Does "iff" make sense in a definition?
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From: Edward Green on 26 Jul 2010 18:41 I ran across a definition in Wikipedia of the form "A is B iff C", and I can't quite get my head around this. Does adding "and only if" really add something to "if" when we are stating a definition? Does it even make sense? (Ref: http://en.wikipedia.org/wiki/Totally_bounded_space "Definition for a metric space")
From: FredJeffries on 26 Jul 2010 19:39 On Jul 26, 3:41 pm, Edward Green <spamspamsp...(a)netzero.com> wrote: > I ran across a definition in Wikipedia of the form "A is B iff C", and > I can't quite get my head around this. Does adding "and only if" > really add something to "if" when we are stating a definition? Does it > even make sense? > > (Ref:http://en.wikipedia.org/wiki/Totally_bounded_space"Definition > for a metric space") http://en.wikipedia.org/wiki/If_and_only_if#Definitions
From: Edward Green on 26 Jul 2010 20:13 On Jul 26, 7:39 pm, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 26, 3:41 pm, Edward Green <spamspamsp...(a)netzero.com> wrote: > > > I ran across a definition in Wikipedia of the form "A is B iff C", and > > I can't quite get my head around this. Does adding "and only if" > > really add something to "if" when we are stating a definition? Does it > > even make sense? > > > (Ref:http://en.wikipedia.org/wiki/Totally_bounded_space"Definition > > for a metric space") > > http://en.wikipedia.org/wiki/If_and_only_if#Definitions "In philosophy and logic, "iff" is used to indicate definitions, since definitions are supposed to be universally quantified biconditionals. In mathematics and elsewhere, however, the word "if" is normally used in definitions, rather than "iff". This is due to the observation that "if" in the English language has a definitional meaning, separate from its meaning as a propositional conjunction. This separate meaning can be explained by noting that a definition (for instance: A group is "abelian" if it satisfies the commutative law; or: A grape is a "raisin" if it is well dried) is not an equivalence to be proved, but a rule for interpreting the term defined. (Some authors,[3] nevertheless, explicitly indicate that the "if" of a definition means "iff"!)" Well, my instinct was correct, even if my follow through was poor. I suppose a purist might object that "a grape is a 'raisin' if it is well dried" leaves open the possibility that a grape might also be a 'raisin' if painted blue. Thanks for clearing that up.
From: spudnik on 26 Jul 2010 23:40 iff as definitional seems to make some sort of sense, but what is wrong with "a raisin is a grape, if and only if -- that is to say for short, IFF -- it had been thoroughly dessicated?" I believe that the main definition of iff comes from Liebniz, his definition of proof as satisfying "neccesity & sufficiency," and that is a matter of using the words in a literate manner, in some way. (of course, if you can even prove just one of the two criteria, it is cake, at least in synthetic geometry .-) --les ducs d'oil! http://tarpley.net --Light, A History! http://wlym.com/~animations/fermat/august08-fermat.pdf
From: G. A. Edgar on 27 Jul 2010 07:15
In fact, the short word "iff" was INVENTED (I think by J. L. Kelley, or was it P. R. Halmos?) for use in definitions. Later, others took it over as a synonym for "if and only if" in other contexts as well. So when I write mathematically, I never write "if and only if" for a definition, but try always to write "iff". Contrariwise, in formal published mathematical writing, use "if and only if" when you mean that and reserve "iff" only for definitions. That's my recommendation! My advisor, Garrett Birkhoff, had a pronunciation for this word, with the two Fs pronounced separately, if'f . In article <1f68cf55-9ec8-423a-858f-1a07d6a6c34f(a)f33g2000yqe.googlegroups.com>, Edward Green <spamspamspam3(a)netzero.com> wrote: > I ran across a definition in Wikipedia of the form "A is B iff C", and > I can't quite get my head around this. Does adding "and only if" > really add something to "if" when we are stating a definition? Does it > even make sense? > > (Ref: http://en.wikipedia.org/wiki/Totally_bounded_space "Definition > for a metric space") -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ |