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All functions that represent straight lines on the coordinate plane are linear functions. No linear functions are non-convex functions. A function is either a non-convex fuction or a convex function. All quasi-convex functions are real-valued functions. All convex functions are quasi-convex functions. The maximum of quasiconvex functions is a function. The maximum of quasiconvex functions is a function that represents straight lines on the coordinate plane or it is a convex function or it is not a non-convex function.

∀x (Function(x) ∧ RepresentOn(x, straightLine, coordinatePlane) → LinearFunction(x)) ∀x (LinearFunction(x) → ¬NonConvexFunction(x)) ∀x (Function(x) → NonConvexFunction(x) ⊕ ConvexFunction(x)) ∀x (QuasiConvexFunction(x) → RealValuedFunction(x)) ∀x (ConvexFunction(x) → QuasiConvexFunction(x)) Function(maximumOfQuasiConvexFunction) (Function(maximumOfQuasiConvexFunction) ∧ RepresentOn(maximumOfQuasiConvexFunction, straightLine, coordinatePlane)) ∨ ConvexFunction(maximumOfQuasiConvexFunction) ∨ ¬NonConvexFunction(maximumOfQuasiConvexFunction)

The maximum of quasiconvex functions is not a real-valued function.

¬RealValuedFunction(maximumOfQuasiConvexFunction)

False

All functions that represent straight lines on the coordinate plane are linear functions. No linear functions are non-convex functions. A function is either a non-convex fuction or a convex function. All quasi-convex functions are real-valued functions. All convex functions are quasi-convex functions. The maximum of quasiconvex functions is a function. The maximum of quasiconvex functions is a function that represents straight lines on the coordinate plane or it is a convex function or it is not a non-convex function.

∀x (Function(x) ∧ RepresentOn(x, straightLine, coordinatePlane) → LinearFunction(x)) ∀x (LinearFunction(x) → ¬NonConvexFunction(x)) ∀x (Function(x) → NonConvexFunction(x) ⊕ ConvexFunction(x)) ∀x (QuasiConvexFunction(x) → RealValuedFunction(x)) ∀x (ConvexFunction(x) → QuasiConvexFunction(x)) Function(maximumOfQuasiConvexFunction) (Function(maximumOfQuasiConvexFunction) ∧ RepresentOn(maximumOfQuasiConvexFunction, straightLine, coordinatePlane)) ∨ ConvexFunction(maximumOfQuasiConvexFunction) ∨ ¬NonConvexFunction(maximumOfQuasiConvexFunction)

The maximum of quasiconvex functions is a quasi-convex function or it is not a real-valued function.

QuasiConvexFunction(maximumOfQuasiConvexFunction) ∨ ¬RealValuedFunction(maximumOfQuasiConvexFunction)

True

If two soccer teams score the same number of goals in one UCL final during the regular time, they need to play for the extra time. If two soccer teams score the same number of goals in one UCL final during both regular and extra time, they need to play the penalty shoot-out. Real Madrid and Atlético Madrid both scored one goal in the 2016 UCL final during the regular time. Real Madrid and Atlético Madrid both scored zero goals in the 2016 UCL final during the extra time.

∀w ∀x ∀y ∀z (SoccerTeam(x) ∧ SoccerTeam(y) ∧ NumberOfGoalScored(x, z) ∧ NumberOfGoalScored(y, w) ∧ y=w ∧ During(regularTime) → PlayExtra(x, y)) ∀x ∀y (SoccerTeam(x) ∧ SoccerTeam(y) ∧ SameScore(x, y) ∧ During(regularTime) ∧ During(extraTime) → PlayPenalty(x, y)) SoccerTeam(realMadrid) ∧ SoccerTeam(atleticoMadrid) ∧ SameScore(realMadrid, atleticoMadrid) ∧ During(regularTime) SoccerTeam(realMadrid) ∧ SoccerTeam(atleticoMadrid) ∧ SameScore(realMadrid, atleticoMadrid) ∧ During(extraTime)

Real Madrid and Atlético Madrid needed to play a penalty shoot-out in the 2016 UCL final.

PlayPenalty(realMadrid, atleticoMadrid)

True

If two soccer teams score the same number of goals in one UCL final during the regular time, they need to play for the extra time. If two soccer teams score the same number of goals in one UCL final during both regular and extra time, they need to play the penalty shoot-out. Real Madrid and Atlético Madrid both scored one goal in the 2016 UCL final during the regular time. Real Madrid and Atlético Madrid both scored zero goals in the 2016 UCL final during the extra time.

∀w ∀x ∀y ∀z (SoccerTeam(x) ∧ SoccerTeam(y) ∧ NumberOfGoalScored(x, z) ∧ NumberOfGoalScored(y, w) ∧ y=w ∧ During(regularTime) → PlayExtra(x, y)) ∀x ∀y (SoccerTeam(x) ∧ SoccerTeam(y) ∧ SameScore(x, y) ∧ During(regularTime) ∧ During(extraTime) → PlayPenalty(x, y)) SoccerTeam(realMadrid) ∧ SoccerTeam(atleticoMadrid) ∧ SameScore(realMadrid, atleticoMadrid) ∧ During(regularTime) SoccerTeam(realMadrid) ∧ SoccerTeam(atleticoMadrid) ∧ SameScore(realMadrid, atleticoMadrid) ∧ During(extraTime)

Real Madrid and Atlético Madrid did not need to play a penalty shoot-out in the 2016 UCL final.

¬PlayPenalty(realMadrid, atleticoMadrid)

False

System 7 is a UK-based electronic dance music band. Steve Hillage and Miquette Giraudy formed System 7. Steve Hillage and Miquette Giraudy are former members of the band Gong. Electric dance music bands are bands. System 7 has released several club singles. Club singles are not singles.

BasedIn(system7, uk) ∧ ElectronicDanceMusicBand(system7) Form(stevehillage, system7) ∧ Form(miquettegiraudy, system7) FormerMemberOf(stevehillage, gong) ∧ FormerMemberOf(miquettegiraudy, gong) ∀x (ElectronicDanceMusicBand(x) → Band(x)) ∃x (ClubSingle(x) ∧ Release(system7, x)) ∀x (ClubSingle(x) → ¬Single(x))

System 7 was formed by former members of Gong.

∃x (Form(x, system7) ∧ FormerMemberOf(x, gong))

True

System 7 is a UK-based electronic dance music band. Steve Hillage and Miquette Giraudy formed System 7. Steve Hillage and Miquette Giraudy are former members of the band Gong. Electric dance music bands are bands. System 7 has released several club singles. Club singles are not singles.

BasedIn(system7, uk) ∧ ElectronicDanceMusicBand(system7) Form(stevehillage, system7) ∧ Form(miquettegiraudy, system7) FormerMemberOf(stevehillage, gong) ∧ FormerMemberOf(miquettegiraudy, gong) ∀x (ElectronicDanceMusicBand(x) → Band(x)) ∃x (ClubSingle(x) ∧ Release(system7, x)) ∀x (ClubSingle(x) → ¬Single(x))

System 7 has released several singles.

∃x (Single(x) ∧ Release(system7, x))

Uncertain

System 7 is a UK-based electronic dance music band. Steve Hillage and Miquette Giraudy formed System 7. Steve Hillage and Miquette Giraudy are former members of the band Gong. Electric dance music bands are bands. System 7 has released several club singles. Club singles are not singles.

BasedIn(system7, uk) ∧ ElectronicDanceMusicBand(system7) Form(stevehillage, system7) ∧ Form(miquettegiraudy, system7) FormerMemberOf(stevehillage, gong) ∧ FormerMemberOf(miquettegiraudy, gong) ∀x (ElectronicDanceMusicBand(x) → Band(x)) ∃x (ClubSingle(x) ∧ Release(system7, x)) ∀x (ClubSingle(x) → ¬Single(x))

System 7 is not a band.

¬Band(system7)

False

A summarization model is always faithful if it uses content from the input documents. Extractive models are summarization models. An extractive model can only use content from the input documents.

∀x (Model(x) ∧ Summarization(x) ∧ OnlyUseInputDocument(x) → Faithful(x)) ∀x (Model(x) ∧ Extractive(x) → Model(x) ∧ Summarization(x)) ∀x (Model(x) ∧ Extractive(x) → OnlyUseInputDocument(x))

Extractive models are always faithful.

∀x (Model(x) ∧ Extractive(x) → Faithful(x))

True

A summarization model is always faithful if it uses content from the input documents. Extractive models are summarization models. An extractive model can only use content from the input documents.

∀x (Model(x) ∧ Summarization(x) ∧ OnlyUseInputDocument(x) → Faithful(x)) ∀x (Model(x) ∧ Extractive(x) → Model(x) ∧ Summarization(x)) ∀x (Model(x) ∧ Extractive(x) → OnlyUseInputDocument(x))

Extractive models are not always faithful.

∃x (Model(x) ∧ Extractive(x) ∧ ¬Faithful(x))

False

If Robin's friends practice coding questions, then they are not studying to go to medical school to become a doctor. If Robin's friends want to work in the software engineering industry, then they practice coding questions. If Robin's friends enjoy healthcare fields and want to help people with medical issues, then they are studying to go to medical school to become a doctor. If Robin's friends grew up with parents who worked as doctors, then they enjoy healthcare fields and want to help people with medical issues. If Robin's friends study hard, then they grew up with parents who worked as doctors. Mark is Robin's friend. If Mark neither enjoys healthcare fields and wants to help people with medical issues nor grew up with parents who worked as doctors, then Mark is either a person who studies hard or grew up with parents who worked as doctors.

∀x (RobinsFriends(x) ∧ Practice(x, codingQuestion) → ¬StudyingToGoToToBecome(x, medicalSchool, doctor)) ∀x (RobinsFriends(x) ∧ WantToWorkIn(x, softwareEngineeringIndustry) → PracticeCodingQuestions(x)) ∀x (RobinsFriends(x) ∧ Enjoy(x, healthcareField) ∧ WantToHelp(x, peopleWithMedicalIssue) → StudyingToGoToToBecome(x, medicalSchool, doctor)) ∀x (RobinsFriends(x) ∧ (∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) → EnjoyHealthcareFields(x) ∧ WantToHelp(x, peopleWithMedicalIssue)) ∀x (RobinsFriends(x) ∧ StudyHard(x) → ∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) RobinsFriends(mark) ¬((Enjoy(x, healthcareField) ∧ WantToHelp(mark, peopleWithMedicalIssues)) ∧ ¬(∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) → StudyHard(mark) ∨ (∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z)))

Mark is Robin's friend and he is a person who studies hard.

RobinsFriends(mark) ∧ StudyHard(mark)

Uncertain

If Robin's friends practice coding questions, then they are not studying to go to medical school to become a doctor. If Robin's friends want to work in the software engineering industry, then they practice coding questions. If Robin's friends enjoy healthcare fields and want to help people with medical issues, then they are studying to go to medical school to become a doctor. If Robin's friends grew up with parents who worked as doctors, then they enjoy healthcare fields and want to help people with medical issues. If Robin's friends study hard, then they grew up with parents who worked as doctors. Mark is Robin's friend. If Mark neither enjoys healthcare fields and wants to help people with medical issues nor grew up with parents who worked as doctors, then Mark is either a person who studies hard or grew up with parents who worked as doctors.

∀x (RobinsFriends(x) ∧ Practice(x, codingQuestion) → ¬StudyingToGoToToBecome(x, medicalSchool, doctor)) ∀x (RobinsFriends(x) ∧ WantToWorkIn(x, softwareEngineeringIndustry) → PracticeCodingQuestions(x)) ∀x (RobinsFriends(x) ∧ Enjoy(x, healthcareField) ∧ WantToHelp(x, peopleWithMedicalIssue) → StudyingToGoToToBecome(x, medicalSchool, doctor)) ∀x (RobinsFriends(x) ∧ (∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) → EnjoyHealthcareFields(x) ∧ WantToHelp(x, peopleWithMedicalIssue)) ∀x (RobinsFriends(x) ∧ StudyHard(x) → ∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) RobinsFriends(mark) ¬((Enjoy(x, healthcareField) ∧ WantToHelp(mark, peopleWithMedicalIssues)) ∧ ¬(∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) → StudyHard(mark) ∨ (∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z)))

Mark is Robin's friend and he practices coding questions and wants to work in the software engineering industry.

