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In other words, a truth value in a mathematical sense is binary, but human communication is fallible and runs the risk of fallacy, error, and distortion.

The idea of being "close enough" to the truth - even when being sincere - is something I think we can all agree exists so that we do not burden each other with unreasonable expectations of accuracy and precision.

If for example, someone asks me if the glass is full of water and it is 99% full, I will say - with no burden on my conscience - that it is indeed full.

Is it reasonable to conjecture that the human conscience contains a mechanism for assessing truth based on a spectrum of approximations rather than an exact set of criteria that must be met?

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    This is more of a neuroscience and psychology than philosophy question. Neurons fire only when the activation potential reaches a threshold, it is reasonable to assume that such threshold operation filters to the higher cognitive functions as well. It is well established in some areas, pattern recognition for example. You should ask on Psychology & Neuroscience SE. – Conifold Jul 23 '18 at 17:39
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Roy Sorensen does think it is reasonable because of the vagueness of the truth predicate in ordinary discourse. He is discussing truth as one of the conditions for knowledge :

Vagueness in the Truth Condition

Since the objects of knowledge are usually considered to be linguistic entities such as sentences or quasi-linguistic entities such as proposi- tions, it is natural to expect the objects of knowledge to be vague. Since knowledge implies truth, the vagueness of truth-bearers sometimes creates unclarity as to whether the truth condition is satisfied. For example, Nick asks Nancy 'When will you arrive for lunch?' Nancy answers 'Noonish.' Nick watches an accurate clock as he waits in order to be sure whether Nancy arrives noonish. Both Nick and Nancy understand 'noonish' to mean 'near noon.' If Nick sees Nancy arrive at 12:01, then Nick knows Nancy arrived noonish. Likewise, Nick knows that Nancy arrived noonish if she arrives at 12:02, 12:03, or 12:04. But as we add minutes, uncertainty about noonishness grows. Since we cannot specify the last noonish moment, we cannot specify the last time at which Nick knows. Our uncertainty as to whether he knows that time t is noonish can be traced to the vagueness of 'noon-ish' and thus the vagueness of the object of his belief.

Reference

Roy A. Sorensen, 'The Vagueness of Knowledge', Canadian Journal of Philosophy, Vol. 17, No. 4 (Dec., 1987), pp. 767-804 : 768.

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