I might be confused about something fundamental here.
Why is this single notion of implication generating two very different Venn diagrams?
One is diagram of a subset relationship, which makes sense. $P\rightarrow Q$ means every member of $P$ is a member of $Q$. So on the diagram it would be $P$ inside $Q.$
Second is the diagram of the logical equivalent to implication: $\thicksim P \space \text{or} \space Q,$ which looks obviously very different.
How do I reconcile the two different Venn diagrams?
Best Answer
To compare two Venn diagrams for a Boolean combination of P,Q one needs for each to shade (or check off) the regions of each diagram for which the Boolean is true. In case of $P \implies Q,$ this means three regions: (P and Q), ((not P) and Q), and ((not P and (not Q)). That is, all regions checked except a region (if any exists in the diagram) for (P and (not Q)).
In the first diagram, with P inside Q, there isn't a region corresponding to (P and (not Q)), but the remaining three regions should be checked off, as noted above.
In the second (usual) Venn diagram, there are also exactly three regions to check off: All regions except that lying inside P and outside Q.
So in each diagram three regions are checked, and the same labels for each in the two diagrams. A confusing part in this may be that in the P inside Q case all regions are checked (there being only three such regions) but in the second "usual" Venn there are four regions with only three checked off. But the two diagrams correspond.