Question about terminology of formal logic concerning the word “axiom”

I am writing a book (with intent to publish ) about a system of logic I am developing.
I have already defined the central “axiom” of my whole system of logic and named it and I have named it the “entropic axiom”.

From that axiom, I have deduced certain useful conclusions that are useful because they, in turn, rather like the entropic axiom, can be used as a premise for further deductions leading to may other useful conclusions. The problem is, I want to give those conclusions that can be very often used as a premise for further deductions and conclusions reasonably short specific NAMES so that I don't have to extremely tediously explain them over and over again from scratch each and every time I want to refer to them in my book. I have a very large number of them to name and see a need to find a systematic and consistent method of naming them else what we will have is totally unacceptable total chaos.

Now, I understand that an “axiom” is a premise for a starting point for a deduction from which you derive conclusions. But, still, would it really be an abuse in terminology of formal logic to call these conclusions from my “entropic axiom” to call same/all of them: something - “axiom”? I am, for example, very tempted to call a particular one I am working on right at this very moment in my book the “clone median axiom” which is deduced from the entropic axiom (by an extremely long complicated series of deductions ) but I am rather concerned that I might be severely criticized for doing so because I am not sure if that is really allowed in formal logic.

Whether I am allowed to do that or not, is there a special technical/conventional word that I could use to help name a conclusion from an axiom that can be or is used as a premise for further deductions? So I can name them; something-“X” where X is whatever that word is?

Originally posted by humy
I am writing a book (with intent to publish ) about a system of logic I am developing.
I have already defined the central “axiom” of my whole system of logic and named it and I have named it the “entropic axiom”.

From that axiom, I have deduced certain useful conclusions that are useful because they, in turn, rather like the entropic axiom, can be used as a ...[text shortened]... se for further deductions? So I can name them; something-“X” where X is whatever that word is?

This might be to basic for you, but this link is a Stanford university course in basic logic, free online course:

https://www.coursera.org/course/intrologic

The word you want is theorem.

Originally posted by sonhouse
This might be to basic for you, but this link is a Stanford university course in basic logic, free online course:

https://www.coursera.org/course/intrologic

I am slightly surprised it is free for anyone. Being online explains most of that but, I think, surely, it must still cost something to employ the instructors to help and advise students? Unless those expert instructors volunteer i.e. do it for free with no extra wages!?
-just curious.

In any case, I will consider it 🙂

Originally posted by humy
I am slightly surprised it is free for anyone. Being online explains most of that but, I think, surely, it must still cost something to employ the instructors to help and advise students? Unless those expert instructors volunteer i.e. do it for free with no extra wages!?
-just curious.

In any case, I will consider it 🙂

There's a lot of free stuff online nowadays. In many cases it's just lecturers putting stuff online they have already offered their own students.

Originally posted by KazetNagorra
There's a lot of free stuff online nowadays. In many cases it's just lecturers putting stuff online they have already offered their own students.

So they don't have to do anything new, no new recordings, etc.

Originally posted by DeepThought
The word you want is theorem.

arr, somehow I didn't think of that. So, for example, I deduce the “clone median theorem” (not the “clone median axiom” ) from the "entropic axiom". At first it seemed to me a bit eccentric to call something a "theorem" when, as in this case, the axiom it deduced from is true-by-definition as it merely expresses what is meant by something thus all that is deduced from it is just a tautology including the “clone median theorem”. This is because when I hear the word "theorem", I illogically think it sounds to me that it is implying it is an unproven "theory" as opposed to a proven fact, and yet I think we will regard the “clone median theorem” as a proven fact (once I get my book published ). But I keep forgetting that "theorem", in formal logic, normally DOES mean a deduced proven fact!
So I think I will use that one! Thanks 🙂

Originally posted by humy
arr, somehow I didn't think of that. So, for example, I deduce the “clone median theorem” (not the “clone median axiom” ) from the "entropic axiom". At first it seemed to me a bit eccentric to call something a "theorem" when, as in this case, the axiom it deduced from is true-by-definition as it merely expresses what is meant by something thus all that is deduc ...[text shortened]... rmal logic, normally DOES mean a deduced proven fact!
So I think I will use that one! Thanks 🙂

Everyone occasionally needs something they already know pointed out to them. Don't forget about the word lemma as well.

You may also want to take a look at the entries on logic on the Stanford philosophy website. It contains a lot of information about various alternative logics (Intuitionist, Linear, Connexive, Fuzzy, ...) and help you relate your system to the various others, which will be important in trying to get it accepted.

I'm yet to read it, as the articles are definitely not light reading, but linear logic looks fun. Propositions are regarded as resources which are exhausted after use, so a given proposition can only be used once in a proof.

http://plato.stanford.edu