Talks at Appropriate Times
Posted by Neal on April 7, 2011
Doug’s report card came home last week, and on the list of nonacademic, behavioral characteristics, he had a minus for “Talks at appropriate time.” I knew from the conference with his teacher last month that Doug had no problem speaking up at appropriate times. What he does have a problem with is not talking at inappropriate times. I tweeted about the grade:
Glen tweeted in response:
Good point! In other words, take the conditional statement (1):
(1) If it’s an appropriate time, Doug talks.
Even if (1) is true, (2) doesn’t have to be true:
(2) If it’s not an appropriate time, Doug doesn’t talk.
Alternatively, Glen could have noted that a statement doesn’t imply its logical converse. That is, even if (1) is true, (3) doesn’t have to be true:
(3) If Doug talks, it’s an appropriate time.
In fact, (2) and (3) are logically equivalent. However, if you negate both clauses and reverse them — in other words, if you get the inverse of a statement and then get the converse of that — the resulting statement is still true. That’s known as the statement’s contrapositive:
(4) If Doug doesn’t talk, it’s not an appropriate time.
So a statement and its contrapositive are logically equivalent, and its inverse and converse are logically equivalent. This is easier to take in as a diagram (click to view full-size). Read p → q as “if p, then q,” and ¬ p as “not p” (or if you want “it is not the case that p“). The logically equivalent statements are in same-colored boxes; the statements in green are contrapositives of each other, as are the statements in blue. The arrows go two ways because the inverse of an inverse is the original statement, and likewise for the converse of a converse.
There’s one little problem in all this. The report card didn’t phrase the behavioral desideratum as a conditional. It didn’t say, “If it’s not an appropriate time, Doug will not talk.” It simply said (with an understood subject), “Talks at appropriate times.” Can that be accurately rephrased as a conditional in the first place? Take a sentence like:
Dagwood sings in the shower.
Does that mean that if Dagwood’s in the shower, he sings? Or does it mean that if Dagwood’s singing, he’s in the shower? Of course, generic sentences like this can have exceptions, but what is supposed to be more surprising: Dagwood taking a shower and not singing, or Dagwood singing in someplace other than the shower? Or is it supposed to be a biconditional: That Dagwood sings if and only if he’s in the shower? Any of those readings could be the intended one, based on context and intonation.
So in the same way, (Doug) talks at appropriate times could be interpreted either (1) or (2) (or to be complete, (3) or (4) as well). It’s only the context of that parent-teacher conference we had last month, and the general knowledge of how American kids are expected to behave in school, that tells me that (Doug) talks at appropriate times is to be interpreted as both (1) and (2), i.e. as the biconditional (5):
(5) Doug talks if and only if it’s an appropriate time.
Logicians have occasion to talk about biconditionals so much that they have abbreviated if and only if into iff. I don’t know exactly how that’s pronounced to distinguish it from if (probably just by saying “if and only if”), but in text it works, once you’ve realized it’s not a typo. But how can we get that biconditional reading unambiguously in the report card criterion, which isn’t phrased as a conditional? I tweeted back to Glen: