Barutin Cove is a cove named after the Bulgarian settlement of Barutin. Barutin Cove is on the southwest coast of Snow Island. Snow Island, Greenwich Island, and Deception Island are located in the South Shetland Islands. Antarctica is located on the South Shetland Islands. If place A is located in place B and place B is located in place C, then place A is located in place C.

Cove(barutinCove) ∧ NamedAfter(barutinCove, barutinSettlement) ∧ LocatedIn(barutinSettlement, bulgaria) ∧ LocatedIn(barutinCove, snowIsland) ∧ LocatedIn(snowIsland, southShetlandIslands) ∧ LocatedIn(greenwichIsland, southShetlandIslands) ∧ LocatedIn(deceptionIsland, southShetlandIslands) ∧ LocatedIn(antarctica, southShetlandIslands) ∧ ∀x ∀y ∀z (LocatedIn(x, y) ∧ LocatedIn(y, z) → LocatedIn(x, z))

Two corrections made: (1) Premise 4: the original states 'Antarctica is located on the South Shetland Islands' — 'LocatedIn(southShetlandIslands, antarctica)'. However, this is geographically inverted: the South Shetland Islands are located IN Antarctica, not the other way around. The NL premise is factually incorrect as stated. The FOL faithfully represents the NL as written ('LocatedIn(southShetlandIslands, antarctica)'), which is the correct direction for the logical chain to work (Barutin Cove → Snow Island → South Shetland Islands → Antarctica). Retained as-is since the FOL correctly mirrors the NL, but flagged as a real-world factual inversion — in reality it is the South Shetland Islands that are located in Antarctica, which is what the FOL encodes and what is needed for the conclusions to be derivable. The NL should have read 'The South Shetland Islands are located in Antarctica' rather than 'Antarctica is located on the South Shetland Islands'. (2) Premise 1: the original FOL includes 'LocatedIn(barutinSettlement, bulgaria)' which is not stated in the NL premise — the NL only says 'named after the Bulgarian settlement of Barutin', implying the settlement is Bulgarian but not explicitly asserting LocatedIn(barutinSettlement, bulgaria) as a FOL fact. However this grounding is necessary for conclusion 119 to be True; it is retained as a reasonable inference from 'Bulgarian settlement' and flagged.

Cove(barutinCove) ∧ NamedAfter(barutinCove, barutinSettlement) ∧ LocatedIn(barutinSettlement, bulgaria)
LocatedIn(barutinCove, snowIsland)
LocatedIn(snowIsland, southShetlandIslands) ∧ LocatedIn(greenwichIsland, southShetlandIslands) ∧ LocatedIn(deceptionIsland, southShetlandIslands)
LocatedIn(southShetlandIslands, antarctica)
∀x ∀y ∀z ((LocatedIn(x, y) ∧ LocatedIn(y, z)) → LocatedIn(x, z))


Barutin Cove is named after all islands in Antarctica.

∀x (LocatedIn(x, antarctica) → NamedAfter(barutinCove, x))

The formalization can be considered correct. However, it lacks any referement to the concept of 'island'. If required, it should be put both in the story and in the conclusions.


There is at least one cove in Antarctica named after a place in Bulgaria.

∃x ∃y (Cove(x) ∧ LocatedIn(x, antarctica) ∧ NamedAfter(x, y) ∧ LocatedIn(y, bulgaria))

The original FOL uses 'NameAfter(x, y)' — missing the 'd' suffix, making it inconsistent with 'NamedAfter' used throughout the premises and conclusion 118. Corrected to 'NamedAfter(x, y)'.

∃x ∃y (Cove(x) ∧ LocatedIn(x, antarctica) ∧ NameAfter(x, y) ∧ LocatedIn(y, bulgaria))


Barutin Cove is not located in Antarctica.


Some affection is love. Some love is positive.

∃x (Affection(x) ∧ Love(x)) ∧ ∃x (Love(x) ∧ Positive(x))

∃x (Affection(x) ∧ Love(x))
∃x (Love(x) ∧ Positive(x))


If something can transport multiple passengers, then it is not a one-seater. All Tesla Model 3 can transport multiple passengers. All single-seat electric cars are one-seaters. All Solo EV cars are single-seat electric cars. If John's car is not a Tesla Model 3, then John's car is not a Tesla Model 3 or a single-seat electric car.

∀x (∃y ∃z (Passenger(y) ∧ Passenger(z) ∧ Transport(x, y) ∧ Transport(x, z) ∧ ¬(y=z)) → ¬OneSeater(x)) ∧ ∀x (TeslaModel3(x) → ∃y ∃z (Passenger(y) ∧ Passenger(z) ∧ Transport(x, y) ∧ Transport(x, z) ∧ ¬(y=z))) ∧ ∀x (SingleSeat(x) ∧ ElectricCar(x) → OneSeater(x)) ∧ ∀x (SoloEV(x) ∧ Car(x) → SingleSeat(x) ∧ ElectricCar(x)) ∧ (¬TeslaModel3(johnsCar) → ¬TeslaModel3(johnsCar) ∧ ¬(SingleSeat(johnsCar) ∧ ElectricCar(johnsCar))) ∧ Car(johnsCar)

Four corrections made: (1) Premises 1 and 2: the original has 'Transport(x, y) ∧ Transport(x, y)' — the second conjunct is a duplicate of the first (both use y), meaning the two-passenger condition is never actually enforced. (2) Premise 2: the original has an unclosed parenthesis after the existential block; corrected by adding the missing closing parenthesis. (3) Premise 5: the original uses a free variable 'x' in 'SingleSeat(x) ∧ ElectricCar(johnsCar)' (4) Added the contextual information that John's car is a car.

∀x (∃y ∃z (Passenger(y) ∧ Passenger(z) ∧ Transport(x, y) ∧ Transport(x, y) ∧ ¬(y=z)) → ¬OneSeater(x))
∀x (TeslaModel3(x) → (∃y ∃z (Passenger(y) ∧ Passenger(z) ∧ Transport(x, y) ∧ Transport(x, y) ∧ ¬(y=z)))
∀x (SingleSeat(x) ∧ ElectricCar(x) → OneSeater(x))
∀x (SoloEV(x) ∧ Car(x) → SingleSeat(x) ∧ ElectricCar(x))
¬TeslaModel3(johnsCar) → ¬(TeslaModel3(johnsCar) ∨ (SingleSeat(x) ∧ ElectricCar(johnsCar)))


The label 'Uncertain' is correct: nothing in the premises positively asserts TeslaModel3(johnsCar), nor is it refutable. The premises only establish conditional relationships. Under the open-world assumption, John's car could or could not be a Tesla Model 3. Note however that if TeslaModel3(johnsCar) were false, then by the corrected premise 5, ¬(SingleSeat(johnsCar) ∧ ElectricCar(johnsCar)) would hold, and by premise 4, ¬(SoloEV(johnsCar) ∧ Car(johnsCar)) would follow. The truth value of TeslaModel3(johnsCar) itself remains undetermined.


If we translate it as in the original 'SoloEV(johnsCar) ∧ Car(johnsCar)', then it was not derivable from the premises since in the previous translation there was nothing that implied Car(johnsCar), hence the label should be 'Uncertain'. Now it is fixed in this regard.

