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Same way as I can say something is obviously a heap of sand without pointing out exactly at which point it turned into one from it's beginnings as some grains of sand (Sorites paradox)But that's not the situation here. You're saying that there is some scale at which something is "obviously wrong" implying a convergent consensus of observance. Tying it to sorites, it would be like you claiming that there's a fuzziness beyond which the heap should be considered "obviously too big" by everyone, while to others there's no such concept as a heap that's too big (or even merely disagree that you've entered "too big" territory yet). You can't argue that it's too big, without pulling in something external that they might share.
> Regarding moral relativism, I don't buy it
It's not that morals are "relative", and I agree that's a terrible label, but more that morals are held as axiomatic. There is no and cannot be justification of morals themselves, they simply are held (or are not held) by a person as the foundational root clauses from which all their other justifications are derived.
So there's no possible way to argue or justify root morals to others who simply don't share them. You'd need to link them to morals that the other does hold for there to be any worthwhile concept conveyed at all. Otherwise it's just yelling that others are not following your particular flying spaghetti monster. If one fundamentally believes that animal suffering matters, and another fundamentally believes that it does not, what possible justification or persuasion would cross that boundary?
> You are right that pure moral arguments don't change anything, but I don't think the reason is that people don't share morals, but rather have to do with people being extremely good at rationalizing away the immorality, or never questioning it in the first place.
Re-read what you said, and consider it in the initial discussion of animal rights. You're saying that there is a morality held in a person that is invoked in an observed situation that is being rationalized away, implying that they fundamentally subscribe to the same morals as you. I think this convergence (and I'd simply call it a projection) is an extreme claim, going against the observed variety among cultures & schools of thought around the world and throughout history. That's simply not an accurate picture of humanity.
In particular, sensitivity and callousness to specific situations of harm are very trained, and degrees of harm in varied directions exist in absolutely everything, all the way down to entropy. Just look at governmental overreach to see how the concept of "harm" can be all encompassing, bringing a justification for involvement wherever they may choose, from economics to surveillance.