# Thread: Logical Vs Material equivalence

1. ## Logical Vs Material equivalence

Hi

I have a conceptual question about the logical equivalence. Suppose we are asked
to prove

$p\equiv q$

Is it same as proving

$p\Leftrightarrow q$ ?

The reason I am asking this is that I have read at few places on this forum
that they are equivalent. following post is an example.

but the wikipedia article on logical equivalence says that logical equivalence is
different from material equivalence. How so ? I couldn't understand the reasoning
given there.

thanks

2. ## Re: Logical Vs Material equivalence

Originally Posted by issacnewton
the wikipedia article on logical equivalence says that logical equivalence is
different from material equivalence
I've never seen such a distinction in any textbook or article I can recall. What source is the Wikipedia article based on?

I do agree, though, with the Wikipedia article that there is a distinction between the syntactical (G |- P<->Q) and the semantical (G |= P<->Q), though, of course, for first order we have G |- P<->Q if and only if G |= P<->Q.

But, unless there is some special context for the use of the symbols, generally, the triple bar symbol and the double arrow symbol are just different typographics for the same thing - material equivalence.

Of course, though, one might wish to use a different symbol for material equivalence at the object level from material equivalence at the meta-level.

3. ## Re: Logical Vs Material equivalence

Hi Moeblee

I think there might be difference in philosophical logic. So as far as mathematics is concerned , $p\equiv q$ is same as $p\Leftrightarrow q$ , right ?

4. ## Re: Logical Vs Material equivalence

In ordinary symbolic logic, in ordinary mathematical logic, and in ordinary mathematics, I'd take those two symbols (the triple bar and the double arrow) to stand for the same thing.

Of course, in a given context, an author may stipulate some special usage. But, ordinarily, if I'm reading a book where the author uses the triple bar and another book where the author uses the double arrow, then I understand them both to refer to the same thing.

5. ## Re: Logical Vs Material equivalence

thanks for clarification. when you go to the wikipedia article on "Logical equivalence" , you see the links to the article on that subject in French and German. I don't understand
either of the languages but looking at the mathematical words and the length of the article , it seems to me that those articles are better written than the one in English...

6. ## Re: Logical Vs Material equivalence

I think the Wikipedia article on "Logical equivalence" makes sense. I am not sure if p ↔ q is called material implication and p ⇔ q is called logical implication. The important difference is between the object language and the metalanguage. The object language is the set of well-formed formulas. The metalanguage is our ordinary mathematical language, which we use to reason about the object language (for example, to say that a certain formula is true in a particular interpretation). Two formulas p and q are equivalent on the metalevel, denoted by p ⇔ q, if they are true in the same interpretations. Thus, p ⇔ q is a statement of the metalanguage. It talks about all interpretations. In contrast, p ↔ q is a formula of the object language, i.e., a sequence of characters. It is true in some interpretation I if p and q have the same truth value in I. So, p ⇔ q iff p ↔ q is a tautology.

The notation for equivalence in the object language and the metalanguage (↔, ⇔, ≡) is not universal. The distinction between object- and metalevel arises only in logic, which explicitly studies formal languages.

7. ## Re: Logical Vs Material equivalence

emakarov, thanks for the input. i don't quite understand the terms 'metalanguage' or 'object language' but I think as far as mathematics is concerned $p\leftrightarrow q$ , $p\Leftrightarrow q$ and $p\equiv q$ are same. that's what moebless said......

8. ## Re: Logical Vs Material equivalence

There's material equivalence at the object level and material equivalence at the meta-level. Personally, I've never heard of a terminology in which 'logical equivalence' is used for the meta-level and 'material equivalence' for the object level, but I guess there might be some text in which that terminology is used.