RobinsFriends(mark) ∧ Practice(mark, codingQuestion) ∧ WantToWorkIn(mark, softwareEngineeringIndustry)

False

If Robin's friends practice coding questions, then they are not studying to go to medical school to become a doctor. If Robin's friends want to work in the software engineering industry, then they practice coding questions. If Robin's friends enjoy healthcare fields and want to help people with medical issues, then they are studying to go to medical school to become a doctor. If Robin's friends grew up with parents who worked as doctors, then they enjoy healthcare fields and want to help people with medical issues. If Robin's friends study hard, then they grew up with parents who worked as doctors. Mark is Robin's friend. If Mark neither enjoys healthcare fields and wants to help people with medical issues nor grew up with parents who worked as doctors, then Mark is either a person who studies hard or grew up with parents who worked as doctors.

∀x (RobinsFriends(x) ∧ Practice(x, codingQuestion) → ¬StudyingToGoToToBecome(x, medicalSchool, doctor)) ∀x (RobinsFriends(x) ∧ WantToWorkIn(x, softwareEngineeringIndustry) → PracticeCodingQuestions(x)) ∀x (RobinsFriends(x) ∧ Enjoy(x, healthcareField) ∧ WantToHelp(x, peopleWithMedicalIssue) → StudyingToGoToToBecome(x, medicalSchool, doctor)) ∀x (RobinsFriends(x) ∧ (∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) → EnjoyHealthcareFields(x) ∧ WantToHelp(x, peopleWithMedicalIssue)) ∀x (RobinsFriends(x) ∧ StudyHard(x) → ∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) RobinsFriends(mark) ¬((Enjoy(x, healthcareField) ∧ WantToHelp(mark, peopleWithMedicalIssues)) ∧ ¬(∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) → StudyHard(mark) ∨ (∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z)))

Mark is Robin's friend and he neither practices coding questions nor works to work in the software engineering industry.

RobinsFriends(mark) ∧ ¬(Practice(mark, codingQuestion) ∨ WantToWorkIn(mark, softwareEngineeringIndustry))

True

People who work at Jess's company and go to the spa frequently are not people who are miserly and need to save a large portion of their income. People who work at Jess's company are either miserly and need to save a large portion of their income, or frivolously spend a lot of money. If people who work at Jess's company frivolously spend a lot of money when they go out, then they value quality manufacturing and luxury items. If people who work at Jess's company value quality manufacturing and luxury items, then they enjoy shopping for materialistic items in their free time. Thomas works at Jess's company. If Thomas is not miserly and needs to save a large portion of his income, then Thomas does not value quality manufacturing and luxury items. Thomas values quality manufacturing and luxury items or he is not miserly.

∀x (WorkAt(x, jesssCompany) ∧ GoToSpafrequently(x) → ¬(Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome))) ∀x (WorkAt(x, jesssCompany) → Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome)⊕ SpendFrivolously(x, aLotOfMoney)) ∀x (WorkAt(x, jesssCompany) ∧ SpendFrivolously(x, aLotOfMoney) → Value(x, qualityManufacturing) ∧ Value(x, luxuryItem)) ∀x (WorkAt(x, jesssCompany) ∧ Value(x, qualityManufacturing) ∧ Value(x, luxuryItem) → Enjoy(x, shopping, materialisticItem)) WorkAt(thomas, jesssCompany) ¬(Miserly(thomas) ∧ NeedToSave(thomas, aLargePortionOfIncome)) → ¬((Value(thomas, qualityManufacturing) ∧ Value(thomas, luxuryItem))) (Value(thomas, qualityManufacturing) ∧ Value(thomas, luxuryItem)) ∨ ¬(Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome))

Thomas frivolously spends a lot of money.

SpendFrivolously(thomas, aLotOfMoney)

False

People who work at Jess's company and go to the spa frequently are not people who are miserly and need to save a large portion of their income. People who work at Jess's company are either miserly and need to save a large portion of their income, or frivolously spend a lot of money. If people who work at Jess's company frivolously spend a lot of money when they go out, then they value quality manufacturing and luxury items. If people who work at Jess's company value quality manufacturing and luxury items, then they enjoy shopping for materialistic items in their free time. Thomas works at Jess's company. If Thomas is not miserly and needs to save a large portion of his income, then Thomas does not value quality manufacturing and luxury items. Thomas values quality manufacturing and luxury items or he is not miserly.

∀x (WorkAt(x, jesssCompany) ∧ GoToSpafrequently(x) → ¬(Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome))) ∀x (WorkAt(x, jesssCompany) → Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome)⊕ SpendFrivolously(x, aLotOfMoney)) ∀x (WorkAt(x, jesssCompany) ∧ SpendFrivolously(x, aLotOfMoney) → Value(x, qualityManufacturing) ∧ Value(x, luxuryItem)) ∀x (WorkAt(x, jesssCompany) ∧ Value(x, qualityManufacturing) ∧ Value(x, luxuryItem) → Enjoy(x, shopping, materialisticItem)) WorkAt(thomas, jesssCompany) ¬(Miserly(thomas) ∧ NeedToSave(thomas, aLargePortionOfIncome)) → ¬((Value(thomas, qualityManufacturing) ∧ Value(thomas, luxuryItem))) (Value(thomas, qualityManufacturing) ∧ Value(thomas, luxuryItem)) ∨ ¬(Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome))

Thomas either enjoys shopping for materialistic items in his free time or, if he does not, then he goes to the spa frequently.

Enjoy(thomas, shopping, materialisticItem) ⊕ GoToSpaFrequently(thomas)

True

People who work at Jess's company and go to the spa frequently are not people who are miserly and need to save a large portion of their income. People who work at Jess's company are either miserly and need to save a large portion of their income, or frivolously spend a lot of money. If people who work at Jess's company frivolously spend a lot of money when they go out, then they value quality manufacturing and luxury items. If people who work at Jess's company value quality manufacturing and luxury items, then they enjoy shopping for materialistic items in their free time. Thomas works at Jess's company. If Thomas is not miserly and needs to save a large portion of his income, then Thomas does not value quality manufacturing and luxury items. Thomas values quality manufacturing and luxury items or he is not miserly.

∀x (WorkAt(x, jesssCompany) ∧ GoToSpafrequently(x) → ¬(Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome))) ∀x (WorkAt(x, jesssCompany) → Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome)⊕ SpendFrivolously(x, aLotOfMoney)) ∀x (WorkAt(x, jesssCompany) ∧ SpendFrivolously(x, aLotOfMoney) → Value(x, qualityManufacturing) ∧ Value(x, luxuryItem)) ∀x (WorkAt(x, jesssCompany) ∧ Value(x, qualityManufacturing) ∧ Value(x, luxuryItem) → Enjoy(x, shopping, materialisticItem)) WorkAt(thomas, jesssCompany) ¬(Miserly(thomas) ∧ NeedToSave(thomas, aLargePortionOfIncome)) → ¬((Value(thomas, qualityManufacturing) ∧ Value(thomas, luxuryItem))) (Value(thomas, qualityManufacturing) ∧ Value(thomas, luxuryItem)) ∨ ¬(Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome))

If Thomas either enjoys shopping for materialistic items in his free time or, if he does not, then he goes to the spa frequently, then Thomas neither values quality manufacturing and luxury items nor goes to the spa frequently.

Enjoy(thomas, shopping, materialisticItem) ⊕ GoToSpaFrequently(thomas) → ¬((Value(x, qualityManufacturing) ∧ Value(x, luxuryItem)) ∨ GoToSpaFrequently(thomas))

False

The indie pop band Phoenix has released six albums. Phoenix's album "Wolfgang Amadeus Phoenix" sold over 500,000 copies. A certified gold album or single is one which sold over half a million copies. "1901" is a single from Phoenix's album "Wolfgang Amadeus Phoenix." Over 400,000 copies of "1901" have been sold.

AlbumsReleased(phoenix, 6) Album(wolfgangamadeusphoenix) ∧ IsAlbumOf(wolfgangamadeusphoenix, phoenix) ∧ SoldOver(wolfgangamadeusphoenix, 500,000) ∀x ((Album(x) ∨ Single(x)) ∧ SoldOver(x, 500,000) → CertifiedGold(x)) Single(1901) ∧ From(1901, wolfgangamadeusphoenix) ∧ By(1901, phoenix) SoldOver(l1901, 400,000)

The album "Wolfgang Amadeus Phoenix" is a certified gold album.

CertifiedGold(wolfgangamAdeusPhoenix)

True

The indie pop band Phoenix has released six albums. Phoenix's album "Wolfgang Amadeus Phoenix" sold over 500,000 copies. A certified gold album or single is one which sold over half a million copies. "1901" is a single from Phoenix's album "Wolfgang Amadeus Phoenix." Over 400,000 copies of "1901" have been sold.

AlbumsReleased(phoenix, 6) Album(wolfgangamadeusphoenix) ∧ IsAlbumOf(wolfgangamadeusphoenix, phoenix) ∧ SoldOver(wolfgangamadeusphoenix, 500,000) ∀x ((Album(x) ∨ Single(x)) ∧ SoldOver(x, 500,000) → CertifiedGold(x)) Single(1901) ∧ From(1901, wolfgangamadeusphoenix) ∧ By(1901, phoenix) SoldOver(l1901, 400,000)

The single "1901" is a certified gold single.

CertifiedGold(1901)

Uncertain

Peter Parker is either a superhero or a civilian. The Hulk is a destroyer. The Hulk wakes up when he is angry. If the Hulk wakes up, then he will break a bridge. Thor is a god. Thor will break a bridge when he is happy. A god is not a destroyer. Peter Parker wears a uniform when he is a superhero. Peter Parker is not a civilian if a destroyer is breaking a bridge. If Thor is happy, the Hulk is angry.

Superhero(peterParker) ⊕ Civilian(peterParker) Destroyer(theHulk) Angry(theHulk) → WakesUp(theHulk) WakesUp(theHulk) → Breaks(theHulk, bridge) God(thor) Happy(thor) → Breaks(thor, bridge) ∀x (God(x) → ¬Destroyer(x)) Superhero(peter) → Wears(peter, uniform) ∀x ((Destroyer(x) ∧ Breaks(x,bridge)) → ¬Civilian(peter)) Happy(thor) → Angry(theHulk)

If the Hulk does not wake up, then Thor is not happy.

¬WakesUp(theHulk) → ¬Happy(thor)

True

Peter Parker is either a superhero or a civilian. The Hulk is a destroyer. The Hulk wakes up when he is angry. If the Hulk wakes up, then he will break a bridge. Thor is a god. Thor will break a bridge when he is happy. A god is not a destroyer. Peter Parker wears a uniform when he is a superhero. Peter Parker is not a civilian if a destroyer is breaking a bridge. If Thor is happy, the Hulk is angry.

Superhero(peterParker) ⊕ Civilian(peterParker) Destroyer(theHulk) Angry(theHulk) → WakesUp(theHulk) WakesUp(theHulk) → Breaks(theHulk, bridge) God(thor) Happy(thor) → Breaks(thor, bridge) ∀x (God(x) → ¬Destroyer(x)) Superhero(peter) → Wears(peter, uniform) ∀x ((Destroyer(x) ∧ Breaks(x,bridge)) → ¬Civilian(peter)) Happy(thor) → Angry(theHulk)

If Thor is happy, then Peter Parker wears a uniform.

