Problem 18 - Thinking Mathematically

Question

Use Euler diagrams to determine whether each argument is valid or invalid. All thefts are immoral acts. \(\underline{\text { Some thefts are justifiable. }}\) Therefore, some immoral acts are justifiable.

Step-by-Step Solution

Answer

The argument is valid. The premises logically entail the conclusion in the Euler diagram.

Step 1: Drawing the Initial Euler Diagram

First, draw three overlapping circles, each labeled respectively as 'thefts', 'immoral acts' and 'justifiable acts'. From 'All thefts are immoral acts', you denote 'thefts' as a set completely contained within 'immoral acts', indicating that all thefts are immoral.

Step 2: Adding the Second Premise to the Diagram

From 'Some thefts are justifiable', you denote an intersection between 'thefts' and 'justifiable acts'. This indicates that there is at least one member of the 'thefts' set that is also in the 'justifiable acts' set.

Step 3: Evaluating the Conclusion

With these relations depicted, you should now evaluate the conclusion of the argument: 'Therefore, some immoral acts are justifiable'. You can see from the diagram that the intersection of 'thefts' and 'justifiable acts' falls within the 'immoral acts' circle, indicating that at least some immoral acts are justifiable. Therefore, the conclusion follows logically from the premises, and the argument is valid.

Key Concepts

Validity of Arguments

Understanding the validity of arguments is crucial in logical reasoning. An argument is considered valid if the conclusion logically follows from its premises. That means, if the premises are true, the conclusion must also be true. In the given exercise, we examine an argument using a tool called an Euler diagram. This visual representation helps determine if there is a logical connection that supports the argument's conclusion.

As outlined in the solution, you start by mapping each category as a circle and then consider how they overlap based on the given statements. Since all thefts are depicted within the boundary of immoral acts, and some thefts intersect with justifiable acts, we are led to see the overlap between justifiable acts and immoral acts. This overlap confirms the conclusion that some immoral acts can indeed be justifiable, illustrating the argument's validity.

Logical Reasoning

The essence of logical reasoning lies in evaluating information in a structured and critical way to arrive at a sound conclusion. It involves the application of deductive and inductive reasoning principles. In the context of the exercise, we apply deductive reasoning: a top-down approach that moves from the general to the specific. By starting with a general premise ('All thefts are immoral acts.') and moving to a specific one ('Some thefts are justifiable.'), we assess whether the specific case ('Therefore, some immoral acts are justifiable.') follows.

It’s crucial to remember, however, that logical reasoning isn’t just about the structure of the argument, but also the truth of the premises. In real-world scenarios, a valid argument may still lead to a factually incorrect conclusion if the initial statements are false. Therefore, both the form and the content of an argument are important in logical reasoning.

Set Theory

Set theory is a mathematical concept that deals with the collection of objects, known as elements or members, which are grouped together in a set. This theory forms the basis of several fields within mathematics, including the study of probability, statistics, and algebra. Euler diagrams, which we use in the given exercise, are a graphical representation that can help in understanding set theory concepts such as union, intersection, and subset.

In our exercise, 'thefts', 'immoral acts', and 'justifiable acts' are considered as sets. When we say 'All thefts are immoral acts', we are essentially describing a subset relationship where the set of thefts is within the set of immoral acts. Similarly, when 'Some thefts are justifiable' is represented, it demonstrates an intersection between two sets—theft and justifiable acts. These visual cues from Euler diagrams are powerful aids in comprehending complex set relationships.