I recently discovered I have an Erdos Number of 3, i.e. I’ve published with someone who is 2 removed from publishing with Paul Erdos, the “Oddballs oddball.”

The links are as follows:

John Beverley coauthored with Yang Wang
Yang Wang coauthored with Frank Hsu
Frank Hsu coauthored with Paul Erdös

The distribution of Erdös numbers looks like (from “Facts about Erdos Numbers”):

Erdös number 0 ---      1 person
      Erdös number 1 ---    504 people
      Erdös number 2 ---   6593 people
      Erdös number 3 --- 33605 people +1 John Beverley
      Erdös number 4 --- 83641 people
      Erdös number 5 --- 87760 people
      Erdös number 6 --- 40014 people
      Erdös number 7 --- 11591 people
      Erdös number 8 ---   3146 people
      Erdös number 9 ---    819 people
      Erdös number 10 ---    244 people
      Erdös number 11 ---     68 people
      Erdös number 12 ---     23 people
      Erdös number 13 ---      5 people

I’ve also discovered that I’ve a Dijkstra Number of 4, through the following links:

John Beverley coauthored with Gilbert Omenn
Gilbert Omenn coauthored with Todd Smith
Todd Smith coauthored with Jayadev Misra
Jayadev Misra coauthored with Edsger Dijkstra

The above information all being pulled from co-authors.net.

Question

One sunny afternoon John asks Sally how she's doing, and Sally responds that she isn't happy. Sally then quotes the song by the band Garbage - "I'm only happy when it rains." John then claims Sally has committed a fallacy. Is John correct?

Answer

To get a feel for the question, consider whether you think the following is a good inference:

(i) S is only happy if it rains
(ii) It does not rain
(iii) Hence, S is not happy

Let "It rains" be denoted by Q and "S is happy" by P. If you think (i) has the form (Q->P) then this will be invalid. You might think this because "if" on its own seems to be introducing "It rains" as the antecedent of a material conditional. Then (i) is equivalent to: "It rains, only if S is happy." That is:

(1) It rains only if S is happy (Q->P)
(2) It does not rain (~Q)
(3) Hence, S is not happy (~P)

On the other hand, if you think (i) has the form (P->Q), then the argument is valid. You might think this because "only if" is typically taken to introduce the consequent of a material conditional, and "S is only happy if..." is plausibly read as "S is happy only if...". Then (i) is equivalent to "S is happy only if it rains." That is:

(4) S is happy only if it rains (P->Q)
(5) It does not rain (~Q)
(6) Hence, S is not happy (~P)

There are two further options as the setup permits another reading:

(iv) S is only happy if it rains
(v) S is not happy
(vi) Hence, it does not rain

So that taking "if" to introduce the antecedent, we have the valid:

(7) It rains only if S is happy (Q->P)
(8) S is not happy (~P)
(9) Hence, it does not rain (~Q)

And if you take "only happy if" to introduce the consequent, we have the invalid:

(10) S is happy only if it rains (P->Q)
(11) S is not happy (~P)
(12) Hence, it does not rain (~Q)

So, whether a fallacy has been committed depends on whether Sally intended to infer "it does not rain" or "Sally is not happy, as well as whether "Sally is only happy if it rains" should be read as "Sally is happy only if it rains" or "It rains only if Sally is happy."

Concerning the first, I take Sally to be supporting the claim "I'm not happy" by appealing to the fact that it is not raining, which is common knowledge. If this is correct, then the second premise of the argument should be "It does not rain." So we can rule out setup (vi)-(vi).

Concerning the second, I take Sally to be claiming that if she's happy, then it rains. That is, Sally is making a valid argument and John is incorrect. This reading is also consistent with Sally not being happy, but it raining nevertheless. In other words, all we know is that if Sally is happy then it's raining, and if it's not raining then Sally isn't happy. This is also a bit sadder. Sally isn't even happy all and only those times it rains. Rather, Sally is only happy some of the times it rains.