Common Invalid Proofs

A Proof Can Look Plausible and Still Fail

A proof is a chain of reasoning from stated assumptions to a conclusion. Each step must follow from the assumptions, definitions, or results already established. An argument may contain correct calculations and still fail if one transition is unjustified. In this tutorial, an invalid proof means an argument that does not establish the claim it is offered to prove. The claim itself might be true or false; validity concerns whether the reasoning guarantees the conclusion.

This distinction is important. A true conclusion does not make every argument for it valid, and one counterexample to a proposed argument does not by itself show that the conclusion is false. When checking a proof, ask two separate questions: Is the statement true? And, assuming the premises of this argument, does the conclusion necessarily follow?

The earlier tutorials Converse Inverse and Contrapositive and Logical Connectives provide tools for tracking conditional statements. Here we use those ideas to examine common errors in mathematical arguments. Each error has a recognizable shape, but the remedy is not merely to memorize a label. We must identify the inference that fails and show why the premises can hold while the claimed conclusion does not.

A counterexample to an argument is not always a counterexample to its conclusion. To expose an invalid inference, find a situation in which the argument's premises are true and its conclusion is false. That shows the premises do not force the conclusion.

Two Invalid Inferences Involving Conditionals

Suppose $P$ and $Q$ are propositions. Knowing $P\to Q$ means that whenever $P$ holds, $Q$ holds. It does not say that $P$ is the only way for $Q$ to hold, nor that $Q$ can hold only when $P$ holds. The following two patterns mistakenly draw one of those stronger conclusions.

Affirming the consequent.
From $P\to Q$ and $Q$, conclude $P$. The conclusion does not follow: $Q$ might be true for another reason.

Denying the antecedent.
From $P\to Q$ and $\neg P$, conclude $\neg Q$. The conclusion does not follow: $Q$ may still be true when $P$ is false.

These are not merely unreliable strategies that sometimes happen to work. Each is an invalid inference form. To establish this precisely, we give a truth assignment where the premises are true but the conclusion is false. Such an assignment is enough to show that the premises do not logically guarantee the conclusion.

Worked Example: Why Affirming the Consequent Is Invalid

Proposition. The inference from $P\to Q$ and $Q$ to $P$ is invalid.

Proof. Assign $P$ the value false and $Q$ the value true. Under this assignment, $P\to Q$ is true: a conditional with a false antecedent is true. The second premise, $Q$, is true by the assignment. But the proposed conclusion $P$ is false. Thus both premises hold while the conclusion fails, so the inference is invalid.

The failure is not that a conditional has been used incorrectly. The issue is that $Q$ does not identify the reason it is true. The implication permits other situations in which $Q$ holds, including ones where $P$ does not.

Worked Example: Why Denying the Antecedent Is Invalid

Proposition. The inference from $P\to Q$ and $\neg P$ to $\neg Q$ is invalid.

Proof. Again assign $P$ the value false and $Q$ the value true. The conditional $P\to Q$ is true because its antecedent is false. The premise $\neg P$ is true, but the proposed conclusion $\neg Q$ is false because $Q$ is true. The premises therefore do not guarantee the conclusion, and the inference is invalid.

In a particular problem, the conclusion $\neg Q$ might follow from additional information. It does not follow from these two premises alone. Valid reasoning must use hypotheses that are actually present, rather than information that would make the desired conclusion more likely.

The truth-assignment proofs above are short because the claims concern inference patterns, not a particular number system. The same patterns appear in ordinary mathematical language. The next examples show how they can be hidden inside claims about numbers.

When a Conditional Is Reversed Without Justification

A theorem with the form “if $P$, then $Q$” does not automatically give “if $Q$, then $P$.” The second statement is the converse, and it requires its own proof. In Converse Inverse and Contrapositive, we established the relevant logical distinction; the example below applies it to a concrete claim.

Worked Example: A Square Does Not Determine Its Root's Sign

Claim under review. “If $x=3$, then $x^2=9$. We know $x^2=9$, so $x=3$.”

The first sentence states a true conditional: substituting $x=3$ gives $x^2=3^2=9$. The proposed argument then affirms the consequent. To check it, take $x=-3$. Then $$ x^2=(-3)^2=9, $$ so the premise $x^2=9$ is true, but $x=3$ is false. Therefore the argument is invalid.

The flaw is not a mistake in calculating the square. It is the assumption that the stated condition $x^2=9$ can arise only from the value $x=3$. A correct conclusion needs to account for every real number satisfying the premise. In fact, both $3$ and $-3$ satisfy it, so the asserted conclusion is too restrictive.

A similar check works for denial of the antecedent. For instance, the conditional “if an integer is divisible by $6$, then it is even” does not imply that every integer not divisible by $6$ is odd. The integer $10$ is not divisible by $6$, but it is even. The conditional remains true; it is the attempted inference to oddness that fails.

Division by a Quantity That Might Be Zero

Algebraic transformations can also introduce an invalid step. Dividing both sides of an equation by the same number is permitted only when that number is nonzero. If a proof divides by an expression that could equal zero, it may discard a necessary case or assert a false conclusion. A cancellation step is division in another form and has the same requirement.

