# Role of classical logic in studying logic

1. Aug 13, 2013

### HJ Farnsworth

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=$\neg$A. A and B are object language statements, and the underlined statement is of the form $\neg$B, ie., $\neg$$\neg$A. If I use classical logic in the object language, I might use A$\vee$$\neg$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$\vee$$\neg$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

2. Aug 14, 2013

### verty

Logic works in this way: one chooses a truth function that will assign a value of true or false (or possibly more values or even continuously many values) to a statement in the object language. If you change the truth function, you change the semantics of the object language.

I don't quite know what it would mean to study logic without appealing to classical logic. You would still define the same truth function and it would function in the expected way.

3. Aug 14, 2013

### Stephen Tashi

I'm not a logician either. My thoughts are:

I don't know if "logic used in the metalanguage" is an apt expression. One reasons about the metalanguage so one uses logic upon the metalanguage. As far as I know the logic one uses to reason about the metalanguage is same logic that is used to reason about any mathematics. I don't know if "classical logic" has a standard definition. If you are talking about deductions like "if A and B are both true then A is true". That sort of logic is used. It's used because it's regarded as effective. The distinction between the metalanguage and the language isn't, as far as I know, a distinction in the type of logic that is used. It's a distinction in the subject matter.

You can find articles about people developing "metalanguages" to discuss "logics" for quantum mechanics and things like that. But the metalanguage discusses the language as a system of symbols and formulas. One reasons about formal systems of symbols and formulas in the ordinary way.

I think your guess is correct. Can we imagine a different situation? It is easy to imagine things like modal logics, fuzzy logic etc. However, when we reason about these topics we make statements about them that are neither modal or fuzzy. It's amusing to consider how one would do something as simple as write a definition without resorting to "crisp" statements. For example, in ordinary fuzzy logc the membership relation of an element in a fuzzy set is a real number in the interval [0,1]. Would it be useful to describe a different fuzzy logic by saying that "the membership relation of an element in a fuzzy set" is a member to the degree of 0.93 in the set of mathematical systems where the membership of an element in a set is a number in the interval [0,1]"?

I wouldn't write off the idea of a "non-classical" logic being applied to study a non-classical logic. However, the traditional way to study a logic is to study it as a formal system - as if it were strings of symbols written according to formal syntax rules and manipulated by certain algorithms. It seems unlikely that any sort of alternative logic would be better at describing that situation than ordinary logic is. To find a justification for using a non-classical logic to study a logic, I think you would have to ditch the idea that the logic being studied is represented in such a way.

Most of mathematics seems like an effort to represent things as strings of symbols and manipulate them in certain ways. I wonder if it can do anything else.

You could imagine a kind of unreliable writing tablet where symbols your wrote spontaneously changed shape and moved about. Can you study such a tablet by writing about it on a similar tablet?

4. Aug 15, 2013

### HJ Farnsworth

Thanks for responding, verty and Stephen Tashi!

Based on what both of you are saying, I would guess that the answer to my question, "has logic ever been studied without a 'final appeal' to classical logic?" is "no".

Furthermore, if I ask, "Is it possible to study logic without a 'final appeal' to classical logic?", I would guess that the answer is also "no", although I am still unsure as to whether this is a limitation in human (or at least my) ability, or more of an "inherently real" limitation. (I know I'm starting to sound a bit pseudo-philosophical here, please forgive me).

Well put. It would be an interesting exercise to try to study a formal system (i.e., manipulate the symbols in that system) using an alternative logic and see what happens. Verty, regarding your comment,

I guess what it would mean to study logic without appealing to classical logic would simply be that the truth functions would not function in the expected ways, since an alternative logic is being used to manipulate the formal system, in which the truth functions are expressed as meaningless combinations of symbols. However, if I understand you correctly, Stephen, then I think you are right in that that sort of formulation of logic is probably unfeasible.

But maybe I'll try it anyway sometime and see what happens. Or, does anyone know of any literature where someone has already done this kind of thing?

Thanks again for the help.

-HJ Farnsworth

5. Aug 15, 2013

### Stephen Tashi

We could equally well ask if it is possible to study anything without a final appeal to classical logic. I think this depends on how we define "study". The mathematical way to study something is to study statements that are definitely true. So this leads naturally to the logic of "propositions". I think this is a limitation of human beings only insofar as it is a limitation in what interests human beings.

The purpose of logic is to give methods that are completely reliable in deducing truths from other truths. So if you want an alternative to (classical) logic, you either need an alternative method that accomplishes the same goals or you need different goals.

If you found an alternative method that accomplished substantially the same goals as classical logic then it would be unremarkable unless there were some proposition that was "undecided" in one logic but proven or disproven in the other.

Changing the goal is hard to imagine. You would have to "study" a phenomena using something other than "propositions". The goal would not be to proceed from certainties to other certainties or to make comments about the process that were certain. You'd have to "comment" about the process in some way that you felt was useful or interesting but the comments wouldn't be propositions.

Last edited: Aug 15, 2013
6. Aug 21, 2013

### atyy

I found this remark in Rautenberg's "A Concise Introduction to Mathematical Logic" (page xiv):"The logical means of the metatheory are sometimes allowed or even explicitly required to be different from those of the object language. But in this book the logic of object languages, as well as that of the metalanguage, are classical, two-valued logic.".

I don't know whether he meant there is an example where the metalanguage is nonclassical but the object language is classical.

7. Nov 21, 2013

### gufiguer

Beautiful philosophical discussion!!
The human reason is behind all and Logic was born to describe it. If it was possible to forget Logic and to think without it, we would be no-humans. If there was a way to think without (classical) logic, then this way of thinking would be a part of logic and will not be separated of it, it would have been addressed by the logic at its emergence.
Maybe the crazy people has some way of thinking without logic or with another logic.