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Social Sciences and Humanities

SSH 105

David Hunter

Fall

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Chapter 5 – Reasoning About
Alternatives and Necessary And
Sufficient Conditions
Quick Recap – Chapter 3 we studied what it is for our evidence to be sufficient, and in Chapter
4 we studied what it is for our evidence to be acceptable. Really helped me to understand the
difference between necessary and sufficient:
It is necessary to have $1 Million to be a millionaire. If you have $50 Million you are a
millionaire, having $50 Million is sufficient to be a millionaire, but not necessary because
you could have $20 Million and be a millionaire, or $1.2 Million and you would still be a
millionaire.
• Reasoning About Alternatives
o We always decide what to believe or what to do when faced with several
alternative possibilities from among which we have to choose, and we try to
reason to a conclusion by ruling some of them out. For example Jones can
continue his education by going to either law school or medical school, or he can
follow his bliss by pursuing an acting career. I
o When we reason among alternatives, we usually formulate them using a
disjunction. A disjunction is just a statement using the word “or”. Remember that
a disjunction does not assert either disjunct, it asserts the relationship between
the two, that the solution is one or the other.
A powerful reasoning tool is Denying a Disjunct. This reasoning starts by
listing a series of possibilities in the form of a disjunction, and then
denying one or more of the disjuncts, concluding that the remaining
disjunct must be true. Using the False Premise Test in Chapter 3, we see
that the premises are dependent. If any of the premises are false, the
conclusion could not be supported. Therefore we see that denying a
disjunct is a VALID form of reasoning because fi the premise are true, the
conclusion must be true too.
• Think of a classic “Sherlock Holmes” Murder Case. P or Q or R
committed the crime
Ex: Either P or Q or R is the case
But it is not the case that P And it is not the case that R
So, it must be the case that Q
• Ex 2: The patient either has a viral infection or cancer. The doctor
is able to rule out cancer, therefore the patient has a viral
infection.
False Disjunctions
• Several ways reasoning about alternatives can go wrong. We want our reasons to
support our conclusion (valid) but we also want our reasons to be acceptable.
o This means we want our disjunctions and our claims denying a disjunct to be true
or at least acceptable
• The Mistake of False Disjunction (sometimes called Mistake of False Dilemma) is
when an argument has a false disjunction as a premise.
o Although a valid argument with false premises might still have a true conclusion,
this is not the case in disjunctions. If the disjunction is false – neither disjunct is
true – then reasoning by denying the disjunct will inevitably have a false result.
To summarize, reasoning by denying a disjunct will yield a false conclusion when
the disjunction itself is false.
Example: The car will not start. Either its battery is dead or it is out of gas.
I just checked and there is plenty of gas. So, the battery must be dead
• Suppose that the disjunction is false, and the battery is fine and
the car has lots of gas. This results in the arguments conclusion,
that the battery is dead, to be false too.
When are Disjunctions Acceptable?
• One way to know if a disjunction is true is to know which one of the disjuncts is true. But
if we knew this, we would not need to reason about alternatives. We need to make sure
that our disjunctions are true, even if we do not know which disjunct is true.
• We need to make our disjunction exhaustive. A disjunction is exhaustive when it
includes all the possibilities that have not yet been ruled out. If our alternative reasoning
includes all the possible alternatives, we will be guaranteed to have a true disjunction.
Exhaustive disjunctions help us avoid the Mistake of False Disjunction.
o They also help us to avoid a related second mistake, the Mistake Of The Lucky
Disjunction.
For example, consider the doctor previously mentioned overlooked other
possible conditions that have the exact same symptoms, such as a vitamin deficiency, bladder infection, etc. Although it was in fact a viral
infection causing issues for the patient, we would say that she did not
know the patient had a viral infection, she just got lucky. She should have
done a more exhaustive study, and relied on an exhaustive disjunction.
So, exhaustive disjunctions are preferable not just because they are
guaranteed to be true (Mistake of False Disj), they also eliminate the
need to rely on luck in our reasoning (Mistake of Lucky Disj)
• How can we know that a disjunction is exhaustive? Maybe we just need a great mind like
that of Einstein’s to think outside of the box and conceive of a new possibility for a set of
symptoms.
o Believing that there are no possibilities on the grounds that one does not know or
cannot think of any more is to commit a special version of the Mistake of
Appealing To Ignorance: it is a mistake to believe that something is the case
simply because it has not been proven otherwise.
Ex: James must have spilled the milk. It was either him or the cat who did
it. If the cat did it, she would be covered in milk, but she is dry. This is why
I am sure James did it.
The author could not think of other reasons, is not sufficient reason to
