Numerousness is a higher-order property. "Men are numerous" means "the set of men has many members." You're describing the set and not the members of the set so you can't infer that a man is numerous.
From de Swart's Philosophical and Mathematical Logic:
"It is interesting to note that the notions of ‘finite’ and of ‘non-enumerable’, which
are not first-order properties, can be formulated in second-order logic. In second-order logic, one is allowed to quantify not only over individual
variables, but also over function variables and predicate variables."
From the Stanford Encyclopedia of Philosophy's article on properties:
"Properties are those entities that can be predicated of things or, in other words, attributed to them. Thus, properties are often called predicables."
"This duplicity grounds the common distinction between different orders or types of properties: first-order ones are properties of things that are not themselves predicables; second-order ones are properties of first-order properties; and so on. Even though the formal and ontological issues behind this terminology are controversial, it is widely used and is often connected to the subdivision between first-order and higher-order logics"
@damionhonegan
Well done!
Set theory for beginners:
In the set “men are mortal”, the mortality applies to each member of the set, and therefore also to Socrates. Hence, a valid inference.
Whereas, in the set “men are numerous”, the numerosity applies to the set, and therefore not to Socrates. Hence, a fallacious inference.
