# Are all iff s definitions?

1. Jun 16, 2009

### lolgarithms

are all "iff"s definitions?

are all statements of the form "p if and only if q" definitions or equivalences? can there be any iff statements that are not statements of equivalence?

2. Jun 16, 2009

### SW VandeCarr

Re: are all "iff"s definitions?

In logic there's material implication and strict implication which might be related to the notion of equivalence as follows:

In material implication we have P implies Q iff (P^Q)or(~P^Q)or(~P^~Q).

In strict implication we have P implies Q iff (P^~Q) is not possible. This is a modal logic which deals with the concepts of possibility and necessity.

My understanding is that 'equivalence' is more appropriate to saying that at least one formula can be substituted for another in some formal language which in this example would suggest material implication.

In strict implication, there is only one formula with no equivalent formulas for P=>Q Therefore I would think that this would be a case where 'iff 'that does not represent 'equivalence'. Strict implication is stronger than equivalence.

As for what a definition is, I think the basic truth tables of formal logic(s) are essentially axioms which conform to our intuition regarding some application.

Last edited: Jun 16, 2009
3. Jun 16, 2009

### honestrosewater

Re: are all "iff"s definitions?

"If and only if" is normally interpreted as a kind of logical equivalence. An equivalence states that two things are the same in some way. The precise way in which they are the same depends on the context and your definitions. What is your definition of iff? What do you think is the difference between a definition and a logical equivalence? Definitions aren't usually defined as precisely as logical equivalences are, but that two terms have "the same meaning" usually means that they are at least logically equivalent.
Sure, if you interpret iff to mean something other than equivalence.