Thursday, April 14, 2011

syntax for definitions

How should one write a definition? The mathematician Douglas West (The Grammar According to West) provides the following advice:

Definitions. Words being defined should be distinguished by italics (or perhaps boldface in a textbook context). When italics are used to indicate a word being defined, it is unnecessary to use "called" or "said to be"; the use of italics announces that this is the term being defined and replaces these words.

Many definitions are phrased as "An object has property
italicized term if condition holds." We use just "if" even though subsequently it is understood that an object has the property if and only if the defining condition holds. The italicization alerts the reader to this situation. The convention can be justified by saying that the property or object does not actually exist until the definition is complete, so one does not yet in the definition say that the named property implies the condition.

This seems confused to me for a number of reasons---especially the justification for the convention: "the property or object does not actually exist until the definition is complete". What does that mean?
Definition. An object x is a firath if and only if x is a female giraffe.
Obviously, firaths existed long before I defined "firath"! They just were not so-called.

But what I mostly don't like is that the "definition" schema appears to merely give a sufficient condition for falling under the definiendum. The following is not a good definition of "firath".
Definition. An object x is a firath if x is a giraffe.

Does his advice seem reasonable or not? Presumably, a lot of mathematicians are adhering to his grammar suggestions.


Wolfgang Schwarz said...

Maybe the thought is that before the definition, the word 'firath' did not have any legitimate usage. The definition now makes it true that it may be legitimately used for female giraffes. Hence afterwards the word may be legitimately used for female giraffes and nothing else.

To be pedantic, I guess definitions should be stated as meta-linguistic imperatives: let us henceforth use 'firath' as synonymous to 'female giraffe'!

Tristan Haze said...

The thing about the object/property not existing prior to the definition indeed seems wrong or incoherent.

Regarding the 'if'/'if and only if' point: sure, the schema *appears* (to us philosophers at least) to merely give a sufficient condition, but it's clear that West is saying this is a convention for giving necessary and sufficient conditions, and indeed I've seen it a few times. So saying that an instance of the schema is 'not a good definition', if this is meant in a logical sense, seems unwarranted. Rather, one should say it is a good definition put in terms which may make it look like a bad one.

Since merely giving sufficient conditions is never good definitional practice, West could argue that the 'if' for 'if and only if' convention is appropriate, in line with the principle of charity. Not being a mathematician, I'd rather stick to 'iff' (pronounced 'if and only if'), even in conversation.