Law of Excluded Middle

The law of excluded middle is a fundamental principle of propositional logic stating that every proposition must be either true or its negation must be true. There is no third middle ground or intermediate possibility. 

Formally, if we represent any given proposition with the letter p, the principle is expressed as follows:

p ∨ ¬p

It is read: "p or not p." Although this formula uses the inclusive disjunction connective (∨), it covers all logical possibilities for p: being true (p) or being false (¬p). The two options are mutually exclusive.

An everyday example is the state of a light switch: "the light is on or the light is not on." These two options exhaust the possible physical states of the system, leaving no room for a third condition.

Proof

The statement p ∨ ¬p is a tautology, meaning it is a compound proposition that is true in every possible case, regardless of the truth values of its components. The clearest way to verify this is through a truth table:

As shown, the final column (p ∨ ¬p) is always true (T). This confirms that, for any proposition p, the statement "p or not p" holds true.

This principle is also known in Latin as tertium non datur, which literally translates to "a third is not given." This naming highlights the same concept: between an assertion and its negation, no third option or intermediate state exists.

The law of excluded middle is one of the three fundamental laws of classical bivalent logic (two-valued: true and false), alongside:

• The law of identity: formally, p → p, which states that every proposition is identical to itself.
• The law of non-contradiction (LNC): formally, ¬(p ∧ ¬p), which asserts that a proposition cannot be both true and false at the same time.

It is important not to confuse the law of excluded middle with the law of non-contradiction, despite their close relationship. The law of non-contradiction states that a proposition and its negation cannot both be true simultaneously, thus excluding the conjunction or overlap of both values. In contrast, the law of excluded middle states that at least one of the two (the proposition or its negation) must be true, thereby excluding any intermediate possibility or third truth value.

Examples

To better understand how this law operates, consider the following examples:

• The state of a digital file: "this computer file is open or it is not open." There is no intermediate state for this specific condition.
• Tossing a coin: "the coin lands on heads or it does not land on heads (that is, it lands on tails)." These two options exhaust all possible outcomes.
• Membership status: "John is a club member or he is not a club member." There is no third category outside this logical disjunction.
• In elementary arithmetic: "the integer 7 is even or it is not even (that is, it is odd)." This statement is necessarily true.
• In basic geometry, evaluating a specific shape: "this triangle is equilateral or it is not equilateral." The figure either satisfies the definition or it does not.

Distinguishing between contrary terms and contradictory terms is essential for applying this principle correctly. The law of excluded middle applies strictly to contradictory pairs, where the truth of one directly implies the falsehood of the other. In contrast, contrary terms allow for intermediate possibilities.

Everyday language frequently uses contrary pairs, which can lead to confusion. For instance, the statement "the dress is white or it is black" is not an application of this law, because if the dress is not white, it could be blue, red, gray, etc., and not necessarily black. The proper formulation under bivalent logic would be "the dress is white or it is not white." 

Classical logic does not permit "gray areas" or intermediate states for an atomic proposition: a statement must be either true or false. Thus, even if one says "the sky is partly cloudy," propositional logic ultimately evaluates the claim "the sky is cloudy" as strictly true or false based on a defined threshold.

Applications

The law of excluded middle provides the logical framework supporting a foundational mathematical method: proof by contradiction (also referred to as indirect proof or reductio ad absurdum).

This technique is used to prove that a proposition p is true. It begins by assuming the contrary: that p is false, which means its negation (¬p) is assumed to be true. From this assumption ¬p, deductive steps proceed using established axioms and theorems until reaching a contradiction.

The resulting contradiction (for example, deriving that a statement q and its negation ¬q are simultaneously true) shows that the initial assumption (¬p) is untenable, as it violates the law of non-contradiction. 

Under the law of excluded middle, if ¬p leads to a logical impossibility and therefore cannot be true, the only remaining option is that the other side of the disjunction (p ∨ ¬p) must be true. Consequently, p must necessarily be true, completing the proof.