Truth Functional Rules and PatternsBy Mike Monge

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

Using letters and symbols to represent arguments can help reveal the skeletal structure of an argument, and examining its structure allows us to determine the validity of the argument. With that in mind consider the following arguments:

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

Each of these arguments deals with very different subject matters, but each one has the same form or structure. The validity of the argument is solely a function of its structure, and not the content of the argument.

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

Let’s concentrate on the first argument.

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

Let’s concentrate on the first argument.

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

First, we will highlight each of the simple statements.

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

First, we will highlight each of the simple statements.

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

First, we will highlight each of the simple statements.

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

First, we will highlight each of the simple statements.

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

First, we will highlight each of the simple statements.

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

Then, we will replace the simple statements with letters.

Truth Functional Rules and Patterns

1. If , then .

not . Therefore, not .

Then, we will replace the simple statements with letters.

S LL

S

Truth Functional Rules and Patterns

1. If , then .

not . Therefore, not .

Next, we replace the logical words with symbols.

S LL

S

Truth Functional Rules and Patterns

1. If , then .

not . Therefore, not .

Next, we replace the logical words with symbols.

S LL

S

~~

Truth Functional Rules and Patterns

1.

Now, with all replacements made, we can move everything into position.

S LL

S

~~

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

Now, with all replacements made, we can move everything into position.

S LL

S

~~

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

Now, with all replacements made, we can move everything into position.

S LLS

~~

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

This basic pattern is so common and useful that is has its own name. It is called modus tollens. Any argument which exhibits the pattern of modus tollens is a valid argument.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

1. If the sample were an acid, then the litmus paper would have turned red. The litmus paper did not turn red. Therefore, the sample is not an acid.

Let’s go back to the other arguments, one at a time.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

Let’s go back to the other arguments, one at a time.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

We need to find the simple statements again.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

We need to find the simple statements again.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

We need to find the simple statements again.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

We need to find the simple statements again.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

We need to find the simple statements again.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet only if it has cleared its orbital path. Ceres has not cleared its orbital path, which means that Ceres is not a planet.

Notice that the words “only if” are replaced by the horseshoe, and that “which means that” is a conclusion indicator.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet it has cleared its orbital path. Ceres has not cleared its orbital path,

Ceres is not a planet.

Notice that the words “only if” are replaced by the horseshoe, and that “which means that” is a conclusion indicator.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

2. Ceres is a planet it has cleared its orbital path. Ceres has not cleared its orbital path,

Ceres is not a planet.

Let’s look at the last argument. There is one wrinkle we haven’t yet discussed.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

Let’s look at the last argument. There is one wrinkle we haven’t yet discussed.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

As before, we will identify the simple statements.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

As before, we will identify the simple statements.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

As before, we will identify the simple statements.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

As before, we will identify the simple statements.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

As before, we will identify the simple statements.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness would imply that he doesn’t celebrate birthdays. My brother celebrates birthdays. So, my brother is not a devout Jehovah’s witness.

We can also fill in the symbols.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdays.

My brother celebrates birthdays. my brother is not a devout Jehovah’s witness.

We can also fill in the symbols.

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdays.

My brother celebrates birthdays. my brother is not a devout Jehovah’s witness.

There is something very different about this argument. In this case, the second part of the conditional, the consequent, is being negated, and the second premise is affirmative. We need to adjust our structure…

S LLS

~~

Modus Tollens

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdays.

My brother celebrates birthdays. my brother is not a devout Jehovah’s witness.

There is something very different about this argument. In this case, the second part of the conditional, the consequent, is being negated, and the second premise is affirmative. We need to adjust our structure…

S LL

S

~

~

Modus Tollens ?

Truth Functional Rules and Patterns

3. My brother being a devout Jehovah’s Witness he doesn’t celebrate birthdays.

My brother celebrates birthdays. my brother is not a devout Jehovah’s witness.

This argument still fits the pattern of modus tollens. The second premise asserts the opposite of the consequent, while the conclusion asserts opposite of the antecedent.

S LL

S

~

~

Modus Tollens

Truth Functional Rules and Patterns

Click on the circle below an argument if the argument is an example of modus tollens.

p q~q~p

(A B) ~C~C A B

That’s right!That’s not right. Look closely.

A (~B C)~(~B C) ~A

That’s right!

(A B) ~CA B ~C

That’s not right. Look closely.

Truth Functional Rules and Patterns

For those of you who don’t trust me yet, here is a truth table to prove that modus tollens is a valid argument pattern.

A B A B ~B ~A

T T T F F

T F F T F

F T T F T

F F T T T