“Subtraction [as defined from pairs of natural numbers to natural numbers] sometimes produces undefined quantities” is wrong (or at best phrased in a sloppy way

“Subtraction [as defined from pairs of natural numbers to natural numbers] sometimes produces undefined quantities” is wrong (or at best phrased in a sloppy way which you in one part interpret in a way that would make it wrong). Instead, for some inputs, there is no defined/prescribed output. Anyone who thinks that example you gave is a true contradiction in-the-ordinary-sense is simply confused about about what they mean by those things. (Though, again, if you are using “contradiction” in a way other than the ordinary sense, this doesn’t apply to you) This confusion would presumably come from being sloppy with phrasing. If one simultaneously interprets “produces an undefined quantity” as synonymous with “does not have a defined quantity” (which is a true statement) and synonymous with “produces a quantity which doesn’t exist, [thereby establishing the existence of something that doesn’t exist]” (which is a false statement), it is no wonder that one would conclude that there is a true contradiction, and this is precisely the mistake. Thinking that two things (e.g. statements) are the same thing (and therefore have the same properties, e.g. if they are statements, that if one is true so is the other) simply because one has given them the same name, is a mistake, and is a somewhat common source of confusion. Now, in some cases, everyone (or nearly everyone) considering a topic will be confused about something in a way that is at least kind of similar to that way of being confused (or perhaps even in exactly that way). In these cases, one way that people might come to realize that they are confused is if they seem to be able to derive a contradiction. But this doesn’t mean that the contradiction is true, but that either (perhaps due to being confused) at least one of their assumptions was wrong, or one or more of their inference steps were invalid (e.g. unintentionally conflating two distinct things/ideas). There are no true contradictions, only people (such as myself) who are wrong ( and/or confused ) about something.