Introduction to Reasoning and Proof

You already know how to reason. Every time you figure out that it must have rained because the sidewalk is wet, or conclude that your friend is upset based on their short text replies, you are using the same logical thinking that mathematicians use. The difference is that mathematicians have developed a precise language and structure for this kind of reasoning, one that allows them to be absolutely certain about their conclusions. In this lesson, you will learn that language, and you will discover that the logical thinking you already do naturally can be sharpened into a powerful tool for proving mathematical truths.

Core Concepts

Two Ways of Thinking: Inductive and Deductive Reasoning

Imagine you are at a new restaurant. You try their pasta dish and it is delicious. You come back and try their pizza, also delicious. You try their salad, their soup, their dessert. All delicious. Based on this pattern, you conclude: “Everything at this restaurant is delicious.”

That is inductive reasoning. You observed specific examples and made a general conclusion based on those observations. Inductive reasoning is powerful and useful. It is how scientists form hypotheses and how we navigate daily life. But notice something important: no matter how many dishes you try, you cannot be 100% certain that every dish is delicious. There might be one you have not tried yet that is terrible.

Now consider a different scenario. You know that all items on the lunch special menu are $10. You also know that the pasta is on the lunch special menu. Therefore, the pasta costs $10. This conclusion is not just probable, it is absolutely certain (assuming your starting facts are true).

That is deductive reasoning. You started with general rules and applied them to reach a specific, guaranteed conclusion. Deductive reasoning is the backbone of mathematical proof. When mathematicians prove something, they are not saying “this is probably true based on patterns.” They are saying “this must be true, with complete certainty, because it follows logically from things we already know.”

Reasoning Type	Direction	Certainty	Example
Inductive	Specific to general	Probable, not certain	“I’ve seen 100 swans, all white, so all swans are white”
Deductive	General to specific	Certain (if premises are true)	“All squares have 4 sides. This is a square. Therefore, it has 4 sides.”

Conjectures and Counterexamples

When you use inductive reasoning to form a general statement, that statement is called a conjecture. A conjecture is an educated guess, a claim you believe might be true based on evidence, but which has not yet been proven.

Here is a conjecture: “The sum of any two odd numbers is even.”

Is this true? Let us test some examples:

• $1 + 3 = 4$ (even)
• $5 + 7 = 12$ (even)
• $11 + 15 = 26$ (even)

Every example works. But remember, inductive reasoning cannot give us certainty. To truly prove this conjecture, we would need deductive reasoning (which we will explore soon). However, we can use inductive reasoning another way: to disprove conjectures.

A counterexample is a single example that proves a conjecture false. You only need one. Consider this conjecture: “All prime numbers are odd.”

Let us think about this. Is 2 a prime number? Yes, its only factors are 1 and itself. Is 2 odd? No, it is even. That is it. The single counterexample of 2 completely destroys the conjecture. No matter how many odd primes exist (3, 5, 7, 11, 13…), the existence of just one even prime proves the conjecture false.

This is a beautiful asymmetry in mathematics:

• To prove a conjecture true, you often need careful deductive reasoning
• To prove a conjecture false, you just need one counterexample

Conditional Statements: The Language of Logic

Mathematical reasoning relies heavily on “if-then” statements, called conditional statements. These have a precise structure:

“If it is raining, then the ground is wet.”

The part after “if” is called the hypothesis (or premise). The part after “then” is called the conclusion.

• Hypothesis: It is raining
• Conclusion: The ground is wet

We often use variables to represent these parts. If we call the hypothesis $p$ and the conclusion $q$, we can write the conditional as “If $p$, then $q$” or symbolically as $p \rightarrow q$.

Understanding this structure is essential because mathematical proofs are chains of conditional statements. Each step says “if this is true, then that must be true,” building toward a final conclusion.

Converse, Inverse, and Contrapositive

Every conditional statement has three related statements that you should know. Let us start with our original:

Original (Conditional): If it is raining, then the ground is wet. $(p \rightarrow q)$

The converse switches the hypothesis and conclusion:

Converse: If the ground is wet, then it is raining. $(q \rightarrow p)$

Notice something important: the converse is not necessarily true even if the original is true. The ground could be wet because someone watered the lawn, or because of morning dew, or because a fire hydrant burst. The original statement and its converse are logically independent.

The inverse negates both parts:

Inverse: If it is not raining, then the ground is not wet. $(\neg p \rightarrow \neg q)$

Again, this is not necessarily true. It might not be raining right now, but the ground could still be wet from earlier rain.

The contrapositive both switches and negates:

Contrapositive: If the ground is not wet, then it is not raining. $(\neg q \rightarrow \neg p)$

Here is the key insight: the contrapositive is always logically equivalent to the original statement. If the original is true, the contrapositive must be true, and vice versa. Think about it: if rain always makes the ground wet, then finding dry ground means it cannot be raining.

