People in this club who perform in school talent shows often attend and are very engaged with school events. People in this club either perform in school talent shows often or are inactive and disinterested community members. People in this club who chaperone high school dances are not students who attend the school. All people in this club who are inactive and disinterested members of their community chaperone high school dances. All young children and teenagers in this club who wish to further their academic careers and educational opportunities are students who attend the school. Bonnie is in this club and she either both attends and is very engaged with school events and is a student who attends the school or is not someone who both attends and is very engaged with school events and is not a student who attends the school.

∀x (InThisClub(x) ∧ PerformOftenIn(x, schoolTalentShow) → Attend(x, schoolEvent) ∧ VeryEngagedWith(x, schoolEvent)) ∧ ∀x (InThisClub(x) → PerformOftenIn(x, schoolTalentShow) ⊕ (InActive(x) ∧ Disinterested(x) ∧ MemberOf(x, community))) ∧ ∀x (InThisClub(x) ∧ Chaperone(x, highSchoolDance) → ¬(Student(x) ∧ AttendSchool(x))) ∧ ∀x (InThisClub(x) ∧ (InActive(x) ∧ Disinterested(x) ∧ MemberOf(x, community)) → Chaperone(x, highSchoolDance)) ∧ ∀x (InThisClub(x) ∧ (YoungChild(x) ∨ Teenager(x)) ∧ WishToFurther(x, academicCareer) ∧ WishToFurther(x, educationalOpportunity) → (Student(x) ∧ AttendSchool(x))) ∧ InThisClub(bonnie) ∧ ((Attend(bonnie, schoolEvent) ∧ VeryEngagedWith(bonnie, schoolEvent) ∧ Student(bonnie) ∧ AttendSchool(bonnie)) ⊕ (¬(Attend(bonnie, schoolEvent) ∧ VeryEngagedWith(bonnie, schoolEvent)) ∧ ¬(Student(bonnie) ∧ AttendSchool(bonnie)))

1. Spelling: Corrected 'Studen' to 'Student' and 'bonne' to 'bonnie'. 2. Linkage: 'highSchoolDances' changed to 'highSchoolDance' in Premise 4. 3. Missing Parenthesis & Operators: Fixed the missing opening parenthesis for the implication in Premise 5. Changed 'YoungChildren' to 'YoungChild' and XOR '⊕' to inclusive OR '∨' in Premise 5 to match 'young children and teenagers'. Added missing 'educationalOpportunity' to WishToFurther to fully capture the NL. 4. Premise 6: Removed hallucinated outer parentheses added by the previous reviewer and properly bounded the variables to 'bonnie'.

∀x (InThisClub(x) ∧ PerformOftenIn(x, schoolTalentShow) → Attend(x, schoolEvent) ∧ VeryEngagedWith(x, schoolEvent))
∀x (InThisClub(x) → PerformOftenIn(x, schoolTalentShow) ⊕ (InActive(x) ∧ Disinterested(x) ∧ MemberOf(x, community)))
∀x (InThisClub(x) ∧ Chaperone(x, highSchoolDance) → ¬(Studen(x) ∧ AttendSchool(x)))
∀x (InThisClub(x) ∧ (InActive(x) ∧ Disinterested(x) ∧ MemberOf(x, community)) → Chaperone(x, highSchoolDances))
∀x (InThisClub(x) ∧ (YoungChildren(x) ⊕ Teenager(x)) ∧ WishToFurther(x, academicCareer)) → Studen(x) ∧ AttendSchool(x))
InThisClub(bonnie) ∧ ¬((Attend(x, schoolEvent) ∧ VeryEngagedWith(bonnie, schoolEvent)) ⊕ (Studen(bonne) ∧ AttendSchool(bonnie)))


Bonnie performs in school talent shows often.

1. Predicate Match: Corrected 'Perform' to 'PerformOftenIn' to match the premises. 2. Removed Extraneous Conjunction: The original included 'InThisClub(bonnie)' which is not stated in the conclusion NL. The conclusion only asserts the performance.

InThisClub(bonnie) ∧ (Perform(bonnie, schoolTalentShow))


If Bonnie is either both a young child or teenager who wishes to further her academic career and educational opportunities and chaperones high school dances or neither is a young child nor teenager who wishes to further her academic career and educational opportunities, then Bonnie is either a student who attends the school or is an inactive and disinterested member of the community.

(((YoungChild(bonnie) ∨ Teenager(bonnie)) ∧ WishToFurther(bonnie, academicCareer) ∧ WishToFurther(bonnie, educationalOpportunity) ∧ Chaperone(bonnie, highSchoolDance)) ⊕ (¬(YoungChild(bonnie) ∨ Teenager(bonnie)) ∧ ¬WishToFurther(bonnie, academicCareer) ∧ ¬WishToFurther(bonnie, educationalOpportunity))) → ((Student(bonnie) ∧ AttendSchool(bonnie)) ⊕ (InActive(bonnie) ∧ Disinterested(bonnie) ∧ MemberOf(bonnie, community)))

1. Syntax & Unbalanced Parentheses: Completely rebuilt the formula to fix severe syntax errors and unmatched parentheses. 2. Variable Scope: Replaced the unbound 'x' with 'bonnie' in 'Student'. 3. Predicate Consistency: Corrected 'Studen' to 'Student' and 'YoungChildren' to 'YoungChild'. 4. Consequent Match: Rewrote the consequent to accurately reflect 'either a student... or an inactive... member' instead of the hallucinated repetition of the antecedent found in the old FOL. Added 'educationalOpportunity' to fully map the NL.

¬((YoungChildren(bonnie) ⊕ Teenager(bonnie)) ∧ WishToFurther(bonnie, academicCareer)) ⊕ Chaperone(bonnie, highSchoolDance)) → (Studen(x) ∧ AttendSchool(x)) ⊕ (YoungChildren(bonnie) ⊕ Teenager(bonnie)) ∧ WishToFurther(bonnie, academicCareer))


If Bonnie either chaperones high school dances or, if she does not, she performs in school talent shows often, then Bonnie is both a young child or teenager who wishes to further her academic career and educational opportunities and an inactive and disinterested member of the community.

(Chaperone(bonnie, highSchoolDance) ⊕ PerformOftenIn(bonnie, schoolTalentShow)) → ((YoungChild(bonnie) ∨ Teenager(bonnie)) ∧ WishToFurther(bonnie, academicCareer) ∧ WishToFurther(bonnie, educationalOpportunity) ∧ InActive(bonnie) ∧ Disinterested(bonnie) ∧ MemberOf(bonnie, community))

Predicate Match: Corrected 'Perform' to 'PerformOftenIn' to properly link with the premises. Missing Entities: Added 'WishToFurther(bonnie, educationalOpportunity)' to strictly align with the NL.

(Chaperone(bonnie, highSchoolDance) ⊕ Perform(bonnie, schoolTalentShow)) → (YoungChildren(bonnie) ⊕ Teenager(bonnie)) ∧ WishToFurther(bonnie, academicCareer)) ∧ (InActive(bonnie) ∧ Disinterested(bonnie) ∧ MemberOf(bonnie, community))


All employees who schedule a meeting with their customers will go to the company building today. Everyone who has lunch in the company building schedules meetings with their customers. Employees have lunch either in the company building or at home. If an employee has lunch at home, they are working remotely from home. All employees who are in other countries work remotely from home. No managers work remotely from home. James will appear in the company today if and only if he is a manager.

∀x ((Employee(x) ∧ Schedule(x, meeting, customers)) → AppearIn(x, company)) ∧ ∀x (HasLunch(x, company) → Schedule(x, meeting, customers)) ∧ ∀x (Employee(x) → (HasLunch(x, company) ⊕ HasLunch(x, home))) ∧ ∀x ((Employee(x) ∧ HasLunch(x, home)) → Work(x, home)) ∧ ∀x ((Employee(x) ∧ (¬In(x, homeCountry))) → Work(x, home)) ∧ ∀x (Manager(x) → ¬Work(x, home)) ∧ ¬(Manager(james) ⊕ AppearIn(james, company))

1. 'Everyone who has lunch in the company' is relativized to 'everyone that is an employee', but this restriction is not present in NL

∀x ((Employee(x) ∧ Schedule(x, meeting, customers)) → AppearIn(x, company))
∀x ((Employee(x) ∧ HasLunch(x, company)) → Schedule(x, meeting, customers))
∀x (Employee(x) → (HasLunch(x, company) ⊕ HasLunch(x, home)))
∀x ((Employee(x) ∧ HasLunch(x, home)) → Work(x, home))
∀x ((Employee(x) ∧ (¬In(x, homeCountry))) → Work(x, home))
∀x (Manager(x) → ¬Work(x, home))
¬(Manager(james) ⊕ AppearIn(james, company))


James does not have lunch in the company.


If James is either a manager or in other countries, then James does not either have lunch at home or work remotely from home. If James either has lunch at home or works remotely from home, then he is neither a manager nor does he work in other countries.

((Manager(james) ∨ ¬In(james, homeCountry)) → ¬(HasLunch(james, home) ∨ Work(james, home))) ∧ ((HasLunch(james, home) ∨ Work(james, home)) → ¬(Manager(james) ∨ ¬In(james, homeCountry)))

Translation Failure: The original FOL 'Manager(james) → ¬Work(james, home)' completely ignored the vast majority of the NL sentence. The NL contains two complex conditionals that act as contrapositives involving 'in other countries' and 'lunch at home'. The FOL has been rewritten to exhaustively translate the English sentence exactly as written. Because the premises do not establish whether James is an employee or his current location, we cannot logically evaluate this complex implication. The label is therefore corrected from True to Uncertain. (2) There is a problem also with other countries that is not really the opposite of homeCountry.


When the Monkeypox virus occurs in a being, it may get Monkeypox. Monkeypox virus can occur in certain animals. Humans are mammals. Mammals are animals. Symptoms of Monkeypox include fever, headache, muscle pains, and tiredness. People feel tired when they get the flu.

∃x (OccurIn(monkeypoxVirus, x) ∧ Get(x, monkeypoxVirus)) ∧ ∃x (Animal(x) ∧ OccurIn(monkeypoxVirus, x)) ∧ ∀x (Human(x) → Mammal(x)) ∧ ∀x (Mammal(x) → Animal(x)) ∧ ∀x ((Fever(x) ∨ Headache(x) ∨ MusclePain(x) ∨ Tiredness(x)) → SymptomOf(x, monkeypoxVirus)) ∧ ∀x (Human(x) ∧ Get(x, flu) → Feel(x, tired))

1. Translation Failure (Premise 5): The original FOL weakly asserted via an existential quantifier ('∃x') that there is at least one symptom that is one of those conditions. The NL 'Symptoms ... include ...' requires a universal implication to denote that all instances of these conditions are symptoms. Corrected to '∀x ((Fever(x) ∨ Headache(x) ∨ MusclePain(x) ∨ Tired(x)) → SymptomOf(x, monkeypoxVirus))'. 2. Typo (Premise 5): Corrected 'SymptonOf' to 'SymptomOf'.

∃x (OccurIn(monkeypoxVirus, x) ∧ Get(x, monkeypoxVirus))
∃x (Animal(x) ∧ OccurIn(monkeypoxVirus, x))
∀x (Human(x) → Mammal(x))
∀x (Mammal(x) → Animal(x))
∃x (SymptonOf(x, monkeypoxVirus) ∧ (Fever(x) ∨ Headache(x) ∨ MusclePain(x) ∨ Tired(x)))
∀x (Human(x) ∧ Get(x, flu) → Feel(x, tired))


There is no referement to humans (in the story it is talked about animals also), but usually 'no one' refer to human beings, so the original formalization can be considered acceptable


∃x (SymptomOf(x, monkeypoxVirus) ∧ Coughing(x))

1. Typo: Corrected 'SymptonOf' to 'SymptomOf'.

∃x (SymptonOf(x, monkeypoxVirus) ∧ Coughing(x))


There are six types of wild turkeys: Eastern wild turkey, Osceola wild turkey, Gould's wild turkey, Merriam's wild turkey, Rio Grande wild turkey, and Ocellated wild turkey. Tom is not an Eastern wild turkey. Tom is not an Osceola wild turkey. Tom is not a Gould's wild turkey. Tom is neither a Merriam's wild turkey nor a Rio Grande wild turkey. Tom is a wild turkey.

∀x (WildTurkey(x) ↔ (EasternWildTurkey(x) ∨ OsceolaWildTurkey(x) ∨ GouldsWildTurkey(x) ∨ MerriamsWildTurkey(x) ∨ RiograndeWildTurkey(x) ∨ OcellatedWildTurkey(x))) ∧ ¬(EasternWildTurkey(tom)) ∧ ¬(OsceolaWildTurkey(tom)) ∧ ¬(GouldsWildTurkey(tom)) ∧ ¬(MerriamsWildTurkey(tom) ∨ RiograndeWildTurkey(tom)) ∧ WildTurkey(tom)

Now inserted a bi-implication instead of a simple implication, since if x is an eastern wild turkey, then it is also a wild turkey.

∀x (WildTurkey(x) → (EasternWildTurkey(x) ∨ OsceolaWildTurkey(x) ∨ GouldsWildTurkey(x) ∨ MerriamsWildTurkey(x) ∨ RiograndeWildTurkey(x) ∨ OcellatedWildTurkey(x)))
¬(EasternWildTurkey(tom))
¬(OsceolaWildTurkey(tom))
¬(GouldsWildTurkey(tom))
¬(MerriamsWildTurkey(tom) ∨ RiograndeWildTurkey(tom))
WildTurkey(tom)


A Japanese game company created the game the Legend of Zelda. All games on the Top 10 list are made by Japanese game companies. If a game sells more than one million copies, then it will be included in the Top 10 list. The Legend of Zelda sold more than one million copies.

Game(theLegendofZelda) ∧ ∃x (Japanese(x) ∧ VideoGameCompany(x) ∧ Created(x, theLegendofZelda)) ∧ ∀x ∀y ((Game(x) ∧ InTop10(x) ∧ Created(y,x)) → (Japanese(y) ∧ VideoGameCompany(y))) ∧ ∀x ((Game(x) ∧ ∃y(GreaterThan(y, oneMillion) ∧ CopiesSold(x, y))) → InTop10(x)) ∧ ∃y(GreaterThan(y, oneMillion) ∧ CopiesSold(theLegendofZelda,y))

1. Translation Failure (Premise 2): The NL says 'Japanese game companies', but the old FOL only asserted 'Japanese(y)'. Corrected the consequent to '(Japanese(y) ∧ VideoGameCompany(y))'. 2. Syntax Error (Premise 3): Removed an unmatched closing parenthesis. 3. Broken Predicate Link (Premise 2 & 3): Standardized 'Top10' to 'InTop10' to resolve the logical chain. 4. Silent Edits Reverted: Removed hallucinated outer parentheses around Premise 1 and restored original spacing in Premises 2, 3, and 4.

