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2 Revised the title.

# Logic:Whatisthe difference between the biconditional iff. and equality = ?

Post Made Community Wiki by Anton Geraschenko♦♦
1

# Logic: difference between the biconditional iff. and = ?

Hello,

I've been used to writing logical transformations using equality, but the other day it struck me that perhaps I should be using the biconditional $\iff$?

So my question is: What is the difference between the biconditional iff. $\iff$ and equality = ? Can they be used interchangeably? And when should one be used instead of the other? Is this matter of style? meaning? or correctness?

Simple Examples to illustrate the point: here p,q are propositions, ^ is "and", v is "or", and ~ is "not":

Using equality:

(1) p ^ (q v r) = (p ^ q) v (p ^ r)

and

(2) p $\Rightarrow$ q = ~ p v q

But is this more correctly written using the biconditional iff.?

(1') p ^ (q v r) $\iff$ (p ^ q) v (p ^ r)

and

(2') p $\Rightarrow$ q $\iff$ ~ p v q

In rendering into English, are these logically equivalent statements? Equal statements (equality being considered as an equivalence relation that then establishes an equivalence class)? Or is it that the very meaning of equality in the context of logical propositions is iff.?