# What is the origin of “law of excluded middle”?

Reading an article I have stumbled across the concept of law of excluded middle.

Wikipedia mentions that original expression is principium tertii exclusi which literally translates to principle of the excluded third. My native language (Romanian) also uses the literally translation.

I am wondering where does middle come from.

Question: What is the origin of "law of excluded middle"? I specifically refer to middle instead of third.

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• This might be better answered at one of the mathematics.SE sites. Even though it is a question about etymology of an English phrase, they are more likely to know the story behind it. – Mitch Aug 22 '18 at 13:19
• It may have originated with that Latin expression, but the English phrase places additional emphasis on the referenced options being mutually exclusive of one another by specifying that there is not even any "middle-ground" between the proposition being true or false, no sort ofs or maybes or compromises, which is how the word "middle" entered in. – Billy Aug 22 '18 at 14:40
• The Stanford philosophy site is a good source of information. Formal systems of argument go back a long way, and they influenced each other over time. So the word middle could have been introduced in several places. plato.stanford.edu/entries/logic-ancient/#ArgPatValInf – Global Charm Aug 22 '18 at 18:06
• @Mitch - yes, that's fine with me. – Alexei Aug 23 '18 at 13:41

OED attests 'excluded middle, third' (in the entry for excluded, adj., .... b.) from 1849, in William Thomson's An Outline of the Necessary Laws of Thought. Although OED does not give a quote, merely the citation, the relevant text is this:

3d Criterion. The principle of the middle being excluded, (lex exclusi medii.) "Either a given judgement must be true of any subject, or its contradictory; there is no middle course."

I did not find any earlier work containing the phrase lex exclusi medii. Some earlier works contain the phrase "excluded third". Among them, the 1815 Encyclopaedia Londinensis mentions the "excluded third" (top of second column):

1. The Principle of the Excluded Third, (principium exclusi medii inter duo contradictoria,) upon which the (logical) necessity of a Knowledge is grounded; namely, that we must judge so, and not otherwise; i.e. that the opposite is false; for Apodictical Judgements.

Together, the context of these uses suggests that early appearances in English of both "the principle of the excluded third" and "the principle of the excluded middle" were associated with the Latin exclusi medii, rather than the Latin exclusi tertii.

The account represented in the 1815 Encyclopaedia Londinensis is drawn from an 1813 translation of Immanuel Kant's Logic by Thomas Wirgman (see page 3, op. cit.). The Latin rendering of the fuller phrase, principium exclusi medii inter duo contradictoria, may have been drawn from Kant directly, or from a Latin translation of Kant.

My simple intuition is that if you assume a bivalent logic system, otherwise called two-valued logic system, it means that every proposition is either true or not true, and there is no "middle" ground, so to speak, or a third option, what you called "the excluded third."
Principle of bivalence

As far as I can tell Aristotle (3rd century BC) first wrote about this. In a slightly different way he writes about "law of non-contradiction", basically meaning that something cannot be and not be at the same time.

And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call "man", and others were to call "not-man"; but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact.
Aristotle and excluded middle

This idea, that something cannot both be and not be is expressed in logical notation as ¬(P ∧ ¬P), said "not (P AND NOT P)". Basically something cannot be and not be at the same time. If this is true, and we a bivalent logic where every proposition is either true or false, then ¬(P ∧ ¬P) leads to (P ∨ ¬P), meaning something has to either be or not be.

Aristotle specifically refers to the "intermediate". He says 'intermediate among two contradictories."

He then proposes that "there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate" (Book IV, CH 7, p. 531). In the context of Aristotle's traditional logic, this is a remarkably precise statement of the law of excluded middle, P ∨ ¬P.
Aristotle and law of excluded middle

So as to why it's called the excluded middle, it may be because Aristotle called it the disallowed "intermediate between contradictories", I'm not sure about this. The term makes intuitive sense to me, if "true" is on one side, and "false" is on the other, neither true nor false would fall in between, the middle, and it's excluded, presuming you accept a two-value logic system.

The article goes on to Leibniz:

"Every judgment is either true or false"

And Bertrand Russell:

1. Law of excluded middle: "Everything must either be or not be."

Also, if I may make a correction, Wikipedia doesn't mention that the original expression is principium tertii exclusi, it says it's "also known as":

The law is also known as the law (or principle) of the excluded third, in Latin principium tertii exclusi.
Law of excluded middle Wikipedia article

It was originally written, or translated, I should say, from Aristotle, being "intermediate between contradictories".

• But what is the origination of the English phrasing using 'middle' instead of the expected translation 'third'? – Mitch Aug 22 '18 at 22:06
• @Mitch The first instance of the rule is written as "intermediate between contradictories" by Aristotle (c. 300 BC). I haven't found "third" in the remainder of the Wikipedia article. It's mentioned at the beginning when it says "is also known as". I searched in the Stanford Encyclopedia of Philosophy articles on "Contradiction" and "Ancient Logic" and the word "third" with regard to this rule doesn't show up at all. So I believe it may be more instructive to instead find out how "excluded third" got into the Wikipedia article and the origin of this term. I might do this later if I get time. – Zebrafish Aug 23 '18 at 6:07
• Correct: Meta, Book IV, part 7, 1011b23 : "Nor indeed can there be any intermediate between contrary statements..." Original Greek μεταξύ : between, amongst, amid. – Mauro ALLEGRANZA Aug 28 '18 at 9:29