All of which brings us to the issue that I am most curious to hear your opinion on. Given the above, can you see any reason that we need to postulate the existence of an "objective reality" before we can arrive at reliable truths, assuming "truth" is defined according to the Correspondence Theory? I would love to hear your thoughts on that!
Very interesting points there! I'll throw out a bit of a warning straight off the bat that I'm a bit out of my depth here. I'll do my best to answer your questions but please feel free to correct me if I've misinterpreted what you're saying or if I make mistakes in my answer. This is a field in which I'm very much a student and not a master!
I definitely think that the more people there are to verify something, the more reliable that claim can be. However, that still leaves the potential for some really bizarre human error. Are you familiar with the Mandela effect? It's essentially collective false memory, wherein a great many people (sometimes thousands) remember a particular event or fact which doesn't actually correspond to reality. For example, the Monopoly man does
not have a monocle, despite a lot of people apparently being certain that he does.
Some people have argued that the Mandela effect is proof of alternative realities. Let's ignore that side of things for the time-being and just work on the assumption that this demonstrates the capacity for large numbers of people to be certain of something that just isn't the case. Using your cat example, a great number of people may see the cat and agree that it exists. However, further down the line they may distinctly remember seeing a dog instead. So the validity of inter-subjective verification for establishing objective truth may also be subject to a sort of decay. In other words, it becomes viable for saying "there is a cat" but much more suspect for saying "there was a cat."
So far, that only suggests that arriving at truths without asserting the existence of objective reality may have an expiration date. It doesn't say that it
can't be done. Going by the correspondence theory you suggested, I don't see why we couldn't assume that we can establish truth for at least a time.
The next bit is mostly waffle and food for thought that you may or may not find interesting.
Now I suggested two axioms for an objective reality. One being that there is a world beyond our perception of it. That's for the sake of the brain in the jar line of reasoning. The second is that others can observe that objective world in a manner roughly similar to oneself (the "roughly" in that line is to accommodate colour blindness and the like). The reason for this one is because of the possibility that some, or all, other people are actually zombies. That is to say, people who look and behave exactly as a conscious being would, without actually being conscious themselves.
It's a similar thought experiment to the old "brain in the jar" in many ways. I bring it up because it's at least arguable that all those people you bring in to see the cat aren't
actually seeing the cat, they're not really conscious. Now I don't know how seriously it's worth taking the zombie theory, it strikes me as something entertaining to contemplate (in an uncanny horror sort of way!) but could be a cop out in discussion. If, for the sake of argument, we do take it seriously, then I would think establishing a couple of axioms would be necessary to then establish reliable truth.
Now, my question would be whether or not establishing axioms by default also postulates the existence of an objective reality? Alternatively, would it simply be a decision based in pragmatism with no need or desire to postulate an objective reality?