Question
Introduction
This is an
olympiad type of question that tests one’s use of logic. Sadly, logic is seldom taught in schools,
except maybe in courses like knowledge inquiry, theory of knowledge, or in selected topics like geometry. Mathematics
is actually very much connected with logic, but it seems to be regarded as
difficult to teach. Being able to use logic
and think critically is very important in our lives in various aspects.
Let us
study how we can analyse conditional
statements using logic.
Analysing Conditional Statement with
Logic
A conditional
statement is a statement of the form “If H then C.”
An example would be: “If the blue litmus paper turns red, then the liquid
is an acid”. The front clause H (the part about the litmus paper
turning red) is called the hypothesis (or premise). The latter clause C (the part about the acid) is called the conclusion. Every conditional statement has three other
related forms: the contrapositive, the converse, and the inverse.
For example, let say there is a NC16-rated movie and you can watch it only if you are at least 16 years old. So H := “watch NC16” and C := “16 years old and above”. The conditional statement means “If you want to watch the movie, then you must be 16 years old and above”. The contrapositive is “If you below 16 years old, then you cannot watch the movie”. The contrapositive always has exactly the same meaning as the original conditional statement. If one is true, so is the other. If one is false, so is the other. The contrapositive is merely phrased in a different way.
For example, let say there is a NC16-rated movie and you can watch it only if you are at least 16 years old. So H := “watch NC16” and C := “16 years old and above”. The conditional statement means “If you want to watch the movie, then you must be 16 years old and above”. The contrapositive is “If you below 16 years old, then you cannot watch the movie”. The contrapositive always has exactly the same meaning as the original conditional statement. If one is true, so is the other. If one is false, so is the other. The contrapositive is merely phrased in a different way.
The converse says “If you are
16 and above, then you will watch the movie.”
The inverse would say “If you don’t want to watch the movie, then
you are not 16 and above.” Note that the
inverse is the contrapositive of the converse.
The inverse always has exactly the same
meaning as the converse. If one is true, so is the other. If one is false, so is the other.
The converse (and the inverse) has a different meaning
from the original statement. If the
original statment is true, the converse may or may not be true. In our
example, “If you want to watch the movie, then you must be 16 years old and
above” is true, but the converse “If you are 16 years old and above, then you
want to watch the movie.” may not be
true, as some people who are 16 years old and above may want to do something
else. Like reading a book. Or playing golf. So you see, there are really two camps: the original conditional statement and its contrapositive in one camp, and the converse and the inverse in the other.
Let us
analyse the given problem using logic.
Analysis
The
statement ‘any card with a letter A on one side always has the number “1” on the other side’ can be rephrased
as ‘If letter A appears, then the
other side is “1”’. Let us analyse each position in order. For (a), since the letter A appears, we definitely need to know
what is behind the card to test whether the claim is true. For (b), note that the claim does not say ‘If
the letter is not A, then the other
side is not “1”’ (inverse)
nor ‘If the number is “1”, then the other side must be an A.’ (converse). It is possible
that A is on the other side and we
are fine. But it is also possible to
have a non-A with the number “1”. The given condition does not prohibit
that. So, if the other side is not an A,
it is still OK. So the other side can be
anything, and it does not matter. For
(c) we definitely have to flip to check the other side in case the other side
is an A, then the claim would be
proven false. For (d), since the card is
not an A, it is irrelevant. The claim is talking about A. It is not talking about B. We can summarise the
analysis in the diagram below:-
Ans: We need to flip (a) and
(c)
Suitable Levels
* Primary School Olympiad
* syllabuses that involve logic and epistemology
