With sophistry their sauce they sweeten

Dictionary of Idioms and Their Origins by Linda Flavell & Roger Flavell. Page 156.

hog: to go the whole hog
to do something thoroughly

A number of theories have been advanced for this phrase and there is also some uncertainty as to whether it was coined in America or England. Although written use occurred first in the US in 1828 and in England soon after, there is no way of knowing on which side of the Atlantic it first gained spoken currency.

Speculation over the country of origin is not clarified by the fact that, in the last century, a hog was a slang term for a ten cent piece in America but also for an Irish shilling, so that, according to one theory, to go the whole hog meant to be willing to spend the whole amount on something. As Brandreth aptly comments, this would make the phrase a close relation of in for a penny, in for a pound.

The poet Cowper apparently enjoyed popularity on both sides of the Atlantic and Funk (1950) suggests that a likely origin is to be found in one of his poems, The Love of the World Reproved: or Hypocrisy Detected (1779), in which he discusses the strictures Muslims placed upon the eating of pork. Mohammed prohibited his followers from eating certain parts of a pig but was singularly unclear about what these were. Muslims were wont to interpret his decree according to their own personal taste so that, between them, the whole hog was devoured:

Had he the sinful part express’d,
They might with safety eat the rest;
But for one piece they thought it hard
From the whole hog to be debar’d;
And set their wit at work to find
What joint the prophet had in mind.
Much controversy straight arose,
These choose the back, the belly those;
By some ’tis confidently said
He meant not to forbid the head;
While others at that doctrine rail,
And piously prefer the tail.
Thus, conscience freed from every clog,
Mahometans eat up the hog.
Each thinks his neighbour makes too free,
Yet likes a slice as well as he:
With sophistry their sauce they sweeten,
Till quite from tail to snout ’tis eaten.

I have had perhaps some hundreds of Muslim friends and acquaintances and certainly a couple of dozen with whom I would have felt comfortable making an inquiry into this idea that there was ambiguity as to what part of a pig might be proscribed.

However, among all my friends and acquaintances, it has never occurred to me to ask. When the proscription has been explained to me, it has always been as a blanket prohibition. There was no ambiguity. And among those whom I have observed, self-identified Muslims either do or do not eat flesh from a pig, there has never been any effort to justify bacon versus pork chops, for example.

And those who do not observe the prohibition have always tended to be more westernized and/or secular.

So did Cowper simply get it wrong? Perhaps there really is a blanket prohibition. Or perhaps there are regional variations. Perhaps in his time (1731 to 1800) customs were different. Perhaps, the prohibition is not universal and there are arguments for some parts versus another. I don't know.

It searching for an answer, I discover that the form of the poem provided by the Flavells is abbreviated. The full poem uses Muslims as an instance but then draws a parallel to similar behaviors in the West.

The Love of the World Reproved: or, Hypocrisy Detected
by William Cowper

Thus says the prophet of the Turk;
Good musselman, abstain from pork!
There is a part in every swine
No friend or follower of mine
May taste, whate'er his inclination,
On pain of excommunication.
Such Mahomet's mysterious charge,
And thus he left the point at large.
Had he the sinful part expressed,
They might with safety eat the rest;
But for one piece they thought it hard
And set their wit at work to find
What joint the prophet had in mind.

Much controversy straight arose,
These choose the back, the belly those;
By some 'tis confidently said
He meant not to forbid the head,
While others at that doctrine rail,
And piously prefer the tail.
Thus, conscience freed from every clog,
Mahometans eat up the hog.

You laugh! — 'tis well, — the tale applied
May make you laugh on t'other side.
Renounce the world, the preacher cries; —
We do, — a multitude replies,
While one as innocent regards
A snug and friendly game at cards;
And one, whatever you may say,
Can see no evil in a play;
Some love a concert or a race,
And others, shooting and the chase.
Reviled and loved, renounced and followed,
Thus bit by bit the world is swallowed;
Each thinks his neighbour makes too free,
Yet likes a slice as well as he,
With sophistry their sauce they sweeten,
Till quite from tail to snout 'tis eaten.