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Represent the following sentences into formulas in predicate logic

  1. John likes all kinds of food
  2. Apples are food
  3. Chicken are food
  4. Anything anyone eats and isn't killed by is food
  5. Bill eats peanuts and is still alive
  6. Sue eats. everything Bill eats.

Certainly! Let's convert each of the given sentences into formulas in predicate logic. We'll define the necessary predicates, constants, and variables to represent the statements accurately.

Definitions

  • Predicates:

    • ( Likes(x, y) ): ( x ) likes ( y ).
    • ( Food(x) ): ( x ) is food.
    • ( Eats(x, y) ): ( x ) eats ( y ).
    • ( KilledBy(x, y) ): ( x ) is killed by ( y ).
    • ( Alive(x) ): ( x ) is alive.

  • Constants:

    • ( John )
    • ( Bill )
    • ( Sue )
    • ( Peanuts )
    • ( Apples )
    • ( Chicken )

Representations in Predicate Logic

  1. John likes all kinds of food [ $$\forall x (Food(x) \rightarrow Likes(John, x))$$ ] This formula states that for all ( x ), if ( x ) is food, then John likes ( x ).

  2. Apples are food [ $$Food(Apples)$$ ] This formula states that apples are food.

  3. Chicken are food [ $$Food(Chicken)$$ ] This formula states that chicken are food.

  4. Anything anyone eats and isn't killed by is food [ $$\forall x \forall y ((Eats(y, x) \land \neg KilledBy(y, x)) \rightarrow Food(x))$$ ] This formula states that for all ( x ) and ( y ), if ( y ) eats ( x ) and ( y ) is not killed by ( x ), then ( x ) is food.

  5. Bill eats peanuts and is still alive [ $$Eats(Bill, Peanuts) \land Alive(Bill)$$ ] This formula states that Bill eats peanuts and Bill is alive.

  6. Sue eats everything Bill eats [ $$\forall x (Eats(Bill, x) \rightarrow Eats(Sue, x))$$ ] This formula states that for all ( x ), if Bill eats ( x ), then Sue eats ( x ).

Summary

To summarize, here are all the sentences represented in predicate logic:

  1. $$( \forall x (Food(x) \rightarrow Likes(John, x)) )$$
  2. $$( Food(Apples) )$$
  3. $$( Food(Chicken) )$$
  4. $$( \forall x \forall y ((Eats(y, x) \land \neg KilledBy(y, x)) \rightarrow Food(x)) )$$
  5. $$( Eats(Bill, Peanuts) \land Alive(Bill) )$$
  6. $$( \forall x (Eats(Bill, x) \rightarrow Eats(Sue, x)) )$$

These formulas capture the logical structure of the given sentences using predicate logic, allowing for precise and formal representation of the knowledge.