In a previous question's answer, @motosubatsu quoted a passage from James Bigg, Esq. stating this as a "legal axiom."
Where does this perception arise from?
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James Bigg, Esq. published his Criminal Law Consolidation (from which the “legal axiom” of the question is quoted) in 1865. At that time, the general view of England’s common law system was that it was grounded in the customs of society as a whole. We might therefore turn Bigg’s axiom on its head: It’s not that every Englishman is presumed to know the law; it’s that the law is presumed to be what every Englishman knows.
On this theory, the ordinary Englishman may not in fact know (or even know of) every precedent and statute that exists, but he will know the customs of his country, and know right from wrong. The root of the law is to behave in a way that any decent, right-thinking Englishman would think fair and proper.
I therefore stile these parts of our law leges non scriptae [unwritten laws], because their original institution and authority are not set down in writing, as acts of parliament are, but they receive their binding power, and the force of laws by long and immemorial usage, and by their universal reception throughout the kingdom.
The English were quite proud of their common law, and resisted attempts to codify it. In his “Proposition Touching Amendment of Laws” (1657), Francis Bacon wrote:
Obj[ection] IV. Labour were better bestowed in bringing the common laws of England to a text law, as the statutes are, and setting both of them down in method and by titles.
Resp[onse] …there are more doubts that rise upon our statutes, which are a text law, than upon the common law, which is no text law. But howsoever that question be determined, I dare not advise to cast the law into a new mould. The work which I propound tendeth to proyning and grafting the law, and not to ploughing up and planting it again; for such a remove I should hold indeed for a perilous innovation.
The thinking appears to be that, unlike under the civil law of continental Europe, an English judge was free to innovate if doing so delivered a ruling that reflected a “common” notion of fairness and justice better than the code of laws could. Of course Blackstone did not consider judges as making law; later on in Section III, on the subject of precedent, he says that a judge is:
…sworn to determine, not according to his own private judgment, but according to the known laws and customs of the land; not delegated to pronounce a new law, but to maintain and expound the old one. Yet this rule admits of exception, where the former determination is most evidently contrary to reason; much more if it be contrary to the divine law. But even in such cases the subsequent judges do not pretend to make a new law, but to vindicate the old one from misrepresentation. For if it be found that the former decision is manifestly absurd or unjust, it is declared, not that such a sentence was bad law, but that it was not law; that is, that it is not the established custom of the realm, as has been erroneously determined.
(Amusingly, there is an uncited claim on Wikipedia that Blackstone’s Commentaries was criticised as a covert attempt to codify the common law. I would love to see evidence for this, but a brief search has not turned up any.)
This thinking persisted into the 20th century. Consider Donoghue v Stevenson, where Lord Atkin used the Bible as his basis for saying that manufacturers owe a duty of care to consumers. In this there is an echo of Blackstone’s declaration of the primacy of divine law, but it was also feasible simply because the UK remained, by and large, a Christian country: the precepts of the Bible were the “established custom of the realm”.
This answer is unlikely to be wholly satisfactory, as I don’t have explicit evidence linking the historic view of the common law to the presumption that everyone knows the law. The evidence I have laid out is also focused on common law, but this presumption extends to statute law as well.
Moreover, in the course of writing this answer, I read that this view is no longer popular among legal scholars or practitioners: neither as an account of how the law developed, nor as a rubric to how it should be applied today.
But it was certainly a mainstay of English legal theory at the time when Bigg wrote of this “legal axiom”, and I therefore offer it as an answer to the question. It complements, but does not replace, the pragmatic argument others have put forth, that it is untenable to let ignorance be a ready defence.
I think it's ultimately a natural extension of the basic concept from Roman Law: ignorantia legis non excusat, i.e "ignorance of the law excuses not" and that in order for that to be workable laws can only truly obtain proper binding force when they are promulgated. In simple terms someone can't claim innocence through ignorance because the knowledge of what the law is has been made easily available to them.
