Some such arguments are called Enthymemes, arguments in which one of the premises, or the conclusion, is not stated but implied and needed for the argument to follow.
Why leave these out?
It depends on the situation, and the shared understanding of those involved in the discussion.
Generally, one part of an argument may be unspoken because it’s assumed by both and doesn’t need to be stated. So these parts will need to be teased out by a third-party analyst of the argument to determine fully what is being argued.
This can be an intellectually honest form of argument, and I offer some examples here.
I’m using here standard form deductive syllogisms—conditionally certain three-part arguments with two premises and a conclusion—for ease of presentation. The first is a 1st order, or unstated major premise argument.
The Magna is a mutant.
So the Magna is radioactive.
With the major premise being:
All mutants are radioactive.
This one has a hidden minor premise, or 2nd order enthymeme structure.
Not giving the proper homage to the Nine Who are One will endanger all our lives.
So we should not fail to give proper homage to the Nine.
With the hidden premise given as:
The Nine would wish us to do that which preserves our lives.
Finally, we have one in which the conclusion is left unstated, of the 3rd order:
We must deal ruthlessly with all freakishly powerful threats.
The Mirus is a freakishly powerful threat.
It’s not hard to see where this one will go… The conclusion, though unstated, should be obvious.
In some situations this sort of argument is less than intellectually honest, when the assumptions are NOT shared, or need to be fully expressed, this may be used to obscure matters as a rhetorical fallacy, as a tactic that hides the meaning of an argument and makes misdirection and confusion easy.
This happens when the aim intended, or not avoided, is thwarting honest critical discussion. I’ll provide an example of this as a fallacy, this one from a hypothetical creationism/evolution debate in which the major premise is obscured:
No fossil meeting my (impossible to satisfy) criterion as a transitional form has ever been found,
So there are no transitional fossils, so evolution is false.
But here is the missing major premise, NOT shared or expressed, and assumed only by the creationist:
To count as transitional, a fossil must be an impossible, half-formed monstrosity combining unlikely features of dissimilar species or ‘kinds,’ like a lizard/bird hybrid with incomplete, useless wings… (or the supposedly impossible ‘crocoduck,’ AKA, dinosaur genus Spinosaurus)
‘Enthymeme’ has also been used to refer to probabilistic arguments, such as those used in inductive logic or in much informal logic with language inextricably bound up with an argument’s content, with the conclusion following from the premises more or less strongly depending on the audience.
One such claim may be “Present-day Continental philosophy is not credible,” which could elicit different responses and have differing levels of credibility depending on the chosen philosophical schools of those hearing or reading it.
As can be seen, some of the very same sort of statements used in ordinary argumentation can be fallacies, and indeed, when informal, their fallacious nature depends on their misuse as argument strategies, not so much the the structure of the argument but more often violations of procedure.
Many informal fallacies are not always such, but even otherwise effective arguments, when they are put to specious use, are pure argumentative poison no matter their rational structure.
Tf. Tk. Tts.
(Updated, Links Removed, Image Added on 2017.06.06)