**Truth Tables for Conditional Sentences**

As you remember, before the midterm, we examined truth tables for disjunctions, conjunctions, and negations.

Today, we will examine the truth table for conditional sentences.

We already know that a conditional sentence consists of the __antecedent__ and a __consequent.__

So in this sentence “IF you take an Advil, your headache will go away”, the antecedent is the taking of an Advil, and the consequent is the headache going away.

Let’s symbolize the antecedent, that is, the proposition “You take an Advil” with the letter A.

Let’s symbolize the consequent, that is, the proposition “Your headache goes away” with the letter G.

In sum:

Proposition A: “You take an Advil.”

Proposition G: “Your headache goes away.”

Combining the two propositions into one conditional sentence will look like this:

A -> G

We already know that this symbolic expression reads: If A then G, meaning if A is true, then G will follow, meaning if you take an Advil, then your headache will go away.

Now we want to verify if this whole sentence is indeed true. Let’s imagine we read the claim “IF you take this Advil, your headache will go away” on a box of pills at a store and we want to make sure the company that produced this advertisement is not lying.

To keep track of our investigation, we will create a truth table for this advertisement about the Advil. So let’s first draw the truth table, like this:

A | G | A -> G |

Now we will imagine 4 distinct scenarios and enter the truth values for each scenario.

- In the first scenario you take the Advil, and then your headache goes away. So the claim “If you take the Advil, your headache will go away” is true. We will enter T (True) in all 3 columns of the first row. Again, you took the Advil, then your headache went away; so the sentence cannot be false because your headache went away after you took the Advil. Thus we enter the letter T in the last column as well.

A | G | A -> G |

T | T | T |

Again, it was True that you took the Advil. It was True that your headache went away after you took the Advil, and so the sentence is true.

- In the second scenario you take the Advil but the headache still persists! So we have to enter F (False) in the second column of the second row because the consequent is now false, meaning that it’s not true that your headache went away. Clearly the Advil does not always work so the entire sentence is a lie! Thus we have to write F in the third column. We will enter the truth-values for this scenario in the second row, like this:

A | G | A -> G |

T
T |
T
F |
T
F |

- In the third situation, you did not take the Advil. So we will enter F in the first column because it is not true that you took the Advil.

But then your headache goes away anyway!

Since your headache went away, we will enter T in the second column, like this:

A | G | A -> G |

T
T |
T
F |
T
F |

F | T | |

Now, what do you think we should enter in the third column this time? Should we write T or F?

Again, your headache went away, but you did not even take the pill!

So is the advertisement false? Would you call up the company and say: “You guys are lying, my headache went away without taking your Advil, so your Advil does not work” ?

To this, the Advil producers would say “But wait a minute. We do not claim that your headache will go away if and only if you take the Advil. So we are not lying at all.”

This is why we should not enter F in the third column! We have to enter T, like this:

A | G | A -> G |

T
T |
T
F |
T
F |

F | T | T |

- In the last scenario, you did not take the Advil and the headache did not go away. So can we accuse the Advil company of lying? No. You did not take the Advil, so what do you expect? It would be silly to call the company and complain that your headache did not go away when you did not even bother to take the pill.

Thus, the two F’s, still result in T, like this:

A | G | A -> G |

T
T |
T
F |
T
F |

F
F |
T
F |
T
T |

Thus, we see that a conditional sentence is false in only one scenario, the second scenario (the second row).

Recall again the sentence: “ If you score 100 points on the exam (H), you will get an A on that exam (A).”

H-> A

This sentence is also false in only one scenario, in the case where the teacher did not give you an A even though you earned the perfect score of 100.

In the first scenario, you scored 100 and you got an A, thus:

True -> True = True

Your turn.

Please enter the truth values for the first scenario:

H | A | H -> A |

T | T | T |

In the second scenario, you scored 100 but then you look at the grade and there is a B+. Well then, the teacher was clearly lying which means the sentence “If you score 100, you will get an A” would be false. Since you met the requirement and then the teacher did not give you a perfect A, so the whole sentence is a lie! The teacher’s claim was false.

True -> False = False

Please enter the truth values for the second scenario

H | A | H -> A |

T | T | T |

T | F | F |

In the third scenario, you scored 99 but you still got an A, thus:

False -> True = True

This means that the first part of the sentence is false since you did not get the perfect 100 score. But then you turn the page and there it is – a perfect A! Thus, the second part of the sentence is true. You got an A on that test!

