I can’t shake the idea that maps should be represented classically and territories should be represented intuitionistically.
But, it seems to me that a map is a representation of a territory. So, your statement “maps should be represented classically and territories should be represented intuitionistically” reduces to “representations of the territory should be intuitionistic, and representations of those intuitionistic representations should be classical”. Is this what you intended, or am I missing something?
Also, I’m not an expert in intuitionistic logic, but this statement from the summary sounds problematic:
classical logic is the logic of making a map accurate by comparing it to a territory, which is why the concept of falsehood becomes an integral part of the formal system
But, the concept of falsehood is integral to both classical and intuitionistic logic. Intuitionistic logic got rid of the principle of the excluded middle but did not get rid of the concept of falsity.
Regarding falsehood: I would say that intuitionistic logic ejects falsehood from its formal system in the specific sense mentioned in my link. I could dig up more references if you want me to. I agree that there are many reasonable interpretations in which it does not do so, but I don’t think those interpretations are relevant to my point. I only intended to argue that proof by contradiction is the strategy of correcting a map as opposed to describing a territory.
Regarding mapping versus description: I agree that my motivations were semantic rather than syntactic. I just wanted to know whether the idea I had made sense to others who know something of intuitionistic logic. I guess I have my answer, but for the sake of clarifying the sense I was going for, here’s the example I posted below:
Suppose you have a proposition like, “There is a red cube.” Next, you learn that this proposition leads to a contradiction. You could say one of two things:
This proves there is no red cube.
This means the context in which that proposition occurs is erroneous.
Does it make sense to say that 1 is the strategy of correcting a map and 2 is the strategy of rejecting a description as inaccurate without seeking to correct something?
Regarding mapping versus description: I agree that my motivations were semantic rather than syntactic. I just wanted to know whether the idea I had made sense to others who know something of intuitionistic logic.
Understood. But, the point that I raised is not merely syntactic. On a fundamental level, a description of the territory is a map, so when you attempt to contrast correcting a map vs rejecting a description of a territory, you are really talking about correcting vs. rejecting a map.
Does it make sense to say that 1 is the strategy of correcting a map and 2 is the strategy of rejecting a description as inaccurate without seeking to correct something?
Yes, in the case of number 1 you have proved via contradiction that there is no red cube, and in #2 you have concluded that one or more of your assumptions is incorrect (i.e. that your map is incorrect). However, this is not a map vs. territory distinction; in both cases you are really dealing with a map. To make this clear, I would restate as:
1 is the strategy of correcting the map and 2 is the strategy of rejecting the map as inaccurate without seeking to correct it.
So, I guess I don’t really have anything additional to add about intuitionistic logic—my point is that when you talk about a description of the territory vs. a map, you are really talking about the same thing.
Thanks. The next thing I was going to say is that the intuitionistic strategy of neutrality with regard to affirming or negating propositions in worlds until proof comes along roughly (i.e. in a sense to be argued for later) differentiates the classical and intuitionistic approaches like so:
The classical approach is good for having one “world” description that is almost certainly inaccurate. This can be gradually updated, making it represent one map.
The intuitionistic approach is good for having multiple world descriptions that are almost certainly incomplete. Their contours are filled in as more information becomes available and rejected as inaccurate when they lead to contradictions, making each one a holistic representation of a possible territory. (Shoehorning the same approach into classical logic is possible, but you have to create a set of conventions to do so. These conventions are not universal, making the approach less natural.)
But, it seems to me that a map is a representation of a territory. So, your statement “maps should be represented classically and territories should be represented intuitionistically” reduces to “representations of the territory should be intuitionistic, and representations of those intuitionistic representations should be classical”. Is this what you intended, or am I missing something?
Also, I’m not an expert in intuitionistic logic, but this statement from the summary sounds problematic:
But, the concept of falsehood is integral to both classical and intuitionistic logic. Intuitionistic logic got rid of the principle of the excluded middle but did not get rid of the concept of falsity.
Thanks.
Regarding falsehood: I would say that intuitionistic logic ejects falsehood from its formal system in the specific sense mentioned in my link. I could dig up more references if you want me to. I agree that there are many reasonable interpretations in which it does not do so, but I don’t think those interpretations are relevant to my point. I only intended to argue that proof by contradiction is the strategy of correcting a map as opposed to describing a territory.
Regarding mapping versus description: I agree that my motivations were semantic rather than syntactic. I just wanted to know whether the idea I had made sense to others who know something of intuitionistic logic. I guess I have my answer, but for the sake of clarifying the sense I was going for, here’s the example I posted below:
Suppose you have a proposition like, “There is a red cube.” Next, you learn that this proposition leads to a contradiction. You could say one of two things:
This proves there is no red cube.
This means the context in which that proposition occurs is erroneous.
Does it make sense to say that 1 is the strategy of correcting a map and 2 is the strategy of rejecting a description as inaccurate without seeking to correct something?
Understood. But, the point that I raised is not merely syntactic. On a fundamental level, a description of the territory is a map, so when you attempt to contrast correcting a map vs rejecting a description of a territory, you are really talking about correcting vs. rejecting a map.
Yes, in the case of number 1 you have proved via contradiction that there is no red cube, and in #2 you have concluded that one or more of your assumptions is incorrect (i.e. that your map is incorrect). However, this is not a map vs. territory distinction; in both cases you are really dealing with a map. To make this clear, I would restate as:
So, I guess I don’t really have anything additional to add about intuitionistic logic—my point is that when you talk about a description of the territory vs. a map, you are really talking about the same thing.
Thanks. The next thing I was going to say is that the intuitionistic strategy of neutrality with regard to affirming or negating propositions in worlds until proof comes along roughly (i.e. in a sense to be argued for later) differentiates the classical and intuitionistic approaches like so:
The classical approach is good for having one “world” description that is almost certainly inaccurate. This can be gradually updated, making it represent one map.
The intuitionistic approach is good for having multiple world descriptions that are almost certainly incomplete. Their contours are filled in as more information becomes available and rejected as inaccurate when they lead to contradictions, making each one a holistic representation of a possible territory. (Shoehorning the same approach into classical logic is possible, but you have to create a set of conventions to do so. These conventions are not universal, making the approach less natural.)
Something like that anyway, but Shramko 2012 has put a lot more thought into this than I have: http://kdpu.edu.ua/shramko/files/2012_Logic_and_Logical_Philosophy_What_is_a_Genueny_Intuitionistic_Notion_of_Falsity.pdf I defer to expert opinion here.