If logic is constructed from axioms, and axioms are depended on observation which in term could be subjective, does this means that logic could be limited to our observation, and not really absolute and fundamental?
Many have argued that logic is empirical, or as you describe it, logic's "axioms are dependent on observation".
Quine, in his paper "Two Dogmas of Empiricism" questioned the analytic-synthetic distinction, and suggested that even analytic propositions were dependent on empirical evidence. Since the rules of logic were analytic propositions par excellence, they too, were ultimately dependent on empirical data, and were not absolute laws.
Birkhoff and Von Neuman proposed in the 1930s that the paradoxes of Quantum Mechanics can be explained if we abandoned classical logic and used some form of Quantum logic instead. Such a Quantum logic would change or abandon all together some of the rules of classical logic, and would be a perfect case of logical axioms arrived at by observation.
Hilary Putnam discussed this in depth in his paper "Is Logic Empirical?", later republished as "The Logic of Quantum Mechanics.". In it he argued that, just as empirical physical results - relativity - forced us to abandon Euclidean geometry, so it is possible that the results of quantum mechanics will force us to abandon classical logic.
Although Quantum logic is still an active field of study up to the present day, it is does not get much attention from most philosophers and had been abandoned completely by physicists. Those who do study the topic view it mainly as a mathematical tool for studying Quantum phenomena, not as some sort of fundamental logic to replace our current classical rules of logic.
The main problem that is faced by Quantum logic (or any such radical revision of logic, empirically justified or otherwise), is that we tend to think and communicate in classical logic. It would be very difficult, or in Kantian fashion, outright impossible for us, to perceive and discuss the world in anything other than classical logic - it seems to be hardwired into our brains. Although the logical atomist program failed as a metaphysical theory, it did show us just how ingrained classical logic is into our linguistic and mental structure. As Wittgenstein stated, the limits of language are the limits of the world: No one can place themselves outside of logic, and then pick among different logics to reason and argue with, even if those alternative logics are justified.
Those non-classical logics which have been successful (fuzzy logic, modal logic, intuitionistic logic) are those that extend classical logic, as opposed to replacing it, or at least respect classical truth tables in the limiting case.
As an after thought, one of my favorite Sci-fi short stories discusses the idea that while logic is indeed subjective, we learn classical logic at a very young age and once we grow into adults, we are incapable of unlearning it. If we were to somehow come across non-classical logics at a very young age, we would be capable of all sorts of superhuman feats. The story is of course, just sci-fi, but I do find the idea compelling.
No, logic is not subjective.
In all mathematical theories all experts obtain consensus about the validity of the theorems of an axiomatized theory. But the theories differ and sometimes the Pros/Cons of a theory are debated. E.g., there are two-valued logics and many-valued logic and fuzzy logic and etc. It's not the question about the validitiy of the theorems of the theory. At most it is a question which axioms should be taken as beginning.
It is an insight of the last 200 years that axioms cannot be derived from any foregoing insights, neither intuitive insights nor from results of scientific discoveries. Which axioms to choose may depend on observation. But axioms do not derive from observation like theorems derive from axioms.
Instead it is often a question of usefulness which axioms should be chosen to develop a mathematical theory which fits as the foundation of a scientific theory. E.g., there was a discussion whether classical 2-valued logic is suitable for the interpretation of measurements on the quantum level.
Logic is not fundamental in the sense that there is excatly one calculus of logic. But it is fundamental in the sense that every rational argumentation, in particular any scientific theory, presupposes a certain calculus of logic.
While the answers above pretty much cover it, the question goes so deep into philosophy that it can be seen from many angles, especially in respect to the meaning of subjectivity itself.While I hesitate to say this, and may get brutally corrected, I would argue that Kant can be interpreted as saying that: Subjectivity itself is a logical system.
As has been pointed out, we can have different logical systems. We can drop Euclid's fifth postulate and generate a different, perfectly coherent logical system. The various systems cannot, so it seems, be reduced to one another. So it might seem that the "subjective" aspect is the choice of axioms. The "subject" can stand, so to speak, inside or outside any given system, choosing axioms.
But things are not quite so simple. What separates all these systems, making them irreducible to one another? The various choices of axioms and applications,... hence subjectivity? Yet again, what makes them all "logical" systems? Something beyond subjectivity? Something common to all possible subjectivities?
This is where Kant's transcendental approach might shine some light. We can think of any individual subject freely "choosing" axioms. Yet such "subjective" intervention is quite strictly limited by acceptance of its "logic" to other subjects, or it is simply a coherent madness... a paranoia.
In truth, we do not have such a thing as "individual subjects" or purely singular homo sapiens. We have "subjectivity," as a kind of evolving, punctuated continuum or discontinuous identity. Which is not unlike Logos evolving and differentiating along axiomatic boundaries or niches into various "logical systems."
So we may choose axioms "subjectively" and operate "inside" or "outside" different logical systems. Meanwhile, these Logoi grow and evolve. They outgrow their own axioms and their own "final conclusions" or "self-proofs." If they were to become "closed" systems they become purely tautological and die. So the systems themselves begin to sound not entirely analytic, but subjective or perhaps "synthetic a priori."
Now, the Kantian subject. Any given subject, such as they are, may choose axioms. But can they "subjectively" choose their way out of all logical systems? Were they to do so, they would disintegrate or close themselves up in a paranoia. They would, in effect, lose their very subjectivity. So, in some sense, "subjectivity" is what all these logical systems have in common... and what subjectivity itself presupposes. And this would be that categorical structure of reason, freedom, and morals proposed by Kant.
The difference is that we cannot identify or reason about some noumenal "axioms" of this metasystem. We are forever "inside" its relational structure. This then is the open system that generates axioms or, we might say, from which axioms are derived. So the answer is: Yes, logical systems are subjective, but subjectivity and sensibility are in turn constrained to a logical structure.
I would argue that logic and mathematics really are subjective, but only at a species level (or to the degree they are simply wrong.)
The way that axioms are dependent upon observation is not the same way that scientific principles or other facts are dependent. Axioms are not so much "discovered", or "worked out", but (as the name means in Greek) "found worthy", by being easily called forth in another person's mind, and appealing to them at a deep, intuitive level. It does not matter whether they occur in external reality because they exist in internal reality.
The purpose of disciplines like this is to isolate what understanding is common across all different ranges of experience. They do so by appealing to intuitive response, and the emotion of 'clarity'. They refine intuition for communicability, but they rely upon the intuition itself as backing. Since the only intuition we can interrogate is our own, we can only determine what is shared among those with whom we can communicate.
What is found to differ in a meaningful way between people is assiduously and ruthlessly removed from these subjects. The parts of language and processing that are more linked to the environment are purposively pushed out of logic into grammar, linguistics and philology, and ultimately to psychology, and the corresponding elements of potential imaginary models are pushed out of mathematics into the other sciences and engineering disciplines.
So logic strives to be non-subjective, but there is no way to test the relative subjectivity except between humans, and we can therefore never be sure.