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Calling this a "contradiction" is, imo, unjustified. It is not the accountant who I am calling (more) confused (though if the accountant believes what they ar

by drdeca 5y ago

Calling this a "contradiction" is, imo, unjustified. It is not the accountant who I am calling (more) confused (though if the accountant believes what they are doing to be "impossible" they are still somewhat confused). It is the person who asserts that the thing which they know exists, "is impossible", who is confused. If you have a partial function that takes in a beardnacle and a frompulin and (where it is defined) returns a brimwich, and one has some conditions under which, in the cases it is defined, certain patterns have been shown to hold, and one has found a number of cases in which, it seems as if, if the function were defined at some of the places it isn't, and the value fit with the patterns which have been shown to hold where it is defined, in that other things are consistent with that, even though no brimwich can actually satisfy those properties, a logician can perfectly well ask "well, can we define a concept which generalizes the concept of a brimwich, in order to make this partial function into a total function, in a way that keeps the patterns we have shown?" . This does not involve any contradictions. There are no contradictions here. And yet, this is doing the same kind of thing as generalizing the natural numbers to the integers. You are either using a notion of "contradiction" which is quite unlike the everyday concept (and, if it differs, also quite far from the mathematical concept), or you are mistaken in thinking that there is a contradiction. Like, what is the alleged contradiction? "I want to subtract these two numbers, but there is no natural number which is the result of that subtraction"? That's not a contradiction in the sense that [people who aren't talking about dialectics] use. If you couldn't use it in order to derive False in a theorem prover that allows the relevant statements to be expressed, then it isn't a contradiction-in-the-ordinary-sense. "John is married" and "John is not married" are contradictory. "Smith has 6 apples" and "Smith has fewer than 3 apples" are contradictory. "John has a banana, but Gerald really hates bananas and wishes John didn't have a banana." is not a contradiction-in-the-ordinary-sense. It may be a conflict, which may be resolved in some way, but it isn't a contradiction.

pphysch 5y ago
The pre-negative-number contradiction here is between the "obviously" true propositions: A) "undefined quantities can't/don't exist" (practically a tautology) B) "subtraction sometimes produces an undefined quantity" The first proposition contradicts any proposition that asserts the existence of undefined/impossible/negative numbers. Keep in mind that, before negative numbers, there would be no valid symbols to express the result of the expression "one minus two"; it is irreducible. The root of the problem is that classical logic doesn't have good tools for dealing with ontological deficits, in this case, the lack of a well-defined concept of "negative numbers". All of Plato's classic "laws of thought" presume the existence of a well-defined universal set of things/ideas/truths that they can operate over, which is usually a flawed assumption. In reality, the baseline is that everyone has their own ontology derived from personal experience and it takes serious collective effort (or pedagogical browbeating) to establish shared ontologies on which systems of communication can be built. On the other hand, a core feature of dialectical reasoning is resolving contradictions through the synthesis of novel orthogonal concepts.
- drdeca 5y ago
  “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.

by drdeca 5y ago

pphysch 5y ago

The pre-negative-number contradiction here is between the "obviously" true propositions: A) "undefined quantities can't/don't exist" (practically a tautology) B) "subtraction sometimes produces an undefined quantity" The first proposition contradicts any proposition that asserts the existence of undefined/impossible/negative numbers. Keep in mind that, before negative numbers, there would be no valid symbols to express the result of the expression "one minus two"; it is irreducible. The root of the problem is that classical logic doesn't have good tools for dealing with ontological deficits, in this case, the lack of a well-defined concept of "negative numbers". All of Plato's classic "laws of thought" presume the existence of a well-defined universal set of things/ideas/truths that they can operate over, which is usually a flawed assumption. In reality, the baseline is that everyone has their own ontology derived from personal experience and it takes serious collective effort (or pedagogical browbeating) to establish shared ontologies on which systems of communication can be built. On the other hand, a core feature of dialectical reasoning is resolving contradictions through the synthesis of novel orthogonal concepts.

drdeca 5y ago
“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.