Is Every If and Only If Statement a Definition or Equivalence?

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The discussion centers on the nature of "if and only if" (iff) statements in logic, specifically whether all such statements are definitions or equivalences. It establishes that while "iff" is typically interpreted as a logical equivalence, there are contexts where it may not represent equivalence, particularly in strict implication. The distinction between material implication and strict implication is highlighted, with strict implication being stronger than equivalence. The conversation concludes that definitions and logical equivalences differ in precision and context, suggesting that not all "iff" statements can be classified as equivalences.

PREREQUISITES
  • Understanding of logical implications, specifically material and strict implication.
  • Familiarity with truth tables and formal logic axioms.
  • Knowledge of modal logic concepts, particularly possibility and necessity.
  • Basic grasp of logical equivalence and its contextual applications.
NEXT STEPS
  • Explore the differences between material implication and strict implication in detail.
  • Study the role of truth tables in formal logic and their application in defining equivalences.
  • Investigate modal logic and its implications for understanding necessity and possibility.
  • Examine various definitions of "iff" in different logical contexts and their implications.
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Logicians, philosophy students, and anyone interested in the foundations of logical reasoning and the nuances of definitions versus equivalences in formal logic.

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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?
 
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lolgarithms said:
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?

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.
 
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lolgarithms said:
are all statements of the form "p if and only if q" definitions or equivalences?
"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.
can there be any iff statements that are not statements of equivalence?
Sure, if you interpret iff to mean something other than equivalence.
 

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