Something that seems not likely to be true because it wasn't justified by deriving it from the axioms and theorems of a logical system is called ad hoc1

Then a statement that isn't ad hoc is something that follows from / can be derived from a theory or logical system and doesn't contradict it.

For example, if our logical system said

For each positive integer n, n - 1 is the positive integer preceding it, if it exists2

Then a statement such as

except 5. 5 - 1 is 8

can be described to be ad hoc (and it is false as well, but ad hoc doesn't necessarily mean that something is wrong1)

And a statement such as

except 1. 1 does not have a positive integer that precedes it

may seem ad hoc, so we'll add that "this statement is not ad hoc and is consistent with the mathematics2"

I'm looking for a single word or a phrase that captures this meaning so that on rephrasing, my sentence above would become "this statement is _______."

  1. This may differ from usage of ad hoc with the meaning - "created or done for a particular purpose as necessary" because it carries a different connotation. See Ad hoc hypothesis.
    Also, this makes my question different from Antonym for "ad hoc".
  2. Here, by "positive integer", I mean the numbers 1, 2, 3, 4, ....

  • I think you need to ask this on a maths/logic forum. – Max Williams Jun 2 '16 at 7:29
  • Yeah. I guess ad hoc isn't used in general English as commonly as I thought. Though you can still use ad hoc outside of formal logic. e.g. "The teacher was unable to come up with a better system to decide which students should get the two free tickets for the museum trip so she declared the ad hoc criteria that the tallest students of class will get the tickets" – Peeyush Kushwaha Jun 2 '16 at 9:09
  • 1
    Yes, outside of maths/logic, "ad hoc" means "made up as you go along", or "improvised", and this is the definition that most people are familiar with (if they are familiar with it at all). – Max Williams Jun 2 '16 at 9:58

Here is my two cents.

If one logically analyzes the statement "... N - 1 is a positive integer..., " it is found that "n - 1" for this system is not possible. Hence, inferring from this, a conclusion can be drawn that this is what is logical.

This statement is a sequitur.

Merriam Webster

The conclusion of an inference

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