Getting your head around deductive rules of inference

Make your arguments unkillable

I really like formal logic. This is probably because it makes a lot of sense to me but everytime I try to teach it I am often met with confusion. I'm trying to figure out why that is the case so this is an attempt to correct this.

Modus Ponens

The most classic of them all. Latin for method of affirming. It looks like this:

\begin{array}{rll} 1. & P \supset Q & \text{Premise}\newline 2. & P & \text{Premise}\newline 3. & Q & \text{Modus Ponens from 1, 2} \end{array}

This is also Modus Ponens:

\begin{array}{rll} 1. & \neg P \supset Q & \text{Premise}\newline 2. & \neg P & \text{Premise}\newline 3. & Q & \text{Modus Ponens from 1, 2} \end{array}

And so is this:

\begin{array}{rll} 1. & \neg P \supset \neg Q & \text{Premise}\newline 2. & \neg P & \text{Premise}\newline 3. & \neg Q & \text{Modus Ponens from 1, 2} \end{array}

This is also Modus Ponens:

\begin{array}{rll} 1. & (P \And Q) \supset (R \And S) & \text{Premise}\newline 2. & (P \And Q) & \text{Premise}\newline 3. & (R \And S) & \text{Modus Ponens from 1, 2} \end{array}

And suppose I really wanted to get freaky with it:

\begin{array}{rll} 1. & (P \supset Q) \supset (R \vee S) & \text{Premise}\newline 2. & (P \supset Q) & \text{Premise}\newline 3. & (R \vee S) & \text{Modus Ponens from 1, 2} \end{array}