Keith Devlin's Coursera course "Introduction to Mathematical Thinking" introduces a precise way to write down logical statements, using quantifiers (for all $\forall$, there exists $\exists$) and boolean operators (and $\wedge$, or $\vee$, not $\neg$).
In some exercises he asks us to negate a statement and write it in "positive form."
However, I can't find any explanation of what "positive form" is – this is my question.
My guess is as follows:
Statement:
$$ (A \implies B) $$
Negation:
$$ \lnot (A \implies B) $$
Positive form of negation:
$$ (A \land \lnot B) $$
I am guessing that "positive form" means doing more than just prepending a statement with "it is not the case that". Is this correct?
Best Answer
Devlin alludes to the meaning of "positive form" in Week $3$: Tutorial for Assignment 5 (timestamp $2$:$30$). See link here.
Specifically, he rewrites the formula $\neg ( \exists x \in \mathbb{N} ) [x^3 = 28]$ as the logically equivalent formula $(\forall x \in \mathbb{N}) \neg [x^3 = 28]$ using DeMorgan's Law for quantifiers, which can be rewritten as $(\forall x \in \mathbb{N}) [x^3 \neq 28]$ by definition of $\neq$.
Although he does not give a precise definition of "positive form," his use of the term suggests that a formula is in positive form whenever it is expressed as a logically equivalent and unnegated formula. So I believe your intuition is correct, that is, the positive form of $\neg (A \to B)$ would be $ A \wedge \neg B$.
Defined in this way, positive form is equivalent to negation normal form. A formula is in negation normal form whenever $(a)$ the negation operator is only applied to variables, and $(b)$ the only other allowed logical operators are conjunction $\wedge$ and disjunction $\vee$.
Note the positive form of a formula tends to make the formula more comprehensible to the reader.