If there are propositions or axioms that imply each other fairly easily under common contextual assumptions, then I think it’s reasonable to consider it not-quite-a-mistake to use the same name for such propositions.
What equivalences do you have in mind when you say “imply each other”?
One of the things I’m arguing is that I’m not convinced that imprecision is enough to render a work “false.”
Are you convinced those mistakes are enough to render this piece false or incoherent?
It’s certainly true that there is a scientific/logical conundrum about how to deal with imprecision. I know a lot about what happens when you tinker with the law of excluded middle, though, and I am not convinced this has any impact on your ability to deal with imprecision.
A succinct way of putting this would be to ask: If I were to swap the phrase “law of the excluded middle” in the piece for the phrase “principle of bivalence” how much would the meaning of it change as well as overall correctness?
Additionally, suppose I changed the phrases in just “the correct spots.” Does the whole piece still retain any coherence?
Actually, here’s something that may be helpful in understanding why the principle of bivalence is distinct from the law of excluded middle:
As I understand you, one of the core points you are making is that you want to be able to entertain incompatible models. So let’s say that you have two models M and W that are incompatible with each other.
For simplicity, let’s say both models share a language L in which they can express propositions, and assign truth values to statements in that language using functions vM,vW mapping statements to truth-values. (For instance, maybe M is a flat-earth approximation, and W is a spherical-earth approximation, so vM(The earth is flat)=⊤ but vW(The earth is flat)=⊥.)
Because these are just models, your points don’t apply within the models; it might be fine for an approximation to say that everything is true or false, as long as we keep in mind that it’s just an approximation and different approximations might lead to different results. As a result, all of the usual principles of logic like bivalence, noncontradiction, excluded middle, etc. apply within the models.
However, outside/between the models, there is a sense that what you are saying applies. For instance we get an apparent contradiction/multiple truth values for vM(The earth is flat)=⊤ vs vW(The earth is flat)=⊥. But these truth values live in separate models, so they don’t really interact, and therefore aren’t really a contradiction.
But you might want to have a combined model where they do interact. We can do this simply by using the 2n-valued approach I mentioned in my prior comment. Define a shared model M×W by the truth-value function vM×W(P)=vM(P)vw(P). So for instance vM×W(The earth is flat)=⊤⊥.
Here you might interpret ⊤⊥ as meaning something along the lines of “true in practice but not in theory” or “true for small things but not for big things”, and you might interpret ⊥⊤ as meaning something along the lines of “technically true but not in practice” or “true for big things but not for small things”. While “The earth is flat” would be an example of a natural statement with the former truth value, “The earth is finite” would be an example of a natural statement with the latter truth value. And then we also have truth values like ⊤⊤ meaning “wholly true” and ⊥⊥ meaning “wholly false”.
M×W still satisfies standard logical principles like excluded middle, law of noncontradiction, etc.. This is basically because it is just two logics running side-by-side and not interacting much. This also makes it kind of boring, because you could just as well work with M and W separately.
We can make them somewhat less boring/more interacting by adding logical operators to the language that makes the different logics interact. For instance, one operator we might add is a definitely/necessarily/wholly operator, denoted □P, which is interpreted to mean that P is true in both M and W. So for instance vM×W(□The earth is flat)=⊥⊥ because the earth isn’t flat in the spherical-earth model. We can also add a sort of/possibly/partly operator, denoted ⋄P, which is interpreted to mean that P is true in at least one of M and W, so for instance vM×W(⋄The earth is finite)=⊤⊤.
These operators allow you to express some of the principles that you have been complaining about. For instance, there is a stronger version of excluded middle (□P)∨(□¬P) which is sort of an internalized version of the principle of bivalence, and this stronger version of excluded middle is false for some statements such as “The earth is flat”. There’s also a stronger version of the law of noncontradiction ¬((⋄P)∧(⋄¬P)) which again is a sort of internalized notion of the principle of bivalence and is also false for the same statements the other one is false for.
So far so good; this seems to give you exactly what you want, right? A way to entertain multiple models at once. But this requires all of the models to use a shared language, and in my experience, this requirement is actually the killer for all of these things. For instance, Newtonian physics doesn’t share a language with Einsteinian physics, as the ontology is fundamentally different. It is nearly impossible to make different logics share a language sufficiently well for this to work in interesting ways. I’d guess that the route to solving this goes through John Wentworth’s Natural Abstractions, but they haven’t been developed enough yet to solve it.
Overall, there’s a bunch of work into this sort of logic, but it isn’t useful because the fundamental problem is in how to fluently switch between ontologies, not in how to entertain multiple things at once.
So suppose I have ~(A and ~A). Rather than have this map to False, I say that “False” is an object that you always bounce off of; It causes you to reverse-course, in the following way:
~(A and ~A) --> False --> A or ~A or (some mysterious third thing). What is this mysterious third thing? Well, if you insist that A and ~A is possible, then it must be an admixture of these two things, but you’d need to show me what it is for that to be allowed. In other words:
~(A and ~A) --> A or ~A or (A and ~A).
