Download Counterexamples 2013

Counterexamples
A simple logic proof method.
“Valid”
• In ordinary language: often used as a
synonym for “Good”.
• Daytime TV is full of people with ‘valid’
opinions who make ‘valid’ points.
• Don’t use “valid” this way (for this class)!
– Try to never use it this way!
• Valid has a very precise technical meaning.
What Valid really means.
• A valid argument is one that guarantees the
truth of the conclusion whenever the
premises are true.
• Example Valid argument.
– All dogs are canines.
– No Cats are canines.
– No dogs are cats.
But, valid is a conditional concept
• Arguments whose premises are false, but
which would guarantee the conclusion if the
premises are true, are still valid.
• Eg: valid argument with false premises.
– All dogs are cats.
– All cats are whales.
– All dogs are whales.
Obviously false. (O.F.)
OF
OF
Think of it this way…
• If all dogs were cats, and
• If all cats were whales
• Then all dogs would be whales.
• “Valid” is a description of arguments,
– not parts of arguments
• (which can be either true or false)
We are not good at “hearing” validity.
• We are often focused on meaning and
“truthfulness”.
• It takes a lot of practise to “hear” good
arguments vs bad.
More odd Valid arguments.
• All dogs are cats.
OF
• All cats are canines. OF
• All dogs are canines. Obviously true.
• If all dogs were cats and all cats were canines,
Then it would be true that all dogs are canines.
• This example illustrates that the falsity of the
premises doesn’t imply the falsity of the
conclusion.
This is a valid argument
• All X are Y
• All Y are Z
• Therefore all X are Z.
• Despite the fact that you don’t know what X, Y or
Z represent.
• Ignore “Actual” truth or falsity when considering
validity, concentrate on the relationship between
the premises and the conclusion.
Examples of invalid.
• All dogs are mammals.
• All cats are mammals.
• All dogs are cats.
O.F.
• All dogs are mammals
• All cats are mammals
• No dogs are cats.
– (all true, but still invalid for reasons as follows…)
How do you know that is invalid?
• Any argument that permits a counterexample
is invalid.
• A counterexample is an argument with the
same form as the original argument, but
which has
– obviously true premises and an
– obviously false conclusion.
Counterexample Example
Original argument
• All x are y
• All z are y
• No x are y.
Counterexample:
All fish are cold-blooded.
All spiders are cold-blooded.
Therefore no fish are spiders.
Using Counterexamples
• Counterexamples are a form of PROOF that an
argument is invalid.
• They are also effective in contexts where
people don’t know their logic.
• Someone who has never studied reasoning
can often be convinced “by ear” that their
argument is flawed with the presentation of a
counter example.
Providing counter examples.
• Take the original argument:
• Identify the form (the structure of the
argument)
• Think of another argument with the same
form but with obviously true premises and an
obviously false conclusion.
• Ex next page.
Original argument
• All Premiers are charismatic men.
• Pallister is a charismatic man,
• Pallister will be Premier.
•
•
•
•
Counterexample: Fill in the blanks….
P1 All _______ are __________
P2. __________ is ____________
C: ____________ will be __________
Fallacies and Counterexamples
• In the next section of the course we will study
“the fallacies”
• We will use a lot of counter-examples to help
understand why these arguments are fallacies.
Logical Fallacy
• Two senses:
– Any argument that fails to adequately support its
conclusion.
• It is impossible to define all the ways you can be wrong.
– Any argument that fits into common patterns of
error in reasoning.
• “The fallacies”.
To be a fallacy…
• To be a fallacy a series of statements must first
be an argument:
• You’re a jerk, therefore you’re wrong.
– Is a fallacy (ad hominem aka fallacy of abuse.)
• You’re a jerk,
– is not a fallacy, it is mere abuse.
Example of a fallacy
• “Childhood obesity has increased now that so
many children are playing video games, so
obviously video games cause obesity in
children.”
• This argument relies on the evident fact that
video games and obesity have occurred
together to conclude one is the cause of the
other.
Cum hoc, ergo procter hoc
• A common causal fallacy, know by its
traditional latin name:
• With this, therefore because of this.
• Counterexample: (to expose the fallacy)
– Video games have been gradually increasing in
popularity since the 80’s, and my hair has been
decreasing since then, so obviously video games
have caused my baldness.
Formal versus Informal fallacies
• Formal fallacies are fallacies that violate some
specific logical rule or law.
– Ex:
All Geese can fly
All ducks can fly
All geese are ducks.
This argument commits the formal rule
regarding the distribution of terms in an
argument.
Formal versus Informal fallacies
• Formal fallacies are fallacies that violate some
specific logical rule or law.
– Ex:
All Geese can fly
All ducks can fly
All geese are ducks.
The category of things that can fly
includes both ducks and geese.
The way this argument refers to flying
things, they can be either ducks or
geese, not both.
This argument commits the formal rule
regarding the distribution of terms in an
argument.
Formal vs informal fallacies
• Informal fallacies are not violations of specific
logical rules
• Instead are errors of reasoning common enough
to be named, recognized, and studied.
• Traditional education in law often focused heavily
on the informal fallacies.
• Informal fallacies are often grouped by category
in various ways.
– There are many different groupings in different texts.
Additional sources for Fallacies
• www.fallacyfiles.org
– Extensive collection of fallacies.
• http://www.nizkor.org/features/fallacies/
– Collection of fallacies relevant to the website:
rebutting holocaust deniers.
• http://onegoodmove.org/fallacy/
– This site indicates a “proof” condition for each fallacy
given.