We say a system is causal if and only if for any $x_1(t) , x_2(t)$ and $t_0 \in \mathbb{R}$ $$x_1(t) = x_2(t) , \ \ \ \forall t\lt t_0 \implies Lx_1(t) = Lx_2(t), \ \ \ \forall t \lt t_0$$ In words, it means that the output of $L$ at any time depends only on past values of input. Note that $x_1,x_2 : \mathbb{R} \to \mathbb{C}$ are complex-valued functions of a real variable. I'm trying to find the contrapositive of the above proposition. I think the answer is
There exists $x_1(t) , x_2(t)$ and $t_0 \in \mathbb{R}$ such that $$\exists t\lt t_0 : \ \ Lx_1(t) \not= Lx_2(t) \implies \ \ \ \exists t\lt t_0 : x_1(t)\not = x_2(t) $$
I don't know if this is really meaningful. Maybe "There exists" should be replaced by "For all". Multiple quantifiers complicate the problem. So what's the correct answer?
The contrapositive of the statement $$\forall t_0\in\mathbb{R},\ \forall x_1,x_2\in C(\mathbb{R}),\quad(\forall t<t_0,x_1(t)=x_2(t))\implies(\forall t<t_0,Lx_1(t)=Lx_2(t))$$ is $$\forall t_0\in\mathbb{R},\ \forall x_1,x_2\in C(\mathbb{R}),\quad(\exists t<t_0,Lx_1(t)\ne x_2(t))\implies(\exists t<t_0,x_1(t)\ne x_2(t))$$
[I'm assuming the functions $x_i(t)$ belong to $C(\mathbb{R})$; of course, this can be changed to whatever space they belong to.]