I show elsewhere, that CLOSURE means that an OPERATION combining ELEMENTS OF A GIVEN TYPE results in A COMBINED ELEMENT OF THE SAME TYPE. Equivalently, given two or more gnomons (BASIC UNITS) of the same kind or type, THEIR COMBINATION IS A GNOMON OF THE SAME TYPE, a means of building a given PATTERN. It's "all in the family".

And I show (in the above hyperlink) how CLOSURE-FAILURE CAN REQUIRE DIFFERENT OR NEW SYSTEMS.

The CLOSURE-CONDITION becomes critical in dealing with MEASUREMENTS. A pop version of this says, "You can't add apples and oranges!" Actually, YOU CAN. We do something like this "all the time". Thus, 2 apples + 3 oranges = 5 pieces of fruit. OK. We did this by changing the TYPE. The type-measurement of apple is a subtype of the type-measurement called "fruit"; and so is orange. Hence, the result. But that's not the MEASURE-CLOSURE-PROBLEM we often face.

I created the word "discrese" to mean "carving a discrete element" from a continuum.

WHENEVER WE CAN DISCRESE ELEMENTS FROM A CONTINUUM -- A CONTINUOUS or "analogic" "BACKGROUND" -- WE CAN OBVIOUSLY COUNT INSTANCES OF THIS DISCRESE-OPERATION. No argument. No problem -- provided we do what we claim and state it correctly. WE HAVE NUMBER-CLOSURE ON THE DISCRETIONS.

In the above case, we discrese an apple and another apple from our "reality"-background. Similarly, with discresing the oranges. And we can COUNT 5 DISCRETIONS -- that is, we have NUMBER-CLOSURE. But we get in trouble when we try to ADD GNOMONS OF DIFFERENT TYPE. The problem disappears when WE TRANSFORM TO GNOMONS OF THE SAME TYPE ("fruit"). We herewith have MEASURE-CLOSURE BECAUSE apple, orange, fruit SATISFY REQUIREMENTS OF THAT SIMPLE MEASUREMENT SCALE KNOWN AS TYPE ("kind", "category"). This invokes MEASURE-CLOSURE. So we can "add apples and oranges" because we can transform to the shared MEASURE OF TYPE ("fruit") and achieve MEASURE-CLOSURE ALONG WITH NUMBER-CLOSURE.

But let's see when this fails. We COUNT out two instances of THE MEASUREMENT SUBSCALE called "length", namely, two instances of "a meter of length". And we COUNT out three instances of the MEASUREMENT SUBSCALE of "weight" -- say, three instances of "kilogram". We clearly have (2 + 3) DISCRETIONS = 5 DISCRETIONS, namely, NUMBER-CLOSURE. But we can't perform this addition to claim "5 meter-kilograms". (Some of you know we can do some similar to that by MULTIPLYING DIFFERENT DIMENSIONS OF MEASUREMENT. But we can't ADD!) Unless we can EMBED LENGTHS AND WEIGHTS INTO A DIMENSION MORE COMPLEX THAN THAT OF DISCRETION, then WE CAN'T PERFORM THIS ADDITION! And we can't. So we can't. FAILURE OF MEASURE-CLOSURE!

MEASURE-CLOSURE for LENGTH (alone) occurs because A LENGTH CONCATENATED WITH A LENGTH CAN BE MATCHED IN PARALLEL BY A SINGLE EXTENSION OF THE SAME LENGTH. (The double-extension is equivalent to the single extension, and vice versa.) So 2 meters + 3 meters = 5 meters. In any case where we need 5 meters of length (wood, steel, whatever), either the 2 meter + 3 meter combination will do as well as a single extension of 5 meters. That's an example of WHAT MEASURE-CLOSURE means!

Similarly with weight, in killograms or pounds or any accepted unit of weight. And other successes can be demonstrated.

But it's easy to show that IQ scoring fails CLOSURE and other requirements for adding measures, as psychometrists do.