Happy(thor) → Wears(peterParker, uniform)

True

Peter Parker is either a superhero or a civilian. The Hulk is a destroyer. The Hulk wakes up when he is angry. If the Hulk wakes up, then he will break a bridge. Thor is a god. Thor will break a bridge when he is happy. A god is not a destroyer. Peter Parker wears a uniform when he is a superhero. Peter Parker is not a civilian if a destroyer is breaking a bridge. If Thor is happy, the Hulk is angry.

Superhero(peterParker) ⊕ Civilian(peterParker) Destroyer(theHulk) Angry(theHulk) → WakesUp(theHulk) WakesUp(theHulk) → Breaks(theHulk, bridge) God(thor) Happy(thor) → Breaks(thor, bridge) ∀x (God(x) → ¬Destroyer(x)) Superhero(peter) → Wears(peter, uniform) ∀x ((Destroyer(x) ∧ Breaks(x,bridge)) → ¬Civilian(peter)) Happy(thor) → Angry(theHulk)

If Thor is not happy, then no bridge will be broken.

¬Happy(thor) → ¬Breaks(thor, bridge)

Uncertain

Diethylcarbamazine is a medication discovered in the year 1947. Diethylcarbamazine can be used to treat river blindness. The only preferred treatment for river blindness is ivermectin. Diethylcarbamazine is not ivermectin.

Medication(diethylcarbamazine) ∧ DiscoversIn(diethylcarbamazine, yr1947) Treats(diethylcarbamazine, riverBlindness) PreferredTreatmentFor(riverBlindness, ivermectin) ¬(Is(diethylcarbamazine, ivermectin))

Diethylcarbamazine is not preferred for the treatment of river blindness.

¬(PreferredTreatmentFor(riverBlindness, diethylcarbamazine))

True

Diethylcarbamazine is a medication discovered in the year 1947. Diethylcarbamazine can be used to treat river blindness. The only preferred treatment for river blindness is ivermectin. Diethylcarbamazine is not ivermectin.

Medication(diethylcarbamazine) ∧ DiscoversIn(diethylcarbamazine, yr1947) Treats(diethylcarbamazine, riverBlindness) PreferredTreatmentFor(riverBlindness, ivermectin) ¬(Is(diethylcarbamazine, ivermectin))

Diethylcarbamazine was often used to treat river blindness.

Treats(diethylcarbamazine, riverBlindness)

Uncertain

Diethylcarbamazine is a medication discovered in the year 1947. Diethylcarbamazine can be used to treat river blindness. The only preferred treatment for river blindness is ivermectin. Diethylcarbamazine is not ivermectin.

Medication(diethylcarbamazine) ∧ DiscoversIn(diethylcarbamazine, yr1947) Treats(diethylcarbamazine, riverBlindness) PreferredTreatmentFor(riverBlindness, ivermectin) ¬(Is(diethylcarbamazine, ivermectin))

Diethylcarbamazine is used in the treatment of filariasis.

Treats(diethylcarbamazine, filariasis)

Uncertain

All prime numbers are natural numbers. All integers are real numbers. All real numbers are complex numbers. One is a prime number or a natural number or both. If one is not a complex number, then one is a prime number and an integer.

∀x (PrimeNumber(x) → NaturalNumber(x)) ∀x (Integer(x) → RealNumber(x)) ∀x (RealNumber(x) → ComplexNumber(x)) PrimeNumber(one) ∨ NaturalNumber(one) ¬ComplexNumber(one) → (PrimeNumber(one) ∧ Integer(one))

One is a real number.

RealNumber(one)

Uncertain

All prime numbers are natural numbers. All integers are real numbers. All real numbers are complex numbers. One is a prime number or a natural number or both. If one is not a complex number, then one is a prime number and an integer.

∀x (PrimeNumber(x) → NaturalNumber(x)) ∀x (Integer(x) → RealNumber(x)) ∀x (RealNumber(x) → ComplexNumber(x)) PrimeNumber(one) ∨ NaturalNumber(one) ¬ComplexNumber(one) → (PrimeNumber(one) ∧ Integer(one))

One is a prime number and a natural number.

PrimeNumber(one) ∧ NaturalNumber(one)

True

All prime numbers are natural numbers. All integers are real numbers. All real numbers are complex numbers. One is a prime number or a natural number or both. If one is not a complex number, then one is a prime number and an integer.

∀x (PrimeNumber(x) → NaturalNumber(x)) ∀x (Integer(x) → RealNumber(x)) ∀x (RealNumber(x) → ComplexNumber(x)) PrimeNumber(one) ∨ NaturalNumber(one) ¬ComplexNumber(one) → (PrimeNumber(one) ∧ Integer(one))

One is either a prime number or a natural number.

PrimeNumber(one) ⊕ NaturalNumber(one)

False

If some diseases require a medical diagnosis, then lab tests or imaging is required. All rare diseases require a medical diagnosis. If a disease is mild, then no lab tests or imaging is required. All blood cancers are rare diseases. All types of leukemia are diseases and blood cancers. Bladder cancer is a disease and is blood cancer or Leukemia.

∀x (Disease(x) ∧ Require(x, medicalDiagnosis) → RequiredFor(labTest, x) ∨ RequiredFor(imaging, x)) ∀x (RareDisease(x) → Require(x, medicalDiagnosis)) ∀x (Disease(x) ∧ Mild(x) → ¬(RequiredFor(labTest, x) ∨ RequiredFor(imaging, x))) ∀x (BloodCancer(x) → RareDiseases(x)) ∀x (Disease(x) ∧ Leukemia(x) → BloodCancer(x)) Disease(bladderCancer) ∧ (BloodCancer(bladderCancer) ∨ Leukemia(bladderCancer))

Bladder cancer is a mild disease.

Mild(bladderCancer)

False

If some diseases require a medical diagnosis, then lab tests or imaging is required. All rare diseases require a medical diagnosis. If a disease is mild, then no lab tests or imaging is required. All blood cancers are rare diseases. All types of leukemia are diseases and blood cancers. Bladder cancer is a disease and is blood cancer or Leukemia.

∀x (Disease(x) ∧ Require(x, medicalDiagnosis) → RequiredFor(labTest, x) ∨ RequiredFor(imaging, x)) ∀x (RareDisease(x) → Require(x, medicalDiagnosis)) ∀x (Disease(x) ∧ Mild(x) → ¬(RequiredFor(labTest, x) ∨ RequiredFor(imaging, x))) ∀x (BloodCancer(x) → RareDiseases(x)) ∀x (Disease(x) ∧ Leukemia(x) → BloodCancer(x)) Disease(bladderCancer) ∧ (BloodCancer(bladderCancer) ∨ Leukemia(bladderCancer))

Bladder cancer is Leukemia.

Leukemia(bladderCancer)

Uncertain

If some diseases require a medical diagnosis, then lab tests or imaging is required. All rare diseases require a medical diagnosis. If a disease is mild, then no lab tests or imaging is required. All blood cancers are rare diseases. All types of leukemia are diseases and blood cancers. Bladder cancer is a disease and is blood cancer or Leukemia.

∀x (Disease(x) ∧ Require(x, medicalDiagnosis) → RequiredFor(labTest, x) ∨ RequiredFor(imaging, x)) ∀x (RareDisease(x) → Require(x, medicalDiagnosis)) ∀x (Disease(x) ∧ Mild(x) → ¬(RequiredFor(labTest, x) ∨ RequiredFor(imaging, x))) ∀x (BloodCancer(x) → RareDiseases(x)) ∀x (Disease(x) ∧ Leukemia(x) → BloodCancer(x)) Disease(bladderCancer) ∧ (BloodCancer(bladderCancer) ∨ Leukemia(bladderCancer))

Bladder cancer is either a rare disease or a mild disease.

RareDisease(bladderCancer) ⊕ Mild(bladderCancer)

True

There are no elements with atomic number between 61-63 that are not scarce in China. Non-rare earth elements are not scarce in China. All elements are either non-rare earth elements or rare earth elements. All rare earth elements can be used for industry. All rare earth elements are essential for exploring future directions of electronics. Lithium is either a non-rare earth element and essential for exploring future directions of electronics, or is not a non-rare earth element and is not essential for exploring future directions of electronics.

∀x ((Element(x) ∧ ∃y(Between(y, num61, num63) ∧ AtomicNumber(x, y))) → ScarceIn(x, china)) ∀x (¬RareEarthElement(x) → ¬ScarceIn(x, china)) ∀x (¬RareEarthElement(x) ⊕ RareEarthElement(x)) ∀x (RareEarthElement(x) → UsedIn(x, industry)) ∀x (RareEarthElement(x) → EssentialFor(x, electronics)) ¬(¬RareEarthElement(lithium) ⊕ EssentialFor(lithium, electronics))

Lithium is a rare earth element.

RareEarthElement(lithium)

Uncertain

There are no elements with atomic number between 61-63 that are not scarce in China. Non-rare earth elements are not scarce in China. All elements are either non-rare earth elements or rare earth elements. All rare earth elements can be used for industry. All rare earth elements are essential for exploring future directions of electronics. Lithium is either a non-rare earth element and essential for exploring future directions of electronics, or is not a non-rare earth element and is not essential for exploring future directions of electronics.

∀x ((Element(x) ∧ ∃y(Between(y, num61, num63) ∧ AtomicNumber(x, y))) → ScarceIn(x, china)) ∀x (¬RareEarthElement(x) → ¬ScarceIn(x, china)) ∀x (¬RareEarthElement(x) ⊕ RareEarthElement(x)) ∀x (RareEarthElement(x) → UsedIn(x, industry)) ∀x (RareEarthElement(x) → EssentialFor(x, electronics)) ¬(¬RareEarthElement(lithium) ⊕ EssentialFor(lithium, electronics))

Lithium is an element with atomic number between 61-63 and is used for batteries.

Element(x) ∧ ∃y(Between(y, num61, num63) ∧ AtomicNumber(x, y)) ∧ UsedFor(lithium, batteries)

False

There are no elements with atomic number between 61-63 that are not scarce in China. Non-rare earth elements are not scarce in China. All elements are either non-rare earth elements or rare earth elements. All rare earth elements can be used for industry. All rare earth elements are essential for exploring future directions of electronics. Lithium is either a non-rare earth element and essential for exploring future directions of electronics, or is not a non-rare earth element and is not essential for exploring future directions of electronics.

∀x ((Element(x) ∧ ∃y(Between(y, num61, num63) ∧ AtomicNumber(x, y))) → ScarceIn(x, china)) ∀x (¬RareEarthElement(x) → ¬ScarceIn(x, china)) ∀x (¬RareEarthElement(x) ⊕ RareEarthElement(x)) ∀x (RareEarthElement(x) → UsedIn(x, industry)) ∀x (RareEarthElement(x) → EssentialFor(x, electronics)) ¬(¬RareEarthElement(lithium) ⊕ EssentialFor(lithium, electronics))

If Lithium is not essential for exploring future directions of electronics or an element with atomic number between 61-63, then Lithium is not a non-rare earth element or usable in industry.

¬(EssentialFor(lithium, electronics) ⊕ (∃y(Between(y, num61, num63) ∧ AtomicNumber(lithium, y)))) → ¬(¬RareEarthMetals(lithium) ∨ UsedIn(lithium, industry))

True

If people don't often clean their homes, then they do not have tidy houses. If people don't prioritize cleaning, then they do not often clean their homes. If people hire a maid or cleaning service, then they have tidy houses. If people don't care about cleanliness, then they do not prioritize cleaning. Either Jack does hire a maid or cleaning service and does not often clean his home, or he does not hire a maid or cleaning service nor often clean his home.

∀x (¬CleanOften(x, home) → ¬Have(x, tidyHouse)) ∀x (¬Prioritize(x, cleaning) → ¬CleanOften(x, home)) ∀x (Hire(x, maid) ∨ Hire(x, cleaningService) → Have(x, tidyHouse)) ∀x (¬CareAbout(x, cleanliness) → ¬Prioritize(x, cleaning)) ¬(Hire(x, maid) ∨ Hire(x, cleaningService)) ⊕ ¬CleanOften(jack, home))

Jack doesn't care about cleanliness.