Form	Structure	From “If raining, then wet”
Conditional	If $p$, then $q$	If raining, then wet
Converse	If $q$, then $p$	If wet, then raining
Inverse	If not $p$, then not $q$	If not raining, then not wet
Contrapositive	If not $q$, then not $p$	If not wet, then not raining

Remember: The conditional and contrapositive are logically equivalent. The converse and inverse are logically equivalent to each other, but not to the original.

Biconditional Statements

Sometimes a statement and its converse are both true. When this happens, we can combine them into a single biconditional statement using the phrase “if and only if.”

Consider this statement about angles: “An angle is a right angle if and only if it measures 90 degrees.”

This means two things are true:

1. If an angle is a right angle, then it measures 90 degrees.
2. If an angle measures 90 degrees, then it is a right angle.

The biconditional is written symbolically as $p \leftrightarrow q$, and it is true only when both the conditional and its converse are true.

Biconditionals often appear as definitions in mathematics. When you see “if and only if” (sometimes abbreviated “iff”), you know the relationship works both ways.

Laws of Logic: Building Chains of Reasoning

Two fundamental laws allow us to connect statements and build arguments.

Law of Detachment (Modus Ponens)

If you know that a conditional statement is true, and you know that the hypothesis is true, then you can conclude that the conclusion is true.

Symbolically: If $p \rightarrow q$ is true, and $p$ is true, then $q$ is true.

Example:

• Statement 1: If a figure is a square, then it has four right angles.
• Statement 2: Figure ABCD is a square.
• Conclusion: Figure ABCD has four right angles.

This might seem obvious, but it is the fundamental logical move that powers all of mathematics.

Law of Syllogism (Chain Rule)

If you have two conditional statements where the conclusion of the first matches the hypothesis of the second, you can chain them together.

Symbolically: If $p \rightarrow q$ is true, and $q \rightarrow r$ is true, then $p \rightarrow r$ is true.

Example:

• Statement 1: If it is Saturday, then I go to the farmer’s market.
• Statement 2: If I go to the farmer’s market, then I buy fresh vegetables.
• Conclusion: If it is Saturday, then I buy fresh vegetables.

This law allows us to build long chains of reasoning, where each link connects to the next.

Introduction to Proof Structure

A mathematical proof is a logical argument that demonstrates why a statement must be true. While there are many proof techniques (which you will explore in later lessons), all proofs share certain characteristics:

1. They start with given information. These are facts or conditions you are allowed to assume.

2. Each step follows logically from previous steps. Every statement you make must be justified, either as something given, a definition, a previously proven theorem, or a logical consequence of earlier statements.

3. They end with the statement to be proven. The final step should be exactly the conclusion you were trying to establish.

4. Every step has a reason. In formal proofs, you explicitly state why each step is valid.

Here is a simple example of proof structure:

Given: $m$ and $n$ are even integers. Prove: $m + n$ is even.

Proof:

Statement	Reason
$m$ and $n$ are even integers	Given
$m = 2a$ for some integer $a$	Definition of even
$n = 2b$ for some integer $b$	Definition of even
$m + n = 2a + 2b$	Substitution
$m + n = 2(a + b)$	Factoring
$a + b$ is an integer	Closure of integers under addition
$m + n$ is even	Definition of even

Each step builds on previous ones, every claim has a reason, and we arrive at exactly what we wanted to prove. This is the structure you will use throughout geometry and beyond.

Notation and Terminology

Term	Meaning	Example
Conjecture	An educated guess based on observations	“All primes greater than 2 are odd”
Counterexample	A single example that disproves a conjecture	2 is prime but not odd
Conditional	An “if-then” statement	If $p$, then $q$
Hypothesis	The “if” part of a conditional	$p$ in “If $p$, then $q$”
Conclusion	The “then” part of a conditional	$q$ in “If $p$, then $q$”
Converse	Switch hypothesis and conclusion	If $q$, then $p$
Inverse	Negate both parts	If not $p$, then not $q$
Contrapositive	Negate and switch	If not $q$, then not $p$
Biconditional	True in both directions	$p$ if and only if $q$
Law of Detachment	From $p \rightarrow q$ and $p$, conclude $q$	Modus Ponens
Law of Syllogism	Chain conditionals together	From $p \rightarrow q$ and $q \rightarrow r$, conclude $p \rightarrow r$
Proof	A logical argument showing why a statement must be true	See examples

Examples

Key Properties and Rules

Truth Values of Related Conditionals

If the original $(p \rightarrow q)$ is…	Then the contrapositive $(\neg q \rightarrow \neg p)$ is…
True	True
False	False

The conditional and its contrapositive are logically equivalent. They always have the same truth value.

