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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? exists z: Q_2(z) -> S(z). forall z: Q_2(z) or ~S(z). | forall z: S(z) -> Q_2(z). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? exists z: Q_2(z) -> S(z). forall z: Q_2(z) or ~S(z).
### Response:
forall z: S(z) -> Q_2(z).
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Richard is driving a car. If Robert listens to a song, then Robert is a police officer. For all x_12, if x_12 were a police officer, then x_12 is listening to a song. If Robert listens to a song, then Richard is not driving a car. | Robert is not listening to a song. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Richard is driving a car. If Robert listens to a song, then Robert is a police officer. For all x_12, if x_12 were a police officer, then x_12 is listening to a song. If Robert listens to a song, then Richard is not driving a car.
### Response:
Robert is not listening to a song.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Linda is poor if and only if it is snowing. Karen is a server or it is snowing. Linda is not poor. | If it doesn't snow, then Karen is a server. If Linda were poor, then it is snowing. If it snows, then Linda is poor. It is not snowing. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Linda is poor if and only if it is snowing. Karen is a server or it is snowing. Linda is not poor.
### Response:
If it doesn't snow, then Karen is a server. If Linda were poor, then it is snowing. If it snows, then Linda is poor. It is not snowing.
| null | null |
Consider the following premises. forall z: T(z) -> Q(z). forall z: Q(z) <-> T(z). ~Q(f_4). Can we infer ~T(f_4) from them? | From forall z: Q(z) <-> T(z) we can infer forall z: Q(z) -> T(z). From forall z: Q(z) -> T(z), forall z: T(z) -> Q(z) we can infer forall z: Q(z) <-> T(z). From forall z: Q(z) <-> T(z) we can infer Q(f_4) <-> T(f_4). From Q(f_4) <-> T(f_4) we can infer T(f_4) -> Q(f_4). Finally, from T(f_4) -> Q(f_4), ~Q(f_4) we can infer ~T(f_4). Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall z: T(z) -> Q(z). forall z: Q(z) <-> T(z). ~Q(f_4). Can we infer ~T(f_4) from them?
### Response:
From forall z: Q(z) <-> T(z) we can infer forall z: Q(z) -> T(z). From forall z: Q(z) -> T(z), forall z: T(z) -> Q(z) we can infer forall z: Q(z) <-> T(z). From forall z: Q(z) <-> T(z) we can infer Q(f_4) <-> T(f_4). From Q(f_4) <-> T(f_4) we can infer T(f_4) -> Q(f_4). Finally, from T(f_4) -> Q(f_4), ~Q(f_4) we can infer ~T(f_4). Therefore, the answer is yes.
| null | null |
Consider the following premises. forall x_8: R(x_8) -> P_8(x_8). forall x_8: P_8(x_8) -> Q_8(x_8). forall x_8: ~P_5(x_8). exists x_8: Q_8(x_8) -> R(x_8). forall x_8: P_5(x_8) <-> R(x_8). Can we infer forall x_8: P_8(x_8) from them? If possible, name the inference rules being used at each step. | From forall x_8: P_8(x_8) -> Q_8(x_8), exists x_8: Q_8(x_8) -> R(x_8) we can infer exists x_8: P_8(x_8) -> R(x_8) via existential transitivity. From forall x_8: P_5(x_8) <-> R(x_8), forall x_8: ~P_5(x_8) we can infer forall x_8: ~R(x_8) via universal biconditional elimination. From forall x_8: R(x_8) -> P_8(x_8), exists x_8: P_8(x_8) -> R(x_8) we can infer exists x_8: R(x_8) <-> P_8(x_8) via existential biconditional introduction. Finally, from exists x_8: R(x_8) <-> P_8(x_8), forall x_8: ~R(x_8) we can infer exists x_8: ~P_8(x_8) via existential biconditional elimination, which contradicts forall x_8: P_8(x_8). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_8: R(x_8) -> P_8(x_8). forall x_8: P_8(x_8) -> Q_8(x_8). forall x_8: ~P_5(x_8). exists x_8: Q_8(x_8) -> R(x_8). forall x_8: P_5(x_8) <-> R(x_8). Can we infer forall x_8: P_8(x_8) from them? If possible, name the inference rules being used at each step.
### Response:
From forall x_8: P_8(x_8) -> Q_8(x_8), exists x_8: Q_8(x_8) -> R(x_8) we can infer exists x_8: P_8(x_8) -> R(x_8) via existential transitivity. From forall x_8: P_5(x_8) <-> R(x_8), forall x_8: ~P_5(x_8) we can infer forall x_8: ~R(x_8) via universal biconditional elimination. From forall x_8: R(x_8) -> P_8(x_8), exists x_8: P_8(x_8) -> R(x_8) we can infer exists x_8: R(x_8) <-> P_8(x_8) via existential biconditional introduction. Finally, from exists x_8: R(x_8) <-> P_8(x_8), forall x_8: ~R(x_8) we can infer exists x_8: ~P_8(x_8) via existential biconditional elimination, which contradicts forall x_8: P_8(x_8). Therefore, the answer is no.
| null | null |
Consider the following premises. u -> ~t. u -> q. q <-> u. t. Can we infer q from them? If possible, name the inference rules being used at each step. | From q <-> u we can infer q -> u via biconditional elimination. From u -> q, q -> u we can infer u <-> q via biconditional introduction. From u -> ~t, t we can infer ~u via modus tollens. Finally, from u <-> q, ~u we can infer ~q via biconditional elimination, which contradicts q. Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. u -> ~t. u -> q. q <-> u. t. Can we infer q from them? If possible, name the inference rules being used at each step.
### Response:
From q <-> u we can infer q -> u via biconditional elimination. From u -> q, q -> u we can infer u <-> q via biconditional introduction. From u -> ~t, t we can infer ~u via modus tollens. Finally, from u <-> q, ~u we can infer ~q via biconditional elimination, which contradicts q. Therefore, the answer is no.
| null | null |
Consider the following premises. forall x_14: R_14(x_14) -> S_14(x_14). w_1. R_14(a_14). Can we infer ~w_1 from them? If possible, name the inference rules being used at each step. | That contradicts the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_14: R_14(x_14) -> S_14(x_14). w_1. R_14(a_14). Can we infer ~w_1 from them? If possible, name the inference rules being used at each step.
### Response:
That contradicts the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_6, if x_6 climbs a mountain, then x_6 is bored. For all x_6, x_6 is excited. There is at least one x_6 for which x_6 is bored if and only if x_6 is excited. | There is at least one x_6 for which if x_6 were bored, then x_6 is excited can be inferred via the existential biconditional elimination rule. There is at least one x_6 for which if x_6 were excited, then x_6 is bored can be inferred via the existential biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_6, if x_6 climbs a mountain, then x_6 is bored. For all x_6, x_6 is excited. There is at least one x_6 for which x_6 is bored if and only if x_6 is excited.
### Response:
There is at least one x_6 for which if x_6 were bored, then x_6 is excited can be inferred via the existential biconditional elimination rule. There is at least one x_6 for which if x_6 were excited, then x_6 is bored can be inferred via the existential biconditional elimination rule.
| null | null |
Consider the following premises. If Thomas plays tennis, then it is cold. Barbara is an electrician if and only if Thomas is playing tennis. Richard is not curious. If it is cold, then Richard is curious. Can we infer the following from them? Barbara is an electrician. | From the fact that if Thomas plays tennis, then it is cold, and that if it is cold, then Richard is curious we can infer that if Thomas plays tennis, then Richard is curious. From the fact that if Thomas plays tennis, then Richard is curious, and that Richard is not curious we can infer that Thomas is not playing tennis. Finally, from the fact that Barbara is an electrician if and only if Thomas is playing tennis, and that Thomas is not playing tennis we can infer that Barbara is not an electrician, which contradicts that Barbara is an electrician. Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. If Thomas plays tennis, then it is cold. Barbara is an electrician if and only if Thomas is playing tennis. Richard is not curious. If it is cold, then Richard is curious. Can we infer the following from them? Barbara is an electrician.
### Response:
From the fact that if Thomas plays tennis, then it is cold, and that if it is cold, then Richard is curious we can infer that if Thomas plays tennis, then Richard is curious. From the fact that if Thomas plays tennis, then Richard is curious, and that Richard is not curious we can infer that Thomas is not playing tennis. Finally, from the fact that Barbara is an electrician if and only if Thomas is playing tennis, and that Thomas is not playing tennis we can infer that Barbara is not an electrician, which contradicts that Barbara is an electrician. Therefore, the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? It is raining if and only if it is late. It is cloudy. It is not raining. If Michael works, then it is late. | It is cloudy and it is not raining. It is not raining and it is cloudy. If it rains, then it is late. If it is late, then it is raining. It is not late. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? It is raining if and only if it is late. It is cloudy. It is not raining. If Michael works, then it is late.
### Response:
It is cloudy and it is not raining. It is not raining and it is cloudy. If it rains, then it is late. If it is late, then it is raining. It is not late.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x_3, x_3 is not a poet. For all x_3, x_3 will climb a mountain. For all x_3, x_3 will not climb a mountain or x_3 is a musician. | Nothing can be inferred from these premises. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x_3, x_3 is not a poet. For all x_3, x_3 will climb a mountain. For all x_3, x_3 will not climb a mountain or x_3 is a musician.
### Response:
Nothing can be inferred from these premises.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_7, x_7 will write a letter. For all x_7, if x_7 were a police officer, then x_7 is a sailor. There is at least one x_7 for which if x_7 were a nurse, then x_7 is a sailor. For all x_7, x_7 is a nurse or x_7 is a police officer. | There is at least one x_7 for which x_7 is a sailor can be inferred via the existential disjunction elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_7, x_7 will write a letter. For all x_7, if x_7 were a police officer, then x_7 is a sailor. There is at least one x_7 for which if x_7 were a nurse, then x_7 is a sailor. For all x_7, x_7 is a nurse or x_7 is a police officer.
### Response:
There is at least one x_7 for which x_7 is a sailor can be inferred via the existential disjunction elimination rule.
| null | null |
Translate the following inference to logic notation: For all x, if x_5 writes a letter, then x_5 is listening to a song. There is at least one x for which if x_5 writes a letter, then x_5 is listening to a song. For all x, if x_5 listens to a song, then x_5 is writing a letter. Therefore there is at least one x for which x_5 will write a letter if and only if x_5 will listen to a song. | forall x: P(x) -> Q(x). exists x: P(x) -> Q(x). forall x: Q(x) -> P(x). Therefore exists x: P(x) <-> Q(x). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Translate the following inference to logic notation: For all x, if x_5 writes a letter, then x_5 is listening to a song. There is at least one x for which if x_5 writes a letter, then x_5 is listening to a song. For all x, if x_5 listens to a song, then x_5 is writing a letter. Therefore there is at least one x for which x_5 will write a letter if and only if x_5 will listen to a song.
### Response:
forall x: P(x) -> Q(x). exists x: P(x) -> Q(x). forall x: Q(x) -> P(x). Therefore exists x: P(x) <-> Q(x).
| null | null |
Consider the following premises. T(a_7) -> W(a_7). forall z: T(z) <-> W(z). Can we infer T(a_7) <-> W(a_7) from them? If possible, name the inference rules being used at each step. | We can infer T(a_7) <-> W(a_7) via the universal instantiation rule. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. T(a_7) -> W(a_7). forall z: T(z) <-> W(z). Can we infer T(a_7) <-> W(a_7) from them? If possible, name the inference rules being used at each step.
