Talk of "self-evident" truths, sometimes called "axioms" (though "axiom" now carries mainly the sense of "not inferred," regardless of the attendant justification), was equated by Thomas Aquinas with what Immanuel Kant would call "analytical" knowledge:
(Aquinas): A thing can be self-evident in either of two ways: on the one hand, self-evident in itself, though not to us; on the other, self-evident in itself, and to us. A proposition is self-evident because the predicate is included in the essence of the subject, as "Man is an animal," for animal is contained in the essence of man. If, therefore the essence of the predicate and subject be known to all, the proposition will be self-evident to all; as is clear with regard to the first principles of demonstration, the terms of which are common things that no one is ignorant of, such as being and non-being, whole and part, and such like. If, however, there are some to whom the essence of the predicate and subject is unknown, the proposition will be self-evident in itself, but not to those who do not know the meaning of the predicate and subject of the proposition. Therefore, it happens, as Boethius says (Hebdom., the title of which is: "Whether all that is, is good"), "that there are some mental concepts self-evident only to the learned, as that incorporeal substances are not in space."
(Kant): ... the predicate B belongs to the subject A, as somewhat which is contained (though covertly) in the conception A ... Analytical judgments (affirmative) are therefore those in which the connection of the predicate with the subject is cogitated through identity ...
One domain of discourse where the allure of foundationalism remains fairly strong is set theory. Though a transfoundational vantage is offered in the now-dominant multiverse superdomain, here, there is still a reputable school of set theorists who are far less commital to that superdomain. Directly or indirectly (I don't recall), Peter Koellner is one of these. But so he offers to reformulate talk of self-evidence with the general relation more-evident-than:
A popular view is that axioms are self-evident truths concerning their domain. The difficulty with this view is that there is wide disagreement in the foundations of mathematics as to which statements are self-evident and if we restrict ourselves to the intersection of the statements that all mathematicians would regard as self-evident then the result will be quite limited in reach, perhaps coinciding with Q [a weak version of arithmetic] or slightly more. ... For these reasons we will not employ the notion of self-evidence.
There is, however, a related notion that we shall employ, namely, the notion of one statement being more evident than another. In contrast to the case of self-evidence, there is generally wide agreement (in the cases that concern us) on the question of what is more evident than what.
The Aquinas/Kant picture is connected to the theory of sense-data as the referents of terms, in that the idea is to find the justification of a "self-evident" assertion in the "essential meaning of" its component terms. This subpicture faces a syntactical objection:
BonJour was quite aware that some classical foundationalists would attempt to avoid the regress problem raised in the previous section by appeal to the idea that some things are simply “given” to us, or that we directly “apprehend” or are “acquainted with” them. Inspired by Sellars (1963), BonJour (1978, 1985: ch. 4) presented the following objection to classical foundationalism, often referred to as the “Sellarsian Dilemma”. Does the awareness or acquaintance that is the alleged source of noninferential justification involve the acceptance of a proposition or thought, or at least the categorization of some sensory item or the application of some concept to experience? If, on the one hand, the acquaintance or awareness is propositional or conceptual in this way, then while such acts or episodes of awareness seem capable, in principle, of justifying other beliefs, they would surely need to be justified themselves. The episode of awareness would involve something like the acceptance of a proposition, or the categorization of experience, and such an attitude or act clearly needs justification if it is to justify anything else. But then, the allegedly foundational belief is not foundational after all. If, on the other hand, we can regard direct awareness as nonpropositional and nonconceptual, then while these acts or states of awareness do not require or even admit of justification they also don’t seem capable on their own of providing a reason or justification for propositional items like beliefs. Therefore, the classical foundationalist’s acquaintance or direct awareness cannot serve as a foundational source of knowledge or justified belief.
Now, using a graph-theoretical approach to epistemic justification, we might represent the self-justified justification of other propositions as an initial node with a subsequent node to which it attaches, the edge being the "is justified by" relation. Then the initial node is like an "element of itself" (the node has an edge that loops back to the node itself, as well as an outgoing edge to other possible theorems). In some sense, this is how logic systems with {A → A} as a valid inferential expression work. On the other hand, dropping the autoset picture from the equation, we can observe that though the initial node would not enter into the "is justified by" relation, yet we should parse that relation further, into "is propositionally justified by" and "is subpropositionally justified by," in which case the epistemic graph will expand to include nodes "prior to" the axiomatic one, which added nodes enter into the subpropositional justification relation with the axiom-node.
