I think the idea is, when you argue from first principles, you are implicitly assuming that you know all of the relevant first principles. Since you're human and imperfect, there is always a chance that you don't. How to know?
Well, empirically, check whether the conclusions you get, seem to hold up to reality. The author's experience was that taking investment $$ was necessary (or at least often useful) in a startup, so this put him on the lookout for what missing first principle would explain this.
It doesn't mean axiomatic logic isn't useful, it means that just because the logic seems sound, doesn't mean the conclusion is reliable, because there could be missing axioms (in this case, that profitability is the objective of a company, when cash flow is a more fundamental fact and profit is often either present or not depending on how you do the accounting).
The logic isn't sound though, that's the issue. Let me try to simplify it even more and annotate
1. Not raising money give you more skin in the game (valid observation)
2. Skin in the game is an advantage (valid observation)
3. There exists at least one advantage of not raising money (valid conclusion)
4. You should not raise money (NOT VALID conclusion)
You can't go from a single argument in favor of something to that thing being favorable overall.
Let me just add to this. Discovering new information (or new "axioms") will not change the truth of your previous conclusions if you did everything correctly, but you may find that the information you believed before was incorrect. In general, I believe it will be better to use a probabilistic model for most real-world cases since it is very difficult to find "axioms" for almost anything.
Thanks Ross for the reply. I believe that this is a misunderstanding of how propositional logic works. If the propositions or axioms that you start with are sound, and if you correctly apply all inference rules, then the propositions that you derive will also be sound. "Missing axioms" that you did not use do no matter, regardless of their soundness.
Of course they matter. We're discussing arguments that apply in the real world, not in math theory.
In math, if you have this axiom:
f(x) > 5 for all x >= 20
you are not then allowed to change it with a later axiom
except when x is divisible by 240
However, in real life this happens a lot. I have a company that is taxed a fixed amount per year... except for the years where I make over 100k euros, in which case things become quite complicated. If I omit the second part (which is not impossible, given that I never made over 100k euros a year with that company), and suddenly get a big payout from someone, the result will be very different from my initial estimation - as sound as it was WITHOUT that additional axiom / assumption / rule.
A "missing" axiom, in my experience, is not truly a missing axiom that otherwise has no impact on other axioms. A "missing" axiom is one that exposes a bad assumption in another axiom currently being relied upon.
For instance. Socrates is a man, all men are mortal, therefore Socrates is mortal.
But then you discover that a couple of eons have passed and Socrates is still alive. Clearly there must be a "missing" axiom. And after some investigation you realize that Socrates is a Venusian man, and Venusians are immortal.
"Socrates is Venusian" is a missing axiom, but really the problem is that "All men are mortal" is actually false, since it had implicit assumptions that "All men are human" (false) and "All humans are mortal" (true).
Again, I am sorry for being direct, but this does not make sense. If a Venusian man is immortal, then the "axiom" (preposition) that all men are mortal is false. In other words, the issue is not that the preposition "Socrates is Venusian" was missing but that the preposition "all men are mortal" is false.
It is possible to develop significant mathematical theory without using some axioms. For example, mathematicians sometimes choose not to use the "axiom of choice" when working with Zermelo–Fraenkel set theory. That does not mean that mathematical theorems proven without using the axiom of choice are invalid, even if you later assume that this axiom is true (or false).
The point is that the axiom that all men are mortal was thought to be true, and then was later discovered to be false. My comment was actually in agreement with your previous comment.
Well, empirically, check whether the conclusions you get, seem to hold up to reality. The author's experience was that taking investment $$ was necessary (or at least often useful) in a startup, so this put him on the lookout for what missing first principle would explain this.
It doesn't mean axiomatic logic isn't useful, it means that just because the logic seems sound, doesn't mean the conclusion is reliable, because there could be missing axioms (in this case, that profitability is the objective of a company, when cash flow is a more fundamental fact and profit is often either present or not depending on how you do the accounting).