### Response:
We can infer T(a_7) <-> W(a_7) via the universal instantiation rule. Therefore, the answer is yes.
| null | null |
Consider the following premises. forall x: U(x) or ~T_3(x). forall x: R_2(x) -> U(x). forall x: W_3(x) or T_3(x). forall x: ~W_3(x) or R_2(x). Can we infer ~U(h) from them? If possible, name the inference rules being used at each step. | From forall x: U(x) or ~T_3(x) we can infer forall x: T_3(x) -> U(x) via universal rewriting. From forall x: W_3(x) or T_3(x), forall x: ~W_3(x) or R_2(x) we can infer forall x: T_3(x) or R_2(x) via universal resolution. Finally, from forall x: T_3(x) -> U(x), forall x: R_2(x) -> U(x), forall x: T_3(x) or R_2(x) we can infer forall x: U(x) via universal disjunction elimination, which contradicts ~U(h). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x: U(x) or ~T_3(x). forall x: R_2(x) -> U(x). forall x: W_3(x) or T_3(x). forall x: ~W_3(x) or R_2(x). Can we infer ~U(h) from them? If possible, name the inference rules being used at each step.
### Response:
From forall x: U(x) or ~T_3(x) we can infer forall x: T_3(x) -> U(x) via universal rewriting. From forall x: W_3(x) or T_3(x), forall x: ~W_3(x) or R_2(x) we can infer forall x: T_3(x) or R_2(x) via universal resolution. Finally, from forall x: T_3(x) -> U(x), forall x: R_2(x) -> U(x), forall x: T_3(x) or R_2(x) we can infer forall x: U(x) via universal disjunction elimination, which contradicts ~U(h). Therefore, the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall y: P_4(y) -> P_1(y). forall y: P_4(y). | forall y: P_1(y) can be inferred via the universal modus ponens rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall y: P_4(y) -> P_1(y). forall y: P_4(y).
### Response:
forall y: P_1(y) can be inferred via the universal modus ponens rule.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x, x will make tea. For all x, if x makes tea, then x is a sailor. Michael is curious. | For all x, x is a sailor. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x, x will make tea. For all x, if x makes tea, then x is a sailor. Michael is curious.
### Response:
For all x, x is a sailor.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_13: U(x_13). forall x_13: Q_4(x_13) <-> T_3(x_13). exists x_13: T_3(x_13). | exists x_13: T_3(x_13) and U(x_13). exists x_13: U(x_13) and T_3(x_13). forall x_13: Q_4(x_13) -> T_3(x_13). forall x_13: T_3(x_13) -> Q_4(x_13). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_13: U(x_13). forall x_13: Q_4(x_13) <-> T_3(x_13). exists x_13: T_3(x_13).
### Response:
exists x_13: T_3(x_13) and U(x_13). exists x_13: U(x_13) and T_3(x_13). forall x_13: Q_4(x_13) -> T_3(x_13). forall x_13: T_3(x_13) -> Q_4(x_13).
| null | null |
Consider the following premises. It is sunny if and only if it is snowing. It is snowing. It is not sunny. Can we infer the following from them? Michael is listening to a song. | We cannot infer that from the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. It is sunny if and only if it is snowing. It is snowing. It is not sunny. Can we infer the following from them? Michael is listening to a song.
### Response:
We cannot infer that from the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_7: P_7(x_7) <-> Q_3(x_7). forall x_7: P_7(x_7) <-> Q_7(x_7). | forall x_7: P_7(x_7) -> Q_3(x_7). forall x_7: P_7(x_7) -> Q_7(x_7). forall x_7: Q_3(x_7) -> P_7(x_7). forall x_7: Q_7(x_7) -> P_7(x_7). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_7: P_7(x_7) <-> Q_3(x_7). forall x_7: P_7(x_7) <-> Q_7(x_7).
### Response:
forall x_7: P_7(x_7) -> Q_3(x_7). forall x_7: P_7(x_7) -> Q_7(x_7). forall x_7: Q_3(x_7) -> P_7(x_7). forall x_7: Q_7(x_7) -> P_7(x_7).
| null | null |
Consider the following premises. For all x_10, x_10 is a musician if and only if x_10 is a soldier. For all x_10, if x_10 were a musician, then x_10 is bored. For all x_10, if x_10 were a soldier, then x_10 is curious. For all x_10, if x_10 were curious, then x_10 is a soldier. For all x_10, if x_10 were bored, then x_10 is a musician. For all x_10, x_10 is not curious. Can we infer the following from them? For all x_10, x_10 is not bored. | From the fact that for all x_10, if x_10 were a soldier, then x_10 is curious, and that for all x_10, if x_10 were curious, then x_10 is a soldier we can infer that for all x_10, x_10 is a soldier if and only if x_10 is curious. From the fact that for all x_10, if x_10 were a musician, then x_10 is bored, and that for all x_10, if x_10 were bored, then x_10 is a musician we can infer that for all x_10, x_10 is a musician if and only if x_10 is bored. From the fact that for all x_10, x_10 is a soldier if and only if x_10 is curious, and that for all x_10, x_10 is not curious we can infer that for all x_10, x_10 is not a soldier. From the fact that for all x_10, x_10 is a musician if and only if x_10 is a soldier, and that for all x_10, x_10 is not a soldier we can infer that for all x_10, x_10 is not a musician. Finally, from the fact that for all x_10, x_10 is a musician if and only if x_10 is bored, and that for all x_10, x_10 is not a musician we can infer that for all x_10, x_10 is not bored. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_10, x_10 is a musician if and only if x_10 is a soldier. For all x_10, if x_10 were a musician, then x_10 is bored. For all x_10, if x_10 were a soldier, then x_10 is curious. For all x_10, if x_10 were curious, then x_10 is a soldier. For all x_10, if x_10 were bored, then x_10 is a musician. For all x_10, x_10 is not curious. Can we infer the following from them? For all x_10, x_10 is not bored.
### Response:
From the fact that for all x_10, if x_10 were a soldier, then x_10 is curious, and that for all x_10, if x_10 were curious, then x_10 is a soldier we can infer that for all x_10, x_10 is a soldier if and only if x_10 is curious. From the fact that for all x_10, if x_10 were a musician, then x_10 is bored, and that for all x_10, if x_10 were bored, then x_10 is a musician we can infer that for all x_10, x_10 is a musician if and only if x_10 is bored. From the fact that for all x_10, x_10 is a soldier if and only if x_10 is curious, and that for all x_10, x_10 is not curious we can infer that for all x_10, x_10 is not a soldier. From the fact that for all x_10, x_10 is a musician if and only if x_10 is a soldier, and that for all x_10, x_10 is not a soldier we can infer that for all x_10, x_10 is not a musician. Finally, from the fact that for all x_10, x_10 is a musician if and only if x_10 is bored, and that for all x_10, x_10 is not a musician we can infer that for all x_10, x_10 is not bored. Therefore, the answer is yes.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall x_15: P_15(x_15) or ~Q_15(x_15). P_15(j_3) -> r_3. forall x_15: Q_15(x_15). | forall x_15: Q_15(x_15) -> P_15(x_15) can be inferred via the universal rewriting rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall x_15: P_15(x_15) or ~Q_15(x_15). P_15(j_3) -> r_3. forall x_15: Q_15(x_15).
### Response:
forall x_15: Q_15(x_15) -> P_15(x_15) can be inferred via the universal rewriting rule.
| null | null |
Consider the following premises. forall x_5: P_3(x_5) and Q_3(x_5). forall x_5: P_5(x_5). forall x_5: P_5(x_5) -> S(x_5). Can we infer forall x_5: S(x_5) and P_3(x_5) from them? | From forall x_5: P_5(x_5) -> S(x_5), forall x_5: P_5(x_5) we can infer forall x_5: S(x_5). From forall x_5: P_3(x_5) and Q_3(x_5) we can infer forall x_5: P_3(x_5). Finally, from forall x_5: S(x_5), forall x_5: P_3(x_5) we can infer forall x_5: S(x_5) and P_3(x_5). Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_5: P_3(x_5) and Q_3(x_5). forall x_5: P_5(x_5). forall x_5: P_5(x_5) -> S(x_5). Can we infer forall x_5: S(x_5) and P_3(x_5) from them?
### Response:
From forall x_5: P_5(x_5) -> S(x_5), forall x_5: P_5(x_5) we can infer forall x_5: S(x_5). From forall x_5: P_3(x_5) and Q_3(x_5) we can infer forall x_5: P_3(x_5). Finally, from forall x_5: S(x_5), forall x_5: P_3(x_5) we can infer forall x_5: S(x_5) and P_3(x_5). Therefore, the answer is yes.
| null | null |
Consider the following premises. exists x_14: P_14(x_14) or S(x_14). forall x_14: ~Q_14(x_14). forall x_14: P_10(x_14). forall x_14: P_10(x_14) -> T_3(x_14). forall x_14: P_14(x_14) -> Q_14(x_14). Can we infer forall x_14: ~(S(x_14) and T_3(x_14)) from them? If possible, name the inference rules being used at each step. | From forall x_14: P_14(x_14) -> Q_14(x_14), forall x_14: ~Q_14(x_14) we can infer forall x_14: ~P_14(x_14) via universal modus tollens. From forall x_14: P_10(x_14) -> T_3(x_14), forall x_14: P_10(x_14) we can infer forall x_14: T_3(x_14) via universal modus ponens. From exists x_14: P_14(x_14) or S(x_14), forall x_14: ~P_14(x_14) we can infer exists x_14: S(x_14) via existential disjunctive syllogism. Finally, from exists x_14: S(x_14), forall x_14: T_3(x_14) we can infer exists x_14: S(x_14) and T_3(x_14) via existential conjunction, which contradicts forall x_14: ~(S(x_14) and T_3(x_14)). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. exists x_14: P_14(x_14) or S(x_14). forall x_14: ~Q_14(x_14). forall x_14: P_10(x_14). forall x_14: P_10(x_14) -> T_3(x_14). forall x_14: P_14(x_14) -> Q_14(x_14). Can we infer forall x_14: ~(S(x_14) and T_3(x_14)) from them? If possible, name the inference rules being used at each step.
### Response:
From forall x_14: P_14(x_14) -> Q_14(x_14), forall x_14: ~Q_14(x_14) we can infer forall x_14: ~P_14(x_14) via universal modus tollens. From forall x_14: P_10(x_14) -> T_3(x_14), forall x_14: P_10(x_14) we can infer forall x_14: T_3(x_14) via universal modus ponens. From exists x_14: P_14(x_14) or S(x_14), forall x_14: ~P_14(x_14) we can infer exists x_14: S(x_14) via existential disjunctive syllogism. Finally, from exists x_14: S(x_14), forall x_14: T_3(x_14) we can infer exists x_14: S(x_14) and T_3(x_14) via existential conjunction, which contradicts forall x_14: ~(S(x_14) and T_3(x_14)). Therefore, the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall y: P(y). forall y: P(y) <-> T(y). | forall y: P(y) -> T(y) can be inferred via the universal biconditional elimination rule. forall y: T(y) -> P(y) can be inferred via the universal biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall y: P(y). forall y: P(y) <-> T(y).
### Response:
forall y: P(y) -> T(y) can be inferred via the universal biconditional elimination rule. forall y: T(y) -> P(y) can be inferred via the universal biconditional elimination rule.
| null | null |
Consider the following premises. forall x_8: ~P_8(x_8). forall x_8: P_8(x_8) <-> Q(x_8). exists x_8: W(x_8) -> Q(x_8). Can we infer forall x_8: W(x_8) from them? If possible, name the inference rules being used at each step. | From forall x_8: P_8(x_8) <-> Q(x_8) we can infer forall x_8: Q(x_8) -> P_8(x_8) via universal biconditional elimination. From forall x_8: Q(x_8) -> P_8(x_8), forall x_8: ~P_8(x_8) we can infer forall x_8: ~Q(x_8) via universal modus tollens. Finally, from exists x_8: W(x_8) -> Q(x_8), forall x_8: ~Q(x_8) we can infer exists x_8: ~W(x_8) via existential modus tollens, which contradicts forall x_8: W(x_8). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_8: ~P_8(x_8). forall x_8: P_8(x_8) <-> Q(x_8). exists x_8: W(x_8) -> Q(x_8). Can we infer forall x_8: W(x_8) from them? If possible, name the inference rules being used at each step.