An even more peculiar option is to take a cofounded hypergraph (the Wikipedia article on hypergraphs goes over this topic, which material was cited on the MathSE in an answer to a question about nodeless but edgeful hypergraphs; I can't find the exact link right now but I'll edit this post if I can find it) so that you have two free-floating edges representing the "is justified by" relation folding in on itself, so to say, without being exhibited in an axiomatic form/state. One might interpret this "object" as an imperative force in the axiomatic domain (which force's existence is recognized by most everyone in the use of premises in imperative form, e.g., "Let A = B," or, "Assume that..."). It is self-justifying by being justification-in-itself, so to say, and again is cofounded by nature, but this might be an intermediary option between (i) propositional self-justification and (ii) propositionally non-justified axioms. On the other hand, if these free-floating edges do not attach to any theorematic nodes "elsewhere" in the universe of epistemic graphs, one wonders about the utility of the imagery.
Specific questions of derivations from an empty vocabulary or an empty sequent have, incidentally, some more or less canonical answers. The SEP article on second-order logic at one point reads:
This shows that second-order logic is not completely axiomatizable by effective means, or decidable even in the empty vocabulary.
The article on proof theory mentions an empty sequence and empty sequent:
The empty sequence will be denoted by ∅ ... A contradiction, i.e., the empty sequent ∅ ⇒ ∅, is not provable.
But to prove (on the second-order level) that something is not first-order provable, is still to prove something; as the SEP article on justification logic puts it:
J0 is the logic of general (not necessarily factive) justifications for an absolutely skeptical agent for whom no formula is provably justified, i.e., J0 does not derive t:F for any t and F. Such an agent is, however, capable of drawing relative justification conclusions of the form
If x:A, y:B, …, z:C hold, then t:F.
The same article elsewhere reads: "The Logical Awareness principle states that logical axioms are justified ex officio: an agent accepts logical axioms as justified (including the ones concerning justifications)." There's also a section on self-referential justifications.
ADDENDUM: see also "A 'paradox' of foundationalism? on this very SE for another higher-order question about how axioms are justified (c.f. a sort of mirror problem faced by coherentisms based on conformity to axiomatic probability theory). One might say that axioms are justified by the regress as a whole, in that once we regress far enough, we do seem to run into elementary distinctions (such as that between intensionality and extensionality, which also (roughly) goes by names like connotation/denotation, sense/reference, or discursion/intuition).
Finally, I would recommend reading Charles Parsons' "Reason and Intuition" for some more particular analysis on talk of self-evident truth vs. propositionally ungrounded/an empty grounding of, but justifying (of other propositions), axioms. (See also about Wittgenstein's phrase "hinge propositions.")
Corollary: consider differences in problem-solving in pretheoretical algebra. Say you are given the equation, "2x = 4," to solve. This can be solved for just using the given numbers. In fact, the initial presentation of the equation can be interpreted as the question, "If 2x = 4, then what does x by itself equal?" with the answer being, "If 2x = 4, then x = 2." And so now it seems as though the answer is justified, but is it justified by inference? That depends on whether dividing both sides of the equation by 2 is considered an act of inference. It is discursive, to be sure, at least if we take the distinction between discursive and nondiscursive judgment to be that between indirect and direct apprehension: since here we have a proposition/question in a sequence with an intermediary syntactic moment (the act of division) and then an assertoric stage after that. And the act of division is a "proposition" of its own, so to say.
Anyway, one might take, "If 2x = 4, then x = 2," to be self-justified, or justified by its "parts" (having pre-propositional justification and entering into a justifying relation for other propositions). But now what about the pretheoretically greater difficulty involved in solving an equation where it seems like you have to expand the base expressions, often in perhaps convoluted ways, before you can start getting at a resolute collapse? Again, is this a matter of inferential justification, or noninferential but propositional justification, or pre-propositional justification, or whatever else along these lines?
Or consider defining the square root of -1 as i. "What is the square root of -1?" is the question, and we could parse it as {x = √-1?}. But setting x to i looks like shuffling from one letter to another more than shuffling from a question to a substantial answer, unless the similarity between the letter "i" and the numeral "1" manages to inspire considerations of the complex plane and the quaternion sphere, etc. But then is {√-1 = i} axiomatic, inferred from itself, pre-axiomatic, or...?
