DSAVlab-UNIUD/FOLIO_validation-curated huggingface dataset:
Sentence: 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.
FOL: ∀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)))
Correction notes: 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'.
Sentence: Bonnie performs in school talent shows often.
FOL: PerformOftenIn(bonnie, schoolTalentShow)
Label: Uncertain
Correction notes: 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.
Sentence: 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.
FOL: (((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)))
Label: True
Correction notes: 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.
Sentence: 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.
FOL: (Chaperone(bonnie, highSchoolDance) ⊕ PerformOftenIn(bonnie, schoolTalentShow)) → ((YoungChild(bonnie) ∨ Teenager(bonnie)) ∧ WishToFurther(bonnie, academicCareer) ∧ WishToFurther(bonnie, educationalOpportunity) ∧ InActive(bonnie) ∧ Disinterested(bonnie) ∧ MemberOf(bonnie, community))
Label: False
Correction notes: Predicate Match: Corrected 'Perform' to 'PerformOftenIn' to properly link with the premises. Missing Entities: Added 'WishToFurther(bonnie, educationalOpportunity)' to strictly align with the NL.
Sentence: 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.
FOL: ∀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))
Correction notes: 1. 'Everyone who has lunch in the company' is relativized to 'everyone that is an employee', but this restriction is not present in NL
Sentence: James has lunch in the company.
FOL: HasLunch(james, company)
Label: Uncertain
Sentence: James does not have lunch in the company.
FOL: ¬HasLunch(james, company)
Label: Uncertain
Sentence: 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.
FOL: ((Manager(james) ∨ ¬In(james, homeCountry)) → ¬(HasLunch(james, home) ∨ Work(james, home))) ∧ ((HasLunch(james, home) ∨ Work(james, home)) → ¬(Manager(james) ∨ ¬In(james, homeCountry)))
Label: Uncertain
Correction notes: 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.
Sentence: 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.
FOL: ∃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))
Correction notes: 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'.
Sentence: There is an animal.
FOL: ∃x (Animal(x))
Label: True
Sentence: No one gets the flu.
FOL: ∀x (Human(x) → ¬Get(x, flu))
Label: Uncertain
Correction notes: 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
Sentence: Symptoms of Monkeypox include coughing.
FOL: ∃x (SymptomOf(x, monkeypoxVirus) ∧ Coughing(x))
Label: Uncertain
Correction notes: 1. Typo: Corrected 'SymptonOf' to 'SymptomOf'.
Sentence: 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.
FOL: ∀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)
Correction notes: 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.
Sentence: Tom is an Ocellated wild turkey.
FOL: OcellatedWildTurkey(tom)
Label: True
Sentence: Tom is an Eastern wild turkey.
FOL: EasternWildTurkey(tom)
Label: False
Sentence: Joey is a wild turkey.
FOL: WildTurkey(joey)
Label: Uncertain
Sentence: 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.
FOL: 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))
Correction notes: 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.
Sentence: The Legend of Zelda is on the Top 10 list.
FOL: InTop10(theLegendofZelda)
Label: True
Correction notes: Standardized the predicate 'Top10' to 'InTop10' to correctly link with the adjusted premises.
Sentence: FIFA 22 is made by a Japanese video game company.
FOL: ∃x(Created(x, fifa22) ∧ Japanese(x) ∧ VideoGameCompany(x))
Label: Uncertain
Sentence: The Legend of Zelda is not on the Top 10 list.
FOL: ¬InTop10(theLegendofZelda)
Label: False
Correction notes: Standardized the predicate 'Top10' to 'InTop10' to correctly link with the adjusted premises.
Sentence: All squares are four-sided. All four-sided things are shapes.
FOL: ∀x (Square(x) → FourSided(x)) ∧ ∀x (FourSided(x) → Shape(x))
Sentence: All squares are shapes.
FOL: ∀x (Square(x) → Shape(x))
Label: True
Sentence: 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.
FOL: ∀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)
Correction notes: 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.
Sentence: Rockie is a turtle.
FOL: Turtle(rockie)
Label: Uncertain
Sentence: Rockie is not a turtle.
FOL: ¬Turtle(rockie)
Label: Uncertain
Sentence: Rockie is a turtle or cute.
