Within Aristotelian Logic, there are two types of inference (deduction): immediate (directly from one proposition to another) or mediate (from two propositions to a third).
Since we start with a single axiom—man acts—mediate inference is not relevant. Immediate inference takes you from a proposition to (a) another proposition that is semantically identical to it, or (b) to a more limited or restricted version of the semantic content of the original proposition.
So, SAP takes you to SEP(complement), by obversion. SAP and SEP (complement) say exactly the same thing, in different linguistic ways.
SAP also takes you to SIP (by alternation). There are not identical propositions but SIP is a more limited version of SAP.
The Square of Opposition also allows us to make certain inferences from a single proposition directly to another proposition. So, from the truth of SAP, we can deduce the falsity of SEP (by the rule of contraries)
Now, if we take the proposition—‘man acts’ or ‘persons act’—then we can deduce certain other propositions from it, such as ‘it is not the case that man does not act’ but we cannot deduce from it by any standard rules of logic any of 1-13 below, with the possible exception of 1, if 1 is taken to be simply a restatement of our axiom.
[Now follows your list of putative implications of the action axiom
axiom: persons act.
The implications, which are reflections based on the premise that person’s act. (at least the first 7 anyway).
1. Only individuals act
2. Action Takes Time
3. The future in uncertain (not completely random, but not completely determined).
4. Action is trying to change an existing situation into a more satisfactory situation
5. Means are Scarce
6. Means are allocated
7. Means are economized (different from being allocated because this implication gives a criteria for allocating, which is allocating based on higher-valued ends and using lesser valued means to achieve these higher-valued ends).
8. Value is subjective.
9. Cost is subjective
10. Profit is subjective
11. Value is imputed to means (imputed according to what ends they will serve).
12. The Laws of Utility (First is the diminishing law of marginal utility, and the second is the law of total utility).
13. The allocation of consumer goods (allocate to get the highest marginal utility).
These are the most fundamental implications of personal and interpersonal action, with 12 and 13 being implications in regards to interpersonal action]
What we can do (in this connection (see the last paragraph of my self-citation below) is to engage in the process of conceptual analysis, that is, to ask what action is. If action is a transition from one state to another then we can conclude that it takes time. And so on. This process of conceptual elucidation is perfectly rational but it is not strictly logical either (in the sense of making use of explicitly formal rules of logical deduction) but it’s not il-logical either.
Here is an excerpt from my Murray Rothbard (Major Conservative and Libertarian Thinkers), Bloomsbury 2013 (2010), pp. 30-32 in which I deal with this issue to a certain extent.
‘The theoretical aspect of the field of the most developed branch of praxeology, economics, is constituted for Rothbard by one fundamental axiom and a few broadly empirical subsidiary postulates. The fundamental axiom is ‘Man acts’: ‘All human beings act by virtue of their existence and their nature as human beings’ (Rothbard 2004, 2). 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. The objects of human action and the human actions directed towards them are at once manifold and varied and yet ordered or at least capable of being ordered. If the objects of human action are so orderable, so too should the human actions directed towards them. There are many particular goods that can be chosen by us and yet it is important to us – that is, it is itself another good – that the selection of particular goods should be such that they do not clash with one another and cancel one another out. The good is sought in every limited and particular good and yet no particularized good can exhaustively express or contain it. There are always more and other goods necessarily excluded by our particular choices. An agent chooses means from his environment to achieve his ends and economises his means by directing them, in ways he deems (accurately or not) to be appropriate, towards his most valued ends. So, from the seemingly obvious and uninteresting axiom that ‘Man acts’, the most surprising consequences follow. According to Rothbard, ‘Some of the immediate logical implications that follow from this premise are: the means-ends relationship, the time-structure of production, time-preference, the law of diminishing marginal utility, the law of optimum returns, etc.’ (Rothbard 1997c, 104).
‘Why is the action axiom an axiom? For Rothbard, it is simply inconceivable that human beings should exist but not act. Attempts to deny this axiom are themselves instances of human action and so self-referentially incoherent. ‘[I]f a man cannot affirm a proposition without employing its negation, he is not only caught in an inextricable self-contradiction; he is conceding to the negation the status of an axiom’ (Rothbard 1997d, 6; see Rothbard 2004, 2). Rothbard appears to distance himself from Mises in his characterization of the action axiom. He regards Mises as defining the action axiom in neo-Kantian terms as a law of thought, whereas he himself, as an adherent of the epistemological- realist tradition of Aristotle and St Thomas, regards it as a law of reality. In the end, Rothbard betrays a certain impatience with the question of the appropriate labelling of the action axiom, holding that ‘the all-important fact is that the axiom is self-evidently true, self-evident to a far greater and broader extent than the other postulates. For this Axiom is true for all human beings, everywhere, at any time, and could not even conceivably be violated’ (Rothbard 1997c, 105). However one labels the action, it is ‘a law of reality that is not conceivably falsifiable, and yet is empirically meaningful and true’ (Rothbard 1997c, 105). Moreover, ‘it rests on universal inner experience, and not simply on external experience, that is, its evidence is reflective rather than physical . . . [and] . . . it is clearly a priori to complex historical events’ (Rothbard 1997c, 105–6; see also Rothbard 1997f, 32–3).10
‘The subsidiary postulates that are to be employed in the theoretical structuring of economics are few in number and even though empirical, functionally unfalsifiable. They are (1) the variety of human and natural resources (this gives rise to the division of labour and the market), and (2) leisure as a consumer good. Rothbard believes that only these two postulates are actually needed to complete the essential framework of theoretical economics. From the fundamental axiom and the two subsidiary postulates, economics can deduce the essential elements of Crusoe-economics, barter, and the monetary economy. ‘Once it is demonstrated that human action is a necessary attribute of the existence of human beings, the rest of praxeology (and its subdivision, economic theory) consists of the elaboration of the logical implications of the concept of action’ (Rothbard 2004, 72). He adds, a little later, that ‘Praxeology asserts the action axiom as true, and from this together with a few empirical axioms – such as the existence of a variety of resources and individuals) [sic ] are deduced, by the rules of logical inference, all the propositions of economics . . .’ (Rothbard 2004, 75).
‘Praxeology is presented by Mises and Rothbard as an axiomatic system with one axiom (the action axiom) providing its foundation and with ordinary verbal logic providing its rules of derivation. But it is difficult to see how anything can be validly deduced by standard logical means from a single axiom that isn’t that axiom stated in another way or simply a more restricted version of that axiom. Rather than think of praxeology as a strict axiomatic system on the model of Euclidean geometry, it might be both more insightful and give fewer hostages to fortune to conceive of it as the systematic conceptual exploration of a web of concepts that mutually imply one another, with the concept of human action ranking first among equals.’
I’m not particularly happy with this formulation but I haven’t been able to work out anything more satisfactory. I know that Dr Peter Preusse has also done some work in this area and his writings might be worth looking at.