DSAVlab-UNIUD/FOLIO_validation-curated huggingface dataset:
Sentence: Machine Learning algorithms can be categorized as supervised learning, unsupervised learning, and reinforcement learning. Unsupervised learning algorithms do not require labeled data. The state-of-the-art text summarization model is trained with machine learning algorithms. Reinforcement learning is not used to train the state-of-the-art text summarization model. The Machine Learning algorithm for training text summarization models requires labeled data.
FOL: ∀x (MachineLearningAlgorithm(x) ↔ (SupervisedLearningAlgorithm(x) ∨ UnsupervisedLearningAlgorithm(x) ∨ ReinforcementLearningAlgorithm(x))) ∧ ∀x (UnsupervisedLearningAlgorithm(x) → ¬Require(x, labeledData)) ∧ (∃x (MachineLearningAlgorithm(x) ∧ TrainedWith(stateOfTheArtTextSummarizationModel, x)) ∧ ∀x (TrainedWith(stateOfTheArtTextSummarizationModel, x) → MachineLearningAlgorithm(x))) ∧ ∀x (ReinforcementLearningAlgorithm(x) → ¬TrainedWith(stateOfTheArtTextSummarizationModel, x)) ∧ ∀x ((MachineLearningAlgorithm(x) ∧ TrainedWith(stateOfTheArtTextSummarizationModel, x)) → Require(x, labeledData))
Correction notes: Two corrections made: (1) Premise 3: the original reads '∀x (TrainedWith(stateOfTheArtTextSummarizationModel, x) → MachineLearningAlgorithm(x))' — this says that anything the model is trained WITH is a machine learning algorithm. While this is a reasonable interpretation, the NL says 'the model IS TRAINED WITH machine learning algorithms', asserting the existence of at least one such algorithm: '∃x (MachineLearningAlgorithm(x) ∧ TrainedWith(stateOfTheArtTextSummarizationModel, x))'. The universal conditional form of the original would not allow deriving that any ML algorithm is actually used — it only constrains what could be used. The existential form is required to derive conclusions 138 and 139. Added the existential form. (2) Premise 1: the original uses ∨ (inclusive disjunction) for the three-way categorisation. The NL lists the three categories with 'and' as an enumeration of possibilities, not an exclusive partition. The inclusive ∨ is retained as the standard reading — an algorithm could theoretically belong to multiple categories, though in practice they are treated as mutually exclusive (this doesn't affect the conclusions). Flagged as ambiguous.
Sentence: Supervised learning is used to train the state-of-the-art text summarization model.
FOL: ∃x (SupervisedLearningAlgorithm(x) ∧ TrainedWith(stateOfTheArtTextSummarizationModel, x))
Label: True
Sentence: Unsupervised learning is used to train the state-of-the-art text summarization model.
FOL: ∃x (UnsupervisedLearningAlgorithm(x) ∧ TrainedWith(stateOfTheArtTextSummarizationModel, x))
Label: False
Sentence: If a film is appropriate for people of all ages, then children can watch it without guidance from parents. If a film contains excessive erotic and violent content, children cannot watch it without guidance from their parents. If a film is rated General Audience, then it is appropriate for people of all ages. All family-friendly animated films are rated General Audience. All movies in the Frozen series are family-friendly animated films. 'Hachi: A Dog's Tale' is a film. 'Hachi: A Dog's Tale' is either a family-friendly animated film or is appropriate for people of all ages.