¬(CareAbout(jack, cleanliness))

False

If people don't often clean their homes, then they do not have tidy houses. If people don't prioritize cleaning, then they do not often clean their homes. If people hire a maid or cleaning service, then they have tidy houses. If people don't care about cleanliness, then they do not prioritize cleaning. Either Jack does hire a maid or cleaning service and does not often clean his home, or he does not hire a maid or cleaning service nor often clean his home.

∀x (¬CleanOften(x, home) → ¬Have(x, tidyHouse)) ∀x (¬Prioritize(x, cleaning) → ¬CleanOften(x, home)) ∀x (Hire(x, maid) ∨ Hire(x, cleaningService) → Have(x, tidyHouse)) ∀x (¬CareAbout(x, cleanliness) → ¬Prioritize(x, cleaning)) ¬(Hire(x, maid) ∨ Hire(x, cleaningService)) ⊕ ¬CleanOften(jack, home))

Jack does care about cleanliness.

CareAbout(jack, cleanliness)

True

If people don't often clean their homes, then they do not have tidy houses. If people don't prioritize cleaning, then they do not often clean their homes. If people hire a maid or cleaning service, then they have tidy houses. If people don't care about cleanliness, then they do not prioritize cleaning. Either Jack does hire a maid or cleaning service and does not often clean his home, or he does not hire a maid or cleaning service nor often clean his home.

∀x (¬CleanOften(x, home) → ¬Have(x, tidyHouse)) ∀x (¬Prioritize(x, cleaning) → ¬CleanOften(x, home)) ∀x (Hire(x, maid) ∨ Hire(x, cleaningService) → Have(x, tidyHouse)) ∀x (¬CareAbout(x, cleanliness) → ¬Prioritize(x, cleaning)) ¬(Hire(x, maid) ∨ Hire(x, cleaningService)) ⊕ ¬CleanOften(jack, home))

Jack has a tidy house.

Have(jack, tidyHouse)

Uncertain

If people don't often clean their homes, then they do not have tidy houses. If people don't prioritize cleaning, then they do not often clean their homes. If people hire a maid or cleaning service, then they have tidy houses. If people don't care about cleanliness, then they do not prioritize cleaning. Either Jack does hire a maid or cleaning service and does not often clean his home, or he does not hire a maid or cleaning service nor often clean his home.

∀x (¬CleanOften(x, home) → ¬Have(x, tidyHouse)) ∀x (¬Prioritize(x, cleaning) → ¬CleanOften(x, home)) ∀x (Hire(x, maid) ∨ Hire(x, cleaningService) → Have(x, tidyHouse)) ∀x (¬CareAbout(x, cleanliness) → ¬Prioritize(x, cleaning)) ¬(Hire(x, maid) ∨ Hire(x, cleaningService)) ⊕ ¬CleanOften(jack, home))

Jack neither lives in the suburbs nor is too busy to clean.

¬(¬CareAbout(jack, cleanliness) ∨ ¬CleanOften(jack, home)

True

If people don't often clean their homes, then they do not have tidy houses. If people don't prioritize cleaning, then they do not often clean their homes. If people hire a maid or cleaning service, then they have tidy houses. If people don't care about cleanliness, then they do not prioritize cleaning. Either Jack does hire a maid or cleaning service and does not often clean his home, or he does not hire a maid or cleaning service nor often clean his home.

∀x (¬CleanOften(x, home) → ¬Have(x, tidyHouse)) ∀x (¬Prioritize(x, cleaning) → ¬CleanOften(x, home)) ∀x (Hire(x, maid) ∨ Hire(x, cleaningService) → Have(x, tidyHouse)) ∀x (¬CareAbout(x, cleanliness) → ¬Prioritize(x, cleaning)) ¬(Hire(x, maid) ∨ Hire(x, cleaningService)) ⊕ ¬CleanOften(jack, home))

Jack is overburdened and lives in the suburbs.

¬Prioritize(jack, cleaning) ∨ ¬CareAbout(jack, cleanliness)

False

The bottle not falling is either standing upright or toppled over. The bottle not falling is not standing upright.

¬Falling(bottle) → (Upright(bottle) ⊕ ToppledOver(bottle)) ¬Falling(bottle) → ¬Upright(bottle)

The bottle not falling is toppled over.

¬Falling(bottle) → ToppleOver(bottle)

True

Everyone who chooses what they want to do with their time has flexible schedules. Everyone with a lot of free time chooses what they want to do with their time. People either have a lot of free time or they invest in a career in which they are willing to spend the rest of their lives. If people invest in a career in which they are willing to spend the rest of their lives, then they are hardworking individuals with high ambitions and goals for the future. If people are hardworking individuals with high ambitions and goals for the future, then they are not short sighted. John is not either a hardworking individual with high ambitions and goals for the future or has a flexible schedule.

∀x (ChooseWhatToDoWith(x, time) → FlexibleSchedule(x)) ∀x (Have(x, lotsOfFreetime) → ChooseWhatToDoWith(x, time)) ∀x (Have(x, lotsOfFreetime) ⊕ (∃y (InvestIn(x, y) ∧ Career(y) ∧ WillingToSpendIn(restOfLife, y)))) ∀x (∃y (InvestIn(x, y) ∧ Career(y) ∧ WillingToSpendIn(restOfLife, y)) → Hardworking(x)) ∀x (Hardworking(x) ∧ HaveFor(x, highAmbition, future) ∧ HaveFor(x, goal, future) → ¬ShortSighted(x)) ¬((Hardworking(john) ∧ HaveFor(john, highAmbition, future) ∧ HaveFor(john, goal, future)) ⊕ FlexibleSchedule(john))

John is short sighted.

Organized(john)

False

Everyone who chooses what they want to do with their time has flexible schedules. Everyone with a lot of free time chooses what they want to do with their time. People either have a lot of free time or they invest in a career in which they are willing to spend the rest of their lives. If people invest in a career in which they are willing to spend the rest of their lives, then they are hardworking individuals with high ambitions and goals for the future. If people are hardworking individuals with high ambitions and goals for the future, then they are not short sighted. John is not either a hardworking individual with high ambitions and goals for the future or has a flexible schedule.

∀x (ChooseWhatToDoWith(x, time) → FlexibleSchedule(x)) ∀x (Have(x, lotsOfFreetime) → ChooseWhatToDoWith(x, time)) ∀x (Have(x, lotsOfFreetime) ⊕ (∃y (InvestIn(x, y) ∧ Career(y) ∧ WillingToSpendIn(restOfLife, y)))) ∀x (∃y (InvestIn(x, y) ∧ Career(y) ∧ WillingToSpendIn(restOfLife, y)) → Hardworking(x)) ∀x (Hardworking(x) ∧ HaveFor(x, highAmbition, future) ∧ HaveFor(x, goal, future) → ¬ShortSighted(x)) ¬((Hardworking(john) ∧ HaveFor(john, highAmbition, future) ∧ HaveFor(john, goal, future)) ⊕ FlexibleSchedule(john))

John chooses what he want to do with his time.

ChooseWhatToDoWith(john, time)

Uncertain

Everyone who chooses what they want to do with their time has flexible schedules. Everyone with a lot of free time chooses what they want to do with their time. People either have a lot of free time or they invest in a career in which they are willing to spend the rest of their lives. If people invest in a career in which they are willing to spend the rest of their lives, then they are hardworking individuals with high ambitions and goals for the future. If people are hardworking individuals with high ambitions and goals for the future, then they are not short sighted. John is not either a hardworking individual with high ambitions and goals for the future or has a flexible schedule.

∀x (ChooseWhatToDoWith(x, time) → FlexibleSchedule(x)) ∀x (Have(x, lotsOfFreetime) → ChooseWhatToDoWith(x, time)) ∀x (Have(x, lotsOfFreetime) ⊕ (∃y (InvestIn(x, y) ∧ Career(y) ∧ WillingToSpendIn(restOfLife, y)))) ∀x (∃y (InvestIn(x, y) ∧ Career(y) ∧ WillingToSpendIn(restOfLife, y)) → Hardworking(x)) ∀x (Hardworking(x) ∧ HaveFor(x, highAmbition, future) ∧ HaveFor(x, goal, future) → ¬ShortSighted(x)) ¬((Hardworking(john) ∧ HaveFor(john, highAmbition, future) ∧ HaveFor(john, goal, future)) ⊕ FlexibleSchedule(john))

John is either a hardworking individual with high ambitions and goals for the future or is short sighted.

(Hardworking(john) ∧ HaveFor(john, highAmbition, future) ∧ HaveFor(john, goal, future)) ⊕ ShortSighted(john)

True

Ableton has an office in Germany. Ableton has an office in the USA. USA and Germany are different countries. Any company that has offices in different countries is a multinational company. Ableton makes music software.

OfficeIn(ableton, germany) OfficeIn(ableton, unitedStates) ¬SameCountry(germany, unitedStates) ∀x ∀y ∀z (OfficeIn(x, y) ∧ OfficeIn(x, z) ∧ (¬SameCountry(y, z)) → MultinationalCompany(x)) MakesMusicSoftware(ableton)

Ableton is a multinational company.

MultinationalCompany(ableton)

True

Ableton has an office in Germany. Ableton has an office in the USA. USA and Germany are different countries. Any company that has offices in different countries is a multinational company. Ableton makes music software.

OfficeIn(ableton, germany) OfficeIn(ableton, unitedStates) ¬SameCountry(germany, unitedStates) ∀x ∀y ∀z (OfficeIn(x, y) ∧ OfficeIn(x, z) ∧ (¬SameCountry(y, z)) → MultinationalCompany(x)) MakesMusicSoftware(ableton)

Ableton makes AI software.

MakesAISoftware(ableton)

Uncertain

Ableton has an office in Germany. Ableton has an office in the USA. USA and Germany are different countries. Any company that has offices in different countries is a multinational company. Ableton makes music software.

OfficeIn(ableton, germany) OfficeIn(ableton, unitedStates) ¬SameCountry(germany, unitedStates) ∀x ∀y ∀z (OfficeIn(x, y) ∧ OfficeIn(x, z) ∧ (¬SameCountry(y, z)) → MultinationalCompany(x)) MakesMusicSoftware(ableton)

Ableton does not have an office in Germany.

¬OfficeIn(ableton, germany)

False

Those who can fly over a vast distance glide in the air. Flightless birds cannot fly over a vast distance. Penguins are flightless birds. Nonflying birds in Antarctica are penguins. Fido is a penguin, or flies over a vast distance.

∀x (FlyOver(x, vastDistance) → GlideInAir(x)) ∀x (Flightless(x) ∧ Bird(x) → ¬FlyOver(x, vastDistance)) ∀x (Penguin(x) → Flightless(x) ∧ Bird(x)) ∀x (NonFlying(x) ∧ Bird(x) ∧ In(x, antarctica) → Penguin(x)) Penguin(fido) ∨ FlyOver(fido, vastDistance)

Fido is a flightless bird

Flightless(fido) ∧ Bird(fido)

Uncertain

Those who can fly over a vast distance glide in the air. Flightless birds cannot fly over a vast distance. Penguins are flightless birds. Nonflying birds in Antarctica are penguins. Fido is a penguin, or flies over a vast distance.

∀x (FlyOver(x, vastDistance) → GlideInAir(x)) ∀x (Flightless(x) ∧ Bird(x) → ¬FlyOver(x, vastDistance)) ∀x (Penguin(x) → Flightless(x) ∧ Bird(x)) ∀x (NonFlying(x) ∧ Bird(x) ∧ In(x, antarctica) → Penguin(x)) Penguin(fido) ∨ FlyOver(fido, vastDistance)

Fido is not a nonflying bird in Antarctica, and he cannot glid in the air.

