Correct way to do logical negation? (cf inverse, opposite, contrapositive)

Question

I am doing Keith Devlin's "Introduction to Mathematical Thinking" Coursera course. It starts with the topic of using English to precisely define mathematical ideas, including implication and negation.

I remain confused - so I wanted to talk through my thinking with the following examples.

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Example 1

Statement: "The apple is red".

This looks like a very simple statement, and one might quickly jump to "The apple is not red" as the negation.

However I am asking myself why the following are not negations?

• "Not apple is red" ... that is, any fruit except apple is red.
• "Not apple is not red" ... any fruit except apple is not red.

I see the original statement as having 3 parts: (A) (is) (B). Then there are several combinations where some or all of these 3 parts are negated.

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Example 2

Statement: "Roses are red and violets are blue".

The intuitive negation is "roses are not red AND violets are not blue".

However, in addition to the options mentioned in example 1, such as "any flower except roses are red, any flower except violets are blue" .. there is the complexity of the logical AND.

Is the negation "Neither roses are red, nor violets are blue", that is, roses are not red AND violets are not blue.

Why do I read some texts discussing the negation of a conjunction AND becoming a disjunction OR?

If (A) (AND) (B) is a 3-part statement, are the following options for its negation?

• (negation of A) (AND) (negation of B)
• (A) (NOT-AND) (B)
• NEITHER (A) NOR (B)
• (negation of A) OR ( negation of B)

Note that (negation of A) has the questions raised in example 1.

I won't include an example where the statement uses OR, but the questions raised will be similar.

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Example 3

Statement: "X is apple implies X is fruit"

Here we have an implication.

Is the negation one of the following?

• "X is apple does not imply X is fruit"
• "X is anything except apple, implies, X is anything but fruit"
• "X is apple implies X is anything but fruit"
• "X is fruit implies X is apple"
• "X is not fruit implies X is not apple"

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I am aware of terminology negation, contrapositive, inverse, opposite. Do these cover the different combinations mentioned above?

Answer 1

You should be asking one question at a time. That said, here are answers to all three.

Devlin's point, which you quote, is to think about

using English to precisely define mathematical ideas.

So that's what you should do, instead of testing out all the places you might insert a "not" into a statement you want to negate.

To negate the statement in Example 1 ask yourself what it would take to make it false. You did that and it's correct. Your two bulleted alternatives are not English and make no sense.

To make the assertion in Example 2 false it would suffice to find a rose that wasn't red or a violet that wasn't blue.

In order for the implication in Example 3 to be false all you need is the existence of an apple that's not a fruit.

Answer 2

Example 1

Statement: "The apple is red".

why the following are not negations?

• "Not apple is red" ... that is, any fruit except apple is red.
• "Not apple is not red" ... any fruit except apple is not red.

I see the original statement as having 3 parts: (A) (is) (B). Then there are several combinations where some or all of these 3 parts are negated.

You are mixing up negating a logic formula (this outputs a logic formula) and taking the complement of a set (this outputs a set). We can also negate a number (this outputs a number).

The given statement has the form $R(a),$ so its negation is simply $\lnot R(a).$ Don't mix up the constant $a$ (the apple) and the set of apples $A;$ both make no sense to negate, but at least we can take the latter's complement.

Example 3

Statement: "x is apple implies x is fruit"

Is its negation "x is apple does not imply x is fruit" ?

Negating a logic formula means to output another truth formula such that the two of them have opposite truth values regardless of what meanings we assign to the symbols.

Since the given statement $$A(x)\Rightarrow F(x)$$ and your suggested answer $$\lnot\big(A(x)\Rightarrow F(x)\big)$$ have opposite truth values no matter what $A(x)$ and $F(x)$ means, they are indeed negations of each other.

Is the negation one of the following?

• "x is anything except apple, implies, x is anything but fruit"
• "x is apple implies x is anything but fruit"
• "x is fruit implies x is apple"
• "x is not fruit implies x is not apple"

No. For example, your last suggestion $$\lnot F(x)\Rightarrow \lnot A(x)$$ has the same truth value as the given statement, so they are not negations of each other.

Example 2

Statement: "Roses are red and violets are blue".

The intuitive negation is "roses are not red AND violets are not blue".

No. If we symbolise the given statement as $$P \land Q,$$ then your suggested answer is $$\lnot P \land \lnot Q;$$ by my previous boldfaced point, these two sentence aren't negations of each other, because they are both False when $P$ stands for 1+1=2 and $Q$ stands for 1+1-3.

.. there is the complexity of the logical AND. Why do I read some texts discussing the negation of a conjunction AND becoming a disjunction OR?

To negate a compound logic formula, that is, to obtain a formula that has the opposite truth value regardless of what meanings we assign to their common symbols, we have to negate the given formula's main connective.