The label 'False' is correct via the following reasoning: suppose SoloEV(johnsCar) ∧ Car(johnsCar) held. By premise 4, SingleSeat(johnsCar) ∧ ElectricCar(johnsCar) would follow, making johnsCar a one-seater (premise 3). But premise 2 gives TeslaModel3(x) → transports multiple passengers, and premise 1 gives multiple passengers → ¬OneSeater. If johnsCar were also a Tesla Model 3, it would be both a one-seater and not a one-seater — contradiction. If johnsCar is not a Tesla Model 3, corrected premise 5 gives ¬(SingleSeat(johnsCar) ∧ ElectricCar(johnsCar)) — contradicting what we derived from SoloEV. In both cases a contradiction arises, so SoloEV(johnsCar) ∧ Car(johnsCar) is refutable.


Not considered as an error, but only as a suggestion for a better translation. Giving the fact that 'John's car is a aìcar', these formulations are equivalent.


All of Peter's pets that are mammals are also animals. All monkeys are mammals. Peter's pets are all either monkeys or birds. Peter's birds can fly. All animals that are Peter's pets can breathe. If Peter's pet can fly, then it has wings. Rock is Peter's pet. Rock can fly, or Rock is a bird, or Rock cannot breathe.

∀x (PetersPet(x) ∧ Mammal(x) → Animal(x)) ∧ ∀x (Monkey(x) → Mammal(x)) ∧ ∀x (PetersPet(x) → Monkey(x) ⊕ Bird(x)) ∧ ∀x (PetersPet(x) ∧ Bird(x) → CanFly(x)) ∧ ∀x (PetersPet(x) ∧ Animal(x) → Breathe(x)) ∧ ∀x (PetersPet(x) ∧ CanFly(x) → ∃y ∃z (¬(y=z) ∧ Wing(y) ∧ Wing(z) ∧ Have(x, y) ∧ Have(x, z))) ∧ PetersPet(rock) ∧ (CanFly(rock) ∨ Bird(rock) ∨ ¬Breathe(rock))

Four corrections made: (1) Premise 4: the original reads '∀x (Bird(x) → CanFly(x))' — this universally asserts ALL birds can fly, not just Peter's birds. (2) Premise 7: the original uses 'PeterSPet(rock)' — capital S mid-word and missing apostrophe handling, inconsistent with 'PetersPet' used in all other premises. Corrected to 'PetersPet(rock)'.

∀x (PetersPet(x) ∧ Mammal(x) → Animal(x))
∀x (Monkey(x) → Mammal(x))
∀x (PetersPet(x) → Monkey(x) ⊕  Bird(x))
∀x (Bird(x) → CanFly(x))
∀x (PetersPet(x) ∧ Animal(x) → Breathe(x))
∀x (PetersPet(x) ∧ CanFly(x) → ∃y ∃z (¬(y=z) ∧ Wing(y) ∧ Wing(z) ∧ Have(x, y) ∧ Have(x, z)))
PeterSPet(rock)
CanFly(rock) ∨ Bird(rock) ∨ ¬Breathe(rock)


∃y ∃z (¬(y=z) ∧ Wing(y) ∧ Wing(z) ∧ Have(rock, y) ∧ Have(rock, z))

The label 'True' is derivable regardless of whether Rock is a monkey or a bird, as shown in conclusion 125's analysis: (Case 1) Rock is a monkey → Breathe(rock) → ¬Breathe(rock) false in premise 8 → CanFly(rock) must hold (since Bird(rock) is false) → PetersPet(rock) ∧ CanFly(rock) + premise 6 → Rock has wings. (Case 2) Rock is a bird → corrected premise 4 → CanFly(rock) → PetersPet(rock) ∧ CanFly(rock) + premise 6 → Rock has wings. In both exhaustive cases (from ⊕ in premise 3) Rock has wings — the conclusion is True.


¬∃y ∃z (¬(y=z) ∧ Wing(y) ∧ Wing(z) ∧ Have(rock, y) ∧ Have(rock, z))

¬(∃y ∃z (¬(y=z) ∧ Wing(y) ∧ Wing(z) ∧ Have(rock, y) ∧ Have(rock, z)))


All people who make wedding plans are people who are engaged. All people who invite others to come to their ceremony make wedding plans. Anyone who has a well-attended wedding invites others to come to their ceremony. Some people that have well-attended weddings have larger families. If John is engaged, then John does not invite friends to their ceremony and make wedding plans. If John has a larger family, then either John has a well-attended wedding or invites friends to their ceremony.

∀x (Make(x, weddingPlan) → Engaged(x)) ∧ ∀x ((∃y InviteToComeTo(x, y, wedding)) → Make(x, weddingPlan)) ∧ ∀x (Have(x, wellAttendedWedding) → ∃y InviteToComeTo(x, y, wedding)) ∧ ∃x (Have(x, wellAttendedWedding) ∧ Have(x, largerFamily)) ∧ (Engaged(john) → (¬(∃y InviteToComeTo(john, y, wedding)) ∧ ¬Make(john, weddingPlan))) ∧ (Have(john, largerFamily) → (Have(john, wellAttendedWedding) ⊕ (∃y (InviteToComeTo(john, y, wedding)))))

Several corrections made: (1) Premise 5: the original has two major errors. First, '¬(y=x)' and '¬(z=x)' use a free variable 'x' instead of 'john' — corrected to '¬(y=john)' and '¬(z=john)'. Second, the original uses a mixed predicate 'Invite(john, z, comeTo, wedding)' (4-ary) inconsistent with 'InviteToComeTo(john, z, wedding)' (3-ary) used in all other premises; unified to 'InviteToComeTo'. Third, the NL reads 'John does not invite friends to their ceremony AND make wedding plans' — interpreted as two separate negative consequences: John neither invites others nor makes wedding plans. Corrected consequent to '¬∃y ∃z (...InviteToComeTo...) ∧ ¬Make(john, weddingPlan)'. (2) Premise 6: the original uses free variable 'x' in the invite consequent ('¬(y=x)', 'Invite(x, y, comeTo, wedding)') instead of 'john'; corrected to use 'john' throughout. Also unified 4-ary 'Invite(x, y, comeTo, wedding)' to 3-ary 'InviteToComeTo(john, y, wedding)'. (3) The ⊕ in premise 6 correctly captures 'either... or...' for the two disjuncts regarding John's well-attended wedding vs inviting friends — these are plausibly exclusive (though see ambiguity note). (4) Premises 2 and 3 encode 'inviting others' as requiring at least two distinct invitees (y ≠ z, both ≠ x) — this is a specific modelling choice for 'multiple' that is retained as-is.

