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Valid Forms of Reasoning

Beth

Valid Forms of Reasoning

Philosophy 170: Introduction to Logic

Worksheet Four

When does the conclusion follow from the premises?

The conclusion follows from the premises when the argument is valid.  But what makes the argument valid?

Consider again this simple argument:

  1. If someone lives in NY, then this person lives in the US. Premise 1
  2. John lives in NY. Premise 2
  3. Therefore, he lives in the US. Conclusion

Now think if it is at all possible that the conclusion is false?  Is it possible that John does not live in the US, given that he lives in NY (he really lives there, for 100%) and that NY is definitely in the US?  No, that’s not possible.

So, when it’s impossible for the conclusion to be false based on given premises, then the argument is valid.

Now think if this conclusion can possibly be false given the two premises:

  1. If someone lives in NY, then this person lives in the US. Premise 1
  2. Ana lives in the US. Premise 2
  3. Therefore, Ana lives in NY. Conclusion

Is it possible that the conclusion is false?  Yes of course! Therefore, the argument is not valid.

When the argument is invalid, we know that the conclusion does not follow from the given premises.

So, today we will look at different types of valid and invalid arguments and we will learn how to symbolize these arguments.

Let’s try to symbolize the first argument.

  1. If someone lives in NY, then this person lives in the US. Premise 1
  2. John lives in NY.  Premise 2
  3. Therefore, John lives in the US.

Let’s say that the letter N will stand for “Someone (anyone) lives in NY.”

Let’s also agree that the letter U will stand for “Someone (that someone, that person) lives in the US.”

We learned last week that we can use an arrow to indicate a conditional sentence, so inserting an arrow between the two letters, N and U, will create this statement: N -> U , which we already know reads as: IF N, then U.

So the first sentence will look like this:

  1. N -> U            Premise 1.

Since John is someone, the second premise can also be symbolized with the letter N, and the conclusion with the letter U.

Altogether:

  1. N -> U            Premise 1.
  2. N                     Premise 2.

_____________________

  1. U                   Conclusion

Let’s try another scenario.  This time we will change the second premise.

  1. If someone lives in NY, then this person lives in the US.
  2. Ana does not live in NY.
  3. Therefore, she does not live in the US.

Here the first premise is again:  N-> U.  But the second premise tells us that Ana (a particular someone) does not live in NY.  So we can use the negation symbol to convey the idea that N is not true in this scenario, and that U is not true (according to the argument).

  1. N -> U Premise 1.
  2. -N   Premise 2.

_____________________

  1. – U Conclusion

So we just turned the entire argument into symbols and we have done this correctly but we know that this argument is actually invalid.

The argument is invalid because it is totally possible that Ana lives in the US, even thought she does not live in NY!

Now let’s decipher what the argument below is saying.

  1. N -> U Premise 1.
  2. U Premise 2.

_____________________

  1. N Conclusion

Again, let’s assume the first premise says:

  1. If someone lives in NY, then this person lives in the US.

What does the second premise assert?

Does the conclusion follow?

Is the argument valid?

No, the argument is invalid.  The second premise asserts that Ana (or some other particular person) lives in the US.

And the conclusion says that person therefore lives in NY.  But it’s possible that this person lives in NJ so the argument in invalid.

Finally, is this argument valid:

  1. N -> U Premise 1.          If someone lives in NY, then he/she lives in the US
  2. -U Premise 2.          Max does not live in the US

_____________________

  1. -N Conclusion           Therefore, Max does not live in New York.

Explain here:

Yes, the argument is valid. The second premise states Max does not live in the US and the conclusion illustrates the person does not live in New York. It is factual that if Max does not live in US, he cannot possibly live in NY. So the claim is valid

So what we have just considered is 4 different scenarios.  Let’s now look at these 4 formulas representing these 4 scenarios.

I.

  1. N -> U Premise 1.
  2. N Premise 2.

_____________________

  1. U Conclusion

II.

  1. N -> U Premise 1.
  2. -U Premise 2.

_____________________

  1. -N Conclusion

III.

  1. N -> U Premise 1.
  2. U Premise 2.

_____________________

  1. N Conclusion

IV.

  1. N -> U Premise 1.
  2. -N Premise 2.

_____________________

  1. -U Conclusion

The first two argument formulas in green are valid, and the two argument forms in red are invalid.

Let’s now use different letters (say A and B) to practice constructing the valid and invalid formulas.