Game(theLegendofZelda) ∧ ∃x (Japanese(x) ∧ VideoGameCompany(x) ∧ Created(x, theLegendofZelda))
∀x ∀y ((Game(x) ∧ InTop10(x) ∧ Created(y,x)) → Japanese(y))
∀x ((Game(x) ∧ ∃y(GreaterThan(y, oneMillion) ∧ CopiesSold(x, y))) → Top10(x)))
∃y(GreaterThan(y, oneMillion) ∧ CopiesSold(theLegendofZelda,y))


The Legend of Zelda is on the Top 10 list.

Standardized the predicate 'Top10' to 'InTop10' to correctly link with the adjusted premises.


FIFA 22 is made by a Japanese video game company.

∃x(Created(x, fifa22) ∧ Japanese(x) ∧ VideoGameCompany(x))


The Legend of Zelda is not on the Top 10 list.


All squares are four-sided. All four-sided things are shapes.

∀x (Square(x) → FourSided(x)) ∧ ∀x (FourSided(x) → Shape(x))

∀x (Square(x) → FourSided(x))
∀x (FourSided(x) → Shape(x))


All rabbits that can be spotted near the campus are cute. Some turtles can be spotted near the campus. The only animals that can be spotted near the campus are rabbits and squirrels. If something is skittish, then it is not calm. All the squirrels that can be spotted near the campus are skittish. Rockie can be spotted near the campus, and it is calm.

∀x (Rabbit(x) ∧ CanBeSpottedNear(x, campus) → Cute(x)) ∧ ∃x (Turtle(x) ∧ CanBeSpottedNear(x, campus)) ∧ ∀x (CanBeSpottedNear(x, campus) → (Rabbit(x) ∨ Squirrel(x))) ∧ ∀x (Skittish(x) → ¬Calm(x)) ∧ ∀x (Squirrel(x) ∧ CanBeSpottedNear(x, campus) → Skittish(x)) ∧ CanBeSpottedNear(rockie, campus) ∧ Calm(rockie)

1. Translation Failure (Premise 3): Corrected the exclusive OR '⊕' to an inclusive OR '∨'. The NL states 'are rabbits and squirrels', which translates logically to the union of those two sets, not strictly an XOR relationship. 2. Silent Edits Reverted (Premises 1, 5, 6): Removed hallucinated inner and outer parentheses that were added by the previous reviewer without documentation.

∀x (Rabbit(x) ∧ CanBeSpottedNear(x, campus) → Cute(x))
∃x (Turtle(x) ∧ CanBeSpottedNear(x, campus))
∀x (CanBeSpottedNear(x, campus) → (Rabbit(x) ⊕ Squirrel(x)))
∀x (Skittish(x) → ¬Calm(x))
∀x (Squirrel(x) ∧ CanBeSpottedNear(x, campus) → Skittish(x))
CanBeSpottedNear(rockie, campus) ∧ Calm(rockie)


If Rockie is not both a turtle and a squirrel, then Rockie is either cute or skittish.

¬(Turtle(rockie) ∧ Squirrel(rockie)) → (Cute(rockie) ⊕ Skittish(rockie))

Added parentheses because of operator precedences

¬(Turtle(rockie) ∧ Squirrel(rockie)) → Cute(rockie) ⊕ Skittish(rockie)


If Rockie is cute and calm, then Rockie is a skittish turtle.

(Cute(rockie) ∧ Calm(rockie)) → (Turtle(rockie) ∧ Skittish(rockie))

Added parentheses (even if not necessary): it is not considered a correction.

Cute(rockie) ∧ Calm(rockie) → Turtle(rockie) ∧ Skittish(rockie)


"Stranger Things" is a popular Netflix show. If a Netflix show is popular, Karen will binge-watch it. If and only if Karen binge-watches a Netflix show, she will download it. Karen does not download "Black Mirror". "Black Mirror" is a Netflix show. If Karen binge-watches a Netflix show, she will share it with Lisa.

NetflixShow(strangerThings) ∧ Popular(strangerThings) ∧ ∀x ((NetflixShow(x) ∧ Popular(x)) → BingeWatch(karen, x)) ∧ ∀x (NetflixShow(x) → (BingeWatch(karen, x) ↔ Download(karen, x))) ∧ ¬Download(karen, blackMirror) ∧ NetflixShow(blackMirror) ∧ ∀x ((NetflixShow(x) ∧ BingeWatch(karen, x)) → ShareWith(karen, x, lisa))

Karen may download things that are not Netflix shows, but the original FOL incorrectly restricted the biconditional to only Netflix shows. The NL states 'If and only if Karen binge-watches a Netflix show, she will download it', which means the biconditional should be between 'BingeWatch(karen, x)' and 'Download(karen, x)' without restricting 'x' to being a Netflix show in the antecedent. Corrected to '∀x (NetflixShow(x) → (BingeWatch(karen, x) ↔ Download(karen, x)))' to accurately reflect the NL.

NetflixShow(strangerThings) ∧ Popular(strangerThings)
∀x ((NetflixShow(x) ∧ Popular(x)) → BingeWatch(karen, x))
∀x ((NetflixShow(x) ∧ BingeWatch(karen, x)) ↔ Download(karen, x))
¬Download(karen, blackMirror)
NetflixShow(blackMirror)
∀x ((NetflixShow(x) ∧ BingeWatch(karen, x)) → ShareWith(karen, x, lisa))


Karen will share "Stranger Things" with Lisa.


Karen will share "Black Mirror" with Lisa.


Beijing is the capital of the People's Republic of China. Beijing is the capital city of the world's most populous nation. Beijing is located in Northern China. Beijing hosted the 2008 Summer Olympics and 2008 Summer Paralympics Games. Beijing has hosted the Summer and Winter Olympics and the Summer and Winter Paralympics. Many of Beijing's 91 universities consistently rank among the best universities in the world.

CapitalOf(beijing, peoplesRepublicOfChina) ∧ ∃x (WorldsMostPopulousNation(x) ∧ CapitalOf(beijing, x)) ∧ LocatedIn(beijing, northernChina) ∧ Hosted(beijing, summerOlympics2008) ∧ Hosted(beijing, summerParalympicGames2008) ∧ Hosted(beijing, summerOlympics) ∧ Hosted(beijing, winterOlympics) ∧ Hosted(beijing, summerParalympicGames)  ∧ Hosted(beijing, winterParalympicGames) ∧ ∃x (University(x) ∧ InBeijing(x) ∧ ConsistentlyRankAmongTheBestIn(x, theWorld))

1. Translation Failure (Premise 2): The original '∃x (CapitalOf(beijing, x) → ...)' is a weak existential implication that evaluates to true if there is literally anything Beijing is NOT the capital of. Corrected to an existential conjunction '∃x (WorldsMostPopulousNation(x) ∧ CapitalOf(beijing, x))' to accurately capture the definite description. 2. Invalid Syntax (Premise 4): FOL constants cannot start with numbers. Renamed '2008SummerOlympics' and '2008SummerParalympicGames' to 'summerOlympics2008' and 'summerParalympicGames2008'. 3. Silent Edits Reverted (Premises 4 and 5): Removed hallucinated outer parentheses added by the previous reviewer.

CapitalOf(beijing, peoplesRepublicOfChina)
∃x (CapitalOf(beijing, x) → WorldsMostPopulousNation(x))
LocatedIn(beijing, northernChina)
Hosted(beijing, 2008SummerOlympics) ∧ Hosted(beijing, 2008SummerParalympicGames)
Hosted(beijing, summerOlympics) ∧ Hosted(beijing, winterOlympics) ∧ Hosted(beijing, summerParalympicGames)  ∧ Hosted(beijing, winterParalympicGames)
∃x (University(x) ∧ InBeijing(x) ∧ ConsistentlyRankAmongTheBestIn(x, theWorld))


Beijing hosted both the 2008 Summer Olympics and the Winter Olympics.

Hosted(beijing, summerOlympics2008) ∧ Hosted(beijing, winterOlympics)

Translation Failure: The old FOL used 'summerOlympics' which refers to the generic games from Premise 5, but the NL specifically calls out the '2008 Summer Olympics'. Updated to use 'summerOlympics2008' to accurately reflect the text and correctly link to Premise 4.

Hosted(beijing, summerOlympics) ∧ Hosted(beijing, winterOlympics)


Beijing is the second largest Chinese city.


All aliens are extraterrestrials. If someone is from Mars, then they are an alien. No extraterrestrials are human. All highly intelligent beings from Earth are humans. Marvin is a highly intelligent being. Marvin is either from Earth and from Mars, or he is from neither. If Marvin is not from Earth, then Marvin is an extraterrestrial.

∀x (Alien(x) → Extraterrestrial(x)) ∧ ∀x (From(x, mars) → Alien(x)) ∧ ∀x (Extraterrestrial(x) → ¬Human(x)) ∧ ∀x (HighlyIntelligentBeing(x) ∧ From(x, earth) → Human(x)) ∧ HighlyIntelligentBeing(marvin) ∧ ¬(From(marvin, earth) ⊕ From(marvin, mars)) ∧ (¬From(marvin, earth) → Extraterrestrial(marvin))

No semantic corrections required. The original FOL accurately translated the NL sentences. The negated XOR '¬(A ⊕ B)' perfectly captures the 'both or neither' phrasing of the NL. Minor additions of grouping parentheses for readability do not constitute a formal correction.

∀x (Alien(x) → Extraterrestrial(x))
∀x (From(x, mars) → Alien(x))
∀x (Extraterrestrial(x) → ¬Human(x))
∀x (HighlyIntelligentBeing(x) ∧ From(x, earth) → Human(x))
HighlyIntelligentBeing(marvin)
¬(From(marvin, earth) ⊕ From(marvin, mars))
¬From(marvin, earth) → Extraterrestrial(marvin)


Marvin is neither a human nor from Mars.


If Marvin is not from Mars, then Marvin is a human.

In the only consistent case (Case B), ¬From(marvin, mars) is true but Human(marvin) is false, making the implication false. Label False is correct.


Everyone at the mixer is a Grand Slam champion or an Oscar-nominated actor. Every Grand Slam champion at the mixer is a professional tennis player. All Oscar-nominated actors at the mixer are celebrities. All professional tennis players at the mixer are athletes. If a person at the mixer is a celebrity, then they are well paid. If a person at the mixer is an athlete, then they are famous. All well-paid people at the mixer live in tax havens. Djokovic is at the mixer: if Djokovic is a famous athlete, then Djokovic is well-paid.

∀x (At(x, mixer) → (GrandSlamChampion(x) ∨ (OscarNominated(x) ∧ Actor(x)))) ∧ ∀x (At(x, mixer) → (GrandSlamChampion(x) → (Professional(x) ∧ TennisPlayer(x)))) ∧ ∀x (At(x, mixer) → ((OscarNominated(x) ∧ Actor(x)) → Celebrity(x))) ∧ ∀x ((At(x, mixer) ∧ Professional(x) ∧ TennisPlayer(x)) → Athlete(x)) ∧ ∀x ((At(x, mixer) ∧ Celebrity(x)) → WellPaid(x)) ∧ ∀x ((At(x, mixer) ∧ Athlete(x)) → Famous(x)) ∧ ∀x ((At(x, mixer) ∧ WellPaid(x)) → ∃y (TaxHaven(y) ∧ LiveIn(x, y))) ∧ At(djokovic, mixer) ∧ ((Famous(djokovic) ∧ Athlete(djokovic)) → WellPaid(djokovic))

Corrections applied: (1) Premises 2 and 3: The original FOL statements contained fatal syntax errors (missing closing parentheses) and a major semantic error (using a conjunction '∧' instead of an implication '→' for a universal claim, which asserted that everyone at the mixer actually *is* a champion/actor rather than setting a condition). Corrected the structure to '∀x (At(x, mixer) → (Condition(x) → Result(x)))' to accurately reflect the NL. (2) Since GrandSlam(x) was alwasy used with also Champion(x), we found no need to separate the predicate symbols, even because there is not a clear meaning behind GrandSlam(x) if not 'x is a Grand Slam champion'. They are unified in the unique predicate symbol GrandSlamChampion(x). (3) TaxHaven should not be considered a constant but a predicate.

∀x (At(x, mixer) → (GrandSlam(x) ∧ Champion(x)) ∨ (OscarNominated(x) ∧ Actor(x)))
∀x (At(x, mixer) ∧ (GrandSlam(x) ∧ Champion(x) → Professional(x) ∧ TennisPlayer(x))
∀x (At(x, mixer) ∧ (OscarNominated(x) ∧ Actor(x) → Celebrity(x))
∀x (At(x, mixer) ∧ Professional(x) ∧ TennisPlayer(x) → Athlete(x))
∀x (At(x, mixer) ∧ Celebrity(x) → WellPaid(x))
∀x (At(x, mixer) ∧ Athlete(x) → Famous(x))
∀x (At(x, mixer) ∧ WellPaid(x) → LiveIn(x, taxHaven))
At(djokovic, mixer) ∧ (Famous(djokovic) ∧ Athlete(djokovic) → WellPaid(djokovic))


Note: label_31 was missing from the original input (label_32 was written twice); label inferred from context as 'Uncertain' based on the value appearing in the field wrongly assigned to conclusion_31. Unified the predicate symbols as written in story_340.


¬ (∃x (TaxHaven(x) ∧ LiveIn(djokovic, x)))


Diamond Mine is a professional wrestling stable formed in WWE. Roderick Strong leads Diamond Mine. Diamond Mine includes the Creed Brothers and Ivy Nile. Imperium has a feud with Diamond Mine.

ProfessionalWrestlingStable(diamondMine) ∧ In(diamondMine, wWE) ∧ Leads(roderickStrong, diamondMine) ∧ Includes(diamondMine, creedBrothers) ∧ Includes(diamondMine, ivyNile) ∧ Feuds(imperium, diamondMine)

ProfessionalWrestlingStable(diamondMine) ∧ In(diamondMine, wWE)
Leads(roderickStrong, diamondMine)
Includes(diamondMine, creedBrothers) ∧ Includes(diamondMine, ivyNile)
Feuds(imperium, diamondMine)


Roderick Strong leads a professional wrestling stable.

∃x (Leads(roderickStrong, x) ∧ ProfessionalWrestlingStable(x))

The constant 'roderickstrong' was written in all lowercase in the original FOL, inconsistently with the camelCase convention used in the premises ('roderickStrong'). Normalised to 'roderickStrong' for consistency.

