Are Statements 1 and 2 Logically Equivalent in All Contexts?

[issacnewton]

Hi

I am little confused about the following statements.

1) \;\left[\exists x P(x)\right]\Rightarrow M(x)

and

2) \forall x \left[P(x)\Rightarrow M(x)\right]

Its obvious that they are not logically equivalent. But let's take some examples.

let P(x) = x is majoring in maths
M(x)= x is mad.

so the statement 2 means that all math majors are mad and
statement 1 means that if there is a math major then he is mad

here it looks like they are equivalent in meaning. so what's happening ?

 

[Susanne217]

Hi

I am little confused about the following statements.

1) \;\left[\exists x P(x)\right]\Rightarrow M(x)

and

2) \forall x \left[P(x)\Rightarrow M(x)\right]

Its obvious that they are not logically equivalent. But let's take some examples.

let P(x) = x is majoring in maths
M(x)= x is mad.

so the statement 2 means that all math majors are mad and
statement 1 means that if there is a math major then he is mad

here it looks like they are equivalent in meaning. so what's happening ?

I am not sure what you are trying to say here??

1) There exists an x and then P(x) results in M(x)? You say something more about x in order to use these symbols. Like there (\exists x \in A(x) \Rightarrow A(x) \mapsto M(x)

 

[issacnewton]

Hi Susan

Statement 1 means that " If there exists x such that x is math major then x is mad"
In other words , "If there exists a maths major then he is mad" But wouldn't this have the
same meaning as statement 2 since we can go find if any x is maths major . Then according to
statement 1 , is x is found to be a maths major then x is mad. So all the maths majors are found to be mad...

 

[Mark44]

The first statement says that for some x, P(x) implies M(x). The second statement says that for every x, P(x) implies M(x).

Using your example, the first statement says that some math majors are mad, while the second says that all math majors are mad.

 

[issacnewton]

Mark , on second thought , I think I have made a mistake. The first statement is not even a proposition. The x in antecedent in the first statement is a bound variable and the x in the consequent is a free variable. So we can't talk about the truth value of the statement unless we define x in M(x). So what you are saying is not correct either.