Valid Forms of Reasoning

  2. A -> B Premise 1.
  3. A Premise 2.

_____________________

  1. B Conclusion

II.

  1. A -> B Premise 1.
  2. -B Premise 2.

_____________________

  1. -A Conclusion

Invalid Forms of Reasoning

III.

  1. A -> B Premise 1.
  2. -A Premise 2.

_____________________

  1. -B Conclusion

IV.

  1. A -> B Premise 1.
  2. B Premise 2.

_____________________

  1. A Conclusion

*I strongly suggest that you take now a piece of paper and write down from memory valid and invalid formulas using different letters.

VALID REASONING

  1. X -> Y
  2. X
  3. Y

INVALID REASONING

  1. X -> Y
  2. Y
  3. X

Recall now the 13 arguments from worksheet 1 (week 2). We shall now convert several of those arguments into symbols.  Notice the use of parenthesis.

* For this exercise you might want to review the symbolic notations for disjunctions and for conjunctions (See worksheet 2, week 4).

Argument 1.

  1. If someone is right now in Botswana or in Chad then that person is in Africa.
  2. John is not in Botswana or Chad right now.
  3. Therefore, John is not in Africa right now.

We know that the first sentence is true: If someone is right now in Botswana or in Chad then that person is in Africa. This is true because these two countries are in Africa.  Premise two says that someone, say John, is not in Botswana or Chad. Then the conclusion says that John is not in Africa.  We can symbolize this argument like that:

  1. ( B v C ) -> A
  2. – ( B v C)

_________________________

  1. – A This conclusion does not follow because John might be in Libya, for example.

There are 54 countries in Africa, so, just because someone is not in Botswana or Chad right now, that does not mean that person is not in Africa for sure.

Argument 2.

  1. If someone is now in Arizona, then that person is not in Brazil.
  2. John actually is now in Arizona.
  3. Therefore, John is not in Brazil.

This argument is valid.  Premise 1 says that if you are right now in Arizona, then you cannot be at the same time in Brazil.  This is true.   Premise 2 says that John happens to be in Arizona. So we can conclude that he is not in Brazil.

We can turn the argument into symbols like this:

  1. A ->  – B
  2. A

_____________

  1. – B              

Now translate the 3 arguments into symbols.  Use parenthesis when necessary.

Argument 3.

  1. If this painting does not have any colors, then this painting is not blue. Premise 1
  2. But this painting has many colors. Premise 2
  3. Therefore, this painting has definitely a blue color in it.                                          Conclusion

 

  1. 1. –C -> -B
  2. C

__________

  1. B This conclusion is not valid because having many colors, does not necessarily assert a blue color is in the painting

Argument 4.

  1. If someone is right now in Rio de Janeiro, then that person is in Brazil.
  2. Ana is in Rio de Janeiro right now.
  3. So Ana is in Brazil.

 

  1. R -> B
  2. R

__________

  1. B The conclusion is valid

Argument 5.

  1. If someone is not in Africa, then that person is not is Sudan or Kenya.
  2. Ana is in Africa right now.
  3. So Ana is in Sudan or Kenya.

 

1.-A -> -(S v K)

  1. A

__________

  1. (S v K) This conclusion is not valid because Ana may be in Africa right now, but not necessarily in Sudan or Kenya. Africa has many countries.

Now create your own argument, first in full sentences, and then translate them into symbols.

  1. If Michael was patient, he could have won the lottery
  2. Michael was patient
  3. Therefore, Michael won the lottery

In symbols:

  1. M -> L
  2. M

_______________

  1. L This conclusion is invalid because he could have been patient and still not won the lottery.

Now, say which of the arguments below are valid and which are invalid:

a)

  1. N -> U Premise 1.
  2. N Premise 2.

_____________________

  1. U Conclusion  VALID

b)

  1. N -> U Premise 1.
  2. -U Premise 2.

_____________________

  1. -N                  Conclusion  VALID

c)

  1. N -> U Premise 1.
  2. U Premise 2.

_____________________

  1. N Conclusion  INVALID

 

  1. d)

 

  1. N -> U Premise 1.
  2. -N Premise 2.

_____________________

  1. -U Conclusion  INVALID

 

  1. e)

 

  1. A -> B Premise 1.
  2. A Premise 2.

_____________________

  1. B Conclusion  VALID

f)

  1. A -> B Premise 1.
  2. -B Premise 2.

_____________________

  1. -A Conclusion  VALID

g)

  1. A -> B Premise 1.
  2. -A Premise 2.

_____________________

  1. -B Conclusion  INVALID

 

  1. h)

 

  1. A -> B Premise 1.
  2. B Premise 2.

_____________________

  1. A Conclusion  INVALID

 

  1. i)

 

  1. ( B v C ) -> A
  2. – ( B v C)

_________________________     INVALID

  1. – A

 

  1. j)

 

  1. A ->  – B
  2. A

_____________  VALID

  1. – B

The two valid forms of reasoning we learned about have names!

