Logical structure of definitions

[Ahmad Kishki]

Is the claim that all definitions are biconditionals, true?
Is the converse statement true as well, that all biconditionals are defintions?

 

[jedishrfu]

Can you provide more context here? What course are you taking? Is this homework?

 

[Ahmad Kishki]

Can you provide more context here? What course are you taking? Is this homework?

Just a general question i am asking within the context of a real analysis course i am doing as self study.

I was wondering with the definition of the limit point: x is a limit point of the set A given that every epsilon-neighborhood of x intersects A at some point different from x.

If i have an epsilon neighborhood of x whose intersection with A is at some point other than x, then i usually assume that x is a limit point - as if the statement was a biconditional.

Is it usually valid to take definitions as biconditionals? Or is it just within the context of math?

I have come up across many biconditionals which i sometimes have to use in proofs. Example of this one which is considered as a theorem: x is a limit point of a set A if and only if there exists some sequence in A that converges to x which has none of its terms equal to x.

I have seen it in many proofs that i have to say if x is a limit point it just means that there exists a sequence... As if the biconditional statement here acts as an alternative definition.

So is the converse statement that all biconditionals are definitions true?

 

[Hornbein]

Just a general question i am asking within the context of a real analysis course i am doing as self study.

I was wondering with the definition of the limit point: x is a limit point of the set A given that every epsilon-neighborhood of x intersects A at some point different from x.

If i have an epsilon neighborhood of x whose intersection with A is at some point other than x, then i usually assume that x is a limit point - as if the statement was a biconditional.

Is it usually valid to take definitions as biconditionals? Or is it just within the context of math?

I have come up across many biconditionals which i sometimes have to use in proofs. Example of this one which is considered as a theorem: x is a limit point of a set A if and only if there exists some sequence in A that converges to x which has none of its terms equal to x.

I have seen it in many proofs that i have to say if x is a limit point it just means that there exists a sequence... As if the biconditional statement here acts as an alternative definition.

So is the converse statement that all biconditionals are definitions true?

I would say though that a "definition" is the assignment of an arbitrary label. If one were to say that A is the definition of B and that B is the definition of A then one would have a circular definition, which would be thought invalid.

So to me a definition is something like the common definition of pi. Pi is just an arbitrary label. If in some other context I wish to define pi as something else, I'm free to do so. I have no such freedom with biconditionals.

 

[HallsofIvy]

Basically, a "definition" just gives a label, or name, to an object or property. In order to be "well defined" it must be clear that this name applies to anything that fits that definition. In that sense, yes, a definition if "biconditional". But, no, not every biconditional statement is a "definition" although it is true that the two parts of any biconditional can be given the same name.

 

[Ahmad Kishki]

Basically, a "definition" just gives a label, or name, to an object or property. In order to be "well defined" it must be clear that this name applies to anything that fits that definition. In that sense, yes, a definition if "biconditional". But, no, not every biconditional statement is a "definition" although it is true that the two parts of any biconditional can be given the same name.

Makes sense. Thank you :)

 

[Stephen Tashi]

Is the claim that all definitions are biconditionals, true?
Is the converse statement true as well, that all biconditionals are defintions?

I'd say yes. A formal definition has the form

<statement-to-be-defined> is true if and only if <defined statement is true>.

So you know that if <defined statement is true> then you can conclude <statement-to-be-defined> is true.

A definition amounts to an assumption and , from another point of view, all assumptions are actually definitions. There is an aspect of chronology to definitions. The <statement-to-be-defined> isn't supposed to be defined several times. If you define a statement , then a different claim about it being equivalent to another statement must be proven as a theorem.

 

[FactChecker]

Usually one biconditional is selected as a definition and any others need to be proven.

 

[HallsofIvy]

I'd say yes. A formal definition has the form

<statement-to-be-defined> is true if and only if <defined statement is true>.

I disagree with this. One does NOT define a "statement". One defines an object, a noun.

So you know that if <defined statement is true> then you can conclude <statement-to-be-defined> is true.

