Antonio Calonego
Notes on negation and contradiction
[1] In the propositional calculus of Logic,
given a proposition "p", it is always allowed to
construe a new proposition "~p" just placing the symbol
of negation "~" before it. As it is well known, the
symbol of negation turns the true of "p" into the
falsity of "~p" and viceversa, i. e. it excludes that
the two propositions can be both true and false together and, so
working, it gives rise to an extreme case, which occurs when the
two propositions are linked each other by a logical conjunction.
A logical contradiction ("p AND ~p") presupposes this
role played by the symbol "~" and, therefor, a very
strong concept of negation as it was a sort of external force, so
to say, which turns upside down the previously assigned
true-value of a given proposition any time it applies to it.
[2] In everyday language the role of negation
doesn’t identify with the one we sketched above.
One can oppose someone else who claims. "This is
white", by saying: "No, this is not white". But it
is wrong to consider the stated answer as if it could only be an
instance of "~p" and had to match so strong a sense of
negation as the logical one. There’s no reason, in
principle, to shut out that "No, this is not white"
could be interpreted as "No, it is not-white", meaning,
for instance, it is black. The negative proposition, then, would
certainly exclude its opponent can be true if it is true, but it
would not exclude the falsity of the two both.
Moreover, to someone saying: "It is sweet" I could
reply "No, it is bitter", giving for granted it
can’t be nor sweet nor bitter (the two propositions
can’t be both false), but not excluding it may be either
sweet either bitter (the two propositions can be both true).
In everyday language the role of negation is context sensitive:
it depends on circumstances and can’t be fixed once forever.
[3] For a proposition like "~ this is
white" to be a negation, in the strong sense, of "this
is white", the term "white" has to maintain the
same sense in both cases. But the "permanence of
meaning", which can be properly achieved only by a
scientific discourse, is, in everyday language, rather
problematic.
Suppose "~ this is white" is now a genuine negation, in
the strong sense, of "this is white".
In everyday language, the sheer presence of a strong sense of
negation is not sufficient to conclude that a contradiction is
there. As long as the two "parties" fighting each other
are thought as excluding alternatives (as aut … aut)
there isn’t any contradiction: we feel we must share one
position, because it is true, and oppose the other because it is
false; we don’t feel entrapped in any contradiction.
It is when we begin to view our experience as from a distance and
take it as a whole (as if an AND now linked what was formerly
separated by an aut) that a contradiction appear.
[4] A contradiction is based on a very peculiar
use of negation and it seems to be there no universal reason to
assign to it a primary position, as far as everyday language is
concerned.
Those who take up everyday language as a paradigm and think it
can represent the last word in Philosophy, will conclude that
also at level where theories are built up, it is wrong to assign
to "contradiction" a primary position.
On the contrary, those who assume that the very task of
intellectual research is to construe or find out an order, fixing
concepts and drawing up hierarchies of meanings where the
everyday man can’t see but a disperse plurality, will not
agree to shut out, at least in principle, a determinate theory
which, for instance, would explain society saying that its basic
and relevant conflicts are contradictions – though they will
likely ask it to define "relevant" and provide evidence
and reasons. (We won’t say, then: "The relevant
conflicts of this society are contradictions"; we'll
say: "Given this theory so and so grounded, the relevant
conflicts of this society are contradictions").
But, in both cases, "contradictions" are not a natural
law of human life nor the dynamic core of a living Totality (for
which the asking for evidence would be senseless).