believe that there are only 2 possibilities. We must have exhaustive
disjunctions, even if they are hard to create, we cannot ignore or overlook
possibilities.
Exclusive Disjunctions
• An exhaustive disjunction is one that includes all possibilities that have not been ruled
out, but a disjunction can also be exclusive.
o A disjunction is exclusive just in case at most one of its disjuncts is true. Some
examples are:
Jane’s baby is a boy or a girl
The soccer team won, tied, or lost
The symptoms are either caused by cancer or by something else.
o We know that a set of possibilities is exclusive (boy or girl) but we still use a
disjunction to state alternatives. This can be misleading, because disjuncts
sometimes allow more than one to be true, therefore if we know that at most one
is true we should say this explicitly. o Consider the case: “Either the maid did it or the butler did it. The butler just
confessed. Therefore the maid is innocent”
This form of reasoning is called affirming a disjunct. Unlike denying a
disjunct, which is always valid, affirming a disjunct is valid only when the
disjunction is exclusive. We can rule out possibilities only when we know
that the possibilities are incompatible/
If we know that the disjunction we are reasoning with is exclusive, this bit
of knowledge is in effect a premise we are relying on when we reason by
affirming a disjunct, so we should make premise an explicit part of our
reasoning. Using the previous example:
• “Either the maid did it or the butler did it. And I know that the
murderer acted alone. And the butler just confessed. Therefore,
the maid is innocent after all.
• Another example using Symbols:
Either P or Q is the case
But only one of them can be the case
P is the case
Therefore Q is not the case.
• Mistakes To Avoid: Affirming A Disjunct
o Affirming a disjunct is reasoning from truth of one disjunct to conclude other
disjuncts false. It is valid only if the disjunction is an exclusive disjunction. If one
knows the disjunction is exclusive, it should add this information as an additional
premise.
To make things more complicated, a disjunction can be partly exclusive or
wholly exclusive, but more on that later
Criticizing Reasoning about Alternatives
• When reading an argument in a newspaper or book, we should ask ourselves whether
the author is committing any mistakes we noted. We also have to be careful not to
ridicule the author’s view by noting possibilities that are extremely unlikely or
unreasonable. Usually there are many possibilities that are simply out of the question,
and are not “live options”.
• For example, perhaps building a land bridge might be an alternative to building a
new bridge or renovating the existing one. But, depending on the discussion it
might simply not be a possibility. This is a case of the Red Herring Fallacy,
where one introduces an irrelevant possibility simply in order to better criticize the author’s position. This can also include charging the author with relying on a non-
exhaustive disjunction when only the feasible possibilities were mentioned.
5.2 Reasoning About Sufficient And Necessary Conditions
• A quick review of conditionals. The part that follows the “if” is the antecedent, and the
part that follows the “then” is the consequent. Neither the antecedent nor consequent is
asserted, rather is claimed that if the antecedent is true then the consequent must be
true too. When asserting a conditional, one is asserting two things:
• It is asserted that the truth of the antecedent is sufficient for the truth of the
consequent
• It is asserting that the truth of the consequent is necessary for the truth of the
antecedent.
Let’s analyze each separately
• Sufficient Conditions – If a conditional is true, then the truth of the antecedent is
sufficient for the truth of the consequent. It is the idea behind the notion of validity.
This simply means is that for the consequent to be true is for the antecedent to be true.
• Ex: One way to become a millionaire is to win the lottery. Winning the lottery is
sufficient for becoming a millionaire.
• If Stephen is the prime minister of Canada, then Stephen is a politician. One way
for Stephen to be a politician is for him to be the prime minister of Canada. Being
prime minister is sufficient for, or guarantees, that one is a politician.
• Necessary Conditions – The second point is that if a conditional is true, the truth of
the consequent is necessary for the truth of the antecedent. In other words, if the
conditional is true, there is no way the antecedent could be true if the consequent were
false.
• Ex: If you did not pass the course, then you could not get an A. If you are not a
millionaire, you did not win the lottery. There are many necessary conditions for
something.
• Necessary and Sufficient Conditions
• A condition that is necessary for something can also be sufficient for it. Jones
being an unmarried male is necessary for his being a bachelor, but it is also
sufficient. To be a bachelor, you must be male and unmarried. Those two
conditions are each necessary, though neither one is sufficient on its own for
being a bachelor, but together they are sufficient. They are individually necessary
and jointly sufficient. • We can state the fact that certain conditions are individually necessary and jointly
sufficient using a conjunction of conditionals:
“If jones is a bachelor, then he is an unmarried man and if he is an
unmarried man, then he is a bachelor.”
We can join conditionals with what we call a “bi-condit

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