- #1
HJ Farnsworth
- 128
- 1
Greetings,
Two or three years ago I studied mathematical logic using Kleene’s “Mathematical Logic”. I was thinking about the subject again recently, and I have a question regarding what research has been done into non-classical logic.
I am having difficulty phrasing my question, so please let me know if it is unclear. Basically, I think what I want to ask is this:
When studying logic, we use one system of logic in the metalanguage to conduct our study of the other system of logic, that used in the object language. My question is:
a. Has logic, classical or not, been studied using anything but classical logic as the logic used in the metalanguage?
b. If so, do any of these studies use, as the metalanguage logic, a logical system that was not itself studied in the object language using classical logic in the metalanguage?
c. And so on, infinite regress.
Basically, what I mean is – has logic ever been studied without some sort of “final appeal” to classical logic? My guess would be that it has not, mainly because, if I try to even think using a logic other than classical logic, I find that I am unable to do so without first analyzing the logic I am trying to think in using classical logic.
Here is an example that I made up of the type of thing that I am curious about. Like the rest of this post, it is phrased very poorly, but that’s okay – it’s just a cheap tool that I am trying to use to illustrate my question:
Somebody tells me that there is a baseball and a box, and that the baseball is not not inside of the box. Let A= “The baseball is inside of the box,” and let B=[itex]\neg[/itex]A. A and B are object language statements, and the underlined statement is of the form [itex]\neg[/itex]B, ie., [itex]\neg[/itex][itex]\neg[/itex]A. If I use classical logic in the object language, I might use A[itex]\vee[/itex][itex]\neg[/itex]A to conclude that the baseball is inside of the box.
However, if I am using a logic other than classical logic to analyze the situation, then it might be a logic such that A[itex]\vee[/itex][itex]\neg[/itex]A does not hold. In such a situation, I may not be able to conclude that the baseball is inside of the box - instead, I can only conclude that it is not not inside of the box.
But wait – in the preceding paragraph, I used classical logic in the metalanguage to determine how the analysis would proceed if I did not use classical logic in the object language to analyze the situation with the baseball and the box. Thus, the “final appeal” was to classical logic.
Well, that’s fixed easily enough. What if I try using a non-classical logic in the metalanguage, to analyze what would happen if I used a non-classical logic in the object language? Well, then blah blah blah.
But wait – again, in the preceding paragraph, I am using classical logic to think about what would happen if I did not use classical logic. Then, once again, the “final appeal” is to classical logic.
And so on. No matter what system of logic I use, I am not sure that I can use it without having first studied it, or studied the logic used to study it, or studied the logic used to study the logic used to study it, or studied…(infinite regress), using classical logic.
Is this some sort of built-in rule of logic, or is it merely a limitation my own capabilities?
So, to repeat/rephrase/elaborate on my question,
a. Has logic ever been studied without a “final appeal” to classical logic?
b. If not, has it been formally shown in any way that the “final appeal” must always be to classical logic, so that my search is hopeless?
c. Either way, please point me to the literature for this research, if you know of it.
Thank you very much for any help that you can give.
-HJ Farnsworth
Two or three years ago I studied mathematical logic using Kleene’s “Mathematical Logic”. I was thinking about the subject again recently, and I have a question regarding what research has been done into non-classical logic.
I am having difficulty phrasing my question, so please let me know if it is unclear. Basically, I think what I want to ask is this:
When studying logic, we use one system of logic in the metalanguage to conduct our study of the other system of logic, that used in the object language. My question is:
a. Has logic, classical or not, been studied using anything but classical logic as the logic used in the metalanguage?
b. If so, do any of these studies use, as the metalanguage logic, a logical system that was not itself studied in the object language using classical logic in the metalanguage?
c. And so on, infinite regress.
Basically, what I mean is – has logic ever been studied without some sort of “final appeal” to classical logic? My guess would be that it has not, mainly because, if I try to even think using a logic other than classical logic, I find that I am unable to do so without first analyzing the logic I am trying to think in using classical logic.
Here is an example that I made up of the type of thing that I am curious about. Like the rest of this post, it is phrased very poorly, but that’s okay – it’s just a cheap tool that I am trying to use to illustrate my question:
Somebody tells me that there is a baseball and a box, and that the baseball is not not inside of the box. Let A= “The baseball is inside of the box,” and let B=[itex]\neg[/itex]A. A and B are object language statements, and the underlined statement is of the form [itex]\neg[/itex]B, ie., [itex]\neg[/itex][itex]\neg[/itex]A. If I use classical logic in the object language, I might use A[itex]\vee[/itex][itex]\neg[/itex]A to conclude that the baseball is inside of the box.
However, if I am using a logic other than classical logic to analyze the situation, then it might be a logic such that A[itex]\vee[/itex][itex]\neg[/itex]A does not hold. In such a situation, I may not be able to conclude that the baseball is inside of the box - instead, I can only conclude that it is not not inside of the box.
But wait – in the preceding paragraph, I used classical logic in the metalanguage to determine how the analysis would proceed if I did not use classical logic in the object language to analyze the situation with the baseball and the box. Thus, the “final appeal” was to classical logic.
Well, that’s fixed easily enough. What if I try using a non-classical logic in the metalanguage, to analyze what would happen if I used a non-classical logic in the object language? Well, then blah blah blah.
But wait – again, in the preceding paragraph, I am using classical logic to think about what would happen if I did not use classical logic. Then, once again, the “final appeal” is to classical logic.
And so on. No matter what system of logic I use, I am not sure that I can use it without having first studied it, or studied the logic used to study it, or studied the logic used to study the logic used to study it, or studied…(infinite regress), using classical logic.
Is this some sort of built-in rule of logic, or is it merely a limitation my own capabilities?
So, to repeat/rephrase/elaborate on my question,
a. Has logic ever been studied without a “final appeal” to classical logic?
b. If not, has it been formally shown in any way that the “final appeal” must always be to classical logic, so that my search is hopeless?
c. Either way, please point me to the literature for this research, if you know of it.
Thank you very much for any help that you can give.
-HJ Farnsworth