Peter Suber, Philosophy Department, Earlham College

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 two-valued 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.