∀x (Make(x, weddingPlan) → Engaged(x))
 ∀x (∃y ∃z (¬(y=x) ∧ ¬(z=x) ∧ ¬(y=z) ∧ InviteToComeTo(x, y, wedding) ∧ InviteToComeTo(x, z, wedding)) → Make(x, weddingPlan))
 ∀x (Have(x, wellAttendedWedding) → ∃y ∃z (¬(y=x) ∧ ¬(z=x) ∧ ¬(y=z) ∧ InviteToComeTo(x, y, wedding) ∧ InviteToComeTo(x, z, wedding)))
 ∃x (Have(x, wellAttendedWedding) ∧ Have(x, largerFamily))
 Engaged(john) → ¬(∃y ∃z (¬(y=x) ∧ ¬(z=x) ∧ ¬(y=z) ∧ InviteToComeTo(john, y, wedding) ∧ Invite(john, z, comeTo, wedding))) ∧ Make(john, weddingPlan)
 Have(john, largerFamily) → Have(john, wellAttendedWedding) ⊕ ∃y ∃z (¬(y=x) ∧ ¬(z=x) ∧ ¬(y=z) ∧ Invite(x, y, comeTo, wedding) ∧ InviteToComeTo(x, z, wedding))


The original FOL uses 'Have(john, largeFamily)' — 'largeFamily' (missing 'r') instead of 'largerFamily' used consistently in the premises. Corrected to 'Have(john, largerFamily)'.

The label 'False' is justified via the following chain: suppose Have(john, largerFamily). By corrected premise 6 (⊕): Have(john, wellAttendedWedding) ⊕ ∃y∃z(...InviteToComeTo(john,...)...). Case A: Have(john, wellAttendedWedding) holds. By premise 3, john invites others → by premise 2, Make(john, weddingPlan) → by premise 1, Engaged(john). By corrected premise 5: Engaged(john) → ¬∃y∃z(...InviteToComeTo...) ∧ ¬Make(john, weddingPlan). But we derived Make(john, weddingPlan) — contradiction. Case B: ∃y∃z(...InviteToComeTo(john,...)...) holds. By premise 2 → Make(john, weddingPlan) → Engaged(john) → ¬∃y∃z(...InviteToComeTo...) — contradiction. Both cases yield contradictions, so Have(john, largerFamily) is refutable.


John neither has a larger family nor does he invite others to his wedding.

¬Have(john, largerFamily) ∧ ¬∃y(InviteToComeTo(john, y, wedding))

The label 'True' is correct for both conjuncts: (1) ¬Have(john, largerFamily) is established via conclusion 129. (2) ¬∃y∃z(...InviteToComeTo(john,...)...): suppose john did invite others. By premise 2 → Make(john, weddingPlan) → by premise 1 → Engaged(john) → by corrected premise 5 → ¬∃y∃z(...InviteToComeTo(john,...)) — contradiction. Therefore john does not invite others. Both conjuncts are True.

¬Have(john, largerFamily) ∧ ¬(∃y ∃z (¬(y=john) ∧ ¬(z=john) ∧ ¬(y=z) ∧ InviteToComeTo(john, y, wedding) ∧ InviteToComeTo(john, z, wedding)))


Some mammals have teeth. Platypuses have no teeth. Platypuses are mammals. Humans have teeth.

∃x(Mammal(x) ∧ Have(x, teeth)) ∧ ¬Have(platypus, teeth) ∧ Mammal(platypus) ∧ Have(humans, teeth)

∃x ∃y (Mammal(x) ∧ Mammal(y) ∧ (¬(x=y)) ∧ Have(x, teeth) ∧ Have(y, teeth))
¬Have(platypus, teeth)
Mammal(platypus)
Have(humans, teeth)


Mammal(platypus) ∧ ¬Have(platypus, teeth)

Mammal(platypus) ∧ (¬Have(platypus, teeth))


Plungers suck. Vacuums suck. Vampires suck. Space is a vacuum. A duster is a household appliance that doesn't suck.

∀x (Plunger(x) → Suck(x)) ∧ ∀x (Vacuum(x) → Suck(x)) ∧ ∀x (Vampire(x) → Suck(x)) ∧ Vacuum(space) ∧ HouseholdAppliance(duster) ∧ ¬Suck(duster)

∀x (Plunger(x) → Suck(x))
∀x (Vacuum(x) → Suck(x))
∀x (Vampire(x) → Suck(x))
Vacuum(space)
HouseholdAppliance(duster) ∧ ¬Suck(duster)


If something is a household appliance, it sucks.

The original FOL uses 'HouseHoldApp(x)' — an abbreviated and inconsistently cased predicate name, different from 'HouseholdAppliance' used in premise 5. Corrected to 'HouseholdAppliance(x)' for consistency.


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.

∀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))

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.

∀x (MachineLearningAlgorithm(x) → SupervisedLearningAlgorithm(x) ∨ UnsupervisedLearningAlgorithm(x) ∨ ReinforcementLearningAlgorithm(x))
∀x (UnsupervisedLearningAlgorithm(x) → ¬Require(x, labeledData))
∀x (TrainedWith(stateOfTheArtTextSummarizationModel, x) → MachineLearningAlgorithm(x))
∀x (ReinforcementLearningAlgorithm(x) → ¬TrainedWith(stateOfTheArtTextSummarizationModel, x))
∀x ((MachineLearningAlgorithm(x) ∧ TrainedWith(stateOfTheArtTextSummarizationModel, x)) → Require(x, labeledData))


Supervised learning is used to train the state-of-the-art text summarization model.

∃x (SupervisedLearningAlgorithm(x) ∧ TrainedWith(stateOfTheArtTextSummarizationModel, x))

Now the label 'True' is corrected (because we change the premises formalization and add the existential info about the algorithm for the state of the art models) but with the previous formalization the label should be 'Uncertain'


Unsupervised learning is used to train the state-of-the-art text summarization model.

∃x (UnsupervisedLearningAlgorithm(x) ∧ TrainedWith(stateOfTheArtTextSummarizationModel, x))


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.

∀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))

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

∀x (Film(x) ∧ Rated(x, adultsOnly) → CanWatchWithout(children, x, guidanceFromParents))
∀x (Film(x) ∧ Contain(x, excessiveEroticContent) ∧ Contain(x, excessiveViolentContent) → ¬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)) ⊕ Rated(hachiAdogsTale, adultsOnly)


'Hachi: A Dog's Tale' is rated General Audience.

The original FOL uses a free variable 'x' ('Rated(x, generalAudience)') instead of the constant 'hachiADogsTale'. Corrected to 'Rated(hachiADogsTale, generalAudience)'.

The label 'Uncertain' requires analysis under the corrected premise 7 (⊕ between FamilyFriendly∧Animated and AppropriateFor). If hachiADogsTale is FamilyFriendly ∧ Animated (first disjunct of ⊕): premise 4 → Rated(hachiADogsTale, generalAudience) — True. If hachiADogsTale is AppropriateFor(..., peopleOfAllAges) only (second disjunct of ⊕): no chain reaches Rated(generalAudience) since premise 3 goes from Rated → Appropriate (not the converse). In this case Rated(generalAudience) is not entailed. Since ⊕ allows either case, the conclusion is Uncertain.


'Hachi: A Dog's Tale' contains excessive erotic and violent content or is in the 'Frozen' series or both.

Contain(hachiADogsTale, excessiveEroticViolentContent) ∨ InFrozenSeries(hachiADogsTale)

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.

(Contain(x, excessiveEroticContent) ∧ Contain(x, excessiveViolentContent)) ∨ InfrozenSeries(excessiveViolentContent)


'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.

(Contain(hachiADogsTale, excessiveEroticViolentContent) ∧ InFrozenSeries(hachiADogsTale)) ⊕ ¬(Contain(hachiADogsTale, excessiveEroticViolentContent) ∧ InFrozenSeries(hachiADogsTale))

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'.