### Response:
From forall x_8: P_8(x_8) <-> Q(x_8) we can infer forall x_8: Q(x_8) -> P_8(x_8) via universal biconditional elimination. From forall x_8: Q(x_8) -> P_8(x_8), forall x_8: ~P_8(x_8) we can infer forall x_8: ~Q(x_8) via universal modus tollens. Finally, from exists x_8: W(x_8) -> Q(x_8), forall x_8: ~Q(x_8) we can infer exists x_8: ~W(x_8) via existential modus tollens, which contradicts forall x_8: W(x_8). Therefore, the answer is no.
| null | null |
Consider the following premises. If Jessica plays tennis, then Robert is a server. Richard is climbing a mountain if and only if Robert is a server. If Richard climbs a mountain, then Jessica is playing tennis. Can we infer the following from them? If we can, name the inference rule being used: Richard is climbing a mountain if and only if Robert is a server. | That is one of the premises. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. If Jessica plays tennis, then Robert is a server. Richard is climbing a mountain if and only if Robert is a server. If Richard climbs a mountain, then Jessica is playing tennis. Can we infer the following from them? If we can, name the inference rule being used: Richard is climbing a mountain if and only if Robert is a server.
### Response:
That is one of the premises. Therefore, the answer is yes.
| null | null |
Consider the following premises. q_1 -> p. p_7 or ~p. p_7 -> q_1. Can we infer p_7 <-> p from them? If possible, name the inference rules being used at each step. | From p_7 or ~p we can infer p -> p_7 via rewriting. From p_7 -> q_1, q_1 -> p we can infer p_7 -> p via transitivity. Finally, from p_7 -> p, p -> p_7 we can infer p_7 <-> p via biconditional introduction. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. q_1 -> p. p_7 or ~p. p_7 -> q_1. Can we infer p_7 <-> p from them? If possible, name the inference rules being used at each step.
### Response:
From p_7 or ~p we can infer p -> p_7 via rewriting. From p_7 -> q_1, q_1 -> p we can infer p_7 -> p via transitivity. Finally, from p_7 -> p, p -> p_7 we can infer p_7 <-> p via biconditional introduction. Therefore, the answer is yes.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? R_2(j) <-> w. forall x_7: R_2(x_7) <-> S_7(x_7). ~w. | R_2(j) -> w. w -> R_2(j). ~R_2(j). forall x_7: R_2(x_7) -> S_7(x_7). forall x_7: S_7(x_7) -> R_2(x_7). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? R_2(j) <-> w. forall x_7: R_2(x_7) <-> S_7(x_7). ~w.
### Response:
R_2(j) -> w. w -> R_2(j). ~R_2(j). forall x_7: R_2(x_7) -> S_7(x_7). forall x_7: S_7(x_7) -> R_2(x_7).
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_5, x_5 will not play squash. For all x_5, if x_5 cooks, then x_5 is writing a letter. For all x_5, x_5 will play squash if and only if x_5 will cook. There is at least one x_5 for which if x_5 writes a letter, then x_5 is cooking. | There is at least one x_5 for which x_5 will write a letter if and only if x_5 will cook can be inferred via the existential biconditional introduction rule. There is at least one x_5 for which x_5 will cook if and only if x_5 will write a letter can be inferred via the existential biconditional introduction rule. For all x_5, if x_5 plays squash, then x_5 is cooking can be inferred via the universal biconditional elimination rule. For all x_5, if x_5 cooks, then x_5 is playing squash can be inferred via the universal biconditional elimination rule. For all x_5, x_5 will not cook can be inferred via the universal biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_5, x_5 will not play squash. For all x_5, if x_5 cooks, then x_5 is writing a letter. For all x_5, x_5 will play squash if and only if x_5 will cook. There is at least one x_5 for which if x_5 writes a letter, then x_5 is cooking.
### Response:
There is at least one x_5 for which x_5 will write a letter if and only if x_5 will cook can be inferred via the existential biconditional introduction rule. There is at least one x_5 for which x_5 will cook if and only if x_5 will write a letter can be inferred via the existential biconditional introduction rule. For all x_5, if x_5 plays squash, then x_5 is cooking can be inferred via the universal biconditional elimination rule. For all x_5, if x_5 cooks, then x_5 is playing squash can be inferred via the universal biconditional elimination rule. For all x_5, x_5 will not cook can be inferred via the universal biconditional elimination rule.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? It is not overcast or Joseph is making tea. If David were a politician, then it is raining. Richard is not a lawyer or Patricia is sleeping. It is overcast or Richard is a lawyer. If Joseph makes tea, then David is a politician. If Patricia sleeps, then it is raining. | If Joseph makes tea, then it is raining. Richard is a lawyer or Joseph is making tea. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? It is not overcast or Joseph is making tea. If David were a politician, then it is raining. Richard is not a lawyer or Patricia is sleeping. It is overcast or Richard is a lawyer. If Joseph makes tea, then David is a politician. If Patricia sleeps, then it is raining.
### Response:
If Joseph makes tea, then it is raining. Richard is a lawyer or Joseph is making tea.
| null | null |
Consider the following premises. forall x_5: Q_5(x_5) -> U_2(x_5). forall x_5: U_2(x_5) <-> Q_4(x_5). forall x_5: U_2(x_5) -> Q_5(x_5). forall x_5: ~Q_4(x_5). Can we infer exists x_5: Q_5(x_5) from them? | From forall x_5: U_2(x_5) -> Q_5(x_5), forall x_5: Q_5(x_5) -> U_2(x_5) we can infer forall x_5: U_2(x_5) <-> Q_5(x_5). From forall x_5: U_2(x_5) <-> Q_4(x_5), forall x_5: ~Q_4(x_5) we can infer forall x_5: ~U_2(x_5). Finally, from forall x_5: U_2(x_5) <-> Q_5(x_5), forall x_5: ~U_2(x_5) we can infer forall x_5: ~Q_5(x_5), which contradicts exists x_5: Q_5(x_5). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_5: Q_5(x_5) -> U_2(x_5). forall x_5: U_2(x_5) <-> Q_4(x_5). forall x_5: U_2(x_5) -> Q_5(x_5). forall x_5: ~Q_4(x_5). Can we infer exists x_5: Q_5(x_5) from them?
### Response:
From forall x_5: U_2(x_5) -> Q_5(x_5), forall x_5: Q_5(x_5) -> U_2(x_5) we can infer forall x_5: U_2(x_5) <-> Q_5(x_5). From forall x_5: U_2(x_5) <-> Q_4(x_5), forall x_5: ~Q_4(x_5) we can infer forall x_5: ~U_2(x_5). Finally, from forall x_5: U_2(x_5) <-> Q_5(x_5), forall x_5: ~U_2(x_5) we can infer forall x_5: ~Q_5(x_5), which contradicts exists x_5: Q_5(x_5). Therefore, the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? t_2 -> r. r <-> t_2. ~r. | ~t_2. r -> t_2. t_2 -> r. ~t_2. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? t_2 -> r. r <-> t_2. ~r.
### Response:
~t_2. r -> t_2. t_2 -> r. ~t_2.
| null | null |
Consider the following premises. forall y: W_3(y) <-> R_1(y). exists y: W_3(y) <-> T(y). Can we infer forall y: ~(R_1(y) -> T(y)) from them? | That contradicts the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall y: W_3(y) <-> R_1(y). exists y: W_3(y) <-> T(y). Can we infer forall y: ~(R_1(y) -> T(y)) from them?
### Response:
That contradicts the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: Jessica is not a mechanic or Linda is taking a plane. Susan is rich if and only if Linda is a musician. Susan is not rich. Linda is a musician or Richard is working. | If Susan were rich, then Linda is a musician can be inferred via the biconditional elimination rule. If Linda were a musician, then Susan is rich can be inferred via the biconditional elimination rule. Linda is not a musician can be inferred via the biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: Jessica is not a mechanic or Linda is taking a plane. Susan is rich if and only if Linda is a musician. Susan is not rich. Linda is a musician or Richard is working.
### Response:
If Susan were rich, then Linda is a musician can be inferred via the biconditional elimination rule. If Linda were a musician, then Susan is rich can be inferred via the biconditional elimination rule. Linda is not a musician can be inferred via the biconditional elimination rule.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: Elisabeth is not a carpenter. For all x_9, x_9 will go running. If Thomas plays squash, then Elisabeth is a carpenter. | Thomas is not playing squash can be inferred via the modus tollens rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: Elisabeth is not a carpenter. For all x_9, x_9 will go running. If Thomas plays squash, then Elisabeth is a carpenter.
### Response:
Thomas is not playing squash can be inferred via the modus tollens rule.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall z_2: Q(z_2) or ~S(z_2). forall z_2: S(z_2) <-> Q(z_2). | forall z_2: S(z_2) -> Q(z_2). forall z_2: S(z_2) -> Q(z_2). forall z_2: Q(z_2) -> S(z_2). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall z_2: Q(z_2) or ~S(z_2). forall z_2: S(z_2) <-> Q(z_2).
### Response:
forall z_2: S(z_2) -> Q(z_2). forall z_2: S(z_2) -> Q(z_2). forall z_2: Q(z_2) -> S(z_2).
| null | null |
Consider the following premises. For all z_4, z_4 will take a plane if and only if z_4 is an office clerk. There is at least one z_4 for which z_4 will take a plane if and only if z_4 is excited. For all z_4, z_4 is not an office clerk. Can we infer the following from them? For all z_4, z_4 is excited. | From the fact that for all z_4, z_4 will take a plane if and only if z_4 is an office clerk we can infer that for all z_4, if z_4 takes a plane, then z_4 is an office clerk. From the fact that for all z_4, if z_4 takes a plane, then z_4 is an office clerk, and that for all z_4, z_4 is not an office clerk we can infer that for all z_4, z_4 will not take a plane. Finally, from the fact that there is at least one z_4 for which z_4 will take a plane if and only if z_4 is excited, and that for all z_4, z_4 will not take a plane we can infer that there is at least one z_4 for which z_4 is not excited, which contradicts that For all z_4, z_4 is excited. Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all z_4, z_4 will take a plane if and only if z_4 is an office clerk. There is at least one z_4 for which z_4 will take a plane if and only if z_4 is excited. For all z_4, z_4 is not an office clerk. Can we infer the following from them? For all z_4, z_4 is excited.
### Response:
From the fact that for all z_4, z_4 will take a plane if and only if z_4 is an office clerk we can infer that for all z_4, if z_4 takes a plane, then z_4 is an office clerk. From the fact that for all z_4, if z_4 takes a plane, then z_4 is an office clerk, and that for all z_4, z_4 is not an office clerk we can infer that for all z_4, z_4 will not take a plane. Finally, from the fact that there is at least one z_4 for which z_4 will take a plane if and only if z_4 is excited, and that for all z_4, z_4 will not take a plane we can infer that there is at least one z_4 for which z_4 is not excited, which contradicts that For all z_4, z_4 is excited. Therefore, the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_13: P_13(x_13) -> R(x_13). exists x_13: P_6(x_13) -> R(x_13). forall x_13: R(x_13) -> P_13(x_13). forall x_13: P_6(x_13) or ~P_13(x_13). | exists x_13: P_6(x_13) -> P_13(x_13). forall x_13: P_13(x_13) -> P_6(x_13). forall x_13: P_13(x_13) <-> R(x_13). forall x_13: R(x_13) <-> P_13(x_13). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_13: P_13(x_13) -> R(x_13). exists x_13: P_6(x_13) -> R(x_13). forall x_13: R(x_13) -> P_13(x_13). forall x_13: P_6(x_13) or ~P_13(x_13).
