You ask: ‘How would you restate the axiom to logically say the same thing as only individual acts?’
Let’s take the axiom in the form in which Rothbard gives it (as in my previous posting) so that ‘Man acts’ =df.‘All human beings act by virtue of their existence and their nature as human beings’ Let’s put this in proper form. Where H: human beings; and T: beings who act by virtue of their existence and their nature as human beings. The proposition then is HAT
If we play around with this by eduction, we find this is equivalent to (saying the same thing as):
HET(complement); No human beings are beings who do not act by virtue of their existence and their nature as human beings
T(complement)EH; No beings who do not act by virtue of their existence and their nature as human beings are human beings
T(complement)AH(complement): All beings who do not act by virtue of their existence and their nature as human beings are non-human beings
If we use the square of opposition, we can deduce from HAT that
HET (No human beings are beings who act by virtue of their existence and their nature as human beings) is false
HIT (Some human beings are beings who act by virtue of their existence and their nature as human beings) is true, and
HOT (Some human beings are not beings who act by virtue of their existence and their nature as human beings) is false.
That’s the sum total of equivalences or implications immediately derivable logically from HAT. As you can see, none of these say that ‘Only individuals act’
You write: ‘Also, when you say we must engage in conceptual analysis, then does that mean that we can’t deduce these implications mediately, through the use of a categorical or hypothetical syllogisms, from the action axiom?
You cannot derive any proposition mediately from a single proposition
You write: ‘And we need to use our basic intuitions to understand the implications of action instead?’
Broadly speaking, yes. We ask—what does it mean to say that human being acts? To answer this, we distinguish between, say, sneezing, which isn’t a human act, as distinct, say, from scratching. What makes scratching to be an act? Well, it is something I choose to do rather than something that simply happens to me; it is something that I do for a purpose (to relieve the itching); the relief of the itching is a future state not present now (if it were, I wouldn’t need to scratch). Putting it with a little more formality, we can see that ‘Action is purposeful behavior directed towards the attainment of ends in some future period involving the fulfilment of wants otherwise remaining unsatisfied and involves the expectation of a less imperfectly satisfied state as a result of the action.’
This explication of the concept of action will get you #2 and #4 on your list: action takes time, and action is trying to change an existing situation into a more satisfactory situation.
What about #1: ‘Only individuals act’?
This takes a bit more work. All I can do here is to sketch the broad outlines of how this might be done. We might ask what is being rejected in making this claim? Well, presumably, something along the lines of the proposition that ‘Non-individuals act’, non-individuals being sub-individuals (parts of a human being) or super-individuals (groups, gangs, societies, etc.) We needn’t delay too long on the sub-individual proposition (I don’t know too many who would make this claim). What is effectively being rejected by the claim that ‘Only individuals act’ is any doctrine of group realism which would make a group to be an agent in precisely the same way as ( or in a better way than) an individual. The strategy here has to be to argue that what appears to be group action is, in the end, reducible to and explicable in terms of the action of individuals.
I hope this helps.