The “Law of Excluded Middle” states that every proposition must be either True or False.
This “law” must be discarded.
Its acceptance hinders the study of meaningful propositions, and is incompatible with the nature of reality and the nature of Truth.

The Quotations given below are in support of
the necessity for a third category.

[…] we know that because of instrumental uncertainty and errors of observation, cases will arise in which we cannot say whether the wave length is greater or less than one of the critical values. This it seems to me is characteristic of most judgments involving physical processes– the law of the excluded middle is not a valid description of our physical experience– there has to be a third category of doubtful, in addition to positive or negative.
P. W. Bridgman; The Nature of Physical Theory; 1936; p38

We saw in the preceding chapters that the judgments we form on the testimony of our eyes concerning extension, figure, and motion are never exactly true. Nonetheless, it must be agreed that they are not altogether false […]
N. Malebranche; (1638-1715); The Search after Truth; 1674/1997; p48

As I said earlier, single words without context are neither true nor false.
J. W. Robbins; Without a Prayer; 1997; p78

Ignorance is neither true nor false.
J. W. Robbins; https://godshammer.wordpress.com
/2009/05/18/john-robbins-quick-quote-2/

A proposition therefore is defined as the meaning of a declarative sentence. Some sentences are not declarative, such as commands in the imperative mood, or exhortations in the well-nigh extinct subjunctive mood. Questions, or interrogative sentences, also are neither true nor false. Only declarative sentences are true or false; and it is this common character that is important for propositions. Of course in English rhetoric there are questions that are intended as propositions. They are called rhetorical questions. They are an embellishment of style. They spruce up a speech. But logically they are propositions. A question that is intended as a question is neither true nor false. It can play no part in an argument.

Let us now return– an exhortation, neither true nor false, but one which it is hoped that the student will follow– to the simplest propositions and the simplest form of arguments.
Gordon Clark; Logic; 1988; p30

Knowledge is the exercise of the mind or reason; without reason there can be neither morality nor righteousness. These require thought. Lacking these, animals are neither righteous nor sinful.
Gordon Clark; In Defense of Theology; Imago Dei; p109; 1984

Whereas words are always meaningful or have significance (otherwise, they would not be words in the lexicon of a language), phrases or sentences may be either significant or insignificant, according to the way in which words are put together in their construction. Even when they are significant, phrases, like words, are neither true nor false. Only significant sentences, and among significant sentences only those which can be construed as assertible propositions, are either true or false. Not all significant sentences are declarative in mood; questions, for example, and commands are neither true nor false. Not even all significant sentences which are declarative in mood can be construed as existential affirmations or denials. Only such sentences and only those which are affirmative involve a word or phrase which, occupying the position of the subject term, must be construed as denoting something that really exists; if it cannot be so construed, the affirmative existential proposition is false.
Mortimer J. Adler; Some Questions About Language; 1993; p32

Tertium Quid in Scripture

But God had come to Laban the Syrian in a dream by night, and said to him, “Be careful that you speak to Jacob neither good nor bad.” Gen. 31:24 & 29

Then Balak said to Balaam, “Neither curse them all, nor bless them all!” Num. 23:25

‘Let me pass through your land; I will keep strictly to the road, and I will turn neither to the right nor to the left.’ Deut. 2:27

Then the woman whose son was living spoke to the king, for she yearned with compassion for her son; and she said, “O my lord, give her the living child, and by no means kill him!” But the other said, “Let him be neither mine nor yours, but divide him.” 1Kings 3:26

For I considered all this in my heart, so that I could declare it all: that the righteous and the wise and their works are in the hand of God. People know neither love nor hatred by anything they see before them. Eccl. 9:1

“They shall neither shave their heads nor let their hair grow long; but they shall keep their hair well trimmed.” Ezek. 44:20

It shall be one day which is known to the Lord - Neither day nor night. But at evening time it shall happen that it will be light. Zech. 14:7

Nouns and verbs, provided nothing is added, are like thoughts without combination or separation; ‘man’ and ‘white’, as isolated terms, are not yet either true or false.
Aristotle; De Interpretatione; 16a, 1.10-15.

Epimenides […] claimed that people from Crete always told lies. This was […] somewhat inexplicable, as he himself was from Crete. If it was true, then what he himself was saying should have been a lie; but if it were a lie, then […] Effectively, the statements are neither true nor false, although they look like they ought to be. Unlike sentences such as say, ‘Hello Amy,’ which do not need to be given a ‘truth value.’
Martin Cohen; 101 Philosophy Problems; 1999; p119

Sensations […] are neither true nor false, they simply are.
William James; (1842-1910); Pragmatism; 1906/1978; p117

In jurisdiction there are various guiding principles: one lays down that the judge has to decide every case; according to another he is free to leave a case undecided. If we attempt to carry through the analogy into language, we notice that Aristotelian two-valued logic corresponds to the principle of deciding every case. But this is not the only possible policy.
There is also the possibility of saying, “I want no decision.” And this tendency would find fulfillment in a language in which it would make sense to say of a given sentence that it is, in certain circumstances, neither true nor false.
Friedrich Waismann; Are There Alternative Logics?; 1946

p 141 - […] a verbal form is neither true nor false unless we are certain that we can prove or disprove it.

p142 - It rained in London on January 1, 1066. There may be historical evidence by which this can be proved true or false, but I am not certain that there is, and therefore, by the doctrine, it is neither true nor false. In like manner, the form of words “it will rain in London tomorrow” will become true or false tomorrow (if uttered today), but is neither true nor false when it is uttered.

p 143 - “The greatest finite integer that will ever have been mentioned is not the greatest finite integer.” This, so far as I can see, is, on finitist principles, forever incapable of proof or disproof, and therefore forever neither true nor false.

p145 - Miss Ambrose, if we are to interpret her literally, must hold that the statement “all men are mortal” is neither true nor false, since we are not “certain of being able to verify it or prove it false.”

