Lesson LOGIC PROBLEMS INVOLVING INVALID OR VALID ARGUMENTS WITH SOLUTION AND EXPLANATION

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PROBLEM

Identify which argument is invalid. 
      
1. Either the panda yawns or she is alert.
The panda did not yawn.
Therefore, she is alert.

2. If the panda yawns, then she is not alert.
The panda is not alert.
Therefore, she yawned.

3. If Tom is cooking, then I am not hungry.
I am hungry.
Therefore, Tom is not cooking.

4. If it is hailing, then I will not go outdoors.
If I will not go outdoors, I will not raise any money for charity.
Therefore, if it is hailing, then I will not raise any money for charity

5. If it is hailing, then I am not going outdoors.
It is hailing.
Therefore, I am not going outdoors.

SOLUTION

1. Either the panda yawns or she is alert.
The panda did not yawn.
Therefore, she is alert.

THIS IS VALID BY THE LAW OF DISJUNCTION

2. If the panda yawns, then she is not alert.
The panda is not alert.
Therefore, she yawned.

THIS IS INVALID BY THE FALLACY OF THE CONVERSE.

3. If Tom is cooking, then I am not hungry.
I am hungry.
Therefore, Tom is not cooking.

THIS IS VALID BY THE LAW OF CONTRA-POSITION.

4. If it is hailing, then I will not go outdoors.
If I will not go outdoors, I will not raise any money for charity.
Therefore, if it is hailing, then I will not raise any money for charity

THIS IS VALID BY THE LAW OF SYLLOGISM.

5. If it is hailing, then I am not going outdoors.
It is hailing.
Therefore, I am not going outdoors.

THIS IS VALID BY THE LAW OF DETACHMENT.

This is how the laws work.

LAW OF DETACHMENT

p -> q
p
therefore q

The statement p -> q means "If p is true, then q is true"
As long as p is true, then the statement p -> q is true only if q is true.
If p is true and q is false, then the statement p -> q is false.

Note that the statement p -> q can also be true if p is false, but that's another story we won't get into right now.  If p is false, the statement p -> q can be true but you can't tell anything about q from it.

Your statement number 5 fits this.

That statement is:

If it is hailing, then I am not going outdoors.
It is hailing.
Therefore, I am not going outdoors.

We let:
p = It is hailing
q = I am not going outdoors.

your statement translates to:

p -> q
p
therefore q

p -> q translates to "if it is hailing then i am not going outdoors"
p translates to "it is hailing"
therefore q translates to "therefore i am not going outdoors"

This is valid by the law of detachment.

LAW OF SYLLOGISM

p -> q
q -> r
therefore p -> r

The statement p -> q is assumed to be true.
The statement q -> r is assumed to be true.
If both statements are true, then the statement p -> r has to be true.

Your statement number 4 fits this.

That statement is:

If it is hailing, then I will not go outdoors.
If I will not go outdoors, I will not raise any money for charity.
Therefore, if it is hailing, then I will not raise any money for charity

We let:

p = It is hailing .
q = I am not going outdoors.
r = I will not raise any money for charity.

The law of syllogism states:

p -> q
q -> r
therefore p -> r

This translates to:

if it is hailing than I will not go outdoors.
If i don't go outdoors then I will not raise any money for charity.
Therefore if it is hailing, I will not raise any money for charity.

Since the first 2 statements are assumed to be true, then the third statement is assumed to also be true.

LAW OF DISJUNCTION

A or B implies that either A is true or B is true.  This means that both cannot be false, otherwise the statement would be false.

If you are given the A or B is true, and you are given that B is false, then you can conclude that A is true.   Alternatively, if you are given that A is false, then you can conclude that B is true.

This fits your statement number 1.

That statement is:

Either the panda yawns or she is alert.
The panda did not yawn.
Therefore, she is alert.

Let A = the panda yawns and let B = the panda is alert.
A or B means the panda yawns or the panda is alert.

At least one of those conditions has to be true in order for the OR statement to be true.

Since A is false (the panda did not yawn), then B must be true (the panda is alert.

LAW OF CONTRA-POSITION

The law of contra-position states that:

If a statement is true, then the contra-position of that statement is also true.
Conversely:
If a statement is false, then the contra-position of that statement is also false.

In other words, the statement and its contra-position are equivalent.

In symbolic form, this would show up as:

p -> q is equivalent to ~q -> ~p

p and q are statements.
~p and ~q are negations of those statements.

This can be proven using truth tables but we won't do that here.  We will, however, apply the law to your statement number 3.

That statement is:

If Tom is cooking, then I am not hungry.
I am hungry.
Therefore, Tom is not cooking.

Let p = Tom is cooking.
Let q = I am not hungry
let ~q = I am hungry (negation of I am not hungry)
let ~p = Tom is not cooking (negation of Tom is cooking).

The statement is:
p -> q
The contra-position to this statement is:
~q -> ~p

Translating, this comes out to be:

The statement is:
If Tom is cooking, then I am not hungry.
The contra-positive to that statement is:
if I am not hungry, then Tom is not cooking.

Statement 3 fits this so it is valid by the law of contra-position.

FALLACY OF THE CONVERSE

Fallacy of the converse means that A implies B is true does not mean that B implies A is true.  B implies A might be true, but B implies A might also not be true (be false).

The two statements are not equivalent.

When you make a statement, there are 3 additional statements that can be made from that.

You have:

the statement
the inverse of the statement
the converse of the statement
the contra-positive of the statement.

In symbolic form, this would be shown as:

the statement:
p -> q
the inverse of the statement:
~p -> ~q
the converse of the statement:
q -> p
the contra-positive of the statement:
~q -> ~p

The rules are:

The statement and the contra-positive of the statement are equivalent.
The inverse of the statement and the converse of the statement are equivalent.

The converse of the statement is not equivalent to the statement.
The inverse of the statement is not equivalent to the statement.

If the statement is true, then:
The converse may or may not be true.
The inverse may or may not be true.
The contra-positive will always be true.

If the statement is false, then:
The converse may or may not be false.
The inverse may or may not be false.
The contra-positive will always be false.

The fallacy of the converse is the assumption that just because the statement is true, then the converse must also be true.

This is not so.

Your statement number 2 fits this form.

That statement is:

If the panda yawns, then she is not alert.
The panda is not alert.
Therefore, she yawned.

Let p = the panda yawns
Let q = The panda is not alert.

The statement becomes:
p -> q which means:
If the panda yawns, than the panda is alert.
The panda is not alert translates to:
q
Therefore, she yawned translates to:
p
You can put the q and the p together to form:
q -> p which means
If the panda is not alert, then she yawns.

What you have now is the statement of:
p -> q
and the converse of:
q -> p

Since the converse is not equivalent to the statement, then these 2 statements cannot be considered to be equivalent and you cannot assume that just because p -> q is true that q -> p is also true.

that's the fallacy of the converse.  
It's the assumption that just because the statement is true that the converse of the statement is also true.

Here's an example where the converse of the statement is not true when the statement is true.

p = an animal is a pig
q = an animal is a mammal
p -> q means that "if an animal is a pig, then it is a mammal"
the converse to that statement is:
q -> p which means that "if an animal is a mammal, then it is a pig"
You can easily see that, just because a pig is a mammal, it does not necessarily mean that a mammal is a pig.   A particular mammal may be a pig, but not all mammals are pigs.