The Logic of Arguments

Mathematical logic begins with statements. A statement is a complete assertion that is either true or false.

Examples include:

• “$17$ is prime.” This is true.
• “$9$ is an even number.” This is false.
• “Every positive even integer can be written as the sum of two primes.” This has not been proved or disproved, but it is still a statement because it must ultimately be either true or false.

A question, command or incomplete expression is not a statement. For example, “Is $17$ prime?”, “Find $x$” and “$x+3$” do not themselves have truth values.

A sentence containing a variable may have different truth values for different values of that variable. For example,

$$x^2=9$$

is true when $x=3$ or $x=-3$, but false for most other real values of $x$. To interpret such an assertion properly, you must know:

• the allowed values of the variable;
• whether the claim concerns every allowed value or only at least one value;
• any conditions placed on the variable.

The domain matters. Consider:

If $x^2=4$, then $x=2$.

This is false when $x$ may be any real number, because $x=-2$ is a counterexample. It becomes true if the context specifies that $x$ is non-negative.

TMUA method: before analysing an argument, identify the domain and rewrite vague wording as a precise statement. Many apparent logic errors are actually caused by ignoring a restriction such as “$x$ is positive”, “$n$ is an integer” or “the triangle is non-degenerate”.

A universal claim can be false because of one exceptional case. An existential claim can be true because of one successful case. This distinction is developed fully in the section on quantifiers.

Statements can be combined using and, or and not.

For statements $A$ and $B$:

• “$A$ and $B$” is true only when both $A$ and $B$ are true.
• “$A$ or $B$” is true when at least one of $A$ and $B$ is true.
• “not $A$” has the opposite truth value from $A$.

In mathematics, or is inclusive unless the wording explicitly says otherwise. Therefore “$A$ or $B$” includes three possibilities:

• $A$ is true and $B$ is false;
• $A$ is false and $B$ is true;
• both $A$ and $B$ are true.

It is false only when both are false.

For example, the statement

$12$ is even or $12$ is divisible by $3$

is true even though both parts are true.

By contrast, phrases such as “either $A$ or $B$, but not both” describe an exclusive choice. That is not the default mathematical meaning of “or”.

Negating combined statements

To negate “$A$ and $B$”, it is enough for at least one part to fail:

not “$A$ and $B$” means “not $A$ or not $B$”.

To negate “$A$ or $B$”, both possibilities must fail:

not “$A$ or $B$” means “not $A$ and not $B$”.

These rules are often called De Morgan’s laws, but the name is less important than the reasoning.

Worked example

Negate the statement:

$x>2$ and $x<7$.

The original says that both inequalities hold. Its negation is:

$x\leq 2$ or $x\geq 7$.

It would be incorrect to write “$x\leq2$ and $x\geq7$”, because that describes values for which both original inequalities fail, rather than values for which at least one fails.

Similarly, the negation of

$n$ is even or $n$ is divisible by $3$

is

$n$ is not even and $n$ is not divisible by $3$.

Common trap: when negating a compound statement, change both the individual statements and the joining word. “And” becomes “or”, while “or” becomes “and”.

A conditional statement has the form:

If $A$, then $B$.

Here $A$ is the condition and $B$ is the conclusion. The statement claims that whenever $A$ is true, $B$ must also be true.

For example:

If an integer is divisible by $4$, then it is even.

This is true because every multiple of $4$ is also a multiple of $2$.

A conditional statement is false only when its condition is satisfied but its conclusion fails. Therefore, to disprove “if $A$, then $B$”, you must find a case in which:

• $A$ is true; and
• $B$ is false.

For the false statement

If an integer is even, then it is divisible by $4$,

$n=6$ is a counterexample: $6$ is even but is not divisible by $4$.

A case where $A$ is false does not disprove the statement. For instance, $n=5$ tells you nothing about whether “if $n$ is even, then $n$ is divisible by $4$” is true, because the condition “$n$ is even” has not been met.

The words if, only if and if and only if must be read carefully.

“$A$ if $B$”

This means:

If $B$, then $A$.

The phrase after “if” gives the condition.

Example:

An integer is even if it is divisible by $4$.

This means:

If an integer is divisible by $4$, then it is even.

“$A$ only if $B$”

This means:

If $A$, then $B$.

The phrase after “only if” gives something that must be true whenever $A$ is true.

Example:

An integer is divisible by $4$ only if it is even.

This means:

If an integer is divisible by $4$, then it is even.

A useful reading is: $A$ cannot happen without $B$.

“$A$ if and only if $B$”

This combines both directions:

If $A$, then $B$, and if $B$, then $A$.

It says that $A$ and $B$ always have the same truth value.

Example:

An integer $n$ is even if and only if $n+1$ is odd.

Both directions are true:

• if $n$ is even, then $n+1$ is odd;
• if $n+1$ is odd, then $n$ is even.