∃x (Leads(roderickstrong, x) ∧ ProfessionalWrestlingStable(x))


Roderick Strong leads the Creed Brothers.

Same casing issue as conclusion 34: 'roderickstrong' → 'roderickStrong' and 'creedbrothers' → 'creedBrothers', to align with camelCase convention used in the premises.


Imperium doesn't have a feud with a professional wrestling stable that includes Ivy Nile.

∀x ((ProfessionalWrestlingStable(x) ∧ Includes(x, ivyNile)) → ¬Feuds(imperium, x))

Casing issue: 'ivynile' → 'ivyNile' to align with camelCase convention used in the premises.

∀x ((ProfessionalWrestlingStable(x) ∧ Includes(x, ivynile)) → ¬Feuds(imperium, x))


Symphony No. 9 is a music piece. Composers write music pieces. Beethoven wrote Symphony No. 9. Vienna Music Society premiered Symphony No. 9. Vienna Music Society is an orchestra. Beethoven leads the Vienna Music Society. Orchestras are led by conductors.

MusicPiece(symphony9) ∧ ∀x(Composer(x) → ∃y(MusicPiece(y) ∧ Write(x, y))) ∧ Write(beethoven, symphony9) ∧ Premiered(viennaMusicSociety, symphony9) ∧ Orchestra(viennaMusicSociety) ∧ Lead(beethoven, viennaMusicSociety) ∧ ∀x (Orchestra(x) → ∃y (Conductor(y) ∧ Lead(y, x)))

Composers write music pieces can be interpreted as: it can mean that all the composers write some music pieces, that only composers write music pieces, .... Here it is translated in the first way

Corrections applied: (1) Premise 3: The original 'Writtenby(symphony9, beethoven)' uses an inconsistent predicate. To properly resolve against Premise 2's 'Write(y, x)', it must be rewritten as 'Write(beethoven, symphony9)'. (2) Premise 7: The original FOL contained a critical scoping error '... (∃y Conductor(y) ∧ Lead(y, x))' leaving 'y' free in the second conjunct. Corrected to '... ∃y (Conductor(y) ∧ Lead(y, x))' to bind both predicates.

MusicPiece(symphony9)
∀x (MusicPiece(x) → ∃y (Composer(y) ∧ Write(y, x)))
Writtenby(symphony9, beethoven)
Premiered(viennaMusicSociety, symphony9)
Orchestra(viennaMusicSociety)
Lead(beethoven, viennaMusicSociety)
∀x (Orchestra(x) → (∃y Conductor(y) ∧ Lead(y, x)))


Label correction: The original label 'True' is invalid in strict formal logic. While Premise 2 states every music piece has some composer who wrote it, and Premise 3 states Beethoven wrote it, there is no rule stating that ONLY composers write music pieces or that a piece has a unique writer. Thus, 'Composer(beethoven)' cannot be definitively proven. Changed label to Uncertain.


∃x ∃y (Orchestra(x) ∧ MusicPiece(y) ∧ Premiered(x, y))


Label correction: The original label 'False' implies we can prove Beethoven IS a conductor. However, just because he leads the orchestra (P6) and a conductor leads the orchestra (P7), without an equality axiom or a rule stating ONLY conductors lead orchestras, we cannot definitively prove he is a conductor, nor can we prove he is not. Changed label to Uncertain.


All of Zaha Hadid's design styles that Max adores have interesting geometries. No brutalist buildings that Max adores have interesting geometries. Every style that Max adores is either Zaha Hadid's design style or Kelly Wearstler's design style. All of Kelly Wearstler's design styles that Max adores are evocative. All of Kelly Wearstler's design styles that Max adores are dreamy. If a design by Max that he adores has interesting geometries, then the design is a brutalist building and evocative.

∀x (Adore(max, x) ∧ ZahaHadid(x) ∧ DesignStyle(x) → InterestingGeometry(x)) ∧ ∀x (Adore(max, x) ∧ BrutalistBuilding(x) → ¬InterestingGeometry(x)) ∧ ∀x (Adore(max, x) → ((ZahaHadid(x) ∧ DesignStyle(x)) ⊕ (KellyWearstler(x) ∧ DesignStyle(x)))) ∧ ∀x (Adore(max, x) ∧ KellyWearstler(x) ∧ DesignStyle(x) → Evocative(x)) ∧ ∀x (Adore(max, x) ∧ KellyWearstler(x) ∧ DesignStyle(x) → Dreamy(x)) ∧ ∀x (Adore(max, x) ∧ Design(x) ∧ ByMax(x) ∧ InterestingGeometry(x) → BrutalistBuilding(x) ∧ Evocative(x))

Two corrections made: (1) Premise 3: unclosed parenthesis in the original fixed. (2) Premise 6: the original uses '∃x (Adore(max, x) ∧ Design(x) ∧ ByMax(x) ∧ InterestingGeometry(x) → BrutalistBuilding(x) ∧ Evocative(x))'. This is a well-known FOL scoping pitfall: '∃x (P → Q)' is not the intended reading for a conditional generalisation — and in fact '∃x (P ∧ Q → R)' is logically equivalent to '∃x (¬(P ∧ Q) ∨ R)', which is almost trivially true and does not capture the intended universal conditional. The NL reads as a general rule ('if a design... then...'), so the correct quantifier is ∀, giving '∀x (Adore(max, x) ∧ Design(x) ∧ ByMax(x) ∧ InterestingGeometry(x) → BrutalistBuilding(x) ∧ Evocative(x))'.

∀x (Adore(max, x) ∧ ZahaHadid(x) ∧ DesignStyle(x) → InterestingGeometry(x))
∀x (Adore(max, x) ∧ BrutalistBuilding(x) → ¬InterestingGeometry(x))
∀x (Adore(max, x) → ((ZahaHadid(x) ∧ DesignStyle(x)) ⊕ (KellyWearstler(x) ∧ DesignStyle(x)))
∀x (Adore(max, x) ∧ KellyWearstler(x) ∧ DesignStyle(x) → Evocative(x))
∀x (Adore(max, x) ∧ KellyWearstler(x) ∧ DesignStyle(x) → Dreamy(x))
∃x (Adore(max, x) ∧ Design(x) ∧ ByMax(x) ∧ InterestingGeometry(x) → BrutalistBuilding(x) ∧ Evocative(x))


A design by Max is a brutalist building.

∃x (Design(x) ∧ ByMax(x) ∧ BrutalistBuilding(x))


A design by Max is evocative and dreamy.

∃x (Design(x) ∧ ByMax(x) ∧ Evocative(x) ∧ Dreamy(x))

Label correction: FOR THE ORIGINAL INTERPRETATION: The original label 'True' commits an existential fallacy. The premises are exclusively universal conditionals (∀x). They establish rules about properties, but never assert that any design actually exists in the domain. Furthermore, there is no rule linking a design ByMax to the Dreamy property (which requires KellyWearstler). Because we cannot prove the existence of such an entity, the label must be Uncertain. IF THE INTERPRETATION IS UNIVERSAL: then however, any premise talks about designs by Max that he adores, so it can be True, since nothing informed us about the objects x that satisfy only the properties Design(x) ∧ ByMax(x)


A design by Max is either evocative or dreamy.

∃x (Design(x) ∧ ByMax(x) ∧ (Evocative(x) ⊕ Dreamy(x)))

Label correction: The original label 'False' is logically invalid. To prove this existential statement False, the solver must prove that NO design by Max is either evocative or dreamy. Because the premises do not explicitly forbid Max from having a design that is solely evocative (e.g., via Premise 6), and because we don't even know if he has designs at all, the solver cannot definitively prove this False. The label must be Uncertain. Same as before, even considering the other interpretation


If someone is ranked highly by the Women's Tennis Association, then they are one of the most active players in major tennis. Everyone who lost to Iga Świątek at Roland Garros 2022 is ranked highly by the Women's Tennis Association. All female tennis players at Roland Garros 2022 lost to Iga Świątek. All tennis players at Roland Garros 2022 are either female or male. All male tennis players at Roland Garros in 2022 lost to Rafael Nadal. If Coco Gauff is ranked highly by the Women's Tennis Association or lost to Rafael Nadal, then Coco Gauff is not a male tennis player at Roland Garros 2022. Coco Gauff is at Roland Garros 2022.

∀x (RankedHighlyBy(x, womensTennisAssociation) → MostActivePlayerIn(x, majorTennis)) ∧ ∀x (LostTo(x, świątek, rolandGarros2022) → RankedHighlyBy(x, womensTennisAssociation)) ∧ ∀x ((FemaleTennisPlayer(x) ∧ At(x, rolandGarros2022)) → LostTo(x, świątek, rolandGarros2022)) ∧ ∀x ((TennisPlayer(x) ∧ At(x, rolandGarros2022)) → (FemaleTennisPlayer(x) ⊕ MaleTennisPlayer(x))) ∧ ∀x ((MaleTennisPlayer(x) ∧ At(x, rolandGarros2022)) → LostTo(x, nadal, rolandGarros2022)) ∧ ((RankedHighlyBy(cocoGauff, womensTennisAssociation) ∨ LostTo(cocoGauff, nadal, rolandGarros2022)) → ¬(MaleTennisPlayer(cocoGauff) ∧ At(cocoGauff, rolandGarros2022))) ∧ TennisPlayer(cocoGauff) ∧ At(cocoGauff, rolandGarros2022)

Critical corrections applied: (1) Predicate Unification: The original FOL mixed 'Female(x) ∧ TennisPlayer(x)' with 'FemaleTennisPlayer(x)'. Standardized to 'FemaleTennisPlayer(x)' and 'MaleTennisPlayer(x)' throughout to repair the broken logical chain. (2) Translation Failure: Premise 5 incorrectly translated 'Rafael Nadal' as 'świątek'. Corrected to 'nadal'. (3) Syntax/Translation: Premise 6 was completely broken, using wrong constants ('świątek' instead of 'nadal') and wrong unary predicates ('AtRolandGarros2022(x)'). Standardized to match the established premises, and added strict grouping parentheses to the antecedent to ensure standard parser precedence. (4) the property 'x losts to y at z' were translated as LostTo(x, y) ∧ At(x, z), that is not the correct formalization. Now it is fixed extending LostTo as a ternary binary.

∀x (RankedHighlyBy(x, womensTennisAssociation) → MostActivePlayerIn(x, majorTennis))
∀x (LostTo(x, świątek) ∧  At(x, rolandGarros2022) → RankedHighlyBy(x, womensTennisAssociation))
∀x (FemaleTennisPlayer(x) ∧ At(x, rolandGarros2022) → LostTo(x, świątek) ∧  At(x, rolandGarros2022))
∀x (TennisPlayer(x) ∧ At(x, rolandGarros2022) → (Female(x) ∧ TennisPlayer(x)) ⊕ (Male(x) ∧ TennisPlayer(x)))
∀x (Male(x) ∧ TennisPlayer(x) ∧ At(x, rolandGarros2022) → LostTo(x, świątek) ∧ At(x, rolandGarros2022))
RankedHighlyBy(cocoGauff, womensTennisAssociation) ∨ (LostTo(cocoGauff, świątek) ∧ LostAt(cocoGauff, rolandGarros2022)) → ¬(Male(cocoGauff) ∧ TennisPlayer(cocoGauff) ∧ AtRolandGarros2022(cocoGauff))
TennisPlayer(cocoGauff) ∧ At(cocoGauff, rolandGarros2022)


Coco Gauff is among the most active major tennis players.

The original NL sentence was 'Coco Gauff is among the most active Grand-Slam players.'. Most active player is mentioned only about 'major tennis' in the story and never in Grand-Slams competition. With the provided ontology is not formalizable correctly. We corrected the original NL_sentence.


Translation correction: (1) The original FOL translated 'Rafael Nadal' as 'świątek'. Corrected to 'nadal'. (2) The original FOL inappropriately appended '∧ At(cocoGauff, rolandGarros2022)', which is a premise, not part of the conclusion's statement. Removed. (3) Used LostTo ternary.

LostTo(cocoGauff, świątek) ∧ At(cocoGauff, rolandGarros2022)


Coco Gauff is not both a player who lost to Iga Świątek at Roland Garros 2022 and one of the most active players in major tennis.

¬LostTo(cocoGauff, świątek, rolandGarros2022) ∨ ¬MostActivePlayerIn(cocoGauff, majorTennis)

¬(LostTo(cocoGauff, świątek) ∧ At(cocoGauff, rolandGarros2022)) ∨ ¬MostActivePlayerIn(cocoGauff, majorTennis)


All cats are mammals. Some pets are not mammals.

∀x (Cat(x) → Mammal(x)) ∧ ∃x (Pet(x) ∧ ¬Mammal(x))

∀x (Cat(x) → Mammal(x))
∃x (Pet(x) ∧ ¬Mammal(x))


There are four seasons in a year: Spring, Summer, Fall, and Winter. All students who want to have a long vacation have summer as their favorite season. Emma's favorite season is summer. Mia's favorite season is not the same as Emma's. James wants to have a long vacation.

Season(spring) ∧ Season(summer) ∧ Season(fall) ∧ Season(winter) ∧ ∀x (Season(x) → (x = spring ∨ x = summer ∨ x = fall ∨ x = winter)) ∧ ∀x (Student(x) ∧ Want(x, longVacation) → Favorite(x, summer)) ∧ Favorite(emma, summer) ∧ ∀x ∀y (Season(x) ∧ Season(y) ∧ Favorite(mia, x) ∧ Favorite(emma, y) → ¬(x=y)) ∧ Want(james, longVacation)

1. Spring, Summer, Fall, and Winter are all seasons (∧ adopted) and the only ones 2. Translation Failure (Premise 2): Added the missing 'Student(x)' predicate to match the NL text 'All students who...'. 4. Broken Linkage (Premise 2 & 5): Corrected casing mismatch from 'longvacation' to 'longVacation'.

Season(spring) ∨ Season(summer) ∨ Season(fall) ∨ Season(winter) ∧ (Season(spring) → ¬Season(summer) ∧ ¬Season(fall) ∧ ¬Season(winter)) ∧ (Season(summer) → ¬Season(spring) ∧ ¬Season(fall) ∧ ¬Season(winter)) ∧ (Season(fall) → ¬Season(spring) ∧ ¬Season(summer) ∧ ¬Season(winter)) ∧ (Season(winter) → ¬Season(spring) ∧ ¬Season(summer) ∧ ¬Season(fall))
∀x (Want(x, longvacation) → Favorite(x, summer))
Favorite(emma, summer)
∀x ∀y (Season(x) ∧ Season(y) ∧ Favorite(mia, x) ∧ Favorite(emma, y) → ¬(x=y))
Want(james, longVacation)


Label corrected from True to Uncertain. Premise 2 restricts the rule to 'All students', but there is no premise explicitly stating that James is a student. Deducing this conclusion without the Student(james) constraint is a logical fallacy.