The label 'True' is correct, but actually the sentence is a tautology (excluded-middle principle). It doesn't test the correct formalization of the premises.

¬((Contain(x, excessiveEroticContent) ∧ Contain(x, excessiveViolentContent)) ⊕ InfrozenSeries(excessiveViolentContent))


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.

∀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))

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

The predicate 'Dead' is used to represent 'extinct' in premise 5 and conclusion 144. This is a modelling simplification — 'extinct' applies to species while 'dead' applies to individuals.

∀x (BreedingBack(x) → (ArtificialSelection(x) ∧ DeliberateSelectiveBreedingOfDomesticAnimals(x)))
∃x ∃y (HeckCattle(x) ∧ BreedingBack(x) ∧ Auroch(y) ∧ Resemble(x, y))
∀x (HeckCattle(x) → Animal(x))
∀x (Auroch(x) → Animal(x))
∃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))))


Some Heck cattle are artificially selected.

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.

Directly derivable: premise 2 gives ∃x (HeckCattle(x) ∧ BreedingBack(x) ∧ ...); call this individual h. BreedingBack(h) + premise 1 → ArtificialSelection(h). HeckCattle(h) ∧ ArtificialSelection(h) → existential introduction yields the conclusion.

∃x ∃y(HeckCattle(x) ∧ ArtificialSelection(x) ∧ (¬(x=y)) ∧ HeckCattle(y) ∧ ArtificialSelection(y))


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.

∀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))

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.

∀x (CanRegisterToVoteIn(x, unitedStates) → CanParticipateIn(x, 2024UnitedStatesElection))
∀x (Have(x, unitedStatesCitizenship) → CanRegisterToVoteIn(x, unitedStates))
∀x (Have(x, unitedStatesCitizenship) ∨ Have(x, taiwaneseCitizenship))
∀x (Russian(x) ∧ FederationOfficial(x) → ¬Have(x, taiwaneseCitizenship))
¬Have(vladimir, taiwaneseCitizenship) ∧ ¬ManagerAt(vladimir, gazprom)
(Russian(ekaterina) ∧ FederationOfficial(ekaterina)) ∨ CanRegisterToVoteIn(ekaterina, unitedStates)


Vladimir is a Russian Federation official.

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)


Vladimir is not a Russian Federation official.

ontology : '¬(Russian(vladimir) ∧ FederationOfficial(vladimir))' unified to '¬RussianFederationOfficial(vladimir)'.


Ekaterina can participate in the 2024 United States presidential election or is a manager at Gazprom.

CanParticipateIn(ekaterina, y2024UnitedStatesElection) ∨ ManagerAt(ekaterina, gazprom)

ontology : Constant '2024UnitedStatesElection' → 'y2024UnitedStatesElection' for FOL naming compliance.

CanParticipateIn(ekaterina, 2024UnitedStatesElection) ∨ ManagerAt(ekaterina, gazprom)


Ekaterina can participate in the 2024 United States presidential election or Vladimir is a manager at Gazprom.

CanParticipateIn(ekaterina, y2024UnitedStatesElection) ∨ ManagerAt(vladimir, gazprom)

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 ∨.

CanParticipateIn(ekaterina, 2024UnitedStatesElection) ⊕ ManagerAt(vladimir, gazprom)


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.

¬(CanRegisterToVoteIn(ekaterina, unitedStates) ∧ CanParticipateIn(vladimir, y2024UnitedStatesElection))

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.

CanRegisterToVoteIn(ekaterina, unitedStates) ∧ CanParticipateIn(vladimir, 2024UnitedStatesElection)


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.

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)

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)


Neapolitan Chronicles is an English book.

Book(neapolitanChronicles) ∧ In(neapolitanChronicles, english)


Harry Potter was published by New Vessel Press.


Palace of Flies was translated from Italian.


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.

∀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)))

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.

∀x (Hydrocarbon(x) → Orgranic(x) ∧ Compound(x))
∀x (Alkane(x) → Hydrocarbon(x))
∀x (Orgranic(x) ∧ Compound(x) → Chemical(x) ∧ Compound(x))
∀x (Orgranic(x) ∧ Compound(x) → Contain(x, carbon))
∀x (Chemical(x) ∧ Compound(x) → ∀y ∀z (¬(Contain(x, y) ∧ Contain(x, z) ∧ (y=z))))
¬((Chemical(mixture) ∧ Compound(mixture)) ⊕ (∀y ∀z (Contain(x, y) ∧ Contain(x, z) ∧ y=z)))


The mixture is an alkane and contains carbon.

Alkane(mixture) ∧ Contain(mixture, carbon)


All humans are mortal. All Greeks are humans.

∀x (Human(x) → Mortal(x)) ∧ ∀x (Greek(x) → Human(x))

∀x (Human(x) → Mortal(x))
∀x (Greek(x) → Human(x))


The label 'Uncertain' is correct even under the corrected formulation: the premises establish that IF any Greek exists, they are mortal (Greek → Human → Mortal by universal chain). However, the premises never assert the existence of any Greek

∃x ∃y (Greek(x) ∧ Greek(y) ∧ Mortal(x) ∧ Mortal(y) ∧ ¬(x=y))


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.

∀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)))

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).

∀x (Student(x) ∧ WorkIn(x, library) → (¬From(x, computerScienceDepartment)))
∀x (Student(x) ∧ Have(x, partTimeJob) ∧ 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)
¬∃x (PartTimeJob(x) ∧ Have(james, x) ∧ OfferedBy(x, computerScienceDepartment)) ∧ ¬WorkIn(james, lab)


James has a part-time job offered by the university.

∃y (PartTimeJob(y) ∧ Have(james, y) ∧ OfferedBy(y, university))


James does not have a part-time job offered by the university.

¬∃y (PartTimeJob(y) ∧ Have(james, y) ∧ OfferedBy(y, university))

¬∃y(PartTimeJob(y) ∧ Have(james, y) ∧ OfferedBy(y, university))


James either takes the database course or has a part-time job offered by the university.

Take(james, databaseCourse) ⊕ (∃y (PartTimeJob(y) ∧ Have(james, y) ∧ OfferedBy(y, university)))

The original FOL has an unclosed parenthesis — added missing closing parenthesis around the full ⊕ expression.


The label 'Uncertain' appears inconsistent with the derivation chain shown in conclusion 159's note: WorkIn(james, lab) (premise 6) + premise 5 → InstructedBy(james, professorDavid) + premise 4 → Take(james, databaseCourse). This chain is valid and Take(james, databaseCourse) should be True, not Uncertain. The label appears to be a dataset error. However, there is a subtle issue: premise 4 requires 'Student(james) ∧ InstructedBy(james, professorDavid)' — Student(james) is established by premise 6, so the chain is complete. The label 'Uncertain' is retained as given but flagged as a likely dataset inconsistency — the correct label should be 'True' given the valid derivation chain.


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.

(LikeMusic(george) → WantToCompose(george)) ∧ (AccessToProgram(george) → CanCompose(george)) ∧ (WantToCompose(george) ∧ CanCompose(george) → WillCompose(george))

LikeMusic(george) → WantToCompose(george)
AccesstoProgram(george) → CanCompose(george)
(WantToCompose(george) ∧ CanCompose(george)) → WillCompose(george)


If George likes music and has access to a program, he will compose.

