First, a question, am I correct in understanding that when you write ~(A and ~A), the first ~ is a typo and you meant to write A and ~A (without the first ~)? Because ¬(A∧¬A) is a tautology and thus maps to true rather than to false.
Secondly, it seems to me that you’d have to severely mutilate your logic to make this nontrivial. For instance, rather than going by your relatively elaborate route, it seems like a far simpler route would be Earth flat and earth ~flat ⇒ Earth flat ⇒ Earth flat or Earth spherical or ….
Of course this sort of proof doesn’t capture the paradoxicalness that you are aiming to capture. But in order for the proof to be invalid, you’d have to invalidate one of (A∧B)⟹A and A⟹(A∨B), both of which seem really fundamental to logic. I mean, what do the operators “and” and “or” even mean, if they don’t validate this?
First, a question, am I correct in understanding that when you write ~(A and ~A), the first ~ is a typo and you meant to write A and ~A (without the first ~)? Because ¬(A∧¬A) is a tautology and thus maps to true rather than to false.
I thought of this shortly before you posted this response, and I think that we are probably still okay (even though strictly speaking yes, there was a typo).
Normally we have that ~A means: ~A --> A --> False. However, remember than I am now saying that we can no longer say that “~A” means that “A is False.”
So I wrote:
~(A and ~A) --> A or ~A or (A and ~A)
And it could / should have been:
~(A and ~A) --> (A and ~A) --> False (can omit) [1]or A or ~A or (A and ~A).
So, because of False now being something that an operator “bounces off of”, technically, we can kind of shorten those formulas.
Of course this sort of proof doesn’t capture the paradoxicalness that you are aiming to capture. But in order for the proof to be invalid, you’d have to invalidate one of (A∧B)⟹A and A⟹(A∨B), both of which seem really fundamental to logic. I mean, what do the operators “and” and “or” even mean, if they don’t validate this?
Well, I’d have to conclude that we no longer consider any rules indispensable, per se. However, I do think “and” and “or” are more indispensable and map to “not not” (two negations) and one negation, respectively.
False can be re-omitted if we were decide, for example, that whatever we just wrote was wrong and we needed to exit the chain there and restart. However, I don’t usually prefer that option.
You write in an extremely fuzzy way that I find hard to understand. This is plausibly related to the motivation for your post; I think you are trying to justify why you don’t need to make your thinking crisper? But if so I think you need to focus on it from the psychology/applications/communication angle rather than from the logic/math angle, as that is more likely to be a crux.
It is probably indeed a crux but I don’t see the reason for needing to scold someone over it.
(That’s against my commenting norms by the way, which I’ll note that so far you, TAG, and Richard_Kennaway have violated, but I am not going to ban anyone over it. I still appreciate comments on my posts at all, and do hope that everyone still participates. In the olden days, it was Lumifer that used to come and do the same thing.)
I have an expectation that people do not continually mix up critique from scorn, and please keep those things separate as much as possible, as well as only applying the latter with solid justification.
You can see that yes, one of the points I am trying to make is that an assertion / insistence on consistency seems to generally make things worse. This itself isn’t that controversial, but what I’d like to do is find better ways to articulate whatever the alternatives to that may be, here.
It’s true that one of the main implications of the post is that imprecision is not enough to kill us (but that precision is still a desirable thing). We don’t have rules that are simply tautologies or simply false anymore.
At least we’re not physicists. They have to deal with things like negative probability, and I’m not even anywhere close to that yet.
You write in an extremely fuzzy way that I find hard to understand.
This does. This is a type of criticism that one can’t easily translate into an update that can be made to one’s practice. You’re not saying if I always do this or just in this particular spot, nor are you saying whether it’s due to my “writing” (i.e. style) or actually using confused concepts. Also, it’s usually not the case that anyone is trying to be worse at communicating, that’s why it sounds like a scold.
You have to be careful using blanket “this is false” or “I can’t understand any of this,” as these statements are inherently difficult to extract from moral judgements.
I’m sorry if it was hard to understand, you are always free to ask more specific questions.
