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.?


(Can't find the Community Wiki tag! someone please add it, thanks.)



What is the difference between the biconditional iff. and equality = ?
Usually the biconditional is denoted by $\leftrightarrow$ and logical equivalence is represented by $\Leftrightarrow$.


Given two compound propositions $P$ and $Q$, the proposition $P \Leftrightarrow Q$ means that $P$ and $Q$ have the same truth value for each possible combination of truth values of the variables of which they are composed. That is, $P \Leftrightarrow Q$ means that $P \leftrightarrow Q$ is a tautology (i.e., a proposition that is always true). It's important to note that $\leftrightarrow$ is a connective, whereas $\Leftrightarrow$ is like an "equals sign" for propositions. More formally, the expression $P \Leftrightarrow Q$ is really a proposition about a proposition, namely that "$P \leftrightarrow Q$ is a tautology", whereas the expression $P \leftrightarrow Q$ is just a compound proposition that may or may not be a tautology.


Likewise, the conditional is usually denoted by $\rightarrow$ and logical implication is represented by $\Rightarrow$. Given two compound proposition $P$ and $Q$, the proposition $P \Rightarrow Q$ means $Q$ is true whenever $P$ is true, i.e., $P \Rightarrow Q$ means that $P \rightarrow Q$ is a tautology.


It seems that you may have confused logical equivalence ($\Leftrightarrow$) with the biconditional connective ($\leftrightarrow$) and logical implication ($\Rightarrow$) with the conditional connective ($\rightarrow$).


