Validity: Long and short truth tables
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Week 10!
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Homework Due
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Review: MP,MT,CA
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Validity: Long truth tables
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Short truth table method
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Evaluations!
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For Next Time: Read Chapter 9 pages 325-334
Review
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We ended last time by
looking at three valid
argument forms:
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Modus Ponens
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Modus Tollens
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Chain Argument
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We used truth tables to
show that each argument
type was valid
Modus Ponens
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1. P > Q
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2. P
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3. :. Q
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Why is this argument form
always valid?
What about affirming the
consequent?
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1. P > Q
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2. Q
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3. :. P
P
Q
P>Q
T
T
T
T
F
F
F
T
T
F
F
T
Modus Tollens
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1. P > Q
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2. ~Q
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3. :. ~P
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Why is Modus Tollens a valid
argument form?
What about denying the
antecedent?
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1. P > Q
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2. ~P
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3. :. ~Q
P
Q
P>Q
T
T
T
T
F
F
F
T
T
F
F
T
Chain Argument
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We also said that chain arguments are valid
argument forms:
Every chain argument has two conditional premises
where the consequent of one conditional premise is
the antecedent of the other
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1. P > Q
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2. Q > R
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3. :. P > R
Chain Argument
P
Q
R
P>Q
Q>R
P>R
T
T
T
T
T
T
T
T
F
T
F
F
T
F
T
F
T
T
T
F
F
F
T
F
F
T
T
T
T
T
F
T
F
T
F
T
F
F
T
T
T
T
F
F
F
T
T
T
Invalid Conditional Arguments
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What's wrong with the following conditional argument?
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1. P > Q
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2. R > Q
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3. :. P > R
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This is an invalid argument form, but why?
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Here's a hint: recall the relationship that must hold
between the consequent and antecedent of conditionals
in a chain argument
Proving Invalidity
P
Q
R
P>Q
R>Q
P>R
T
T
T
T
T
T
T
T
F
T
T
F
T
F
T
F
F
T
T
F
F
F
T
F
F
T
T
T
T
T
F
T
F
T
T
T
F
F
T
T
F
T
Practice
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Given the following argument, can you derive R (by
itself)?
Hint: do not use a truth table, use only MP, MT,
and/or CA
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1. ( P v Q) > ( A > B)
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2. P & A
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3. ~(A > B)
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4. ~(P v Q) > R
Practice
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We can prove that R follows by using Modus Tollens
and Modus Ponens:
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1. ( P v Q) > ( A > B)
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2. P & A
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3. ~(A > B)
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4. ~(P v Q) > R
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5. ~(P v Q)
1, 3 MT
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6. R
4,5 MP
Proving Invalidity
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We have been using truth tables to prove that arguments
were valid and invalid (MP, MT, CA)
How did we do that?
We plotted out all the possible truth values for the
premises and checked to see if a row existed where the
premises were true and the conclusion was false
If this kind of row exists then the argument is invalid
If this kind of row does not exist then the argument is
valid
Examples
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Is the following argument valid or invalid? Prove this using a
truth table
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1. A > (B & C)
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2. ~B v ~C
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3. :. ~A
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Your first step should be to construct a truth table
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Your second step should be to plot all of the truth values into the
table
Finally, check to see if there is a row where the premises are true
and the conclusion is false
Practice
A
B
C
B&C
A > (B & C) ~B v ~C
~A
T
T
T
T
T
F
F
T
T
F
F
F
T
F
T
F
T
F
F
T
F
T
F
F
F
F
T
F
F
T
T
T
T
F
T
F
T
F
F
T
T
T
F
F
T
F
T
T
T
F
F
F
F
T
T
T
Short Truth Table Method
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We could construct a truth table for any argument in
order to determine whether the argument is valid or
invalid
Constructing entire truth tables can be time
consuming however
Thankfully there is a faster way to figure out
whether an argument is valid or invalid using a truth
table
We could use the short truth table method
Short Truth Table
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An argument is invalid when we find a row where
the premises are true and the conclusion is false
When we construct a short truth table we are looking
to find only the row that invalidates
In order to do this, we first assume that the
conclusion is false (assign it an F) and then see if it
is possible to construct a row where the premises are
still true
If we can do this then the argument is invalid
Example
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Let's construct a short truth table for the following
argument:
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1.
A>B
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2.
~B > C
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3. :. ~A > C
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The first thing to do is to make (~A > C) false
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When is the only time that conditionals are false?
Example
A
F
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
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B
C
F
A>B
~B > C
~A>C
F
Conditionals are only false when the antecedent is
true and the consequent false
This means that ~A must be true and C must be false
What about the second premise (~B > C)? If C is
false what must ~B be in order for the entire
conditional to come out true?
Example
A
B
C
A>B
~B > C
F
T
F
T
T
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~A>C
F
If we must make B true then how does this affect our
first premise: A > B?
If B is true and A is false then the first premise is true
We therefore have created a row on the truth table
where the premises are true but the conclusion is false
This argument is invalid
Practice
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Construct a short truth table to prove whether the following
argument is valid or invalid:
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1. A & (B v C)
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2. C > D
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3. A > E
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4. :. D & E
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This is is tricky because the conclusion is a conjunction, there are
three possible ways it can be false
Try to make the premises true first, some of the truth values are
'forced' on us and that makes things easier
Practice
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1. A & (B v C)
A
B
C
D
E
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2. C > D
T
T
F
F
T
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3. A > E
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4. :. D & E
A must be true because the
first premise is a conditional
and in order for a conditional
to be true both conjuncts must
be true
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If A must be true then we know
that E must be true as well in
order for premise 3 to be true
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If we know that E must be true
then we know that D must be
false if the conclusion is false
If D must be false then C must
be false and if C must be false
then B must be true
This argument is INVALID
For Next Time
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For Next Time: Read Chapter 9 pages 330-334
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Bring your books Wednesday!