### Response:
exists x_13: P_6(x_13) -> P_13(x_13). forall x_13: P_13(x_13) -> P_6(x_13). forall x_13: P_13(x_13) <-> R(x_13). forall x_13: R(x_13) <-> P_13(x_13).
| null | null |
Consider the following premises. forall x_10: W(x_10) <-> Q_8(x_10). forall x_10: P_10(x_10) or W(x_10). forall x_10: ~P_10(x_10) or R_10(x_10). exists x_10: ~Q_8(x_10). Can we infer forall x_10: ~R_10(x_10) from them? If possible, name the inference rules being used at each step. | From forall x_10: P_10(x_10) or W(x_10), forall x_10: ~P_10(x_10) or R_10(x_10) we can infer forall x_10: W(x_10) or R_10(x_10) via universal resolution. From forall x_10: W(x_10) <-> Q_8(x_10), exists x_10: ~Q_8(x_10) we can infer exists x_10: ~W(x_10) via existential biconditional elimination. Finally, from forall x_10: W(x_10) or R_10(x_10), exists x_10: ~W(x_10) we can infer exists x_10: R_10(x_10) via existential disjunctive syllogism, which contradicts forall x_10: ~R_10(x_10). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_10: W(x_10) <-> Q_8(x_10). forall x_10: P_10(x_10) or W(x_10). forall x_10: ~P_10(x_10) or R_10(x_10). exists x_10: ~Q_8(x_10). Can we infer forall x_10: ~R_10(x_10) from them? If possible, name the inference rules being used at each step.
### Response:
From forall x_10: P_10(x_10) or W(x_10), forall x_10: ~P_10(x_10) or R_10(x_10) we can infer forall x_10: W(x_10) or R_10(x_10) via universal resolution. From forall x_10: W(x_10) <-> Q_8(x_10), exists x_10: ~Q_8(x_10) we can infer exists x_10: ~W(x_10) via existential biconditional elimination. Finally, from forall x_10: W(x_10) or R_10(x_10), exists x_10: ~W(x_10) we can infer exists x_10: R_10(x_10) via existential disjunctive syllogism, which contradicts forall x_10: ~R_10(x_10). Therefore, the answer is no.
| null | null |
Consider the following premises. forall x_5: Q_5(x_5) -> R_5(x_5). forall x_5: P_5(x_5) -> Q_5(x_5). forall x_5: T(x_5) -> P_5(x_5). exists x_5: U(x_5) -> T(x_5). Can we infer forall x_5: ~(U(x_5) -> R_5(x_5)) from them? If possible, name the inference rules being used at each step. | That contradicts the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_5: Q_5(x_5) -> R_5(x_5). forall x_5: P_5(x_5) -> Q_5(x_5). forall x_5: T(x_5) -> P_5(x_5). exists x_5: U(x_5) -> T(x_5). Can we infer forall x_5: ~(U(x_5) -> R_5(x_5)) from them? If possible, name the inference rules being used at each step.
### Response:
That contradicts the premises. Therefore the answer is no.
| null | null |
Consider the following premises. S_19(a_19) <-> w_2. forall x_19: R_19(x_19) -> S_19(x_19). ~w_2. Can we infer ~R_19(a_19) or u from them? If possible, name the inference rules being used at each step. | From forall x_19: R_19(x_19) -> S_19(x_19) we can infer R_19(a_19) -> S_19(a_19) via universal instantiation. From S_19(a_19) <-> w_2, ~w_2 we can infer ~S_19(a_19) via biconditional elimination. From R_19(a_19) -> S_19(a_19), ~S_19(a_19) we can infer ~R_19(a_19) via modus tollens. Finally, from ~R_19(a_19) we can infer ~R_19(a_19) or u via addition. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. S_19(a_19) <-> w_2. forall x_19: R_19(x_19) -> S_19(x_19). ~w_2. Can we infer ~R_19(a_19) or u from them? If possible, name the inference rules being used at each step.
### Response:
From forall x_19: R_19(x_19) -> S_19(x_19) we can infer R_19(a_19) -> S_19(a_19) via universal instantiation. From S_19(a_19) <-> w_2, ~w_2 we can infer ~S_19(a_19) via biconditional elimination. From R_19(a_19) -> S_19(a_19), ~S_19(a_19) we can infer ~R_19(a_19) via modus tollens. Finally, from ~R_19(a_19) we can infer ~R_19(a_19) or u via addition. Therefore, the answer is yes.
| null | null |
Translate the following inference to logic notation: It is not cold. It is cold if and only if it is snowing. If it is sunny, then it is snowing. Therefore it is not sunny. | ~p. p <-> q. r -> q. Therefore ~r. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Translate the following inference to logic notation: It is not cold. It is cold if and only if it is snowing. If it is sunny, then it is snowing. Therefore it is not sunny.
### Response:
~p. p <-> q. r -> q. Therefore ~r.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? exists x_7: Q_7(x_7) -> R(x_7). forall x_7: W(x_7) -> P_7(x_7). forall x_7: P_7(x_7) <-> Q_7(x_7). | forall x_7: P_7(x_7) -> Q_7(x_7). forall x_7: Q_7(x_7) -> P_7(x_7). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? exists x_7: Q_7(x_7) -> R(x_7). forall x_7: W(x_7) -> P_7(x_7). forall x_7: P_7(x_7) <-> Q_7(x_7).
### Response:
forall x_7: P_7(x_7) -> Q_7(x_7). forall x_7: Q_7(x_7) -> P_7(x_7).
| null | null |
Consider the following premises. forall x_7: T(x_7) -> Q_7(x_7). forall x_7: ~Q_7(x_7). forall x_7: T(x_7) and Q_5(x_7). Can we infer exists x_7: ~U(x_7) from them? | The premises are contradictory and we can infer anything from them. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_7: T(x_7) -> Q_7(x_7). forall x_7: ~Q_7(x_7). forall x_7: T(x_7) and Q_5(x_7). Can we infer exists x_7: ~U(x_7) from them?
### Response:
The premises are contradictory and we can infer anything from them. Therefore, the answer is yes.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: w -> s. u_1 <-> w. u_4 -> u_1. ~s. | ~w can be inferred via the modus tollens rule. u_1 -> w can be inferred via the biconditional elimination rule. w -> u_1 can be inferred via the biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: w -> s. u_1 <-> w. u_4 -> u_1. ~s.
### Response:
~w can be inferred via the modus tollens rule. u_1 -> w can be inferred via the biconditional elimination rule. w -> u_1 can be inferred via the biconditional elimination rule.
| null | null |
Consider the following premises. For all x_8, if x_8 were curious, then x_8 is an astronaut. For all x_8, x_8 is curious or x_8 will read a book. There is at least one x_8 for which if x_8 reads a book, then x_8 is an astronaut. For all x_8, x_8 is happy. Can we infer the following from them? For all x_8, either x_8 is happy or x_8 is an astronaut are not true. | From the fact that for all x_8, if x_8 were curious, then x_8 is an astronaut, that there is at least one x_8 for which if x_8 reads a book, then x_8 is an astronaut, and that for all x_8, x_8 is curious or x_8 will read a book we can infer that there is at least one x_8 for which x_8 is an astronaut. Finally, from the fact that for all x_8, x_8 is happy, and that there is at least one x_8 for which x_8 is an astronaut we can infer that there is at least one x_8 for which x_8 is happy and x_8 is an astronaut, which contradicts that For all x_8, either x_8 is happy or x_8 is an astronaut are not true. Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_8, if x_8 were curious, then x_8 is an astronaut. For all x_8, x_8 is curious or x_8 will read a book. There is at least one x_8 for which if x_8 reads a book, then x_8 is an astronaut. For all x_8, x_8 is happy. Can we infer the following from them? For all x_8, either x_8 is happy or x_8 is an astronaut are not true.
### Response:
From the fact that for all x_8, if x_8 were curious, then x_8 is an astronaut, that there is at least one x_8 for which if x_8 reads a book, then x_8 is an astronaut, and that for all x_8, x_8 is curious or x_8 will read a book we can infer that there is at least one x_8 for which x_8 is an astronaut. Finally, from the fact that for all x_8, x_8 is happy, and that there is at least one x_8 for which x_8 is an astronaut we can infer that there is at least one x_8 for which x_8 is happy and x_8 is an astronaut, which contradicts that For all x_8, either x_8 is happy or x_8 is an astronaut are not true. Therefore, the answer is no.
| null | null |
Consider the following premises. forall z: T_4(z) <-> R(z). forall z: R(z) <-> T_4(z). Can we infer exists z: ~(T_4(z) <-> R(z)) from them? If possible, name the inference rules being used at each step. | From forall z: R(z) <-> T_4(z) we can infer forall z: R(z) -> T_4(z) via universal biconditional elimination. From forall z: T_4(z) <-> R(z) we can infer forall z: T_4(z) -> R(z) via universal biconditional elimination. Finally, from forall z: T_4(z) -> R(z), forall z: R(z) -> T_4(z) we can infer forall z: T_4(z) <-> R(z) via universal biconditional introduction, which contradicts exists z: ~(T_4(z) <-> R(z)). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall z: T_4(z) <-> R(z). forall z: R(z) <-> T_4(z). Can we infer exists z: ~(T_4(z) <-> R(z)) from them? If possible, name the inference rules being used at each step.
### Response:
From forall z: R(z) <-> T_4(z) we can infer forall z: R(z) -> T_4(z) via universal biconditional elimination. From forall z: T_4(z) <-> R(z) we can infer forall z: T_4(z) -> R(z) via universal biconditional elimination. Finally, from forall z: T_4(z) -> R(z), forall z: R(z) -> T_4(z) we can infer forall z: T_4(z) <-> R(z) via universal biconditional introduction, which contradicts exists z: ~(T_4(z) <-> R(z)). Therefore, the answer is no.
| null | null |
Consider the following premises. For all x_8, if x_8 drives a car, then x_8 is taking a plane. For all x_8, if x_8 drives a car, then x_8 is taking a plane. For all x_8, if x_8 takes a plane, then x_8 is driving a car. Can we infer the following from them? If we can, name the inference rule being used: For all x_8, if x_8 drives a car, then x_8 is taking a plane. | That is one of the premises. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_8, if x_8 drives a car, then x_8 is taking a plane. For all x_8, if x_8 drives a car, then x_8 is taking a plane. For all x_8, if x_8 takes a plane, then x_8 is driving a car. Can we infer the following from them? If we can, name the inference rule being used: For all x_8, if x_8 drives a car, then x_8 is taking a plane.
### Response:
That is one of the premises. Therefore, the answer is yes.
| null | null |
Consider the following premises. q. s -> q. q -> s. Can we infer r from them? | We cannot infer that from the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. q. s -> q. q -> s. Can we infer r from them?
### Response:
We cannot infer that from the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_14, x_14 is bored if and only if x_14 is a nurse. If Mary were rich, then Barbara is a nurse. If Joseph listens to a song, then Mary is rich. | If Joseph listens to a song, then Barbara is a nurse can be inferred via the transitivity rule. For all x_14, if x_14 were bored, then x_14 is a nurse can be inferred via the universal biconditional elimination rule. For all x_14, if x_14 were a nurse, then x_14 is bored can be inferred via the universal biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_14, x_14 is bored if and only if x_14 is a nurse. If Mary were rich, then Barbara is a nurse. If Joseph listens to a song, then Mary is rich.
