Part 4: If-then Statements
Why If-then Statements Matter
Imagine you have a membership card to an exclusive club. The rule says:
"If you have the membership card, then you can enter the club."
This is an if-then statement. It connects having a membership card (hypothesis) to being allowed inside (conclusion). If-then statements, also known as conditional statements or implications, are everywhere in daily life and in mathematics. They help us set conditions and figure out consequences—essential for problem-solving and structured thinking.
What Is an If-then Statement?
An if-then statement has two main parts:
1. Hypothesis (if-part): The condition that could be true or false.
2. Conclusion (then-part): What is claimed to follow if the hypothesis holds.
In symbolic form, if p then q is written as \(p \to q\), where: * \(p\) = the hypothesis * \(q\) = the conclusion
Everyday Language Examples
"If you jump off the cliff, then you will break your neck."
Hypothesis: "You jump off the cliff."
Conclusion: "You will break your neck."
"What goes up comes down."
* Can be rephrased as: "If something goes up, then it will come down."
"Bigger engines burn more gas."
* Can be rephrased as: "If an engine is bigger, then it burns more gas."
Synonyms
"\(p \to q\)" can be read as "p implies q," or "Whenever p, q follows," or "p being true forces q to be true."
Why the Hypothesis and Conclusion Might Seem Unrelated
In everyday life, we often expect a cause-and-effect relationship: if you water your plants, they'll grow better. However, in formal logic, \(p \to q\) is only false in one scenario: p is true, but q is false.
Consider these four statements:
If the sky is blue, then the oceans are wet.
\(p\) = "The sky is blue" (true), \(q\) = "The oceans are wet" (true)
True hypothesis, true conclusion → the if-then statement is true in formal logic (though the cause-and-effect link is irrelevant).
If the sky is neon yellow, then the oceans are wet.
\(p\) = "The sky is neon yellow" (false), \(q\) = "The oceans are wet" (true)
The hypothesis is false, so \(p \to q\) is considered true by the logic rule.
If the sky is neon yellow, then the oceans are dry.
\(p\) = "The sky is neon yellow" (false), \(q\) = "The oceans are dry" (false)
Again, the hypothesis is false, so \(p \to q\) is true in formal logic.
If the sky is blue, then the oceans are dry.
\(p\) = "The sky is blue" (true), \(q\) = "The oceans are dry" (false)
Here, the hypothesis is true, but the conclusion is false → the statement must be false.
Key Rule for Implications
\(p \to q\) is false only when \(p\) is true and \(q\) is false.
Truth Table for If-then
Let's confirm this with a truth table. We list all possible truth values for p (the hypothesis) and q (the conclusion):
| \(p\) | \(q\) | \(p \to q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Row 1: If \(p\) is true and \(q\) is true, then \(p \to q\) is true.
Row 2: If \(p\) is true and \(q\) is false, \(p \to q\) is false (the only false case).
Row 3: If \(p\) is false and \(q\) is true, \(p \to q\) is true by definition.
Row 4: If \(p\) is false and \(q\) is false, \(p \to q\) is also true by definition.
Dogs and Cats
Suppose \(p\) = "Dogs can read" (false), and \(q\) = "Cats can sing" (false). Then \(p \to q\) = "If dogs can read, then cats can sing" is true in formal logic, because the hypothesis "dogs can read" is false.
Using If-then in Reasoning
In logic, if \(p \to q\) and \(p\) are both true, we can deduce \(q\). This is known as Modus Ponens—one of the most fundamental rules in logical reasoning:
From "\(p \to q\)" and "\(p\)," infer "\(q\)."
Real-Life Example
\(p\) = "I have a valid ticket"
\(q\) = "I can enter the concert"
If it's true that "Having a valid ticket implies I can enter the concert" (\(p \to q\)), and you indeed have a valid ticket (\(p\)), you can conclude you're allowed in (\(q\)).
Common Misconceptions
"If \(p\) is false, \(p \to q\) must be false."
Reality: If \(p\) is false, the statement \(p \to q\) is true regardless of \(q\).
"An if-then statement must reflect a real-life cause-and-effect."
