The term 'not' is a fundamental logical term, and is used to convey that the statement in which it occurs apart from 'not' is false, not true, a contradiction, or otherwise contrary to fact, in the sense that 'it is not cold' means that it is false that it is cold, or it is not true that it is cold, or that it is cold is a contradiction or contrary to fact.

It is an interesting fact about human cognition and human language that it is able to represent what is not so, for this seems a mental enlargement of the actual possibilities that reality offers: That there is no cheese in the larder may be obvious to a mouse and to a man, but only a man can positively affirm and represent the absence of cheese, simply by formulating the statement 'There is no cheese in the larder', that positively and precisely represents that absence.

Another interesting fact about negation in natural language is that there is in many languages a close syntactical similarity, as in English, for the expressions 'No' (as e.g. used in the one-word sentence that expresses dissent); 'not' (the subject of this lemma); 'nothing' (the term to refer to an absence); and the particle 'none' (as used in quantifying statements like 'none of the accused is guilty').

In formal logic 'not' is often written as '~', '-' or '¬', and rules that are adopted for 'not' may involve such as these:

From (A implies B) and (~B) follows (~A).
From (A implies B) follows (~B implies ~A).
From (A implies B&~B) follows (~A).

Some theorems involving 'not' in standard propositional logic are:

T1. ~(p&~p)
T2. ~pVp
T3. (p --> ~q) --> (q --> ~p)
T4. (pVq) iff ~(~p&~q)
T5. (p&q) iff ~(~pV~q)
T6. ~~p iff p

There are several fundamental logical problems concerning negation.

First, there are the related problems of double negation and disjunction, that may be posed as follows. Are the following two logical formulas always true i.e. tautologies:

(~~A) iff A
(~A)VA

That is: Does 'not not A' (i.e.: 'it is not true that it is not true that A', 'it is false that A is false') and 'A' amount to the same? And is it always true that A is true or A is false?

In standard propositional logic (a.k.a. Classical Propositional Logic or CPL) the answer to both questions is the same and positive, and indeed in CPL the two formulas are interderivable: Assuming the one, you can deduce the other.

In intuitionist propositional logic (a.k.a. IPL) the answer to both questions is also the same but negative: Intuitionists affirm that A implies ~~A but deny that ~~A always implies A, on the ground that if it is established that it is false that A is false it is not thereby established that A is true. Similarly, intuitionists insist that it is not always the case that A is true or A is false, since it may well be the case that A is not true and also A is not false.

There is a widely accepted formalization of IPL due to Heyting, that gives rules and axioms that conform to these notions about negation, and the general result is that in IPL less can be proved than in CPL, while in CPL everything can be proved that can also be proved in IPL.

In brief, CPL is stronger than IPL. The reasons that IPL nevertheless is used are that some find it closer to their intuitions concerning negation; that what can be proved with less or weaker assumptions is better founded than what can only be proved with strong assumptions; and that intuitionists believe that classical mathematics that involves CPL is fundamentally mistaken about what mathematics is.

Second, there are the related problems about the place of negation and whether there are several kinds of negation. This is related to the previous problem, and can be brought out by considering the following two statements:

It is not true Aristotle is a singer.
Aristotle is not a singer.

According to the intuitions of many persons, the former statement is somewhat weaker than the latter, in that the former merely denies that Aristotle is a singer, whereas the latter affirms he is not a singer.

In CPL the two statements are equivalent, but it is possible to distinguish two negations while still retaining only two truth-values. Using notation, the distinction may be rendered thus:

'~(Aristotle is a singer)' for 'It is not true Aristotle is a singer'
'-(Aristotle is a singer)' for 'Aristotle is not a singer'

where the former frontal 'not' is referred to as 'denial' or 'weak negation', and the latter frontal 'not' as 'negation' or 'strong negation', and the formal relation between the two turns around

-(p) --> ~(p)

i.e. if p is false then p is not true, but not conversely.

This also allows the introduction of an operator of uncertainty (a.k.a. undeterminedness or undecidedness) that may be defined and written as follows:

?p =def ~-p & ~p

i.e. p is uncertain precisely if p is not true and p is not false. One perfect candidate for statements that satisfy this construction are future contingents: What will happen tomorrow is often today neither true nor false, or at least one has no sufficient grounds to affirm either, and thereby sufficient grounds to deny both.

The above operator of uncertainty can be matched with an operator '+' that affirms positive truth, and then one may propose equivalences like these:

+p IFF ~-p & ~?p
-p IFF ~+p & ~?p
?p IFF ~-p & ~+p

This can be worked out as a binary logic, with a fundamental table like the following one:

For more, see Extended Propositional Logic (EPL).