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 (a conditional or implication). It connects having a membership card (hypothesis) to being allowed inside (conclusion). If-then statements are everywhere in daily life and 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 parts:
1. Hypothesis (if-part): The condition that might 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 (antecedent)
- \(q\) = the conclusion (consequent)
Everyday Language Examples
-
"If you jump off the cliff, then you will get hurt."
Hypothesis: "You jump off the cliff."
Conclusion: "You will get hurt." -
"What goes up comes down."
(Can be phrased as: "If something goes up, then it comes down.") -
"Bigger engines burn more gas."
(As an implication: "If an engine is bigger, then it burns more gas.")
Other Ways to Say It
"\(p \to q\)" can be read as:
- "\(p\) implies \(q\)."
- "Whenever \(p\), \(q\) follows."
- "\(p\) being true forces \(q\) to be true."
Truth of If-Then: It's Not About Causality
In everyday life, we often expect an if-then to reflect a cause-and-effect relationship (if you water plants, they grow). However, in formal logic, the truth of \(p \to q\) depends purely on the truth values of \(p\) and \(q\), not on actual causation or relevance. By definition, \(p \to q\) is false only in one scenario: when \(p\) is true and \(q\) is false. In every other case, \(p \to q\) is true.
Consider these four examples of "\(p \to q\)":
-
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 (even though one doesn't cause the other). -
If the sky is neon yellow, then the oceans are wet.
\(p\) = "The sky is neon yellow" (false), \(q\) = "The oceans are wet" (true).
Hypothesis is false, so \(p \to q\) is considered true (a false "if" makes the implication true, regardless of \(q\)). -
If the sky is neon yellow, then the oceans are dry.
\(p\) = "The sky is neon yellow" (false), \(q\) = "The oceans are dry" (false).
Hypothesis is false, \(q\) is false, but still \(p \to q\) is true (because a false \(p\) makes \(p \to q\) true, regardless of \(q\)). -
If the sky is blue, then the oceans are dry.
\(p\) = "The sky is blue" (true), \(q\) = "The oceans are dry" (false).
Here, hypothesis is true and conclusion is false → \(p \to q\) is false.
Key Truth Rule for Implications
\(p \to q\) is false only when \(p\) is true and \(q\) is false. In all other cases, \(p \to q\) is true.
Truth Table for If-Then
To confirm the rule, let's see the truth table for a conditional \(p \to q\):
| \(p\) | \(q\) | \(p \to q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Row by row:
- Row 1: \(p\) = T, \(q\) = T → \(p \to q\) = T. (True implies true is true.)
- Row 2: \(p\) = T, \(q\) = F → \(p \to q\) = F. (True implies false is the one false case.)
- Row 3: \(p\) = F, \(q\) = T → \(p \to q\) = T (if-part false makes the implication true, regardless of \(q\)).
- Row 4: \(p\) = F, \(q\) = F → \(p \to q\) = T (hypothesis false again makes the implication true).
Far-Fetched Hypothesis
Let \(p\) = "Dogs can read" (false), and \(q\) = "Cats can sing" (false).
The implication \(p \to q\) is "If dogs can read, then cats can sing." According to the truth table, since \(p\) is false, \(p \to q\) is true in formal logic (even though the content is fanciful). This illustrates that logical truth is about structure, not actual real-world causation.
Using If-Then in Reasoning
In logic, if we know \(p \to q\) is true and we know \(p\) is true, then we can deduce \(q\). This form of reasoning is called Modus Ponens — one of the most fundamental inference rules:
From "\(p \to q\)" and "\(p\)," infer "\(q\)."
Real-Life Example (Modus Ponens)
- \(p\) = "I have a valid ticket."
- \(q\) = "I can enter the concert."
If "Having a valid ticket implies I can enter the concert" (\(p \to q\) is true), and indeed I do have a valid ticket (\(p\) is true), then I can conclude I am allowed in (\(q\) is true).
Another common reasoning pattern is the contrapositive form (covered in Part 5): from \(p \to q\) and \(\neg q\), infer \(\neg p\) (this is Modus Tollens). For example, "If I have a membership card, then I can enter. I could not enter, therefore I must not have had a membership card."
Common Misconceptions
-
"If \(p\) is false, then \(p \to q\) must be false."
Reality: If \(p\) is false, \(p \to q\) is true by the logical definition (no matter what \(q\) is). -
"An if-then statement must reflect a real cause-and-effect."
Reality: Formal logic cares only about truth values, not actual causality or relevance. Even a nonsensical implication can be true in the logical sense if the condition is false or the conclusion is true. -
"If \(p \to q\) is true, then \(p\) causes \(q\)."
Reality: "\(p \to q\)" simply means "if \(p\) is true, then \(q\) is true." It says nothing about causation. (For example, "If 2+2=5, then pigs can fly" is logically true because 2+2=5 is false, but there's no causal link intended.)
Historical Context
The concept of conditional statements dates back to ancient Greek logic. Aristotle was among the first to formalize logical reasoning, and the Stoic philosophers further developed propositional logic including early ideas of 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 Gottlob Frege, Bertrand Russell, and Alfred North Whitehead as part of their work to formalize mathematics. This eventually fed into computer science (where if-then rules are fundamental to programming).
Applications in Different Fields
Mathematics: Mathematical theorems are often stated as conditionals ("If a number is prime and greater than 2, then it is odd"). Mastering if-then helps in understanding proofs and the concept of necessary/sufficient conditions.
Computer Science: Conditional statements (if-then-else) are the backbone of decision-making in algorithms and programs. Knowing logical implications helps programmers avoid bugs in complex conditions.
Law: Many laws are conditional ("If a person commits act X, then consequence Y follows"). Understanding the logic helps in interpreting statutes and constructing legal arguments without logical loopholes.
Medicine: Diagnostic reasoning uses implications ("If the patient shows symptoms X, Y, and Z, then they have condition A"). Recognizing these as if-then rules clarifies why certain tests or treatments are applied.
Exercises
Converting to If-Then Form
Rewrite each statement in a clear if-then form (\(p \to q\)). Identify the hypothesis and conclusion in each case:
4.1. "By getting enough votes you will win the election."
4.2. "On the fourth Thursday in November we 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." (Hint: think of it as "If a person does not have a valid ID, then...")
4.7. "I must pay my rent or I will be evicted." (Hint: rewrite as an implication by contrapositive: "If I don't pay rent, then...")
Identifying Hypotheses and Conclusions
For each implication below, label the hypothesis (if-part) and the conclusion (then-part):
4.8. If it rains today, the picnic will be canceled.
4.9. If Tom is a bachelor, then Tom is unmarried.
4.10. If the alarm is ringing, then a door or window was opened.
4.11. If Alice passes the exam, then Bob will throw a party.
Reasoning with Implications
4.12. Modus Ponens: Given "If I finish my work (\(p\)), then I will go out (\(q\))" and "I finished my work (\(p\))," what can you conclude?
4.13. Modus Tollens: Given "If the cake is made with sugar (\(p\)), then it will ferment (\(q\))" and "The cake did not ferment (\(\neg q\))," what can you conclude about \(p\)?
4.14. Explain in your own words why an implication with a false hypothesis is considered true (use an example different from ones given above).
Next: In Part 5: The Converse and the Contrapositive, we will explore what happens when we flip or negate if-then statements — and why some transformations preserve truth while others do not.