Transitivity fail

More entertainment with syllogisms. This one came up in Facebook discussion about my “twin primes” posting, in which I referred to

the first of two twin prime birthdays: 71 and 73. Previous pair, 59 and 61. Next one, 101 and 103, which I’m not likely to see.

Frank McQuarry then played with this, saying:

You’ll never be a Mersenne prime again either. Unless you manage to make it to 127.

To which I replied:

I haven’t been a Mersenne prime for 40 years!

(31 is the Mersenne prime before 127.)

What Frank introduced here was the playful use of Mersenne prime as denoting a property of human beings. Leading to the possibility of reasoning like the following:

Kim is 31.

31 is a Mersenne prime.

∴ Kim is a Mersenne prime.

At first glance, this looks like a simple syllogism based on the transitivity of identity (related to the transitivity of implication and to the categorical syllogism Barbara of Aristotelian logic — all A are B; all B are C; ∴ all A are C). But of course it turns on the ambiguity of the verb be, which has different senses in the two premises, neither sense being identity.

The moral of the story is the familiar one that syllogisms are generalizations about semantic forms, not syntactic forms; applying syllogistic reasoning to sentences of natural languages requires a “translation” of those sentences that exposes their logical form. And neither “Kim is 31” nor “31 is a prime number” translates into an identity form; instead, they predicate properties (age, of the person Kim; primeness, of the number 31). So the transitivity of identity doesn’t come into the matter — though it makes for a cute play on words.


One Response to “Transitivity fail”

  1. Victor Steinbok Says:

    “Kim is a Mersenne prime.”

    No, no, no… you simply must fix this! Perhaps this is better:

    Kim is in her Mersenne prime.

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