This means that you don’t need the perfect score of 100 to get an A. It turns out that obtaining 99 also qualifies as an A.

Please enter the truth values for the third scenario:

H | A | H -> A |

T | T | T |

T | F | F |

F | T | T |

In the fourth scenario you scored only 40 points and you got a C, meaning – it is not true that you scored 100, so we enter F in the first column, and it is not true that you got an A, so we also enter F in the second column. And yet, the sentence is true, the teacher did not lie. You did not get an A because you scored only 40 points. In this case, you can’t say that the teacher lied to you.

Thus:

False -> False = True

Enter all the truth values this time:

H | A | H -> A |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

It may seem perplexing that F and F gives T. But consider again this sentence: “If we live now in 1946, then next year is 1947”. We clearly do not live in 1946 and next year is definitely not 1947, and yet that hypothetical statement makes perfect sense.

We see once again that a conditional sentence is false in only one scenario.

Let’s summarize. When is a conditional sentence true?

It is true in 3 scenarios and false in one scenario.

- True -> True = True
- True -> False = False
- False -> True = True
- False -> False = True

Now review the 3 truth tables that we already know, and then complete more complex truth tables on the next page.

The truth table for conditional sentences:

A | B | A -> B |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

The truth table for disjunction

A | B | A Ú B |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

The truth table for conjunctions

A | B | A & B |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

The truth table for negation (“not”)

A | – A |

T | F |

F | T |

Your turn:

Complete the truth table:

A | B | A Ú -B |

T | T | T |

T | F | T |

F | T | F |

F | F | T |

A | B | A -> B |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

A | B | -A& B |

T | T | F |

T | F | F |

F | T | T |

F | F | F |

A | B | -A->B |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

*Challenging: (A plus level)

A | B | C | A & C | B-> (A& C) |

T | T | T | T | T |

T | T | F | F | F |

T | F | T | T | T |

T | F | F | F | T |

F | T | T | F | F |

F | T | F | F | F |

F | F | T | F | T |

F | F | F | F | T |

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**Question**

**Week 14 Worksheet**

**Topic: Truth Tables for Conditional Sentences**

As you remember, before the midterm, we examined truth tables for disjunctions, conjunctions, and negations.

Today, we will examine the truth table for conditional sentences.

We already know that a conditional sentence consists of the __antecedent__ and a __consequent.__

So in this sentence “IF you take an Advil, your headache will go away”, the antecedent is the taking of an Advil, and the consequent is the headache going away.

Let’s symbolize the antecedent, that is, the proposition “You take an Advil” with the letter A.

Let’s symbolize the consequent, that is, the proposition “Your headache goes away” with the letter G.

In sum:

Proposition A: “You take an Advil.”

Proposition G: “Your headache goes away.”

Combining the two propositions into one conditional sentence will look like this:

A -> G

We already know that this symbolic expression reads: If A then G, meaning if A is true, then G will follow, meaning if you take an Advil, then your headache will go away.

Now we want to verify if this whole sentence is indeed true. Let’s imagine we read the claim “IF you take this Advil, your headache will go away” on a box of pills at a store and we want to make sure the company that produced this advertisement is not lying.

To keep track of our investigation, we will create a truth table for this advertisement about the Advil. So let’s first draw the truth table, like this:

A | G | A -> G |

Now we will imagine 4 distinct scenarios and enter the truth values for each scenario.

- In the first scenario you take the Advil, and then your headache goes away. So the claim “If you take the Advil, your headache will go away” is true. We will enter T (True) in all 3 columns of the first row. Again, you took the Advil, then your headache went away; so the sentence cannot be false because your headache went away after you took the Advil. Thus we enter the letter T in the last column as well.

A | G | A -> G |

T | T | T |

Again, it was True that you took the Advil. It was True that your headache went away after you took the Advil, and so the sentence is true.

- In the second scenario you take the Advil but the headache still persists! So we have to enter F (False) in the second column of the second row because the consequent is now false, meaning that it’s not true that your headache went away. Clearly the Advil does not always work so the entire sentence is a lie! Thus we have to write F in the third column. We will enter the truth-values for this scenario in the second row, like this:

A | G | A -> G |

T
T |
T
F |
T
F |

- In the third situation, you did not take the Advil. So we will enter F in the first column because it is not true that you took the Advil.

