Subsection 7.5.6 Most
A common kind of everyday claim involves the notion of “most”. Let’s analyze “most” reasoning more carefully since it illustrates one fundamental way in which statistical reasoning differs from the kind of absolute reasoning that we’ve been studying.
We’ve shown that:
∀x (P(x) → Q(x))
∀x (Q(x) → R(x))
∴ ∀x (P(x) → R(x))
So, for example, if all cats are mammals and all mammals have lungs, then all cats have lungs.
But now suppose that we tried to introduce a new quantifier (sort of an upside down M) corresponding to the idea of “most”:
x (P(x) → Q(x))
x (Q(x) → R(x))
∴ x (P(x) → R(x))
If we allowed this reasoning, we’d get some useful things.
Most soda has a lot of sugar.
Most sugary stuff is unhealthy.
∴ Most soda is unhealthy
But we’d also get some junk.
Most American adults can read.
Most adults who can read are not Americans.
∴ Most American adults are not Americans.
This sort of attempt to reason with most can fail even if one of the quantifiers is universal.
All babies are children.
Most children go to school.
∴ Most babies go to school.
We need a statistically based reasoning system to work with problems like these.
Exercises Exercises
1.
Define:
P(x): True if x is an American soldier fighting in Afghanistan.
Q(x): True if x was born in the United States.
R(x): True if x has never been to Afghanistan.
Assume that most American soldiers fighting in Afghanistan were born in the United States and that most people born in the United States have never been to Afghanistan. Consider this argument:
x (P(x) → Q(x))
x (Q(x) → R(x))
∴ x (P(x) → R(x))
In this case, is the conclusion true or false?
2.
Define:
P(x): True if x is a knife.
Q(x): True if x is sharp.
R(x): True if x is dangerous.
Assume that most knives are sharp and that most sharp things are dangerous. Consider this argument:
x (P(x) → Q(x))
x (Q(x) → R(x))
∴ x (P(x) → R(x))
In this case, is the conclusion true or false?
3.
Define:
P(x): True if x is a kitten.
Q(x): True if x is fluffy.
R(x): True if x is cute.
Assume that most kittens are fluffy and that most fluffy things are cute. Consider this argument:
x (P(x) → Q(x))
x (Q(x) → R(x))
∴ x (P(x) → R(x))
In this case, is the conclusion true or false?
4.
Define:
P(x): True if x is a bird.
Q(x): True if x can fly.
R(x): True if x is an insect.
Assume that most birds can fly and that most things that can fly are insects. Consider this argument:
x (P(x) → Q(x))
x (Q(x) → R(x))
∴ x (P(x) → R(x))
In this case, is the conclusion true or false?
5.
Define:
PB(x): True if x is a professional basketball player.
RH(x): True if x is right-handed.
PA(x): True if x is a professional athlete.
Assume that most professional basketball players are right-handed and that most right-handed people are not professional athletes. Consider this argument:
x (PB(x) → RH(x))
x (RH(x) → ¬PA(x))
∴ x (PB(x) → ¬PA(x))
In this case, is the conclusion true or false?