests.by.which. they.may.be.distinguished;.logic,.in.a.direct.way.at.least,.has,.in.the.sense.in. which.I.conceive.the.science,.nothing.to.do ..These.questions.are.partly.not. a.subject.of.science.at.all,.partly.that.of.a.very.different.science ..Whatever. is.known.to.us.by.consciousness,.is.known.beyond.possibility.of.question .. (Mill,.2002:.4) Questions.thus.separate.what.belongs.to.logic.from.what.does.not ..There. are. two. possibilities. for. something's. not. belonging. to. logic:. either. the. question.pertains.to.something.else.(to.a.different.science),.or.there.is.no. question.at.all . At. first. glance,. the. two. passages. cited. seem. to. contradict. one. another ..The.first.speaks.of.questions.which."receive.an.answer.from.direct. consciousness";.the.second.states.that."[w]hatever.is.known.to.us.by.consciousness,.is.known.beyond.possibility.of.question" ..However,.there.is. 211E. RAMHARTER: Making Sense of Questions in Logic and Mathematics a.difference.between.being.the.aim.of.a.question.and.being.questionable .. For.example,.if.I.wonder.whether.my.injured.toe.still.hurts,.and.press.my. finger.on.it.to.find.out,.then.the.result.will.be.an.answer.to.the.question. that.I.receive.by.direct.consciousness;.but.it.will.not.itself.be.questionable. or.in.any.way.dubious ..The.question.we.cannot.ask.in.this.context.is:.Do. I.really.feel.pain.in.my.toe?.(Yet.we.clearly.do.have.these.impressions.of. direct.consciousness,.they.do.exist;.and.so.it.makes.sense.to.say.that.they. do.not.belong.to.logic .) But.besides.these."unposable.questions".there.are,.as.I.have.already. observed,.still.other.questions.whose.answers.do.not.belong.to.logic,.either ..And.I.see.no.reason.to.make.a.distinction.between.logic.and.other. sciences. in. this. respect .. Every. science. has,. according. to. Mill,. its. own. questions.that.express.the.characteristic.interests.of.that.field . The.limiting.case.of.this.statement.is:.no.question,.no.science ..The. questionable.constitutes.the.limits.of.science ..But.what.about:.no.science,. no.question?.Are.there.questions.which.we.may.raise,.and.which.do.not. have.any.answers.in.any.science?.Mill.is.explicit.about.this.when.he.writes. (in.the.passage.quoted.above).of."questions.[that].are.partly.not.a.subject. of.science.at.all .".Let.us.compare.all.this.with.Carnap: By. boundlessness. of. scientific. knowledge. we. mean:. there. is. no. question. which.in.principle.[grundsätzlich].cannot.be.answered.by.science ..(Carnap,. 1998:.254) Carnap.denies.the.existence.of.unanswerable.questions ..(In.what.follows,. he.explains.what.he.means.by.grundsätzlich:.every.question.could.be.answered,. provided. there.was.no. temporal.or. spatial. gap.between.us. and. what.we.wanted.to.know;.or.else,.we.would.be.able.to.answer.the.question. once.the.technical.means.were.developed.and.placed.at.our.disposal .).But. perhaps.one.should.not.take.him.too.literally.here,.and.grant.him.a.more. sympathetic.reading ..For.immediately.before.the.sentence.cited.above.he. states.that.life.has.many.dimensions.beyond.science,.and.compares.scientific.knowledge.to.a.plane:.it.has.no.boundaries.within.itself,.but.something.exists.which.lies.outside it: Sometimes.it.is.said.that.the.answers.to.some.questions.cannot.be.grasped. by.concepts,.cannot.be.expressed ..But.in.such.cases.the.question.cannot.be. expressed.either ..