The first one is called Modus Ponens.

The Second is called Modus Tollens.

Modus Ponens   we will abbreviate as MP

Modus Tollens we will abbreviate as MT

Modus Ponens and Modus Tollens

The Modus Ponens form of inference is when the antecedent is affirmed in another premise. Like this:

  1. If someone lives in NY, then this person lives in the US. Premise 1.
  2. John lives in NY. Premise 2.    Here we are affirming that the

condition is met, meaning we are confirming that John lives in NY.

  1. Therefore, he lives in the US. Conclusion

 

  1. N -> U            Premise 1.
  2. N                     Premise 2.

_____________________

  1. U                   Conclusion        So here we reached the conclusion by MP (Modus Ponens).

Remember, the conclusion is something we obtain/derive/infer from the premises.

So, if we are given these two premises:

  1. A -> B
  2. A

we can derive a conclusion from these two pieces of information like this:

  1. A -> B
  2. A

__________

  1. B

and we can write to the right of your conclusion  MP to let the reader know that we derived the conclusion by the Modus Ponens rule of inference, like this:

  1. A -> B
  2. A

__________

  1. B    1, 2 MP   (I derived the conclusion B, from the two premises by Modus Ponens).

The number 1 and 2 indicate the lines from which we derived line 3, which is here the conclusion line.

The second valid argument is this:

  1. N -> U Premise 1.         If someone lives in NY, then he/she lives in the US
  2. -U Premise 2.          Max does not live in the US.  (I’m denying that Max lives in the US.)

_____________________

  1. -N Conclusion           Therefore, Max does not live in New York.

In this argument we are negating the consequent in lie 2. We are saying in line 2 that Max does not live in the US.

Again: The consequent is the letter U in the conditional sentence in premise 1.   In premise 2, we wrote the sign negative, this means we are saying that Max Doe NOT live in the US.  We negated the U.

 

Whenever we deny the consequent in another premise and we conclude that the antecedent is NOT the case, the argument is called Modus Tollens.

So, whenever you derive or obtain (or infer) a conclusion in an argument by the Modus Tollens form, you will add:  1, 2 MT in line 3, that is in the conclusion, like this:

  1. A -> B
  2. – B

________

  1. -A 1, 2 MT       (I derived negative A from the two premises by means of Modus Tollens).

Now your turn.  Say which argument is an example of Modus Ponens inference and which is an example of Modus Tollens inference:

  1. a)

 

  1. X -> Y Premise 1.
  2. X Premise 2.

_____________________

  1. Y Conclusion   1, 2 MP (I derived the conclusion Y from 1,2 premises by Modus Ponens)

 

  1. b)

 

  1. X -> Y Premise 1.
  2. -Y Premise 2.

_____________________

  1. -X Conclusion  1, 2 (I derived negative X from the two premises by Modus Tollens)

Let’s now translate this argument:

  1. If someone is right now in Rio de Janeiro, then that person is in Brazil.
  2. Ana is in Rio de Janeiro.
  3. So she is in Brazil.

Let’s say the letter R stands for:  “Some person is right now in Rio de Janeiro.”

Where is Rio de Janeiro? In Brazil.

So, if someone is right now in Rio de Janeiro, then that person is clearly in Brazil.

Let’s write this in symbols:   R -> B

Now, it turns out that Ana is in Rio de Janeiro.   What can we conclude from this?

That she is in Brazil.

This makes sense.  The argument is valid because the conclusion follows logically from the two premises.

But is the argument an example of Modus Ponens or Modus Tollens?

In symbols:

  1. R -> B
  2. R

_____________

  1. B

Say here if this is MP or MT: Modus Ponens

Consider the following scenario:

The teachers says:  “Students, if you get 100 points on the midterm (H), then you will an A for that midterm (A).”