Worked Example: A False Conclusion from Cancelling Zero

Claim under review. The following calculation appears to prove that $2=1$. Let $a=b=1$. Then $a^2=ab$, and subtracting $b^2$ from both sides gives $$ a^2-b^2=ab-b^2. $$ Factoring gives $$ (a-b)(a+b)=b(a-b). $$ The argument then cancels $a-b$ and concludes $a+b=b$, or $2=1$.

The initial equation and the subtraction are valid. The factorizations are also valid. But $a-b=1-1=0$. Cancelling that factor means dividing both sides by zero, which is not permitted. Indeed, before the cancellation, both sides of the factored equation are zero: $$ (1-1)(1+1)=0,\qquad 1(1-1)=0. $$ The equation $0=0$ does not imply $2=1$. The false conclusion appears only after the illegal division.

The error is easy to miss because the same factor appears on both sides. Before cancelling an expression, check that it is nonzero under the hypotheses. If it may be zero, separate the zero case from the nonzero case and handle each one with valid reasoning.

Cancellation has a hypothesis. From $uv=uw$, one may conclude $v=w$ when $u\ne0$. If $u=0$, the equation reduces to $0=0$ and gives no such conclusion.

Circular Reasoning and Unsupported Steps

Another error is circular reasoning: using the statement to be proved, or an equivalent unproved statement, as a premise in its own proof. The argument may repeat the claim in different words or hide it inside an intermediate step. A chain of correct implications cannot establish a conclusion if one of its starting points is the very conclusion that needs justification.

For example, suppose someone is asked to prove that a particular real number $r$ is positive and writes: “Since $r>0$, it follows that $r$ is positive.” The last sentence is true, but the argument has assumed $r>0$ at the start. It gives no reason to accept that assumption. The issue is not the truth of the sentence but the lack of an independent justification for the premise.

Not every use of a desired conclusion in planning is circular. In Working Backward From a Conclusion, one may ask what would be sufficient to prove the target. That is a way to discover a proof strategy. The written proof must then establish those sufficient conditions from the original hypotheses. Backward exploration helps plan; it cannot replace the forward justification.

A Systematic Check for Gaps

Invalid arguments often become easier to diagnose when each line is checked against the line before it. Identify what is assumed, what is being concluded, and the exact rule that is supposed to connect them. The following audit is useful for both short calculations and longer proofs.

Separate premises from the target.
Write down exactly what is given and exactly what must be proved. Do not treat a desired conclusion as a known fact.

Check the direction of every implication.
From $P\to Q$, do not infer $P$ from $Q$, or $\neg Q$ from $\neg P$, without an additional argument.

Check restrictions before algebraic operations.
Before dividing or cancelling, verify that the quantity is nonzero. Before using an inequality operation, verify the sign conditions that preserve its direction.

Test the reasoning against a possible counterexample.
For a proposed inference, try to make all its premises true and its conclusion false. For a universal claim, test whether one allowed input can violate it.

Account for every case and return to the claim.
If a quantity may vanish or an argument splits into cases, verify that no case is lost and that the final statement is the original target.

Warning sign	Question to ask	Typical repair
The conclusion resembles a hypothesis with its direction reversed.	Has the converse actually been proved?	Use the original hypothesis directly, or supply an independent proof of the converse.
A factor or expression is cancelled.	Could that expression be zero?	Verify it is nonzero, or handle the zero and nonzero cases separately.
The proof begins with the target.	Was this fact already established independently?	Replace it with the given hypotheses and derive the target step by step.
An example is offered for a universal statement.	Does one example cover every input?	Prove the claim for an arbitrary input, or use a counterexample only to refute the universal claim.

The last row highlights another frequent confusion. Verifying a statement for several numbers can suggest a pattern, but finitely many checks do not prove a claim about every real number or every integer. Conversely, one permitted input that fails is enough to disprove a universal statement. The role of examples depends on the direction of the task: examples can illustrate or disprove, but they do not establish an unrestricted universal claim.

When a proof seems suspicious, do not stop at saying that it “feels wrong.” Locate the first step that does not follow, state the missing condition, and explain whether the conclusion can be repaired. Sometimes a single counterexample shows the inference fails; sometimes adding a justified nonzero condition fixes an algebraic step; sometimes the argument needs to be reorganized from its actual hypotheses.

Proof checking is local and global. Check each transition for validity, then check that the complete argument begins with permitted assumptions and ends with the stated target. A correct calculation inside an invalid chain does not make the chain a proof.

Check Your Understanding

For each question, identify the issue and state what would be needed for a valid argument.

An argument has premises $P\to Q$ and $Q$, and concludes $P$. What is the name of this invalid inference, and what truth assignment demonstrates its failure?
Why does $P\to Q$ together with $\neg P$ fail to establish $\neg Q$?
In the calculation with $a=b=1$, which algebraic operation is invalid, and what is the value of the factor being cancelled?
A proof of $r>0$ begins, “Since $r>0$, we are done.” What makes this circular, and what must the proof establish instead?
Why can one example support a universal statement as an illustration but not prove it for every input? How can one example be used to refute such a statement?
Before cancelling a common factor from both sides of an equation, what condition must be checked?

What You'll Learn