By whatever choice of words it is often argued that because one thing leads to another, adverse consequences must inevitably follow if the opponent's proposed course of action is taken.
By whatever choice of words, the above purely rhetorical "proof" is a complete load of codswallop.
There are many names for an argument of the form that if we do P, then r,s,t ... z must inevitably follow. "The slippery slope" is well known, as is "the thin end of the wedge". Less well known, at least outside of law, are the "floodgates argument" from US and UK law and "the parade of the horribles" from US law. Yes, 'horribles' is a plural noun, even if my spell/grammar checker wants to send me to the gulag for using it as such.
The floodgates argument
This is a favourite of politicians. I use the term loosely - judges aren't supposed to do politics but being human they sometimes forget where they are. The argument is in the form that if a particular claim is allowed then this will open the floodgates to everyone who has similar grounds for a claim. The reality is that most people avoid going to court over trivial matters. Of course, if the matter is far from trivial then it is outrageous that one person should be denied justice on the grounds that if it is granted to one, then others will demand their fair share of it.
Having somewhat impugned our judges I feel obliged to willingly disimpugn where disimpugnment is deserved: the court ignored the "floodgate" argument in Bairstow.
"We were warned by learned Counsel for the Respondents that to allow this appeal( Note to self: must get a life! Reading legal cases by way of a hobby is bad enough, but really! Tax law cases? )
would open the floodgates to appeals against the decisions of the General
Commissioners up and down the country. That would cause me no alarm, if
decisions such as that we have spent some time in reviewing were common
up and down the country."
Edwards (Inspector of Taxes) v Bairstow  UKHL 3 (25 July 1955)
The parade of the horribles
Ben Zimmer has written an excellent article about the origins of this phrase, and who am I to take the bread from his mouth? Where did the Supreme Court get its ‘parade of horribles’?
How an obscure Fourth of July custom from New England spawned a legal-world insult
"... outside of legal and governmental circles, it’s an obscure expression, not yet registered in any major dictionary." For shame! That Mr. Webster has got a lot of explaining to do, to say nothing of that bunch of fascicles from Oxford.
And now - the dead mouse
There are really only two kinds of "one thing leads to another" argument: the parade of horribles and the dead mouse. The parade of horribles is, presumably, to be avoided. On the other hand the floodgates argument assumes that showing something in public leads to everyone wanting it. A well known limerick, by the prolific author Anon, illustrates the fallacy rather nicely.
There once was a person from Crewe.
Who found a dead mouse in his stew.
Said the waiter, "Don't shout,
And wave it about,
Or the rest will be wanting one too!"