Which term best describes the statement given below? If [tex]$p \Rightarrow q$[/tex] and [tex]$q \Rightarrow r$[/tex], then [tex]$p \Rightarrow r$[/tex].
A. Contrapositive statement
B. A syllogism
C. Inverse statement
D. Converse statement
Answer (2)
Define contrapositive, inverse, converse, and syllogism.
Recognize that the given statement 'If p A rr q and q A rrr , then p A rrr ' is a syllogism.
Conclude that the best term to describe the statement is 'A syllogism'.
The final answer is B .
Explanation
Understanding the Problem We need to identify the term that accurately describes the given logical statement: If p A rr q and q A rrr , then p A rrr . We will analyze each option to determine the correct one.
Defining the Options Let's define each of the given options:
Contrapositive statement: The contrapositive of p A rr q is ¬ q A rr ¬ p .
Inverse statement: The inverse of p A rr q is ¬ p A rr ¬ q .
Converse statement: The converse of p A rr q is q A rr p .
Syllogism: A syllogism is a form of reasoning where a conclusion is drawn from two given or assumed propositions (premises). The given statement follows this pattern.
Identifying the Correct Term The given statement, 'If p A rr q and q A rrr , then p A rrr ', is a classic example of a syllogism, specifically known as the law of hypothetical syllogism. This law states that if the first statement implies the second, and the second statement implies the third, then the first statement implies the third.
Conclusion Therefore, the term that best describes the given statement is 'A syllogism'.
Examples
Syllogisms are used in everyday reasoning and problem-solving. For example, if you know that 'If it rains, the ground gets wet' and 'The ground is wet, then the grass grows', you can conclude that 'If it rains, the grass grows'. This type of logical deduction helps us make informed decisions and understand cause-and-effect relationships in various situations.
The statement 'If p ⇒ q and q ⇒ r , then p ⇒ r ' is best described as 'A syllogism'. This is because it follows the logical reasoning pattern of concluding one statement from two premises. Hence, the answer is B.
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