No digital media are analog. Every printed text is analog media. All streaming services are digital media. If an object is a hardcover book, then it is printed text. If 1984 is a streaming service, then 1984 is a hardcover book.

∀x (DigitalMedia(x) → ¬AnalogMedia(x)) ∧ ∀x (PrintedText(x) → AnalogMedia(x)) ∧ ∀x (StreamingService(x) → DigitalMedia(x)) ∧ ∀x (HardcoverBook(x) → PrintedText(x)) ∧ (StreamingService(y1984) → HardcoverBook(y1984))

ontology : Only changed '1984' with 'y1984' to avoid conflict with z3-parser

∀x (DigitalMedia(x) → ¬AnalogMedia(x))
∀x (PrintedText(x) → AnalogMedia(x))
∀x (StreamingService(x) → DigitalMedia(x))
∀x (HardcoverBook(x) → PrintedText(x))
StreamingService(1984) → HardcoverBook(1984)


All Romance languages are Indo-European languages. Romance languages are a language family. All languages within a language family are related to each other. French and Spanish are both Romance languages. German is related to Spanish. Basque is not related to any other language.

∀x (RomanceLanguage(x) → IndoEuropeanLanguage(x)) ∧ ∀x (RomanceLanguage(x) → MemberOf(x, romanceFamily)) ∧ ∀x ∀y ∀z ((MemberOf(x, z) ∧ MemberOf(y, z)) → (Related(x, y) ∧ Related(y, x))) ∧ RomanceLanguage(french) ∧ RomanceLanguage(spanish) ∧ Related(german, spanish) ∧ ∀x (x ≠ basque → ¬Related(basque, x))

1. Broken Linkage (Premise 2): Replaced the generic constant 'languageFamily' with the specific 'romanceFamily'. Using a generic constant would incorrectly assign all language families in the domain to a single object.  2. Translation Failure & Broken Linkage (Premise 6): The original translation introduced an undefined 'Language(x)' predicate. Since 'Language' is never linked to 'RomanceLanguage', it breaks the logical chain (we cannot prove French or Spanish are 'Languages' to trigger the rule). Furthermore, it ignored the word 'other'. Corrected to '∀x (x ≠ basque → ¬Related(basque, x))' to accurately capture 'any other' without hallucinating disconnected predicates.

∀x (RomanceLanguage(x) → IndoEuropeanLanguage(x))
∀x (RomanceLanguage(x) → MemberOf(x, languageFamily))
∀x ∀y ∀z ((MemberOf(x, z) ∧ MemberOf(y, z)) → (Related(x, y) ∧ Related(y, x)))
RomanceLanguage(french) ∧ RomanceLanguage(spanish)
Related(german, spanish)
∀x (Language(x) → ¬Related(basque, x))


The label 'False' is correct: if Basque were a Romance language, it would be a member of romanceFamily (premise 2), and hence related to French and Spanish (premise 3). But premise 6 states Basque is not related to any language, yielding a contradiction. Therefore RomanceLanguage(basque) is refutable.


The label 'Uncertain' is correct: the premises establish that German is related to Spanish, but relatedness is not sufficient to establish Romance language membership — the Related predicate is introduced for German only via premise 5 as a bare fact, not derived from family membership. Nothing in the premises rules out German being a Romance language, nor does anything entail it. The conclusion is therefore neither provable nor refutable.


The label 'True' is directly derivable: premise 4 establishes RomanceLanguage(french), and premise 1 states ∀x (RomanceLanguage(x) → IndoEuropeanLanguage(x)), so IndoEuropeanLanguage(french) follows by universal instantiation and modus ponens.


Robert Lewandowski is a striker. Strikers are soccer players. Robert Lewandowski left Bayern Munchen. If a player leaves a team they no longer play for that team.

Striker(robertLewandowski) ∧ ∀x (Striker(x) → SoccerPlayer(x)) ∧ Left(robertLewandowski, bayernMunchen) ∧ ∀x ∀y (Player(x) ∧ Team(y) ∧ Left(x, y) → ¬PlaysFor(x, y))

Translation Failure (Premise 4): The original FOL '∀x ∀y (Left(x, y) → ¬PlaysFor(x, y))' ignored the explicit domain constraints 'player' and 'team' from the NL text ('If a player leaves a team...'). Corrected to '∀x ∀y (Player(x) ∧ Team(y) ∧ Left(x, y) → ¬PlaysFor(x, y))' to maintain absolute semantic fidelity to the natural language, even if it changes downstream entailment.

Striker(robertLewandowski)
∀x (Striker(x) → SoccerPlayer(x))
Left(robertLewandowski, bayernMunchen)
∀x ∀y (Left(x, y) → ¬PlaysFor(x, y))


Robert Lewandowski plays for Bayern Munchen.

Label corrected from False to Uncertain. With Premise 4 strictly translated to require Player(x) and Team(y), we cannot logically deduce that Lewandowski does not play for Bayern Munchen. There is no premise stating that Bayern Munchen is a 'Team' (Team(bayernMunchen)), nor one stating Lewandowski is a 'Player' (we only know 'SoccerPlayer'). Without these exact predicates established, the implication fails to trigger.


Translation Failure: The NL simply says 'is a star', but the original FOL hallucinated the more specific predicate 'SoccerStar'. Corrected the FOL to the strictly literal 'Star(robertLewandowski)'. The label remains Uncertain.


Billings is a city in the state of Montana in U.S. The state of Montana includes the cities of Butte, Helena, and Missoula. White Sulphur Springs and Butte are cities in the same state in U.S. The city of St Pierre is not in the state of Montana. Any city in Butte is not in St Pierre. A city can only be in one state in U.S. except for Bristol, Texarkana, Texhoma and Union City.

City(billings) ∧ In(billings, montana) ∧ State(montana) ∧ City(butte) ∧ In(butte, montana) ∧ City(helena) ∧ In(helena, montana) ∧ City(missoula) ∧ In(missoula, montana) ∧ ∃x (City(whiteSulphurSprings) ∧ In(whiteSulphurSprings, x) ∧ City(butte) ∧ In(butte, x) ∧ State(x)) ∧ City(stPierre) ∧ ¬(In(stPierre, montana)) ∧ ∀x ((City(x) ∧ City(butte) ∧ In(x, butte)) → ¬(In(x, stPierre))) ∧ ∀x ∀y ((City(x) ∧ State(y) ∧ In(x, y) ∧ ¬(x=bristol) ∧ ¬(x=texarkana) ∧ ¬(x=texhoma) ∧ ¬(x=unionCity)) → ¬∃z (¬(z=y) ∧ State(z) ∧ In(x, z)))

1. Broken Linkage (Premises 4 & 5): Replaced the mismatched constant 'pierre' with 'stPierre' to correctly link with the conclusions. 2. Scope & Syntax Error (Premise 6): The original FOL '∀x ∃y ((City...' contained an unbalanced parenthesis and mis-scoped 'y' as an existential variable, leaving it unbound in the consequent. Corrected the quantifier to '∀y' and removed the rogue parenthesis. 3. Invalid Swaps Reverted: Restored 'City(butte)' in Premise 5, which the previous reviewer invalidly removed as a logically equivalent swap.

City(billings) ∧ In(billings, montana)
City(butte) ∧ In(butte, montana) ∧ City(helena) ∧ In(helena, montana) ∧ City(missoula) ∧ In(missoula, montana)
∃x (City(whitesulphursprings) ∧ In(whitesulphursprings, x) ∧ City(butte) ∧ In(butte, x))
City(pierre) ∧ ¬(In(pierre, montana))
∀x ((City(x) ∧ City(butte) ∧ In(x, butte)) → ¬(In(x, pierre)))
∀x ∃y ((City(x) ∧ (In(x, y) ∧ ¬(x=bristol) ∧ ¬(x=texarkana) ∧ ¬(x=texhoma) ∧ ¬(x=unionCity)) → ¬∃z (¬(z=y) ∧ In(x, z)))


Butte and St Pierre are in the same state.

∃x (State(x) ∧ In(butte, x) ∧ In(stPierre, x))

¬(butte=bristol) ∧ ¬(butte=texarkana) ∧ ¬(butte=texhoma) ∧ ¬(butte=unionCity) ∧ ¬(stPierre=bristol) ∧ ¬(stPierre=texarkana) ∧ ¬(stPierre=texhoma) ∧ ¬(stPierre=unionCity)


St Pierre and Bismarck are in the same state.

∃x (In(stPierre, x) ∧ In(bismarck, x) ∧ State(x))

Broken Linkage: The original FOL used 'pierre', but the conclusions (and updated premises) standardise on 'stPierre'. Corrected to 'stPierre' to fix the broken logical chain. Modified also the ontology State. (2) Removed also the fact that these two are cities (not explicitely said in NL)

∃x (City(pierre) ∧ In(pierre, x) ∧ City(bismarck) ∧ In(bismarck, x))


Montana is home to the city of Missoula.


International students in the US have either an F1 visa or a J1 visa. An international student in the US with an F1 visa needs to apply for CPT or OPT if the student wants to work in the US. Mike is an international student. Mike needs to apply for CPT if he wants to work in the US.

∀x (InternationalStudent(x) ∧ In(x, unitedStates) → F1Visa(x) ⊕ J1Visa(x)) ∧ ∀x (InternationalStudent(x) ∧ In(x, unitedStates) ∧ F1Visa(x) ∧ WantToWorkIn(x, unitedStates) → Apply(x, cpt) ∨ Apply(x, opt)) ∧ InternationalStudent(mike) ∧ WantToWorkIn(mike, unitedStates) → Apply(mike, cpt)

1. Scope Error (Premise 4): The original FOL 'WantToWorkIn(x, unitedStates) → Apply(mike, cpt)' used a free, unbound variable 'x' in the antecedent while referring to the constant 'mike' in the consequent. Corrected to 'WantToWorkIn(mike, unitedStates)' to accurately match the NL text.

∀x (InternationalStudent(x) ∧ In(x, unitedStates) → F1Visa(x) ⊕ J1Visa(x))
∀x (InternationalStudent(x) ∧ In(x, unitedStates) ∧ F1Visa(x) ∧ WantToWorkIn(x, unitedStates) → Apply(x, cpt) ∨ Apply(x, opt))
InternationalStudent(mike)
WantToWorkIn(x, unitedStates) → Apply(mike, cpt)


All Brown Swiss cattle are cows. Some pets are Brown Swiss cattle. All cows are domesticated animals. Alligators are not domesticated animals. Ted is an alligator.

∀x (BrownSwissCattle(x) → Cow(x)) ∧ ∃x (Pet(x) ∧ BrownSwissCattle(x)) ∧ ∀x (Cow(x) → DomesticatedAnimal(x)) ∧ ∀x (Alligator(x) → ¬DomesticatedAnimal(x)) ∧ Alligator(ted)

Typo Correction: The original FOL misspelled 'Alligator' as 'Aligator' in Premises 4 and 5. Corrected to 'Alligator' to maintain strict semantic alignment with the standard spelling in the Natural Language text.

∀x (BrownSwissCattle(x) → Cow(x))
∃x (Pet(x) ∧ BrownSwissCattle(x))
∀x (Cow(x) → DomesticatedAnimal(x))
∀x (Aligator(x) → ¬DomesticatedAnimal(x))
Aligator(ted)


If Ted is a Brown Swiss cattle, then Ted is not a pet.


Yale University is a private Ivy League research university. Yale University moved to New Haven in 1716. Yale University's endowment was valued at $42.3 billion. A list of residential colleges at Yale: Benjamin Franklin College, Berkeley College, Branford College, Davenport College, Ezra Stiles College, Grace Hopper College, Jonathan Edwards College, Morse College, Pauli Murray College, Pierson College, Saybrook College, Silliman College, Timothy Dwight College, and Trumbull College.

PrivateIvyLeagueResearchUniversity(yaleUniversity) ∧ MovedTo(yaleUniversity, newHaven) ∧ MovedIn(yaleUniversity, year1716) ∧ ValuedAt(yaleUniversitysEndowment, usd42point3billion) ∧ ResidentialCollege(benjaminFranklinCollege) ∧ At(benjaminFranklinCollege, yaleUniversity) ∧ ResidentialCollege(berkeleyCollege) ∧ At(berkeleyCollege, yaleUniversity) ∧ ResidentialCollege(branfordCollege) ∧ At(branfordCollege, yaleUniversity) ∧ ResidentialCollege(davenportCollege) ∧ At(davenportCollege, yaleUniversity) ∧ ResidentialCollege(ezraStilesCollege) ∧ At(ezraStilesCollege, yaleUniversity) ∧ ResidentialCollege(graceHopperCollege) ∧ At(graceHopperCollege, yaleUniversity) ∧ ResidentialCollege(jonathanEdwardsCollege) ∧ At(jonathanEdwardsCollege, yaleUniversity) ∧ ResidentialCollege(morseCollege) ∧ At(morseCollege, yaleUniversity) ∧ ResidentialCollege(pauliMurrayCollege) ∧ At(pauliMurrayCollege, yaleUniversity) ∧ ResidentialCollege(piersonCollege) ∧ At(piersonCollege, yaleUniversity) ∧ ResidentialCollege(saybrookCollege) ∧ At(saybrookCollege, yaleUniversity) ∧ ResidentialCollege(sillimanCollege) ∧ At(sillimanCollege, yaleUniversity) ∧ ResidentialCollege(timothyDwightCollege) ∧ At(timothyDwightCollege, yaleUniversity) ∧ ResidentialCollege(trumbullCollege) ∧ At(trumbullCollege, yaleUniversity)

1. Translation Failure / Category Error (Premise 4): The original FOL used a universal conditional that failed to assert the existence of the colleges, making Conclusion 68 unprovable. It also treated specific college entities as unary properties instead of constants. Replaced with a single conjunction of ground facts (e.g., 'ResidentialCollege(piersonCollege) ∧ At(piersonCollege, yale)') to properly ground the facts. 2. Typo (Premise 4): Corrected 'BerkleyCollege' to 'BerkeleyCollege' to match the NL. 4. Syntax Error (Premise 3): The constant '42point3billion' starts with a digit, which violates standard FOL syntax rules (for z3-parser). Renamed to 'usd42point3billion'. 5. Unified 'yale' and 'yaleUniversity'

PrivateIvyLeagueResearchUniversity(yaleUniversity)
MovedTo(yaleUniversity, newHaven) ∧ MovedIn(yaleUniversity, year1716)
ValuedAt(yaleUniversitysEndowment, 42point3billion)
∀x (ResidentialCollege(x) → At(x, yale) ∧ (BenjaminFranklinCollege(x) ⊕ BerkleyCollege(x) ⊕ BranfordCollege(x) ⊕ DavenportCollege(x) ⊕ EzraStilesCollege(x) ⊕ GraceHopperCollege(x) ⊕ JonathanEdwardsCollege(x) ⊕ MorseCollege(x) ⊕ PauliMurrayCollege(x) ⊕ PiersonCollege(x) ⊕ SaybrookCollege(x) ⊕ SillimanCollege(x) ⊕ TimothyDwightCollege(x) ⊕ TrumbullCollege(x)))


A private Ivy League research university moved to New Haven.

