Look at the statement below.
“All dogs are mammals.”
Which of these is a logically equivalent statement?
Answer
A. All dogs are furry.
B. If it is not a dog, it is not a mammal.
C. If it is not a mammal, it is not a dog.
D. All animals that are not dogs are not mammals.
C. If it is not a mammal, it is not a dog.
The logically equivalent statement for the given statement "All dogs are mammals" is:
C. If it is not a mammal, it is not a dog.
To determine which of the given statements is logically equivalent to the statement "All dogs are mammals," we need to understand the logic behind the original statement.
The statement "All dogs are mammals" is of the form "All A are B," which can be represented as "If something is a dog, then it is a mammal." In symbolic logic, this can be written as:
A → B
So, we need to find a statement that has the same logical structure. Let's examine the options:
A. "All dogs are furry" does not have the same logical structure. It does not express the same relationship between dogs and mammals, so it is not the correct answer.
B. "If it is not a dog, it is not a mammal" has the same logical structure as the original statement. It can be written as ¬A → ¬B. This statement is logically equivalent to the original statement because it states that if something is not a dog, then it is not a mammal.
C. "If it is not a mammal, it is not a dog" reverses the relationship between dogs and mammals. It states that if something is not a mammal, then it is not a dog, which is not logically equivalent to the original statement.
D. "All animals that are not dogs are not mammals" does not have the same logical structure. It refers to all animals that are not dogs, rather than focusing solely on dogs.
Therefore, the logically equivalent statement is B. "If it is not a dog, it is not a mammal."