Can somebody point out the flaw in this logic? Its obviously wrong but I am not able to figure out what incorrect assumption was made.
The claim:
A Linear Time Invariant (LTI) system's output, $y(t)$, can be broken down into the Zero Input Response (ZIR) and the Zero State Response (ZSR) like so: $$y(t) = ZIR(t) + ZSR(t)$$
Say we have an LTI $L()$ whose input is $x(t)$ and output is $y(t)$. By definition, if $x(t) = 0, $ $\forall t$, then $$L(x(t)) = ZIR(t)$$.
Now lets say our input is $x(t) = 0 + 0, $ $\forall t$. Then by applying linearity and then the above definition, $$L(x(t)) = L(0 + 0) = L(0) + L(0) = 2ZIR(t)$$ Therefore the $ZIR(t)$ must be $0$ $\forall t$
I do not understand the first paragraph. You will have to define $ZIR(t)$ and $ZSR(t)$ in more detail.
If we start from $$L(0) = ZIR(t)$$ Then your derivation of $$ ZIR(t) = 2ZIR(t) $$ is correct, and the conclusion to draw from this is $$ ZIR(t) = 0 $$ $$ L(0) = 0$$ In other words, the response of an LTI system to zero input is zero output