∃x (PrivateIvyLeagueResearchUniversity(x) ∧ MovedTo(x, newHaven))


Yale University has the largest university endowment of any educational institution.

Broken Linkage: The original FOL uses 'yale' instead of 'yaleUniversity' as the constant. Unified to 'yaleUniversity' to match the premises.


Pierson College is a residential college at Yale.

ResidentialCollege(piersonCollege) ∧ At(piersonCollege, yaleUniversity)

ResidentialCollege(piersonCollege) ∧ At(piersonCollege, yale)


Badults is a British Sitcom series starring members of Pappy's. Badults was piloted in July 2013 on BBC Three. The working title 'The Secret Dude Society' was used for Badults. Andrew Collins was the script editor for Badults.

∃x (BritishSitcom(badults) ∧ Series(badults) ∧ MemberOf(x, pappys) ∧ Star(x, badults)) ∧ PilotedIn(badults, july2013) ∧ PilotedOn(badults, bBCThree) ∧ WorkingTitle(theSecretDudeSociety, badults) ∧ UsedFor(theSecretDudeSociety, badults) ∧ ScriptEditorFor(andrewCollins, badults)

∃x (BritishSitcom(badults) ∧ Series(badults) ∧ MemberOf(x, pappys) ∧ Starring(badults, x))
PilotedIn(badults, july2013) ∧ PilotedOn(badults, bBCThree)
WorkingTitle(theSecretDudeSociety, badults) ∧ UsedFor(theSecretDudeSociety, badults)
ScriptEditorFor(andrewCollins, badults)


Andrew Collins was the script editor for a series with the working title 'The Secret Dude Society'.

∃x (ScriptEditorFor(andrewCollins, x) ∧ Series(x) ∧ WorkingTitle(theSecretDudeSociety, x))


No members of Pappy's have starred in a show piloting on BBC Two or BBC Three.

∀x ∀y (MemberOf(x, pappys) ∧ Star(x, y) → ¬(PilotedOn(y, bBCTwo) ∨ PilotedOn(y, bBCThree)))

∀x ∀y (MemberOf(x, pappys) ∧ Starring(y, x) → ¬(PilotedOn(y, bBCTwo) ∨ PilotedOn(y, bBCThree)))


All growth stocks are bought to earn profits from rapid price appreciation. If a stock is bought to earn profits from rapid price appreciation, then it is not suitable for a retirement fund. Some stocks are growth stocks. All mature stocks are suitable for a retirement fund. KO is a mature stock.

∀x ((Stock(x) ∧ Growth(x)) → BoughtToEarnProfitFrom(x, rapidPriceAppreciation)) ∧ ∀x ((Stock(x) ∧ BoughtToEarnProfitFrom(x, rapidPriceAppreciation)) → ¬SuitableFor(x, retirementFund)) ∧ ∃x (Stock(x) ∧ Growth(x)) ∧ ∀x ((Stock(x) ∧ Mature(x)) → SuitableFor(x, retirementFund)) ∧ Stock(kO) ∧ Mature(kO)

1. Compositional Semantics / Translation Failure: Based on the insight that 'growth stock' and 'mature stock' intrinsically imply being a 'stock', the monolithic predicates 'GrowthStock(x)' and 'MatureStock(x)' have been decomposed into 'Stock(x) ∧ Growth(x)' and 'Stock(x) ∧ Mature(x)'. This ensures the domain constraint 'Stock' is universally active across all premises. 2. Translation Failure (Premise 2): Added the 'Stock(x)' constraint to match 'If a stock is bought...'. 3. Broken Linkage (Premise 2): Fixed the ternary 'BoughtToEarnProfitFrom' by removing the hallucinated 'earnProfit' argument to match Premise 1.

∀x (GrowthStock(x) → BoughtToEarnProfitFrom(x, rapidPriceAppreciation))
∀x (BoughtToEarnProfitFrom(x, earnProfit, rapidPriceAppreciation) → ¬SuitableFor(x, retirementFund))
∃x (Stock(x) ∧ GrowthStock(x))
∀x (MatureStock(x) → SuitableFor(x, retirementFund))
MatureStock(kO)


With the compositional decomposition of 'MatureStock(kO)' into 'Stock(kO) ∧ Mature(kO)' in Premise 5, it is now explicitly established that KO is a stock. The label is therefore corrected from Uncertain to True.


1. Translation Failure: The original FOL '¬GrowthStock(kO)' directly contradicted the positive phrasing of the NL. Corrected to 'Stock(kO) ∧ Growth(kO)' to strictly map the text. 2. Logic Check: The label 'False' is logically sound because assuming KO is a Growth stock triggers a contradiction: Premise 1 forces BoughtToEarnProfitFrom to be true, which (combined with Stock(kO)) forces ¬SuitableFor via Premise 2, directly contradicting the SuitableFor conclusion derived from Premises 4 and 5.


If KO is a growth stock or bought to earn profits from rapid price appreciation, then KO is neither a stock nor is its price volatile.

((Stock(kO) ∧ Growth(kO)) ∨ BoughtToEarnProfitFrom(kO, rapidPriceAppreciation)) → (¬Stock(kO) ∧ ¬VolatilePrice(kO))

1. Translation Failure: Applied the 'Stock ∧ Growth' decomposition. 2. Broken Linkage: Fixed the ternary 'BoughtToEarnProfitFrom' predicate to binary. 3. Translation Failure: Replaced '¬BoughtToEarnProfitFrom' in the consequent with '¬VolatilePrice(kO)' to accurately capture 'nor is its price volatile'. 4. Scope: Added required grouping parentheses around the antecedent. 5. Entailment Logic Evaluation: With the new compositional semantics, Stock(kO) is definitively True, making the consequent definitively False. Because KO is a Stock and is SuitableFor a retirement fund, BoughtToEarnProfitFrom MUST be definitively False (otherwise Premise 2 would trigger a contradiction). By extension, Growth(kO) must also be False. Thus, both sides of the antecedent's disjunction are definitively False, making the material implication vacuously True.

GrowthStock(kO) ∨ BoughtToEarnProfitFrom(kO, earnProfit, rapidPriceAppreciation) → ¬Stock(kO) ∧ ¬BoughtToEarnProfitFrom(kO, rapidPriceAppreciation)


All vehicle registration plates in Istanbul begin with the number 34. Plates that do not begin with the number 34 are not from Istanbul. Joe's vehicle registration plate is from Istanbul. Tom's license plate begins with the number 35. If a license plate begins with the number 35, then it does not begin with the number 34.

∀x (VehicleRegistrationPlateIn(x, istanbul) → BeginWith(x, num34)) ∧ ∀x (¬BeginWith(x, num34) → ¬VehicleRegistrationPlateIn(x, istanbul)) ∧ ∃x (Owns(joe, x) ∧ VehicleRegistrationPlateIn(x, istanbul)) ∧ ∃x ∃y (Owns(tom, x) ∧ VehicleRegistrationPlateIn(x, y) ∧ BeginWith(x, num35)) ∧ ∀x (BeginWith(x, num35) → ¬BeginWith(x, num34))

Broken Linkage (Premise 2): The original FOL used 'FromIstanbul(x)', which disconnected it from the 'VehicleRegistrationPlateIn(x, istanbul)' predicate used in Premise 1, Premise 3, and Conclusion 75. Unified the predicate to 'VehicleRegistrationPlateIn(x, istanbul)' to allow the logical chain to resolve. Note: The previous reviewer correctly identified this fix in their explanation but failed to actually apply it to the FOL string in their output. It has now been applied.

∀x (VehicleRegistrationPlateIn(x, istanbul) → BeginWith(x, num34))
∀x (¬BeginWith(x, num34) → ¬FromIstanbul(x))
∃x (Owns(joe, x) ∧ VehicleRegistrationPlateIn(x, istanbul))
∃x (Owns(tom, x) ∧ BeginWith(x, num35))
∀x (BeginWith(x, num35) → ¬BeginWith(x, num34))


Joe's license plate begins with the number 34.

∃x (Owns(joe, x) ∧ BeginWith(x, num34) ∧ ∃y VehicleRegistrationPlateIn(x, y))

There is no referement to x that is a license plate in the original FOL_sentence: given the provided ontology, we can express this information saying that x is a vehicle registration plate in some place


∃x (Owns(tom, x) ∧ VehicleRegistrationPlateIn(x, istanbul))

∀x ∀y ∀z ∀u ((Owns(tom, x) ∧ VehicleRegistrationPlateIn(x, y) ∧ Owns(tom, z) ∧ VehicleRegistrationPlateIn(z, u)) → (x=z ∧ y=u))


If people at Mary's school work in student jobs on campus, then they need to earn money to help pay for their college tuition. If people at Mary's school order takeout frequently in college, then they work in student jobs on campus. People at Mary's school order takeout frequently in college or enjoy the dining hall meals and recipes. If people at Mary's school enjoy the dining hall meals and recipes, then they are not picky eaters. If people at Mary's school enjoy the dining hall meals and recipes, then they spend a lot of their time eating and catching up with friends in the campus dining halls. Hannah is at Mary's school. Hannah works in student jobs on campus and if she needs to earn money to help pay for her college tuition, then she is neither picky nor needs to earn money to help pay for her college tuition.

∀x ((At(x, marysSchool) ∧ WorkIn(x, studentJob, campus)) → NeedToEarnMoneyToHelpPayFor(x, collegeTuition)) ∧ ∀x ((At(x, marysSchool) ∧ OrderFrequently(x, takeout)) → WorkIn(x, studentJob, campus)) ∧ ∀x (At(x, marysSchool) → (OrderFrequently(x, takeout) ∨ Enjoy(x, diningHallMeal))) ∧ ∀x ((At(x, marysSchool) ∧ Enjoy(x, diningHallMeal)) → ¬PickyEater(x)) ∧ ∀x ((At(x, marysSchool) ∧ Enjoy(x, diningHallMeal)) → (Spend(x, aLotOfTime, eating, diningHall) ∧ Spend(x, aLotOfTime, catchingUpWithFriends, diningHall))) ∧ At(hannah, marysSchool) ∧ WorkIn(hannah, studentJob, campus) ∧ (NeedToEarnMoneyToHelpPayFor(hannah, collegeTuition) → (¬PickyEater(hannah) ∧ ¬NeedToEarnMoneyToHelpPayFor(hannah, collegeTuition)))

1. Scope & Syntax Error (Premise 3): The original lacked an implication '→', asserting that all entities in the universe are at Mary's school. Corrected to '∀x (At(...) → (Order(...) ∨ Enjoy(...)))'. 2. Translation Failure (Premise 7): The NL says 'Hannah works... AND if she needs...', meaning the 'WorkIn' fact is independent of the conditional. The original FOL erroneously folded it into the antecedent. Corrected to 'WorkIn(...) ∧ (Need(...) → ...)'. Applied De Morgan's to the consequent to accurately match the 'neither... nor...' phrasing.

∀x (At(x, marysSchool) ∧ WorkIn(x, studentJob, campus) → NeedToEarnMoneyToHelpPayFor(x, collegeTuition))
∀x (At(x, marysSchool) ∧ OrderFrequently(x, takeout) → WorkIn(x, studentJob, campus))
∀x (At(x, marysSchool) ∧ (OrderFrequently(x, takeout) ∨ Enjoy(x, diningHallMeal)))
∀x (At(x, marysSchool) ∧ Enjoy(x, diningHallMeal) → ¬PickyEater(x))
∀x (At(x, marysSchool) ∧ Enjoy(x, diningHallMeal) → Spend(x, aLotOfTime, eating) ∧ Spend(x, aLotOfTime, catchingUpWithFriends, diningHall))
At(hannah, marysSchool)
(WorkIn(hannah, studentJob, campus) ∧ NeedToEarnMoneyToHelpPayFor(hannah, collegeTuition) → ¬(PickyEater(hannah) ∨ NeedToEarnMoneyToHelpPayFor(hannah, collegeTuition))


Hannah needs to earn money to help pay for her college tuition.

Label corrected from Uncertain to True. The premises contain a logical contradiction: Premise 6 (At) and Premise 7 (WorkIn) trigger Premise 1, which proves NeedToEarnMoney. However, Premise 7 states NeedToEarnMoney → ¬NeedToEarnMoney. This yields NeedToEarnMoney ∧ ¬NeedToEarnMoney. Under the Principle of Explosion in classical logic, contradictory premises allow any statement to be proven True.


Hannah is at Mary's school and she is not a picky eater and spends a lot of her time eating and catching up with friends in the campus dining halls.

At(hannah, marysSchool) ∧ ¬PickyEater(hannah) ∧ Spend(hannah, aLotOfTime, eating, diningHall) ∧ Spend(hannah, aLotOfTime, catchingUpWithFriends, diningHall)

The premises contain a logical contradiction

¬(PickyEater(hannah) ∧ Spend(hannah, aLotOfTime, eating, diningHall) ∧ Spend(hannah, aLotOfTime, catchingUpWithFriends, diningHall))


Hannah is at Mary's school and she either is not a picky eater or, if she is, then she spends a lot of her time eating and catching up with friends in the campus dining halls.

At(hannah, marysSchool) ∧ (¬PickyEater(hannah) ∨ PickyEater(hannah)) ∧  (PickyEater(hannah) → (Spend(hannah, aLotOfTime, eating) ∧ Spend(hannah, aLotOfTime, catchingUpWithFriends, diningHall)))

PickyEater(hannah) ⊕ (Spend(hannah, aLotOfTime, eating) ∧ Spend(hannah, aLotOfTime, catchingUpWithFriends, diningHall))


No baked sweets are spicy. All cupcakes are baked sweets. All mala hotpots are spicy. All products from Baked by Melissa are cupcakes. Dried Thai chilies are spicy or mala hotpots or not baked sweets.