FOL: Turtle(rockie) ∨ Cute(rockie)
Label: True
Sentence: If Rockie is not both a turtle and a squirrel, then Rockie is either cute or skittish.
FOL: ¬(Turtle(rockie) ∧ Squirrel(rockie)) → (Cute(rockie) ⊕ Skittish(rockie))
Label: True
Correction notes: Added parentheses because of operator precedences
Sentence: If Rockie is cute and calm, then Rockie is a skittish turtle.
FOL: (Cute(rockie) ∧ Calm(rockie)) → (Turtle(rockie) ∧ Skittish(rockie))
Label: False
Correction notes: Added parentheses (even if not necessary): it is not considered a correction.
Sentence: "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.
FOL: 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))
Correction notes: 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.
Sentence: Karen will share "Stranger Things" with Lisa.
FOL: ShareWith(karen, strangerThings, lisa)
Label: True
Sentence: "Black Mirror" is popular.
FOL: Popular(blackMirror)
Label: False
Sentence: Karen will share "Black Mirror" with Lisa.
FOL: ShareWith(karen, blackMirror, lisa)
Label: Uncertain
Sentence: 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.
FOL: 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))
Correction notes: 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.
Sentence: Beijing hosted both the 2008 Summer Olympics and the Winter Olympics.
FOL: Hosted(beijing, summerOlympics2008) ∧ Hosted(beijing, winterOlympics)
Label: True
Correction notes: 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.
Sentence: Beijing is located in southern China.
FOL: LocatedIn(beijing, southernChina)
Label: Uncertain
Sentence: Beijing is the second largest Chinese city.
FOL: SecondLargestChineseCity(beijing)
Label: Uncertain
Sentence: 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.
FOL: ∀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))
Correction notes: 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.
Sentence: Marvin is an alien.
FOL: Alien(marvin)
Label: Uncertain
Sentence: Marvin is neither a human nor from Mars.
FOL: ¬Human(marvin) ∧ ¬From(marvin, mars)
Label: True
Sentence: If Marvin is not from Mars, then Marvin is a human.
FOL: ¬From(marvin, mars) → Human(marvin)
Label: False
Sentence: 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.
FOL: ∀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))
Correction notes: 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.
Sentence: Djokovic is a Grand Slam champion.
FOL: GrandSlamChampion(djokovic)
Label: Uncertain
Correction notes: 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.
Sentence: Djokovic lives in a tax haven.
FOL: ∃x (TaxHaven(x) ∧ LiveIn(djokovic, x))
Label: True
Correction notes: TaxHaven is a predicate now
Sentence: Djokovic does not live in a tax haven.
FOL: ¬ (∃x (TaxHaven(x) ∧ LiveIn(djokovic, x)))
Label: False
Sentence: 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.
FOL: ProfessionalWrestlingStable(diamondMine) ∧ In(diamondMine, wWE) ∧ Leads(roderickStrong, diamondMine) ∧ Includes(diamondMine, creedBrothers) ∧ Includes(diamondMine, ivyNile) ∧ Feuds(imperium, diamondMine)
Sentence: Roderick Strong leads a professional wrestling stable.
FOL: ∃x (Leads(roderickStrong, x) ∧ ProfessionalWrestlingStable(x))
Label: True
Correction notes: 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.
Sentence: Roderick Strong leads the Creed Brothers.
FOL: Leads(roderickStrong, creedBrothers)
Label: Uncertain
Correction notes: Same casing issue as conclusion 34: 'roderickstrong' → 'roderickStrong' and 'creedbrothers' → 'creedBrothers', to align with camelCase convention used in the premises.
Sentence: Imperium doesn't have a feud with a professional wrestling stable that includes Ivy Nile.
FOL: ∀x ((ProfessionalWrestlingStable(x) ∧ Includes(x, ivyNile)) → ¬Feuds(imperium, x))
Label: False
Correction notes: Casing issue: 'ivynile' → 'ivyNile' to align with camelCase convention used in the premises.
Sentence: 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.
FOL: 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)))
Correction notes: 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.
Sentence: Beethoven is a composer.
FOL: Composer(beethoven)
Correction notes: 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.
Sentence: Some orchestras premiered music pieces.
FOL: ∃x ∃y (Orchestra(x) ∧ MusicPiece(y) ∧ Premiered(x, y))
Label: True