consider the following well-known \"proof\" that all finite collections of indiv
ID: 1943209 • Letter: C
Question
consider the following well-known "proof" that all finite collections of individuals are of the same gender: We argue by induction, let P(n) be the statement that all collections of exactly n individuals are of the same gender. Clearly P(l) is true. Suppose then that the statement is true for all collections of size n and let S be a collection of size n+1. Choose any two individuals a and b from S and let c be an individual from S who is distinct from a and b. let Sa = S -{a} and let Sb = S -{b}. Then both Sa and Sb have exactly n elements and both contain c. By the induction hypothesis, all the individuals in each of the sets Sa and Sb are of the same gender. Thus a and c of the same gender and also b and c. Hence a and b are of the same gender. But a and b were chosen arbitrarily from S, so all members of S are of the same gender, and the induction hypothesis is advanced. Of course the argument must be wrong. But why?Explanation / Answer
The fallacy lies in the fact that how do you move from n=1 to n=2. Of course P(1) is true, but how can we conclude that P(2) is true using P(1)? In a set of 2 persons, and given an individual in that set is of a particular gender (say female) gives us no clue how to conclude that the second will be of the same gender. That is, assuming P(1), we cannot prove P(2). This is where the fault is. And this is usually concealed. Note that the proof of induction given above assumes that there are at least two persons. Clearly, this induction step does not work from going 1 to 2, since you have only one person to start with. I hope you get the reasoning. Reply, if you need more explanation.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.