I think p61 makes it perfectly clear that an area which is Hindering Terrain blocks movement or damages those who enter it.
I don't think it makes it clear that any area that blocks movement or damages those who enter it is Hindering Terrain.
All dogs have four legs. This cat has four legs. Therefore this cat is a dog?
-Hyp.
This would be true if hindering terrain were not a label to be applied to different terrain effects but instead a specific instance.
You do not put down "hindering terrain" in your dungeons. When your players ask what the green thing is, you do not say "oh, its hindering terrain" when its "a pool of acid".
By your reckoning there is nothing that can ever be classified as "hindering terrain" because there is never any statement that says "x is hindering terrain" only "hindering terrain is x" and there is nothing that is hindering terrain by its own merits.
The problem is seen further by your explanation of the logical fallacy. However, that is not the form that is being used. The first statement(and premise) is not "all dogs have 4 legs", the first statement(and premise) is, "all things that have 4 legs are dogs".
Such, the new logical construct reads
Premise: "All things that have 4 legs are dogs"
Premise: "all cats have 4 legs"
Inference: "Therefore all cats are dogs"
Which, while not being cogent is still valid.
A better way to put it would be this
Premise: something is a mammal if it contains the "following properties"
Premise: dogs contain the "following properties"
Inference: Therefore all dogs are mammals
This is the situation which we are dealing with. Not the statement "dogs are" but the statement "mammals are". You are defining the top set of possibilities and not the bottom.