I've recently been fascinated by the axioms that underly various political ideologies, that people never address because it's near impossible to change someone's axioms in any form of debate.

One of them that I was just thinking about is a disagreement over the equivalence of action and inaction (and therefore linguistic negation) that is often core to disputes between anarchists and anarcho-capitalists.

On the one hand, anarchists believe that inaction can be an act of aggression. For example, a corporation that has enough money to feed a homeless person but chooses not to, would be perceived as unethical in a Marxist system.

Ancaps and probably most capitalists on the other hand would say that only explicit action can be aggressive action. For example, the government showing up at your doorstep and demanding taxes at gunpoint, would be perceived as unethical to ancaps.

I arrived at this duality from politics, but it seems to run far deeper. What even is an action? Does lack of a specific action directly imply another? Does negation of a term imply its direct opposite or rather the set of all elements that exclude the term?

Non-aggressive inaction, that is the lack of explicitly acting non-aggressively, becomes an aggressive action to the anarchist. However, this same non-aggressive inaction simply leaves the actual action ambiguous or to be defined upon further information to the ancap.

The anarchists seem to treat negation as a mapping from the set of non-aggressive actions to the set of aggressive actions, while the ancaps seem to treat negation as the complement of a non-aggressive action within the set of all actions.

This seems like a genuine linguistic case of ambiguity. When a waiter offers to grind pepper on my soup, and I tell them "no pepper", I'm negating as a set complement—that is, I'm simply stating that pepper is not desired and information about what is actually desired is left undefined. When I'm at Subway and they have salt and pepper and I tell them "no pepper", it seems to imply that I do still want the opposite, salt. Though I'm not sure whether that's simply because the set we're referring to becomes (salt, pepper) or because of a genuine mapping from pepper to its commonly understood negative.

Regardless, the nature of negation itself is taken axiomatically in most systems of logic, but it seems that in the context of set theory, it has multiple interpretations.