(Carnap,.1998:.254) In.any.case,.there.is.no.extralinguistic approach.to.questions: To.recognize.this,.we.will.further.investigate.what.the.answering.of.a.question.consists.in ..In.a.strictly.logical.sense,.posing.a.question.means.giving.a. statement.and.setting.the.task.of.establishing.that.either.the.statement.or.its. negation.is.true ..(Carnap,.1998:.254) 212 Prolegomena 5 (2) 2006 In.his.later.book.Logische Syntax der Sprache.(1968),.Carnap.elaborates.a.formal.concept.of.the.question: A.yes-no.question.consists.in.the.request.to.either.affirm.or.negate.a.certain. proposition.S1,.that.is,.to.express.either.S1.or.~S1 ..(Carnap,.1968:.222) Yes-no.questions.do.not.cause.any.difficulty,. fitting.perfectly. into.what. Carnap.had. said. in. the.Aufbau. (1998) ..But.how.does.Carnap.bring.his. treatment.of.questions.in.the.Aufbau into.accord.with.that.part.of.the.theory.in Logische Syntax.where.he.formalizes.questions.like."When.is.John. Stuart.here?".–.that.is,.questions.which.do.not.simply.consist.of.a.given. statement.that.is.to.be.affirmed.or.denied?.He.proposes.that.such.a.question.be.understood.as.a.proposition.like."John.Stuart.is.here.at.t",.where.t. is.a.variable.which.is.free.or.bound.by.a.question.operator ..To.answer.such. a.question. is,. therefore,. to.divide.all.propositions.of. this. form.into. true. and.false.ones ..Hence.the.theory.of.wh-questions.could.be.seen.as.a.generalization.of.the.theory.of.yes-no.questions ..Yet.despite.appearances.it.is,. all.the.same,.a.restriction,.a.commitment.to.the.basic.idea.of.yes-no.questions ..Carnap.is.forced.–.or.thinks.he.is.forced.–.by.his.formal.theory.of. logic.to.limit.his.understanding.of.questions.in.this.way ..Mill's.approach. is,.of.course,.free.of.all.such.requirements . 3. Questions and Answers There.is.also.a.more.structural.connection.between.questions.and.propositions.for.both.Mill.and.Carnap: The.answer. to.every.question.which. it. is.possible. to. frame,.must.be.contained.in.a.Proposition,.or.Assertation ..[...].[T]o.know.the.import.of.all.possible. propositions,. would. be. to. know. all. questions. that. can. be. raised,. all. matters.which.are.susceptible.of.being.either.believed.or.disbelieved ..(Mill,. 2002:.12) Questions.and.the.propositions.answering.them.are.theoretically.related.to.one.another ..Mill's.suggestion.that.knowing.all.possible.answers. means.knowing.all.questions.is.one.of.the.standpoints.in.erotetic.logic.–. the.contemporary.logic.of.questions ..Its.various.proposals.are.as.follows: (a).The.sets.of.possible.answers.determine.the.questions . (b).The.sets.of.possible.answers.are.the.questions . (c).The.possible.answers.are.more.loosely.connected.to.the.questions .1 1.A.fourth.possibility.would.be:."Questions.determine.the.possible.answers" ..However,.this.would.require.a.(formal).system.that.starts.with.questions.and.derives.propositions.from.them ..As.far.as.I.know,.no.such.account.has.yet.been.developed . 213E. RAMHARTER: Making Sense of Questions in Logic and Mathematics Mill.is.obviously.a.proponent.of.either.(a).or.(b),.though.which.one. cannot.be.determined ..Carnap.holds.that.(b).is.the.right.view ..In.the.example.of.the.when-question.above,.it.is.clear.that."John.Stuart.is.here.at.t". represents.the.essence.of.the.question.as.well.as.of.its.answers ..This.is.the. first.point.where.interpreting.Mill's.and.Carnap's.work.has.implications. for.the.presuppositions.of.erotetic.logic . Although.a.strong.theoretical.connection.between.questions.and.answers.is.common.to.the.writings.of.Mill.and.Carnap.alike,.for.Mill.questions.are.prior.to.answers.in."reality".