From Logic

A reason for postulating a third truth-value 'indeterminate' is the problem of vagueness. Vagueness exists in the real world, because the real world is not a set of discrete categories - it is a continuum. Therefore, attempts to apply deductive logic to the real world are always as imperfect as attempts to apply geometrical patterns like squares and circles to a world filled with ovals and rectangles.

Let us suppose a specific object, such as a rock. We can then say "This rock is this rock." But in the time it takes to utter that sentence, the rocks temperature may have changed due to its absorption of sunlight and slightly expanded. Or consider a larger time scale: the rock may erode over a long time. It may even shatter into two rocks. On an even smaller level, the molecules of the rock have all shifted their positions slightly. On the quantum level, the electrons of the rock's atoms are best described by probability distributions: are you then making solid statements about probabilities? Where do you draw the line? For whatever properties one draws lines for, more questions can be asked.

Or consider a color spectrum between red and orange. Let us also call the statement, 'It is red here', 'p'. Now, it is obvious that there are "clear enough" cases where 'p' is true (the red case) and "clear enough" cases where 'p' is false (the orange case). However, between the two extremes there seems to be a large class of colors where we just cannot say whether 'p' is true or false. Hence, some have suggested that in such cases 'p' is neither true nor false and that a third truth-value — indeterminate — is needed. Such a suggestion would rule out bivalence but retain the law of excluded middle. The best book on this distinction and the problem of vagueness is Timothy Williamson's book Vagueness.

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