### Response:
If Joseph listens to a song, then Barbara is a nurse can be inferred via the transitivity rule. For all x_14, if x_14 were bored, then x_14 is a nurse can be inferred via the universal biconditional elimination rule. For all x_14, if x_14 were a nurse, then x_14 is bored can be inferred via the universal biconditional elimination rule.
| null | null |
Consider the following premises. forall y_3: Q_3(y_3). ~Q_3(a_12) or r. Can we infer r from them? | We cannot infer that from the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall y_3: Q_3(y_3). ~Q_3(a_12) or r. Can we infer r from them?
### Response:
We cannot infer that from the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x_2, x_2 will listen to a song or x_2 will not cook. For all x_2, x_2 will cook or x_2 will drink water. There is at least one x_2 for which if x_2 drinks water, then x_2 is listening to a song. | For all x_2, if x_2 cooks, then x_2 is listening to a song. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x_2, x_2 will listen to a song or x_2 will not cook. For all x_2, x_2 will cook or x_2 will drink water. There is at least one x_2 for which if x_2 drinks water, then x_2 is listening to a song.
### Response:
For all x_2, if x_2 cooks, then x_2 is listening to a song.
| null | null |
Consider the following premises. ~r_4. t -> r. u_3 <-> r_4. r -> t. r -> u_3. Can we infer ~t from them? | From t -> r, r -> t we can infer t <-> r. From u_3 <-> r_4, ~r_4 we can infer ~u_3. From r -> u_3, ~u_3 we can infer ~r. Finally, from t <-> r, ~r we can infer ~t. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. ~r_4. t -> r. u_3 <-> r_4. r -> t. r -> u_3. Can we infer ~t from them?
### Response:
From t -> r, r -> t we can infer t <-> r. From u_3 <-> r_4, ~r_4 we can infer ~u_3. From r -> u_3, ~u_3 we can infer ~r. Finally, from t <-> r, ~r we can infer ~t. Therefore, the answer is yes.
| null | null |
Consider the following premises. forall x_6: P_6(x_6) -> S_4(x_6). forall x_6: Q_6(x_6) -> S_4(x_6). exists x_6: P_6(x_6) -> Q_6(x_6). forall x_6: S_4(x_6) -> P_6(x_6). Can we infer forall x_6: ~(S_4(x_6) <-> P_6(x_6)) from them? If possible, name the inference rules being used at each step. | From exists x_6: P_6(x_6) -> Q_6(x_6), forall x_6: Q_6(x_6) -> S_4(x_6) we can infer exists x_6: P_6(x_6) -> S_4(x_6) via existential transitivity. Finally, from forall x_6: S_4(x_6) -> P_6(x_6), forall x_6: P_6(x_6) -> S_4(x_6) we can infer forall x_6: S_4(x_6) <-> P_6(x_6) via universal biconditional introduction, which contradicts forall x_6: ~(S_4(x_6) <-> P_6(x_6)). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_6: P_6(x_6) -> S_4(x_6). forall x_6: Q_6(x_6) -> S_4(x_6). exists x_6: P_6(x_6) -> Q_6(x_6). forall x_6: S_4(x_6) -> P_6(x_6). Can we infer forall x_6: ~(S_4(x_6) <-> P_6(x_6)) from them? If possible, name the inference rules being used at each step.
### Response:
From exists x_6: P_6(x_6) -> Q_6(x_6), forall x_6: Q_6(x_6) -> S_4(x_6) we can infer exists x_6: P_6(x_6) -> S_4(x_6) via existential transitivity. Finally, from forall x_6: S_4(x_6) -> P_6(x_6), forall x_6: P_6(x_6) -> S_4(x_6) we can infer forall x_6: S_4(x_6) <-> P_6(x_6) via universal biconditional introduction, which contradicts forall x_6: ~(S_4(x_6) <-> P_6(x_6)). Therefore, the answer is no.
| null | null |
Translate the following inference to logic notation: Jennifer is not working. If James were sad, then Jennifer is working. James is sad. Therefore Susan is driving a car. | ~p. q -> p. q. Therefore r. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Translate the following inference to logic notation: Jennifer is not working. If James were sad, then Jennifer is working. James is sad. Therefore Susan is driving a car.
### Response:
~p. q -> p. q. Therefore r.
| null | null |
Consider the following premises. exists x_10: S_3(x_10) or ~Q_10(x_10). forall x_10: S_3(x_10) <-> Q_10(x_10). Can we infer exists x_10: Q_10(x_10) -> S_3(x_10) from them? | From forall x_10: S_3(x_10) <-> Q_10(x_10) we can infer forall x_10: S_3(x_10) -> Q_10(x_10). Finally, from exists x_10: S_3(x_10) or ~Q_10(x_10) we can infer exists x_10: Q_10(x_10) -> S_3(x_10). Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. exists x_10: S_3(x_10) or ~Q_10(x_10). forall x_10: S_3(x_10) <-> Q_10(x_10). Can we infer exists x_10: Q_10(x_10) -> S_3(x_10) from them?
### Response:
From forall x_10: S_3(x_10) <-> Q_10(x_10) we can infer forall x_10: S_3(x_10) -> Q_10(x_10). Finally, from exists x_10: S_3(x_10) or ~Q_10(x_10) we can infer exists x_10: Q_10(x_10) -> S_3(x_10). Therefore, the answer is yes.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all y, y will read a book. There is at least one y for which y is a server. For all y, y is a poet. | For all y, y will read a book and y is a poet. For all y, y is a poet and y will read a book. Some servers will read a book. Some servers are poets. There is at least one y for which y will read a book and y is a server. Some poets are servers. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all y, y will read a book. There is at least one y for which y is a server. For all y, y is a poet.
### Response:
For all y, y will read a book and y is a poet. For all y, y is a poet and y will read a book. Some servers will read a book. Some servers are poets. There is at least one y for which y will read a book and y is a server. Some poets are servers.
| null | null |
Consider the following premises. q_4 or t. p_4 <-> q_4. t -> p_4. Can we infer p from them? If possible, name the inference rules being used at each step. | We cannot infer that from the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. q_4 or t. p_4 <-> q_4. t -> p_4. Can we infer p from them? If possible, name the inference rules being used at each step.
### Response:
We cannot infer that from the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x_10, x_10 is a lawyer if and only if x_10 is a nurse. For all x_10, x_10 is a lawyer or x_10 is a poet. There is at least one x_10 for which x_10 is not a poet. For all x_10, x_10 is not a nurse. | Since the premises are contradictory, we can infer anything from them. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x_10, x_10 is a lawyer if and only if x_10 is a nurse. For all x_10, x_10 is a lawyer or x_10 is a poet. There is at least one x_10 for which x_10 is not a poet. For all x_10, x_10 is not a nurse.
### Response:
Since the premises are contradictory, we can infer anything from them.
| null | null |
Consider the following premises. For all x_6, if x_6 drives a car, then x_6 is a writer. For all x_6, if x_6 goes running, then x_6 is a bartender. There is at least one x_6 for which if x_6 were a writer, then x_6 is sleeping. For all x_6, if x_6 sleeps, then x_6 is running. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, the fact that x_6 drives a car does not imply that x_6 is a bartender. | That contradicts the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_6, if x_6 drives a car, then x_6 is a writer. For all x_6, if x_6 goes running, then x_6 is a bartender. There is at least one x_6 for which if x_6 were a writer, then x_6 is sleeping. For all x_6, if x_6 sleeps, then x_6 is running. Can we infer the following from them? If we can, name the inference rule being used: For all x_6, the fact that x_6 drives a car does not imply that x_6 is a bartender.
### Response:
That contradicts the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? ~W(i). u <-> q. q <-> U_1(i). forall z: W(z) <-> U_1(z). | u -> q. q -> U_1(i). q -> u. U_1(i) -> q. forall z: W(z) -> U_1(z). forall z: U_1(z) -> W(z). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? ~W(i). u <-> q. q <-> U_1(i). forall z: W(z) <-> U_1(z).
### Response:
u -> q. q -> U_1(i). q -> u. U_1(i) -> q. forall z: W(z) -> U_1(z). forall z: U_1(z) -> W(z).
| null | null |
Translate the following inference to logic notation: Susan is a lawyer if and only if it is raining. Susan is not a lawyer. If James works, then it is raining. If it rains, then James is working. Therefore James is not working. | p <-> q. ~p. r -> q. q -> r. Therefore ~r. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Translate the following inference to logic notation: Susan is a lawyer if and only if it is raining. Susan is not a lawyer. If James works, then it is raining. If it rains, then James is working. Therefore James is not working.
### Response:
p <-> q. ~p. r -> q. q -> r. Therefore ~r.
| null | null |
Consider the following premises. forall x_14: P(x_14) -> R(x_14). forall x_14: S_3(x_14). forall x_14: S_3(x_14) -> P(x_14). Can we infer r from them? | We cannot infer that from the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_14: P(x_14) -> R(x_14). forall x_14: S_3(x_14). forall x_14: S_3(x_14) -> P(x_14). Can we infer r from them?
### Response:
We cannot infer that from the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? There is at least one x_5 for which x_5 is not a server. For all x_5, if x_5 drives a car, then x_5 is an artist. For all x_5, x_5 is a server or x_5 will drive a car. | There is at least one x_5 for which x_5 will drive a car. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? There is at least one x_5 for which x_5 is not a server. For all x_5, if x_5 drives a car, then x_5 is an artist. For all x_5, x_5 is a server or x_5 will drive a car.
### Response:
There is at least one x_5 for which x_5 will drive a car.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall z: R(z) <-> U(z). exists z: R(z) -> U(z). forall z: U(z) -> R(z). | exists z: R(z) <-> U(z) can be inferred via the existential biconditional introduction rule. exists z: U(z) <-> R(z) can be inferred via the existential biconditional introduction rule. forall z: R(z) -> U(z) can be inferred via the universal biconditional elimination rule. forall z: U(z) -> R(z) can be inferred via the universal biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall z: R(z) <-> U(z). exists z: R(z) -> U(z). forall z: U(z) -> R(z).
### Response:
exists z: R(z) <-> U(z) can be inferred via the existential biconditional introduction rule. exists z: U(z) <-> R(z) can be inferred via the existential biconditional introduction rule. forall z: R(z) -> U(z) can be inferred via the universal biconditional elimination rule. forall z: U(z) -> R(z) can be inferred via the universal biconditional elimination rule.
| null | null |
Consider the following premises. For all x_12, x_12 is a mechanic or x_12 is a janitor. For all x_12, if x_12 were a mechanic, then x_12 is a poet. For all x_12, x_12 is not a taxi driver. For all x_12, x_12 is a taxi driver or x_12 is a soldier. There is at least one x_12 for which if x_12 were a janitor, then x_12 is a poet. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, either x_12 is a poet or x_12 is a soldier are not true. | From the fact that for all x_12, if x_12 were a mechanic, then x_12 is a poet, that there is at least one x_12 for which if x_12 were a janitor, then x_12 is a poet, and that for all x_12, x_12 is a mechanic or x_12 is a janitor we can infer that there is at least one x_12 for which x_12 is a poet via existential disjunction elimination. From the fact that for all x_12, x_12 is a taxi driver or x_12 is a soldier, and that for all x_12, x_12 is not a taxi driver we can infer that for all x_12, x_12 is a soldier via universal disjunctive syllogism. Finally, from the fact that there is at least one x_12 for which x_12 is a poet, and that for all x_12, x_12 is a soldier we can infer that some poets are soldiers via existential conjunction, which contradicts that For all x_12, either x_12 is a poet or x_12 is a soldier are not true. Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_12, x_12 is a mechanic or x_12 is a janitor. For all x_12, if x_12 were a mechanic, then x_12 is a poet. For all x_12, x_12 is not a taxi driver. For all x_12, x_12 is a taxi driver or x_12 is a soldier. There is at least one x_12 for which if x_12 were a janitor, then x_12 is a poet. Can we infer the following from them? If we can, name the inference rule being used: For all x_12, either x_12 is a poet or x_12 is a soldier are not true.
