I think moral values are demonstrably subjective. Here is my simple argument:
1. For something to be morally valuable is for it to be being valued.
2. Only a subject can value something
3. Therefore, for something to be morally valuable is for it to be being valued by a subject. — Bartricks
To present your argument, this argument needs to be reworded, as it is rather sloppy as presented.
1. For something to be morally valuable, that something needs to be morally valued.
2. Only a subject can value something.
3. Therefore, for something to be morally valuable, that something must be morally valued by a subject.
(Reason for improvement in wording: A spoon can be valued at 44 cents, so valuing in and by itself is not sufficient. It needs to be valued MORALLY.)
You say the premise and the logic are infallible, so your argument is infallible.
But they are not. Placing a moral value on something by a subject could be universal, that is, absolute or else the placing could be subjective. The stipulation in 1 and 2 do not exclude the possibility that the moral value the subject places on something is universal. NOT every subject must place a moral value on a universal moral value; it is sufficient if only some subjects place a moral value on a universally valuable morality.
Think of it this way: Two blind chickens are pecking at the ground, picking up small pebbles, hoping that one of the pebbles will be an edible piece of grain. One will find grain pieces repeatedly, the other will never find the grain. (Say, within a time period of an hour.) Do the grain exist, and its nutritional value exist? Yes. Does the one chicken that finds it see it justified to believe that the grain and its nutritional value exist? Yes. Does the second one find that? No. Yet the grain exists.
Two people valuate their actions for morals. Neither of them is on the opinion that universal morals exist. They just blindly, haphazardly keep changing behaviour, trying to find the ultimately absolute moral behaviour. One repeatedly finds this behaviour; he will feel good, he will feel that he acted morally. The other never finds this behaviour. He will not have the experience of feeling good due to moral behaviour. But absolute morality, despite not having been found by the second subject, exists.
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P.s. I don't believe in absolute morals. But I am also on the opinion that your argument is not fool proof or irrefutable, as I have shown it above.