For anyone that would like to raise some questions or challenges to help in the chewing process, I've started a thread on the newsgroup humanities.philosophy.objectivism that addresses some problems that have come up there in a discussion I've been having with a Kant enthusiast. My aim is to come to grips with a first-hand understanding of how Objectivism treats the issue of necessity vs. how Hume and Kant treat it, so I believe it covers the same sort of ground covered in Peikoff's "Analytic-Synthetic Dichotomy" essay, just from my own context and best understanding. Below I repost what I put in the first posting of that thread (linked above), with a minor revision in brackets.

The post:

[...] I put a
little thought into the matter of definitions with some clarifying aid
from ITOE. I'm quoting from the page cited in the -Ayn Rand Lexicon-
(p.117):

"A definition is a statement that identifies the nature of the units
subsumed under a concept."

and then, later:

"The purpose of a definition is to distinguish a concept from all
other concepts and thus keep its units differentiated from all other
existents." (ITOE, 52)

Now, keep in mind that Rand also said that "essence" is
epistemological. From p. 118 of the ARL:

"A definition must identify the *nature* of the units, i.e., the
*essential* characteristics without which the units would not be the
kind of existents they are. (ITOE, 55)

How does all this make sense, if for Rand, the *nature* of a thing
(all its characteristics) is not to be confused with its *essential*
(distinguishing) characteristics?

Because while the definition captures what is true of necessity for
all the units -- what is common to the nature of *all* of the units
subsumed under the concept -- the purpose of a definition is to
further identify what is necessarily the case for all the units of the
concept, and to *distinguish* it from all other units for which this
is not necessarily the case.

IOW, the function of a definition is to identify the *nature* of the
units subsumed under the concept in a way that *distinguishes* them
from other sorts of units.

Now, true enough, the *nature* of one apple is such that it is red,
while the nature of another is that it is green. These are the kinds
of differences that a *definition* is not intended to capture as they
do not distinguish apples from non-apples. They are, nonetheless,
differences that are noted and captured under under the *concept*
"apple" in a very open-ended fashion. A concept, in other words, can
subsume vast amounts of information -- only that our means of
economically referencing it is by means of a symbol (term) that we use
to *define* the concept.

The definition only states what by necessity *distinguishes* the
units, but the vast amount of information that can be referenced under
the concept can also capture things that are by necessity true for all
the units.

In the case of ice: the capacity for floating on liquid water does not
*distinguish* ice from all other units, because after all, there are
other, non-ice things that can float on liquid water. The capacity is
therefore not stated in the definition. It is nonetheless contained
in the great amount of information that the concept references, that
ice must float on liquid water in order to be ice.

So it is not the case, per Kant, that by adding "capable of floating
on liquid water" to the term "ice" is adding something new that's not
already contained in the concept of ice. Concepts capture what is
true by necessity for the units subsumed under the concept in an open-
ended, contextual fashion; the only kind of "adding" being done is the
adding that was done *historically* as our knowledge of ice expanded;
given what we *now* know about ice that we may not have known before
*historically speaking*, "floats on liquid water" does not *add*
anything to our concept of ice that isn't already contained in it. We
are not "going outside" of the concept to state truths about ice that
aren't captured in the concept alone. It is merely not captured in
the definition alone.

In sum: there are things necessarily true with regard to a concept and
captured in the concept that are not captured in the definition
alone. The definition only distinguishes -- saying [that which] is
necessarily true *only of these units* and not of other units. The
placement and use of the term "only" here is crucial; people have
gotten confused and thought that a definition states "only [that which] is
necessary to the units." But the difference should be clear.