Give formula proof show that argument forms are valid If the rain doesn't come The crops are ruined and the people will starve The crops are not ruined and the people willnot starve Therefore,the rain comes
To prove the validity of the argument form, we can use propositional logic and truth tables.
Step 1: Assign variables to represent the propositions.
Let's assign R for "The rain comes," C for "The crops are ruined," and P for "The people will starve."
Step 2: Express the premises in logical form.
The premises can be written as:
1. ¬R → C ∧ P (If the rain doesn't come, the crops are ruined, and the people will starve.)
2. ¬C ∧ ¬P (The crops are not ruined and the people will not starve.)
Step 3: Express the conclusion in logical form.
The conclusion can be written as: R (The rain comes).
Step 4: Construct the truth table.
Construct a truth table that includes all the relevant truth values for the premises and conclusion.
| R | C | P | ¬R → C ∧ P | ¬C ∧ ¬P | R |
|---|---|---|------------|----------|---|
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 0 | 1 |
Step 5: Check the truth value combinations for the premises.
Look for rows in the truth table where all premises are true. If in all such rows the conclusion is also true, then the argument is considered valid.
From the truth table, we can see that when ¬R → C ∧ P (premise 1) is true and ¬C ∧ ¬P (premise 2) is true, the conclusion R is also true in all cases.
Since the argument is valid in all truth value combinations, we can conclude that the argument form is valid.