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The “Law of Excluded Middle” states that every proposition must be
Quotations on 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. Bridgman, P.W.; The Nature of Physical Theory, 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 . . . Malebranche, N.; The Search after Truth, p48 As I said earlier, single words without context are neither true nor false. Robbins, J. W.; Without a Prayer, 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.’ Cohen, Martin; 101 Philosophy Problems, p119 Sensations . . . are neither true nor false, they simply are. James, W.; Pragmatism, p117 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. Randall, J.H. & Buchler, J.; Philosophy: An Introduction, p117 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. Geisler, Norman L. & Brooks, Ronald M.; Come Let Us Reason, p134 ... we may have a consistent deductive system in which there is neither truth nor falsity. Randall, J.H. & Buchler, J.; Philosophy: An Introduction, 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! Hersh, R.; What is Mathematics, Really? p310 ... if mathematics is merely the formal manipulation of symbols, truth and falsity in the ordinary sense have nothing to [do] with it. Bunch, B.; Mathematical Fallacies and Paradoxes, p159 We will say that the propositions of arithmetic are neither true nor false, but only compatible or non-compatible with certain conventions. Waismann, Friedrich - Introduction to Mathematical Thinking - p120 Arithmetic is not bivalent. Charles Sayward-- Four Views of Arithmetical Truth - Philos. Qrtly. Vol.40, No.159, 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. Kline, M.; Mathematics: The Loss of Certainty, 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. Dummett, Michael; Truth and Other Enigmas, p155-156
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. Kasner, E. & Newman, J.; Mathematics and the Imagination, 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”? Bell, Eric Temple; The Search for Truth, p246
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