The converse $(q \rightarrow p)$ and inverse $(\neg p \rightarrow \neg q)$…
Are logically equivalent to each other
May or may not have the same truth value as the original

Law of Detachment

If you know:

1. $p \rightarrow q$ is true (the conditional)
2. $p$ is true (the hypothesis)

Then you can conclude:

• $q$ is true (the conclusion)

Warning: You cannot use the Law of Detachment if you only know $q$ is true. Knowing the conclusion does not tell you anything about the hypothesis.

Law of Syllogism

If you know:

1. $p \rightarrow q$ is true
2. $q \rightarrow r$ is true

Then you can conclude:

• $p \rightarrow r$ is true

This can be extended to any number of linked conditionals: $p \rightarrow q$, $q \rightarrow r$, $r \rightarrow s$, … leads to $p \rightarrow$ (final conclusion)

Biconditional Statements

A biconditional $p \leftrightarrow q$ is true only when:

• $p \rightarrow q$ is true, AND
• $q \rightarrow p$ is true

If either direction is false, the biconditional is false.

Proving vs. Disproving

To prove a universal statement (“for all…”)	To disprove a universal statement
Must use deductive reasoning	Just need one counterexample
Must cover all possible cases	One exception is enough
Often requires careful proof	Often requires creative thinking

Real-World Applications

Legal Reasoning

Contracts are full of conditional statements. “If the buyer fails to make payment within 30 days, then the seller may terminate the agreement.” Legal arguments often involve analyzing whether the hypothesis of such a statement has been met, and therefore whether the conclusion applies. Lawyers use the Law of Detachment constantly: they establish that a conditional rule exists, then argue that the hypothesis is satisfied in their case.

The contrapositive also appears in legal reasoning. If a contract states “If you are an employee, then you must sign the confidentiality agreement,” the contrapositive tells us “If someone has not signed the confidentiality agreement, then they are not an employee.” This kind of reasoning helps establish facts.

Medical Diagnosis

Doctors use both inductive and deductive reasoning. Inductively, they notice patterns: “Several patients with these symptoms had this disease.” This leads to conjectures about diagnosis. Deductively, they apply known medical facts: “If a patient has symptom X and symptom Y, then they should be tested for condition Z.”

Medical reasoning also relies heavily on the contrapositive. If a certain disease always causes a particular marker in blood tests, then the absence of that marker rules out the disease.

Computer Programming

Programming is built on conditional logic. Every “if-then” statement in code is a conditional:

if (user_logged_in) {
    show_dashboard();
}

Programmers must think carefully about all four related forms. What happens if the user is not logged in (inverse)? What if someone is viewing the dashboard (converse)? Should someone not seeing the dashboard be assumed not logged in (contrapositive)?

The Law of Syllogism appears as chained conditionals: if A then B, if B then C, therefore if A then C. This is fundamental to how programs flow.

Scientific Method

Science uses inductive reasoning to form hypotheses based on observations. But a hypothesis is just a conjecture until it is tested. Scientists then try to find counterexamples. If they cannot disprove a hypothesis despite many attempts, it gains credibility. But unlike mathematics, scientific knowledge remains provisional. New evidence could always provide a counterexample.

The logical structure “If this theory is correct, then we should observe X” drives experimental design. Scientists use the contrapositive: if we do not observe X, the theory (in its current form) is incorrect.

Self-Test Problems

Summary

• Inductive reasoning uses specific observations to form general conclusions. It is useful for making conjectures but cannot provide certainty.

• Deductive reasoning uses general principles to reach specific conclusions. When the premises are true, the conclusion is guaranteed.

• A conjecture is an educated guess. A single counterexample is enough to disprove it.

• Conditional statements have the form “If $p$, then $q$,” where $p$ is the hypothesis and $q$ is the conclusion.

• Every conditional has three related statements:

  • Converse: Switch hypothesis and conclusion
  • Inverse: Negate both
  • Contrapositive: Switch and negate

• The conditional and contrapositive are logically equivalent. The converse and inverse are equivalent to each other but independent of the original.

• A biconditional ($p$ if and only if $q$) is true when both the conditional and its converse are true.

• The Law of Detachment says: from $p \rightarrow q$ and $p$, conclude $q$.

• The Law of Syllogism says: from $p \rightarrow q$ and $q \rightarrow r$, conclude $p \rightarrow r$.

• Proofs are structured logical arguments where every step has a reason, building from given information to a conclusion.

These tools of logical reasoning are the foundation for everything else in geometry. When you understand how to construct and analyze logical arguments, you are ready to prove theorems, not just memorize them. You are learning to think like a mathematician.