Let’s write what the teacher said in symbols:  H -> A

Now it turns out Ana got hundred points!  We can conclude that she got an A.

This makes sense.  The argument is valid because the conclusion follows logically from the two premises.

In symbols:

  1. H -> A
  2. H

_____________

  1. A

Is this MP or MT? Modus Ponens

Now let’s look at this argument:

  1. R -> B
  2. -B

_____________

  1. -R

Is it MP or MT? Modus Tollens

Again, it is true that if someone is right now in Rio de Janeiro, then that person is clearly in Brazil. Let’s write this in symbols:    R -> B

Now, it turns out that John is NOT in Brazil (- B).   What can we conclude from this?

Can we conclude that therefore he is definitely not in Rio de Janeiro?   Yes (- R).

This makes sense.  The argument is valid because the conclusion follows logically from the two premises.

Finally:

  1. H -> A
  2. – A

_______

  1. – H

Is this MP or MT? Modus Tollens

Again, we know that:  “If you get 100 points on the midterm (H), then you will an A for that midterm (A).”

Let’s write what the teacher said in symbols:   H -> A

Now it turns out that John did NOT get an A.  Can we therefore conclude that he did not obtain 100 points?   Yes.

This makes sense.  The argument is valid because the conclusion follows logically from the two premises.

 

If Ana did not get an A, that means it’s impossible that the conclusion is false.

But what if we deny the antecedent like this:

  1. A -> B Premise 1.
  2. -A Premise 2.

_____________________

  1. -B Conclusion.

Well, this is Not Modus Ponens.  This is simply invalid.   This particular invalid argument is called
denying the antecedent.

Invalid arguments:  Denying the Antecedent

  1. A -> B Premise 1.
  2. -A Premise 2.

_____________________

  1. -B Conclusion.   This conclusion does not follow the two premises and so the argument is invalid.

This logical fallacy is called the fallacy of denying the antecedent.

If you do not understand why this argument is invalid, imagine again that the letter A stands for “Someone lives in NY” and the letter B stands for: “He/she lives in the US,” and the conditional expression A -> B  reads: If someone lives in NY, then that person lives in the US.”

Thus, we read:

  1. If someone lives in NY, then she/he lives in the US. Is this true? Yes.    Does this match

the pattern  A -> B?    Yes, it does.

 

Now, let’s say we know for sure that John does NOT live in NY.  Thus:

  1. – A (John does not live in NY.)     Here we are denying/negating the antecedent “A.”

Can I conclude that John does not live in the US?   No.   This is why the conclusion -B does not follow from the premises.  The argument is invalid.

Now give your own example of an invalid argument:

  1. If I live in Uganda, then I live in Africa
  2. Mark does not live in Uganda
  3. Therefore, Mark does not live in Africa

Are there other invalid arguments?  Yes.

Invalid Argument: Affirming the Consequent

  1. A -> B Premise 1.
  2. B Premise 2.

_____________________

  1. A Conclusion.  This conclusion does not follow the premises.

This logical fallacy is called the fallacy of affirming the consequent.

If you do not understand why this is so, imagine again that the A stands for “Someone lives in NY”, and the B stands for: “He/she lives in the US,” and the conditional expression A ® B  reads: If someone lives in NY, then that person lives in the US.”

Thus, we read:

  1. If someone lives in NY, then she/he lives in the US. Is this true? Yes.    Does this match   the symbols  A -> B?    Yes.

Now, let’s say that we know for sure that Ana lives in the US.  Thus:

  1. B (Ana lives in the US.)     We are affirming the consequent.

Can I logically conclude from the two premises that Ana lives in NY?   No.   This is why the conclusion A does not follow the premises.  The argument is invalid.

Finally, consider once again the argument we looked at last week:

  1. All living things need water.
  2. Dogs need water.
  3. Therefore, dogs are living things.

Can you now see what is wrong with this argument?

If not, compare it to the argument below:

  1. All birds need oxygen.
  2. Dogs need oxygen.
  3. Therefore, dogs are birds.

Now explain in your own words why the first argument about dogs cannot be possibly valid.  Keep in mind the second argument about dogs being birds.

The first argument is invalid because it affirms the consequent, but it does not necessarily mean the living thing is a dog. Similarly, the second argument is invalid because the second premise affirms the consequent, but it is not necessarily the antecedent. 

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Question 

Worksheet Four                          

Philosophy 170: Introduction to Logic

When does the conclusion follow from the premises?

