As you say, it's a side issue. I wouldn't have been unhappy with using "deduce" either, even though I don't think it's quite right.
The latter is right (and induce is wrong) when you say "It's a number from 1 to 8, but it isn't 8". I therefore deduce that it's "a number from 1 to 7".
I "induce" the general case that the silence was because he had two tails, because his statement "at least one of them is a head" after his previous silence acts in a different logical way on the previous silence after a coin toss from the way the statement "but it isn't 8" acts on the previous statement "it's a number from 1 to 8".
It was possible, for example, that the man was operating under a rule of which we were unaware (but which was not the rule that, if he threw two tails, he remained silent and tossed again) that required him to toss the coins again.
Therefore I felt that I was not using mathematical deduction in the Holmesian sense (eliminating all cases until whatever remained, must, however improbable, be the truth).
I was, instead, inducing a state of affairs in the world from a single instance.
The unfortunate fact that "induction" has other medical associations is an irritation, but I don't think I should allow that to discourage me from using it in the sense which does exist.
Re: 1 In 4, Baby, 4 In 1
The latter is right (and induce is wrong) when you say "It's a number from 1 to 8, but it isn't 8". I therefore deduce that it's "a number from 1 to 7".
I "induce" the general case that the silence was because he had two tails, because his statement "at least one of them is a head" after his previous silence acts in a different logical way on the previous silence after a coin toss from the way the statement "but it isn't 8" acts on the previous statement "it's a number from 1 to 8".
It was possible, for example, that the man was operating under a rule of which we were unaware (but which was not the rule that, if he threw two tails, he remained silent and tossed again) that required him to toss the coins again.
Therefore I felt that I was not using mathematical deduction in the Holmesian sense (eliminating all cases until whatever remained, must, however improbable, be the truth).
I was, instead, inducing a state of affairs in the world from a single instance.
The unfortunate fact that "induction" has other medical associations is an irritation, but I don't think I should allow that to discourage me from using it in the sense which does exist.
PJ