For my part, I hold that, as soon as I know what is meant by “men” and what by “mortal,” I know what is meant by “all men are mortal,” and I know quite certainly that either this statement is true or some man is immortal. I am led to reject finitism because (1) it rests on what seems to me an untenable general principle, that what  cannot be proved or disproved is neither true nor false […]

p 147 - The empiricist, therefore, must include among his premises the trustworthiness of memory (with the limitations demanded by common sense), in spite of the fact that neither now nor hereafter can we find any evidence for the truth of this premise.

The Limits of Empiricism; Bertrand Russell; 1936; Proceedings of the Aristotelian Society.
[9/21/25]

The long belief in the universal validity of the principle of the excluded third in mathematics is considered by intuitionism as a phenomenon of civilization of the same kind as the old-time belief in the rationality of π […]
L. E. J. Brouwer; (1881-1966); Consciousness, Philosophy, and Mathematics; 1940

Usually [read: Always], inductive conclusions cannot be called universally true […] because they are generalizations, and exceptions are always possible. Rather than being true or false, they are more or less probable. They involve degrees of probability.
Norman L. Geisler & Ronald M. Brooks; Come Let Us Reason; 2005; p134

[…] there are many non-classical logics in which excluded middle is abandoned as a general principle […]
Hartry Field; (1946-); Indeterminacy, Degree of Belief, and Excluded Middle;
Noûs; 2000; v34; n1; p12

“We choose as axioms propositions that are as evident as possible.” If properly understood, this statement is acceptable; but if the reader is not told also of the definitional character of an axiom, he may not see that an axiom as such is neither true nor false.
Henry Blumberg; Review: Einleitung in die Mengenlehre by A. Fraenkel; 1919; The American Mathematical Monthly; 1923; p200

[…] we may have a consistent deductive system in which there is neither truth nor falsity.
J. H. Randall & J. Buchler; Philosophy: An Introduction;  1957; p136

According to the law of Excluded Middle, every meaningful statement is true or false. According to Frege’s Basic Law Five, the statement “The Monster is a Russell set” is meaningful. It must be true or false. However, Russell discovered, it can’t be true, and it can’t be false!
R. Hersh; (1927-2020); What is Mathematics, Really?; 1997; p310

Real-life examples of 3-valued logic:
In Voting:
1- FOR; 2- AGAINST; 3- ABSTAIN

In Verdicts:
1- Guilty; 2- Not Guilty; 3- Innocent

[…] if mathematics is merely the formal manipulation of symbols, truth and falsity in the ordinary sense have nothing to [do] with it.
Bryan Bunch; Mathematical Fallacies and Paradoxes; 1982; p159

We will say that the propositions of arithmetic are neither true nor false, but only compatible or non-compatible with certain conventions.
Friedrich Waismann; (1896-1959); Introduction to Mathematical Thinking; 2003; p120

Arithmetic is not bivalent.
Charles Sayward; (1937-2023);
Four Views of Arithmetical Truth;
Philosophical Qrtly.; v40; n159; 1990; p157

The axioms and theorems of these geometries are neither empirical nor a priori truths. They are neither true nor false any more than the use of polar coordinates rather than rectangular is true or false. Poincaré called them conventions.
M. Kline; Mathematics: The Loss of Certainty; 1980; p343

In the classical two-valued logic, the truth value of a complex statement is determined by the truth-values of its constituents; but this is inessential to the realist position. […] This would involve the use of three-valued truth-tables […] in this sense we could then say that a statement might be neither true nor false. This kind of rejection of the law of excluded middle does not reflect any divergence from realism.
Michael Dummett; (1925-2011); Truth and Other Enigmas; 1978; p155-156

1. the number of Xs is > the number of Ys.
2. the number of Ys is > the number of Zs.
        therefore
3. the number of Xs is > the number of Zs.
Here the statements are neither true nor false.
It is in these forms of statements that the mathematician is interested, and not in what the statements are about, i.e. not in their subject-matter.
J. H. Randall Jr.; Philosophy: An Introduction; 1957; p62

Strictly speaking, mathematical propositions are neither true nor false; they are merely implied by the axioms which we assume. If we accept these premises and employ legitimate logical arguments, we obtain legitimate propositions.
The postulates are not characterized by being true or false; we simply agree to abide by them.
E. Kasner & J. Newman; Mathematics and the Imagination; 1940; p219

Might it not be possible to devise a system in which there is a third label in addition to the two labels “true” and “false”?
[…] perfectly good workable logics or deductive systems were created in which a proposition can have either the value “true” or “not-true,” or any one of any given number of values different from these.
[…]  the “laws of thought” to which habit has accustomed us for 2300 years are no more “necessary” for a consistent description and correlation of our experiences than was Euclidean geometry.
Eric Temple Bell; The Search for Truth; 1946; p246

CLARIFICATION
Multi-valued logic must not be confused with polylogism. Polylogism is the theory that each class within a society has its own logic.