FOL: ∀x (Film(x) ∧ AppropriateFor(x, peopleOfAllAges) → CanWatchWithout(children, x, guidanceFromParents)) ∧ ∀x (Film(x) ∧ Contain(x, excessiveEroticViolentContent) → ¬CanWatchWithout(children, x, guidanceFromParents)) ∧ ∀x (Film(x) ∧ Rated(x, generalAudience) → AppropriateFor(x, peopleOfAllAges)) ∧ ∀x (Film(x) ∧ FamilyFriendly(x) ∧ Animated(x) → Rated(x, generalAudience)) ∧ ∀x (Film(x) ∧ InFrozenSeries(x) → FamilyFriendly(x) ∧ Animated(x)) ∧ Film(hachiADogsTale) ∧ ((FamilyFriendly(hachiADogsTale) ∧ Animated(hachiADogsTale)) ⊕ AppropriateFor(hachiADogsTale, peopleOfAllAges))
Correction notes: Several corrections made: (1) Premise 1: the original uses 'Rated(x, adultsOnly)' as the antecedent condition, but the NL says 'appropriate for people of all ages' — not 'rated adults only'. The original FOL has the wrong antecedent condition entirely. Corrected to 'AppropriateFor(x, peopleOfAllAges)' to match the NL. (2) Premise 5: the original uses 'InfrozenSeries' (lowercase 'f') — corrected to 'InFrozenSeries' for camelCase consistency. (3) Premise 7: the original reads '(FamilyFriendly(hachiADogsTale) ∧ Animated(hachiADogsTale)) ⊕ Rated(hachiADogsTale, adultsOnly)'. The NL says 'either a family-friendly animated film or is appropriate for people of all ages' — the second disjunct should be 'AppropriateFor(hachiADogsTale, peopleOfAllAges)', not 'Rated(hachiADogsTale, adultsOnly)'. The original uses the wrong predicate in the second disjunct. Corrected accordingly. (4) Constant naming: 'hachiAdogsTale' normalised to 'hachiADogsTale' for camelCase consistency. (5) Since they always appear altoghether (and to avoid any possible misinterpretations, erotic and violent content are clashed in the unique symbol excessiveEroticViolentContent
Sentence: 'Hachi: A Dog's Tale' is rated General Audience.
FOL: Rated(hachiADogsTale, generalAudience)
Label: Uncertain
Correction notes: The original FOL uses a free variable 'x' ('Rated(x, generalAudience)') instead of the constant 'hachiADogsTale'. Corrected to 'Rated(hachiADogsTale, generalAudience)'.
Sentence: 'Hachi: A Dog's Tale' contains excessive erotic and violent content or is in the 'Frozen' series or both.
FOL: Contain(hachiADogsTale, excessiveEroticViolentContent) ∨ InFrozenSeries(hachiADogsTale)
Label: False
Correction notes: Three major errors in the original: (1) Both occurrences of 'x' are free variables — neither is bound to 'hachiADogsTale'. Corrected to use 'hachiADogsTale' throughout. (2) The second disjunct 'InfrozenSeries(excessiveViolentContent)' uses 'excessiveViolentContent' as the argument to InFrozenSeries — a category error; 'excessiveViolentContent' is a content type constant, not a film. Corrected to 'InFrozenSeries(hachiADogsTale)'. (3) 'InfrozenSeries' → 'InFrozenSeries' for camelCase consistency.
Sentence: 'Hachi: A Dog's Tale' either contains excessive erotic and violent content and is a film in the 'Frozen' series, or it neither contains excessive erotic and violent content nor is a film in the 'Frozen' series.
FOL: (Contain(hachiADogsTale, excessiveEroticViolentContent) ∧ InFrozenSeries(hachiADogsTale)) ⊕ ¬(Contain(hachiADogsTale, excessiveEroticViolentContent) ∧ InFrozenSeries(hachiADogsTale))
Label: True
Correction notes: Same errors as conclusion 141: (1) free variable 'x' replaced with 'hachiADogsTale' throughout. (2) 'InfrozenSeries(excessiveViolentContent)' — category error corrected to 'InFrozenSeries(hachiADogsTale)'. (3) Casing: 'InfrozenSeries' → 'InFrozenSeries'.
Sentence: Breeding back is a form of artificial selection by the deliberate selective breeding of domestic animals. Heck cattle were bred back in the 1920s to resemble the aurochs. Heck cattle are animals. Aurochs are animals. Some animals to be bred back resemble extinct animals.
