Confused about logical equivalence of some statements

• Jul 13th 2011, 02:10 AM
issacnewton
Confused about logical equivalence of some statements
Hi

I am little confused about the following statements.

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

and

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

Its obvious that they are not logically equivalent. But lets take some examples.

let P(x) = x is majoring in maths

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 whats happening ?
• Jul 13th 2011, 02:47 AM
abhishekkgp
Re: Confused about logical equivalence of some statements
Quote:

Originally Posted by issacnewton
Hi

I am little confused about the following statements.

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

and

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

Its obvious that they are not logically equivalent. But lets take some examples.

let P(x) = x is majoring in maths

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
no, statement 1 means there is at least one math major who is mad
here it looks like they are equivalent in meaning. so whats happening ?

...
• Jul 13th 2011, 02:51 AM
emakarov
Re: Confused about logical equivalence of some statements
Quote:

Originally Posted by issacnewton
$\displaystyle 1) \;\left[\exists x P(x)\right]\Rightarrow M(x)$

This formula is not a proposition (either true or false) because x in M occurs freely, and so the formula's truth value depends on the value of x. In contrast, $\displaystyle \forall x \left[P(x)\Rightarrow M(x)\right]$ is a proposition. So, the concept of equivalence does not apply to them.
• Jul 13th 2011, 03:02 AM
issacnewton
Re: Confused about logical equivalence of some statements
Hi abhishek,

ok. so the key word is at least one. But I am still having difficulty
understanding the difference between the two statements. Statement 1 guarantees that if we have even one math major then he is mad , am I right ? But does statement 1 mean that we can have 1 mad math major and rest of the math majors are not necessarily mad ?
• Jul 13th 2011, 03:12 AM
issacnewton
Re: Confused about logical equivalence of some statements
Quote:

Originally Posted by emakarov
This formula is not a proposition (either true or false) because x in M occurs freely, and so the formula's truth value depends on the value of x. In contrast, $\displaystyle \forall x \left[P(x)\Rightarrow M(x)\right]$ is a proposition. So, the concept of equivalence does not apply to them.

Oh... but what if x is a mad person who is not a math major ? then there does not exist any math major (antecedent is false) but there is a mad person
(consequent is true). by rules of implication, the statement is true. is it not ?
• Jul 13th 2011, 03:56 AM
emakarov
Re: Confused about logical equivalence of some statements
Quote:

Oh... but what if x is a mad person who is not a math major ? then there does not exist any math major (antecedent is false)
I see no reason why ∃x P(x) is false.

Quote:

but there is a mad person
(consequent is true). by rules of implication, the statement is true. is it not ?
Yes. I don't see the connection between [∃x P(x)] => M(x) being true for some particular x and the equivalence of [∃x P(x)] => M(x) and ∀x [P(x) => M(x)] ("equivalence" is not applicable to these two formulas).

Quote:

Originally Posted by issacnewton
Statement 1 guarantees that if we have even one math major then he is mad , am I right ?

No, it says that if we have even one math major, then x is mad, where x is undefined so far.
• Jul 13th 2011, 04:15 AM
issacnewton
Re: Confused about logical equivalence of some statements
Quote:

Originally Posted by emakarov
No, it says that if we have even one math major, then x is mad, where x is undefined so far.

I think I am beginning to see some sense after all. The problem is , x in antecedent is bound variable and x in the consequent is free variable, right ?

I have to read again about these variables. They are so confusing. Any online source would you recommend ?
• Jul 13th 2011, 04:31 AM
emakarov
Re: Confused about logical equivalence of some statements
The PlanetMath article is a little formal but not too hard. This presentation [PDF] seems nice. See also this page written by some very qualified people.
• Jul 13th 2011, 04:44 AM
issacnewton
Re: Confused about logical equivalence of some statements
haha, I was reading that page at cnx.org at the time you replied.......