To attempt to clarify it a bit more, I’m not trying to say that worse is better. It’s that you can’t consider rules (i.e. yes / no conditionals) to be absolutely indispensable.
Well, to use your “real world” example, isn’t that just the definition of a manifold (a space that when zoomed in far enough, looks flat)?
I think it satisfies the either-or-”mysterious third thing” formulae.
~(Earth flat and earth ~flat) --> Earth flat (zoomed in) or earth spherical (zoomed out) or (earth more flat-ish the more zoomed in and vice-versa).
First, a question, am I correct in understanding that when you write ~(A and ~A), the first ~ is a typo and you meant to write A and ~A (without the first ~)? Because ¬(A∧¬A) is a tautology and thus maps to true rather than to false.
Secondly, it seems to me that you’d have to severely mutilate your logic to make this nontrivial. For instance, rather than going by your relatively elaborate route, it seems like a far simpler route would be Earth flat and earth ~flat ⇒ Earth flat ⇒ Earth flat or Earth spherical or ….
Of course this sort of proof doesn’t capture the paradoxicalness that you are aiming to capture. But in order for the proof to be invalid, you’d have to invalidate one of (A∧B)⟹A and A⟹(A∨B), both of which seem really fundamental to logic. I mean, what do the operators “and” and “or” even mean, if they don’t validate this?
I thought of this shortly before you posted this response, and I think that we are probably still okay (even though strictly speaking yes, there was a typo).
Normally we have that ~A means: ~A --> A --> False. However, remember than I am now saying that we can no longer say that “~A” means that “A is False.”
So I wrote:
~(A and ~A) --> A or ~A or (A and ~A)
And it could / should have been:
~(A and ~A) --> (A and ~A) --> False (can omit) [1]or A or ~A or (A and ~A).
So, because of False now being something that an operator “bounces off of”, technically, we can kind of shorten those formulas.
Well, I’d have to conclude that we no longer consider any rules indispensable, per se. However, I do think “and” and “or” are more indispensable and map to “not not” (two negations) and one negation, respectively.
False can be re-omitted if we were decide, for example, that whatever we just wrote was wrong and we needed to exit the chain there and restart. However, I don’t usually prefer that option.
You write in an extremely fuzzy way that I find hard to understand. This is plausibly related to the motivation for your post; I think you are trying to justify why you don’t need to make your thinking crisper? But if so I think you need to focus on it from the psychology/applications/communication angle rather than from the logic/math angle, as that is more likely to be a crux.
It is probably indeed a crux but I don’t see the reason for needing to scold someone over it.
(That’s against my commenting norms by the way, which I’ll note that so far you, TAG, and Richard_Kennaway have violated, but I am not going to ban anyone over it. I still appreciate comments on my posts at all, and do hope that everyone still participates. In the olden days, it was Lumifer that used to come and do the same thing.)
I have an expectation that people do not continually mix up critique from scorn, and please keep those things separate as much as possible, as well as only applying the latter with solid justification.
You can see that yes, one of the points I am trying to make is that an assertion / insistence on consistency seems to generally make things worse. This itself isn’t that controversial, but what I’d like to do is find better ways to articulate whatever the alternatives to that may be, here.
It’s true that one of the main implications of the post is that imprecision is not enough to kill us (but that precision is still a desirable thing). We don’t have rules that are simply tautologies or simply false anymore.
At least we’re not physicists. They have to deal with things like negative probability, and I’m not even anywhere close to that yet.
What makes you say I’m scolding you?
This does. This is a type of criticism that one can’t easily translate into an update that can be made to one’s practice. You’re not saying if I always do this or just in this particular spot, nor are you saying whether it’s due to my “writing” (i.e. style) or actually using confused concepts. Also, it’s usually not the case that anyone is trying to be worse at communicating, that’s why it sounds like a scold.
You have to be careful using blanket “this is false” or “I can’t understand any of this,” as these statements are inherently difficult to extract from moral judgements.
I’m sorry if it was hard to understand, you are always free to ask more specific questions.
To attempt to clarify it a bit more, I’m not trying to say that worse is better. It’s that you can’t consider rules (i.e. yes / no conditionals) to be absolutely indispensable.