FOL: ∀x (BreedingBack(x) → ArtificialSelection(x) ∧ DeliberateSelectiveBreedingOfDomesticAnimals(x)) ∧ ∃x ∃y ∃z ∃w (HeckCattle(x) ∧ BreedingBack(x) ∧ HeckCattle(y) ∧ BreedingBack(y) ∧ ¬(x=y) ∧ Auroch(z) ∧ Resemble(x, z) ∧ Auroch(w) ∧ Resemble(y, w) ∧ ¬(w=z)) ∧ ∀x (HeckCattle(x) → Animal(x)) ∧ ∀x (Auroch(x) → Animal(x)) ∧ ∃x ∃y ∃w ∃z (Animal(x) ∧ Animal(y) ∧ ¬(x=y) ∧ BreedingBack(x) ∧ BreedingBack(y) ∧ Dead(w) ∧ Resemble(x, w) ∧ ¬(w=z) ∧ Dead(z) ∧ Resemble(y, z))
Correction notes: Two corrections made: (1) Premise 5: the original FOL is severely malformed: '∃x ∃y (Animal(x) ∧ Animal(y) ∧ ¬(x=y) ∧ BreedingBack(x) ∧ BreedingBack(y) ∧ (∃w(Dead(w) ∧ Resemble(x, w)) ∧ (¬(w=z)) ∧ (∃z(Dead(z) ∧ Resemble(y, z))))'. Multiple errors: (a) The variable 'z' appears free in '¬(w=z)' — z is introduced in the inner ∃z but the inequality ¬(w=z) appears before z is quantified, making it a scope error. (b) The variable 'w' leaks outside its ∃w scope in the ¬(w=z) conjunct. (2) Premise 1: original correctly uses conjunction in the consequent; parentheses clarified. (3) Premises (2) with the two-witness pattern to be consistent with the conclusion_143
Sentence: Some Heck cattle are artificially selected.
FOL: ∃x ∃y (HeckCattle(x) ∧ ArtificialSelection(x) ∧ ¬(x=y) ∧ HeckCattle(y) ∧ ArtificialSelection(y))
Label: True
Correction notes: The original FOL uses the two-witness pattern '∃x ∃y (HeckCattle(x) ∧ ArtificialSelection(x) ∧ ¬(x=y) ∧ HeckCattle(y) ∧ ArtificialSelection(y))' — but not in Premise 2 (hence with the translations provided, no entailment in the FOL version'. Now, we used the two-witness pattern also in the premises.
Sentence: Aurochs are extinct.
FOL: ∀x (Auroch(x) → Dead(x))
Label: Uncertain
Sentence: Everyone who can register to vote in the United States can participate in the 2024 United States presidential election. If someone has United States citizenship, then they can register to vote in the United States. A person either has United States citizenship or Taiwanese citizenship. No Russian Federation officials hold Taiwanese citizenship. Vladimir neither holds Taiwanese citizenship nor is he a manager at Gazprom. Ekaterina can register to vote in the United States, or she is a Russian Federation official.
FOL: ∀x (CanRegisterToVoteIn(x, unitedStates) → CanParticipateIn(x, y2024UnitedStatesElection)) ∧ ∀x (Have(x, unitedStatesCitizenship) → CanRegisterToVoteIn(x, unitedStates)) ∧ ∀x (Have(x, unitedStatesCitizenship) ⊕ Have(x, taiwaneseCitizenship)) ∧ ∀x (RussianFederationOfficial(x) → ¬Have(x, taiwaneseCitizenship)) ∧ (¬Have(vladimir, taiwaneseCitizenship) ∧ ¬ManagerAt(vladimir, gazprom)) ∧ (CanRegisterToVoteIn(ekaterina, unitedStates) ∨ RussianFederationOfficial(ekaterina))
Correction notes: Five suggestions made (but they are not to be considered as corrections): (1) Add an 'y' to the constant '2024UnitedStatesElection' that began with a digit consistently throughout all premises and conclusions. (2) Premise 3: the original uses inclusive ∨ for 'either has United States citizenship or Taiwanese citizenship'. We use exclusive. (3) Premise 4: the original splits the Russian Federation official condition into two separate predicates 'Russian(x) ∧ FederationOfficial(x)', which is an awkward decomposition. Unified to a single predicate 'RussianFederationOfficial(x)' consistent with the NL 'Russian Federation officials'. Applied consistently to premise 6 as well. (4) Premise 6: the original has the disjuncts in reverse order relative to the NL ('Russian ∧ FederationOfficial ∨ CanRegisterToVote') — reordered to match NL order: 'CanRegisterToVoteIn(ekaterina, unitedStates) ∨ RussianFederationOfficial(ekaterina)'. Logically equivalent but more readable. (5) Conclusions 145 and 146: 'Russian(x) ∧ FederationOfficial(x)' unified to 'RussianFederationOfficial(x)' for consistency.
