Donald and Daisy Duck took their nephews, age 4, 5, and 6, on an outing. Each bo

Question

Donald and Daisy Duck took their nephews, age 4, 5, and 6, on an outing. Each boy wore a tee-shirt with a different design on it and of a different color. You are also given the following information: Huey is younger than the boy in the green tee-shirt. The 5-year-old wore the tee-shirt with the camel design. Dewey’s tee-shirt was yellow. Louie’s tee-shirt bore the giraffe design. The panda design was not featured on the white tee-shirt. And, Use the language containing the constants huey, dewey, louie, 4, 5, 6, green, yellow, white, camel, giraffe, and panda; the function symbols age/1, color/1, and design/1; and the predicate symbols Nephew/1, = /2 and < /2. Questions: (a) Represent these facts as sentences in FOL. (b) Using your formalization, is it possible to conclude the age of each boy together with the color and design of the tee-shirt he is wearing? Show semantically how you determined your answer. (c) If your answer was “no,” indicate what further sentences you would need to add so that you could conclude the age of each boy together with the color and design of the tee-shirt he is wearing. Reference https://www.physicsforums.com/threads/first-order-logic.460227/

Explanation / Answer

a) Representing facts as sentences in FOL.

(1) x [(color (Shirt (x)) =green) (age (huey) < age (x))]

(2) x [(age(x) = 5) (design (shirt(x))= camel)]

(3) Shirt(dewey) = yellow

(4) Ey louie (y) / (design, shirt (y) = giraffe)

(5)x [color(shirt(x)) = white -> design(shirt(x)) =/ Panda]

b) We can conclude age of the boys together with the color and design of their shirt through these facts:

By statement 1, we know that huey is not wearing a green shirt:

(1) x [(color (Shirt (x)) =green) (age (huey) < age (x))]

Statement 3 implies that he is not wearing the yellow shirt:

(3) Shirt(dewey) = yellow Then statement 4&5 imply that huey’s shirt bore the camel design:

(4) Ey louie (y) / (design, shirt (y) = giraffe)

(5)x [color(shirt(x)) = white -> design(shirt(x)) =/ Panda]

Statement 2 implies that huey is 5:

(2) x [(age(x) = 5) (design (shirt(x))= camel)]

Hence: huey, 5, white, camel By Statement 1, since Huey is younger than the boy in the green t-shirt,

this implies that the boy in the green shirt is 6.

Hence by Statement 3, Louie is 6(Age) and Dewey is 4(Age). Hence: Louie, 6, green, giraffe and Dewey, 4, yellow, panda.

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