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.

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.