Sentence: Vladimir is a Russian Federation official.
FOL: RussianFederationOfficial(vladimir)
Label: Uncertain
Correction notes: ontology : 'Russian(vladimir) ∧ FederationOfficial(vladimir)' unified to 'RussianFederationOfficial(vladimir)' for consistency with the unified predicate. -> needed because the intended meaning is that Vladimir is an ufficial of the Russian Federation (not that he is russian and he is also a federation official)
Sentence: Vladimir is not a Russian Federation official.
FOL: ¬RussianFederationOfficial(vladimir)
Label: Uncertain
Correction notes: ontology : '¬(Russian(vladimir) ∧ FederationOfficial(vladimir))' unified to '¬RussianFederationOfficial(vladimir)'.
Sentence: Ekaterina can participate in the 2024 United States presidential election or is a manager at Gazprom.
FOL: CanParticipateIn(ekaterina, y2024UnitedStatesElection) ∨ ManagerAt(ekaterina, gazprom)
Label: True
Correction notes: ontology : Constant '2024UnitedStatesElection' → 'y2024UnitedStatesElection' for FOL naming compliance.
Sentence: Ekaterina can participate in the 2024 United States presidential election or Vladimir is a manager at Gazprom.
FOL: CanParticipateIn(ekaterina, y2024UnitedStatesElection) ∨ ManagerAt(vladimir, gazprom)
Label: True
Correction notes: The original FOL uses ⊕ (exclusive disjunction) but the NL says 'or' without any indication of exclusivity. As shown in conclusion 147's derivation, CanParticipateIn(ekaterina, y2024UnitedStatesElection) is provable in both cases — meaning the first disjunct is always True. If the first disjunct is always True, then under ⊕ the conclusion would require ManagerAt(vladimir, gazprom) to be False for ⊕ to hold. Premise 5 establishes ¬ManagerAt(vladimir, gazprom), so under ⊕: True ⊕ False = True. Under ∨: True ∨ False = True. Both give True, but ⊕ is not warranted by the NL. Corrected to ∨.
Sentence: It is not true that Ekaterina can register to vote in the United States and Vladimir can participate in the 2024 United States presidential election.
FOL: ¬(CanRegisterToVoteIn(ekaterina, unitedStates) ∧ CanParticipateIn(vladimir, y2024UnitedStatesElection))
Label: False
Correction notes: Two corrections: (1) The original FOL omits the negation '¬' entirely — it reads 'CanRegisterToVoteIn(ekaterina, unitedStates) ∧ CanParticipateIn(vladimir, y2024UnitedStatesElection)', which is a positive conjunction. But the NL says 'It is NOT true that...', requiring the negation.
Sentence: New Vessel Press is a publishing house specializing in translating foreign literature into English. All of New Vessel Press's published books are in English. Neapolitan Chronicles is a book published by New Vessel Press. Neapolitan Chronicles was translated from Italian. Palace of Flies is a book published by New Vessel Press.
FOL: PublishingHouse(newVesselPress) ∧ SpecializesInTranslatingIntoEnglish(newVesselPress, foreignLiterature) ∧ ∀x (Book(x) ∧ PublishedBy(x, newVesselPress) → In(x, english)) ∧ Book(neapolitanChronicles) ∧ PublishedBy(neapolitanChronicles, newVesselPress) ∧ TranslatedFrom(neapolitanChronicles, italian) ∧ Book(palaceOfFlies) ∧ PublishedBy(palaceOfFlies, newVesselPress)
Sentence: Neapolitan Chronicles is an English book.