What this statement means in semantic terms is: Suppose you give me a contradiction. Rather than simply try really hard to believe it, or throw everything away entirely, I have a choice between believing A, believing ~A, or believing a synthesis between these two things.
The most important feature of this construction is that I am no longer faced with simply concluding “false” and throwing it all away.
Two examples:
Suppose we have the statement 1 = 2[1]. In most default contexts, this statement simply maps to “false,” because it is assumed that this statement is an assertion that the two symbols to the left and right of the equals sign are indistinguishable from one another.
But what I’m arguing is that “False” is not the end-all, be-all of what this statement can or will be said to mean in all possible universes forever unto eternity. “False” is one possible meaning which is also valid, but it cannot be the only thing that this means.
So, using our formula from above:
1 = 2 -->[2] 1 or 2 or (1 and 2). So if you tell me “1 = 2”, in return I tell you that you can have either 1, either 2, or either some mysterious third thing which is somehow both 1 and 2 at the same time.
So you propose to me that (1 and 2) might mean something like 2 (1/2), that is, two halves, which mysteriously are somehow both 1 and 2 at the same time when put together. Great! We’ve invented the concept of 1⁄2.
Second example:
We don’t know if A is T and thus that ~A is F or vice-versa. Therefore we do not know if A and ~A is TF or FT. Somehow, it’s got to be mysteriously both of these at the same time. And it’s totally fine if you don’t get what I’m about to say because I haven’t really written it anywhere else yet, but this seems to produce two operators, call them “S” (for swap) and “2″ (for 2), each duals of one another.
S is the Swaperator, and 2 is the Two...perator. These also buy you the concept of 1⁄2 as well. But all that deserves more spelling out, I was just excited to bring it up.
It is arguably appropriate to use 1 == 2 as well, but I want to show that a single equals sign “=” is open to more interpretations because it is more basic. This also has a slightly different meaning too, which is that the symbols 1 and 2 are swappable with one another.
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.
If I were to swap the phrase “law of the excluded middle” in the piece for the phrase “principle of bivalence” how much would the meaning of it change
A lot. At least I associate multi-valued logic with a different sphere of research than intuitionism.
as well as overall correctness?
My impression is that a lot of people have tried to do interesting stuff with multi-valued logic to make it handle the sorts of things you mention, and they haven’t made any real progress, so I would be inclined to say that it is a dead-end.
Though arguably objections like “It introduces the concept of “actually true” and “actually false” independent of whether or not we’ve chosen to believe something.” also apply to multi-valued logic so idk to what extent this is even the angle you would go on it.
What equivalences do you have in mind when you say “imply each other”?
It’s certainly true that there is a scientific/logical conundrum about how to deal with imprecision. I know a lot about what happens when you tinker with the law of excluded middle, though, and I am not convinced this has any impact on your ability to deal with imprecision.
A succinct way of putting this would be to ask: If I were to swap the phrase “law of the excluded middle” in the piece for the phrase “principle of bivalence” how much would the meaning of it change as well as overall correctness?
Additionally, suppose I changed the phrases in just “the correct spots.” Does the whole piece still retain any coherence?
Actually, here’s something that may be helpful in understanding why the principle of bivalence is distinct from the law of excluded middle:
As I understand you, one of the core points you are making is that you want to be able to entertain incompatible models. So let’s say that you have two models M and W that are incompatible with each other.
For simplicity, let’s say both models share a language L in which they can express propositions, and assign truth values to statements in that language using functions vM,vW mapping statements to truth-values. (For instance, maybe M is a flat-earth approximation, and W is a spherical-earth approximation, so vM(The earth is flat)=⊤ but vW(The earth is flat)=⊥.)
Because these are just models, your points don’t apply within the models; it might be fine for an approximation to say that everything is true or false, as long as we keep in mind that it’s just an approximation and different approximations might lead to different results. As a result, all of the usual principles of logic like bivalence, noncontradiction, excluded middle, etc. apply within the models.
However, outside/between the models, there is a sense that what you are saying applies. For instance we get an apparent contradiction/multiple truth values for vM(The earth is flat)=⊤ vs vW(The earth is flat)=⊥. But these truth values live in separate models, so they don’t really interact, and therefore aren’t really a contradiction.
But you might want to have a combined model where they do interact. We can do this simply by using the 2n-valued approach I mentioned in my prior comment. Define a shared model M×W by the truth-value function vM×W(P)=vM(P)vw(P). So for instance vM×W(The earth is flat)=⊤⊥.
Here you might interpret ⊤⊥ as meaning something along the lines of “true in practice but not in theory” or “true for small things but not for big things”, and you might interpret ⊥⊤ as meaning something along the lines of “technically true but not in practice” or “true for big things but not for small things”. While “The earth is flat” would be an example of a natural statement with the former truth value, “The earth is finite” would be an example of a natural statement with the latter truth value. And then we also have truth values like ⊤⊤ meaning “wholly true” and ⊥⊥ meaning “wholly false”.