### Response:
From the fact that for all x_12, if x_12 were a mechanic, then x_12 is a poet, that there is at least one x_12 for which if x_12 were a janitor, then x_12 is a poet, and that for all x_12, x_12 is a mechanic or x_12 is a janitor we can infer that there is at least one x_12 for which x_12 is a poet via existential disjunction elimination. From the fact that for all x_12, x_12 is a taxi driver or x_12 is a soldier, and that for all x_12, x_12 is not a taxi driver we can infer that for all x_12, x_12 is a soldier via universal disjunctive syllogism. Finally, from the fact that there is at least one x_12 for which x_12 is a poet, and that for all x_12, x_12 is a soldier we can infer that some poets are soldiers via existential conjunction, which contradicts that For all x_12, either x_12 is a poet or x_12 is a soldier are not true. Therefore, the answer is no.
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Consider the following premises. There is at least one y for which if y writes a letter, then y is a poet. For all y, if y were a musician, then y is listening to a song. For all y, if y were excited, then y is writing a letter. For all y, if y listens to a song, then y is excited. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which if y were a musician, then y is a poet. | From the fact that for all y, if y were a musician, then y is listening to a song, and that for all y, if y listens to a song, then y is excited we can infer that for all y, if y were a musician, then y is excited via universal transitivity. From the fact that for all y, if y were excited, then y is writing a letter, and that there is at least one y for which if y writes a letter, then y is a poet we can infer that there is at least one y for which if y were excited, then y is a poet via existential transitivity. Finally, from the fact that for all y, if y were a musician, then y is excited, and that there is at least one y for which if y were excited, then y is a poet we can infer that there is at least one y for which if y were a musician, then y is a poet via existential transitivity. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. There is at least one y for which if y writes a letter, then y is a poet. For all y, if y were a musician, then y is listening to a song. For all y, if y were excited, then y is writing a letter. For all y, if y listens to a song, then y is excited. Can we infer the following from them? If we can, name the inference rule being used: There is at least one y for which if y were a musician, then y is a poet.
### Response:
From the fact that for all y, if y were a musician, then y is listening to a song, and that for all y, if y listens to a song, then y is excited we can infer that for all y, if y were a musician, then y is excited via universal transitivity. From the fact that for all y, if y were excited, then y is writing a letter, and that there is at least one y for which if y writes a letter, then y is a poet we can infer that there is at least one y for which if y were excited, then y is a poet via existential transitivity. Finally, from the fact that for all y, if y were a musician, then y is excited, and that there is at least one y for which if y were excited, then y is a poet we can infer that there is at least one y for which if y were a musician, then y is a poet via existential transitivity. Therefore, the answer is yes.
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Consider the following premises. forall y: P(y) -> U(y). exists y: ~U(y). forall y: U(y) <-> P(y). Can we infer forall y: P(y) from them? | From forall y: U(y) <-> P(y) we can infer forall y: U(y) -> P(y). From forall y: U(y) -> P(y), forall y: P(y) -> U(y) we can infer forall y: U(y) <-> P(y). Finally, from forall y: U(y) <-> P(y), exists y: ~U(y) we can infer exists y: ~P(y), which contradicts forall y: P(y). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall y: P(y) -> U(y). exists y: ~U(y). forall y: U(y) <-> P(y). Can we infer forall y: P(y) from them?
### Response:
From forall y: U(y) <-> P(y) we can infer forall y: U(y) -> P(y). From forall y: U(y) -> P(y), forall y: P(y) -> U(y) we can infer forall y: U(y) <-> P(y). Finally, from forall y: U(y) <-> P(y), exists y: ~U(y) we can infer exists y: ~P(y), which contradicts forall y: P(y). Therefore, the answer is no.
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Consider the following premises. For all x_11, if x_11 plays tennis, then x_11 is drinking water. For all x_11, if x_11 were a doctor, then x_11 is playing tennis. For all x_11, x_11 will cook if and only if x_11 will read a book. For all x_11, x_11 will drink water if and only if x_11 will read a book. For all x_11, x_11 will not cook. Can we infer the following from them? If we can, name the inference rule being used: For all x_11, x_11 is not a doctor. | From the fact that for all x_11, x_11 will cook if and only if x_11 will read a book, and that for all x_11, x_11 will not cook we can infer that for all x_11, x_11 will not read a book via universal biconditional elimination. From the fact that for all x_11, x_11 will drink water if and only if x_11 will read a book, and that for all x_11, x_11 will not read a book we can infer that for all x_11, x_11 will not drink water via universal biconditional elimination. From the fact that for all x_11, if x_11 plays tennis, then x_11 is drinking water, and that for all x_11, x_11 will not drink water we can infer that for all x_11, x_11 will not play tennis via universal modus tollens. Finally, from the fact that for all x_11, if x_11 were a doctor, then x_11 is playing tennis, and that for all x_11, x_11 will not play tennis we can infer that for all x_11, x_11 is not a doctor via universal modus tollens. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_11, if x_11 plays tennis, then x_11 is drinking water. For all x_11, if x_11 were a doctor, then x_11 is playing tennis. For all x_11, x_11 will cook if and only if x_11 will read a book. For all x_11, x_11 will drink water if and only if x_11 will read a book. For all x_11, x_11 will not cook. Can we infer the following from them? If we can, name the inference rule being used: For all x_11, x_11 is not a doctor.
### Response:
From the fact that for all x_11, x_11 will cook if and only if x_11 will read a book, and that for all x_11, x_11 will not cook we can infer that for all x_11, x_11 will not read a book via universal biconditional elimination. From the fact that for all x_11, x_11 will drink water if and only if x_11 will read a book, and that for all x_11, x_11 will not read a book we can infer that for all x_11, x_11 will not drink water via universal biconditional elimination. From the fact that for all x_11, if x_11 plays tennis, then x_11 is drinking water, and that for all x_11, x_11 will not drink water we can infer that for all x_11, x_11 will not play tennis via universal modus tollens. Finally, from the fact that for all x_11, if x_11 were a doctor, then x_11 is playing tennis, and that for all x_11, x_11 will not play tennis we can infer that for all x_11, x_11 is not a doctor via universal modus tollens. Therefore, the answer is yes.
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Consider the following premises. For all x_12, x_12 is a scientist. There is at least one x_12 for which if x_12 were a lawyer, then x_12 is not a scientist. For all x_12, x_12 is a lawyer if and only if x_12 will make tea. Can we infer the following from them? For all x_12, x_12 will make tea. | From the fact that for all x_12, x_12 is a lawyer if and only if x_12 will make tea we can infer that for all x_12, if x_12 makes tea, then x_12 is a lawyer. From the fact that there is at least one x_12 for which if x_12 were a lawyer, then x_12 is not a scientist, and that for all x_12, x_12 is a scientist we can infer that there is at least one x_12 for which x_12 is not a lawyer. Finally, from the fact that for all x_12, if x_12 makes tea, then x_12 is a lawyer, and that there is at least one x_12 for which x_12 is not a lawyer we can infer that there is at least one x_12 for which x_12 will not make tea, which contradicts that For all x_12, x_12 will make tea. Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_12, x_12 is a scientist. There is at least one x_12 for which if x_12 were a lawyer, then x_12 is not a scientist. For all x_12, x_12 is a lawyer if and only if x_12 will make tea. Can we infer the following from them? For all x_12, x_12 will make tea.
### Response:
From the fact that for all x_12, x_12 is a lawyer if and only if x_12 will make tea we can infer that for all x_12, if x_12 makes tea, then x_12 is a lawyer. From the fact that there is at least one x_12 for which if x_12 were a lawyer, then x_12 is not a scientist, and that for all x_12, x_12 is a scientist we can infer that there is at least one x_12 for which x_12 is not a lawyer. Finally, from the fact that for all x_12, if x_12 makes tea, then x_12 is a lawyer, and that there is at least one x_12 for which x_12 is not a lawyer we can infer that there is at least one x_12 for which x_12 will not make tea, which contradicts that For all x_12, x_12 will make tea. Therefore, the answer is no.
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Consider the following premises. For all x_4, if x_4 were a police officer, then x_4 is playing squash. There is at least one x_4 for which x_4 will take a plane or x_4 is a police officer. For all x_4, x_4 will not take a plane or x_4 is a mechanic. For all x_4, x_4 will not play squash. Can we infer the following from them? There is at least one x_4 for which x_4 is a mechanic. | From the fact that for all x_4, if x_4 were a police officer, then x_4 is playing squash, and that for all x_4, x_4 will not play squash we can infer that for all x_4, x_4 is not a police officer. From the fact that there is at least one x_4 for which x_4 will take a plane or x_4 is a police officer, and that for all x_4, x_4 will not take a plane or x_4 is a mechanic we can infer that there is at least one x_4 for which x_4 is a police officer or x_4 is a mechanic. Finally, from the fact that there is at least one x_4 for which x_4 is a police officer or x_4 is a mechanic, and that for all x_4, x_4 is not a police officer we can infer that there is at least one x_4 for which x_4 is a mechanic. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_4, if x_4 were a police officer, then x_4 is playing squash. There is at least one x_4 for which x_4 will take a plane or x_4 is a police officer. For all x_4, x_4 will not take a plane or x_4 is a mechanic. For all x_4, x_4 will not play squash. Can we infer the following from them? There is at least one x_4 for which x_4 is a mechanic.
### Response:
From the fact that for all x_4, if x_4 were a police officer, then x_4 is playing squash, and that for all x_4, x_4 will not play squash we can infer that for all x_4, x_4 is not a police officer. From the fact that there is at least one x_4 for which x_4 will take a plane or x_4 is a police officer, and that for all x_4, x_4 will not take a plane or x_4 is a mechanic we can infer that there is at least one x_4 for which x_4 is a police officer or x_4 is a mechanic. Finally, from the fact that there is at least one x_4 for which x_4 is a police officer or x_4 is a mechanic, and that for all x_4, x_4 is not a police officer we can infer that there is at least one x_4 for which x_4 is a mechanic. Therefore, the answer is yes.
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Consider the following premises. For all x_7, x_7 will climb a mountain if and only if x_7 will read a book. Patricia is climbing a mountain or it is cloudy. Patricia is not reading a book. Can we infer the following from them? It is sunny. | We cannot infer that from the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. For all x_7, x_7 will climb a mountain if and only if x_7 will read a book. Patricia is climbing a mountain or it is cloudy. Patricia is not reading a book. Can we infer the following from them? It is sunny.
### Response:
We cannot infer that from the premises. Therefore the answer is no.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall z: T_3(z) <-> R(z). ~T_3(e_2) -> s. ~R(e_2). | forall z: T_3(z) -> R(z) can be inferred via the universal biconditional elimination rule. forall z: R(z) -> T_3(z) can be inferred via the universal biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall z: T_3(z) <-> R(z). ~T_3(e_2) -> s. ~R(e_2).
### Response:
forall z: T_3(z) -> R(z) can be inferred via the universal biconditional elimination rule. forall z: R(z) -> T_3(z) can be inferred via the universal biconditional elimination rule.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? ~p or t. s -> p_1. p or s. ~p_1. | ~s. s or t. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? ~p or t. s -> p_1. p or s. ~p_1.
### Response:
~s. s or t.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_6: U(x_6) -> S(x_6). forall x_6: S(x_6) -> ~Q_6(x_6). forall x_6: Q_6(x_6). exists x_6: U(x_6) <-> W_1(x_6). | forall x_6: ~S(x_6). exists x_6: U(x_6) -> W_1(x_6). exists x_6: W_1(x_6) -> U(x_6). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_6: U(x_6) -> S(x_6). forall x_6: S(x_6) -> ~Q_6(x_6). forall x_6: Q_6(x_6). exists x_6: U(x_6) <-> W_1(x_6).
