Not: In
logic: Term for negation, denial,
rejection. 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 oneword 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
truthvalues. 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:

+p 
p 
?p 
++p 
+p 
+?p 
+p 
p 
?p 
~+p 
~p 
~?p 
+p 
T 


T 



T 
T 

T 
T 
p 

T 


T 

T 

T 
T 

T 
?p 


T 


T 



T 
T 

For more, see Extended Propositional Logic (EPL).