(in.the.practice.of.research) ..This.is. not.clear.in.every.case.with.Carnap,.as.we.shall.see . Having.explained.what.counts.as.an.answer.to.a.question,.we.must. consider. what. makes. an. answer. a. right. answer .. Roughly. speaking,. the. basic.propositions.in.Carnap's.Konstitutionstheorie.are.obtained.in.a.manner.quite.similar.to.(empirical).propositions.in.Mill's.theory.–.namely,.by. induction ..Moreover,.both.authors.share.a.rather.specific.understanding.of. induction:."In.short,.Carnap.appears.to.be.in.substantial.agreement.with. J ..S ..Mill's.view.that.the.fundamental.type.of.inductive.reasoning.is.'from. particulars.to.particulars'".(Nagel,.1963:.802) .2 Induction.need.not.always.lead.from.a.particular.case.to.the.general. case,.or.via the general case.to.another.particular.case ..It.may.be.that.no. universal.proposition.is.concluded ..For.example,.if.one.has.seen.a.number. of.white.swans.and.no.black.ones,.then.it.would.be.reasonable.to.conclude. that.the.next.swan.will.also.be.white,.even.though.one.does.not.have.the. slightest.reason.to.suppose.that.only.white.swans.exist . Apart.from.empirical.propositions,.however,.there.are.also.answers.to. questions.which.are.confirmed.as.true.in.a.different.way:.logical.theorems .. According.to.Carnap,.these.are.true.because.we.can.deduce.them.in.the. formal.system.we.are.using.(or,.in.late.Carnap,.they.are.semantically.valid. in.that.system);.according.to.Mill,.because.they.have.evidence.(see.above:. "[O]ther.[questions],.if.resolved.at.all,.can.only.be.resolved.by.evidence .. Logic.is.concerned.with.these.last .") ..Mill.does.not.really.explain.what.he. means.by."evidence" ..He.simply.opposes.it.to.consciousness.or.intuition. (using. these. terms. indiscriminately),.whereby.we.know.what.we.know. directly. (for. example,.our.knowledge.of.being,.or.having.been,.hungry. at.a.given.moment),.and.states.that.what.we.know.by.reasoning.must.be. known.through.evidence ..However,.I.will.not.deal.with.this.quite.complicated.topic.any.further,.as.it.has.no.relevance.to.my.argumentation.below. (on.Mill's.deductive.logic,.see.e .g ..Jackson,.1941) . The.main.points.above.may.be.summarized.by.answering.the.following.question:. In.which. respects.do.Mill's.and.Carnap's.views.coincide,. 2.This.is.what.Aristotle.calls.paradeigma,.as.opposed.to.epagoge . 214 Prolegomena 5 (2) 2006 and.in.which.do.they.differ?.They.both.(a).limit.meaningful.questions.by. the.propositions. that.might. answer. them,. (b). locate. the.question. in. the. general.context.of.scientific.inquiry,.and.(c).distinguish.between.two.sorts. of.reasons.which.make.an.answer.true.and,.therefore,.between.two.sorts. of.questions ..However,.(a).whereas.Carnap.places.empirical.reasons.and. analytical.connections.in.opposition,.Mill's.distinction.is.one.of.evidence. and.consciousness,.and. (b). for.Mill. there.are.questions. that.are.outside. science,.but.for.Carnap.(if.taken.literally).there.are.not . 