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  1. In discussions of quantity, the concept of infinity has been explicitly excluded. This is a mathematical grey area where the quantity property of mutual exclusivity does not necessarily hold. For instance, an infinite quantity added by an infinite quantity is an infinite quantity, or x+x=x, x != 0. Intuition tells us that a quantity x, added to a positive quantity y, will always be greater than x. So we have a situation where x + x >= x, defying the mutual exclusivity principle.






  2. Even the concept of quantity arithmetic is under debate, as discussed later.






  3. This leads into the question of if something obtains meaning if people believe it does, which is beyond the scope of this paper.






  4. This too is beyond the scope of the paper.






  5. The common rule is to assign the highest frequency string to the least complex/shortest encoding, and proceed to assign less common strings to more complex encodings.






  6. Also, one will notice that Ellis asks for a numeral, whereas Campbell asks for a number. The distinction will be discussed in a later section.






  7. This statement as well will be justified at a later section.






  8. Again, to be discussed later. Mass and volume are considered direct measurements, even though a common method of obtaining their quantities is through an instrument or indirect means.






  9. Take a moment to digest that before proceeding.






  10. This is often done to obtain greater precision and accuracy than what is identifiable by human senses. This thought is developed later.






  11. For elaboration, please refer to the work. Kyburg discusses the proposition from (62).






  12. Please note that this counterexample only works against a claim of always having a logical relation. A consideration of whether we are right most of the time is beyond (continued) this paper, but suffice it to say that humans are exponentially more accurate in relational judgments as the size of the discrepancy grows larger than our error "bounds."