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Non Contradiction

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Non-Contradiction and Excluded Middle
Peter Suber, Philosophy Department, Earlham College



Three principles
Denying one or more of these principles



Rivals to these principles



Kinds of logical opposition

Three principles
The Principle of Non-Contradiction (PNC) and Principle of Excluded Middle
(PEM) are frequently mistaken for one another and for a third principle which
asserts their conjunction.
Given a statement and its negation, p and ~p, the PNC asserts that at most
one is true. The PEM asserts that at least one is true. The PNC says "not both"
and the PEM "not neither". Together, and only together, they assert that
exactly one is true.
Let us call the principle that asserts the conjunction of the PNC and PEM, the
Principle of Exclusive Disjunction for Contradictories (PEDC). Surprisingly, this
important principle has acquired no particular name in the history of logic.
PNC

at most one is true; both can be
false

PEM

at least one is true; both can be
true

PEDC

exactly one is true, exactly one is
false

Clearly the PEDC is not identical to either the PNC or the PEM, and the latter
two are not identical to one another.
The PEM is simple inclusive disjunction for p and ~p. The PNC is the denial of
their conjunction. Conjoining these gives us exclusive disjunction: at least one
of the contradictories is true (PEM) and not both are true (PNC).
Because the PEDC is logically equivalent to the conjunction of the PNC and
PEM, that is, because PEDC
(PNC · PEM), whatever implies the PEDC implies
the two other principles as well. Many logical principles, axioms, and
unacknowledged choices imply the PEDC. From ordinary logic, the principle of
double negation (p
~~p) implies the PEDC. So does the initial selection of a
two-valued logic that requires every proposition to take exactly one of two
truth values.

In a standard two-valued logic, then, one should not be surprised that the
statements of the PNC and PEM are equivalent to one another and to the
PEDC.
PNC

~(p · ~p)

PEM

p ~p

PEDC

(p ~p) · ~(p · ~p), or p

~p

These are logically equivalent because they are all tautologies, and all
tautologies are logically equivalent. This equivalence does not mean that the
principles are the same, however. They bear the same truth-value, not the
same meaning. Under De Morgan's theorem, the PNC can be transformed into
the PEM and vice versa, but this only shows that De Morgan's Theorem
presupposes the PEDC. (Logics that deny the PEM must deny some forms of De
Morgan's theorem.)
The PNC and PEM need not be equivalent in n-valued logics when n > 2,
although the principles must be reformulated for those logics and could look
very different. Even in two-valued logics these three formulas are distinct as
soon as we replace p and ~p with (for example) p and q. The relations they
assert are only equivalent in the special case when the relations are asserted
of contradictories.
If we use a standard two-valued logic, the three principles are already present
even if they do not appear as axioms. The three principles can be proved in
such a logic, but any such proof would be viciously circular.

Denying one or more of these principles
The PEDC is central to ordinary notions of consistency, but it may consistently
be denied. The PEDC may be denied by denying one or both of its conjuncts,
which gives us three cases:
PNC

PEM

Case 1

true

false

Case 2

false

true

Case 3

false

false

Case 1. If the PNC were true of the world, and the PEM false, then there
would be some pairs of contradictories for which neither member was true.
The world would be underdetermined. The world would be thinner and more
abstract than the PEDC would have it.
Case 2. If the PEM were true of the world and the PNC false, then there would
be some pairs of contradictories for which both members were true. The

world would be overdetermined; it would be richer and more concrete (in
Hegel's sense, more articulated or differentiated and more dense and
continuous) than the PEDC would allow.
Case 3. If both were false, the world would be underdetermined in some
respects and overdetermined in others. These are the three ways in which the
world may be said to be "inconsistent". Consistent logics can be developed
that enable us to describe these inconsistent states of affairs; see e.g.
Rescher and Brandom, Logic of Inconsistency, Rowman and Littlefield, 1979.
Another way to legitimate a kind of inconsistency has been introduced by
Graham Priest under the name "dialetheism". (See Graham Priest,
"Contradiction, Belief, and Rationality," Proceedings of the Aristotelian
Society, 86 (1986) 99-116.) In standard logics, contradictions are always false;
for dialetheism, contradictions are both true and false. Hence, dialetheism
affirms the PEM but denies the PNC (hence, it also denies the PEDC). Standard
logic gets nowhere against dialetheism by insisting that contradictions are
always false, for dialetheism admits this but adds that they are true too. If
standard logic insists that contradictions are nothing but false, then it must
justify this or else beg the question against dialetheism; but as we have seen,
it is difficult to produce such a justification that is not viciously circular by
presupposing the standard PEDC and hence the PNC that dialetheism has
rejected. Dialetheism accepts the principles that (1) if p is true, then ~p is
false, and (2) if p is false, then ~p is true. These two principles are as
compatible with dialetheism as with standard logic. When p is a contradiction
that is both true and false, then these two principles imply that ~p is also a
contradiction that is both true and false.
For most people, truth and falsehood map onto acceptance and rejection. But
if one wishes to accept all truths and reject all falsehoods, then dialetheism
cannot be followed, for it holds that some propositions are both true and
false; presumably it is impossible to accept and reject the same proposition at
the same time. However, dialetheism is as easy to put into practice as
standard logic if one vows only to accept all truth, without adding the vow
that one should reject all falsehood. If truth and falsehood "come inextricably
intermingled" as Priest says, then this can be rational. (See also Graham Priest
et al., Paraconsistent Logics, Philosophia Verlag, 1986.)
Reversing dialetheism by denying the PEM but affirming the PNC is the
intuitionistic school of mathematics, inspired by the work of L.E.J. Brouwer
(1881-1967). Intuitionists do not deny the PEM in all contexts, but do reject it
in reasoning about infinite sets. It follows that for such reasoning they also
reject the PEDC and will not make use of indirect proofs that affirm a
proposition merely because its negation leads to a contradiction. This
essentially cuts off most of the mathematics of the infinite for them, which is
as they wish. On the whole they demand constructive proofs that exhibit the
existence of posited entities or provide effective methods for constructing
them.