(LikeMusic(george) ∧ AccessToProgram(george)) → WillCompose(george)

(LikeMusic(george) ∧ AccesstoProgram(george)) → WillCompose(george)


If George will not compose, George cannot compose.


A hawk never swims. Some birds are hawks.

∀x (Hawk(x) → ¬Swim(x)) ∧ ∃x (Bird(x) ∧ Hawk(x))

∀x (Hawk(x) → ¬Swim(x))
∃x ∃y (Bird(x) ∧ Bird(y) ∧ Hawk(x) ∧ Hawk(y) ∧ ¬(x=y))


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.

∃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)

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.

∃x (Show(x) ∧ AiringAtOn(x, 8PMMonday) ∧ GivenOutOn(x, rose, tV))
∀x (Show(x) ∧ GivenOutOnAt(rose, tV, x) → TheBachelor(x))
∀x (TheBachelor(x) → Portray(x, lifeOfRealPeople))
∀x (Portray(x, liveOfRealPeople) → RealityTVShow(x))
Show(breakingBad)
¬RealityTVShow(breakingBad)


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.


Breaking Bad is a TV show in which roses are given out on Monday at 8 pm.

Show(breakingBad) ∧ GivesOutOnTV(breakingBad, rose) ∧ AiringAt(breakingBad, eightPMMonday)

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)'.


If roses are given out during Breaking Bad, then it is on Monday at 8 pm.

GivesOutOnTV(breakingBad, rose) → AiringAt(breakingBad, eightPMMonday)

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)'.


Heinrich Schmidt was a German politician. Heinrich Schmidt was also a member of the Prussian State Parliament and the Nazi Reichstag.

German(heinrichSchmidt) ∧ Politician(heinrichSchmidt) ∧ Member(heinrichSchmidt, prussianStateParliament) ∧ Member(heinrichSchmidt, naziReichstag)

German(heinrichSchmidt) ∧ Politician(heinrichSchmidt)
Member(heinrichSchmidt, prussianStateParliament) ∧ Member(heinrichSchmidt, naziReichstag)


Heinrich Schmidt was German or Russian or both.


Some German politician was part of both the Prussian State Parliament and the Nazi Reichstag.

∃x (German(x) ∧ Politician(x) ∧ Member(x, prussianStateParliament) ∧ Member(x, naziReichstag))


No politicians are part of the Nazi Reichstag.

∀x (Politician(x) → ¬Member(x, naziReichstag))


If something is a deadly disease, then it comes with a low survival rate. Severe cancers are deadly diseases. Bile duct cancer is a severe form of cancer. All Cholangiocarcinoma is bile duct cancer. Mild flu comes with a low survival rate. Colorectal cancer is not both a bile duct cancer and with a low survival rate.

∀x (DeadlyDisease(x) → ComeWith(x, lowSurvivalRate)) ∧ ∀x (SevereCancer(x) → DeadlyDisease(x)) ∧ ∀x (BileDuctCancer(x) → SevereCancer(x)) ∧ ∀x (Cholangiocarcinoma(x) → BileDuctCancer(x)) ∧ ∀x (MildFlu(x) → ComeWith(x, lowSurvivalRate)) ∧ ¬(BileDuctCancer(colorectalCancer) ∧ ComeWith(colorectalCancer, lowSurvivalRate))

Three corrections made: (1) Predicate naming: the original uses 'DeadlyDiseases(x)' (plural) in premises 1 and 2 — corrected to singular 'DeadlyDisease(x)' to match standard predicate naming conventions and the NL 'a deadly disease'. (2) Premise 5: the original reads '∀x (MildFlu(x) → ¬ComeWith(x, lowSurvivalRate))' — this says mild flu does NOT come with a low survival rate, which is the correct biological reading (mild flu has high survival). However, the NL premise states 'Mild flu comes with a low survival rate', which is the opposite.

∀x (DeadlyDiseases(x) → ComeWith(x, lowSurvivalRate))
∀x (SevereCancer(x) → DeadlyDiseases(x))
∀x (BileDuctCancer(x) → SevereCancer(x))
∀x (Cholangiocarcinoma(x) → BileDuctCancer(x))
∀x (MildFlu(x) → ¬ComeWith(x, lowSurvivalRate))
¬(BileDuctCancer(colorectalCancer) ∧ ComeWith(colorectalCancer, lowSurvivalRate))


Colorectal cancer is a kind of severe cancer.


If colorectal cancer is a kind of bile duct cancer or a form of Cholangiocarcinoma, then colorectal cancer is a kind of bile duct cancer and a kind of mild flu.

(BileDuctCancer(colorectalCancer) ∨ Cholangiocarcinoma(colorectalCancer)) → (BileDuctCancer(colorectalCancer) ∧ MildFlu(colorectalCancer))

The label 'True' is correct via ex falso.

¬(BileDuctCancer(colorectalCancer) ∨ Cholangiocarcinoma(colorectalCancer)) ∨ (BileDuctCancer(colorectalCancer) ∧ MildFlu(colorectalCancer))


Colorectal cancer is a form of Cholangiocarcinoma and it is a kind of mild flu or a kind of bile duct cancer, or all of the above.

Cholangiocarcinoma(colorectalCancer) ∧ (MildFlu(colorectalCancer) ∨ BileDuctCancer(colorectalCancer))


All people who eat salads regularly are very conscious about their health and eating habits. All people who grew up in health-conscious childhood homes eat salads regularly. All people who fulfill their daily nutritional intakes grew up in health-conscious childhood homes. All people who disregard their physical well-being are not very conscious about their health and eating habits. If people visit the gym at least once a day, then they always fulfill their daily nutritional intakes. Taylor either grew up in a health-conscious childhood home and disregards her physical well-being, or she did neither.

∀x (RegularlyEat(x, salad) → ConsciousAbout(x, health) ∧ ConsciousAbout(x, eatingHabit)) ∧ ∀x (GrewUpIn(x, healthConsciousChildhoodHome) → RegularlyEat(x, salad)) ∧ ∀x (Fulfill(x, dailyNutritionalIntake) → GrewUpIn(x, healthConsciousChildhoodHome)) ∧ ∀x (Disregard(x, physicalWellbeing) → ¬(ConsciousAbout(x, health) ∧ ConsciousAbout(x, eatingHabit))) ∧ ∀x (VisitDaily(x, gym) → Fulfill(x, dailyNutritionalIntake)) ∧ ¬(GrewUpIn(taylor, healthConsciousChildhoodHome) ⊕ Disregard(taylor, physicalWellbeing))

Five corrections made: (1) Constant naming: the original uses 'health-consciousChildhoodHome' with a hyphen — hyphens are not suggested as FOL constant identifiers as they can be parsed as subtraction operators. Corrected to 'healthConsciousChildhoodHome' (camelCase) throughout all premises and conclusions. (2) Premise 1: the original uses 'ConsciousAbout(x, eatinggHabit)' — double 'g' typo, causing a mismatch. Corrected to 'ConsciousAbout(x, eatingHabit)'. (3) Premise 6: the original uses 'GrowUpIn' (present tense) instead of 'GrewUpIn' (past tense) used in premises 2 and 3. Unified to 'GrewUpIn' throughout

∀x (RegularlyEat(x, salad) → ConsciousAbout(x, health) ∧ ConsciousAbout(x, eatinggHabit))
∀x (GrewUpIn(x, health-consciousChildhoodHome) → RegularlyEat(x, salad))
∀x (Fulfill(x, dailyNutritionalIntake) → GrewUpIn(x, health-consciousChildhoodHome))
∀x (Disregard(x, physicalWellbeing) → ¬(ConsciousAbout(x, health) ∧ ConsciousAbout(x, eatingHabit)))
∀x (VisitDaily(x, gym) → Fulfill(x, dailyNutritionalIntake))
¬(GrowUpIn(taylor, health-consciousChildhoodHome) ⊕ Disregard(taylor, physicalWellbeing))


Taylor visits the gym at least once a day.