Valid Forms of Reasoning

The conclusion follows from the premises when the argument is valid.  But what makes the argument valid?

Consider again this simple argument:

  1. If someone lives in NY, then this person lives in the US. Premise 1
  2. John lives in NY. Premise 2
  3. Therefore, he lives in the US. Conclusion

Now think if it is at all possible that the conclusion is false?  Is it possible that John does not live in the US, given that he lives in NY (he really lives there, for 100%) and that NY is definitely in the US?  No, that’s not possible.

So, when it’s impossible for the conclusion to be false based on given premises, then the argument is valid.

Now think if this conclusion can possibly be false given the two premises:

  1. If someone lives in NY, then this person lives in the US. Premise 1
  2. Ana lives in the US. Premise 2
  3. Therefore, Ana lives in NY. Conclusion

Is it possible that the conclusion is false?  Yes of course! Therefore, the argument is not valid.

When the argument is invalid, we know that the conclusion does not follow from the given premises.

So, today we will look at different types of valid and invalid arguments and we will learn how to symbolize these arguments.

Let’s try to symbolize the first argument.

  1. If someone lives in NY, then this person lives in the US. Premise 1
  2. John lives in NY.  Premise 2
  3. Therefore, John lives in the US.

Let’s say that the letter N will stand for “Someone (anyone) lives in NY.”

Let’s also agree that the letter U will stand for “Someone (that someone, that person) lives in the US.”

We learned last week that we can use an arrow to indicate a conditional sentence, so inserting an arrow between the two letters, N and U, will create this statement: N -> U , which we already know reads as: IF N, then U.

So the first sentence will look like this:

  1. N -> U            Premise 1.

Since John is someone, the second premise can also be symbolized with the letter N, and the conclusion with the letter U.

Altogether:

  1. N -> U            Premise 1.
  2. N                     Premise 2.

_____________________

  1. U                   Conclusion

Let’s try another scenario.  This time we will change the second premise.

  1. If someone lives in NY, then this person lives in the US.
  2. Ana does not live in NY.
  3. Therefore, she does not live in the US.

Here the first premise is again:  N-> U.  But the second premise tells us that Ana (a particular someone) does not live in NY.  So we can use the negation symbol to convey the idea that N is not true in this scenario, and that U is not true (according to the argument).

  1. N -> U Premise 1.
  2. -N   Premise 2.

_____________________

  1. – U Conclusion

So we just turned the entire argument into symbols and we have done this correctly but we know that this argument is actually invalid.

The argument is invalid because it is totally possible that Ana lives in the US, even thought she does not live in NY!

Now let’s decipher what the argument below is saying.

  1. N -> U Premise 1.
  2. U Premise 2.

_____________________

  1. N Conclusion

Again, let’s assume the first premise says:

  1. If someone lives in NY, then this person lives in the US.

What does the second premise assert?

Does the conclusion follow?

Is the argument valid?

No, the argument is invalid.  The second premise asserts that Ana (or some other particular person) lives in the US.

And the conclusion says that person therefore lives in NY.  But it’s possible that this person lives in NJ so the argument in invalid.

Finally, is this argument valid:

  1. N -> U Premise 1.          If someone lives in NY, then he/she lives in the US
  2. -U Premise 2.          Max does not live in the US

_____________________

  1. -N Conclusion           Therefore, Max does not live in New York.

Explain here:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

So what we have just considered is 4 different scenarios.  Let’s now look at these 4 formulas representing these 4 scenarios.

 

I.

 

  1. N -> U Premise 1.
  2. N Premise 2.

_____________________

 

  1. U Conclusion

 

 

 

 

II.

 

  1. N -> U Premise 1.
  2. -U Premise 2.

_____________________

 

  1. -N Conclusion

 

 

 

 

III.

 

  1. N -> U Premise 1.
  2. U Premise 2.

_____________________

 

  1. N Conclusion

 

 

 

IV.

 

  1. N -> U Premise 1.
  2. -N Premise 2.

_____________________

 

  1. -U           Conclusion

 

 

 

The first two argument formulas in green are valid, and the two argument forms in red are invalid.

 

 

 

 

Let’s now use different letters (say A and B) to practice constructing the valid and invalid formulas.

 

 

Valid Forms of Reasoning

 

 

  1. A -> B Premise 1.
  2. A Premise 2.

_____________________

 

  1. B Conclusion

 

 

 

 

II.