∀x (BakedSweet(x) → ¬Spicy(x)) ∧ ∀x (Cupcake(x) → BakedSweet(x)) ∧ ∀x (MalaHotpot(x) → Spicy(x)) ∧ ∀x (Product(x) ∧ From(x, bakedByMelissa) → Cupcake(x)) ∧ (Spicy(driedThaiChili) ∨ MalaHotpot(driedThaiChili)∨ ¬BakedSweet(driedThaiChili))

∀x (BakedSweet(x) → ¬Spicy(x))
∀x (Cupcake(x) → BakedSweet(x))
∀x (MalaHotpot(x) → Spicy(x))
∀x (Product(x) ∧ From(x, bakedByMelissa) → Cupcake(x))
Spicy(driedThaiChili) ∨ MalaHotpot(driedThaiChili)∨ ¬BakedSweet(driedThaiChili)


Dried Thai chilies are products of Baked by Melissa.

Product(driedThaiChili) ∧ From(driedThaiChili, bakedByMelissa)


Dried Thai chilies are not products of Baked by Melissa.

¬(Product(driedThaiChili) ∧ From(driedThaiChili, bakedByMelissa))


Dried Thai chilies are neither products of Baked by Melissa nor baked sweets.

¬(Product(driedThaiChili) ∧ From(driedThaiChili, bakedByMelissa)) ∧ ¬BakedSweet(driedThaiChili)


Dried Thai chilies are cupcakes and products of Baked by Melissa.

Cupcake(driedThaiChili) ∧ Product(driedThaiChili) ∧ From(driedThaiChili, bakedByMelissa)


If a restaurant is listed in Yelp's recommendations, then the restaurant has not received many negative reviews. All restaurants with a rating greater than four are listed in Yelp's recommendations. Some restaurants that do not provide take-out service receive many negative reviews. All restaurants that are popular among local residents have ratings greater than four. The Hamden Plaza Subway store has a rating greater than four, or it is popular among local residents.

∀x (ListedIn(x, yelpRecommendation) → ¬ReceiveManyNegativeReviews(x)) ∧ ∀x (∃y (HaveRating(x, y) ∧ GreaterThan(y, four)) → ListedIn(x, yelpRecommendation)) ∧ ∃x (¬Provide(x, takeOutService) ∧ ReceiveManyNegativeReviews(x)) ∧ ∀x (PopularAmong(x, localResidents) → ∃y (HaveRating(x, y) ∧ GreaterThan(y, four))) ∧ (∃y (HaveRating(hamdenPlazaSubway, y) ∧ GreaterThan(y, four)) ∨ PopularAmong(hamdenPlazaSubway, localResidents))

(1) Since FOL is not suitable to express ambiguous quantity as 'many' that should be treated as 'at least one' with an existential, it is replaced the formulation '∃y (NegativeReview(y) ∧ Receive(x, y))' with 'ReceiveManyNegativeReviews(x)'. Furthermore, the original formulation  '∀x ∃y (ListedIn(x, yelpRecommendation) → NegativeReview(y) ∧ ¬Receive(x, y))' is malformed since there can be a restaurant listed in the yelp reccomandation even when it has negative reviews if we consider as y a negative review that is not received by that specific restaurant.(2) Adjusted the intepretation of 'some' as at least two different entities (two-witness interpretation). (2) Premise 2: the original '∀x ∀y (HaveRating(x, y) ∧ GreaterThan(y, 4) → ListedIn(x, yelpRecommendation))' universally quantifies over y, but a more correct reading is existential (even because it is not specified that there is only on rating). (3) Premise 3: the original '∃x ∃y (¬Provide(x, takeOutService) → NegativeReview(y) ∧ Receive(x, y))' uses a conditional inside existential quantifiers, giving 'there exist x and y such that if x doesn't provide takeout then y is a negative review received by x' — nearly vacuously true, beacuse satisfies even if there are no restaurants than don't provide take-out service with negative reviews (because for holding, we only need to have a restaurant x that provide takeOutService).(4) Premise 5: the original uses ⊕ (exclusive disjunction) but the NL says 'has a rating greater than four, or it is popular among local residents' — a simple inclusive disjunction with no exclusivity implied. Also, the original wraps the left disjunct as '∀x (HaveRating(hamdenPlazaSubway, x) ∧ GreaterThan(x, 4) ⊕ ...)' using universal quantification, which is incorrect — the rating is an existential property. Corrected to inclusive ∨ with existential: '(∃y (HaveRating(hamdenPlazaSubway, y) ∧ GreaterThan(y, 4)) ∨ PopularAmong(hamdenPlazaSubway, localResidents))'.
  Replaced '4' with 'four' to not create conflict with z3-parser

∀x ∃y (ListedIn(x, yelpRecommendation) → NegativeReview(y) ∧ ¬Receive(x, y))
∀x ∀y (HaveRating(x, y) ∧ GreaterThan(y, 4) → ListedIn(x, yelpRecommendation))
∃x ∃y (¬Provide(x, takeOutService) → NegativeReview(y) ∧ Receive(x, y))
∀x ∀y (PopularAmong(x, localResidents) → HaveRating(x, y) ∧ GreaterThan(y, 4))
∀x (HaveRating(hamdenPlazaSubway, x) ∧ GreaterThan(x, 4) ⊕ PopularAmong(hamdenPlazaSubway, localResidents))


If the Hamden Plaza Subway store provides take-out service and receives many negative reviews, then its rating is greater than 4 and it does not provide take-out service.

(Provide(hamdenPlazaSubway, takeOutService) ∧ ReceiveManyNegativeReviews(hamdenPlazaSubway)) → (∃y (HaveRating(hamdenPlazaSubway, y) ∧ GreaterThan(y, four)) ∧ ¬Provide(hamdenPlazaSubway, takeOutService))

Three issues in the original FOL: (1) Unclosed parenthesis: the original '∃x ((TakeOutService(...) ∧ ... → ...)' has a misplaced opening parenthesis after ∃x, incorrectly scoping the entire conditional under the existential quantifier. The existential should only scope over the negative review variable x, not over the whole implication. Corrected by restructuring as a top-level conditional with ∃x scoped only over the NegativeReview conjunct. (2) The original consequent uses 'PopularAmong(hamdenPlazaSubway, localResidents)' but the NL says 'its rating is greater than 4' — these are entirely different properties. Corrected to 'RatingGreaterThan(hamdenPlazaSubway, 4)' to match the NL. (3) The original antecedent uses 'TakeOutService(hamdenPlazaSubway)' as a unary predicate (the store IS a takeout service) while the NL and consequent use 'Provide(hamdenPlazaSubway, takeOutService)' as a binary predicate (the store PROVIDES takeout service). Unified to use 'Provide(hamdenPlazaSubway, takeOutService)' in the antecedent for consistency with the consequent and the NL.

∃x ((TakeOutService(hamdenPlazaSubway) ∧ NegativeReview(x) ∧ Receive(hamdenPlazaSubway, x) → PopularAmong(hamdenPlazaSubway, localResidents) ∧ ¬Provide(hamdenPlazaSubway, takeOutService))


The Hamden Plaza Subway store provides take-out service or does not receive many negative reviews.

Provide(hamdenPlazaSubway, takeOutService) ∨ ¬ReceiveManyNegativeReviews(hamdenPlazaSubway)

ontology : two issues in the original: (1) The original wraps the entire disjunction under '∃x (...)', making it 'there exists a review x such that (Provide ∨ (NegativeReview(x) ∧ ¬Receive(...)))'. The existential is misplaced — 'Provide(hamdenPlazaSubway, takeOutService)' does not depend on x, and the second disjunct should express 'does not receive many negative reviews' as a negated existential rather than a positive existential with ¬Receive. Corrected to a top-level disjunction: 'Provide(hamdenPlazaSubway, takeOutService) ∨ ¬∃x (NegativeReview(x) ∧ Receive(hamdenPlazaSubway, x))'. (2) The label 'True' is derivable: premise 5 (corrected) gives either HaveRating > 4 or PopularAmong, both of which via premises 4→2 or directly via premise 2 yield ListedIn(hamdenPlazaSubway, yelpRecommendation), which by corrected premise 1 gives ¬∃y (NegativeReview(y) ∧ Receive(hamdenPlazaSubway, y)) — satisfying the second disjunct of the conclusion.

∃x (Provide(hamdenPlazaSubway, takeOutService) ∨ (NegativeReview(x) ∧ ¬Receive(hamdenPlazaSubway, x)))


The Hamden Plaza store does not provide take-out service.

The label 'Uncertain' is correct: nothing in the premises establishes whether or not Hamden Plaza Subway provides take-out service. Premise 3 asserts that SOME restaurants without take-out service receive negative reviews, but this does not constrain Hamden Plaza Subway specifically. The conclusion is neither provable nor refutable.


In superhero movies, the good guys always win. The Surprising Adventures of Sir Digby Chicken Caesar is a superhero movie. Good guys fight bad guys and vice versa. Sir Digby fights his nemesis. If a superhero movie is named after a character, that character is a good guy. The Surprising Adventures of Sir Digby Chicken Caesar is named after Sir Digby. If somebody wins a fight, the person they are fighting does not win. If a superhero movie is named after a character, that character is in the movie.

∀x ∀y (SuperheroMovie(x) ∧ In(y, x) ∧ GoodGuy(y) → Wins(y)) ∧ SuperheroMovie(theSurprisingAdventuresOfSirDigbyChickenCaesar) ∧ ∀x ∀y ((GoodGuy(x) ∧ BadGuy(y)) → (Fight(x, y) ∧ Fight(y, x))) ∧ Fight(sirDigby, sirDigbysNemesis) ∧ ∀x ∀y (SuperheroMovie(x) ∧ NamedAfter(x, y) → GoodGuy(y)) ∧ NamedAfter(theSurprisingAdventuresOfSirDigbyChickenCaesar, sirDigby) ∧ ∀x ∀y (Fight(x, y) ∧ Wins(x) → ¬Wins(y)) ∧ ∀x ∀y (SuperheroMovie(x) ∧ NamedAfter(x, y) → In(y, x))

(1) The original translation for Premise 3 is '∀x ∀y (GoodGuy(x) ∧ Fight(x, y) ↔ BadGuy(y) ∧ Fight(y, x))', that holds also when there are a good guy and a bad guy that don't fight each other (both the parts of the bi-implication are false); it is replaced with '∀x ∀y ((GoodGuy(x) ∧ BadGuy(y)) → (Fight(x, y) ∧ Fight(y, x)))', intended as ALL good guys fight bad guys (usually this is the menaning retained by this construct. (The conclusion labels are not affected by this formalization choice, even if some information that may be conceived by the author (like that if a good guy is fighting with another guy, then the latter is a bad guy) can lack with this formalization. [NOTE: if a different formalization doesn't affect the conclusion, this means that this formalization is not tested by the FOLIO approach!] (2) Premise 7: the original uses two different predicate names — 'Fights(x, y)' in the antecedent and 'Wins(y)' in the consequent, while premise 1 uses 'Wins(y)' and premise 4 uses 'Fight(x, y)' (without the trailing 's'). The predicate 'Fights' is a typo/inconsistency; corrected to 'Fight(x, y)' to unify with the predicate used in premises 3 and 4. (3) Premise 7: the original uses 'Win(x)' in the antecedent but 'Wins(y)' in the consequent — two different predicate names for the same concept. Unified to 'Wins' throughout for consistency with premise 1. (4) Constant casing: the original uses 'theSurprisingAdventuresofSirDigbyChickenCaesar' (lowercase 'of') in some places; normalised to 'theSurprisingAdventuresOfSirDigbyChickenCaesar' (camelCase 'Of') consistently throughout.

∀x ∀y (SuperheroMovie(x) ∧ In(y, x) ∧ GoodGuy(y) → Wins(y))
SuperheroMovie(theSurprisingAdventuresofSirDigbyChickenCaesar)
∀x ∀y (GoodGuy(x) ∧ Fight(x, y) ⟷ BadGuy(y) ∧ Fight(y, x))
Fight(sirDigby, sirDigbysNemesis)
∀x ∀y (SuperheroMovie(x) ∧ NamedAfter(x, y) → GoodGuy(y))
NamedAfter(theSurprisingAdventuresofSirDigbyChickenCaesar, sirDigby)
∀x ∀y (Fights(x, y) ∧ Win(x) → ¬Wins(y))
∀x ∀y (SuperheroMovie(x) ∧ NamedAfter(x, y) → In(y, x))


The original uses '¬Win(sirDigbysNemesis)' with the predicate 'Win' (without trailing 's'), inconsistent with 'Wins' used in premises 1 and 7. Corrected to '¬Wins(sirDigbysNemesis)' for consistency.


Books contain tons of knowledge. When a person reads a book, that person gains knowledge. If a person gains knowledge, they become smarter. Harry read the book 'Walden' by Henry Thoreau.

∀x (Book(x) → Contains(x, knowledge)) ∧ ∀x ∀y (Book(x) ∧ ReadBook(y, x) → Gains(y, knowledge)) ∧ ∀x (Gains(x, knowledge) → Smarter(x)) ∧ ReadBook(harry, walden) ∧ Book(walden)

Added Book(x) in the seecond premises. One consideration: premise 4 in the original reads 'ReadBook(harry, walden) ∧ Book(walden)' — while 'Book(walden)' is not explicitly stated as a separate premise in the NL ('Harry read the book Walden' implies walden is a book), it is a reasonable and necessary grounding fact for the chain of reasoning to work (premise 1 requires Book(x) to fire). The original FOL correctly includes it as a conjunct in premise 4, so it is retained. However, note that premise 2 uses 'ReadBook(x, y)' without requiring Book(y) in the antecedent — meaning any ReadBook fact triggers Gains(x, knowledge) regardless of whether y is a book. This is a minor semantic looseness inherited from the original but not corrected as it does not affect the conclusions.

∀x (Book(x) → Contains(x, knowledge))
∀x ∀y (ReadBook(x, y) → Gains(x, knowledge))
∀x (Gains(x, knowledge) → Smarter(x))
ReadBook(harry, walden) ∧ Book(walden)


Significant NL/FOL mismatch in the original.

The original FOL 'Gains(harry, knowledge)' would also be True (derivable from premise 4 via premise 2), but it corresponds to conclusion 89's content rather than conclusion 88's NL. The two conclusions appear to have had their FOL formulas swapped or mislabelled in the original dataset.