M×W still satisfies standard logical principles like excluded middle, law of noncontradiction, etc.. This is basically because it is just two logics running side-by-side and not interacting much. This also makes it kind of boring, because you could just as well work with M and W separately.
We can make them somewhat less boring/more interacting by adding logical operators to the language that makes the different logics interact. For instance, one operator we might add is a definitely/necessarily/wholly operator, denoted □P, which is interpreted to mean that P is true in both M and W. So for instance vM×W(□The earth is flat)=⊥⊥ because the earth isn’t flat in the spherical-earth model. We can also add a sort of/possibly/partly operator, denoted ⋄P, which is interpreted to mean that P is true in at least one of M and W, so for instance vM×W(⋄The earth is finite)=⊤⊤.
These operators allow you to express some of the principles that you have been complaining about. For instance, there is a stronger version of excluded middle (□P)∨(□¬P) which is sort of an internalized version of the principle of bivalence, and this stronger version of excluded middle is false for some statements such as “The earth is flat”. There’s also a stronger version of the law of noncontradiction ¬((⋄P)∧(⋄¬P)) which again is a sort of internalized notion of the principle of bivalence and is also false for the same statements the other one is false for.
So far so good; this seems to give you exactly what you want, right? A way to entertain multiple models at once. But this requires all of the models to use a shared language, and in my experience, this requirement is actually the killer for all of these things. For instance, Newtonian physics doesn’t share a language with Einsteinian physics, as the ontology is fundamentally different. It is nearly impossible to make different logics share a language sufficiently well for this to work in interesting ways. I’d guess that the route to solving this goes through John Wentworth’s Natural Abstractions, but they haven’t been developed enough yet to solve it.
Overall, there’s a bunch of work into this sort of logic, but it isn’t useful because the fundamental problem is in how to fluently switch between ontologies, not in how to entertain multiple things at once.
So suppose I have ~(A and ~A). Rather than have this map to False, I say that “False” is an object that you always bounce off of; It causes you to reverse-course, in the following way:
~(A and ~A) --> False --> A or ~A or (some mysterious third thing). What is this mysterious third thing? Well, if you insist that A and ~A is possible, then it must be an admixture of these two things, but you’d need to show me what it is for that to be allowed. In other words:
~(A and ~A) --> A or ~A or (A and ~A).
What this statement means in semantic terms is: Suppose you give me a contradiction. Rather than simply try really hard to believe it, or throw everything away entirely, I have a choice between believing A, believing ~A, or believing a synthesis between these two things.
The most important feature of this construction is that I am no longer faced with simply concluding “false” and throwing it all away.
Two examples:
Suppose we have the statement 1 = 2[1]. In most default contexts, this statement simply maps to “false,” because it is assumed that this statement is an assertion that the two symbols to the left and right of the equals sign are indistinguishable from one another.
But what I’m arguing is that “False” is not the end-all, be-all of what this statement can or will be said to mean in all possible universes forever unto eternity. “False” is one possible meaning which is also valid, but it cannot be the only thing that this means.
So, using our formula from above:
1 = 2 -->[2] 1 or 2 or (1 and 2). So if you tell me “1 = 2”, in return I tell you that you can have either 1, either 2, or either some mysterious third thing which is somehow both 1 and 2 at the same time.
So you propose to me that (1 and 2) might mean something like 2 (1/2), that is, two halves, which mysteriously are somehow both 1 and 2 at the same time when put together. Great! We’ve invented the concept of 1⁄2.
Second example:
We don’t know if A is T and thus that ~A is F or vice-versa. Therefore we do not know if A and ~A is TF or FT. Somehow, it’s got to be mysteriously both of these at the same time. And it’s totally fine if you don’t get what I’m about to say because I haven’t really written it anywhere else yet, but this seems to produce two operators, call them “S” (for swap) and “2″ (for 2), each duals of one another.
S is the Swaperator, and 2 is the Two...perator. These also buy you the concept of 1⁄2 as well. But all that deserves more spelling out, I was just excited to bring it up.
It is arguably appropriate to use 1 == 2 as well, but I want to show that a single equals sign “=” is open to more interpretations because it is more basic. This also has a slightly different meaning too, which is that the symbols 1 and 2 are swappable with one another.
You could possibly say “--> False or 1 or 2 or …”, too, but then you’d probably not select False from those options, so I think it’s okay to omit it.
You should use a real-world example, as that would make the appropriate logical tools clearer.
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.
A lot. At least I associate multi-valued logic with a different sphere of research than intuitionism.
My impression is that a lot of people have tried to do interesting stuff with multi-valued logic to make it handle the sorts of things you mention, and they haven’t made any real progress, so I would be inclined to say that it is a dead-end.
Though arguably objections like “It introduces the concept of “actually true” and “actually false” independent of whether or not we’ve chosen to believe something.” also apply to multi-valued logic so idk to what extent this is even the angle you would go on it.