FOL: Book(neapolitanChronicles) ∧ In(neapolitanChronicles, english)
Label: True
Sentence: Harry Potter was published by New Vessel Press.
FOL: PublishedBy(harryPotter, newVesselPress)
Label: Uncertain
Sentence: Palace of Flies was translated from Italian.
FOL: TranslatedFrom(palaceOfFlies, italian)
Label: Uncertain
Sentence: All hydrocarbons are organic compounds. All alkanes are hydrocarbons. All organic compounds are chemical compounds. All organic compounds contain carbon. No chemical compounds contain only one element. The mixture either both is a chemical compound and contains only one element, or neither.
FOL: ∀x (Hydrocarbon(x) → OrganicCompound(x)) ∧ ∀x (Alkane(x) → Hydrocarbon(x)) ∧ ∀x (OrganicCompound(x) → ChemicalCompound(x)) ∧ ∀x (OrganicCompound(x) → Contain(x, carbon)) ∧ ∀x (ChemicalCompound(x) → (¬∃y (Contain(x, y) ∧ ∀z (Contain(x, z) → z = y)))) ∧ ¬(ChemicalCompound(mixture) ⊕ ∃y (Contain(mixture, y) ∧ ∀z (Contain(mixture, z) → z = y)))
Correction notes: Multiple corrections made: (1) Predicate naming inconsistency: the original uses both 'Orgranic(x) ∧ Compound(x)' (typo 'Orgranic', and splits organic compound into two predicates) and 'Chemical(x) ∧ Compound(x)' (same split). Unified to single predicates 'OrganicCompound(x)' and 'ChemicalCompound(x)' throughout, matching the NL 'organic compounds' and 'chemical compounds' as single concepts. (2) Premise 1: 'Orgranic' typo corrected to 'OrganicCompound'. (3) The fact that 'a compound contains only one element' was expressed as ∀y ∀z (¬(Contain(x, y) ∧ Contain(x, z) ∧ (y=z))). But this is satisfiable also if the compound doesn't contain any element. (4) Premise 6: the original has a free variable 'x' in the second disjunct of the ⊕ ('∀y ∀z (Contain(x, y) ∧ Contain(x, z) ∧ y=z)') — x is unbound. Corrected to use 'mixture' consistently. Also the 'only one element' condition is reformulated consistently as '∀y ∀z (Contain(mixture, y) ∧ Contain(mixture, z) → y = z)' — meaning all elements contained are the same.
Sentence: The mixture contains carbon.
FOL: Contain(mixture, carbon)
Label: Uncertain
Sentence: The mixture is an alkane and contains carbon.
FOL: Alkane(mixture) ∧ Contain(mixture, carbon)
Label: False
Sentence: All humans are mortal. All Greeks are humans.
FOL: ∀x (Human(x) → Mortal(x)) ∧ ∀x (Greek(x) → Human(x))
Sentence: Some Greeks are mortal.
FOL: ∃x (Greek(x) ∧ Mortal(x))
Label: Uncertain
Sentence: None of the students who currently work in the library are from the computer science department. All of the students with part-time jobs offered by the university currently work in the library. All of the students who are taking the database course are from the computer science department. All of the students taking a class with Professor David are taking the database course. All students working in the lab are taking a class with Professor David. James is a student working in the lab. James doesn't work in the lab or have a part-time job offered by the university.