The original FOL uses 'GainKnowledge(x)' as a unary predicate, which is inconsistent with the binary predicate 'Gains(x, knowledge)' used throughout the premises. The NL 'has gained knowledge' corresponds directly to 'Gains(x, knowledge)'; corrected to '∀x (Smarter(x) → Gains(x, knowledge))' for consistency.

The label 'Uncertain' is correct: the premises only establish Smarter as a consequence of Gains (premise 3: Gains → Smarter), not the converse.


A natural language processing task can be categorized as either a language generation task or a language understanding task. If the output of a natural language processing task is a text sequence, then it is a language generation task. Machine translation is a natural language processing task. The output of machine translation is a text sequence.

∀x (NaturalLanguageProcessingTask(x) ↔ LanguageGenerationTask(x) ⊕ LanguageUnderstandingTask(x)) ∧ ∀x (NaturalLanguageProcessingTask(x) ∧ OutputSequence(x, text) → LanguageGenerationTask(x)) ∧ NaturalLanguageProcessingTask(machineTranslation) ∧ OutputSequence(machineTranslation, text)

Ontology. One correction made: premise 2 in the original uses 'NaturalLanguageProcessingTasks(x)' (plural, with trailing 's') as the predicate name, inconsistent with 'NaturalLanguageProcessingTask(x)' (singular) used in premises 1, 3 and the conclusions. This is a typo that would make premise 2 refer to a completely different predicate, disconnecting the reasoning chain. Corrected to 'NaturalLanguageProcessingTask(x)' throughout.

∀x (NaturalLanguageProcessingTask(x) → LanguageGenerationTask(x) ⊕ LanguageUnderstandingTask(x))
∀x (NaturalLanguageProcessingTasks(x) ∧ OutputSequence(x, text) → LanguageGenerationTask(x))
NaturalLanguageProcessingTask(machineTranslation)
OutputSequence(machineTranslation, text)


Machine translation is a language generation task.


Machine translation is a language understanding task.

The label 'False' is correct under the exclusive disjunction (⊕) reading of premise 1.


Pets are allowed in some managed buildings. A deposit is required to rent an apartment in a managed building. The security deposit can be either equal to the monthly rent at a managed building or more. Fluffy is Tom's cat. Cats are pets. The Olive Garden is a managed building. The monthly rent at the Olive Garden is $2000. $2000 is more than $1500. Tom will rent an apartment in a managed building if he is allowed to move in with Fluffy, and the security deposit is no more than $1500. If a managed building allows pets, then people are allowed to move in with a pet.

∃x (ManagedBuilding(x) ∧ AllowPet(x)) ∧ ∀x ∀y (ManagedBuilding(x) ∧ RentApartmentIn(y, x) → DepositRequired(y)) ∧ ∀x ∀y ∀z ((SecurityDeposit(x, z) ∧ ManagedBuilding(z) ∧ MonthlyRentAt(y, z)) → (MoreThan(x, y) ⊕ Equal(x, y))) ∧ Cat(fluffy) ∧ BelongTo(fluffy, tom) ∧ ∀x (Cat(x) → Pet(x)) ∧ ManagedBuilding(oliveGarden) ∧ MonthlyRentAt(uSD2000, oliveGarden) ∧ MoreThan(uSD2000, uSD1500) ∧ ∀x ∀y (ManagedBuilding(x) ∧ AllowedToMoveInWith(tom, x, fluffy) ∧ SecurityDeposit(y, x) ∧ ¬MoreThan(y, uSD1500) → RentApartmentIn(tom, x)) ∧ ∀x ∀y ∀z (ManagedBuilding(x) ∧ AllowPet(x) ∧ Pet(z) → AllowedToMoveInWith(y, x, z))

(1) It is used the two-witness choice to indicate 'some buildings': this makes the instance uniform to the other similar instances that used this grammatical construct (in the original dataset, the construct is treated inconsistently, one time with the two-witness choice, and another time with the single one). This doesn't affect the conclusions labels, so it is not considered as a correction. (2) Fixed the unclosed parenthesis in the original. (3) Premise 3 also implicitly treats SecurityDeposit as a unary predicate (one argument), but conclusion 95 uses it as a binary predicate 'SecurityDeposit(x, oliveGarden)'. This inconsistency is resolved by considering it as a binary predicate also in the premises.

either_or, 'Tom will rent an apartment in a managed building if he is allowed to move in with Fluffy, and the security deposit is no more than $1500.' should be treated as an implication of sufficient conditions or we mean that if he is not allowed to move in with Fluffy then he will not rent the house? This affect the concl_166_93

∃x (ManagedBuilding(x) ∧ AllowPet(x))
∀x ∀y (ManagedBuilding(x) ∧ RentApartmentIn(y, x) → DepositRequired(y))
∀x ∀y ∀z (SecurityDeposit(x) ∧ ManagedBuilding(z) ∧ MonthlyRentAt(y, z) → (MoreThan(x, y) ⊕ Equal(x, y))
Cat(fluffy) ∧ BelongTo(fluffy, tom)
∀x (Cat(x) → Pet(x))
ManagedBuilding(oliveGarden)
MonthlyRentAt(uSD2000, oliveGarden)
MoreThan(uSD2000, uSD1500)
∀x ∀y (ManagedBuilding(x) ∧ AllowedToMoveInWith(tom, x, fluffy) ∧ SecurityDeposit(y) ∧ ¬MoreThan(y, uSD1500) → RentApartmentIn(tom, x))
∀x ∀y ∀z (ManagedBuilding(x) ∧ AllowPet(x) ∧ Pet(z) → AllowedToMoveInWith(y, x, z))


Tom will rent an apartment in The Olive Garden.

see correction_explanation of story_166 that explains the label mismatch


Tom is allowed to move into an apartment in The Olive Garden with Fluffy.


The security deposit at the Olive Garden is either $2000 or more.

∀x (SecurityDeposit(x, oliveGarden) → (MoreThan(x, uSD2000) ⊕ Equal(x, uSD2000)))

The binary form is more precise and appropriate for conclusion 95 (it specifies the deposit AT Olive Garden), and is arguably the better modelling choice throughout.

∀x (SecurityDeposit(x, oliveGarden) → MoreThan(x, uSD2000) ⊕ Equal(x, uSD2000))


All business organizations are legal entities. All companies are business organizations. All private companies are companies. All legal entities are created under law. All legal entities have legal obligations. If the Harvard Weekly Book Club is created under law, then it is not a private company.

∀x (BusinessOrganization(x) → LegalEntity(x)) ∧ ∀x (Company(x) → BusinessOrganization(x)) ∧ ∀x (PrivateCompany(x) → Company(x)) ∧ ∀x (LegalEntity(x) → CreatedUnderLaw(x)) ∧ ∀x (LegalEntity(x) → LegalObligation(x)) ∧ (CreatedUnderLaw(harvardWeeklyBookClub) → ¬PrivateCompany(harvardWeeklyBookClub))

∀x (BusinessOrganization(x) → LegalEntity(x))
∀x (Company(x) → BusinessOrganization(x))
∀x (PrivateCompany(x) → Company(x))
∀x (LegalEntity(x) → CreatedUnderLaw(x))
∀x (LegalEntity(x) → LegalObligation(x))
CreatedUnderLaw(harvardWeeklyBookClub) → ¬PrivateCompany(harvardWeeklyBookClub)


The Harvard Weekly Book Club has legal obligations.


The Harvard Weekly Book Club has legal obligations and is a private company.

The label 'False' is correct: as established in the story note, PrivateCompany(harvardWeeklyBookClub) is refutable via the contradiction chain (PrivateCompany → ... → CreatedUnderLaw → ¬PrivateCompany). Since the second conjunct is False, the conjunction is False regardless of the truth value of LegalObligation(harvardWeeklyBookClub).


If the Harvard Weekly Book Club is a private company, then it either has legal obligations or is created under law.

PrivateCompany(harvardWeeklyBookClub) → (LegalObligation(harvardWeeklyBookClub) ⊕ CreatedUnderLaw(harvardWeeklyBookClub))

The original FOL omits the outer parentheses around the full implication, which could cause precedence ambiguity with the ⊕ operator potentially binding more tightly than →. Added explicit parentheses to scope the implication unambiguously

PrivateCompany(harvardWeeklyBookClub) → LegalObligation(harvardWeeklyBookClub) ⊕ CreatedUnderLaw(harvardWeeklyBookClub)


Animals are either invertebrates or vertebrates. All animals with backbones reproduce by male-and-female mating. All vertebrate animals have a backbone. All bees do not reproduce by male-and-female mating. All queen bees are bees. Harry is a bee.

∀x (Animal(x) → Invertebrate(x) ⊕ Vertebrate(x)) ∧ ∀x (Animal(x) ∧ HasBackbone(x) → ReproduceBy(x, maleAndFemaleMating)) ∧ ∀x (Animal(x) ∧ Vertebrate(x) → HasBackbone(x)) ∧ ∀x (Bee(x) → ¬ReproduceBy(x, maleAndFemaleMating)) ∧ ∀x (QueenBee(x) → Bee(x)) ∧ Bee(harry) ∧ ∀x (Bee(x) →  Animal(x))

Renamed only constants and predicate symbols (no correction, but suggestion). Added the formula '∀x (Bee(x) →  Animal(x))' because needed for the conclusions 100 and 101 to work with the label given by the authors.

∀x (Animal(x) → Invertebrate(x) ⊕ Vertebrate(x))
∀x (Animal(x) ∧ With(x, backbone) → ReproduceBy(x, male-and-femaleMating))
∀x (Animal(x) ∧ Vertebrate(x) → With(x, backbone))
∀x (Bee(x) → ¬ReproduceBy(x, male-and-femaleMating))
∀x (QueenBee(x) → Bee(x))
Bee(harry)


Harry is an invertebrate or a queen bee.

The label 'True' is only derivable if Bee(x) → Animal(x) is assumed as a background axiom (not stated in the premises). If this assumption is made, then Invertebrate(harry) follows and the disjunction is True. Without it, the label should arguably be 'Uncertain'. This missing axiom is a gap in the premise set and we added it in the premises.


If Harry is either both a vertebrate and an animal with a backbone, or neither a vertebrate nor an animal with a backbone, then Harry is neither an invertebrate nor a queen bee.

((Vertebrate(harry) ∧ HasBackbone(harry)) ∨ (¬Vertebrate(harry) ∧ ¬HasBackbone(harry))) → (¬Invertebrate(harry) ∧ ¬QueenBee(harry))

Three issues in the original: (1) The original FOL '¬(Invertebrate(harry) ⊕ With(harry, backbone))' uses Invertebrate instead of Vertebrate in the antecedent — a predicate substitution error inconsistent with the NL which refers to 'vertebrate'. Corrected to '¬(Vertebrate(harry) ⊕ HasBackbone(harry))'. (2) The predicate 'With(harry, backbone)' corrected to 'HasBackbone(harry)' as per the premise corrections. (3) Added outer parentheses for unambiguous scoping of the full conditional.

The label 'False' requires analysis: the antecedent ¬(Vertebrate(harry) ⊕ HasBackbone(harry)) is true when Harry is either both a vertebrate and has a backbone, or neither. As established in conclusion 100 (assuming Animal(harry)), Harry is an invertebrate — meaning ¬Vertebrate(harry) and, since HasBackbone implies being reproduced by male-female, then it can't be. This means Vertebrate and HasBackbone are both false, satisfying the 'neither' case, making the antecedent true. The consequent requires ¬Invertebrate(harry) ∧ ¬QueenBee(harry). But Invertebrate(harry) holds (from the Animal(harry) assumption chain), making ¬Invertebrate(harry) false, and thus the consequent false. A true antecedent with a false consequent makes the conditional False — consistent with the label.

¬(Invertebrate(harry) ⊕ With(harry, backbone)) → ¬Invertebrate(harry) ∧ ¬QueenBee(harry)


People who are born in a multiple birth with siblings spend a lot of time hanging out with and playing with their siblings. If people have siblings who were born together, then they were born in a multiple birth. If people complain often about annoying siblings, then they have siblings who were born together. If people live at home, then they do not live with strangers. If people spend a lot of time hanging out with and playing with their siblings, then they often live at home. Luke either is a baby born in a multiple birth and lives with strangers, or is not a baby born in a multiple birth and does not live with strangers.

∀x (BornIn(x, multipleBirth) → SpendTimePlayingWith(x, siblings)) ∧ ∀x (∃y (Sibling(x, y) ∧ BornTogether(x, y)) → BornIn(x, multipleBirth)) ∧ ∀x (ComplainAboutOften(x, annoyingSiblings) → ∃y (Sibling(x, y) ∧ BornTogether(x, y))) ∧ ∀x (LiveAt(x, home) → ¬LiveWith(x, strangers)) ∧ ∀x (SpendTimePlayingWith(x, siblings) → LiveAt(x, home)) ∧ ¬(BornIn(luke, multipleBirth) ⊕ LiveWith(luke, strangers))

Four corrections made: (1) Premise 1: the original uses 'SpendTimePlayingWith(x, sibling)' (singular 'sibling') while premise 5 uses 'SpendTimeplayingWith(x, sibling)' (lowercase 'p' in 'playing'). Both refer to the same relation; unified to 'SpendTimePlayingWith(x, siblings)' with consistent camelCase and plural 'siblings' to match the NL 'their siblings'. (2) Premise 5: 'SpendTimeplayingWith' (lowercase 'p') corrected to 'SpendTimePlayingWith' (uppercase 'P') for camelCase consistency. (3) BornTogether has now arity 2 and not 1.

∀x (BornIn(x, multipleBirth) → SpendTimePlayingWith(x, sibling))
∀x (∃y (Sibling(x, y) ∧ BornTogether(y)) → BornIn(x, multipleBirth))
∀x (ComplainAboutOften(x, annoyingSiblings) → ∃y (Sibling(x, y) ∧ BornTogether(y)))
∀x (LiveAt(x, home) → ¬LiveWith(x, strangers))
∀x (SpendTimeplayingWith(x, sibling) → LiveAt(x, home))
¬(BornIn(luke, multipleBirth) ⊕ LiveWith(luke, strangers))


Luke spends a lot of time hanging out and playing with his siblings.


Luke complains often about annoying siblings.

The label 'False' is correct via the following chain: if ComplainAboutOften(luke, annoyingSiblings) held, then by premise 3, ∃y (Sibling(luke, y) ∧ BornTogether(y)), and by premise 2, BornIn(luke, multipleBirth). By premise 1, SpendTimePlayingWith(luke, siblings). By premise 5, LiveAt(luke, home). By premise 4, ¬LiveWith(luke, strangers). But premise 6 (XNOR) with BornIn(luke, multipleBirth) = true requires LiveWith(luke, strangers) = true — contradiction. Therefore ComplainAboutOften(luke, annoyingSiblings) is refutable.


