Validity: Long and short truth tables

Sign In! Week 10! Homework Due Review: MP,MT,CA Validity: Long truth tables Short truth table method Evaluations! For Next Time: Read Chapter 9 pages 325-334

Review

We ended last time by looking at three valid argument forms:

Modus Ponens Modus Tollens Chain Argument We used truth tables to

show that each argument type was valid

Modus Ponens

1. P > Q

2. P

3. :. Q

Why is this argument form always valid?

What about affirming the consequent?

1. P > Q

2. Q

3. :. P

P Q P > Q

T T T

T F F

F T T

F F T

Modus Tollens

1. P > Q

2. ~Q

3. :. ~P

Why is Modus Tollens a valid argument form?

What about denying the antecedent?

1. P > Q

2. ~P

3. :. ~Q

P Q P > Q

T T T

T F F

F T T

F F T

Chain Argument

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

1. P > Q 2. Q > R 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

What's wrong with the following conditional argument? 1. P > Q 2. R > Q 3. :. P > R This is an invalid argument form, but why? 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

F F F T T T

Practice

Given the following argument, can you derive R (by itself)?

Hint: do not use a truth table, use only MP, MT, and/or CA

1. ( P v Q) > ( A > B) 2. P & A 3. ~(A > B) 4. ~(P v Q) > R

Practice

We can prove that R follows by using Modus Tollens and Modus Ponens:

1. ( P v Q) > ( A > B) 2. P & A 3. ~(A > B) 4. ~(P v Q) > R 5. ~(P v Q) 1, 3 MT 6. R 4,5 MP

Proving Invalidity

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

Is the following argument valid or invalid? Prove this using a truth table

1. A > (B & C)

2. ~B v ~C

3. :. ~A

Your first step should be to construct a truth table

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

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

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

Let's construct a short truth table for the following argument:

1. A > B 2. ~B > C 3. :. ~A > C The first thing to do is to make (~A > C) false When is the only time that conditionals are false?

Example

A B C A > B ~B > C ~ A > C

F F 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 ~ A > C

F T F T T 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

Construct a short truth table to prove whether the following argument is valid or invalid:

1. A & (B v C)

2. C > D

3. A > E

4. :. D & E

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

A B C D E

T T F F T

1. A & (B v C)

2. C > D

3. A > E

4. :. D & E

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

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

If A must be true then we know that E must be true as well in order for premise 3 to be true

For Next Time

For Next Time: Read Chapter 9 pages 330-334 Bring your books Wednesday!