Problem 17
Question
Use Euler diagrams to determine whether each argument is valid or invalid. Some people enjoy reading. Some people enjoy TV. \(\overline{\text { Therefore, some people who enjoy reading enjoy TV. }}\)
Step-by-Step Solution
Verified Answer
The argument is invalid because the conclusion does not logically follow from the given premises.
1Step 1: Understand and analyze the argument
The argument contains two premises and a conclusion. The premises are 'Some people enjoy reading' and 'Some people enjoy TV'. These premises suggest two overlapping sets, one for people who enjoy reading and another for people who enjoy TV. The conclusion is 'Therefore, some people who enjoy reading enjoy TV.' This conclusion suggests that there is an intersection between the sets of people who enjoy reading and who enjoy TV.
2Step 2: Construct the Euler diagram
An Euler diagram can be drawn with two overlapping circles, one representing people who enjoy reading, and the other representing people who enjoy TV. The overlap represents people who enjoy both.
3Step 3: Analyze the validity of the conclusion
The conclusion 'Therefore, some people who enjoy reading enjoy TV.' suggests a common subset of people who enjoy both reading and TV. The conclusion assumes that this overlap exists. However, the premises do not provide evidence for this. They simply state that there are people who enjoy reading and people who enjoy TV, but do not specify whether these groups overlap. Therefore, based on the premises, the conclusion is invalid.
4Step 4: Determining validity
An argument is valid if the conclusion logically follows from the premises. In this case, the argument is invalid. Although it's possible that some people who enjoy reading also enjoy TV, the premises do not provide sufficient evidence to conclude this definitively. Therefore, using Euler diagrams, the argument is determined to be invalid.
Key Concepts
Logical reasoningValid argumentVenn diagrams
Logical reasoning
Logical reasoning is the process of using a rational, systematic series of steps based on sound mathematical procedures and given statements to arrive at a conclusion. In the context of our exercise, logical reasoning is used to determine whether the conclusion that 'some people who enjoy reading enjoy TV' logically follows from the premises given.
When engaging in logical reasoning, one must carefully analyze the statements provided and consider all possible scenarios before drawing a conclusion. Equipped with critical thinking, one should examine the premises for their structure and content, and then, using deductive or inductive reasoning, establish whether the conclusion is a logical result of these statements.
In the case of the exercise, though it might seem plausible to assume an overlap between people who enjoy reading and TV, logical reasoning requires evidence from the premises. Since the premises do not explicitly state an overlap, reaching the conclusion without this evidence would be a leap beyond the bounds of logical reasoning.
When engaging in logical reasoning, one must carefully analyze the statements provided and consider all possible scenarios before drawing a conclusion. Equipped with critical thinking, one should examine the premises for their structure and content, and then, using deductive or inductive reasoning, establish whether the conclusion is a logical result of these statements.
In the case of the exercise, though it might seem plausible to assume an overlap between people who enjoy reading and TV, logical reasoning requires evidence from the premises. Since the premises do not explicitly state an overlap, reaching the conclusion without this evidence would be a leap beyond the bounds of logical reasoning.
Valid argument
A 'valid argument' in logic refers to a situation where if the premises are true, the conclusion must also be true. In other words, the validity of an argument is determined by its logical structure, not the actual truth of the premises or the conclusion.
In the exercise, we are presented with two premises about different sets of people: those who enjoy reading and those who enjoy TV. For the argument to be valid, the conclusion that some people enjoy both must be inescapable given the premises. However, the exercise shows that there is no necessary overlap between the two sets based on the given information, hence the argument is identified as invalid.
It's important for learners to distinguish between 'validity' and 'truth'. An argument can be valid but have false premises and/or a false conclusion, and an argument can have all true statements but still be invalid if the conclusion does not logically follow from the premises.
In the exercise, we are presented with two premises about different sets of people: those who enjoy reading and those who enjoy TV. For the argument to be valid, the conclusion that some people enjoy both must be inescapable given the premises. However, the exercise shows that there is no necessary overlap between the two sets based on the given information, hence the argument is identified as invalid.
It's important for learners to distinguish between 'validity' and 'truth'. An argument can be valid but have false premises and/or a false conclusion, and an argument can have all true statements but still be invalid if the conclusion does not logically follow from the premises.
Venn diagrams
Venn diagrams are visual representations of mathematical or logical relationships between different groups (sets). They consist of circles that overlap in various ways to illustrate how sets intersect or differ.
In the context of our example, a Venn diagram helps to visualize the relationships between people who enjoy reading and those who enjoy TV. If we draw two overlapping circles, with one circle for each activity, the area where they intersect would represent people who enjoy both. However, simply showing two circles does not confirm an intersection; this would require additional information or premises.
Despite their simplicity, Venn diagrams are powerful tools in understanding and solving logical problems. They allow a visual test of the validity of an argument by clearly presenting the relationships between sets. For students working with logical reasoning and arguments, mastering the use of Venn diagrams is therefore an invaluable skill.
In the context of our example, a Venn diagram helps to visualize the relationships between people who enjoy reading and those who enjoy TV. If we draw two overlapping circles, with one circle for each activity, the area where they intersect would represent people who enjoy both. However, simply showing two circles does not confirm an intersection; this would require additional information or premises.
Despite their simplicity, Venn diagrams are powerful tools in understanding and solving logical problems. They allow a visual test of the validity of an argument by clearly presenting the relationships between sets. For students working with logical reasoning and arguments, mastering the use of Venn diagrams is therefore an invaluable skill.
Other exercises in this chapter
Problem 16
Let \(p\) and \(q\) represent the following simple statements: \(p\) : This is an alligator. \(q\) : This is a reptile. Write each compound statement in symboli
View solution Problem 16
Form the negation of each statement. It is snowing.
View solution Problem 17
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 ar
View solution Problem 17
Use De Morgan's laws to write a statement that is equivalent to the given statement. It is not the case that the United States has eradicated poverty or racism.
View solution