Where does extension collide?

In logic and the philosophy of language, the extension of a term is the set of all things to which the term correctly applies: the extension of planet is the collection of the actual planets, and the extension of is red is every red thing. It is contrasted with the intension, the concept or meaning that decides which things belong, so that creature with a heart and creature with a kidney can share one extension while differing in intension. Carnap's 1947 Meaning and Necessity made this pairing precise, treating the extension as what a term picks out and the intension as the rule for picking it out. Here the referent is a set of objects that a word denotes, and it has nothing to do with enlarging a number field, straightening a limb at a joint, or carrying research to farmers.[1]

Unit

The set of all objects a term applies to

Recovery means

Not applicable

Measured by

Fixed by specifying which objects fall under the term, and set against the term's intension

In algebra, a field extension is a larger field that contains a given field as a subfield, so that the smaller field's addition and multiplication are just the restrictions of the larger one's: the complex numbers extend the real numbers, which extend the rationals. Studying which numbers a field must gain in order to solve a given equation is the heart of Galois theory, and the size of an extension is captured by its degree, the dimension of the larger field viewed as a vector space over the smaller. Steinitz's 1910 Algebraische Theorie der Körper founded the abstract theory of fields and their extensions, classifying how one field can sit inside another. Here the referent is one algebraic structure enlarging another, and it is unrelated to the collection of things a word denotes, a movement that straightens a limb, or an advisory service for farmers.[2]

Unit

A larger field containing a base field, scaled by its degree over that base

Recovery means

Not applicable

Measured by

Characterised by its degree, by whether it is algebraic or transcendental, and by its Galois group

In rural development and adult education, extension is the system of out-of-school teaching that carries research-based knowledge from universities and research stations to farmers and rural communities, helping them adopt improved practices. Cooperative and agricultural extension services employ field agents who advise, demonstrate, and train, turning findings into changed practice on the farm. Anderson and Feder's 2004 Agricultural Extension: Good Intentions and Hard Realities reviewed how such services are organised and why they so often struggle to deliver. Here the referent is an educational and advisory institution, and it shares nothing with the objects a word denotes, a larger number field, or the straightening of a limb at a joint.[3]

Unit

An advisory service linking research to farmers, per programme or field agent

Recovery means

Not applicable

Measured by

Assessed by farmer adoption of recommended practices, by contact and reach, and by effects on yields and welfare