What is the specifier (tamyīz)?

Something in the sentence is left vague, and one word settles it. That word is the تَمْيِيز.

اِشْتَرَيْتُ عِشْرِينَ كِتَابًا (ishtaraytu 'ishrīna kitāban) — I bought twenty books. Without كِتَابًا, twenty of what? The number is the vagueness; the tamyīz removes it.

Tamyīz of a thing clears up a vague quantity; tamyīz of a relation clears up a vague statement.

It comes in two kinds. تَمْيِيز ذَات clears up a vague thing — after numbers, weights, measures: رِطْلًا زَيْتًا, a pound of oil. تَمْيِيز نِسْبَة clears up a vague relationship in the sentence as a whole: طَابَ مُحَمَّدٌ نَفْسًا (ṭāba muḥammadun nafsan), Muhammad was good — in respect of his soul.

The tamyīz is always indefinite, and always manṣūb.

Do not confuse the تَمْيِيز with the حَال.

Both are indefinite and both are manṣūb, which is why students mix them.

A حَال answers in what state? — it describes a condition at the time of the action: جَاءَ زَيْدٌ رَاكِبًا.

A تَمْيِيز answers in respect of what? — it removes an ambiguity, usually about a quantity or a relation: طَابَ مُحَمَّدٌ نَفْسًا.

A useful test: a tamyīz can usually be rewritten with مِنْ — عِشْرُونَ كِتَابًا, twenty of books. A ḥāl cannot.