FOL: ∀x (Student(x) ∧ WorkIn(x, library) → ¬From(x, computerScienceDepartment)) ∧ ∀x (Student(x) ∧ ∃y (PartTimeJob(y) ∧ Have(x, y) ∧ OfferedBy(y, university)) → WorkIn(x, library)) ∧ ∀x (Student(x) ∧ Take(x, databaseCourse) → From(x, computerScienceDepartment)) ∧ ∀x (Student(x) ∧ InstructedBy(x, professorDavid) → Take(x, databaseCourse)) ∧ ∀x (Student(x) ∧ WorkIn(x, lab) → InstructedBy(x, professorDavid)) ∧ (Student(james) ∧ WorkIn(james, lab)) ∧ (¬WorkIn(james, lab) ∧ ¬∃y (PartTimeJob(y) ∧ Have(james, y) ∧ OfferedBy(y, university)))
Correction notes: Four corrections made: (1) Premise 2: the original '∀x (Student(x) ∧ Have(x, partTimeJob) ∧ OfferedBy(y, university) → WorkIn(x, library))' has a free variable 'y' in 'OfferedBy(y, university)' — y is unbound, making the formula ill-formed. (2) Premise 7: the original reads '¬∃x (PartTimeJob(x) ∧ Have(james, x) ∧ OfferedBy(x, computerScienceDepartment)) ∧ ¬WorkIn(james, lab)'. Two errors: (a) 'OfferedBy(x, computerScienceDepartment)' uses 'computerScienceDepartment' as the offerer, but the NL says 'offered by the university' — corrected to 'university'. (b) a remark more than an error: the premise should not to be considered ambiguous and should be treated as 'James neither works in lab, nor have a part-time job...' (notice the verb 'have' in the premises that goes with 'doesn't'). The problem with this formulation (that is the same given originally by the authors) is that the premises are contradictories, hence any conclusion must be True (and we set it so).
Sentence: James has a part-time job offered by the university.
FOL: ∃y (PartTimeJob(y) ∧ Have(james, y) ∧ OfferedBy(y, university))
Label: True
Sentence: James does not have a part-time job offered by the university.
FOL: ¬∃y (PartTimeJob(y) ∧ Have(james, y) ∧ OfferedBy(y, university))
Label: True
Sentence: James either takes the database course or has a part-time job offered by the university.
FOL: Take(james, databaseCourse) ⊕ (∃y (PartTimeJob(y) ∧ Have(james, y) ∧ OfferedBy(y, university)))
Label: True
Correction notes: The original FOL has an unclosed parenthesis — added missing closing parenthesis around the full ⊕ expression.
Sentence: James takes the database course.
FOL: Take(james, databaseCourse)
Label: True
Sentence: If George likes music, he wants to compose. If George has access to a program, he can compose. If George can compose and wants to compose, he will compose.
FOL: (LikeMusic(george) → WantToCompose(george)) ∧ (AccessToProgram(george) → CanCompose(george)) ∧ (WantToCompose(george) ∧ CanCompose(george) → WillCompose(george))
Correction notes: ontology
Sentence: If George likes music and has access to a program, he will compose.
FOL: (LikeMusic(george) ∧ AccessToProgram(george)) → WillCompose(george)
Label: True
Sentence: If George will not compose, George cannot compose.
FOL: ¬WillCompose(george) → ¬CanCompose(george)
Label: Uncertain
Sentence: A hawk never swims. Some birds are hawks.
FOL: ∀x (Hawk(x) → ¬Swim(x)) ∧ ∃x (Bird(x) ∧ Hawk(x))
Sentence: All birds swim.
FOL: ∀x (Bird(x) → Swim(x))
Label: False
Sentence: Some show airing at 8 pm on Monday gives out roses on TV. If a show gives out roses on TV, then the show is an episode of The Bachelor. The Bachelor portrays the lives of real people. All shows portraying the lives of real people are reality TV shows. Breaking Bad is a show. Breaking Bad is not a reality TV show.
