Wolfgang Schwarz

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A Quiz (about "or" in English and in philosophers' English)

Posted on Saturday, 13 Jan 2007.

<update 2007-01-18>The poll is closed. The results are pretty much as I expected.</update>


I've found this question in one of the papers Kai von Fintel mentioned in the comments to the previous entry, and I'm interested in how philosophers would answer it. There's another question after this one, and then a survey of the current results and a few comments.

Please answer without using pen and paper.

Suppose you know the following about a specific hand of cards:
  1. a. If there is a king in the hand then there is an ace in the hand,
    or else
    b. if there isn't a king in the hand then there is an ace in the hand.
  2. There is a king in the hand.
What, if anything, follows?
It follows that there is an ace in the hand.
It follows that there is no ace in the hand.
Neither of these follows.

Thanks, second question:

Suppose you know the following about a specific hand of cards:
  1. Either
    a. if there is a king in the hand then there is an ace in the hand,
    or
    b. if there is a king in the hand then there is a queen in the hand.
  2. There is a king in the hand.
What follows?
It follows that there is an ace in the hand.
It follows that there is a queen in the hand.
Neither of these follows.

Results:

Question 1:
There is an ace in the hand.
There is no ace in the hand.
Neither.

Question 2:
There is an ace in the hand.
There is a queen in the hand.
Neither.

The original question is from Johnson-Laird and Savary 1996, "Illusory inferences about probabilities". Apparently almost everyone answers that there is an ace in the hand, despite the fact that the first premise is a disjunction of conditionals and this conclusion only follows on the first disjunct (together with the second premise). Here it is again:

Suppose you know the following about a specific hand of cards:
  1. a. If there is a king in the hand then there is an ace in the hand,
    or else
    b. if there isn't a king in the hand then there is an ace in the hand.
  2. There is a king in the hand.
What, if anything, follows?

When I read the question, I thought that "neither" is clearly the correct answer. I'm not sure if that's because I've been trained to parse awkward sentences like 1 and to read "or" disjunctively, or if it's merely because I was reading a paper on disjunction and was therefore alert to disjunctions.

Bart Geurts, in the paper I'm reading, argues that 1, despite the "or", actually expresses a conjunction of conditionals, so that 1 alone entails that there is an ace in the hand. We sometimes do use disjunctions of conditionals in this way, but it seems to me that then the if-clause should always be somehow parenthetical:

Fred will be at the pub by now (if he got the message) or else (if he didn't get it) he'll still be in the office.

By comparison, the "or" seems a little out of place here, and should better be replaced by "and" or "on the other hand", I think (but again, this could be due to philosophical distortion):

If Fred got the message, he will be at the pub by now. Or, if he didn't get the message, he'll still be in the office.

I think sentence 1 is easily read conjunctively due to the "or else" and because the if-clauses are exclusive and exhaustive, and because the content on the conjunctive reading is easy to understand (no matter what, there is an ace in the hand), whereas the disjunctive reading is quite elusive (if the conditionals are material, the disjunctive content is empty). That's why I created the second example, which lacks all these features. I'd expect that in this case, the disjunctive reading is more common than the conjunctive one.

Comments

# on 13 January 2007, 23:40

since you said you were interested in how philosphers would answer your questions, I thought I ought to mention that I answered them too (and as you know I'm not a philosopher). I picked answer a) for Question 1 and both answers a) and b) for question 2. I think that 'neither' is definitely the wrong answer to question 1, as a hand of cards will always either contain an ace or not. Maybe the correct answer should be "who knows? Anything is possible." or so.

# on 14 January 2007, 02:54

oh dear, thanks ju. "Neither" was meant to be "neither of these follow", not "it follows that neither of these are true". If you thought the correct answer is "who knows? anything is possible", does that mean you agree it doesn't really follow that there's an ace in the hand?

Since this poll is thoroughly unscientific anyway, I've changed the wording now to make the intended meaning more clear. For the record, the results so far were:

Question 1:
17/23 There is an ace in the hand.
0/23 There is no ace in the hand.
6/23 Neither.