4. Questions in Mathematics At.a.symposium.held.around.1940,.Carnap.asked:."Are.we.now.back.with. John.Stuart.Mill?".He.alleged.that.Mill.considered.mathematical.propositions.to.be.empirical.ones ..That.this.is.what.Carnap.really.thought.is.clear. from.the.context.in.which.he.voiced.this.question:.he.had.been.attacked. by.von.Mises,.Tarski.and.Quine.for.having.stated.that.mathematical.truths. are.non-factual.and.non-empirical .3 In.some.sense,.Mill.and.Carnap.may.be.seen.as.holding.two.extreme. positions.in.the.discussion.on.the.ontological.and.epistemic.status.of.mathematical. propositions .. For. Carnap,. mathematical. propositions. lack. any. empirical.content;.for.Mill,.all.propositions,.and.particularly.mathematical. theorems,.derive.some.of.their. justification.from.empirical.grounds .. But.does.Mill.really.claim.that.mathematical.propositions.are.empirical. ones? In. Mill's. time,. the. modern. understanding. of. mathematics. existed. mainly.in.the.field.of.geometry ..Thus.if.we.wish.to.know.what.the.epistemological.status.of.mathematical.theorems.is.and.to.understand.mathematics.as.a.deductive.science.on.an.axiomatic.basis,.we.should.study.Mill's. thoughts.on.geometry ..There.is.no.doubt.that.Mill.regards.mathematics.as. being.founded.on.real,.empirical.objects: We.are.thinking,.all.the.time,.of.precisely.such.objects.as.we.have.seen.and. touched,.and.with.all.the.properties.which.naturally.belong.to.them;.but,.for. scientific.convenience,.we.feign.them.to.be.invested.of.all.properties,.except. those.which.are.material.to.our.purpose,.and.in.regard.to.which.we.design.to. consider.them ..(Mill,.2002:.148) The.first.non-empirical.element.of.mathematical.theorems.is.their.restriction.to.a.certain.aspect,.an."insofar" ..This.can.be.approached.from.two. sides:.either.by.ignoring.the.inexactness.of.a.real.object,.or.by.considering. only.such.objects.as.are.sufficiently.exact.for.the.given.purpose ..There.is. evidence.in.Mill's.text.for.both.possibilities: 3.For.Carnap's.standpoint,.see.Carnap,.1963:.46f . 215E. RAMHARTER: Making Sense of Questions in Logic and Mathematics In.this.relation,.of.course,.the.derivative.truths.of.every.deductive.science. must.stand.to.the.inductions,.or.assumptions,.on.which.the.science.is.founded,. and.which,.whether.true.or.untrue,.certain.or.doubtful.in.themselves,.are.always.supposed.certain.for.the.purposes.of.the.particular.science ..And.therefore.the.conclusions.of.all.deductive.sciences.were.said.by.the.ancients.to.be. necessary.propositions ..(Mill,.2002:.149) A. second. and. even. more. important. non-empirical. element. is. included. here:.supposing,.or. the.will. to.suppose ..This.decisive.or.normative.element.is.a.very.modern.constituent.of.Mill's.conception.of.mathematical. statements4.(think.of.the.late.Wittgenstein,.for.example): In. all. propositions. concerning. numbers,. a. condition. is. implied,. without. which.none.of.them.would.be.true.[...] ..The.condition.is,.that.1=1;.that.all. the.numbers.are.numbers.of.the.same.or.of.equal.units ..Let.this.be.doubtful,. and.not.one.of.the.propositions.of.arithmetic.will.hold.to.be.true ..[...].How. can.we.know.that.a.forty-horse.power.is.always.equal.to.itself,.unless.we. assume.that.all.horses.are.of.equal.strength?.[...].What.is.commonly.mathematical.certainty,.therefore,.which.comprises.the.twofold.conception.of.unconditional.truth.and.perfect.accuracy,.is.not.an.attribute.of.all.mathematical. truths,.but.of.those.only.which.relate.to.pure.Number,.as.distinguished.from. Quantity.in.the.more.enlarged.sense;.and.only.so.long.as.we.abstain.from. supposing.that. the.numbers.are.a.precise. index.to.actual.quantities .. (Mill,. 