David: Let’s take it that your argument is valid as it stands. That being so, if the premises are true, the conclusion would have to be true also. The critical premise, as you note, is premise 3. This, in one form or another, is at the heart of what has come to be called the Perverted Faculty Argument (henceforth PFA).
It’s clear that Natural Law (NL) theorists are committed to a teleological account of ethics. Put very simply, human beings are certain kinds of beings (not just formless masses of cellular material) and we can give a broad account of those activities that genuinely contribute to human flourishing and those that don’t. So, a life lived without friends is, in general, not one that could be said to be flourishing. So too a life with a near complete absence of physiological necessities – foods, drink. However, even if all this is so, there is no single prescription for the good human life.
Now all NL theorists invest human nature with a normative dimension. Earlier NL theorists tend to make use of the PFA in some form or another. However the difficulties with this arguments are well known. If it applies to sexual functions, does it apply to all other functions? If not, why not? What’s so special about sexuality? A hammer may have as its function the driving of nails but there doesn’t seem to be anything essentially problematic about using it to prepare steak for frying. And so on, and so on. The scholarly journals and the web are replete with discussions and examples.
The New NL theorists have attempted to give an account of NL theory that is not open to the usual objections but, as much criticism of their position has indicated, it’s often not clear to what extent they retain a conception of human nature as normative. If you liberate ethics completely from the teleology of our embodied condition, it seems as if there is nothing to prevent us from characterising our acts in any way that we choose. (I’m not saying, by that way, that this criticism is correct or that the New NL theorists don’t have a response….)
Despite all this, many people, myself included, retain a core intuition that as we are essentially embodied creatures, the teleology of our physical natures cannot but be relevant to the fulfilment of our ends and hence ethically relevant.
In my earlier academic life, I published an analysis of St Thomas’s account of the morality of action in the context of lying which pointed to some difficulties in that account. You can find this at: http://www.ucd.ie/philosophy/staff/gerardcasey/casey/unityinStT.pdf
Some years later, I published another piece, again analysing ethics from a broadly teleological perspective. You can find this at: http://www.ucd.ie/philosophy/staff/gerardcasey/casey/EthHumNat.pdf
So, where does that leave us or, more specifically, me? The PFA, while it gives expression to the basic intuition that our embodied condition and its intrinsic teleology is ethically significant, does so in a way that leaves it open to strong, perhaps insuperable, objections. The New NL theory isn’t open to these particular objections but has its own problems. If all this is so, can any reasonable case be made for the preservation of traditional sexual ethics that escapes the PFA objections and what tradition NL theories see as the limitations of the new NL theory? [I should say here that I have the utmost respect for those, such as Edward Feser and Janet Smyth, who have attempted to give reasoned accounts that are, broadly speaking, within the traditional NL camp.]
Short answer – I don’t know, but it’s worth trying. Here you can find someone attempting to do just this. https://anscombe.stanford.edu/sites/default/files/A%20Conjugal%20Theory%20of%20Sexual%20Morality%20%5BFinal%5D.pdf
I don’t necessarily endorse this particular argument but I simply offer it to you as an example of where the discussion has moved to.
I realise that you may well find all that I have said most unsatisfactory. I don’t find it particularly satisfactory myself and would like to be able to offer you a knock-down drag-out argument that settled everything conclusively. However, it should be remembered that the conclusion of an invalid or unsound argument isn’t necessarily false – it just hasn’t been proved by that argument. Moreover, we believe many things to be so without our being able to provide any argument for them at all.
This discussion takes us well outside the scope of logic which is primarily concerned with the establishment of validity/invalidity. But it also illustrates clearly the value of logic in enabling us to focus our critical attention on those parts of an argument that are essential to its soundness.