Reality: Formal logic only cares about truth values, not actual causality.
"If \(p \to q\) is true, that means \(p\) causes \(q\)."
Reality: "\(p \to q\)" only says "if \(p\) is true, then \(q\) is true"—not that \(p\) directly causes \(q\).
Historical Context
The concept of conditional statements dates back to ancient Greek logic. Aristotle was among the first to formalize logical reasoning, while the Stoics further developed propositional logic including the if-then statement.
In modern symbolic logic, the implication operator (\(\to\)) was standardized in the late 19th and early 20th centuries by logicians like Frege, Russell, and Whitehead as part of their work to formalize mathematical reasoning.
Applications in Different Fields
Mathematics: Theorems are often stated as conditionals ("If a triangle is equilateral, then it is equiangular")
Computer Science: Conditional statements (if-then-else) are fundamental constructs in programming languages
Law: Legal rules are frequently phrased as conditionals ("If a person commits act X, then punishment Y applies")
Medicine: Diagnostic reasoning uses conditionals ("If the patient has symptoms X, Y, and Z, then they likely have condition A")
Exercises
Converting to If-then Form
Rewrite each statement below in a clear if-then form (\(p \to q\)). Identify the hypothesis and the conclusion for each.
4.1. "By getting enough votes you will win the election."
4.2. "On the fourth Thursday in November we will eat turkey."
4.3. "I am happy when I am studying calculus."
4.4. "People taller than six feet weigh more than three pounds."
4.5. "I will attend school in the Fall if I am accepted at Stanford."
4.6. "A valid ID is required for admission."
4.7. "I must pay my rent or I will be evicted."
Identifying Hypotheses and Conclusions
In each of the statements below, pinpoint the hypothesis (if-part) and the conclusion (then-part).
4.8. "If it rains in the city, the streets get wet."
4.9. "You will have no money if you buy the car."
4.10. "If you go 80 MPH, then you will get there in time."
4.11. "If wishes were horses, dreamers would ride."
4.12. "You may go see a movie if you finish your homework."
4.13. "If California is east of Columbia, then Thailand is west of Turkey."
4.14. "You'll break your neck if you fall out of that tree!"
4.15. "If you can't run with the big dogs, then stay on the porch."
True or False Conditionals
Assume: * "Pigs can fly" (\(p\)) is false. * "Horses can talk" (\(q\)) is false. * "Goats can climb trees" (\(r\)) is true. * "Math is fun" (\(s\)) is true.
Determine whether each if-then statement below is true or false using the truth-table definition:
4.16. If horses can talk (\(q\)) then pigs can fly (\(p\)).
4.17. If math is fun (\(s\)) then goats can climb trees (\(r\)).
4.18. If pigs can fly (\(p\)) then math is fun (\(s\)).
4.19. If goats can climb trees (\(r\)) then horses can talk (\(q\)).
Analyzing If-then Statements
4.20. "If it's Monday, then I have a meeting." Suppose it's not Monday—does the statement become false or remain true in logic? Explain.
4.21. "If I wear my lucky socks, then my team will win." If your team wins but you aren't wearing your lucky socks, is the statement still true? Why or why not?
4.22. Make up two if-then statements that seem unrelated in real life, and explain why they are considered true in formal logic when their hypotheses are false.
Practical Applications
4.23. Find three examples of if-then statements in legal documents, contracts, or terms of service. Identify the hypothesis and conclusion in each. 4.24. Write a conditional statement that could be used in a computer program to solve a real-world problem. 4.25. Create a logical argument using at least two connected if-then statements to reach a conclusion.
Final Thoughts
If-then statements serve as the backbone of logical reasoning and deduction. They allow us to express relationships between conditions and consequences in a precise way. By understanding when a conditional statement is true or false according to formal logic, you gain powerful tools for:
- Analyzing arguments and detecting logical fallacies
- Writing clear specifications and requirements
- Designing efficient decision processes
- Understanding the logical structure of mathematical proofs
In the next sections, we'll explore related concepts like converses, contrapositives, and biconditional statements that will further enhance your logical toolkit.