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
Modus Ponens | p→q p ∴q | Elimination | a. p ∨ q ~q ∴p b. p ∨ q ~p ∴q |
Modus Tollens | p→q ~p ∴~q
| Transitivity | p→q q→r ∴p→r
|
Generalization | a.p ∴ p ∨ q b.q ∴ p ∨ q | Proof by Division into Cases | p∨q p→r q→r ∴r |
Specialization | a. p ∧q ∴p b. p ∧q ∴q | ||
Conjunction | p q ∴p∧q | Contradiction Rule | ^p →c ∴p |
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.
b. If my glasses are on the kitchen table, then I saw them at breakfast.
c. I did not see my glasses at breakfast.
d. I was reading the newspaper in the living room or I was reading the newspaper in the kitchen.
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
b. q →r
c. p ∧ s → t
d. ~r
e. ~q → u ∧ s
f. ∴ t
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