Valid and Invalid Reasoning

The Big Idea

A correct answer can be reached for the wrong reason, and a logically valid argument can begin with a mistaken premise. To evaluate reasoning, we must separate the truth of individual statements from the structure connecting them.

A deductive argument is valid when there is no possible case in which all its premises are true and its conclusion is false.

An argument is invalid when such a case is possible. Validity asks whether the reasoning preserves truth. It does not ask whether the premises happen to be true in the current situation.

Arguments Have Premises and a Conclusion

An argument is a group of propositions in which some propositions—the premises—are offered as reasons for accepting another proposition—the conclusion.

For example:

1. Every integer divisible by \(4\) is even.
2. \(20\) is divisible by \(4\).
3. Therefore, \(20\) is even.

The first two statements are premises. The final statement is the conclusion. Words such as because, since, and given that often introduce premises; words such as therefore, thus, and hence often introduce conclusions.

Truth and Validity Describe Different Things

A proposition is not “valid,” and an argument is not ordinarily “true.” Keeping the vocabulary separate helps us locate the actual problem.

Validity Is a No-Counterexample Standard

Suppose an argument has premises \(P_1,P_2,\ldots,P_n\) and conclusion \(C\). It is valid exactly when

\[ (P_1\land P_2\land\cdots\land P_n)\to C \]

is always true.

To show invalidity, find a countermodel: an interpretation or assignment in which every premise is true and the conclusion is false. A case with a false premise does not demonstrate invalidity because validity asks what happens when all premises hold.

Soundness Adds True Premises

A deductive argument is sound when:

1. the argument is valid, and
2. every premise is true.

Every sound argument has a true conclusion. Validity supplies the reliable structure, and true premises supply the correct starting information.

An argument may be valid but unsound. Consider:

1. If \(12\) is divisible by \(6\), then \(12\) is odd.
2. \(12\) is divisible by \(6\).
3. Therefore, \(12\) is odd.

The modus-ponens form is valid, but the first premise is false. The argument is unsound, and its conclusion is false.

A True Conclusion Does Not Repair Invalid Reasoning

Consider:

1. If an integer is divisible by \(4\), then it is even.
2. \(8\) is even.
3. Therefore, \(8\) is divisible by \(4\).

Every statement happens to be true, but the reasoning is invalid. The form reverses the conditional:

\[ p\to q,\qquad q \qquad\therefore\qquad p. \]

If we replace \(8\) with \(6\), both premises remain true while the conclusion becomes false. This countermodel exposes the invalid structure.

Modus Ponens

Modus ponens applies a conditional by establishing its hypothesis:

\[ p\to q,\qquad p \qquad\therefore\qquad q. \]

If differentiability implies continuity at a point and a function is differentiable there, then it is continuous there. The second premise activates the promise made by the first.

Modus Tollens

Modus tollens uses the contrapositive:

\[ p\to q,\qquad\neg q \qquad\therefore\qquad\neg p. \]

If every multiple of \(4\) is even and \(15\) is not even, then \(15\) is not a multiple of \(4\). A false necessary consequence rules out the proposed sufficient condition.

Hypothetical Syllogism

Conditional statements can be chained:

\[ p\to q,\qquad q\to r \qquad\therefore\qquad p\to r. \]

For example:

1. If an integer is divisible by \(8\), then it is divisible by \(4\).
2. If an integer is divisible by \(4\), then it is even.
3. Therefore, if an integer is divisible by \(8\), then it is even.

This valid form is called hypothetical syllogism. It organizes implication chains and subset relationships.

Disjunctive Syllogism

When at least one of two alternatives must hold and one is eliminated, the other follows:

\[ p\lor q,\qquad\neg p \qquad\therefore\qquad q. \]

If an integer is positive or nonpositive, and it is not positive, then it is nonpositive.

Standard logical “or” is inclusive. Disjunctive syllogism remains valid because ruling out \(p\) leaves \(q\) as the only way for \(p\lor q\) to be true.

Building and Using Conjunctions

Two simple but useful rules manage statements joined by “and.”

Conjunction introduction

\[ p,\qquad q \qquad\therefore\qquad p\land q. \]

Conjunction elimination

\[ p\land q \qquad\therefore\qquad p \]

and similarly \(p\land q\therefore q\). Proofs often establish several facts separately, combine them, and later extract whichever component is needed.

Common Valid Inference Forms

Invalid Form: Affirming the Consequent

The form

\[ p\to q,\qquad q \qquad\therefore\qquad p \]

is invalid. The conclusion \(q\) may have more than one sufficient cause or condition.

“If a figure is a square, then it has four sides. This figure has four sides. Therefore it is a square” fails because rectangles, rhombi, and other quadrilaterals also have four sides.

