8/9 8. Reasoning Errors and Informal Fallacies

1. Quantifiers, Predicates, and Domains

A quantified statement makes a claim about objects in a specified domain. Predicate logic represents a property with a predicate such as $P(x)$, where $x$ is a variable.

The two basic quantifiers are:

• The universal quantifier $\forall$, meaning “for every,” “all,” or “each.”

• The existential quantifier $\exists$, meaning “there exists,” “some,” or “at least one.”

For example, if the domain is all integers and $E(x)$ means “$x$ is even,” then $\forall x\,E(x)$ says that every integer is even, while $\exists x\,E(x)$ says that at least one integer is even.

The domain of discourse matters because it determines which objects are considered. The statement $\forall x\,(x^2\ge 0)$ is true over the real numbers, whereas $\forall x\,(x>0)$ is false over the integers because $0$ and negative integers are included.

A statement such as “Every student passed” can be written in either of these forms:

or, using a restricted domain,

“Some student passed” requires both properties of the same object:

The conditional form $\exists x\,(Student(x)\rightarrow Passed(x))$ is usually too weak, because it is automatically true for any object that is not a student.

Takeaway: Always identify the domain and distinguish “every” from “at least one” before evaluating a quantified claim.

2. Scope and Variable Binding

The scope of a quantifier is the formula governed by that quantifier. Use parentheses or brackets to show the scope clearly:

In this formula, $x$ is a bound variable because its occurrence lies within the scope of $\forall x$. By contrast, in

$x$ is bound but $y$ is free. A formula containing a free variable is generally an open sentence whose truth depends on what object is assigned to that variable.

Scope can change meaning. Compare:

with

In the first formula, both predicates occur inside the scope of the quantifier. In the second, only $P(x)$ is quantified; $Q(x)$ lies outside that scope and may contain a free variable.

When translating English, make the restriction explicit. “Every student passed” is not merely $\forall x\,Passed(x)$ unless the domain itself has already been restricted to students. With a general domain, use $\forall x\,(Student(x)\rightarrow Passed(x))$.

Takeaway: Parentheses identify which conditions a quantifier governs and prevent a change in meaning caused by ambiguous scope.

3. Negating Quantified Statements

Negating a quantified statement requires two changes: switch the quantifier and negate the predicate.

Thus, “Not everything has property $P$” means “Something does not have property $P$,” while “Nothing has property $P$” means “Everything lacks property $P$.”

For a restricted universal statement:

For example, the negation of “Every laptop in the room is charged” is “At least one laptop in the room is not charged”:

The negation of “Some student submitted the assignment” is “No student submitted the assignment”:

A common mistake is to negate “Every student passed” as “No student passed.” The correct negation says only that at least one student did not pass. The original statement can be false even when nearly every student passed.

Takeaway: To negate a quantified claim, switch $\forall$ and $\exists$, then negate the condition while preserving its logical structure.

4. Nested Quantifiers and Order

With nested quantifiers, quantifier order determines whether an object may vary from case to case or must remain the same.

Let $L(x,y)$ mean “person $x$ likes person $y$.” Then:

means that everyone likes at least one person. The person liked may be different for each individual.

By contrast,

means that there is one person whom everyone likes. The same individual must work for every $x$, so this statement is generally stronger.

The same distinction appears in ordinary language:

• “For every customer, there is a representative” allows different representatives for different customers.

• “There is one representative for every customer” requires a single representative who serves everyone.

Do not assume that $\forall x\,\exists y\,P(x,y)$ and $\exists y\,\forall x\,P(x,y)$ are equivalent. Reversing their order can replace a modest claim with a much stronger one.

Takeaway: When quantifiers are nested, ask whether the object satisfying the existential condition can vary or must be one fixed object.

5. Evidence for Universal and Existential Claims

Universal and existential claims require different kinds of evidence.

Universal claims

A universal claim has the form

One valid counterexample is enough to refute it: find an object $a$ in the domain for which $P(a)$ is false. For example, the claim “Every prime number is odd” is refuted by $2$, which is prime but not odd.

Testing several examples does not by itself prove a universal claim. A proof must apply to an arbitrary member of the domain, often using a definition, theorem, algebraic argument, or structural argument. To show that the sum of two even integers is even, let $m=2a$ and $n=2b$, where $a$ and $b$ are integers:

Because $a+b$ is an integer, the sum has the form $2k$ and is even.

Existential claims

An existential claim has the form

One valid witness establishes it. For “Some integer is both positive and prime,” $5$ is a witness because it is positive and prime.

To refute an existential claim, show that no object in the domain satisfies the condition. Equivalently, prove:

Finding several failed examples does not refute an existential claim unless every possible object has been ruled out.

Takeaway: Refute a universal claim with one valid counterexample, establish an existential claim with one valid witness, and use a proof or complete check when examples do not settle the issue.

6. A Practical Method for Avoiding Reasoning Errors

Use the following procedure whenever you evaluate a quantified claim:

1. Identify the domain. Determine what objects the variables range over.

2. Define the predicates. State precisely what each predicate means.

3. Mark the scope. Add parentheses or brackets to show what each quantifier governs.

4. Translate carefully. Distinguish “all,” “some,” “none,” “at least one,” and “exactly one.”

5. Test a universal claim for a counterexample. The object must belong to the domain, satisfy the restriction, and violate the conclusion.

6. Test an existential claim for a witness. The object must satisfy every required condition.

7. Use a proof or complete finite check when examples are insufficient. Repeatedly observing a pattern is not automatically a proof.

8. Negate structurally. Switch $\forall$ and $\exists$, then negate the predicate.

9. Check nested order. Ask whether the existentially chosen object may vary or must be the same in every case.

These checks prevent several recurring errors:

• Concluding that a universal claim is true because the first few examples work.

• Using an object outside the relevant category as a counterexample.

• Confusing “not all” with “none.” In symbols, $\neg\forall x\,P(x)$ means $\exists x\,\neg P(x)$, not $\forall x\,\neg P(x)$.

• Inferring $\forall x\,P(x)$ from $\exists x\,P(x)$.

• Ignoring the domain. “Every number has a reciprocal” is false over the real numbers because $0$ has no reciprocal, but the corresponding claim may be true over nonzero real numbers.

• Misplacing a negation. The negation of $\forall x\,[P(x)\rightarrow Q(x)]$ is $\exists x\,[P(x)\land \neg Q(x)]$, not $\exists x\,[\neg P(x)\rightarrow \neg Q(x)]$.

Final takeaway: Reliable reasoning about quantified statements depends on precise domains, explicit scope, correct negation, appropriate evidence, and careful attention to quantifier order.