Simplifying Arguments

For the rest of this course, we will deal with real-life examples of reasoning. To do this, we need to understand logical operators. 

You have seen some logical operators in this course:   

¬ = not
∧  = and
∨ = or
→ = if … then …

We use these operators to simplify an argument. That can help us to judge the validity or strength of the argument. The following tables show us how.

The Not Operator (¬):

P	¬ P
True	False
False	True

Let’s use an example to see what this means:

P = It is rainy.
¬ P = it is not rainy.

If P is true, then ¬ P must be false. In other words, when it is rainy, then it is false to say that it is not rainy.

Similarly, if P is false, then ¬ P must be true. When it is not true that it is rainy, then it is not rainy. 

The And Operator (∧):

P	Q	P ∧ Q
True	True	True
True	False	False
False	True	False
False	False	False

Here’s an example:

P = It is rainy.
Q = I have an umbrella.

Notice that there is only one True entry for P ^ Q above, when P is true and Q is true. This is pretty straightforward: 1) if it is rainy and 2) if I have an umbrella, then I can say it is rainy and I have an umbrella.

In the third row from the top (P=True and Q=False), 1) it is rainy, but 2) I don’t have an umbrella, so I cannot say that it is rainy and I have an umbrella. That would be false.

We need to satisfy both conditions P and Q to say that  P ∧ Q is true. 

The Or Operator (∨):

P	Q	P ∨ Q
True	True	True
True	False	True
False	True	True
False	False	False

Let’s take the same example:

P = It is rainy.
Q = I have an umbrella.

There is only one False entry for P ∨ Q in the table above. It is in the last row when P is false and when Q is false. P ∨ Q is only false when P and Q are both false. In other words, 1) when it is not rainy and 2) I don’t have an umbrella, then it is false to say it is rainy or I have an umbrella.

If one of the conditions P or Q is true, then  P ∨ Q is true.  

If … then … Operator (→):

It is easier to show how this operator works in examples rather than in a table. This will be a review, as you have seen this earlier in the course.

Let’s take this example:

P = It is rainy.
Q = There are clouds in the sky.

In the example above, P → Q is true. If it is rainy, there must be clouds in the sky because rain comes from clouds. Let’s look at this argument (which you’ve seen before):

P → Q 
P
∴ Q (The symbol ∴ means therefore.)

This argument is true (or valid). Premise #1 tells us if it is rainy, then there are clouds in the sky. Premise #2 tells us that it is rainy. Therefore, there must be clouds in the sky.

The above rule holds even when P → Q seems to be false. Take this example:

P = My sisters are hungry.
Q = My sisters dance.

Now Premise #1 becomes this: if my sisters are hungry, they dance. Premise #2 says my sisters are hungry. So the conclusion is true – they will dance.

How about this argument:

P = It is rainy.
Q = There are clouds in the sky.

P → Q 
Q
∴P x

The argument above is false (invalid). Premise #1 tells us if it is rainy, then there are clouds in the sky. Premise #2 tells us it is cloudy. The conclusion tells us it must be rainy. However, we can have cloudy weather without rain, so this argument is false (or invalid).

Interestingly, this argument is true, or valid:

P → Q 
¬ Q
∴  ¬ P

To use our example above, Premise #1 says if it is rainy, then it is cloudy. Premise #2 says it is not cloudy. Rain comes from clouds, so if there are no clouds there cannot be rain. So, the conclusion is correct: it is not rainy.

Finally, this argument is not valid:

P → Q 
¬ P
∴ Q x 

This argument says if it is not rainy, then it will be cloudy. However, when it is not rainy, it could be sunny.

We can do a lot with these few logical operators. We often combine them. Let’s look at a story:

Bart forgot to bring his lunch to school, so he went to the nearby convenience store for a big sandwich. If Bart has a credit card or enough money, he can buy one. Luckily, Bart had enough money in his sock, so he bought a big sandwich. 

Next, let’s find the basic argument in the story. Note that we do not need to put some of the information into our argument. The first sentence, for example: “Bart forgot to bring his lunch to school, so he went to the nearby convenience store for a big sandwich.” This is important background information for the story, but we do not need it in our argument.