¬(NonFlying(fido) ∧ Bird(fido) ∧ In(fido, antarctica)) ∧ ¬GlideInAir(fido)

False

Those who can fly over a vast distance glide in the air. Flightless birds cannot fly over a vast distance. Penguins are flightless birds. Nonflying birds in Antarctica are penguins. Fido is a penguin, or flies over a vast distance.

∀x (FlyOver(x, vastDistance) → GlideInAir(x)) ∀x (Flightless(x) ∧ Bird(x) → ¬FlyOver(x, vastDistance)) ∀x (Penguin(x) → Flightless(x) ∧ Bird(x)) ∀x (NonFlying(x) ∧ Bird(x) ∧ In(x, antarctica) → Penguin(x)) Penguin(fido) ∨ FlyOver(fido, vastDistance)

If Fido either can fly over a vast distance or cannot fly over a vast distance, then Fido is a nonflying bird in Antartica.

(FlyOver(fido, vastDistance) ⊕ ¬FlyOver(fido, vastDistance)) → (NonFlying(fido) ∧ Bird(fido) ∧ In(fido, antarctica))

True

All members of the university faculty are professors. All principal investigators are members of the university faculty. No professor is also an undergraduate student. Anyone pursuing a bachelor's degree is an undergraduate student. Leon is not pursuing a bachelor's degree, and he is not a principal investigator. If Leon is not pursuing a bachelor's degree, then he is a professor.

∀x (MemberOf(x, universityFaculty) → Professor(x)) ∀x (PrincipalInvestigator(x) → MemberOf(x, universityFaculty)) ∀x (Professor(x) → ¬UndergraduateStudent(x)) ∀x (Pursuing(x, bachelor) → UndergraduateStudent(x)) ¬(Pursuing(leon, bachelor) ⊕ PrincipalInvestigator(leon)) ¬Pursuing(leon, bachelor) → Professor(leon)

Leon is a member of university faculty.

MemberOf(leon, universityFaculty)

Uncertain

All members of the university faculty are professors. All principal investigators are members of the university faculty. No professor is also an undergraduate student. Anyone pursuing a bachelor's degree is an undergraduate student. Leon is not pursuing a bachelor's degree, and he is not a principal investigator. If Leon is not pursuing a bachelor's degree, then he is a professor.

∀x (MemberOf(x, universityFaculty) → Professor(x)) ∀x (PrincipalInvestigator(x) → MemberOf(x, universityFaculty)) ∀x (Professor(x) → ¬UndergraduateStudent(x)) ∀x (Pursuing(x, bachelor) → UndergraduateStudent(x)) ¬(Pursuing(leon, bachelor) ⊕ PrincipalInvestigator(leon)) ¬Pursuing(leon, bachelor) → Professor(leon)

Leon is neither an undergraduate student nor a principal investigator.

¬UndergraduateStudent(leon) ∧ ¬PrincipalInvestigator(leon)

True

All members of the university faculty are professors. All principal investigators are members of the university faculty. No professor is also an undergraduate student. Anyone pursuing a bachelor's degree is an undergraduate student. Leon is not pursuing a bachelor's degree, and he is not a principal investigator. If Leon is not pursuing a bachelor's degree, then he is a professor.

∀x (MemberOf(x, universityFaculty) → Professor(x)) ∀x (PrincipalInvestigator(x) → MemberOf(x, universityFaculty)) ∀x (Professor(x) → ¬UndergraduateStudent(x)) ∀x (Pursuing(x, bachelor) → UndergraduateStudent(x)) ¬(Pursuing(leon, bachelor) ⊕ PrincipalInvestigator(leon)) ¬Pursuing(leon, bachelor) → Professor(leon)

If leon is not a principal investigator, then Leon is an undergraduate student.

¬PrincipalInvestigator(leon) → UndergraduateStudent(leon)

False

A cutman is responsible for preventing and treating physical damage to a fighter. Cutmen appear in boxing matches, kickboxing matches, or mixed martial arts matches bout. Cutmen handle swelling, nosebleeds and lacerations. Jack is a cutman.

∀x (Cutman(x) → Prevent(x, physicalDamageToAFighter) ∧ Treat(x, physicalDamageToAFighter)) ∀x (Cutman(x) → AppearIn(x, boxingMatch) ∨ AppearIn(x, kickboxingMatch) ∨ AppearIn(x, mixedMartialArtsMatchBout)) ∀x (Cutman(x) → Handle(x, swelling) ∧ Handle(x, nosebleed) ∧ Handle(x, laceration)) Cutman(jack)

No cutmen appear in boxing matches.

¬(∃x (Cutman(x) ∧ AppearIn(x, boxingMatch)))

Uncertain

A cutman is responsible for preventing and treating physical damage to a fighter. Cutmen appear in boxing matches, kickboxing matches, or mixed martial arts matches bout. Cutmen handle swelling, nosebleeds and lacerations. Jack is a cutman.

∀x (Cutman(x) → Prevent(x, physicalDamageToAFighter) ∧ Treat(x, physicalDamageToAFighter)) ∀x (Cutman(x) → AppearIn(x, boxingMatch) ∨ AppearIn(x, kickboxingMatch) ∨ AppearIn(x, mixedMartialArtsMatchBout)) ∀x (Cutman(x) → Handle(x, swelling) ∧ Handle(x, nosebleed) ∧ Handle(x, laceration)) Cutman(jack)

If someone is not a cutman, then they cannot handle nosebleeds.

∀x (¬Cutman(x) → ¬Handle(x, nosebleed))

Uncertain

A cutman is responsible for preventing and treating physical damage to a fighter. Cutmen appear in boxing matches, kickboxing matches, or mixed martial arts matches bout. Cutmen handle swelling, nosebleeds and lacerations. Jack is a cutman.

∀x (Cutman(x) → Prevent(x, physicalDamageToAFighter) ∧ Treat(x, physicalDamageToAFighter)) ∀x (Cutman(x) → AppearIn(x, boxingMatch) ∨ AppearIn(x, kickboxingMatch) ∨ AppearIn(x, mixedMartialArtsMatchBout)) ∀x (Cutman(x) → Handle(x, swelling) ∧ Handle(x, nosebleed) ∧ Handle(x, laceration)) Cutman(jack)

Jack is responsible for treating physical damage to a fighter.

Treat(jack, physicalDamageToAFighter)

True

The Mona Lisa is a world's best-known painting. The Mona Lisa is a portrait painted by Leonardo da Vinci. Leonardo da Vinci was a scientist and painter. Painting genres can be history, portrait, animal, landscape, and still life.

Painting(monaLisa) ∧ TheWorldsBestKnown(monaLisa) PaintedBy(monaLisa, leonardodaVinci) ∧ Portrait(monaLisa) Scientist(leonardodaVinci) ∧ Painter(leonardodaVinci) ∀x (Painting(x) → (History(x) ∨ Portrait(x) ∨ Animal(x) ∨ Landscape(x) ∨ StillLife(x)))

A world's best-known artwork is painted by a scientist.

∃x ∃y (Painting(x) ∧ TheWorldsBestKnown(x) ∧ PaintedBy(x, y) ∧ Scientist(y))

True

The Mona Lisa is a world's best-known painting. The Mona Lisa is a portrait painted by Leonardo da Vinci. Leonardo da Vinci was a scientist and painter. Painting genres can be history, portrait, animal, landscape, and still life.

Painting(monaLisa) ∧ TheWorldsBestKnown(monaLisa) PaintedBy(monaLisa, leonardodaVinci) ∧ Portrait(monaLisa) Scientist(leonardodaVinci) ∧ Painter(leonardodaVinci) ∀x (Painting(x) → (History(x) ∨ Portrait(x) ∨ Animal(x) ∨ Landscape(x) ∨ StillLife(x)))

Leonardo da Vinci has artworks in the landscape genre.

∃x (PaintedBy(x, leonardodaVinci) ∧ Landscape(x))

Uncertain

The Mona Lisa is a world's best-known painting. The Mona Lisa is a portrait painted by Leonardo da Vinci. Leonardo da Vinci was a scientist and painter. Painting genres can be history, portrait, animal, landscape, and still life.

Painting(monaLisa) ∧ TheWorldsBestKnown(monaLisa) PaintedBy(monaLisa, leonardodaVinci) ∧ Portrait(monaLisa) Scientist(leonardodaVinci) ∧ Painter(leonardodaVinci) ∀x (Painting(x) → (History(x) ∨ Portrait(x) ∨ Animal(x) ∨ Landscape(x) ∨ StillLife(x)))

No world's best-known artworks are portraits.

∀x (WorldsBestKnown(x) → ¬Portrait(x))

False

No professional tennis umpires are professional tennis players. If you are a World Tour player, then you are a professional tennis player. All Grand Slam champions are World Tour players. All Grand Slam umpires are professional tennis umpires. Nadal is a World Tour player or a Grand Slam champion

∀x (ProfessionalTennisUmpire(x) → ¬ProfessionalTennisPlayer(x)) ∀x (WorldTourPlayer(x) → ProfessionalTennisPlayer(x)) ∀x (GrandSlamChampion(x) → WorldTourPlayer(x)) ∀x (GrandSlamUmpire(x) → ProfessionalTennisUmpire(x)) WorldTourPlayer(nadal) ∨ GrandSlamChampion(nadal)

Nadal is a Grand Slam umpire.

GrandSlamUmpire(nadal)

False

No professional tennis umpires are professional tennis players. If you are a World Tour player, then you are a professional tennis player. All Grand Slam champions are World Tour players. All Grand Slam umpires are professional tennis umpires. Nadal is a World Tour player or a Grand Slam champion

∀x (ProfessionalTennisUmpire(x) → ¬ProfessionalTennisPlayer(x)) ∀x (WorldTourPlayer(x) → ProfessionalTennisPlayer(x)) ∀x (GrandSlamChampion(x) → WorldTourPlayer(x)) ∀x (GrandSlamUmpire(x) → ProfessionalTennisUmpire(x)) WorldTourPlayer(nadal) ∨ GrandSlamChampion(nadal)

Nadal is not a Grand Slam umpire.

¬GrandSlamUmpire(nadal)

True

No professional tennis umpires are professional tennis players. If you are a World Tour player, then you are a professional tennis player. All Grand Slam champions are World Tour players. All Grand Slam umpires are professional tennis umpires. Nadal is a World Tour player or a Grand Slam champion

∀x (ProfessionalTennisUmpire(x) → ¬ProfessionalTennisPlayer(x)) ∀x (WorldTourPlayer(x) → ProfessionalTennisPlayer(x)) ∀x (GrandSlamChampion(x) → WorldTourPlayer(x)) ∀x (GrandSlamUmpire(x) → ProfessionalTennisUmpire(x)) WorldTourPlayer(nadal) ∨ GrandSlamChampion(nadal)

Nadal is a Grand Slam champion.

GrandSlamChampion(nadal)

Uncertain

No professional tennis umpires are professional tennis players. If you are a World Tour player, then you are a professional tennis player. All Grand Slam champions are World Tour players. All Grand Slam umpires are professional tennis umpires. Nadal is a World Tour player or a Grand Slam champion

∀x (ProfessionalTennisUmpire(x) → ¬ProfessionalTennisPlayer(x)) ∀x (WorldTourPlayer(x) → ProfessionalTennisPlayer(x)) ∀x (GrandSlamChampion(x) → WorldTourPlayer(x)) ∀x (GrandSlamUmpire(x) → ProfessionalTennisUmpire(x)) WorldTourPlayer(nadal) ∨ GrandSlamChampion(nadal)

Nadal is neither a Grand Slam umpire nor a professional tennis umpire.

¬(GrandSlamUmpire(nadal) ∨ ProfessionalTennisUmpire(nadal))

True

No professional tennis umpires are professional tennis players. If you are a World Tour player, then you are a professional tennis player. All Grand Slam champions are World Tour players. All Grand Slam umpires are professional tennis umpires. Nadal is a World Tour player or a Grand Slam champion

∀x (ProfessionalTennisUmpire(x) → ¬ProfessionalTennisPlayer(x)) ∀x (WorldTourPlayer(x) → ProfessionalTennisPlayer(x)) ∀x (GrandSlamChampion(x) → WorldTourPlayer(x)) ∀x (GrandSlamUmpire(x) → ProfessionalTennisUmpire(x)) WorldTourPlayer(nadal) ∨ GrandSlamChampion(nadal)

If Nadal is a professional tennis umpire, then Nadal is a Grand Slam Umpire.