 

  1. A -> B Premise 1.
  2. -B Premise 2.

_____________________

 

  1. -A Conclusion

 

 

 

 

 

 

Invalid Forms of Reasoning

 

 

 

 

 

III.

 

  1. A -> B Premise 1.
  2. -A Premise 2.

_____________________

 

  1. -B Conclusion

 

 

 

 

 

 

 

IV.

 

  1. A -> B Premise 1.
  2. B Premise 2.

_____________________

 

  1. A Conclusion

 

 

 

 

 

 

 

 

*I strongly suggest that you take now a piece of paper and write down from memory valid and invalid formulas using different letters.

 

 

 

 

 

 

 

Recall now the 13 arguments from worksheet 1 (week 2). We shall now convert several of those arguments into symbols.  Notice the use of parenthesis.

 

 

* For this exercise you might want to review the symbolic notations for disjunctions and for conjunctions (See worksheet 2, week 4).

 

 

 

 

Argument 1.

 

 

  1. If someone is right now in Botswana or in Chad then that person is in Africa.
  2. John is not in Botswana or Chad right now.
  3. Therefore, John is not in Africa right now.

 

 

We know that the first sentence is true: If someone is right now in Botswana or in Chad then that person is in Africa. This is true because these two countries are in Africa.  Premise two says that someone, say John, is not in Botswana or Chad. Then the conclusion says that John is not in Africa.  We can symbolize this argument like that:

 

 

  1. ( B v C ) -> A
  2. – ( B v C)

_________________________

  1. – A This conclusion does not follow because John might be in Libya, for example.

There are 54 countries in Africa, so, just because someone is not in Botswana or Chad right now, that does not mean that person is not in Africa for sure.

 

 

 

 

Argument 2.

 

  1. If someone is now in Arizona, then that person is not in Brazil.
  2. John actually is now in Arizona.
  3. Therefore, John is not in Brazil.

 

 

 

 

 

 

 

This argument is valid.  Premise 1 says that if you are right now in Arizona, then you cannot be at the same time in Brazil.  This is true.   Premise 2 says that John happens to be in Arizona. So we can conclude that he is not in Brazil.

We can turn the argument into symbols like this:

 

  1. A ->  – B
  2. A

_____________

  1. – B              

 

 

 

 

 

 

 

Now translate the 3 arguments into symbols.  Use parenthesis when necessary.

 

 

Argument 3.

 

  1. If this painting does not have any colors, then this painting is not blue. Premise 1
  2. But this painting has many colors.                                                       Premise 2
  3. Therefore, this painting has definitely a blue color in it.                                          Conclusion

 

1.

2.

__________

 

 

 

 

Argument 4.

 

  1. If someone is right now in Rio de Janeiro, then that person is in Brazil.
  2. Ana is in Rio de Janeiro right now.
  3. So Ana is in Brazil.

 

1.

2.

__________

 

 

 

 

 

Argument 5.

 

  1. If someone is not in Africa, then that person is not is Sudan or Kenya.
  2. Ana is in Africa right now.
  3. So Ana is in Sudan or Kenya.

 

 

1.

2.

__________

 

 

 

 

 

Now create your own argument, first in full sentences, and then translate them into symbols.

 

1.

2.

3.

 

 

 

 

In symbols:

1.

2.

_______________

 

 

 

 

 

 

 

 

Now, say which of the arguments below are valid and which are invalid:

a)

 

 

  1. N -> U Premise 1.
  2. N Premise 2.

_____________________

 

  1. U Conclusion

 

 

 

b)

 

  1. N -> U Premise 1.
  2. -U Premise 2.

_____________________

 

  1. -N Conclusion

 

 

 

 

 

c)

 

 

  1. N -> U Premise 1.
  2. U Premise 2.

_____________________

 

  1. N Conclusion

 

 

 

 

 

 

  1. d)

 

  1. N -> U Premise 1.
  2. -N Premise 2.

_____________________

 

  1. -U Conclusion

 

 

 

 

  1. e)

 

  1. A -> B Premise 1.
  2. A Premise 2.

_____________________

 

  1. B Conclusion

 

 

 

 

f)

 

  1. A -> B Premise 1.
  2. -B Premise 2.

_____________________

 

  1. -A Conclusion

 

 

 

 

g)

 

  1. A -> B Premise 1.
  2. -A Premise 2.

_____________________

 

  1. -B Conclusion

 

 

 

 

  1. h)

 

  1. A -> B Premise 1.
  2. B Premise 2.

_____________________

 

  1. A Conclusion

 

 

 

 

  1. i)

 

 

  1. ( B v C ) -> A
  2. – ( B v C)

_________________________

  1. – A

 

 

 

 

 

 

 

 

  1. j)
  2. A ->  – B
  3. A

_____________

  1. – B

 

 

 

 

 

 

 

 

 

 

The two valid forms of reasoning we learned about have names!

