Refer to the picture of Tarski’s world given in Example. Let Above(x, y) mean that x is ab...

Refer to the picture of Tarski’s world given in Example. Let Above(x, y) mean that x is above y (but possibly in a different column). Determine the truth or falsity of each of the following statements. Give reasons for your answers.

a.  ∀u, Circle(u)  → Gray(u).

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b. ∀u, Gray(u) → Circle(u).

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c.  ∃y such that Square(y) A Above(y, d).

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d. ∃z such that Triangle(z) A Above(f, z).

Example

Investigating Tarski’s World

The program for Tarski’s World provides pictures of blocks of various sizes, shapes, and colors, which are located on a grid. Shown in Figure is a picture of an arrangement of objects in a two-dimensional Tarski world. The configuration can be described using logical operators and—for the two-dimensional version—notation such as Triangle(x), meaning “x is a triangle,” Blue(y), meaning “y is blue,” and RightOf(x, y), meaning “x is to the right of y (but possibly in a different row).” Individual objects can be given names such as a, b, or c.

Figure

Determine the truth or falsity of each of the following statements. The domain for all variables is the set of objects in the Tarski world shown above.

a.  Triangle(t)→Blue(t).

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b.  Blue(x)→Triangle(x).

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c.

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d.

Solution

a. This statement is true: All the triangles are blue.

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b. This statement is false. As a counterexample, note that e is blue and it is not a triangle.

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c. This statement is true because e and h are both square and d is to their right.

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d. This statement is false: All the squares are either blue or black.