### Response:
forall x_6: ~S(x_6). exists x_6: U(x_6) -> W_1(x_6). exists x_6: W_1(x_6) -> U(x_6).
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Consider the following premises. Sarah is a writer if and only if it is sunny. Charles is a doctor if and only if it is sunny. Sarah is not a writer. Can we infer the following from them? If we can, name the inference rule being used: Mary is playing a game. | We cannot infer that from the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. Sarah is a writer if and only if it is sunny. Charles is a doctor if and only if it is sunny. Sarah is not a writer. Can we infer the following from them? If we can, name the inference rule being used: Mary is playing a game.
### Response:
We cannot infer that from the premises. Therefore the answer is no.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_12, x_12 will play a game or x_12 will not drink water. For all x_12, x_12 will take a plane if and only if x_12 is a carpenter. There is at least one x_12 for which if x_12 plays a game, then x_12 is taking a plane. | For all x_12, if x_12 drinks water, then x_12 is playing a game can be inferred via the universal rewriting rule. For all x_12, if x_12 takes a plane, then x_12 is a carpenter can be inferred via the universal biconditional elimination rule. For all x_12, if x_12 were a carpenter, then x_12 is taking a plane can be inferred via the universal biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: For all x_12, x_12 will play a game or x_12 will not drink water. For all x_12, x_12 will take a plane if and only if x_12 is a carpenter. There is at least one x_12 for which if x_12 plays a game, then x_12 is taking a plane.
### Response:
For all x_12, if x_12 drinks water, then x_12 is playing a game can be inferred via the universal rewriting rule. For all x_12, if x_12 takes a plane, then x_12 is a carpenter can be inferred via the universal biconditional elimination rule. For all x_12, if x_12 were a carpenter, then x_12 is taking a plane can be inferred via the universal biconditional elimination rule.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall x_12: R_12(x_12) -> S_12(x_12). S_12(j) -> R_12(j). ~R_12(j). | Nothing can be inferred from these premises. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall x_12: R_12(x_12) -> S_12(x_12). S_12(j) -> R_12(j). ~R_12(j).
### Response:
Nothing can be inferred from these premises.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall x_16: Q_9(x_16) -> R(x_16). forall x_16: Q_16(x_16) -> R_16(x_16). forall x_16: S(x_16). forall x_16: R(x_16) -> U(x_16). forall x_16: P_16(x_16) -> R_16(x_16). forall x_16: R_16(x_16) -> Q_9(x_16). forall x_16: P_16(x_16) or Q_16(x_16). | forall x_16: Q_9(x_16) -> U(x_16) can be inferred via the universal transitivity rule. forall x_16: Q_16(x_16) -> Q_9(x_16) can be inferred via the universal transitivity rule. forall x_16: P_16(x_16) -> Q_9(x_16) can be inferred via the universal transitivity rule. forall x_16: R_16(x_16) -> R(x_16) can be inferred via the universal transitivity rule. forall x_16: R_16(x_16) can be inferred via the universal disjunction elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall x_16: Q_9(x_16) -> R(x_16). forall x_16: Q_16(x_16) -> R_16(x_16). forall x_16: S(x_16). forall x_16: R(x_16) -> U(x_16). forall x_16: P_16(x_16) -> R_16(x_16). forall x_16: R_16(x_16) -> Q_9(x_16). forall x_16: P_16(x_16) or Q_16(x_16).
### Response:
forall x_16: Q_9(x_16) -> U(x_16) can be inferred via the universal transitivity rule. forall x_16: Q_16(x_16) -> Q_9(x_16) can be inferred via the universal transitivity rule. forall x_16: P_16(x_16) -> Q_9(x_16) can be inferred via the universal transitivity rule. forall x_16: R_16(x_16) -> R(x_16) can be inferred via the universal transitivity rule. forall x_16: R_16(x_16) can be inferred via the universal disjunction elimination rule.
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Consider the following premises. exists x: T(x). Can we infer forall x: ~(T(x) or Q(x)) from them? If possible, name the inference rules being used at each step. | That contradicts the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. exists x: T(x). Can we infer forall x: ~(T(x) or Q(x)) from them? If possible, name the inference rules being used at each step.
### Response:
That contradicts the premises. Therefore the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? P_9(a) -> R(a). forall x_9: P_9(x_9) or ~R(x_9). | forall x_9: R(x_9) -> P_9(x_9). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? P_9(a) -> R(a). forall x_9: P_9(x_9) or ~R(x_9).
### Response:
forall x_9: R(x_9) -> P_9(x_9).
| null | null |
Consider the following premises. forall x_8: P_3(x_8) and Q_3(x_8). forall x_8: Q_8(x_8) -> U(x_8). forall x_8: P_8(x_8) -> U(x_8). forall x_8: P_8(x_8) or Q_8(x_8). Can we infer exists x_8: ~(U(x_8) and P_3(x_8)) from them? | From forall x_8: P_8(x_8) -> U(x_8), forall x_8: Q_8(x_8) -> U(x_8), forall x_8: P_8(x_8) or Q_8(x_8) we can infer forall x_8: U(x_8). From forall x_8: P_3(x_8) and Q_3(x_8) we can infer forall x_8: P_3(x_8). Finally, from forall x_8: U(x_8), forall x_8: P_3(x_8) we can infer forall x_8: U(x_8) and P_3(x_8), which contradicts exists x_8: ~(U(x_8) and P_3(x_8)). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_8: P_3(x_8) and Q_3(x_8). forall x_8: Q_8(x_8) -> U(x_8). forall x_8: P_8(x_8) -> U(x_8). forall x_8: P_8(x_8) or Q_8(x_8). Can we infer exists x_8: ~(U(x_8) and P_3(x_8)) from them?
### Response:
From forall x_8: P_8(x_8) -> U(x_8), forall x_8: Q_8(x_8) -> U(x_8), forall x_8: P_8(x_8) or Q_8(x_8) we can infer forall x_8: U(x_8). From forall x_8: P_3(x_8) and Q_3(x_8) we can infer forall x_8: P_3(x_8). Finally, from forall x_8: U(x_8), forall x_8: P_3(x_8) we can infer forall x_8: U(x_8) and P_3(x_8), which contradicts exists x_8: ~(U(x_8) and P_3(x_8)). Therefore, the answer is no.
| null | null |
Consider the following premises. forall x_11: T(x_11) or Q(x_11). forall x_11: P_11(x_11) <-> Q_11(x_11). forall x_11: ~P_11(x_11). exists x_11: T(x_11) <-> Q_11(x_11). Can we infer forall x_11: ~Q(x_11) from them? | From forall x_11: P_11(x_11) <-> Q_11(x_11), forall x_11: ~P_11(x_11) we can infer forall x_11: ~Q_11(x_11). From exists x_11: T(x_11) <-> Q_11(x_11), forall x_11: ~Q_11(x_11) we can infer exists x_11: ~T(x_11). Finally, from forall x_11: T(x_11) or Q(x_11), exists x_11: ~T(x_11) we can infer exists x_11: Q(x_11), which contradicts forall x_11: ~Q(x_11). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_11: T(x_11) or Q(x_11). forall x_11: P_11(x_11) <-> Q_11(x_11). forall x_11: ~P_11(x_11). exists x_11: T(x_11) <-> Q_11(x_11). Can we infer forall x_11: ~Q(x_11) from them?
### Response:
From forall x_11: P_11(x_11) <-> Q_11(x_11), forall x_11: ~P_11(x_11) we can infer forall x_11: ~Q_11(x_11). From exists x_11: T(x_11) <-> Q_11(x_11), forall x_11: ~Q_11(x_11) we can infer exists x_11: ~T(x_11). Finally, from forall x_11: T(x_11) or Q(x_11), exists x_11: ~T(x_11) we can infer exists x_11: Q(x_11), which contradicts forall x_11: ~Q(x_11). Therefore, the answer is no.
| null | null |
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: It is late if and only if it is sunny. If it is sunny, then Jessica is playing tennis. Jessica is playing tennis if and only if Karen is taking a plane. It is not late. If Jessica plays tennis, then it is sunny. | It is sunny if and only if Jessica is playing tennis can be inferred via the biconditional introduction rule. Jessica is playing tennis if and only if it is sunny can be inferred via the biconditional introduction rule. If it is late, then it is sunny can be inferred via the biconditional elimination rule. If Jessica plays tennis, then Karen is taking a plane can be inferred via the biconditional elimination rule. If it is sunny, then it is late can be inferred via the biconditional elimination rule. If Karen takes a plane, then Jessica is playing tennis can be inferred via the biconditional elimination rule. It is not sunny can be inferred via the biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: It is late if and only if it is sunny. If it is sunny, then Jessica is playing tennis. Jessica is playing tennis if and only if Karen is taking a plane. It is not late. If Jessica plays tennis, then it is sunny.
### Response:
It is sunny if and only if Jessica is playing tennis can be inferred via the biconditional introduction rule. Jessica is playing tennis if and only if it is sunny can be inferred via the biconditional introduction rule. If it is late, then it is sunny can be inferred via the biconditional elimination rule. If Jessica plays tennis, then Karen is taking a plane can be inferred via the biconditional elimination rule. If it is sunny, then it is late can be inferred via the biconditional elimination rule. If Karen takes a plane, then Jessica is playing tennis can be inferred via the biconditional elimination rule. It is not sunny can be inferred via the biconditional elimination rule.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x_10, if x_10 were a taxi driver, then x_10 is a nurse. For all x_10, x_10 is a taxi driver. There is at least one x_10 for which x_10 is sad and x_10 will play a game. | For all x_10, x_10 is a nurse. There is at least one x_10 for which x_10 is sad. There is at least one x_10 for which x_10 will play a game. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? For all x_10, if x_10 were a taxi driver, then x_10 is a nurse. For all x_10, x_10 is a taxi driver. There is at least one x_10 for which x_10 is sad and x_10 will play a game.
### Response:
For all x_10, x_10 is a nurse. There is at least one x_10 for which x_10 is sad. There is at least one x_10 for which x_10 will play a game.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? p_10 <-> s. s -> w_4. s <-> w_4. | p_10 -> s. s -> w_4. s -> p_10. w_4 -> s. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? p_10 <-> s. s -> w_4. s <-> w_4.
### Response:
p_10 -> s. s -> w_4. s -> p_10. w_4 -> s.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall z: W(z) or S_4(z). forall z: Q(z) -> T(z). forall z: W(z) <-> T(z). forall z: S_4(z) -> Q(z). | forall z: S_4(z) -> T(z) can be inferred via the universal transitivity rule. forall z: W(z) -> T(z) can be inferred via the universal biconditional elimination rule. forall z: T(z) -> W(z) can be inferred via the universal biconditional elimination rule. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? Name the inference rule being used: forall z: W(z) or S_4(z). forall z: Q(z) -> T(z). forall z: W(z) <-> T(z). forall z: S_4(z) -> Q(z).
### Response:
forall z: S_4(z) -> T(z) can be inferred via the universal transitivity rule. forall z: W(z) -> T(z) can be inferred via the universal biconditional elimination rule. forall z: T(z) -> W(z) can be inferred via the universal biconditional elimination rule.
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Translate the following inference to logic notation: It is raining or it is not late. If Patricia makes tea, then it is late. It is not raining. Therefore Patricia is not making tea. | p or ~q. r -> q. ~p. Therefore ~r. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Translate the following inference to logic notation: It is raining or it is not late. If Patricia makes tea, then it is late. It is not raining. Therefore Patricia is not making tea.
