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An argument for some sort of 'principles' in the Aristotelian sense is as follows:

1.) If we can think something about something, we must be able to think at least the most simple of propositions about such things.

2.) Since it is self-contradictory to say that we cannot think something about something, there must exist the most simple of propositions that can be said of things.

3.) These propositions must be universal since the simplest thing that can be said of something can be said of anything, since anything is in its most simple sense something.

4.) These propositions we call principles.

As such, if we admit that we can say something about something, must we admit principles? Can they be rejected?

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Is this not what Descartes tried to demonstrate with Cogito ergo sum? Similar to your second point: "it is self-contradictory to say that we cannot think something about something", Descartes argued:

[W]e cannot doubt of our existence while we doubt

Later, Charles Sanders Peirce's pragmatic criticism of Descartes does not remove principles, but instead enshrines them:

We cannot begin with complete doubt. We must begin with all the prejudices which we actually have when we enter upon the study of philosophy. … A person may, it is true, in the course of his studies, find reason to doubt what he began by believing; but in that case he doubts because he has a positive reason for it, and not on account of the Cartesian maxim. Let us not pretend to doubt in philosophy what we do not doubt in our hearts.

In a sense, you could say that Peirce is arguing for more principles, and that as we go about our studies, if we have to undo some of our principles along the way, so be it. Peirce's principles then wouldn't be "true" principles, but "working" principles.

So, if anything has happened to the discourse of principles along the way, it is that we have decided that two extremes -- 1) We can conclusively determine self-evident principles and 2) We cannot assume any principles -- are not useful. We can assume certain principles for the sake of a foundation, and then later on if we need to revise or reject a principle, we can revisit it at that time.

Reference: Cogito ergo sum - Wikipedia

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Principles, or arche is what Aristotle in line with his predecessors take as the principles by which Nature is understood (this is to be contrasted with his own concept, which is nature or entelechy).

Some are prior to others; so there are the simplest such, in that they cannot be reduced to any more basic; he says that there is more than one and less than three.

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