Negating a statement with an arbitrary variable

Question

Consider the statement $P(a)$, where $a$ is an arbitrary element from $D$.

Which of these is its negation?

• $\neg P(a)$, where $a$ is an arbitrary element from $D$

• $\neg P(a)$, where $a$ is some element from $D$

Answer 1

Based on previous answers given, I think I have come to a conclusion:

Let $P(x)$ be the propositional function "$x$ is an even number".

Then the proposition "For an arbitrary integer $a$, $P(a)$" states "For an arbitrary integer $a$, $a$ is an even number.

The negation of this proposition can be interpreted in two ways:

If I negate the proposition like this "For an arbitrary integer $a$, $\neg P(a)$", I am stating "For an arbitrary integer $a$, it is not the case that $a$ is an even number."

If I negate the proposition like this "$\neg$ (For an arbitrary integer $a$, $P(a)$)", I am stating "It is not the case that for an arbitrary integer $a$, $a$ is an even number." Or in other words, there exist integers that are not even numbers.

Answer 2

Good question. I think I see what you mean.

You could put it this way: in order to be a proposition, the expression $P(a)$ has to have all of its arguments filled in.

For example, if $a$ already has a specific value such as "the sky" then $P(a)$ is a proposition like "The sky is blue." It is a proposition because all of the arguments have been filled in. Its negation is $\neg P(a)$, "The sky is not blue."

In addition to propositions, you can also talk about the generic expression $P(\cdot)$, which means something like "______ is blue." This is not a proposition, because we have not yet filled in all of the arguments. It is a function with arguments that you can plug in; by plugging in different values, you get different propositions $P(a)$, $P(b)$, $P(c)$.

Sometimes instead of writing the function like $P(\cdot)$, we name its argument and call it, say, $P(x)$, where $x$ is just a placeholder for something you could fill in later. This way of writing it is potentially confusing, because just by glancing at the page, you are not sure if $x$ refers to a specific thing, or if it is just the name we've given to a function's argument.

If $P(x)$ is just a function and its arguments have not yet been filled in, then $P(x)$ is not a proposition. It does not mean "A specific thing $x$ is blue", or "Every $x$ is blue." It is a form with a blank in it; it is "____ is blue." You can tell because inside those parentheses, $x$ means "the name of $P$'s function argument" and outside of those parentheses, $x$ has not been defined.

Some people write functions like $(x \mapsto P(x))$ so they have a different way of writing it than $P(a)$.

• When you already have defined $a$ ("the sky"), you can negate a proposition like $P(a)$ ("The sky is blue"). When you negate it, you get $\neg P(a)$ ("The sky is not blue.").
• You can negate a proposition like $\forall a. P(a)$ ("Every $a$ in the universe is blue.") When you negate it, you get $\neg \forall a. P(a)$ ("Not every $a$ in the universe is blue").
• You can negate a proposition like $\exists a. P(a)$ ("Some $a$ in the universe is blue."). When you negate it, you get $\neg \exists a. P(a)$. ("No $a$ in the universe is blue.")
• When you have not already defined $x$, then $P(x)$ is a function whose argument is named $x$. You technically cannot negate a function, because it is not a proposition. But if you have one function $(x \mapsto P(x))$ you can consider the related function $(x\mapsto \neg P(x))$, which corresponds to "_____ is not blue".

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Extra note: In lambda calculus, this sort of difference between functions and propositions is made precise and concrete as the difference between $\lambda x. fx$ and $(\lambda x. fx)(a)$.