### Response:
p or ~q. r -> q. ~p. Therefore ~r.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? James is a lawyer or Jessica is not a taxi driver. James is a lawyer if and only if it is cold. Jessica is not drinking water or it is cold. For all x_16, x_16 will drink water or x_16 is a taxi driver. | If Jessica were a taxi driver, then James is a lawyer. If James were a lawyer, then it is cold. If it is cold, then James is a lawyer. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? James is a lawyer or Jessica is not a taxi driver. James is a lawyer if and only if it is cold. Jessica is not drinking water or it is cold. For all x_16, x_16 will drink water or x_16 is a taxi driver.
### Response:
If Jessica were a taxi driver, then James is a lawyer. If James were a lawyer, then it is cold. If it is cold, then James is a lawyer.
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Consider the following premises. exists x_8: P_8(x_8) -> U(x_8). forall x_8: U(x_8) -> P_8(x_8). forall x_8: P_8(x_8) -> U(x_8). forall x_8: ~U(x_8). Can we infer exists x_8: ~P_8(x_8) from them? | From exists x_8: P_8(x_8) -> U(x_8), forall x_8: U(x_8) -> P_8(x_8) we can infer exists x_8: P_8(x_8) <-> U(x_8). From exists x_8: P_8(x_8) <-> U(x_8) we can infer exists x_8: U(x_8) -> P_8(x_8). From exists x_8: U(x_8) -> P_8(x_8), forall x_8: P_8(x_8) -> U(x_8) we can infer exists x_8: U(x_8) <-> P_8(x_8). Finally, from exists x_8: U(x_8) <-> P_8(x_8), forall x_8: ~U(x_8) we can infer exists x_8: ~P_8(x_8). Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. exists x_8: P_8(x_8) -> U(x_8). forall x_8: U(x_8) -> P_8(x_8). forall x_8: P_8(x_8) -> U(x_8). forall x_8: ~U(x_8). Can we infer exists x_8: ~P_8(x_8) from them?
### Response:
From exists x_8: P_8(x_8) -> U(x_8), forall x_8: U(x_8) -> P_8(x_8) we can infer exists x_8: P_8(x_8) <-> U(x_8). From exists x_8: P_8(x_8) <-> U(x_8) we can infer exists x_8: U(x_8) -> P_8(x_8). From exists x_8: U(x_8) -> P_8(x_8), forall x_8: P_8(x_8) -> U(x_8) we can infer exists x_8: U(x_8) <-> P_8(x_8). Finally, from exists x_8: U(x_8) <-> P_8(x_8), forall x_8: ~U(x_8) we can infer exists x_8: ~P_8(x_8). Therefore, the answer is yes.
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Consider the following premises. forall x_5: Q_8(x_5) -> S_5(x_5). forall x_8: P_8(x_8) or Q_8(x_8). forall x_8: ~P_8(x_8). Can we infer r from them? | We cannot infer that from the premises. Therefore the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. forall x_5: Q_8(x_5) -> S_5(x_5). forall x_8: P_8(x_8) or Q_8(x_8). forall x_8: ~P_8(x_8). Can we infer r from them?
### Response:
We cannot infer that from the premises. Therefore the answer is no.
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Consider the following premises. exists x_10: Q_10(x_10) -> P_2(x_10). forall x_10: ~P_10(x_10). forall x_10: P_2(x_10) -> Q_10(x_10). forall x_10: P_10(x_10) <-> Q_10(x_10). Can we infer forall x_10: P_2(x_10) from them? If possible, name the inference rules being used at each step. | From forall x_10: P_10(x_10) <-> Q_10(x_10), forall x_10: ~P_10(x_10) we can infer forall x_10: ~Q_10(x_10) via universal biconditional elimination. From exists x_10: Q_10(x_10) -> P_2(x_10), forall x_10: P_2(x_10) -> Q_10(x_10) we can infer exists x_10: Q_10(x_10) <-> P_2(x_10) via existential biconditional introduction. From exists x_10: Q_10(x_10) <-> P_2(x_10) we can infer exists x_10: P_2(x_10) -> Q_10(x_10) via existential biconditional elimination. Finally, from exists x_10: P_2(x_10) -> Q_10(x_10), forall x_10: ~Q_10(x_10) we can infer exists x_10: ~P_2(x_10) via existential modus tollens, which contradicts forall x_10: P_2(x_10). Therefore, the answer is no. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. exists x_10: Q_10(x_10) -> P_2(x_10). forall x_10: ~P_10(x_10). forall x_10: P_2(x_10) -> Q_10(x_10). forall x_10: P_10(x_10) <-> Q_10(x_10). Can we infer forall x_10: P_2(x_10) from them? If possible, name the inference rules being used at each step.
### Response:
From forall x_10: P_10(x_10) <-> Q_10(x_10), forall x_10: ~P_10(x_10) we can infer forall x_10: ~Q_10(x_10) via universal biconditional elimination. From exists x_10: Q_10(x_10) -> P_2(x_10), forall x_10: P_2(x_10) -> Q_10(x_10) we can infer exists x_10: Q_10(x_10) <-> P_2(x_10) via existential biconditional introduction. From exists x_10: Q_10(x_10) <-> P_2(x_10) we can infer exists x_10: P_2(x_10) -> Q_10(x_10) via existential biconditional elimination. Finally, from exists x_10: P_2(x_10) -> Q_10(x_10), forall x_10: ~Q_10(x_10) we can infer exists x_10: ~P_2(x_10) via existential modus tollens, which contradicts forall x_10: P_2(x_10). Therefore, the answer is no.
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Consider the following premises. Robert is a scientist or Elisabeth is writing a letter. For all x_14, if x_14 cooks, then x_14 is a police officer. If Elisabeth writes a letter, then Jennifer is cooking. If Robert were a scientist, then Jennifer is cooking. Can we infer the following from them? Jennifer is a police officer. | From the fact that for all x_14, if x_14 cooks, then x_14 is a police officer we can infer that if Jennifer cooks, then Jennifer is a police officer. From the fact that if Robert were a scientist, then Jennifer is cooking, that if Elisabeth writes a letter, then Jennifer is cooking, and that Robert is a scientist or Elisabeth is writing a letter we can infer that Jennifer is cooking. Finally, from the fact that if Jennifer cooks, then Jennifer is a police officer, and that Jennifer is cooking we can infer that Jennifer is a police officer. Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. Robert is a scientist or Elisabeth is writing a letter. For all x_14, if x_14 cooks, then x_14 is a police officer. If Elisabeth writes a letter, then Jennifer is cooking. If Robert were a scientist, then Jennifer is cooking. Can we infer the following from them? Jennifer is a police officer.
### Response:
From the fact that for all x_14, if x_14 cooks, then x_14 is a police officer we can infer that if Jennifer cooks, then Jennifer is a police officer. From the fact that if Robert were a scientist, then Jennifer is cooking, that if Elisabeth writes a letter, then Jennifer is cooking, and that Robert is a scientist or Elisabeth is writing a letter we can infer that Jennifer is cooking. Finally, from the fact that if Jennifer cooks, then Jennifer is a police officer, and that Jennifer is cooking we can infer that Jennifer is a police officer. Therefore, the answer is yes.
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Translate the following inference to logic notation: There is at least one x for which y_1 will not go running. For all x, if y_1 cooks, then y_1 is sleeping. For all x, if y_1 goes running, then y_1 is cooking. For all x, if y_1 sleeps, then y_1 is running. Therefore there is at least one x for which y_1 will not cook. | exists x: ~P(x). forall x: Q(x) -> R(x). forall x: P(x) -> Q(x). forall x: R(x) -> P(x). Therefore exists x: ~Q(x). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Translate the following inference to logic notation: There is at least one x for which y_1 will not go running. For all x, if y_1 cooks, then y_1 is sleeping. For all x, if y_1 goes running, then y_1 is cooking. For all x, if y_1 sleeps, then y_1 is running. Therefore there is at least one x for which y_1 will not cook.
### Response:
exists x: ~P(x). forall x: Q(x) -> R(x). forall x: P(x) -> Q(x). forall x: R(x) -> P(x). Therefore exists x: ~Q(x).
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? If Charles climbs a mountain, then Michael is a soldier. For all x_12, x_12 is sad if and only if x_12 will climb a mountain. For all z, z is not a soldier. | For all x_12, if x_12 were sad, then x_12 is climbing a mountain. For all x_12, if x_12 climbs a mountain, then x_12 is sad. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? If Charles climbs a mountain, then Michael is a soldier. For all x_12, x_12 is sad if and only if x_12 will climb a mountain. For all z, z is not a soldier.
### Response:
For all x_12, if x_12 were sad, then x_12 is climbing a mountain. For all x_12, if x_12 climbs a mountain, then x_12 is sad.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall z: T(z) <-> Q(z). ~Q(d_4). T(d_4) or ~s. | s -> T(d_4). forall z: T(z) -> Q(z). forall z: Q(z) -> T(z). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall z: T(z) <-> Q(z). ~Q(d_4). T(d_4) or ~s.
### Response:
s -> T(d_4). forall z: T(z) -> Q(z). forall z: Q(z) -> T(z).
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Translate the following inference to logic notation: David is cooking. It is snowing if and only if David is cooking. Therefore it is snowing or William is a cashier. | p. q <-> p. Therefore q or r. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Translate the following inference to logic notation: David is cooking. It is snowing if and only if David is cooking. Therefore it is snowing or William is a cashier.
### Response:
p. q <-> p. Therefore q or r.
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What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_1: ~Q(x_1). exists x_1: Q(x_1) or ~R(x_1). forall x_1: Q(x_1) -> R(x_1). | exists x_1: R(x_1) -> Q(x_1). | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
What can be inferred from the following premises in a single inference step (ignoring inferences that add new predicates or constants)? forall x_1: ~Q(x_1). exists x_1: Q(x_1) or ~R(x_1). forall x_1: Q(x_1) -> R(x_1).
### Response:
exists x_1: R(x_1) -> Q(x_1).
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Consider the following premises. exists x_5: T(x_5) -> W_3(x_5). forall x_5: U(x_5) -> W_3(x_5). forall x_5: U(x_5) or T(x_5). forall x_5: W_3(x_5) -> U(x_5). Can we infer exists x_5: W_3(x_5) from them? | From forall x_5: U(x_5) -> W_3(x_5), forall x_5: W_3(x_5) -> U(x_5) we can infer forall x_5: U(x_5) <-> W_3(x_5). From forall x_5: U(x_5) <-> W_3(x_5) we can infer forall x_5: U(x_5) -> W_3(x_5). Finally, from forall x_5: U(x_5) -> W_3(x_5), exists x_5: T(x_5) -> W_3(x_5), forall x_5: U(x_5) or T(x_5) we can infer exists x_5: W_3(x_5). Therefore, the answer is yes. | LogicInference Dataset e | ### Instruction:
Answer the following.
### Input:
Consider the following premises. exists x_5: T(x_5) -> W_3(x_5). forall x_5: U(x_5) -> W_3(x_5). forall x_5: U(x_5) or T(x_5). forall x_5: W_3(x_5) -> U(x_5). Can we infer exists x_5: W_3(x_5) from them?
### Response:
From forall x_5: U(x_5) -> W_3(x_5), forall x_5: W_3(x_5) -> U(x_5) we can infer forall x_5: U(x_5) <-> W_3(x_5). From forall x_5: U(x_5) <-> W_3(x_5) we can infer forall x_5: U(x_5) -> W_3(x_5). Finally, from forall x_5: U(x_5) -> W_3(x_5), exists x_5: T(x_5) -> W_3(x_5), forall x_5: U(x_5) or T(x_5) we can infer exists x_5: W_3(x_5). Therefore, the answer is yes.
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