# Logical null question

Is there any real difference between the Principal of Reciprocal Non-Identity and the Dictum de Nullo? If so, how do you explain the difference?

Principle of Reciprocal Non-Identity: If $a=b$ and $b \not = c$, then $a \not = c$

Dictum de Nullo: If all $A$'s are $B$'s, and no $B$'s are $C$'s, then no $A$'s are $C$'s

Well, they are certainly similar, especially if we express them like this:

Principle of Reciprocal Non-Identity: If $a$ is $b$ and $b$ is not $c$, then $a$ is not $c$

Dictum de Nullo: If every $A$ is a $B$ and no $B$ is a $C$, then no $A$ is a $C$

However, one important difference is that the 'is' in the Principe of Non-Reciprocal Non-Identity is a diffrent kind of 'is' than in the Dictum de Nullo: The former expresses equality (of specific objects) while the latter does not (and instead expresses the inclusion of one class of objects inside another). Thus, if we switch the left and right side of the 'is' in the Principle, it still holds, but this does not work for the Dictum.

Example:

If $b=a$ and $b \not = c$, then $a \not = c$ ... Still works!

If every $B$ is a $A$, and no $B$ is a $C$, then no $A$ is a $C$ ... No longer works!

Expressing them symbolically may make the distrinction clearer.

Principle of Reciprocal Non-Identity: If $a=b$ and $b≠c$, then $a≠c$

$$a=b~,~ a\neq c ~\vdash~ a\neq c$$

Dictum de Nullo: If all $A$'s are $B$'s, and no $B$'s are $C$'s, then no $A$'s are $C$'s.

$$A\subseteq B~,~B\subseteq C^\complement~\vdash~ A\subseteq C^\complement\\ \text{or}\\\forall a~(a\in A\to a\in B)~,~\forall a~(a\in B\to a\notin C)~\vdash~\forall a~(a\in A\to a\notin C)$$