A set of premises and a conclusion are given. Use the valid argument forms listed in Table...

A set of premises and a conclusion are given. Use the valid argument forms listed in Table to deduce the conclusion from the premises, giving a reason for each step as in Example . Assume all variables are statement variables.

Table

Valid Argument Forms

Example

Application: A More Complex Deduction

You are about to leave for school in the morning and discover that you don’t have your

glasses. You know the following statements are true:

a. If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table.

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b. If my glasses are on the kitchen table, then I saw them at breakfast.

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c. I did not see my glasses at breakfast.

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d. I was reading the newspaper in the living room or I was reading the newspaper in the kitchen.

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e. If I was reading the newspaper in the living room then my glasses are on the coffee table.

Where are the glasses?

Solution

Let RK = I was reading the newspaper in the kitchen.

GK = My glasses are on the kitchen table.

SB = I saw my glasses at breakfast.

RL = I was reading the newspaper in the living room.

GC = My glasses are on the coffee table.

Here is a sequence of steps you might use to reach the answer, together with the rules of

inference that allow you to draw the conclusion of each step:

1. RK → GK  by (a)

GK → SB  by (b)

∴ RK → SB  by transitivity

2. RK → SB  by the conclusion of (1)

∼SB  by (c)

∴ ∼R by modus tollens

3. RL ∨ RK by (d)

∼RK by the conclusion of (2)

∴ RL by elimination

4. RL → GC by (e)

RL by the conclusion of (3)

∴ GC by modus ponens

Thus the glasses are on the coffee table.

a. p ∨ q

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b. q →r

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c. p ∧ s → t

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d. ~r

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e. ~q → u ∧ s

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f. ∴ t