Is illogical = not logical?

Question

I think law of excluded middle makes sense to mean that a statement should be either logical or illogical but in this case I don't assume "not logical" = "illogical" since the author didn't say "illogical", the author said "not logical."

Can we formalize logic about logic in a way so that a statement such as "That's not logical" is formal when it's now used in a non-formal way while I interpret it as "metalogic" - logic about logic and just a negation. Logic doesn't formalize itself, logic formalizes statements, so what does logic mean? Is logic itself also just a statement similarly to how an axiom is also just a statement and a rule is also just a statement so what we could agree on is what type of statement is meant with just a negation so that "nothing exists" i.e.

• "It's not technical." this statement doesn't even say that there is anything technical
• "It's not accounting" ..and therefore not economics which might as well be the empty set
• "It's not a detail" ..ergo what is meant is "the big picture" and only if I understand correctly
• "No prisoner escaped during the night" - this statement doesn't even say that there exists any prisoners or prisons when "All prisoners were prevented from escaping" at least tells us that there was something instead of nothing
• "It's not logical/It's not rational" - What is meant is not even "illogical" or "irrational" since I can't assume that "not logical" = "illogical" so again it's the empty set and even a statement about the system itself which a mathematical system like theoretical philosophical logic is so a meaning with meta level still not ruling out that it's the empty set. And there are mathematical truths that are not logical for instance the so-called Gabriel's horn that has infinite area and finite volume which is not logical and still mathematically correct
• "I don't understand." - The statement neither says what you do when you don't understand nor does it define what it means to understand and again the statement is on the negation form "There is nothing..."

So how can I rule out that all these statements that are just a negation indeed are about something instead of nothing / the empty set or undefined?

So that the statement A is neither logical nor illogical it is simply not logical as how it is perceived as "not logical" or "not a logical sequence" disctinct from being an illogical sequence where a logical sequence typically would be "A causes B" like cause and effect while a statement that is about logic and a negation also can be true, false, a negation or provable.

Answer 1

There are several factors at play in your question.

It appears that you have (re-)discovered the distinction between implicational and non-implicational negation (also sometimes known as "choice negation" and "exclusion negation").

The literature on this topic goes back to ancient times: for example, Indian logic (both Buddhist and Nyāya) draws a distinction between prasajya negation (i.e., "This is not a brahmin") and paryudāsa negation (i.e., "This is a nonbrahmin"). In the former case, we are negating a proposition; in the latter case we are negating a term.

So, when we say "The number seven is not green", we are not implying that it is some other color.

Note that this is completely orthogonal to the subject of "meta-logic."

In other words, your choice of the word "illogical" as an example seems to be leading you to second-order logic (which may be your goal), but it is not necessarily linked to your questions about negation (which also apply to first-order logic.)

I'd recommend a good undergraduate introductory logic textbook, but I'm afraid I don't know what's in current use these days.

Answer 2

Please excuse the double-post, but this is a completely separate stream of thought.

I am a programmer, so deal with logic all day. So I thought about this question in those terms.

For those with no coding background, a boolean variable is either true or false. No other value is possible, it's a binary value.

So, referring to a boolean variable x:

1 - (a!=x) a not equal to x - this is a comparison between two values.

2 - (a=!x) a equal to not x - this assigns the value opposite to x, to a.

statement 2 results in a having a boolean value, opposite to x.

statement 1 however, implies nothing about a at all, except that it is not the same as x (assuming the comparison is true, implying the equality is false). It COULD be the opposite value (I want to not go), but it could be anything else at all (I don't care if I go, I want to go elsewhere, I would rather not go but will if you make me, or even Let them eat cake - ie a doesn't need to be related to x in any way at all!)

I would equate statement 2 to "that is not logical", and statement 1 to "that is illogical".

Rich

Answer 3

To me, the answer here is similar to something I say quite often:

"I don't want to go" - this is not the same as "I want to not go".

We often use the first syntax in general speech, but we usually mean the second.

That's not logical = there's no logical way to come to that conclusion. That's illogical = there's a logical way to come to the opposite conclusion (disprove that conclusion).

They're similar, but not the same, in my opinion.