TMUA translation shortcut:

• after if comes the condition;
• before only if comes the condition;
• if and only if requires both directions.

When the wording is complicated, rewrite it immediately as one or two “if ..., then ...” statements before judging it.

Start with the statement:

If $A$, then $B$.

Its converse is formed by reversing the two parts:

If $B$, then $A$.

The converse does not automatically have the same truth value as the original statement.

For example:

Original: If an integer is divisible by $4$, then it is even.

Converse: If an integer is even, then it is divisible by $4$.

The original is true, but the converse is false because $6$ is even and not divisible by $4$.

Its contrapositive is formed by reversing the two parts and negating both:

If not $B$, then not $A$.

For the same example:

If an integer is not even, then it is not divisible by $4$.

This is true. An odd integer cannot be a multiple of $4$.

A conditional statement and its contrapositive are logically equivalent. They always have the same truth value. This is because both are broken by exactly the same type of case: $A$ true and $B$ false.

The converse is different. Reversing a correct implication may create an incorrect one.

Worked example

Consider the statement, for real $x$:

If $x=3$, then $x^2=9$.

• Converse: If $x^2=9$, then $x=3$. This is false because $x=-3$ is a counterexample.
• Contrapositive: If $x^2\neq9$, then $x\neq3$. This is true and equivalent to the original.

Efficient use in the TMUA

When checking a proposed deduction:

1. identify the original “if ..., then ...” direction;
2. check whether the argument uses that direction or silently reverses it;
3. if a direct argument feels awkward, consider the contrapositive.

A frequent invalid deduction has the form:

If $A$, then $B$. $B$ is true. Therefore $A$ is true.

This incorrectly assumes the converse. For example, knowing that a shape is a rectangle does not prove that it is a square, even though every square is a rectangle.

Another invalid deduction is:

If $A$, then $B$. $A$ is false. Therefore $B$ is false.

The original statement gives no conclusion when $A$ is false.

The valid forms are:

If $A$, then $B$. $A$ is true. Therefore $B$ is true.

and, using the contrapositive,

If $A$, then $B$. $B$ is false. Therefore $A$ is false.

The words necessary and sufficient describe the two roles in a conditional statement.

Suppose:

If $A$, then $B$.

Then:

• $A$ is sufficient for $B$;
• $B$ is necessary for $A$.

$A$ is sufficient for $B$ because knowing $A$ is true gives enough information to conclude that $B$ is true.

$B$ is necessary for $A$ because $A$ cannot be true unless $B$ is also true.

Using the statement

If an integer is divisible by $4$, then it is even,

we can say:

• divisibility by $4$ is sufficient for being even;
• being even is necessary for divisibility by $4$.

Being even is not sufficient for divisibility by $4$, because $6$ is even but not divisible by $4$. Divisibility by $4$ is not necessary for being even, because some even integers are not divisible by $4$.

A condition can be:

• necessary but not sufficient;
• sufficient but not necessary;
• both necessary and sufficient;
• neither necessary nor sufficient.

Necessary and sufficient means both directions hold. It is equivalent to “if and only if”.

For real $x$:

$x=0$ is necessary and sufficient for $x^2=0$.

This means both:

• if $x=0$, then $x^2=0$;
• if $x^2=0$, then $x=0$.

Language conversions

The following say the same thing:

• $A$ is sufficient for $B$;
• $B$ is necessary for $A$;
• if $A$, then $B$;
• $B$ if $A$;
• $A$ only if $B$.

The following also say the same thing:

• $A$ is necessary and sufficient for $B$;
• $B$ is necessary and sufficient for $A$;
• $A$ if and only if $B$.

Common trap: students often reverse “necessary” and “sufficient”. Ask two separate questions:

• Does this condition guarantee the result? If yes, it is sufficient.
• Must this condition hold whenever the result holds? If yes, it is necessary.

Worked example

Let $n$ be an integer. Consider the condition “$n$ is divisible by $6$” and the result “$n$ is divisible by $3$”.

Divisibility by $6$ guarantees divisibility by $3$, so it is sufficient. Divisibility by $6$ is not necessary, because $9$ is divisible by $3$ but not by $6$. Conversely, divisibility by $3$ is necessary for divisibility by $6$, but not sufficient.

A quantifier tells you how many values are being claimed to satisfy a statement.

The universal quantifier appears in phrases such as:

• for all;
• for every;
• for any.

These mean that the statement must hold for every allowed value.

Example:

For every real number $x$, $x^2\geq0$.

To prove a universal statement, the reasoning must apply to an arbitrary allowed value. Checking several examples is not enough when infinitely many values are possible.

To disprove a universal statement, one counterexample is enough.

Example:

For every integer $n$, $n^2>n$.

