"I think I see the problem.  The Taldane word 'implies' probably means all sorts of vague things besides... anyways.  Let's use material implication for the particular kind 'implies' we used here.  Now, we're going to have to erase this wall soon, but let's look back at the blue circles.  In particular, let's look at this blue circle containing a large red triangle, a large blue square, a small blue square, and a large red square.  The way I define material implication, we can take the statement 'For all z, z being triangular, materially implies z being red, and say that it's true of every object z, including the ones that aren't triangles.  We could look at this small blue square, and say of it truthfully, 'if a small blue square is triangular, then a small blue square is red' - the way we're defining material implication, that symbol I wrote like this," Keltham points to a -> symbol, "that would be a true thing to say.  Why define it that way?  So that the statement over here," Keltham points to \ h. red(h) -> large(h), "can be true when we evaluate it at every object h could refer to, including the objects that aren't red at all.  If we said that 'red h materially implies large h' was false whenever h wasn't red, putting a blue square in the world would mean we could not say of it, 'for every object in the world, the redness of that object materially implies its largeness'."

"Now, wanna take another shot at 'if p materially implying q materially implies p, then p'?  True across all possible worlds, or false in some of them?"

"I think I see the problem.  The Taldane word 'implies' probably means all sorts of vague things besides... anyways.  Let's use 'material implication' to narrowly denote for the particular kind of 'implies' I used here.  Now, we're going to have to erase this wall soon, but let's look back at the blue circles.  In particular, let's look at this blue circle containing a large red triangle, a large blue square, a small blue square, and a large red square.  The way I define material implication, we can take the statement 'For all z, z being triangular, materially implies z being red, and say that it's true of every object z, including the ones that aren't triangles.  We could look at this small blue square, and say of it truthfully, 'if a small blue square is triangular, then a small blue square is red' - the way we're defining material implication, that symbol I wrote like this," Keltham points to a -> symbol, "that would be a true thing to say.  Why define it that way?  So that the statement over here," Keltham points to \ h. red(h) -> large(h), "can be true when we evaluate it at every object h could refer to, including the objects that aren't red at all.  If we said that 'red h materially implies large h' was false whenever h wasn't red, putting a blue square in the world would mean we could not say of it, 'for every object in the world, the redness of that object materially implies its largeness'."

"Now, wanna take another shot at 'if p materially implying q materially implies p, then p'?  True across all possible worlds, or false in some of them?"

"I think I see the problem.  The Taldane word 'implies' probably means all sorts of vague things besides... anyways.  Let's use 'material implication' to narrowly denote the particular kind of 'implies' I used here.  Now, we're going to have to erase this wall soon, but let's look back at the blue circles.  In particular, let's look at this blue circle containing a large red triangle, a large blue square, a small blue square, and a large red square.  The way I define material implication, we can take the statement 'For all z, z being triangular, materially implies z being red, and say that it's true of every object z, including the ones that aren't triangles.  We could look at this small blue square, and say of it truthfully, 'if a small blue square is triangular, then a small blue square is red' - the way we're defining material implication, that symbol I wrote like this," Keltham points to a -> symbol, "that would be a true thing to say.  Why define it that way?  So that the statement over here," Keltham points to \ h. red(h) -> large(h), "can be true when we evaluate it at every object h could refer to, including the objects that aren't red at all.  If we said that 'red h materially implies large h' was false whenever h wasn't red, putting a blue square in the world would mean we could not say of it, 'for every object in the world, the redness of that object materially implies its largeness'."

"Now, wanna take another shot at 'if p materially implying q materially implies p, then p'?  True across all possible worlds, or false in some of them?"