I have heard people say that one given idea presupposes another, and I myself might have used this language before, though I really meant that to believe in one idea which is unverifiable requires, at least if one is consistent, that one believe some preceding, prerequisite idea at well. Otherwise, I meant that the nature of a certain idea hinges on another. The idea that stars exist, if true, logically requires that matter exists. The existence of matter does not logically require, or in other words, necessitate, that stars exist. No, even in my rationalistic embrace of sensory skepticism, I do not think that stars do not exist, for this too is unprovable. I only mean that of these ideas, one intrinsically requires the other in order to be true, but not the other way around.
As for people rather than ideas, a person could assume an idea is true or believe it without even having any sincere commitment to its ramifications, presupposing it on faith in sheer epistemological stupidity. People can presuppose/assume that an idea is correct if they allow themselves to, but an idea can only logically require another idea. The necessary laws of logic and the concepts they govern do not believe in anything because they are not conscious entities; logical necessity and true concepts are true in themselves whether or not they are grasped or believed, and it is an objective fact that an idea necessitated logically by another idea has that relationship even if the "preceding" idea is false.
However, a person is indeed in error anytime, no matter the subject or the extent of their fallacious persuasion, that he or she believes something without logical proof. A thing can be both true and demonstrable and still be merely assumed by a fool who has put no effort into avoiding assumptions or attempting to verify the issue in question, starting with the metaphysically self-necessary, epistemologically self-evident truths of logical axioms on which all else depends. No idea can presuppose another even then because necessary truths do not think. They are grasped by the mind. Necessary truths do not depend on the mind, for they are true in themselves with or without other metaphysical things, which means they exist independent of all else--since logical axioms being false still entails that they are true (for instance, it cannot be true that truth does not exist, and so it exists necessarily), logic is more foundational and transcendent than even God [1].
Someone might use the wording of one concept presupposing another, but ideas do not perceive, and so they cannot assume anything; they are abstract concepts true or false regardless of belief or perception. It is people who can be rational or irrational, who can discover, celebrate, neglect, deny, or oppose logical truths. It is people who can hold the notion that things are true, and in doing so, if they are not rationalistic, they are making assumptions. Hence, they can presuppose philosophies, but their their philosophies do not presuppose, though they logically require either preceding ideas up to a point or ramifications that would follow from them necessarily if true. Ideas require, and people presuppose. The latter do not have to, of course. It is merely easier to for people who are accustomed to belief based on intuition, social acceptance, personal convenience, emotional persuasion, and so on to continue looking to them.
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