Return to BS5

Sometimes, you return to something that you worked hard on in the past and an obvious aspect leaps out at you that wasn't obvious at the time.  In The Fans of Plantinga's BS5, I was arguing against the claim:

This was primarily because this is used to claim that, since we cannot prove that (a maximally great) god does not exist, it is possible that (a maximally great) god does exist, furthermore since we cannot prove that (a maximally great) god is does not exist necessarily it is possible that (a maximally great) god does exist necessarily and it therefore follows that (a maximally great) god does exist (necessarily).  Sounds like BS, right?

Anyway, I also argued against a more complex version of the argument (raised by cpdavey):

1. ◊□A (Assumption for Conditional Proof (CP))

2. ~□A (Assumption for Reductio Ad Absurdum (RAA))

3. ~~◊~A (2 E2)

4. ◊~A (3 Double Negation (DN))

5. □| ◊~A (4 S5-reit)

6. □◊~A (5 nec intro)

7. □~□~~A (6 E1)

8. □~□A (7 DN)

9. ~◊~~□A (8 E2)

10. ~◊□A (9 DN)

11. ◊□A & ~◊□A (1,10 Conjunction)

12. ~~□A (1-11 RAA)

13. □A (12 DN)

14. ◊□A -> □A Q.E.D. (1-13 CP)

I concluded that the Reductio Ad Absurdum wasn't actually necessary, because you can do it like this:

1. ◊□A (Initial position)
2. A <--> ~B (Assumption for Logical Fiddle)

3. ◊□~B (Substitution of 2 into 1)

4. ~~◊~~□~B (3 double DN)

5. ~□~□~B (4 E2)

6. ~□~□A (Substitution of 2 into 5)

7. □A (6 n2)

8. ◊□A -> □A Q.E.D. (1-7)

But here we can see that the step between (6) and (7) is not valid (because it's reversed) - so there's a problem which is being disguised by the process of going through the Reductio Ad Absurdum.

I pretty much left it at that, inviting the reader to point out where the problem is, if they could see it.  That was five years ago.

So, what was the obvious thing that recently leaped out at me?  The proof put together, or at least presented by cpdavey relies rather heavily on the introduction of double negatives and swapping from ~□~ to ◊ and from ~◊~ to □.  Both of these latter two moves can be reversed (see Classical Modal Logic).  What it doesn't rely on, is the value of A.  Which means we can do this:

1. ◊~G (Initial weakish atheism position)
2. ~□~~G (1 classical modal logic)

3. ~~□~□~~G (2 n2, right way around)

4. □~□G (3 DN removed)

5. ~◊□G (4 classic modal logic)

This means we arrive at the same conclusion whether we assume that it is possible that there is no god (◊~G) or that it is not necessary that there is a god (~□G), namely that it is not possible that there is necessarily a god (~◊□G), which is precisely what we would expect.

Effectively all cpdavey was arguing was that if his presumption was correct, then assuming the truth of the opposite of his presumption would lead to a conclusion that his presumption was incorrect.  Which it did and which we would expect.  That's why there's no great problem within his "Reductio Ad Absurdum"  but there is when we run his proof with no RAA.  His whole argument came down to a "heads I'm right, tails you're wrong" sort of con.

The error is quite clear in Step 12 above, where cpdavey writes "12. ~~□A (1-11 RAA)".  Clearly the RAA started at Step 2, not Step 1.

Sometimes we just don't see things that are staring us in the face.  Perhaps I was trying too hard to be clever.  But then again, so was cpdavey, since the result that he needed to get his RAA to work was precisely the one he was aiming at, via use of the RAA, ie the fallacious claim that, given ◊□A, □A.

A Little to Do with Nothing

There’s an interesting claim made in this Vox article, not so much about nothing as about zero.  Namely that young humans have a delay in processing a specific claim about a zero set, a set with nothing in it.

For example, using their image:

A child is asked (according to the Vox article) “to pick the card with the fewest number of objects”.  That’s a strange thing to ask, “fewest number”, but I guess it works, working from the top left and working clockwise, the answers would be 2, 4, 8 and 1.  By this I mean that there are two numbers of objects in the top right card, 1 and 2 (because there are two, there is also one, making two numbers).  Looking at the bottom left, there is only one number of objects, namely zero, but this would mean that a set of zero objects and a set of one object would both have one number of objects (0 and 1, respectively).

But let’s look past that and assume that the authors meant, but perhaps didn’t quite want to say “fewest objects”.  Looking at the second test, which card has the fewest objects?  The card with no objects on it, or the card with 8 objects on it?

My gut feeling is that it’s the one with 8 objects.  In a sense, the other one doesn’t have zero specific objects on it, it has no objects on it at all.  The logic here is to think, question 1: which of the cards has objects on it, and question 2: which of those cards has fewest objects?

Think of instructing people to go into a room, find the table with the “fewest number of cats on it”, and report back with that number.  If there are seven cats and eight tables, would you really expect them to come back every time with the number zero, or would you expect them to eliminate the empty tables from consideration and to count only the tables with cats on them?

If we are forced to consider such empty tables in terms of what are not on them, then the list of what is there (in lots of zero) is indefinitely long – there are no cats, no dogs, no chickens, no horses, no parrots, no rats, no cacti, etc, etc, etc.  That’s not a rational way to go about categorising things.  (Which table is it on, dear? – The one in the hallway, with the vase on it, and no cats, no dogs, no chickens, no horses, no parrots, no rats, no cacti …)

I don’t find it strange that large numbers of children don’t count a card with no objects on it as being the card with fewest objects on it, because it doesn’t qualify as being a card with objects on it.  I don’t find it strange to think that it takes a tiny bit more cognitive effort to consider whether you are being asked for the lowest non-zero value or the lowest value including zero, or to recognise O. (What is that, noting that it's in another font, is it the letter "o" or the number zero?)

That all said, I do have to say that I nevertheless appreciate the value of nothing, the use of which makes so much possible.

The Ol' Cat and Table Trick

Say you walk into a room with two tables in it and five cats.  One looks a bit like this:

The other looks a bit like this:

Which table has fewest cats on it?

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It really depends, doesn't it?