ProfessionalTennisUmpire(nadal) → GrandSlamUmpire(nadal)

True

No professional tennis umpires are professional tennis players. If you are a World Tour player, then you are a professional tennis player. All Grand Slam champions are World Tour players. All Grand Slam umpires are professional tennis umpires. Nadal is a World Tour player or a Grand Slam champion

∀x (ProfessionalTennisUmpire(x) → ¬ProfessionalTennisPlayer(x)) ∀x (WorldTourPlayer(x) → ProfessionalTennisPlayer(x)) ∀x (GrandSlamChampion(x) → WorldTourPlayer(x)) ∀x (GrandSlamUmpire(x) → ProfessionalTennisUmpire(x)) WorldTourPlayer(nadal) ∨ GrandSlamChampion(nadal)

If Nadal is a Grand Slam umpire or a professional tennis player, then Nadal is a Grand Slam umpire.

GrandSlamUmpire(nadal) ∨ ProfessionalTennisPlayer(nadal) → GrandSlamUmpire(nadal)

False

Businesses are either sanctioned or unsanctioned. Sanctioned businesses are limited. Unsanctioned businesses are free. The Crude Oil Data Exchange is a business that isn't free.

∀x (Buisness(x) → Sanctioned(x) ⊕ ¬Sanctioned(x)) ∀x (Buisness(x) ∧ Sanctioned(x) → Limited(x)) ∀x (Buisness(x) ∧ ¬Sanctioned(x) → Free(x)) Buisness(crudeOilDataExchange) ∧ ¬Free(crudeOilDataExchange)

Crude Oil Data Exchange is sanctioned.

Sanctioned(crudeOilDataExchange)

True

Businesses are either sanctioned or unsanctioned. Sanctioned businesses are limited. Unsanctioned businesses are free. The Crude Oil Data Exchange is a business that isn't free.

∀x (Buisness(x) → Sanctioned(x) ⊕ ¬Sanctioned(x)) ∀x (Buisness(x) ∧ Sanctioned(x) → Limited(x)) ∀x (Buisness(x) ∧ ¬Sanctioned(x) → Free(x)) Buisness(crudeOilDataExchange) ∧ ¬Free(crudeOilDataExchange)

Crude Oil Data Exchange is unsanctioned.

¬Sanctioned(crudeOilDataExchange)

False

Businesses are either sanctioned or unsanctioned. Sanctioned businesses are limited. Unsanctioned businesses are free. The Crude Oil Data Exchange is a business that isn't free.

∀x (Buisness(x) → Sanctioned(x) ⊕ ¬Sanctioned(x)) ∀x (Buisness(x) ∧ Sanctioned(x) → Limited(x)) ∀x (Buisness(x) ∧ ¬Sanctioned(x) → Free(x)) Buisness(crudeOilDataExchange) ∧ ¬Free(crudeOilDataExchange)

Crude Oil Data Exchange is limited.

Limited(crudeOilDataExchange)

True

When something is depressing, it is sad. The end of a relationship is depressing.

∀x (Depressing(x) → Sad(x)) Depressing(v)

The end of a relationship is invigorating

Invigorating(v)

Uncertain

Palstaves are a type of early bronze axe. Palstaves are found in northern, western, and southwestern Europe and are cast in molds. John Evans is an archeologist who popularized the term "palstave." Paalstabs are not a type of axe but rather a digging shovel.

EarlyBronzeAge(palstave) ∧ Axe(palstave) FoundIn(palstave, northernEurope) ∨ FoundIn(palstave, westernEurope) ∨ FoundIn(palstave, southWesternEurope)) ∧ CastIn(palstave, molds) Archeologist(johnEvans) ∧ Popularize(johnEvans, termPalstave) ¬Axe(paalstab) ∧ DiggingShovel(paalstab)

John Evans Popularized the term paalstab.

Popularized(johnEvans, termPalstave)

Uncertain

Palstaves are a type of early bronze axe. Palstaves are found in northern, western, and southwestern Europe and are cast in molds. John Evans is an archeologist who popularized the term "palstave." Paalstabs are not a type of axe but rather a digging shovel.

EarlyBronzeAge(palstave) ∧ Axe(palstave) FoundIn(palstave, northernEurope) ∨ FoundIn(palstave, westernEurope) ∨ FoundIn(palstave, southWesternEurope)) ∧ CastIn(palstave, molds) Archeologist(johnEvans) ∧ Popularize(johnEvans, termPalstave) ¬Axe(paalstab) ∧ DiggingShovel(paalstab)

There is an axe that is found in Western Europe.

∃x (Axe(x) ∧ FoundIn(x, westernEurope))

Uncertain

Palstaves are a type of early bronze axe. Palstaves are found in northern, western, and southwestern Europe and are cast in molds. John Evans is an archeologist who popularized the term "palstave." Paalstabs are not a type of axe but rather a digging shovel.

EarlyBronzeAge(palstave) ∧ Axe(palstave) FoundIn(palstave, northernEurope) ∨ FoundIn(palstave, westernEurope) ∨ FoundIn(palstave, southWesternEurope)) ∧ CastIn(palstave, molds) Archeologist(johnEvans) ∧ Popularize(johnEvans, termPalstave) ¬Axe(paalstab) ∧ DiggingShovel(paalstab)

Archeologists haven't popularized anything.

∀x ∀y (Archeologist(x) → ¬Popularize(x, y))

False

Koei Tecmo is a Japanese video game and anime holding company. Holding companies hold several companies. Tecmo was disbanded in Japan, while Koei survived but was renamed. Video game holding companies are holding companies.

Japanese(koeitecmo) ∧ VideoGameHoldingCompany(koeitecmo) ∧ AnimeHoldingCompany(koeitecmo) ∧ HoldingCompany(x) ∀x (HoldingCompany(x) → ∃y(Company(y) ∧ Holds(x, y))) DisbandsIn(tecmo, japan) ∧ Survives(koei) ∧ Renames(koei) ∀x (VideoGameHoldingCompany(x) → HoldingCompany(x))

Koei Tecmo holds another company.

∃x (Company(x) ∧ Holds(koeitecmo, x))

True

Koei Tecmo is a Japanese video game and anime holding company. Holding companies hold several companies. Tecmo was disbanded in Japan, while Koei survived but was renamed. Video game holding companies are holding companies.

Japanese(koeitecmo) ∧ VideoGameHoldingCompany(koeitecmo) ∧ AnimeHoldingCompany(koeitecmo) ∧ HoldingCompany(x) ∀x (HoldingCompany(x) → ∃y(Company(y) ∧ Holds(x, y))) DisbandsIn(tecmo, japan) ∧ Survives(koei) ∧ Renames(koei) ∀x (VideoGameHoldingCompany(x) → HoldingCompany(x))

Tecmo holds another company.

∃x (Company(x) ∧ Holds(tecmo, x))

Uncertain

Koei Tecmo is a Japanese video game and anime holding company. Holding companies hold several companies. Tecmo was disbanded in Japan, while Koei survived but was renamed. Video game holding companies are holding companies.

Japanese(koeitecmo) ∧ VideoGameHoldingCompany(koeitecmo) ∧ AnimeHoldingCompany(koeitecmo) ∧ HoldingCompany(x) ∀x (HoldingCompany(x) → ∃y(Company(y) ∧ Holds(x, y))) DisbandsIn(tecmo, japan) ∧ Survives(koei) ∧ Renames(koei) ∀x (VideoGameHoldingCompany(x) → HoldingCompany(x))

Koei Tecmo holds anime.

AnimeHoldingCompany(koeitecmo)

Uncertain

The PlayStation EyeToy is a camera accessory for the PlayStation 2 system. The PlayStation Eye is a camera accessory for the PlayStation 3 system. The PlayStation Camera is a camera accessory for the PlayStation 4 and the PlayStation 5 systems. Camera accessories for a system are compatible with that system. Playstation 2, 3,4, and 5 are all different. Only the PlayStation Camera camera system is compatible with different systems.

System(playStation2) ∧ CameraAccessoryFor(playStationEyeToy, playStation2) System(playStation3) ∧ CameraAccessoryFor(playStationEye, playStation3) System(playStation4) ∧ System(playStation5) ∧ CameraAccessoryFor(playStationCamera, playStation4) ∧ CameraAccessoryFor(playStationCamera, playStation5) ∀x ∀y (CameraAccessoryFor(x, y) ∧ System(y) → CompatibleWith(x, y)) ¬(playStation2=playStation3) ∧ ¬(playStation2=playStation4) ∧ ¬(playStation2=playStation5) ∧ ¬(playStation3=playStation4) ∧ ¬(playStation3=playStation5) ∧ ¬(playStation4=playStation5) ∀x ∃y ∃z (System(y) ∧ System(z) ∧ ¬(y=z) ∧ CompatibleWith(x, y) ∧ CompatibleWith(x, z) → x=playstationCamera)

The Playstation Eye is compatible with the PlayStation 2 and the PlayStation 3.

Compatible(playStationEye, playStation2) ∧ Compatible(playStationEye, playStation3)

False

The PlayStation EyeToy is a camera accessory for the PlayStation 2 system. The PlayStation Eye is a camera accessory for the PlayStation 3 system. The PlayStation Camera is a camera accessory for the PlayStation 4 and the PlayStation 5 systems. Camera accessories for a system are compatible with that system. Playstation 2, 3,4, and 5 are all different. Only the PlayStation Camera camera system is compatible with different systems.

System(playStation2) ∧ CameraAccessoryFor(playStationEyeToy, playStation2) System(playStation3) ∧ CameraAccessoryFor(playStationEye, playStation3) System(playStation4) ∧ System(playStation5) ∧ CameraAccessoryFor(playStationCamera, playStation4) ∧ CameraAccessoryFor(playStationCamera, playStation5) ∀x ∀y (CameraAccessoryFor(x, y) ∧ System(y) → CompatibleWith(x, y)) ¬(playStation2=playStation3) ∧ ¬(playStation2=playStation4) ∧ ¬(playStation2=playStation5) ∧ ¬(playStation3=playStation4) ∧ ¬(playStation3=playStation5) ∧ ¬(playStation4=playStation5) ∀x ∃y ∃z (System(y) ∧ System(z) ∧ ¬(y=z) ∧ CompatibleWith(x, y) ∧ CompatibleWith(x, z) → x=playstationCamera)

The Playstation EyeToy is compatible with the PlayStation 2.

Compatible(playStationEyeToy, playStation2)

True

The PlayStation EyeToy is a camera accessory for the PlayStation 2 system. The PlayStation Eye is a camera accessory for the PlayStation 3 system. The PlayStation Camera is a camera accessory for the PlayStation 4 and the PlayStation 5 systems. Camera accessories for a system are compatible with that system. Playstation 2, 3,4, and 5 are all different. Only the PlayStation Camera camera system is compatible with different systems.

System(playStation2) ∧ CameraAccessoryFor(playStationEyeToy, playStation2) System(playStation3) ∧ CameraAccessoryFor(playStationEye, playStation3) System(playStation4) ∧ System(playStation5) ∧ CameraAccessoryFor(playStationCamera, playStation4) ∧ CameraAccessoryFor(playStationCamera, playStation5) ∀x ∀y (CameraAccessoryFor(x, y) ∧ System(y) → CompatibleWith(x, y)) ¬(playStation2=playStation3) ∧ ¬(playStation2=playStation4) ∧ ¬(playStation2=playStation5) ∧ ¬(playStation3=playStation4) ∧ ¬(playStation3=playStation5) ∧ ¬(playStation4=playStation5) ∀x ∃y ∃z (System(y) ∧ System(z) ∧ ¬(y=z) ∧ CompatibleWith(x, y) ∧ CompatibleWith(x, z) → x=playstationCamera)

The Playstation Camera can be used for all Playstation consoles.

