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Chapter 3: Problem 79

Write an original argument in words for the transitive reasoning form.

Short Answer

Expert verified
By applying transitive reasoning, we can deduce that if A (a pizza) is equal to B (a food item) and B (a food item) is also equal to C (something edible), then A (a pizza) must be equal to C (something edible). This is a clear, logical demonstration of the transitive reasoning form.

Step by step solution

01

Define the Terms

In any argument, it's important to clearly define the terms. So, taking three elements - A Pizza, A Food item, and A Edible item. Here, 'Pizza' can serve as A, 'Food item' as B, and 'Edible item' as C.
02

Establish Relationships

Connect A to B, and B to C. Understand the relationships between these terms. In this scenario, a pizza is a type of food item (A = B), and a food item is something that can be consumed or edible (B = C).
03

Apply Transitive Reasoning

By applying transitive reasoning to these relationships, one can deduce that A is also C. In this case, if a pizza is a type of food item (A = B) and a food item is something edible (B = C), using transitive reasoning, we can conclude that a pizza is also something edible (A = C).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logical Reasoning
Logical reasoning is a process of using structured steps to analyze information and reach conclusions. This type of reasoning involves systematically analyzing statements or problems to understand their logical connections and implications. In the context of the original exercise, logical reasoning helps us break down relationships between different elements, such as the connection between a pizza, a food item, and an edible item.

Here鈥檚 how logical reasoning plays a role:
  • Identifying Relationships: It helps recognize the connection between the elements, such as A = B and B = C.
  • Establishing Consistency: Ensures that conclusions follow logically from the premises given.
  • Clarifying Definitions: Precisely defines what terms like 'pizza', 'food item', and 'edible item' mean to avoid confusion.
Using logical reasoning ensures that the conclusions we reach are based on clear and consistent relationships.
Deductive Reasoning
Deductive reasoning is a specific approach to logical thinking that moves from general premises to a specific conclusion. It's like following a structured path from established facts to a logical outcome. In our exercise, we use deductive reasoning to figure out an argument in the transitive reasoning form.

Here's how it works:
  • Start with General Statements: Begin with broad premises such as "A pizza is a food item" and "A food item is an edible item."
  • Progress through Logical Steps: Use these premises to infer a specific conclusion that logically follows.
  • Reach a Conclusion: Deduce that "A pizza is an edible item."
Deductive reasoning is powerful because it guarantees that the conclusion is true if the premises are true.
Mathematical Argumentation
Mathematical argumentation involves constructing a logical argument using mathematical principles. In the exercise, mathematical argumentation underpins the entire process by requiring a strict adherence to logical steps and precision. This ensures that every step is justified and leads to a valid conclusion.

Consider these points in mathematical argumentation:
  • Begin with Definitions: Clearly defining terms is crucial, as in defining 'pizza', 'food item', and 'edible item.'
  • Establish a Framework: Set up a logical sequence, just like connecting A = B and B = C.
  • Validate Each Step: Ensure that each conclusion logically follows from the previous statements.
This approach fosters a deep understanding of the logical structure behind mathematical statements and their implications, which is essential in proving arguments like those seen in transitive reasoning.

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Most popular questions from this chapter

Determine whether each argument is valid or invalid. All natural numbers are whole numbers, all whole numbers are integers, and \(-4006\) is not a whole number. Thus, \(-4006\) is not an integer.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made Euler diagrams for the premises of an argument and one of my possible diagrams illustrated the conclusion, so the argument is valid.

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. The writers of My Mother the Car were told by the network to improve their scripts or be dropped from prime time. The writers of My Mother the Car did not improve their scripts. Therefore, ...

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If it is hot and humid, I complain. It is not hot or it is not humid. \(\therefore\) I am not complaining.

Use Euler diagrams to determine whether each argument is valid or invalid. All actors are artists. Sean Penn is an artist. Therefore, Sean Penn is an actor.
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