If we aren't careful, we might find that everything is one or the other, and the distinction will mean nothing. If everything is subjective or everything is objective, what good is the distinction? What are we going to use the distinction for? It would be a pointless exercise.
What we are looking for is a distinction under which some things are objective and everything else is subjective.
First, let's admit that everything we know is known through our own faculties of reason and sensation, both of which are known to be imperfect. If susceptibility to such imperfection defines the subjective, then everything would be subjective (so we rule out that definition).
Second, let's also admit that the subjective nature of our individual faculties might be a matter of objective fact. So, if the distinction is to be made, I think we ought to be able to make it whether or not every fact in the universe could be objectively known.
I think that these two admissions lead me to propose the following definition:
An attribute of a thing is subjective when it cannot be determined that the attribute is a feature of the thing itself rather than a consequence of our mental faculties.Note that, under this definition, if we don't have evidence of the faculty-independence of an attribute, we regard that attribute as subjective.
So, what constitutes evidence of faculty-independence?
I think that evidence of faculty-independence comes in the form of high-precision, alternative comparators. To explain, let's look at why physical mass is regarded as objective.
Technically, humans aren't sensitive to mass, but to force. So we're asking, is "how heavy a thing feels" just a subjective perception? In absolute terms, weight is subjective - a small child will find a 100 pound weight immovable, but an average adult won't. However, what we're really asking is whether relative weight is a measure of something about the weighed objects, or whether it is just about how we are perceiving them.
The way to test this is to find something other than personal sensation of force to use as a comparator. A balance scale, for example. Based on how comparatively heavy two objects feel, you can pretty accurately predict which way the scale will tip when the scale is used as a comparator. You can also do the reverse, and, based on the scale, predict which will feel heavier. You can do this independently of the visual size, shape or composition of the objects.
I think a lack of an acceptable external comparator is what makes a field subjective. If I have two rooms, one containing an ugly person, and another containing a beautiful person, there's no 'beauty-meter' I can poke through the keyholes to tell me which room is which. If I could build such a device, it could work only by emulating me (e.g., through training against my tastes).
Likewise, there's no justice machine that tells us whether one act is more just than another. Any such machine would have to be trained or conditioned against our tastes (e.g., through the use of juries and legal precedent).
Can another person act as an external comparator? I think the answer is "no," and that precision is the vital clue here.
In objective fields, human-trained machines can outperform their trainers. For example, based on my experience with some 1-pound weights, I can devise a machine (such as a balance scale) that can compare two masses weighing hundreds of pounds. More importantly, this machine can be far more sensitive than me when it comes to weighing very small masses. So, I can use the machine to extrapolate beyond my senses and get higher precision, and verify that this sensitivity is real. I can weigh all the grains of rice in a sack, and verify that the heaviest 25% of the grains weigh noticeably more than the lightest 25% of the grains. I can verify that the machine is better than I am at weighing things.
In contrast, human-trained machines can't outperform their trainers in subjective fields. I don't think that a committee or a technical analysis is a more precise judge of beauty or gastronomic taste than the "best" human judge. Likewise, I don't think that the justice system or a religious authority has demonstrated higher moral precision than a good individual human. I think this is a clue that our inventions in these spaces are mere approximations to our subjective feelings.
Similarly, in a subjective field, I cannot verify that another person can work better as external comparator. Suppose you wish to use your friend Plato as an external comparator for the beauty of women. Suppose that 90% of the time, Plato agrees with you. When Plato disagrees, how will you settle the argument? There's no mechanism you can use to verify that Plato is right or wrong.
The same goes for morality. No other person can act as a verifiable external comparator. If God is taken to be an external comparator, then not only must you have faith in his existence, but you must also have (double) faith that he is an accurate external comparator because you certainly can't verify that his is right and you are wrong.
Based on my definition, we are forced to conclude that morality and aesthetics are subjective, whereas mathematics and physics are objective.
Can we find an alternate definition that will hold morality objective, but maintain the subjective-objective distinction? Well, I can't rule out that possibility, but I can't think of a viable alternative. This is because I think that one can't claim objectivity of an attribute if it's observer-dependent, so the tests I have outlined appear to be minimum requirements of objectivity.