Neither True
Nor False




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


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

Our rejection of a third division represents only our penchant for simplicity.

C. I. Lewis; (1883-1964); A Pragmatic Conception of the a priori; 1923


Does the artist communicate? … The work of art conveys no literal meaning– no fact or law is asserted, and the work is not true or false.

J. H. Randall & J. Buchler; Philosophy: An Introduction; 1957; p117

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 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-); What is Mathematics, Really?; 1997; p310

… 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-); 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.
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





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