Directions: Define the necessary symbols, rewrite the argument in symbolic form, and us a truth table to determine whether the argument is valid. If the argument is invalid, interpret the specific circumstances that cause the argument to be invalid.
Problem:
1. All pesticides are harmful to the environment.
2. No fertilizer is a pesticide.
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Therefore, no pesticide is harmful to the environment.
To solve this problem, we need to define the necessary symbols and rewrite the argument in symbolic form. Then we can use a truth table to determine whether the argument is valid or not.
Let's define the symbols:
- P: Pesticides
- H: Harmful to the environment
- F: Fertilizer
Now let's rewrite the argument in symbolic form:
1. ∀x (P(x) → H(x)) (All pesticides are harmful to the environment)
2. ∀x (F(x) → ¬P(x)) (No fertilizer is a pesticide)
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Therefore, ¬∃x (P(x) ∧ H(x)) (No pesticide is harmful to the environment)
To proceed, we need to construct a truth table for these statements. Since there are two variables (P and F), we need four rows in the truth table to cover all possible combinations.
Here is the truth table:
| P | H | F | P(x) → H(x) | F(x) → ¬P(x) | P(x) ∧ H(x) | ¬∃x (P(x) ∧ H(x)) |
|---|---|---|-------------|--------------|-------------|----------------------|
| T | T | T | T | T | T | F |
| T | T | F | T | F | T | F |
| F | T | T | T | T | F | T |
| F | T | F | T | T | F | T |
In the last column, the symbol "T" represents "True," and "F" represents "False". The truth table shows the possible combinations of truth values for the premises and conclusion.
Now, let's interpret the truth table:
- When there is at least one case where both P(x) and H(x) are true (top two rows), the conclusion ¬∃x (P(x) ∧ H(x)) is false. This means that when there is a pesticide that is harmful to the environment, the argument is invalid. The specific circumstances that cause the argument to be invalid are when there is a pesticide that is harmful to the environment.
- On the other hand, when there is no case where both P(x) and H(x) are true, the conclusion ¬∃x (P(x) ∧ H(x)) is true. This means that when there is no pesticide that is harmful to the environment, the argument is valid.
Therefore, the argument is invalid based on the possibility of there being pesticides that are harmful to the environment.
Step 1: Define the necessary symbols:
Let P represent "Pesticides are harmful to the environment."
Let F represent "Fertilizer is a pesticide."
Step 2: Rewrite the argument in symbolic form:
1. ∀x(P(x)) (All pesticides are harmful to the environment)
2. ¬∃x(F(x)) (No fertilizer is a pesticide)
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3. ¬∃x(P(x)) (Therefore, no pesticide is harmful to the environment)
Step 3: Create a truth table to determine the validity of the argument:
For the truth table, we need to consider all possible combinations of truth values for P(x) and F(x):
| P(x) | F(x) | ¬∃x(F(x)) | ¬∃x(P(x)) |
|------|------|-----------|-----------|
| T | T | F | F |
| T | F | T | F |
| F | T | F | T |
| F | F | T | T |
From the truth table, we can see that there are situations where both premises are true (row 1) while the conclusion is false. In this case, the argument is invalid.
Step 4: Interpret the specific circumstances that cause the argument to be invalid:
The specific circumstance that causes the argument to be invalid is when there are pesticides that are harmful to the environment (P(x) is true) and there are fertilizers that are not pesticides (F(x) is false). In this scenario, both premises will be true, but the conclusion will be false, contradicting the argument.