We can form an argument using the important information from the story:

If Bart has a credit card or enough money, he can buy a big sandwich.
Bart had enough money.
Therefore, he bought a big sandwich.

If this argument sounds valid to you, then you are correct. Let’s put the argument into symbols and use our logical arguments. That will help us to clearly see that the argument is valid. 

P = Bart has a credit card.
Q = Bart has enough money.
R = Bart bought a sandwich.

P ∨ Q → R
Q
∴ R

We need two pieces of information to judge if this is valid: the Or Operator table and the rules on conditionals (if… then…). 

The story does not tell us that Bart has a credit card, so we will assume that he doesn’t. In other words, P is false. When we do not have information, that is the safest thing to do. We know that Q is true because Bart has enough money. Now if we look at the Or Operator table, we can see that our story fits the 4th row (in blue):

P	Q	P ∨ Q
True	True	True
True	False	True
False	True	True
False	False	False

And we can see that P ∨ Q is true. That is helpful!

Now, look at the If… then… Operator rules again, and you can find that this is similar to our argument:

P → Q 
P
∴ Q

P in the rule above is the same as P ∨ Q in our argument about Bart. We know the 2nd P above is true (because Bart has enough money), so the reasoning is valid.

Let’s look at a second example:

Liza hates to get up in the morning. If sets her alarm clock and asks her mother to call her, then she can wake up. If she doesn’t, she can’t wake up. Liza had a test today, so she set her alarm clock but forgot to ask her mother to call her last night. As a result, she was able to wake up this morning.

Is this argument valid? Let’s use the important information to form an argument:

If Liza does not set her alarm clock and ask her mother to call her, she cannot wake up.
Liza set her alarm clock last night.
Liza did not ask her mother to call her last night. 
Thus, Liza woke up this morning.

Again, we’ll put this argument into symbols:

P = Liza sets her alarm clock.
Q = Liza asks her mother to call her.
R = Liza wakes up.

Now let’s form the argument with logical operators:

P ∧ Q → R
P
¬ Q
∴ R

If we look at the And Operator table, we can see that the first three lines of our argument fit the 3rd row, where P is true and Q is false (in blue):

P	Q	P ∧ Q
True	True	True
True	False	False
False	True	False
False	False	False

Now we can see that for our argument,  P ∧ Q is false.

From the If… then… Operator rules, we can see that our argument is similar to this pattern:

P → Q 
¬ P
∴ Q x

Therefore, the above argument is not valid, and the story is not true. Liza needs to both set her alarm and ask her mother to call her to wake up in the morning.

Many arguments do not use the words “if… then…”, but they still have a condition; in other words, we can use the If…then… Operator (→). Here’s an example:  

Isabella wanted grass in her backyard, but she didn’t want to water the grass because that is not good for the environment. To grow grass that doesn’t need watering, Isabella needed native grass seed and fertilizer. Luckily, she found some of both at the local garden centre. Now she has a nice lawn that does not need watering.

In this argument, the expression “To grow grass that doesn’t need watering, Isabella needed native grass seed and fertilizer” is equivalent to “If Isabella had native grass seed and fertilizer, she wouldn’t have to water her grass”. Now let’s try to form an argument from the story above:

If Isabella had native grass seed and fertilizer, she could grow grass that does not need water.
Isabella found native grass seed and fertilizer.
Thus, Isabella grew grass that does not need water.

Next, let’s use symbols:

P = Isabella has grass seed.
Q = Isabella has fertilizer.
R = Isabella can grow grass that doesn’t need water.

And we can form this argument:

P ∧ Q → R
P ∧ Q 
∴ R 

This argument is very similar to this valid argument in the If… then… Operator rules:

P → Q 
P
∴ Q 

Therefore, this is a valid argument.

Review:

• We use logical operators to help us understand real-life examples of reasoning.
• Logical operators:
  • ¬ = not
  • ∧  = and
  • ∨ = or
  • → = if … then …
  • ∴ = therefore
• Operator tables and If… then.. Operator rules can help us determine if an argument is valid or sound.
• Some arguments use If… then… conditionals even though they do not use the words “if” and “then”.