The label 'False' is correct: suppose VisitDaily(taylor, gym). By premise 5 → Fulfill(taylor, dailyNutritionalIntake) → by premise 3 → GrewUpIn(taylor, healthConsciousChildhoodHome). But the 'neither' case of premise 6 gives ¬GrewUpIn(taylor, healthConsciousChildhoodHome) — contradiction. Therefore VisitDaily(taylor, gym) is refutable under the closed-world assumption.


Taylor neither grew up in a health-conscious childhood home nor does she visit the gym at least once a day.

¬GrewUpIn(taylor, healthConsciousChildhoodHome) ∧ ¬VisitDaily(taylor, gym)

'GrowUpIn' → 'GrewUpIn' for tense consistency; added missing closing parenthesis.

¬GrowUpIn(taylor, health-consciousChildhoodHome) ∧ ¬VisitDaily(taylor, gym))


The summer Olympic games is a sporting event. The last summer Olympic games was in Tokyo. The United States won the most medals in Tokyo.

SportingEvent(olympics) ∧ LastSummerOlympics(tokyoolympics) ∧ MostMedals(unitedStates, tokyoolympics)

SportingEvent(olympics)
LastSummerOlympics(tokyo)
MostMedals(unitedStates, tokyo)


The world championships is a sporting event.


The last summer Olympic games were not in Tokyo.


The United States won the most medals in the last summer Olympic games.

∃x (LastSummerOlympics(x) ∧ MostMedals(unitedStates, x))


All horses have hooves. No humans have hooves.

∀x (Horse(x) → ∃y ∃z (¬(y=z) ∧ Hoof(y) ∧ Hoof(z) ∧ Have(x, y) ∧ Have(x, z))) ∧ ∀x (Human(x) → ¬∃y (Hoof(y) ∧ Have(x, y)))

Two corrections made: (1) Premise 1: the original encodes 'all horses have hooves' as having at least two distinct hooves — '∃y ∃z (¬(y=z) ∧ Hoof(y) ∧ Hoof(z) ∧ Have(x, y) ∧ Have(x, z))'. While horses biologically have four hooves, the NL 'have hooves' (plural) could reasonably be interpreted as at least two. This encoding is retained as a plausible formalisation of the plural NL. (2) Premise 2: the original '∀x (Human(x) → ¬(∃y ∃z (¬(y=z) ∧ Hoof(y) ∧ Hoof(z) ∧ Have(x, y) ∧ Have(x, z))))' says humans do not have two distinct hooves — but this would allow humans to have exactly one hoof, which is not the intended meaning. The NL 'no humans have hooves' means humans have NO hooves at all. Corrected to '∀x (Human(x) → ¬∃y (Hoof(y) ∧ Have(x, y)))' — no hoof whatsoever.

∀x (Horse(x) → ∃y ∃z (¬(y=z) ∧ Hoof(y) ∧ Hoof(z) ∧ Have(x, y) ∧ Have(x, z)))
∀x (Human(x) → ¬(∃y ∃z (¬(y=z) ∧ Hoof(y) ∧ Hoof(z) ∧ Have(x, y) ∧ Have(x, z))))


The original FOL uses the two-witness pattern '∃x (Human(x) ∧ Human(y) ∧ Horse(x) ∧ Horse(y) ∧ ¬(x=y))' — additionally, 'y' appears free (no ∃y quantifier is present). This is ill-formed (free variable y).


Susan flies to LGA airport. The departure and arrival cannot be at the same airport. John flies from LGA airport.

FlyTo(susan, lgaAirport) ∧ ∀x ∀y ¬(FlyFrom(x, y) ∧ FlyTo(x, y)) ∧ FlyFrom(john, lgaAirport)

One correction made: premise 2 in the original reads '∀x ∀y (FlyFrom(x, y) ⊕ FlyTo(x, y))' — universally quantifying over both x (person) and y (airport). This asserts that for ALL persons x and ALL airports y, x either flies from y or flies to y (but not both). This is far too strong — it would mean every person either flies from or flies to every airport, which is nonsensical. The intended meaning of 'departure and arrival cannot be at the same airport' is that no person both departs from and arrives at the same airport, i.e. for any person x and airport y, they cannot both FlyFrom(x, y) AND FlyTo(x, y). The correct encoding is: '∀x ∀y ¬(FlyFrom(x, y) ∧ FlyTo(x, y))' — equivalently, FlyFrom(x, y) → ¬FlyTo(x, y) and vice versa.

FlyTo(susan, lgaAirport)
∀x ∀y (FlyFrom(x, y) ⊕ FlyTo(x, y))
FlyFrom(john, lgaAirport)


No one nice to animals is also mean to animals. Some grumpy people are mean to animals. All animal lovers are nice to animals. All pet owners love animals. Tom is a pet owner.

∀x (NiceTo(x, animal) → ¬MeanTo(x, animal)) ∧ ∃x (Grumpy(x) ∧ MeanTo(x, animal)) ∧ ∀x (AnimalLover(x) → NiceTo(x, animal)) ∧ ∀x (PetOwner(x) → AnimalLover(x)) ∧ PetOwner(tom)

∀x (NiceTo(x, animal) → ¬MeanTo(x, animal))
∃x (Grumpy(x) ∧ MeanTo(x, animal))
∀x (AnimalLover(x) → NiceTo(x, animal))
∀x (PetOwner(x) → AnimalLover(x))
PetOwner(tom)


Tom is not both a grumpy person and mean to animals.


If Tom is not both a grumpy person and mean to animals, then Tom is neither a grumpy person nor an animal lover.

(¬(Grumpy(tom) ∧ MeanTo(tom, animal)) → ¬Grumpy(tom) ∧ ¬AnimalLover(tom))

The original FOL has an unclosed parenthesis in the consequent — added missing opening parenthesis to wrap the full conditional correctly.


A podcast is not a novel. If a person is born in an American city, the person is American. If a book is a novel and it is written by a person, then the person is a novel writer. Dani Shapiro is an American writer. Family History is written by Dani Shapiro. Family History is a novel written in 2003. Dani Shapiro created a podcast called Family Secrets. Boston is an American city.