FOL: ∃x (Show(x) ∧ AiringAt(x, eightPMMonday) ∧ GivesOutOnTV(x, rose)) ∧ ∀x (Show(x) ∧ GivesOutOnTV(x, rose) → TheBachelor(x)) ∧ ∀x (TheBachelor(x) → Portray(x, livesOfRealPeople)) ∧ ∀x (Portray(x, livesOfRealPeople) → RealityTVShow(x)) ∧ Show(breakingBad) ∧ ¬RealityTVShow(breakingBad)
Correction notes: Several corrections made: (1) Premise 1: the original uses 'GivenOutOn(x, rose, tV)' (passive, 3-ary with capitalised 'tV') while premise 2 uses 'GivenOutOnAt(rose, tV, x)' (different argument order and predicate name) and conclusions use 'GivenOutOn(y, tv)' (2-ary). The predicate is used with inconsistent names, arities and argument orders across premises and conclusions. Unified to a consistent 3-ary predicate 'GivesOutOn(x, rose, tv)' where x is the show, matching the NL 'a show gives out roses on TV'. (2) Premise 1: 'AiringAtOn(x, 8PMMonday)' — the constant '8PMMonday' begins with a digit; replaced with 'eightPMMonday'. (3) Premise 4: the original uses 'liveOfRealPeople' (missing 's') while premise 3 uses 'lifeOfRealPeople' — both are inconsistent with each other and with the NL 'lives of real people'. Unified to 'livesOfRealPeople' throughout.
Sentence: Breaking Bad is on Monday at 8 pm.
FOL: AiringAt(breakingBad, eightPMMonday)
Label: Uncertain
Correction notes: The original FOL reads '∀x (MondayAt8PM(x) ∧ On(breakingBad, x))' — a universal statement asserting that everything that is a Monday-8PM slot has Breaking Bad on it, which is a much stronger and stranger claim than the NL; furthermore, there is a symbol mismatch. The NL simply says 'Breaking Bad is on Monday at 8 pm', which is a ground atom. Corrected to 'AiringAt(breakingBad, eightPMMonday)' using the predicate from the corrected premises.
Sentence: Breaking Bad is a TV show in which roses are given out on Monday at 8 pm.
FOL: Show(breakingBad) ∧ GivesOutOnTV(breakingBad, rose) ∧ AiringAt(breakingBad, eightPMMonday)
Label: False
Correction notes: The original FOL reads '∀x ∀y (MondayAt8PM(x) ∧ Rose(y) ∧ GivenOutOn(y, tv) ∧ On(tv, x) ∧ From(y, breakingBad))' — a universal statement with multiple undefined predicates and a category error (From(y, breakingBad) treats y as a rose coming from Breaking Bad). Rewritten to use consistent predicate vocabulary: 'Show(breakingBad) ∧ GivesOutOn(breakingBad, rose, tv) ∧ AiringAt(breakingBad, eightPMMonday)'.
Sentence: If roses are given out during Breaking Bad, then it is on Monday at 8 pm.
FOL: GivesOutOnTV(breakingBad, rose) → AiringAt(breakingBad, eightPMMonday)
Label: True
Correction notes: The original FOL reads '∀x ∀y ((Rose(y) ∧ GivenOutOn(y, tv) ∧ On(tv, x) ∧ From(y, breakingBad)) → MondayAt8PM(x))' — using undefined predicates and a universally quantified structure. Simplified to a ground conditional using consistent predicate vocabulary: 'GivesOutOn(breakingBad, rose, tv) → AiringAt(breakingBad, eightPMMonday)'.
Sentence: Heinrich Schmidt was a German politician. Heinrich Schmidt was also a member of the Prussian State Parliament and the Nazi Reichstag.
FOL: German(heinrichSchmidt) ∧ Politician(heinrichSchmidt) ∧ Member(heinrichSchmidt, prussianStateParliament) ∧ Member(heinrichSchmidt, naziReichstag)
Sentence: Heinrich Schmidt was German or Russian or both.
FOL: German(heinrichSchmidt) ∨ Russian(heinrichSchmidt)
Label: True
Sentence: Some German politician was part of both the Prussian State Parliament and the Nazi Reichstag.
FOL: ∃x (German(x) ∧ Politician(x) ∧ Member(x, prussianStateParliament) ∧ Member(x, naziReichstag))
Label: True
Sentence: No politicians are part of the Nazi Reichstag.
FOL: ∀x (Politician(x) → ¬Member(x, naziReichstag))
Label: False