Compatible(playStationCamera, playStation2) ∧ Compatible(playStationCamera, playStation3) ∧ Compatible(playStationCamera, playStation4) ∧ Compatible(playStationCamera, playStation5)

Uncertain

Adam Buska is a European football player. If a European plays football, they play what Americans call soccer.

FootballPlayer(adamBuska) ∧ European(adamBuska) ∀x (FootballPlayer(x) ∧ European(x) → ∃y (Call(american, y, soccer) ∧ Play(x, y)))

Adam Buska plays what Americans call soccer.

∃y (Call(american, y, soccer) ∧ Play(adamBuska, y))

True

If a game is one of the top-3 best selling video-games, then it is multiplatform. If a game has sold more than 100 million copies, then it is one of the top-3 best-selling video games. Some games that support Windows are developed by Nintendo. All multiplatform games can be played on a wide range of devices. Pokemon Diamond version is neither developed by Nintendo nor can be played on a wide range of devices.

∀x (ATop3BestSellingVideoGame(x) → Multiplatform(x)) ∀x (SoldMoreThan100MillionCopies(x) → ATop3BestSellingVideoGame(x)) ∃x ((SupportsWindows(x) ∧ AGameDevelopedByNintendo(x))) ∀x (Multiplatform(x) → CanBePlayedOnAWideRangeOfDevices(x)) ¬(DevelopedByNintendo(PokemonDiamond) ∨ CanBePlayedOnAWideRangeOfDevices(PokemonDiamond))

Pokemon Diamond version supports Windows.

Game(PokemonDiamond) ∧ SupportsWindows(PokemonDiamond)

Uncertain

If a game is one of the top-3 best selling video-games, then it is multiplatform. If a game has sold more than 100 million copies, then it is one of the top-3 best-selling video games. Some games that support Windows are developed by Nintendo. All multiplatform games can be played on a wide range of devices. Pokemon Diamond version is neither developed by Nintendo nor can be played on a wide range of devices.

∀x (ATop3BestSellingVideoGame(x) → Multiplatform(x)) ∀x (SoldMoreThan100MillionCopies(x) → ATop3BestSellingVideoGame(x)) ∃x ((SupportsWindows(x) ∧ AGameDevelopedByNintendo(x))) ∀x (Multiplatform(x) → CanBePlayedOnAWideRangeOfDevices(x)) ¬(DevelopedByNintendo(PokemonDiamond) ∨ CanBePlayedOnAWideRangeOfDevices(PokemonDiamond))

Pokemon Diamond version supports Windows and has sold more than 100 million copies.

(Game(PokemonDiamond) ∧ SupportsWindows(PokemonDiamond)) ∧ (Game(PokemonDiamond) ∧ SoldMoreThan100MillionCopies(PokemonDiamond))

False

If a game is one of the top-3 best selling video-games, then it is multiplatform. If a game has sold more than 100 million copies, then it is one of the top-3 best-selling video games. Some games that support Windows are developed by Nintendo. All multiplatform games can be played on a wide range of devices. Pokemon Diamond version is neither developed by Nintendo nor can be played on a wide range of devices.

∀x (ATop3BestSellingVideoGame(x) → Multiplatform(x)) ∀x (SoldMoreThan100MillionCopies(x) → ATop3BestSellingVideoGame(x)) ∃x ((SupportsWindows(x) ∧ AGameDevelopedByNintendo(x))) ∀x (Multiplatform(x) → CanBePlayedOnAWideRangeOfDevices(x)) ¬(DevelopedByNintendo(PokemonDiamond) ∨ CanBePlayedOnAWideRangeOfDevices(PokemonDiamond))

If Pokemon Diamond version either supports Windows or has sold more than 100 million copies, then Pokemon Diamond version either is both multiplatform and one of the top-3 best selling video games, or is neither multiplatform nor one of the top-3 best selling video games.

((Game(PokemonDiamond) ∧ SupportsWindows(PokemonDiamond)) ⊕ ((Game(PokemonDiamond) v (SoldMoreThan100MillionCopies(PokemonDiamond))) → (Multiplatform(PokemonDiamond) ∧ (Game(PokemonDiamond) ∧ ATop3BestSellingVideoGame(PokemonDiamond))) ⊕ (¬Multiplatform(PokemonDiamond) ∧ ¬(Game(PokemonDiamond) ∧ ATop3BestSellingVideoGame(PokemonDiamond)))

True

China is one of the BRICS, and its economy is emerging. If someone is from China, then they are from a country of BRICS. India is one of the BRICS, and its economy is emerging. If someone is from India, then they are in a country of BRICS. All people from China are Chinese people. All people from India are Indian people. There is a person from India.

∃x (BRIC(x) ∧ ¬(x=china) ∧ BRIC(china) ∧ Emerging(chinaEconomy)) ∀x (From(x, china) → From(x, bric)) BRIC(india) ∧ Emerging(indiaEconomy) ∀x (From(x, india) → From(x, bric)) ∀x (From(x, china) → Chinese(x)) ∀x (From(x, india) → Indian(x)) ∃x (From(x, india))

No people from BRICS are Indian people.

∀x (From(x, countryOfBRICS) → ¬IndianPeople(x))

False

China is one of the BRICS, and its economy is emerging. If someone is from China, then they are from a country of BRICS. India is one of the BRICS, and its economy is emerging. If someone is from India, then they are in a country of BRICS. All people from China are Chinese people. All people from India are Indian people. There is a person from India.

∃x (BRIC(x) ∧ ¬(x=china) ∧ BRIC(china) ∧ Emerging(chinaEconomy)) ∀x (From(x, china) → From(x, bric)) BRIC(india) ∧ Emerging(indiaEconomy) ∀x (From(x, india) → From(x, bric)) ∀x (From(x, china) → Chinese(x)) ∀x (From(x, india) → Indian(x)) ∃x (From(x, india))

India's economy is not emerging.

EmergingEconomy(india)

False

China is one of the BRICS, and its economy is emerging. If someone is from China, then they are from a country of BRICS. India is one of the BRICS, and its economy is emerging. If someone is from India, then they are in a country of BRICS. All people from China are Chinese people. All people from India are Indian people. There is a person from India.

∃x (BRIC(x) ∧ ¬(x=china) ∧ BRIC(china) ∧ Emerging(chinaEconomy)) ∀x (From(x, china) → From(x, bric)) BRIC(india) ∧ Emerging(indiaEconomy) ∀x (From(x, india) → From(x, bric)) ∀x (From(x, china) → Chinese(x)) ∀x (From(x, india) → Indian(x)) ∃x (From(x, india))

There is an Indian people from BRICS.

∃x (IndianPeople(x) ∧ From(x, countryOfBRICS))

True

Daveed Diggs is an actor and film producer. Daveed Diggs played two roles in the musical Hamilton on Broadway. One of the actors from Hamilton won the best actor award. The actor playing Thomas Jefferson won the best actor award. Daveed Diggs played Thomas Jefferson. Musicals on Broadway are not films.

Actor(daveedDiggs) ∧ FilmProducer(daveedDiggs) ∃x ∃y(PlaysIn(daveedDiggs, x, hamilton) ∧ (¬(x=y)) ∧ PlaysIn(daveedDiggs, y, hamilton)) ∧ OnBroadway(hamilton) ∧ Musical(hamilton) ∃x ∃y(Actor(x) ∧ PlaysIn(x, y, hamilton) ∧ Wins(x, bestActorAward)) ∃x (Actor(x) ∧ PlaysIn(x, thomasJefferson, hamilton) ∧ Wins(x, bestActorAward)) Plays(daveedDiggs, thomasJefferson) ∀x ((Musical(x) ∧ OnBroadway(x)) → ¬Film(x))

Hamilton is a film.

Film(hamilton)

False

Daveed Diggs is an actor and film producer. Daveed Diggs played two roles in the musical Hamilton on Broadway. One of the actors from Hamilton won the best actor award. The actor playing Thomas Jefferson won the best actor award. Daveed Diggs played Thomas Jefferson. Musicals on Broadway are not films.

Actor(daveedDiggs) ∧ FilmProducer(daveedDiggs) ∃x ∃y(PlaysIn(daveedDiggs, x, hamilton) ∧ (¬(x=y)) ∧ PlaysIn(daveedDiggs, y, hamilton)) ∧ OnBroadway(hamilton) ∧ Musical(hamilton) ∃x ∃y(Actor(x) ∧ PlaysIn(x, y, hamilton) ∧ Wins(x, bestActorAward)) ∃x (Actor(x) ∧ PlaysIn(x, thomasJefferson, hamilton) ∧ Wins(x, bestActorAward)) Plays(daveedDiggs, thomasJefferson) ∀x ((Musical(x) ∧ OnBroadway(x)) → ¬Film(x))

Daveed Diggs won the best actor award.

Wins(daveedDiggs, bestActorAward)

True

Daveed Diggs is an actor and film producer. Daveed Diggs played two roles in the musical Hamilton on Broadway. One of the actors from Hamilton won the best actor award. The actor playing Thomas Jefferson won the best actor award. Daveed Diggs played Thomas Jefferson. Musicals on Broadway are not films.

Actor(daveedDiggs) ∧ FilmProducer(daveedDiggs) ∃x ∃y(PlaysIn(daveedDiggs, x, hamilton) ∧ (¬(x=y)) ∧ PlaysIn(daveedDiggs, y, hamilton)) ∧ OnBroadway(hamilton) ∧ Musical(hamilton) ∃x ∃y(Actor(x) ∧ PlaysIn(x, y, hamilton) ∧ Wins(x, bestActorAward)) ∃x (Actor(x) ∧ PlaysIn(x, thomasJefferson, hamilton) ∧ Wins(x, bestActorAward)) Plays(daveedDiggs, thomasJefferson) ∀x ((Musical(x) ∧ OnBroadway(x)) → ¬Film(x))

Hamilton won two awards.

∃x ∃y(Wins(hamilton, x) ∧ (¬(x=y)) ∧ Wins(hamilton, y))

Uncertain

Ernest Pohl was a Polish football player. A football player in the Polish First Division has scored over 180 goals. Ernest Pohl scored more than 180 goals in the Polish First Division. Górnik Zabrze's stadium was named after a soccer player from Ruda Śląska. Ernest Pohl is from Ruda Śląska.

Polish(ernestPohl) ∧ FootballPlayer(ernestPohl) ∃x (FootballPlayer(x) ∧ In(x, polishFirstDivision) ∧ ScoredOver(x, 180Goals)) In(ernestPohl, polishFirstDivision) ∧ ScoredOver(ernestPohl, 180Goals) ∃x ∃y (GornikZabrzes(x) ∧ Stadium(x) ∧ NamedAfter(x, y) ∧ SoccerPlayer(y) ∧ From(y, rudaŚląska)) From(ernestPohl, rudaŚląska))

Ernest Pohl has not scored more than 180 goals.

¬ScoredOver(ernestPohl, 180Goals)

False

Ernest Pohl was a Polish football player. A football player in the Polish First Division has scored over 180 goals. Ernest Pohl scored more than 180 goals in the Polish First Division. Górnik Zabrze's stadium was named after a soccer player from Ruda Śląska. Ernest Pohl is from Ruda Śląska.

Polish(ernestPohl) ∧ FootballPlayer(ernestPohl) ∃x (FootballPlayer(x) ∧ In(x, polishFirstDivision) ∧ ScoredOver(x, 180Goals)) In(ernestPohl, polishFirstDivision) ∧ ScoredOver(ernestPohl, 180Goals) ∃x ∃y (GornikZabrzes(x) ∧ Stadium(x) ∧ NamedAfter(x, y) ∧ SoccerPlayer(y) ∧ From(y, rudaŚląska)) From(ernestPohl, rudaŚląska))

Górnik Zabrze's stadium was named after Ernest Pohl.