2002:.170) Certainty.in.mathematics.depends,.accordingly,.on.the.presupposition.of. mathematical. principles ..We. are. free. to. suppose. these. principles,. or. to. neglect.them;.yet.only.so.long.as.we.suppose.them,.and.do.not.let.the.truth. of.a.mathematical.proposition.depend,.for.example,.on.physical.facts,.can. a.proposition.possess.the.specific.certainty.of.mathematics ..However,.in. order. to. be. useful. and. meaningful,. mathematical. principles. need. to. be. obtained.by.induction.from.empirical.facts ..This.does.not.necessarily.signify.a.loss.of.normativity ..The.idea.of.induction.is.also.normative,.yet.not. analytic.(not."verbal",.in.Mill's.terms) ..To.be.more.precise,.it.is.not.induction.but.a.reliance.on.causal.relations.in.nature.(cf ..Scarre,.1998).which. constitutes.the.normative.element.in.Mill's.theory.of.inductive.reasoning . Let.us.now.consider.the.role.of.questions.aimed.at.mathematical.or. logical.truths ..How.did.this.problem.arise,.and.how.there.can.be."new". theorems.in.mathematics.when.all.mathematical.theorems.are.true.a pri- ori?.A. conflict. only. occurs. when. we. consider. the. enterprise. of. finding. such. theorems,. i .e .. research.work ..Therefore,.questions are. the.point.at. which.the.origin.of.mathematical.truth.becomes.relevant . 4.The.normative.impact.of.Mill's.considerations.is.discussed.in.Skorupski,.1998:.53 . 216 Prolegomena 5 (2) 2006 Following. Carnap,. one. can. pose. such. questions,. formulating. them. in. a. syntactically. correct.way ..Hence. these. are. no. Scheinfragen,. in. the. sense.of.metaphysical.questions;.yet.in.another.respect.they.are,.however,. meaningless,.as.their.answers.are.set.in.advance ..A.mathematical.proposition.is.true.or.false.due.to.syntactical.reasons.(or.formal-semantical.ones),. and.does.not.require.any.empirical.information ..Thus.a.question.inquiring. about. the. truth.or.falsity.of.such.a.proposition.is.not.a.real.question,.at. least.not. in. logic.(although.perhaps. it. is.one. in.psychology) ..Following. Mill,.of.course,.such.questions.make.perfect.sense,.if.only.because.their. empirical.grounds.have.to.be.evaluated . 5. Consequences of Choosing a System of Erotetic Logic Questions. in. mathematics. are. the. crucial. point. in. discussions. about. rival. theories.of.questions,.with. intensional. conceptions.on. the.one. side,. and.those.which.operate.with.bound.or.unbound."query.variables".on.the. other . Aqvist.(1965),.Hintikka.(1976),.Kubinski.(1980),.Lewis.and.Lewis. (1975).and.several.other.authors.argue.that.a.question.aimed.at.the.truth.of. a.sentence.p.means.a.proposition.including.a.certain.modal.operator,.such. as:."Let.it.turn.out.to.be.the.case.that.I.know.that.p .".(In.formal.notation:. ?p:=!Kp,.where."!".means."Let.it.turn.out.to.be.the.case".and."K".means. "I.know.that" .).Some.variants.are:."Bring.it.about.that.I.know...",."Bring. it.about.that.I.believe...",.or."Tell.me.truly.if..." .5 For.Carnap,.such.intensional.approaches.are.quite.indispensable,.because asking a question like ?(2+3=5) or ?(p → p), where "?(...)" means an.arbitrary.formalization.of.that.question.within.a.formal.system,.would. be.pointless.if.there.were.no.difference.between.being.an.answer.and.being. known.as.an.answer.–.and.this.would.be.the.case.for.all.mathematical.questions ..Therefore,.in.order.to.make.the.formal.expression.of.a.question.in. mathematics.meaningful,.one.must.introduce.a.formal.equivalent.for.the. person.who.wants.to.know ..This.can.be.done.by.using.a.modal.operator . For.Mill,.on.the.other.hand,.these.questions.might.have.three.aims: (1).the.empirical.basis, (2).the.adequateness.of.restriction.to.a.certain.aspect,.and (3).acceptance.of.the.underlying.norms . Consider.the.example:."Is.there.an.angle.in.a.triangle.greater.than.180°?". In.this.instance.