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The professor in my Discrete Mathematics class gave us the following example for the negation of a proposition.

A: "All math majors are male" is the proposition.

(negation)A: "It is not the case that all math majors are male" OR

(negation)A: "There exists at least one math major who is female"

My question is: Isn't the negation of proposition A a partial negation? To be honest, is there such a thing as partial negation?

If I was to negate A, I would say: "All math majors are female"

I relate this to: 4 + x = 13 (this is a proposition)

Its negation would be: 4 + x != 13

It's not:

4 + x may not be equal to 13 OR

4 + x is not 13 for at least one value of x.

The negation is definite. It is rigid. In the example given at the very top, I cannot get the "feel" of what negation really is.

Help me understand this please.

Saunved Mutalik
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3 Answers3

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"All majors are female" is (one of) the opposite of "All majors are male"

The negation of a sentence is another proposition that makes your sentence "look like" false.

The way to go is to replace the universal quantifier "All" with its negation "There exist one", and the solutions your professor gave are indeed correct.

Think at the sentence "All swans are white", then the negation would be a sort of counter-example: "There exist a swan that is not white"

To make this more schematic, your sentence can be written as

$$\forall x, P(x)$$ where $x=$"one major" and $P(x)$="is a male". Then the negation is :

$$ \exists x: \bar{P(x)}$$ where $\bar{P(x)}$ is the negation of $P(x)$, $\bar{P(x)}$="is a female"

mariob6
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  • Then what is the difference between "opposite" and "negation"? Is my solution appropriate? I have always looked at "negation" as the exact opposite of any proposition. Is this outlook wrong? – Saunved Mutalik Jun 29 '16 at 18:19
  • Use this concept: the negation of a sentence is true only if my sentence is false. Then consider your example: "All majors are male" if there are 99 males and 1 female, the sentence is false but also your negation "All majors are female" is false! – mariob6 Jun 29 '16 at 18:22
  • Perfect. Thanks a lot :) – Saunved Mutalik Jun 29 '16 at 18:23
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The negation of a statement $\varphi$, semantically, is a statement $\neg\varphi$ which is true iff $\varphi$ is false.

Now in your case, if "all mathematicans are male" is true then "all mathematicans are female" is false. However is in not the case that if "all mathematicians are male" is false, then, all mathematicians are female. Indeed the statement can be false because there is a female mathematician, and a male one.

So (provided some erroneous background assumptions about ascription of gender) the negation of "all mathematicians are male" is equivalent to the statement "there is a female mathematician".

The error you are making is to do with negating quantifiers. The statements "all $P$s are $Q$" has negation "there is a $P$ which is not a $Q$" rather than "all $P$s are not $Q$". For a mathematical example the statement "all primes are odd" is false, but it is not the case that all primes are even! It is false because there is a prime which is even.

I should expand a little on the mathematical implications of the third paragraph. Its important to note that, if you replace "$x$ is not male" with "$x$ is a female" then you are using some extra assumptions about the properties "male" and "female". I imagine it would be uncontroversial that tables are not male, but they aren't female either.

James
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Let us look at the specific case of two mathmajors. There are the following three possibilities:

$\{m,f\},\{m,m\},\{f,f\}$

Of these, only the one outcome would make the phrase "All math majors are male" true, namely where it was $\{m,m\}$. On the other hand, both of the phrases $\{m,f\}$ and $\{f,f\}$ would make the phrase "All math majors are male" false.

When we negate a proposition, all outcomes which formerly were true will now be false and all outcomes which were formerly false will now be true. In other words, the negation of the statement should have both $\{m,f\}$ and $\{f,f\}$ return true. With this in mind, you should recognize that "All math majors are female" would leave $\{m,f\}$ as still a false statement, implying that it is not the correct negation.

In general, try to think of whether the proposition is worded as an existential or as a universal statement. The negation of universal statements become existential and vice versa.

$\forall x, P(x)$ is negated to be $\exists x, \neg P(x)$. Your "all math majors are women" mistake is as though you negated $\forall x, P(x)$ as $\forall x, \neg P(x)$ instead.

Longer statements with multiple quantifiers can be more exotic, but still follow similar rules.

(definition of $f(x)$ is continuous at $c$)

$\forall \epsilon>0, \exists \delta>0, \forall x(|x-c|<\delta\Rightarrow |f(x)-f(c)|<\epsilon)$

(negation of the above, i.e. $f(x)$ is not continuous at $c$)

$\exists \epsilon >0, \forall \delta>0, \exists x (|x-c|<\delta~\wedge~|f(x)-f(c)|\geq \epsilon)$

Your second example, $4+x=13$, it is unclear whether the statement is intended to be read as $\forall x, 4+x=13$ (a false statement), $\exists x, 4+x=13$ (a true statement), or simply as $4+x=13$ (a conditional statement which is either true or false depending on the specific value of $x$). Without more context, it should be assumed to be the third.

The negation of $\forall x, 4+x=13$ is $\exists x, 4+x\neq 13$

The negation of $\exists x, 4+x=13$ is $\forall x, 4+x\neq 13$

The negation of $4+x=13$ is $4+x\neq 13$

Note that all values of $x$ which made $4+x=13$ true make its negation false and all values of $x$ which made $4+x=13$ false will make its negation true.

JMoravitz
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