In exercise, reorder the premises in each of the arguments to show that the conclusion fol...

In exercise, reorder the premises in each of the arguments to show that the conclusion follows as a valid consequence from the premises. It may be helpful to rewrite the statements in if-then form and replace some statements by their contrapositives. Exercise refer to the kinds of Tarski worlds discussed in Example 1 and 2.

Example 1

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. ∀t, Triangle(t)→Blue(t).

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b. ∀x, Blue(x)→Triangle(x).

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c. ∃y such that Square(y) ;∧; RightOf(d, y).

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d. ∃z such that Square(z) ;∧; Gray(z).

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.

Example 2

Truth of a ∀∃ Statement in a Tarski World

Consider the Tarski world shown in Figure.

Figure

Show that the following statement is true in this world:

For all triangles x, there is a square y such that x and y have the same color.

Solution

The statement says that no matter which triangle someone gives you, you will be able to find a square of the same color. There are only three triangles, d, f, and i. The following table shows that for each of these triangles a square of the same color can be found.

Exercise

1. If an object is above all the triangles, then it is above all the blue objects.

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2. If an object is not above all the gray objects, then it is not a square.

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3. Every black object is a square.

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4. Every object that is above all the gray objects is above all the triangles.

∴ If an object is black, then it is above all the blue objects.