Luke is neither born in a multiple birth nor complains often about annoying siblings.

¬BornIn(luke, multipleBirth) ∧ ¬ComplainAboutOften(luke, annoyingSiblings)

Two issues in the original: (1) The original FOL uses '¬(BornIn(luke, multipleBirth) ∨ ComplainAbout(luke, often, annoyingSiblings))' — equivalent by De Morgan to '¬BornIn ∧ ¬ComplainAbout', which correctly represents 'neither... nor...'. However, the predicate name 'ComplainAbout(luke, often, annoyingSiblings)' is a ternary decomposition inconsistent with the binary 'ComplainAboutOften(x, annoyingSiblings)' used in premise 3 and conclusion 103. Corrected to use the binary predicate.

¬(BornIn(luke, multipleBirth) ∨ ComplainAbout(luke, often, annoyingSiblings) )


It costs $205 to take the GRE test, which is cheaper than $300. ETS provides financial aid to those GRE applicants who prove economic hardship. Those living in single-parent families or having few resources available to them can prove economic hardship. Tom lives in a single-parent family. Tom's dad has been out of work, and Tom has few resources available to him. Tom is applying to take the GRE test.

Cost(gre, uSD205) ∧ Cheaper(uSD205, uSD300) ∧ ∀x (ApplicantOf(x, gre) ∧ Prove(x, economicHardship) → ProvideTo(ets, financialAid, x)) ∧ ∀x (LivingIn(x, singleParentFamily) ∨ AvailableTo(fewResources, x) → Prove(x, economicHardship)) ∧ LivingIn(tom, singleParentFamily) ∧ OutOfWork(tomsDad) ∧ AvailableTo(fewResources, tom) ∧ ApplicantOf(tom, gre)

Some corrections made: (1) Constant naming: the original uses bare numerals '205' and '300' as FOL constants: changed to 'uSD205' and 'uSD300' with a prefix to comply with FOL syntax. Similarly 'gRE' uses an inconsistent capitalisation pattern; normalised to 'gre' throughout. (2) Premise 2: the original uses 'ProvideTo(ets, financialAid, x)' as a ternary predicate, while conclusion 105 uses 'ProvidesFinancialAidTo(eTS, tom)' as a binary predicate — two different predicates for the same relation. Unified to 'ProvideTo(ets, financialAid, x)' in the premises (ternary form retained as it encodes both the provider, the aid type, and the recipient), and conclusion 105 corrected accordingly.

Cost(gRE, 205) ∧ Cheaper(205, 300)
∀x (ApplicantOf(x, gre) ∧ Prove(x, economicHardship) → ProvideTo(ets, financialAid, x))
∀x (LivingIn(x, singleParentFamily) ∨ AvailableTo(fewResources, x) → Prove(x, economicHardship))
LivingIn(tom, singleParentFamily)
OutOfWork(tomsDad) ∧ AvailableTo(fewResources, tom)
ApplicantOf(tom, gre)


The original FOL uses 'ProvidesFinancialAidTo(eTS, tom)' — a binary predicate inconsistent with 'ProvideTo(ets, financialAid, x)' used in premise 2. Corrected to 'ProvideTo(ets, financialAid, tom)' for consistency. Also 'eTS' → 'ets' for consistent casing.


It costs below US $300 to take the GRE test.

Two issues in the original: (1) The original uses a free variable 'x' in 'Cost(gRE, x) ∧ Cheaper(x, 300)' without any quantifier — x is unbound, making the formula not a well-formed closed FOL sentence. Corrected to an existential: '∃x (Cost(gre, x) ∧ Cheaper(x, uSD300))'.


No one taking the GRE test has financial aid provided to them by something.

¬∃x ∃y (ApplicantOf(x, gre) ∧ ProvideTo(y, financialAid, x))

Three issues in the original: (1) The original uses 'Applicant(x, gRE)' — a different predicate name from 'ApplicantOf(x, gre)' used in premises 2 and 6. Corrected to 'ApplicantOf(x, gre)'. (2) The original uses 'ProvidesFinancialAidTo(y, x)' — binary predicate inconsistent with the ternary 'ProvideTo(ets, financialAid, x)' in premise 2. Corrected to 'ProvideTo(y, financialAid, x)' to match the premise structure while keeping y as a variable for the provider. (3) 'gRE' → 'gre' for casing consistency.

The label 'False' is correct: as established in conclusion 105, ProvideTo(ets, financialAid, tom) is derivable, and ApplicantOf(tom, gre) is a premise. Therefore ∃x ∃y (ApplicantOf(x, gre) ∧ ProvideTo(y, financialAid, x)) holds (with x = tom, y = ets), directly refuting the universal negation.

¬(∃x ∃y (Applicant(x, gRE) ∧ ProvidesFinancialAidTo(y, x)))


If people spill a lot of food on their clothing, then they are not notably tidy. All clumsy foodies who go out frequently to find new food restaurants to try are people who spill a lot of food on their clothing. If people are cleanly, then they are notably tidy. If people value order and spotlessness, then they are cleanly. All people whose families prioritize order and spotlessness themselves value order and spotlessness. Peter is either a person who spills a lot of food on his clothing and is cleanly, or a person who does not spill a lot of food on his clothing and is not cleanly.

∀x (SpillOn(x, aLotOfFood, clothing) → ¬NotablyTidy(x)) ∧ ∀x (ClumsyFoodie(x) ∧ GoOutFrequentlyToFindToTry(x, newFoodRestaurant) → SpillOn(x, aLotOfFood, clothing)) ∧ ∀x (Cleanly(x) → NotablyTidy(x)) ∧ ∀x (Value(x, order) ∧ Value(x, spotlessness) → Cleanly(x)) ∧ ∀x (∃y (FamilyOf(y, x) ∧ Prioritize(y, order) ∧ Prioritize(y, spotlessness)) → Value(x, order) ∧ Value(x, spotlessness)) ∧ ¬(SpillOn(peter, aLotOfFood, clothing) ⊕ Cleanly(peter))

Four corrections made: (1) Premise 1: the original uses 'Spill(x, aLotOfFood, clothing)' as the predicate, while premise 2 uses 'SpillOn(x, aLotOfFood, clothing)'. These must refer to the same relation; unified to 'SpillOn(x, aLotOfFood, clothing)' throughout for consistency. (2) Premise 5: the original has an unclosed parenthesis. (3) Premise 6: the original uses 'Spill(peter, aLotOfFood, clothing)' — again unified to 'SpillOn(peter, aLotOfFood, clothing)'.

∀x (Spill(x, aLotOfFood, clothing) → ¬NotablyTidy(x))
∀x (ClumsyFoodie(x) ∧ GoOutFrequentlyToFindToTry(x, newFoodRestaurant) → SpillOn(x, aLotOfFood, clothing))
∀x (Cleanly(x) → NotablyTidy(x))
∀x (Value(x, order) ∧ Value(x, spotlessness) → Cleanly(x))
∀x ((∃y (FamilyOf(y, x) ∧ Prioritize(y, order) ∧ Prioritize(y, spotlessness)) → Value(x, order) ∧ Value(x, spotlessness))
¬(Spill(peter, aLotOfFood, clothing) ⊕ Cleanly(peter))


Peter is either a clumsy foodie who goes out frequently to find new food restaurants to try or he is someone whose family prioritizes order and spotlessness.

(ClumsyFoodie(peter) ∧ GoOutFrequentlyToFindToTry(peter, newFoodRestaurant)) ⊕ (∃y (FamilyOf(y, peter) ∧ Prioritize(y, order) ∧ Prioritize(y, spotlessness)))

The original formalization lacked the information 'who goes out frequently to find new food restaurants to try'. It is also used ⊕ to express the either ... or, even if it doesn't change the label of the reason why the label 'false' is right.

The label 'False' is correct under the following reasoning: as established in conclusion 108's note, case (b) holds — ¬SpillOn(peter) ∧ ¬Cleanly(peter). If ClumsyFoodie(peter) ∧ GoOutFrequentlyToFindToTry(peter, newFoodRestaurant) held, premise 2 would give SpillOn(peter, aLotOfFood, clothing), contradicting ¬SpillOn(peter). So ClumsyFoodie(peter) is refutable. Similarly, if ∃y (FamilyOf(y, peter) ∧ Prioritize(y, order) ∧ Prioritize(y, spotlessness)) held, premise 5 would give Value(peter, order) ∧ Value(peter, spotlessness), then premise 4 would give Cleanly(peter), contradicting ¬Cleanly(peter). Therefore both disjuncts are refutable, making the disjunction False.

ClumsyFoodie(peter) ∨ (∃y (FamilyOf(y, peter) ∧ Prioritize(y, order) ∧ Prioritize(y, spotlessness)))


Peter is either both a clumsy foodie who goes out frequently to find new food restaurants to try and is someone whose family prioritizes order and spotlessness, or he neither is a clumsy foodie who goes out frequently to find new food restaurants to try nor does his family prioritize order and spotlessness.

¬((ClumsyFoodie(peter) ∧ GoOutFrequentlyToFindToTry(peter, newFoodRestaurant)) ⊕ (∃y (FamilyOf(y, peter) ∧ Prioritize(y, order) ∧ Prioritize(y, spotlessness))))

The original formalization lacked the information 'who goes out frequently to find new food restaurants to try'.

The label 'True' is correct: the NL expresses the XNOR pattern ('both... or neither...'), correctly encoded as ¬(P ⊕ Q). As established in conclusion 109, both ClumsyFoodie(peter) and ∃y (FamilyOf(y, peter) ∧ Prioritize(y, order) ∧ Prioritize(y, spotlessness)) are refutable — meaning both P and Q are false. Under XNOR, P = false and Q = false satisfies ¬(P ⊕ Q) since neither holds (the 'neither' case of 'both or neither'). The conclusion is therefore True.

¬(ClumsyFoodie(peter) ⊕ (∃y (FamilyOf(y, peter) ∧ Prioritize(y, order) ∧ Prioritize(y, spotlessness))))


All fir trees are evergreens. Some objects of worship are fir trees.

∀x (FirTree(x) → Evergreen(x)) ∧ ∃x (ObjectOfWorship(x) ∧ FirTree(x))

∀x (FirTree(x) → Evergreen(x))
∃x ∃y (ObjectOfWorship(x) ∧ ObjectOfWorship(y) ∧ FirTree(x) ∧ FirTree(y) ∧ ¬(x=y))


Some evergreens are not objects of worship.

∃x ∃y (Evergreen(x) ∧ Evergreen(y) ∧ ¬ObjectOfWorship(x) ∧ ¬ObjectOfWorship(y) ∧ ¬(x=y))


The Picuris Mountains are a mountain range in New Mexico or Texas. Juan de Onate visited the Picuris Mountains. The Harding Pegmatite Mine, located in the Picuris Mountains, was donated. There are no mountain ranges in Texas that have mines that have been donated.

MountainRange(thePicurisMountains) ∧ (In(thePicurisMountains, newMexico) ∨ In(thePicurisMountains, texas)) ∧ Visited(juanDeOnate, thePicurisMountains) ∧ In(hardingPegmatiteMine, thePicurisMountains) ∧ Mine(hardingPegmatiteMine) ∧ Donated(hardingPegmatiteMine) ∧ ∀x ∀y (Mine(x) ∧ Donated(x) ∧ In(x, y) ∧ MountainRange(y) → ¬In(y, texas))

MountainRange(thePicurisMountains) ∧ (In(thePicurisMountains, newMexico) ⊕ In(thePicurisMountains, texas))
Visited(juanDeOnate, thePicurisMountains)
In(hardingPegmatiteMine, thePicurisMountains) ∧ Mine(hardingPegmatiteMine) ∧ Donated(hardingPegmatiteMine)
∀x ∀y (Mine(x) ∧ Donated(x) ∧ In(x, y) ∧ MountainRange(y) → ¬In(y, texas))


Juan de Onate visited a mountain range in New Mexico.

∃x (Visited(juanDeOnate, x) ∧ MountainRange(x) ∧ In(x, newMexico))


Juan de Onate visited a mountain range in Texas.

∃x (Visited(juanDeOnate, x) ∧ MountainRange(x) ∧ In(x, texas))


The Harding Pegmatite Mine is not located in a mountain range in New Mexico.

∀x (In(hardingPegmatiteMine, x) → ¬(MountainRange(x) ∧ In(x, newMexico)))


If a man is taller than another man, the taller one can block the other's shooting. Michael is a man who is taller than everyone else in his class. If person x is taller than person y, and person y is taller than person z, then x is taller than z. Peter is a man who is taller than Michael. Michael can block any shooting from a person as long as the person does not jump when shooting. Michael cannot block Windy's shooting. Every shooter who can jump when shooting is a great shooter.

∀x ∀y (Man(x) ∧ Man(y) ∧ Taller(x, y) → CanBlock(x, y)) ∧ Man(michael) ∧ SameClass(michael, michael) ∧ ∀x (Man(x) ∧ SameClass(x, michael) ∧ ¬(x = michael) → Taller(michael, x)) ∧ ∀x ∀y ∀z (Taller(x, y) ∧ Taller(y, z) → Taller(x, z)) ∧ Man(peter) ∧ Taller(peter, michael) ∧ ∀x (¬JumpWhenShooting(x) → CanBlock(michael, x)) ∧ ¬CanBlock(michael, windy) ∧ ∀x (JumpWhenShooting(x) → GreatShooter(x))

Two corrections made: (1) Premise 2: the original does not assert Man(michael) or SameClass(michael, michael) — Michael's own class membership and being a man are implied by the NL but never stated. Added but not needed for the conclusions.

∀x ∀y (Man(x) ∧ Man(y) ∧ Taller(x, y) → CanBlock(x, y))
∀x (Man(x) ∧ SameClass(x, michael) ∧ ¬(x=michael) → Taller(michael, x))
∀x ∀y ∀z ((Taller(x, y) ∧ Taller(y, z)) → Taller(x, z))
Man(peter) ∧ Taller(peter, michael)
∀x (¬JumpWhenShooting(x) → CanBlock(michael, x))
¬CanBlock(michael, windy)
∀x (JumpWhenShooting(x) → GreatShooter(x))


Peter is shorter than a man in Michael's class.

∃x (Man(x) ∧ SameClass(x, michael) ∧ Taller(x, peter))

F := 'A shorter than B' is not really equivalent to G := 'B not taller than A' since G holds also when A and B have the same size, but F not. So, it should be better to formalize it in the proposed way

∃x (Man(x) ∧ SameClass(x, michael) ∧ ¬Taller(peter, x))