For example, Goldbach's famous conjecture states that every even number
(except 2) is the sum of two primes. For two centuries it has tantalized
mathematicians because, while its assertion is simple, it has never been
proved or disproved. Non-intuitionists would accept the conjecture as
disproved if it implied a contradiction. Intuitionists would accept it as
disproved only if one could actually produce a counterexample: an even
number that is not the sum of two primes.
More generally, intuitionists will admit p ~p (PEM) as a theorem of a system
only if p or ~p, that is, only if we have already proved p or ~p. In the world of
metamathematics, the intuitionists are not at all exotic, despite the
centrality of the PEDC (hence the PEM) to the ordinary sense of consistency.
Their opponents do not scorn them as irrationalists but, if anything, pity them
for the scruples that do not permit them to enjoy some "perfectly good"
mathematics.

Rivals to these principles
Let us call the principle of dialectic (PD) the principle that neither the PNC
nor the PEM is true. In dialectical logics "truth" may be defined coherently so
that neither the PNC nor PEM is true in it, even if they have some "provisional"
applications. Note that under the PD, if the PEDC is false it may also be true;
this is not impossible once one has denied the PEDC.
Now note that between the PEDC and the PD there is a contradiction. There is
also a contradiction between the PNC (and the PEM) and the PD. Let us focus
on the former. The PD says of this contradiction that both principles (PD and
PEDC) may be true; the PEDC says that exactly one is true, namely, itself.
Each claims to preserve its truth in the face of its contradictory principle, but
the PEDC does this in a way that appears viciously circular. Truth under the
PEDC is exclusive of opposition; truth under the PD is inclusive.
Aristotle's indirect proof of the PNC does not refute the PD. Aristotle argues
that any denial of the PNC presupposes the PNC, for it wishes to be the denial
and not also the affirmation of the PNC. A naive denial of the PNC that did not
also affirm the PNC would be vulnerable to Aristotle's argument. But the PD
understands that the falsehood of the PEDC is consistent (in the PD's own
sense of consistency) with the truth of the PEDC; only the PEDC itself would
forbid this. PD, then, both affirms and denies the PNC and thereby avoids
Aristotle's argument.

Kinds of logical opposition
"Contradictories" are statements that are negations of one another in a twovalued logic, that is, under the PEDC. Under the PEDC, contradictories cannot
both be true and cannot both be false. Hence, exactly one is true. But
contradictories are not the only kind of opposing or conflicting statements.

"Contraries" are statements that can both be false, but that cannot both be
true; for example (1) all S is P, and (2) no S is P. "Subcontraries" are
statements that can both be true, but that cannot both be false; for example
(3) some S is P, and (4) some S is not P.
Just to round things out, there are pairs of statements that can both be true
or both be false, such that one member of the pair implies the other but not
vice versa; they are called "alternatives". Within such a pair, the
"superalternative" implies the "subalternative". Propositions (1) and (3),
above, are alternatives, as are (2) and (4). Observe that propositions (1) and
(4) are contradictories, as are (2) and (3).
Contraries, subcontraries, and subalternatives only possess the properties
ascribed to them here in a non-empty universe (in which there is at least one
S to be or not be P.

This file is an electronic hand-out for the course, Logical Systems.
The logic symbols in this file are GIFs. See my Notes on Logic Notation on the
Web.
Peter Suber, Department of Philosophy, Earlham College, Richmond,
Indiana, 47374, U.S.A.
[email protected] Copyright © 1997, Peter Suber.

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