But then your headache goes away anyway!

Since your headache went away, we will enter T in the second column, like this:

A | G | A -> G |

T
T |
T
F |
T
F |

F | T | |

Now, what do you think we should enter in the third column this time? Should we write T or F?

Again, your headache went away, but you did not even take the pill!

So is the advertisement false? Would you call up the company and say: “You guys are lying, my headache went away without taking your Advil, so your Advil does not work” ?

To this, the Advil producers would say “But wait a minute. We do not claim that your headache will go away if and only if you take the Advil. So we are not lying at all.”

This is why we should not enter F in the third column! We have to enter T, like this:

A | G | A -> G |

T
T |
T
F |
T
F |

F | T | T |

- In the last scenario, you did not take the Advil and the headache did not go away. So can we accuse the Advil company of lying? No. You did not take the Advil, so what do you expect? It would be silly to call the company and complain that your headache did not go away when you did not even bother to take the pill.

Thus, the two F’s, still result in T, like this:

A | G | A -> G |

T
T |
T
F |
T
F |

F
F |
T
F |
T
T |

Thus, we see that a conditional sentence is false in only one scenario, the second scenario (the second row).

Recall again the sentence: “ If you score 100 points on the exam (H), you will get an A on that exam (A).”

H-> A

This sentence is also false in only one scenario, in the case where the teacher did not give you an A even though you earned the perfect score of 100.

In the first scenario, you scored 100 and you got an A, thus:

True -> True = True

Your turn.

Please enter the truth values for the first scenario:

H | A | H -> A |

In the second scenario, you scored 100 but then you look at the grade and

there is a B+. Well then, the teacher was clearly lying which means the sentence “If you score 100, you will get an A” would be false. Since you met the requirement and then the teacher did not give you a perfect A, so the whole sentence is a lie! The teacher’s claim was false.

True -> False = False

Please enter the truth values for the second scenario

H | A | H -> A |

In the third scenario, you scored 99 but you still got an A, thus:

False -> True = True

This means that the first part of the sentence is false since you did not get the perfect 100 score. But then you turn the page and there it is – a perfect A! Thus, the second part of the sentence is true. You got an A on that test!

This means that you don’t need the perfect score of 100 to get an A. It turns out that obtaining 99 also qualifies as an A.

Please enter the truth values for the third scenario:

H | A | H -> A |

In the fourth scenario you scored only 40 points and you got a C, meaning – it is not true that you scored 100, so we enter F in the first column, and it is not true that you got an A, so we also enter F in the second column. And yet, the sentence is true, the teacher did not lie. You did not get an A because you scored only 40 points. In this case, you can’t say that the teacher lied to you.

Thus:

False -> False = True

Enter all the truth values this time:

H | A | H -> A |

It may seem perplexing that F and F gives T. But consider again this sentence: “If we live now in 1946, then next year is 1947”. We clearly do not live in 1946 and next year is definitely not 1947, and yet that hypothetical statement makes perfect sense.

We see once again that a conditional sentence is false in only one scenario.

Let’s summarize. When is a conditional sentence true?

It is true in 3 scenarios and false in one scenario.

- True -> True = True
- True -> False = False
- False -> True = True
- False -> False = True

Now review the 3 truth tables that we already know, and then complete more complex truth tables on the next page.

The truth table for conditional sentences:

A | B | A -> B |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

The truth table for disjunction

A | B | A Ú B |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

The truth table for conjunctions

A | B | A & B |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

The truth table for negation (“not”)

A | – A |

T | F |

F | T |

Your turn:

Complete the truth table:

A | B | A Ú -B |

T | T | |

T | F | |

F | T | |

F | F |

A | B | A -> B |

T | T | |

T | F | |

F | T | |

F | F |

A | B | -A& B |

T | T | |

T | F | |

F | T | |

F | F |

A | B | -A->B |

T | T | |

T | F | |

F | T | |

F | F |

*Challenging: (A plus level)

A | B | C | A & C | B-> (A& C) |

T | T | T | ||

T | T | F | ||

T | F | T | ||

T | F | F | ||

F | T | T | ||

F | T | F | ||

F | F | T | ||

F | F | F |