∀x (Podcast(x) → ¬Novel(x)) ∧ ∀x (∃y (BornIn(x, y) ∧ City(y) ∧ LocatedIn(y, america)) → American(x)) ∧ ∀x ∀y (Novel(x) ∧ WrittenBy(x, y) → WritesNovel(y)) ∧ American(daniShapiro) ∧ Writer(daniShapiro) ∧ WrittenBy(familyHistory, daniShapiro) ∧ Novel(familyHistory) ∧ WrittenIn(familyHistory, yr2003) ∧ Podcast(familySecrets) ∧ CreatedBy(familySecrets, daniShapiro) ∧ City(boston) ∧ LocatedIn(boston, america)

Several corrections made: (1) Constant naming: the original uses underscores in constant names ('dani_Shapiro', 'family_History', 'family_Secrets') — while underscores are technically valid in some FOL notations, the consistent convention throughout this dataset is camelCase. Normalised to 'daniShapiro', 'familyHistory', 'familySecrets'. (2) Premise 2: the original has an unclosed parenthesis — '∀x((∃y(BornIn(x, y) ∧ City(y) ∧ LocatedIn(y, america)) → American(x))' missing closing parenthesis; corrected. (3) Premise 8: the original uses 'City(boston) ∧ American(boston)' — but 'American(boston)' treats Boston as an entity that is American using the same predicate applied to persons (American(daniShapiro)). The NL says 'Boston is an American city', meaning Boston is a city located in America. Corrected to 'City(boston) ∧ LocatedIn(boston, america)' to use the location predicate consistent with premise 2, and removed the conflation of 'American' as applying to both cities and people.

∀x (Podcast(x) → ¬Novel(x))
∀x((∃y(BornIn(x, y) ∧ City(y) ∧ LocatedIn(y,america)) → American(x))
∀x ∀y ((Novel(x) ∧ WrittenBy(x, y)) → WritesNovel(y))
American(dani_Shapiro) ∧ Writer(dani_Shapiro)
WrittenBy(family_History, dani_Shapiro)
Novel(family_History) ∧ WrittenIn(family_History, yr2003)
Podcast(family_Secrets) ∧ CreatedBy(family_Secrets, dani_Shapiro)
City(boston) ∧ American(boston)


Two corrections: (1) Predicate name 'IsNovel' → 'Novel' to match premise usage. (2) Constant 'family_Secrets' → 'familySecrets' for camelCase consistency.


Some professional basketball players are not American nationals. All professional basketball players can do jump shots. If someone can do jump shots, they leap straight into the air. If someone leaps straight into the air, they activate their leg muscles. Yuri does not activate his leg muscles.

∃x (Professional(x) ∧ BasketballPlayer(x) ∧ ¬AmericanNational(x)) ∧ ∀x (Professional(x) ∧ BasketballPlayer(x) → CanDo(x, jumpShot)) ∧ ∀x (CanDo(x, jumpShot) → LeapStraightIntoAir(x)) ∧ ∀x (LeapStraightIntoAir(x) → Activate(x, legMuscle)) ∧ ¬Activate(yuri, legMuscle)

∃x (Professional(x) ∧ BasketballPlayer(x) ∧ ¬AmericanNational(x))
∀x (Professional(x) ∧ BasketballPlayer(x) → CanDo(x, jumpShot))
∀x (CanDo(x, jumpShot) → LeapStraightIntoAir(x))
∀x (LeapStraightIntoAir(x) → Activate(x, legMuscle))
¬Activate(yuri, legMuscle)


Yuri is not an American professional basketball player.

¬(AmericanNational(yuri) ∧ Professional(yuri) ∧ BasketballPlayer(yuri))


Yuri is an American professional basketball player.

AmericanNational(yuri) ∧ Professional(yuri) ∧ BasketballPlayer(yuri)


If Yuri does not leap straight into the air, then Yuri is an American professional basketball player.

¬LeapStraightIntoAir(yuri) → (AmericanNational(yuri) ∧ Professional(yuri) ∧ BasketballPlayer(yuri))


If Yuri is not an American professional basketball player, then Yuri is a professional basketball player.

¬(AmericanNational(yuri) ∧ Professional(yuri) ∧ BasketballPlayer(yuri)) → (Professional(yuri) ∧ BasketballPlayer(yuri))


Events are either happy or sad. At least one event is happy.

∀x (Event(x) → Happy(x) ⊕ Sad(x)) ∧ ∃x (Event(x) ∧ Happy(x))

∀x (Event(x) → Happy(x) ⊕ Sad(x))
∃x (Event(x) ∧ Happy(x))


The winner of the 1992 du Maurier Classic was Steinhauer. Steinhauer participated in the 1992 du Maurier Classic. One six-way tie was on the leaderboard, and one person in the six-way tie was from Belgium. Descampe is from Belgium and is on the leaderboard of the 1992 du Maurier Classic. All people on the leaderboard of the 1992 du Maurier Classic participated in the 1992 du Maurier Classic.

WinnerOf(steinhauer, duMaurierClassic, year1992) ∧ Participate(steinhauer, duMaurierClassic, year1992) ∧ ∃x ∃y (OnLeaderboardOf(x, duMaurierClassic) ∧ SixWayTie(x) ∧ In(y, x) ∧ From(y, belgium)) ∧ From(descampe, belgium) ∧ OnLeaderboardOf(descampe, duMaurierClassic) ∧ ∀x (OnLeaderboardOf(x, duMaurierClassic) → Participate(x, duMaurierClassic, year1992)) ∧ ∀x ∀y ∀z (OnLeaderboardOf(x, y) ∧ In(z, x) → OnLeaderboardOf(z, y))

Several corrections made: (1) Premise 2: the original uses 'Participate(duMaurierClassic, steinhauer, year1992)' with the event as the first argument and the person as the second — reversed from the NL 'Steinhauer participated in the 1992 du Maurier Classic' where the person is the subject. Corrected to 'Participate(steinhauer, duMaurierClassic, year1992)' for consistent person-first argument order. (2) Premise 3: the original has a misplaced closing parenthesis — '∃x ∃y (OnLeaderBoardOf(x, duMaurierClassic) ∧ SixWayTie(x)) ∧ In(y, sixWayTie) ∧ Belgium(y))' causes y to fall outside the ∃y scope. Corrected to '∃x ∃y (OnLeaderboardOf(x, duMaurierClassic) ∧ SixWayTie(x) ∧ In(y, x) ∧ From(y, belgium))' — y is now properly scoped, and 'In(y, sixWayTie)' is corrected to 'In(y, x)' since x is the six-way tie entity (not a separate constant 'sixWayTie'). Also 'Belgium(y)' corrected to 'From(y, belgium)' to match the predicate used in premise 4. (3) Predicate casing: 'OnLeaderBoardOf' (capital B) normalised to 'OnLeaderboardOf' throughout. (4) Premise 5: the original reads '∀x (LeaderBoard(maurier, x) → Participate(maurier, x))' — uses an undefined binary predicate 'LeaderBoard(maurier, x)' with 'maurier' as a new constant not introduced elsewhere, and 'Participate(maurier, x)' with the wrong argument order and wrong subject. Corrected to '∀x (OnLeaderboardOf(x, duMaurierClassic) → Participate(x, duMaurierClassic, year1992))' to match the NL 'all people on the leaderboard of the 1992 du Maurier Classic participated in the 1992 du Maurier Classic'. (5) Added (at the end of the premises) the implicitely assumed fact that any person in the tie in the leaderbord is in the leaderbord itself (not necessary for the conclusions).

yes: premises 3 implicitly states that there is a unique six-way tie or only one person in it from Belgium? (No implications in the conclusions labels)

WinnerOf(steinhauer, duMaurierClassic, year1992)
Participate(duMaurierClassic, steinhauer, year1992)
∃x ∃y (OnLeaderBoardOf(x, duMaurierClassic) ∧ SixWayTie(x)) ∧ In(y, sixWayTie) ∧ Belgium(y))
From(descampe, belgium) ∧ OnLeaderBoardOf(descampe, duMaurierClassic)
∀x (LeaderBoard(maurier, x) → Participate(maurier, x))


Steinhauer was not the winner of the 1992 du Maurier Classic.


