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

Preamble

A symbolic sentence can be negated by plonking the symbol ¬ in front of it; correspondingly, it is false thatblah blah blah is a literal negation of the statement blah blah blah.

Now, by definition, to negate a statement is to logically flip its truth value. Thus,

1. a statement and its negation have opposite truth values in every possible world (as we vary the context and definitions);
2. a statement's negations are logically equivalent to one another.

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Statement 1: "The apple is red"

One might quickly jump to "The apple is not red" as the negation.

Yes, Red(a)'s negation is ¬Red(a).

Statement 2: "Roses are red and violets are blue"

Why are some texts discussing the negation of a conjunction 'AND' becoming a disjunction 'OR'?

By De Morgan's laws, ¬(P ∧ Q) is logically equivalent to ¬P ∨ ¬Q.

Statement 3: "x is apple implies x is fruit"

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

Have you overlooked the given statement's implicit universal quantification? It is written out in full as for every object x, [Apple(x) ⇒ Fruit(x)]. We can then write its negation as

• this is false: every apple is a fruit (the four canonical categorical propositions)
• some apple isn't a fruit
• for some object x, [Apple(x) ∧ ¬Fruit(x)].

Your suggested negation x is an apple does not imply that x is a fruit is ambiguous: did you mean for every object x, ¬[Apple(x) ⇒ Fruit(x)] or it is not the case that for every object x, [Apple(x) ⇒ Fruit(x)]? Only the latter is a correct answer!

Ultimately, regarding your main question, the most useful advice is to just keep sense-making; Ethan's advice, "To negate the statement, ask yourself what it would take to make it false," circular as it may be, is apropos!

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Statement 1: "The apple is red"

1. "Not apple is red", that is, "Any fruit except apple is red"
2. "Not apple is not red", that is, "Any fruit except apple is not red"

Statement 2: "Roses are red and violets are blue"

3. "Roses are not red AND violets are not blue"

Statement 3: "x is apple implies x is fruit"

4. "x is anything except apple, implies, x is anything but fruit"
5. "x is apple implies x is anything but fruit"
6. "x is fruit implies x is apple"
7. "x is not fruit implies x is not apple"

The above negations are all incorrect, because each violates my preamble's theorem, "a statement and its negation have opposite truth values regardless of the context." For example,

  (3) The original sentence P ∧ Q and your suggestion ¬P ∧ ¬Q are both false when P and Q are redefined as "1+1=2" and "1+1=3", respectively.

  (5) The original sentence for every object x, [A(x) ⇒ F(x)] and your suggestion for every object x, [A(x) ⇒ ¬F(x)] are both vacuously true when A(x) and F(x) are redefined as "x is a unicorn" and "x is fertile", respectively.

Answer 3

There is an important distinction between the negation (complement) of a term within a statement (like your "not apple" example) and the negation of a statement as a whole. They yield different meanings.

Historically, the branch of Mathematics/Logic where this distinction was first studied is term logic, founded by Aristotle of Stagira and algebraized by Gottfried Leibniz (of Calculus fame). (It was proved that term logic can be modelled as a fragment of Boolean algebra, so the alternative explanation I offer here is fully justified within the scope of your question.)

I will illustrate this distinction with some examples in term logic:

1. "Leibniz is human":   $l \in h$.

Here, each of the terms $\{l,h\}$ is either an individual or a set.

The entire statement is constructed around the copula (or relation) $\in$.

2. "Every human is mortal":   $h\mathbf{A}m$,   or   $h\subseteq m$.

Here, each of the terms $\{h,m\}$ is a set.

The entire statement is constructed around the copula (or relation) $\mathbf{A}$ (or $\subseteq$).

From the statements 1 and 2, we conclude that

3. "Leibniz is mortal":   $l \in m$.

Now, let's play with the negation of the predicative term $m$. The entire statement 2 is equivalent to

• "Every human is _not immortal_":   $h\mathbf{A}(m')'$,   or   $h\subseteq (m')'$,   or

• "No human is immortal":   $h\mathbf{E}m'$.

In these equivalent statements, we can see that the set/term immortal is the negation (or complement) of mortal. That is, in a universe of discourse $I$ where every individual is either mortal or immortal, we have:

$m \cup m' = I\\ m \cap m' = \varnothing.$

This, by the way, is the definition of complement in Boolean algebra.

We negated a set/term. Now let's play with the negation of a relation.

4. "Leibniz is not a woman":   $l\not\in w$.

Here, the relation $\in$ (previously employed in statement 1) was negated. We neither negated the term $l$ nor the term $w$ – only the relation $\in$ ("is").

One could rewrite this statement as

• "It is not the case that: Leibniz is a woman":   $\overline{l\in w}$,   or   $l\overline\in w$.

Let's move to another example that illustrates the negation of an entire statement.

5. "At least one human is not a woman" [$h\mathbf{O}w$, or $h\nsubseteq w$].

(Namely, Leibniz. Statement 5 is the conclusion from combining the statements 1 and 4.)

This statement is equivalent to

• "It is not the case that: Every human is a woman":   $\overline{h\mathbf{A}w}$,   or   $h\mathbf{\overline{A}}w$,   or   $h\overline{\subseteq} w$.

Thus, one can see that the relation $\mathbf{O}$ (or $\nsubseteq$) is the negation of $\mathbf{A}$ (or $\subseteq$), and vice-versa.

We have seen, therefore, that in term logic one may negate terms/sets ($m\rightarrow m'$) and statements/relations ($\mathbf{A}\rightarrow\mathbf{\overline{A}}$, or $\subseteq\rightarrow\nsubseteq$).