This is false because $n=1$ gives $n^2=n$, not $n^2>n$.

The existential quantifier appears in phrases such as:

• for some;
• for at least one;
• there exists.

These mean that one or more allowed values satisfy the statement.

Example:

There exists a real number $x$ such that $x^2=2$.

To prove an existential statement, it is enough to exhibit one valid example or prove that at least one must exist.

To disprove an existential statement, you must show that no allowed value works.

In mathematical language, some means “at least one”. It does not imply “some but not all”. Therefore, if every member of a set has a property, it is also true that some member has that property, provided the set is non-empty.

The order of quantifiers matters

Compare:

For every real number $x$, there exists a real number $y$ such that $y>x$.

This is true: after $x$ is chosen, take $y=x+1$.

Now compare:

There exists a real number $y$ such that, for every real number $x$, $y>x$.

This is false: there is no largest real number.

The same words appear, but changing their order changes the meaning.

Restricted universal statements

A statement such as

For every integer $n$, if $n$ is divisible by $4$, then $n$ is even

only makes a demand when the condition “$n$ is divisible by $4$” holds. To find a counterexample, you need an integer that is divisible by $4$ but not even. Values not divisible by $4$ cannot disprove it.

TMUA method: underline the quantifier and the domain before testing a statement. Then ask:

• Is the claim universal, so one counterexample would defeat it?
• Is the claim existential, so one successful example would establish it?
• Does one variable depend on a value chosen earlier?

The negation of a statement describes exactly the cases in which the original statement is false. A correct negation must cover all such cases and no others.

Negating simple comparisons

Use the complementary relation:

• not $x=a$ means $x\neq a$;
• not $x\neq a$ means $x=a$;
• not $x>a$ means $x\leq a$;
• not $x\geq a$ means $x<a$;
• not $x<a$ means $x\geq a$;
• not $x\leq a$ means $x>a$.

Do not merely reverse the inequality sign. Equality must be included when the original uses a strict inequality.

Negating and/or statements

• not “$A$ and $B$” means “not $A$ or not $B$”;
• not “$A$ or $B$” means “not $A$ and not $B$”.

Example:

Original: $x>0$ and $x<5$.

Negation: $x\leq0$ or $x\geq5$.

Negating a conditional statement

The statement

If $A$, then $B$

fails precisely when $A$ is true and $B$ is false. Therefore its negation is:

$A$ and not $B$.

It is not “if $A$, then not $B$”.

Example:

Original: If an integer is divisible by $4$, then it is even.

Negation: There is an integer that is divisible by $4$ and is not even.

When a conditional statement is presented as a universal claim, its negation normally asserts the existence of a counterexample.

Negating “if”, “only if” and “if and only if”

First translate the wording.

• “$A$ if $B$” means “if $B$, then $A$”, so its negation is “$B$ and not $A$”.
• “$A$ only if $B$” means “if $A$, then $B$”, so its negation is “$A$ and not $B$”.
• “$A$ if and only if $B$” says that both directions hold. It is false exactly when one statement is true and the other is false.

Thus the negation of “$A$ if and only if $B$” is:

$A$ and not $B$, or $B$ and not $A$.

Negating universal and existential statements

The negation of a universal statement changes “every” into “there exists” and negates the property:

Original: For every allowed $x$, property $P$ holds.

Negation: There exists an allowed $x$ for which property $P$ does not hold.

Example:

Original: Every integer is positive.

Negation: There exists an integer that is not positive.

The negation of an existential statement changes “there exists” into “for every” and negates the property:

Original: There exists an allowed $x$ for which property $P$ holds.

Negation: For every allowed $x$, property $P$ does not hold.

Example:

Original: There exists a real number $x$ such that $x^2=-1$.

Negation: For every real number $x$, $x^2\neq-1$.

Negating nested quantifiers

Negate one layer at a time, changing each quantifier as you pass through it.

Consider:

For every integer $n$, there exists an integer $m$ such that $m>n$.

Its negation is:

There exists an integer $n$ such that, for every integer $m$, $m\leq n$.

The original is true and its negation is false.

Now consider:

For every integer $n$, there exists an integer $m$ such that $m^2=n$.

Its negation is:

There exists an integer $n$ such that, for every integer $m$, $m^2\neq n$.

This negation is true; for example, $n=2$ works.

Negation checklist

When negating a statement:

1. preserve the domain;
2. change “for all” to “there exists”, or “there exists” to “for all”;
3. change “and” to “or”, or “or” to “and”;
4. negate each simple condition accurately;
5. replace a conditional statement by “condition true and conclusion false”;
6. read the result as ordinary language and check that it describes exactly how the original could fail.

Formal truth tables and symbolic logic notation are not required for this topic. The fastest TMUA approach is usually to translate the sentence into precise words, identify its failure case, and then test that case carefully.