∀x (GornikZabrzes(x) ∧ Stadium(x) → NamedAfter(x, ernestPohl))

Uncertain

Ann J. Land was a member of the Philadelphia City Council and the Democratic Party. Ann J. Land ran unopposed for the Philadelphia City Council in 1980. People who run unopposed for the Philadelphia City Council are elected to the positions they run for in the same year. Michael Nutter was a political challenger. Ann J. Land defeated Michael Nutter and ran for the Philadelphia City Council in 1987.

MemberOf(annJLand, philadelphiaCityCouncil) ∧ MemberOf(annJLand, democraticParty) RunUnopposedFor(ann, philadelphiaCityCouncil, year1980) ∀x ∀y (RunUnopposedFor(x, philadelphiaCityCouncil, y) → ElectedTo(x, philadelphiaCityCouncil, y)) PoliticalChallenger(michaelNutter) Defeat(annJLand, michaelNutter) ∧ RunFor(annJLand, philadelphiaCityCouncil, year1987)

Ann J. Land was elected to the Philadelphia City Council in 1980.

ElectedTo(ann, philadelphiaCityCouncil, year1980)

True

Ann J. Land was a member of the Philadelphia City Council and the Democratic Party. Ann J. Land ran unopposed for the Philadelphia City Council in 1980. People who run unopposed for the Philadelphia City Council are elected to the positions they run for in the same year. Michael Nutter was a political challenger. Ann J. Land defeated Michael Nutter and ran for the Philadelphia City Council in 1987.

MemberOf(annJLand, philadelphiaCityCouncil) ∧ MemberOf(annJLand, democraticParty) RunUnopposedFor(ann, philadelphiaCityCouncil, year1980) ∀x ∀y (RunUnopposedFor(x, philadelphiaCityCouncil, y) → ElectedTo(x, philadelphiaCityCouncil, y)) PoliticalChallenger(michaelNutter) Defeat(annJLand, michaelNutter) ∧ RunFor(annJLand, philadelphiaCityCouncil, year1987)

Ann J. Land was elected to the Philadelphia City Council in 1987.

ElectedTo(ann, philadelphiaCityCouncil, year1987)

Uncertain

Ann J. Land was a member of the Philadelphia City Council and the Democratic Party. Ann J. Land ran unopposed for the Philadelphia City Council in 1980. People who run unopposed for the Philadelphia City Council are elected to the positions they run for in the same year. Michael Nutter was a political challenger. Ann J. Land defeated Michael Nutter and ran for the Philadelphia City Council in 1987.

MemberOf(annJLand, philadelphiaCityCouncil) ∧ MemberOf(annJLand, democraticParty) RunUnopposedFor(ann, philadelphiaCityCouncil, year1980) ∀x ∀y (RunUnopposedFor(x, philadelphiaCityCouncil, y) → ElectedTo(x, philadelphiaCityCouncil, y)) PoliticalChallenger(michaelNutter) Defeat(annJLand, michaelNutter) ∧ RunFor(annJLand, philadelphiaCityCouncil, year1987)

There was some member of the Democratic Party elected to the Philadelphia City Council in 1980.

∃x (MemberOf(x, democraticParty) ∧ ElectedTo(x, philadelphiaCouncil, year1980))

True

Aberdeen won the cup in the 2013 final. Rangers won the cup in the 2014 final. Aberdeen and Rangers are different teams. Different teams cannot win the cup in the same year's final.

WonCup(aberdeen, year2013Final) WonCup(rangers, year2014Final) ¬(aberdeen=rangers) ∀x ∀y ∀z ∀w (¬(x=y) ∧ WonCup(x, z) ∧ WonCup(y, w) → ¬(z=w))

Rangers won the cup in 2015.

WonCup(rangers, year2015Final)

Uncertain

Aberdeen won the cup in the 2013 final. Rangers won the cup in the 2014 final. Aberdeen and Rangers are different teams. Different teams cannot win the cup in the same year's final.

WonCup(aberdeen, year2013Final) WonCup(rangers, year2014Final) ¬(aberdeen=rangers) ∀x ∀y ∀z ∀w (¬(x=y) ∧ WonCup(x, z) ∧ WonCup(y, w) → ¬(z=w))

Rangers won the cup in 2013.

WonCup(rangers, year2013Final)

False

Aberdeen won the cup in the 2013 final. Rangers won the cup in the 2014 final. Aberdeen and Rangers are different teams. Different teams cannot win the cup in the same year's final.

WonCup(aberdeen, year2013Final) WonCup(rangers, year2014Final) ¬(aberdeen=rangers) ∀x ∀y ∀z ∀w (¬(x=y) ∧ WonCup(x, z) ∧ WonCup(y, w) → ¬(z=w))

Aberdeen has once won a cup.

∃x (WonCup(aberdeen, x))

True

All young working professionals who have regular 9-5 jobs have stable jobs. Some people living in Manhattan are young professionals with regular 9-5 jobs. All people who have stable jobs are people who work regularly. People who work regularly do not frequently disobey their bosses. Mary either frequently disobeys her bosses and works regularly, or that she neither frequently disobeys her bosses nor works regularly.

∀x (YoungWorkingProfessional(x) ∧ Have(x, regular9To5Job) → Have(x, stableJob)) ∃x (LiveIn(x, manhattan) ∧ YoungWorkingProfessional(x) ∧ Have(x, regular9To5Job)) ∀x (Have(x, stableJob) → WorkRegularly(x)) ∀x (WorkRegularly(x) → ¬DisobeyFrequently(x, boss)) ¬(DisobeyFrequently(mary, boss) ⊕ WorkRegularly(mary))

Mary lives in Manhattan.

LiveIn(mary, manhattan)

Uncertain

All young working professionals who have regular 9-5 jobs have stable jobs. Some people living in Manhattan are young professionals with regular 9-5 jobs. All people who have stable jobs are people who work regularly. People who work regularly do not frequently disobey their bosses. Mary either frequently disobeys her bosses and works regularly, or that she neither frequently disobeys her bosses nor works regularly.

∀x (YoungWorkingProfessional(x) ∧ Have(x, regular9To5Job) → Have(x, stableJob)) ∃x (LiveIn(x, manhattan) ∧ YoungWorkingProfessional(x) ∧ Have(x, regular9To5Job)) ∀x (Have(x, stableJob) → WorkRegularly(x)) ∀x (WorkRegularly(x) → ¬DisobeyFrequently(x, boss)) ¬(DisobeyFrequently(mary, boss) ⊕ WorkRegularly(mary))

Mary lives in Manhattan and is a young working professional who has a regular 9-5 job.

LiveIn(mary, manhattan) ∧ YoungWorkingProfessional(mary) ∧ Have(mary, regular9-5Job)

False

All young working professionals who have regular 9-5 jobs have stable jobs. Some people living in Manhattan are young professionals with regular 9-5 jobs. All people who have stable jobs are people who work regularly. People who work regularly do not frequently disobey their bosses. Mary either frequently disobeys her bosses and works regularly, or that she neither frequently disobeys her bosses nor works regularly.

∀x (YoungWorkingProfessional(x) ∧ Have(x, regular9To5Job) → Have(x, stableJob)) ∃x (LiveIn(x, manhattan) ∧ YoungWorkingProfessional(x) ∧ Have(x, regular9To5Job)) ∀x (Have(x, stableJob) → WorkRegularly(x)) ∀x (WorkRegularly(x) → ¬DisobeyFrequently(x, boss)) ¬(DisobeyFrequently(mary, boss) ⊕ WorkRegularly(mary))

If Mary is a young working professional who has a regular 9-5 job, then Mary does not live in Manhattan.

YoungWorkingProfessional(mary) ∧ Have(mary, regular9-5Job) → ¬LiveIn(mary, manhattan)

True

All brain study designs are either block designs or event-related designs. All event-related brain study designs are brain image acquisition. All brain image acquisition in brain study designs is preceded by data processing. Nothing in brain study designs preceded by data processing analyzes data. Picture memory is a type of brain study design that is not either event-related or analyzing data.

∀x (BrainStudy(x) → (BlockDesign(x) ⊕ Event-relatedDesign(x))) ∀x ((BrainStudy(x) ∧ EventRelatedDesign(x)) → BrainImageAcquisition(x)) ∀x ((BrainStudy(x) ∧ BrainImageAcquisition(x)) → PrecededBy(x, dataProcessing)) ∀x ((BrainStudy(x) ∧ PrecededBy(x, dataProcessing)) → ¬Analyze(x, data)) BrainStudy(pictureMemory) ∧ (¬(EventRelatedDesign(pictureMemory) ⊕ AnalyzingData(pictureMemory)))

Picture memory is preceded by data processing.

PrecededBy(pictureMemory, dataProcessing)

Uncertain

All brain study designs are either block designs or event-related designs. All event-related brain study designs are brain image acquisition. All brain image acquisition in brain study designs is preceded by data processing. Nothing in brain study designs preceded by data processing analyzes data. Picture memory is a type of brain study design that is not either event-related or analyzing data.

∀x (BrainStudy(x) → (BlockDesign(x) ⊕ Event-relatedDesign(x))) ∀x ((BrainStudy(x) ∧ EventRelatedDesign(x)) → BrainImageAcquisition(x)) ∀x ((BrainStudy(x) ∧ BrainImageAcquisition(x)) → PrecededBy(x, dataProcessing)) ∀x ((BrainStudy(x) ∧ PrecededBy(x, dataProcessing)) → ¬Analyze(x, data)) BrainStudy(pictureMemory) ∧ (¬(EventRelatedDesign(pictureMemory) ⊕ AnalyzingData(pictureMemory)))

Picture memory is a block design.

BlockDesign(pictureMemory)

True

All brain study designs are either block designs or event-related designs. All event-related brain study designs are brain image acquisition. All brain image acquisition in brain study designs is preceded by data processing. Nothing in brain study designs preceded by data processing analyzes data. Picture memory is a type of brain study design that is not either event-related or analyzing data.

∀x (BrainStudy(x) → (BlockDesign(x) ⊕ Event-relatedDesign(x))) ∀x ((BrainStudy(x) ∧ EventRelatedDesign(x)) → BrainImageAcquisition(x)) ∀x ((BrainStudy(x) ∧ BrainImageAcquisition(x)) → PrecededBy(x, dataProcessing)) ∀x ((BrainStudy(x) ∧ PrecededBy(x, dataProcessing)) → ¬Analyze(x, data)) BrainStudy(pictureMemory) ∧ (¬(EventRelatedDesign(pictureMemory) ⊕ AnalyzingData(pictureMemory)))

Picture memory is either a block design or analyzing data.

BlockDesign(pictureMemory) ⊕ Analyze(pictureMemory, data)

True

All brain study designs are either block designs or event-related designs. All event-related brain study designs are brain image acquisition. All brain image acquisition in brain study designs is preceded by data processing. Nothing in brain study designs preceded by data processing analyzes data. Picture memory is a type of brain study design that is not either event-related or analyzing data.

∀x (BrainStudy(x) → (BlockDesign(x) ⊕ Event-relatedDesign(x))) ∀x ((BrainStudy(x) ∧ EventRelatedDesign(x)) → BrainImageAcquisition(x)) ∀x ((BrainStudy(x) ∧ BrainImageAcquisition(x)) → PrecededBy(x, dataProcessing)) ∀x ((BrainStudy(x) ∧ PrecededBy(x, dataProcessing)) → ¬Analyze(x, data)) BrainStudy(pictureMemory) ∧ (¬(EventRelatedDesign(pictureMemory) ⊕ AnalyzingData(pictureMemory)))

If picture memory is not analyzing data, then picture memory is a block design and analyzing data.

¬Analyze(pictureMemory, data) → (BlockDesign(pictureMemory) ∧ Analyze(pictureMemory, data))

False