Question 2:
3/21 There is an ace in the hand.
3/21 There is a queen in the hand.
16/21 Neither.

(Apparently two people have given up after the first question. Or maybe the form doesn't work in certain browsers?)

# on 14 January 2007, 21:40

for your tec question: I failed to go to the 2nd question with FF 1.5 without Java, IE 5.5 with Java did not fail.

JU's point is interesting, since it (not sure of legimatelty) brings up a "realistic" perspective, not an interpretation focused on implicature.
I went for "neither" in both cases, since I took the epistemlogical interpretation and reasoned that from contradiction nothing follows (which is of course strange to say ...).
Interesting quiz for sure, the semantical pitfalls discovered looks like one should have a look at the theories that tell us how we get from natural language sentences/utterances to "logical form" (taken in the philosophers sense of the word). What is your background assumption on this?

M.

# on 15 January 2007, 18:04

> does that mean you agree it doesn't really follow that there's an ace in the hand?

only after reading your comments. I don't think I understood the significance of 'or else' - I just thought 'that's a funny way of putting it'...

# on 16 January 2007, 19:10

I definitely read the 'or else' as a conjunction; it's fairly natural to read it as one given that the antecedents of the the two conditionals exhaust the logical possibilities. The way I read 'or else if ...' might be roughly parsed as 'and, in the alternate case where there isn't a King in the hand ...'

# on 18 January 2007, 01:03

Wow, look how dramatically the ratio of A to C answers has decreased since you changed the wording in response to ju's comment. It was 17/6 = 2.8 and now it's 80/61 = 1.3! Of course, the total number of answers was quite small before.

Would you mind sharing the original wording?

# on 18 January 2007, 13:04

I've closed the vote now.

The original wording was as it still is in the results: "There is an ace in the hand/There is no ace in the hand/Neither".

I don't have an interesting theory about this case. I still believe (unlike Geurts) the "or" is good old boolean disjunction here, as it is by most people's light in the second question.

As I said in the posting, there are related sentences where the 'conjunctive' reading is very natural, but that's because the if-clauses there are somehow parenthetical remarks about the disjuncts.

What puzzled me is that even though I can see the conjunctive reading in the original case (on which "there is an ace in the hand" follows), the other reading (on which nothing follows) seemed more natural to me. So I was puzzled that Johnson-Laird and Savary, who believe that "neither" is the correct answer, report that "only one person among the many distinguished cognitive scientists to whom we have given the problem made the correct response. Several hundred individuals at public lectures from Stockholm to Seattle have made the same error, just one person asserted that nothing follows from the premises" (p.219, quoted on p.399 in the Geurts paper). It looks like the results would have been significantly different if they had asked their colleagues in the philosophy department.

# on 18 January 2007, 14:24

btw, here are some things I don't understand:

- I can understand that the total number of replies to question 2 is larger than the number of participants (seems like I wasn't the only one who chose two answers instead of just one), but why does option a) have one participant less (138) than options b) and c)(139)?

- For question 1, the total number or replies is 149, but the total number of participants is 150. How come?

# on 18 January 2007, 15:32

The second on is simple: you could tick any number of checkboxes before submitting, including 0. The 138-139 difference is stranger. The three checkboxes get their state transmitted to the server independently, by three different GET requests. So it is theoretically possible that only two of the requests make it through. The relevant requests were made from a computer in China (61.51.122.227), and for some reason, its requests took a very long time to reach my server: the three ones for the first question span 14 seconds. Maybe the great Chinese firewall interfered, blocking this site because of all my political propaganda (or because I'm using votes)?

# on 19 January 2007, 14:37

ah, that would make sense.

Actually, the thing that surprised me most is how many people read your blog! :-)

# on 19 January 2007, 21:22

and let me add to Ju: Who is reading it ... impressive

M.

# on 19 January 2007, 21:46

2nd thoughts on 2nd reading: What wording (if any) did you change?
I know that I voted for "neither"/"neither", but now I would rather go for (1) Ace, and (2) Ace and Queen .... so much for the stability of semantic intuitions ...

M.

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