A definition amounts to an assumption and , from another point of view, all assumptions are actually definitions. There is an aspect of chronology to definitions. The <statement-to-be-defined> isn't supposed to be defined several times. If you define a statement , then a different claim about it being equivalent to another statement must be proven as a theorem.

 

[Stephen Tashi]

I disagree with this. One does NOT define a "statement". One defines an object, a noun.

A formal definition must define a statement. Otherwise it wouldn't be a logical equivalence. For example, it wouldn't make sense to say " Real number if and only if ...". The definition must say something like "X is a real number if and only if ...".

 

[HallsofIvy]

No, a definition would be "A real number is ...".

 

[Stephen Tashi]

No, a definition would be "A real number is ...".

The problem with that approach is that you must specify a mathematical definition for "is". The common language definition of "is" is symmetric. (i.e. if A "is" B then we have that B "is" A) By common language if "7 is a real number" the "real number" should be 7. So you are employing a definition of "is" that implies the relation "is a member of". But the phrase "A real number is a member of ..." isn't a noun.

 

[TeethWhitener]

Practically, a biconditional will suffice. But I don't think we want to say that definitions and biconditionals are strictly equivalent (pardon the sloppy language).

The issue is that there are two notions of definition: extensional and intensional. An intensional definition is a list of properties: "A Heegner number is a number d for which the quadratic field ℚ(√(-d)) is uniquely factorable into the factors a + b√(-d)." An extensional definition is a list of objects: "A Heegner number is a member of the set {1, 2, 3, 7, 11, 19, 43, 67, 163}."

Typically, mathematics works by showing that two (intensionally defined) objects have the same extension. For instance, a holomorphic function is (intensionally) defined as a function that is complex differentiable at every point on a domain. An analytic function is (intensionally) defined as a function that has a convergent power series at every point on its domain. As it turns out, the above definitions of holomorphic functions and analytic functions pick out identical sets of functions, but we have to prove this. Depending on your interpretation of the ZF axiom of extensionality, you can say that these two identical sets are actually the same set, and we can say that all analytic functions are holomorphic: ∀f(anal(f) ⇔ holo(f)). However, I think it would be a mistake to say that an analytic function can now be defined as a complex differentiable function at every point on its domain. That loses important information about the relations between the intensions.

 

[HallsofIvy]

The problem with that approach is that you must specify a mathematical definition for "is". The common language definition of "is" is symmetric. (i.e. if A "is" B then we have that B "is" A) By common language if "7 is a real number" the "real number" should be 7. So you are employing a definition of "is" that implies the relation "is a member of". But the phrase "A real number is a member of ..." isn't a noun.

"7 is a real number is real number" is NOT a definition!

 

[Stephen Tashi]

"7 is a real number is real number" is NOT a definition!

I agree. I didn't say that statement was a definition. My point concerns the form of your definition and what your definition means by "is". If you intend to define "A real number" by saying what that real number "is identucal to" or "is equal to", you need more words to avoid giving a definition that defines exactly one real number (such as 7).

Do you disagree that definitions have the form of "biconditionals"? A biconditional (i.e. logical equivalence) involves two statements.

 

[Stephen Tashi]

Practically, a biconditional will suffice. But I don't think we want to say that definitions and biconditionals are strictly equivalent (pardon the sloppy language).

I agree. And we should also observe that meaningful statements can be false. A statement can have the form of a biconditional and be a false statement. As FactChecker observed, only one biconditional involving a statement is taken as its definition. That biconditional must relate a previously undefined statement to an already defined statement. The definition amounts to an assumption that the biconditional is true. Other biconditionals relating the statement to other statements must be proven as theorems before they are accepted as true.

To formally write the definition ""A Heegner number is a member of the set {1, 2, 3, 7, 11, 19, 43, 67, 163}.", it could be rewritten as biconditional using relations that are defined in set theory ( "is a member of" and "equal" (as equality is defined for sets)). To write an informal definition involving membership in sets one may say "is a member of" without meaning the relation $\in$. However, in a formal definition, I think that should be avoided.