Introduction to Invalid Deductions

Welcome to one of the most powerful tools in your Critical Thinking toolkit! In this chapter, we are looking at deductive reasoning. Unlike some arguments that are just "strong" or "weak," deductive arguments aim to be 100% certain. However, people often make specific logical mistakes that look like good reasoning but are actually "invalid."

By the end of these notes, you will be able to spot two of the most common traps in the Thinking Skills exam: Affirming the Consequent and Denying the Antecedent. Don't worry if those names sound a bit intimidating—they are just fancy ways of describing "backwards logic."

The Building Blocks: If and Then

Before we look at the mistakes, we need to understand the "If... then..." structure. Most deductive arguments use a conditional statement:

"If it is raining, then the ground is wet."

In logic, we label these two parts:

1. The Antecedent: This is the "If" part (the condition). In our example, the antecedent is "it is raining." Let's call this \(P\).
2. The Consequent: This is the "then" part (the result). In our example, the consequent is "the ground is wet." Let's call this \(Q\).

The logical formula is: \(P \rightarrow Q\) (If \(P\), then \(Q\)).

1. Affirming the Consequent

This is a very common flaw where someone looks at the result (the "then" part) and assumes the cause (the "if" part) must be true.

How it looks:

1. If \(P\), then \(Q\).
2. \(Q\) is true.
3. Therefore, \(P\) must be true. (This is the error!)

Example:

"If a person is a professional athlete, they are physically fit. Sarah is physically fit. Therefore, Sarah is a professional athlete."

Why is it invalid?

Even though the first statement is generally true, Sarah could be fit for other reasons. She might be a firefighter, a dedicated amateur runner, or just someone who enjoys the gym. Having the "result" (being fit) doesn't prove the specific "cause" (being a pro athlete) mentioned in the argument.

Quick Tip: If you see an argument that says, "Result X happened, so Cause Y must have happened," check if there are other ways Result X could have occurred. If there are, the argument is affirming the consequent.

2. Denying the Antecedent

This flaw happens when someone says that because the cause (the "if" part) didn't happen, the result (the "then" part) definitely won't happen either.

How it looks:

1. If \(P\), then \(Q\).
2. \(P\) is NOT true.
3. Therefore, \(Q\) is NOT true. (This is the error!)

Example:

"If you study for 10 hours a day, you will pass the exam. You did not study for 10 hours a day. Therefore, you will not pass the exam."

Why is it invalid?

The first statement says that 10 hours of study guarantees a pass. However, it doesn't say it's the only way to pass. You might be naturally gifted, or you might have studied for 5 very high-quality hours. Just because you didn't do the specific thing mentioned in the "If" clause doesn't mean the outcome is impossible.

Key Takeaway: Denying the antecedent wrongly treats a "sufficient condition" (one way to get a result) as if it were a "necessary condition" (the only way to get a result).

Summary Comparison Table

To help you distinguish between valid and invalid moves, look at this table based on the statement: "If you jump into the pool (\(P\)), you will get wet (\(Q\))."

The Move: Confirming the "If" (\(P\) is true)
Status: VALID (Modus Ponens)
Logic: You jumped in, so you are wet.

The Move: Denying the "Then" (\(Q\) is false)
Status: VALID (Modus Tollens)
Logic: You are not wet, so you couldn't have jumped in.

The Move: Affirming the Consequent (\(Q\) is true)
Status: INVALID (FALLACY)
Logic: You are wet, so you must have jumped in. (Wrong! It might be raining).

The Move: Denying the Antecedent (\(P\) is false)
Status: INVALID (FALLACY)
Logic: You didn't jump in, so you cannot be wet. (Wrong! Someone might have poured water on you).

How to Spot These in the Exam

When you are evaluating reasoning in Paper 2 or Paper 4, follow these steps:

1. Find the conditional: Look for "If... then..." or "Whenever X happens, Y follows."
2. Identify the parts: Label the "If" as \(P\) and the "then" as \(Q\).
3. Check the conclusion: Is the author concluding that \(P\) is true because they saw \(Q\)? (Affirming the Consequent). Or are they concluding that \(Q\) is false because \(P\) didn't happen? (Denying the Antecedent).
4. Explain the flaw: Don't just name the fallacy. Explain that "the author fails to consider that there may be other reasons for [the result] to occur."

Common Mistakes to Avoid

Mistake 1: Confusing the two fallacies.
Remember: Affirming starts with the Aftermath (the result). Denying starts with Discarding the first part (the cause).

Mistake 2: Thinking the conclusion is definitely "false."
Just because an argument is invalid doesn't mean the conclusion is definitely a lie. It just means the reasoning doesn't prove it. If Sarah is fit, she might be a pro athlete, but the argument hasn't proven it logically.

Quick Review

  • Invalid Deduction: A logical error where the conclusion does not follow with certainty from the reasons.
  • Affirming the Consequent: If \(P \rightarrow Q\), then \(Q \rightarrow P\). (False logic: "She's crying, so she must be sad." — She could be peeling onions!)
  • Denying the Antecedent: If \(P \rightarrow Q\), then not \(P \rightarrow\) not \(Q\). (False logic: "It's not snowing, so the ground isn't white." — It could be covered in white paint!)

Note: For more on how these relate to other flaws, see the chapter on "Recognise and evaluate informal fallacies."