 

The first one is called Modus Ponens.

The Second is called Modus Tollens.

 

Modus Ponens   we will abbreviate as MP

Modus Tollens we will abbreviate as MT

 

 

Modus Ponens and Modus Tollens

 

 

 

The Modus Ponens form of inference is when the antecedent is affirmed in another premise. Like this:

 

 

  1. If someone lives in NY, then this person lives in the US. Premise 1.
  2. John lives in NY. Premise 2. Here we are affirming that the

condition is met, meaning we are confirming that John lives in NY.

 

  1. Therefore, he lives in the US. Conclusion

 

 

 

  1. N -> U            Premise 1.
  2. N                     Premise 2.

_____________________

  1. U                   Conclusion        So here we reached the conclusion by MP (Modus Ponens).

 

 

Remember, the conclusion is something we obtain/derive/infer from the premises.

 

 

So, if we are given these two premises:

 

 

  1. A -> B
  2. A

 

 

we can derive a conclusion from these two pieces of information like this:

 

  1. A -> B
  2. A

__________

  1. B

 

 

and we can write to the right of your conclusion  MP to let the reader know that we derived the conclusion by the Modus Ponens rule of inference, like this:

 

  1. A -> B
  2. A

__________

  1. B    1, 2 MP   (I derived the conclusion B, from the two premises by Modus Ponens).

 

The number 1 and 2 indicate the lines from which we derived line 3, which is here the conclusion line.

 

 

 

 

The second valid argument is this:

 

  1. N -> U Premise 1.         If someone lives in NY, then he/she lives in the US
  2. -U Premise 2.          Max does not live in the US.  (I’m denying that Max lives in the US.)

_____________________

 

  1. -N Conclusion           Therefore, Max does not live in New York.

 

 

In this argument we are negating the consequent in lie 2. We are saying in line 2 that Max does not live in the US.

 

Again: The consequent is the letter U in the conditional sentence in premise 1.   In premise 2, we wrote the sign negative, this means we are saying that Max Doe NOT live in the US.  We negated the U.

 

Whenever we deny the consequent in another premise and we conclude that the antecedent is NOT the case, the argument is called Modus Tollens.

 

So, whenever you derive or obtain (or infer) a conclusion in an argument by the Modus Tollens form, you will add:  1, 2 MT in line 3, that is in the conclusion, like this:

 

 

  1. A -> B
  2. – B

________

 

  1. -A 1, 2 MT       (I derived negative A from the two premises by means of Modus Tollens).

 

 

 

 

Now your turn.  Say which argument is an example of Modus Ponens inference and which is an example of Modus Tollens inference:

 

  1. a)

 

  1. X -> Y Premise 1.
  2. X Premise 2.

_____________________

 

  1. Y Conclusion

 

 

 

 

 

 

  1. b)

 

  1. X -> Y Premise 1.
  2. -Y Premise 2.

_____________________

 

  1. -X Conclusion

 

 

 

Let’s now translate this argument:

 

  1. If someone is right now in Rio de Janeiro, then that person is in Brazil.
  2. Ana is in Rio de Janeiro.
  3. So she is in Brazil.

 

Let’s say the letter R stands for:  “Some person is right now in Rio de Janeiro.”

 

Where is Rio de Janeiro? In Brazil.

 

So, if someone is right now in Rio de Janeiro, then that person is clearly in Brazil.

 

Let’s write this in symbols:   R -> B

 

Now, it turns out that Ana is in Rio de Janeiro.   What can we conclude from this?

That she is in Brazil.

This makes sense.  The argument is valid because the conclusion follows logically from the two premises.

 

But is the argument an example of Modus Ponens or Modus Tollens?

 

In symbols:

 

  1. R -> B
  2. R

_____________

  1. B

 

Say here if this is MP or MT:

 

 

 

Consider the following scenario:

The teachers says:  “Students, if you get 100 points on the midterm (H), then you will an A for that midterm (A).”

