The Law of Excluded Middle

The law of excluded middle (LEM) is one of three fundamental laws of thought discovered by Aristotle. It can be expressed informally as follows: for a statement to be meaningful it must be either true or false, or be composed of parts that are either true or false. Formally, this is written (P ∨ ¬P). The law is disputed in the context of many-valued logics, where rejection of LEM is thought to be a concession necessary for their realisation. I argue that many-valued logics need not (and must not) violate the law of excluded middle, and that the relevant concession involves an incorrect interpretation of the law.

In the words of Aristotle (Metaphysics, 1011b), “there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate.” Aristotle clarifies that this does not relate to properties or differences which are only nominally regarded as opposite, for example Black & White, with the intermediate terms being shades of grey (ibid.), but only to a special category of formal contradictories such as True & False or Exist & Not-Exist (De Interpretatione, 9): “if every affirmation or negation is true or false it is necessary for everything either to be the case or not to be the case.” In case of natural properties, which are simply differences, irrespective of whether we regard them as contradictory or whether there is a range of steps between them, we must either affirm or deny any one property, be it white, black, or any particular shade of grey. As such, classical logic accepts only two truth values (True or False). There are obviously cases where it is impossible to determine whether something is true or false, but not knowing the truth-value of something does not entail an additional truth-value (at least not a truth-value of the same type), just like the absence of something (a predicate) does not have the same logical structure as ‘the presence of absence’ (a predicate of a predicate). When we speak of a something having two possible logical predicates, we speak an object-language (or base-language), that is, the object of a sentence is distinct from the predicates in terms of which we speak about the object; when we speak in terms of predicates about a predicate we speak a meta-language, where meta-predicates refer to predicates from the object-level (base) in the logical hierarchy.

Integrating different types of truth-values to get a quasi-intermediate result can be useful in some circumstances, but does not negate the law of excluded middle: at every level of the logical hierarchy the law of excluded middle still strictly applies. Any system of logic which posits intermediate truth values, be it 3-valued logic or Gödel logics, relates a degree of uncertainty ‘about’ predicates (a higher order predicate) to the base predicates (true/false), allowing for their relation to be quantified without inconsistency, insofar as the relation is understood as belonging to a meta-language and does not constitute a relation between terms of the object-language, anything else would be a category mistake. “Recognizing a type-distinction is always a way of disallowing exceptions to the Law of Excluded Middle.” (Geach 1956)

It may be further postulated that any violation of LEM negates any system of logic which would purport to accomodate it. When we declare an object or a property, anything that is identifiable (X), we are implicitly asserting the law of identity: at time t, there exists only one X which in every respect is identical with X. If we would then assert that X does not fully exist at t (a LEM violation), then the X that exists is not identical with the X that does not fully exist. Another way, if LEM is not true then it is possible for X to be less real or less true than itself, which violates the law of identity and the law of non-contradiction. Therefore no identity, and if no identity, then no logic. We can infer from this that every law of thought is conditional on the other two: identity presupposes the excluded middle and non-contradiction, the excluded middle presupposes identity and non-contradiction, non-contradiction presupposes identity and the excluded middle.

As a possible challenge to the above argument let us posit something called X-ness: a property that can vary in degree, for example, transparency. For the law of identity to hold we cannot say that a homogenous object X has a range of X-ness, unless we just mean that X is identical to the range of X-ness, which is then trivially true of X-ness. We must instead posit the range of X-ness as a property which is distinct from the identity X, which, at time t, can have only a definite amount and distribution of X-ness. If we can identify X and a definite degree of its X-ness, then we must say that X with a definite degree of X-ness fully exists (or does not exist at all), or that a definite degree of X-ness fully exists. It would not make sense to equate X-ness with degrees of existence; this would amount to saying that ‘a specific degree of existence’ exists, what would be either trivially true or violate the law of identity with respect to ‘existence’. The same can be said in terms of predicates True and False: that ‘a specific degree of truth is true’ is at best a tautology, or it conflates two different meanings of ‘truth’, one being a matter of degree, the other binary.

In order to demonstrate the necessity of LEM it may be sufficient to simply ask whether any logic which rejects the necessity of LEM is proven to be consistent. If the answer is anything but Yes it implies a No, so answering the question with a Yes demonstrates commitment to LEM. If the respondent does not know the answer then the proof of consistency remains in doubt and question remains unanswered. The same question could be asked about any specific case where LEM is violated, to any truth-value in any non-classical logic - ‘is this truth-value correct?’ - thus requiring a commitment to LEM (Yes or No) to prove the contrary, therefore contradiction. Perhaps the source of confusion in non-classical logics stems simply from mislabeling ‘values’ (that are subject to LEM) as ‘truth-values’. Simply labelling something a ‘truth-value’ does not mean it is valid as a truth-value. That a truth-value can be questioned about its logical validity implies that it is not a truth-value.

In conclusion, the law of excluded middle remains one of the fundamental laws of logic, indispensable to construction of meaning or logical sense, and the several formal systems purporting to evade this law are a result of its incorrect application, interpreting statements of object- and meta-languages as objects of the same logical type. This does not mean that the affected systems are inconsistent, but only that insofar as they are consistent they do not violate LEM despite ostensibly rejecting it.

Geach, P. T., and W. F. Bednarowski. Symposium: The Law of Excluded Middle. Proceedings of the Aristotelian Society, 1956.