This error treats a necessary condition as though it were sufficient.

Invalid Form: Denying the Antecedent

The form

\[ p\to q,\qquad\neg p \qquad\therefore\qquad\neg q \]

is invalid. Failing one sufficient condition does not rule out every other way to obtain the conclusion.

“If an integer is divisible by \(4\), then it is even. The integer \(6\) is not divisible by \(4\). Therefore \(6\) is not even” has true premises and a false conclusion.

This error treats a sufficient condition as though it were necessary.

Invalid Form: Eliminating an Inclusive Alternative

From \(p\lor q\) and \(p\), we cannot generally conclude \(\neg q\):

\[ p\lor q,\qquad p \qquad\not\therefore\qquad\neg q. \]

Inclusive “or” permits both propositions to be true. If exclusivity is intended, it must be supplied as an additional premise.

For example, \(12\) is even or divisible by \(3\), and \(12\) is even. We cannot conclude that \(12\) is not divisible by \(3\); both properties hold.

Testing Validity with a Truth Table

To test modus ponens, inspect the rows where both premises \(p\to q\) and \(p\) are true.

Only the first row makes all premises true, and the conclusion is true there. No true-premise, false-conclusion row exists, so the form is valid.

Countermodels Demonstrate Invalidity

To show that an argument form is invalid, one countermodel is enough. For affirming the consequent, choose:

\[ p=F,\qquad q=T. \]

Then \(p\to q\) is true and \(q\) is true, but the proposed conclusion \(p\) is false.

A countermodel does not need to match the original topic. It exposes the logical form. A concrete mathematical example can then make the same failure easier to recognize.

Reasoning with Quantifiers

Quantified arguments require careful handling of arbitrary objects and witnesses.

From “some integer is even,” we may not conclude “\(8\) is even” unless we already know that \(8\) is the witness. Existence guarantees an object, not a particular object of our choosing.

Valid Logic Still Requires Legal Mathematics

A proof can use a recognizable logical form and still fail because an algebraic step is not reversible or permitted.

Common problems include:

• dividing by an expression that may equal zero,
• squaring both sides and forgetting to check extraneous solutions,
• taking a square root without considering both signs,
• canceling terms across addition,
• using a theorem outside its domain, and
• changing the meaning of a symbol during the argument.

Valid reasoning requires both sound logical inference and mathematically justified transformations.

Circular Reasoning and Hidden Assumptions

An argument is circular when it assumes, directly or indirectly, the conclusion it is supposed to establish.

Restating a claim in different words is not a proof. Nor is using a theorem whose proof depends on the very result under discussion unless the dependency has been independently resolved.

Hidden assumptions can also create gaps. A diagram may look like a right angle without one being given; a variable may be treated as positive without a stated reason; a function may be assumed invertible before injectivity is established.

Good reasoning makes the dependency chain visible.

Deductive Validity and Inductive Strength

The strict labels valid and invalid apply most directly to deductive arguments. Inductive arguments are usually described as strong or weak, because their premises support rather than guarantee the conclusion.

A strong inductive argument can still reach a false conclusion. A valid deductive argument with true premises cannot.

This distinction matters when mathematics interacts with data, probability, modeling, and science. The kind of reasoning determines the kind of confidence the conclusion can carry.

Formal Errors and Informal Fallacies

A formal fallacy has a defective logical structure, such as affirming the consequent. An informal fallacy depends on language, relevance, evidence, or context.

Attacking the person instead of the argument, misrepresenting an opponent’s claim, shifting the meaning of a key word, or presenting two options when more exist are informal reasoning failures.

Mathematical writing is less exposed to some rhetorical fallacies, but ambiguity, equivocation, hidden assumptions, and false alternatives can still appear. Formal notation helps only when the translation into notation is accurate.

An Argument-Checking Routine

1. List the premises and conclusion separately.
2. Clarify definitions, domains, and quantifiers.
3. Translate the logical form.
4. Ask whether true premises could coexist with a false conclusion.
5. Search for a countermodel if the form looks doubtful.
6. Name the inference rule when possible.
7. Check every mathematical transformation.
8. Look for reversed implications, hidden assumptions, and circularity.
9. After validity, check the premises for truth.
10. Classify the result. Valid, sound, invalid, or—if inductive—strong or weak?

This routine separates two quality-control questions: Does the reasoning work, and is the starting information correct?

Common Confusions

Working Summary

A deductive argument is valid when true premises cannot lead to a false conclusion. It is sound when it is valid and all its premises are true. A true conclusion alone does not establish validity, and valid form alone does not establish soundness.

Modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, and conjunction rules preserve truth. Affirming the consequent, denying the antecedent, and eliminating an inclusive alternative do not. Truth tables and countermodels make these differences testable.

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