Let’s write what the teacher said in symbols:  H -> A

Now it turns out Ana got hundred points!  We can conclude that she got an A.

This makes sense.  The argument is valid because the conclusion follows logically from the two premises.

 

In symbols:

 

  1. H -> A
  2. H

_____________

  1. A

 

Is this MP or MT?

 

 

 

Now let’s look at this argument:

 

 

  1. R -> B
  2. -B

_____________

  1. -R

 

Is it MP or MT?

 

 

Again, it is true that if someone is right now in Rio de Janeiro, then that person is clearly in Brazil. Let’s write this in symbols:    R -> B

 

Now, it turns out that John is NOT in Brazil (- B).   What can we conclude from this?

Can we conclude that therefore he is definitely not in Rio de Janeiro?   Yes (- R).

This makes sense.  The argument is valid because the conclusion follows logically from the two premises.

 

 

 

 

 

 

Finally:

 

  1. H -> A
  2. – A

_______

  1. – H

 

Is this MP or MT?

Again, we know that:  “If you get 100 points on the midterm (H), then you will an A for that midterm (A).”

 

Let’s write what the teacher said in symbols:   H -> A

Now it turns out that John did NOT get an A.  Can we therefore conclude that he did not obtain 100 points?   Yes.

This makes sense.  The argument is valid because the conclusion follows logically from the two premises.

 

If Ana did not get an A, that means it’s impossible that the conclusion is false.

 

 

 

 

 

But what if we deny the antecedent like this:

 

 

  1. A -> B Premise 1.
  2. -A Premise 2.

_____________________

 

  1. -B          Conclusion.

 

 

Well, this is Not Modus Ponens.  This is simply invalid.   This particular invalid argument is called
denying the antecedent.

 

 

 

Invalid arguments:  Denying the Antecedent

 

  1. A -> B Premise 1.
  2. -A Premise 2.

_____________________

 

  1. -B Conclusion.   This conclusion does not follow the two premises and so the argument is invalid.

 

 

This logical fallacy is called the fallacy of denying the antecedent.

 

If you do not understand why this argument is invalid, imagine again that the letter A stands for “Someone lives in NY” and the letter B stands for: “He/she lives in the US,” and the conditional expression A -> B  reads: If someone lives in NY, then that person lives in the US.”

 

 

Thus, we read:

 

  1. If someone lives in NY, then she/he lives in the US. Is this true? Yes.    Does this match

the pattern  A -> B?    Yes, it does.

 

Now, let’s say we know for sure that John does NOT live in NY.  Thus:

 

  1. – A (John does not live in NY.)     Here we are denying/negating the antecedent “A.”

Can I conclude that John does not live in the US?   No.   This is why the conclusion -B does not follow from the premises.  The argument is invalid.

 

 

 

 

Now give your own example of an invalid argument:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Are there other invalid arguments?  Yes.

 

Invalid Argument: Affirming the Consequent

 

 

  1. A -> B Premise 1.
  2. B Premise 2.

_____________________

 

  1. A Conclusion.  This conclusion does not follow the premises.

 

 

This logical fallacy is called the fallacy of affirming the consequent.

If you do not understand why this is so, imagine again that the A stands for “Someone lives in NY”, and the B stands for: “He/she lives in the US,” and the conditional expression A ® B  reads: If someone lives in NY, then that person lives in the US.”

 

Thus, we read:

  1. If someone lives in NY, then she/he lives in the US. Is this true?  Yes.    Does this match   the symbols  A -> B?    Yes.

 

 

Now, let’s say that we know for sure that Ana lives in the US.  Thus:

 

  1. B (Ana lives in the US.)     We are affirming the consequent.

 

Can I logically conclude from the two premises that Ana lives in NY?   No.   This is why the conclusion A does not follow the premises.  The argument is invalid.

 

 

 

 

 

 

Finally, consider once again the argument we looked at last week:

 

 

 

  1. All living things need water.
  2. Dogs need water.
  3. Therefore, dogs are living things.

 

 

Can you now see what is wrong with this argument?

If not, compare it to the argument below:

 

  1. All birds need oxygen.
  2. Dogs need oxygen.
  3. Therefore, dogs are birds.

 

 

 

Now explain in your own words why the first argument about dogs cannot be possibly valid.  Keep in mind the second argument about dogs being birds.

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