Lecture Notes: General Statements, Counterexamples, and Proof by Failure to Find a Counterexample

Introduction

Focus on understanding three key terms:
- General Statements
- Counterexamples
- Proof by Failure to Find a Counterexample

Definition: A general statement is about all things of a certain kind.
- Example: "All dogs have four legs" is a general statement.
Characteristics:
- Often, general statements aren't perfectly accurate.
- Made about all members of a set or category.
Examples of General Statements:
- "No human being has six arms." (Refers to all humans)
- "If a girl is 17 years old, then she's a teenager." (Refers to all 17-year-old girls)
- "All birds except ostriches can fly." (Refers to all birds that are not ostriches)
Implications:
- Making general statements can lead to problems in formal settings, like a court of law.
- Common sense may accept general statements without challenge in informal settings.

Definition: A counterexample is a specific example that disproves a general statement.
- Must be a concrete, factual example from the same subject matter.
Example:
- General Statement: "All birds can fly."
- Counterexample: "Penguins cannot fly." (Penguins are birds)
Importance:
- A counterexample can successfully disprove a general statement.
- Hypothetical scenarios do not qualify as counterexamples.

Concept:
- Refers to assuming something is true simply because no counterexample has been found.
Explanation:
- Not finding evidence against a claim doesn't prove the claim.
Examples:
- Easter egg hunt analogy: Just because you don't find all the eggs doesn't mean they don't exist.
- Two cousins with a Bible: Failing to find a specific verse doesn’t prove it doesn’t exist.
Logical Fallacy:
- Assumes that inability to find a counterexample equates to proof of correctness.

Avoid making general statements without evidence.
Use counterexamples to refute general statements.
Be wary of accepting conclusions based on failure to find counterexamples, as this is a logical fallacy.