Thus the law can operate on the presumption that the people it applies to are aware of them. Indeed there have been occasional instances in history where someone could not have been reasonably expected to know that what they did was against the law and this has been taken into account.
Let's assume for one moment that no person is assumed to know what the law is, because that is the inverse of the premise. This also means that the old brocard of Ignoratia legis non excusat & ignorantia legis neminem excusat (ignorance of the law isn't an excuse/is an excuse for nobody) is to be inverted: Ignorance of the law becomes an excuse for having broken the law.
If ignorance of the law is an excuse to have broken it, every person gets a "free shot" at whatever they want to do, because you can only presume someone knows it because they are put under verifiable notice of the law. Under this setup, there is no way to have an effective legal system.
Only if perfect knowledge of the law is presumed on everybody can the legal system be both effective and fair: no matter if you actually do know or not know, the same verdict for the same act will be reached, if mens rea is not a factor.
First, the laws are generally modeled on a core sense of "right and wrong", and it is expected that every parent, school and foster system will make great effor to teach this to every child. Separately to this, is it modeled on "Do unto others as you would wish them do unto you". So if you reasonably wouldn't want people stealing your car, don't steal cars. If you don't want to see streets covered in litter, don't litter. See also the "reasonable man" metric.
This allows most people to generally navigate life without inadvertently committing criminal offense. It seems to work pretty well.
Of course, life branches into 10,000 different specialties. Say you buy a narrowboat. The onus is on you to learn the laws of operating inside your specialty. If your specialty is commonly accessed by consumers, like owning a narrowboat, there is plenty of stuff to help you, from government education and certifications to private lessons and of course, the Internet. Enforcement is also wise to the earnest but oblivious novice, and will be non-cruel and give you advice. You better listen.
Of course once you start operating at a commercial specialty, like operating an oil tanker or cruise ship, or building a windmill farm or job-matching website, you are expected to retain the counsel of experts - as in hire them, starting with lawyers. Your experts' professional job is to understand the craft-specific laws which no layman would have any reason to know. No excuse is accepted. That gets small business owners in trouble when their business grows dramatically and gains scope their diligence wasn't prepared for - when Port Wynn "match-a-ride" becomes Cornwall Match-a-Ride becomes UK Match-a-Ride and you're still running it with the same in-town accountant and lawyer you started with.
In Kiriri Cotton Co. v. Dewani,  UKPC 27, p. 3-4 Lord Denning said:
It is not correct to say that everyone is presumed to know the law. The true proposition is that no man can excuse himself from doing his duty by saying that he did not know the law on the matter. Ignorantia juris neminem excusat.
It is simply the case that application of the law to you does not require your prior awareness of it.
In a jury trial, it is critical that the judge explain the applicable law correctly to the jury. Failure to instruct the jury on a critical point of law can leave a conviction open to appeal—jurors, like everyone, are not presumed to know the law.
I just spent a year in jail because I didn't know it was a serious crime to change the tires on your car on a Wednesday evening between 8pm and 9pm. Ignorance of the law does not excuse, right? Well, that doesn't happen.
Laws make it illegal for you to do illegal things to others, but it also makes it illegal for others to do illegal things to you. If there is an action that you wouldn't want to suffer from others, and that you think they shouldn't be allowed to do, then you shouldn't yourself do this to others. Even if you don't know for sure that a specific law exists, that cannot be an excuse if it is something that you think should be illegal for others to do. And that's from a normal healthy person's point of view, a sociopath might think otherwise.
Now, ignorance of facts will often be an excuse. Not knowing that it is illegal to exceed the speed limit is not an excuse. Not knowing the speed limit at some bit of public road can be an excuse, depending on the situation. You are supposed to watch out for speed limit signs, so not knowing the speed limit is often not an excuse. But if the sign was invisible because it was behind a tree, that is likely an excuse.
In the end, you don't have to know the laws. You just need to not violate them. If you do out of ignorance, that's just tough for you.