Descampe is in the six-way tie in the leaderboard of the 1992 du Maurier Classic.

The original FOL 'SixWayTie(descampe)' treats 'SixWayTie' as a unary predicate applying directly to Descampe — asserting Descampe herself is a six-way tie, which is a category error. The NL says 'Descampe is IN the six-way tie', meaning Descampe is a participant in a six-way tie entity. Corrected to '∃x (SixWayTie(x) ∧ OnLeaderboardOf(x, duMaurierClassic) ∧ In(descampe, x))' — there exists a six-way tie x on the leaderboard such that Descampe is in it.


Jane does not like any animals that bark. All dogs are animals that bark. Jane likes all animals that jump. If an animal has legs, then it jumps. If an animal is terricolous, then it has legs. KiKi is an animal. If KiKi neither jumps nor has legs, then KiKi is terricolous or has legs.

∀x (Animal(x) ∧ Bark(x) → ¬Like(jane, x)) ∧ ∀x (Dog(x) → Animal(x) ∧ Bark(x)) ∧ ∀x (Animal(x) ∧ Jump(x) → Like(jane, x)) ∧ ∀x (Animal(x) ∧ HasLegs(x) → Jump(x)) ∧ ∀x (Animal(x) ∧ Terricolous(x) → HasLegs(x)) ∧ Animal(kiki) ∧ (¬Jump(kiki) ∧ ¬HasLegs(kiki) → Terricolous(kiki) ∨ HasLegs(kiki))

Three corrections made: (1) Premises 4 and 5 and premise 7: the original encodes 'has legs' using the two-witness existential '∃y ∃z (¬(y=z) ∧ Leg(y) ∧ Leg(z) ∧ Have(x, y) ∧ Have(x, z))' — asserting at least two distinct leg objects. While most animals have more than one leg, the NL simply says 'has legs' as a property. This encoding is unnecessarily complex and creates asymmetric treatment compared to other predicates. More importantly, using this complex existential in premise 7's antecedent and consequent makes the formula hard to reason with. Simplified to a unary predicate 'HasLegs(x)' throughout for clarity and consistency. (2) Premise 7: the original antecedent uses '¬Jump(kiki) ∧ ¬(∃y ∃z (...))' which is the negation of the two-witness leg condition. Under the simplified HasLegs predicate, this becomes '¬Jump(kiki) ∧ ¬HasLegs(kiki)'. The consequent 'Terricolous(kiki) ∨ ∃y ∃z (...)' similarly simplified to 'Terricolous(kiki) ∨ HasLegs(kiki)'. (3) In premises 4 and 5, we should add the relativization to animals (add Animal(x) in the assumptions of the implications)

∀x (Animal(x) ∧ Bark(x) → ¬Like(jane, x))
∀x (Dog(x) → Animal(x) ∧ Bark(x))
∀x (Animal(x) ∧ Jump(x) → Like(jane, x))
∀x (∃y ∃z (¬(y=z) ∧ Leg(y) ∧ Leg(z) ∧ Have(x, y) ∧ Have(x, z)) → Jump(x))
∀x (Terricolous(x) → ∃y ∃z (¬(y=z) ∧ Leg(y) ∧ Leg(z) ∧ Have(x, y) ∧ Have(x, z)))
Animal(kiki)
¬Jump(kiki) ∧ ¬(∃y ∃z (¬(y=z) ∧ Leg(y) ∧ Leg(z) ∧ Have(kiki, y) ∧ Have(kiki, z))) → Terricolous(kiki) ∨ ∃y ∃z (¬(y=z) ∧ Leg(y) ∧ Leg(z) ∧ Have(kiki, y) ∧ Have(kiki, z))


The label 'False' was incorrect, since no information about the fact that Kiki is a dog (or it's barking) is mentioned in the premises.


Ailton Silva, born in 1995, is commonly known as Ailton. Ailton is a football player who was loaned out to Braga. Ailton Silva is a Brazilian footballer who plays for Náutico. Náutico is a football club along with Braga. Fluminense is a football club.

BornIn(ailtonSilva, year1995) ∧ CommonlyKnownAs(ailtonSilva, ailton) ∧  ailton = ailtonSilva ∧ FootballPlayer(ailton) ∧ LoanedTo(ailton, braga) ∧ Brazilian(ailtonSilva) ∧ FootballPlayer(ailtonSilva) ∧ PlayFor(ailtonSilva, nautico) ∧ FootballClub(nautico) ∧ FootballClub(braga) ∧ FootballClub(fluminense)

Three corrections made: (1) Premise 3: the original uses 'Footballplayer(ailtonSilva)' (lowercase 'p') inconsistent with 'FootballPlayer(ailton)' (uppercase 'P') in premise 2. Unified to 'FootballPlayer(ailtonSilva)'. (2) The story uses two distinct constants 'ailton' and 'ailtonSilva' for the same person — 'Ailton' is explicitly stated to be the common name for 'Ailton Silva' via CommonlyKnownAs(ailtonSilva, ailton). This means premises about 'ailton' (premises 2) and premises about 'ailtonSilva' (premises 1, 3) refer to the same individual but through different constants. For full logical interoperability, an identity axiom ailton = ailtonSilva or a bridging axiom would be needed. This is flagged as a modelling gap — without it, properties of 'ailton' (e.g. LoanedTo(ailton, braga)) cannot be directly transferred to 'ailtonSilva' via standard FOL.

BornIn(ailtonSilva, year1995) ∧ CommonlyKnownAs(ailtonSilva, ailton)
FootballPlayer(ailton) ∧ LoanedTo(ailton, braga)
Brazilian(ailtonSilva) ∧ Footballplayer(ailtonSilva) ∧ PlayFor(ailtonSilva, nautico)
FootballClub(nautico) ∧ FootballClub(braga)
FootballClub(fluminense)


No one playing for Nautico is Brazilian.

∀x (PlayFor(x, nautico) → ¬Brazilian(x))


Ailton Silva does not play for a football club.

∀x (FootballClub(x) → ¬PlayFor(ailtonSilva, x))


Ailton was not loaned out to a football club.

∀x (FootballClub(x) → ¬LoanedTo(ailton, x))


Ailton Silva was loaned out to a football club.

∃x (FootballClub(x) ∧ LoanedTo(ailtonSilva, x))

The label was 'Uncertain' (it reflects the two constants identity gap), but it should be True, since ailtonSilva = ailton.