(1).and.(2).seem.quite.clear,.while.(3).may.be.stated.as. follows:.Do.we.accept.the.presuppositions.and.means.leading.to.the.con5.For.an.overview.of.these.accounts,.see.e .g ..Harrah,.2002 . 217E. RAMHARTER: Making Sense of Questions in Logic and Mathematics clusion.that.there.is.no.such.angle?.Fulfillment.of.these.three.requirements. gives.an.exact.characterization.of.the.usual.manner.of.justification.of.and. in.formal.systems ..(We.judge.a.mathematical.proof.to.be.correct.if.we.accept.the.premises.and.each.step.[i .e ..the.method].leading.to.the.result .) Let.us.now.turn.to.someone.we.might.not.expect.here: To.ask.a.question.means.to.bring.into.the.open ..The.openness.of.what.is.in. question.consists.in.the.fact.that.the.answer.is.not.settled ..(Gadamer,.1975:. 326) The.context.in.which.this.passage.from.Truth and Method.is.located.offers. many.interesting.connections.with.formal.logic;.much.of.what.Gadamer. says.can.even.be.translated.into.meta-logical.statements ..But.here.I.will. only.borrow.his.idea.of."openness".for.my.purposes ..If.a.question.is.to.be. open,.then.?(2+3=5),.as.Carnap.understands.it,.is.not.a.question;.we.must. introduce.an.intensional.operator.(such.as."knowing.that").into.the.logical.system.in.order.to.express.the.idea.of.openness ..(But.one.could.argue. that.in.doing.so.we.lose.this.very.same.notion.of.openness .).None.of.these. problems.arise.with.Mill,.since.for.him.there.is.simply.no.question.without. openness ..Whether.we.are.asking.about.an.empirical.fact.or.a.mathematical.theorem,.there.are.always.questionable.ingredients:.empirical.grounds,. norms,.or.at.least.the.acceptance.of.some.evidence . But.now.what.about.formalization?.Mill's.standpoint.saves.us.from. the.problem.by.spreading.it.to.anything.and.everything ..By.putting.a.question.one.also.questions.part.of.the.framework,.the.formal.system.in.which. the.question. is. located ..The.concept.of. a. "question.within. a. fixed. (formal).framework".is.simply.impossible.from.Mill's.point.of.view,.even.in. mathematics .. In. this. respect,.questions.do.not.differ. from.propositions;. however,.questions.need.openness.in.order.to.be.meaningful,.while.propositions.do.not ..Choosing.the.means.of.formalization.always.constitutes.an. essential.part.of.the.subject.of.a.question ..Thus.mathematical.questions.do. not.generate.a.need.for.extensions.of.erotetic.logic.like.those.proposed.by. Hintikka,.Kubinski.and.others,.since.all.questions,.including.mathematical.ones,.have.a.subject.(which.is.not.set.in.advance),.namely,.the.system. or.framework.in.which.they.are.expressed ..Therefore,.the.distinction.between."being.an.answer".and."being.known.as.an.answer".is.not.crucial. in this context .6.Thus. being. "back. with. John. Stuart. Mill". would. mean. surpassing.some.of.the.presuppositions.of.modern.erotetic.logic . 6.In.line.with.Mill's.thinking,.one.may.introduce.the.formalization.of.questions.into.a. logical.system,.which.may.be.regarded.as.a.formal.component,.and.does.not itself.require. openness.in.any.sense . 218 Prolegomena 5 (2) 2006 References Aqvist,.L ..1965 ..A New Approach to the Logical Theory of Interrogatives (Uppsala:.Almqvist.&.Wiksell) . 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