The Deepest of Mysteries


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Evaluating Deductive Arguments

Although I'm having you treat every argument you can't otherwise identify as a deductive argument, "deductive" arguments are actually a very specific kind of argument, precisely defined and beloved by logic geeks everywhere. So, if you'll forgive a little bit of geekery, here's the skinny on deductive arguments.

A deductive argument is one that relies on the purported truth of its premises and on the purported fact that it is impossible for those premises to be true if the conclusion is false. (Any argument that isn't "deductive" is "inductive.")

A deductive argument that has good logical form is called "valid," one that doesn't is called "invalid." Invalid deductive arguments are no good. Pshaw! They're crap. (And they know it, the stinkers.) By the way, this is something that only applies to arguments. Only arguments can be valid or invalid. Statements can be true or false, but they can't be valid or invalid.

A valid deductive argument with true premises is called "sound." A sound argument has a true conclusion. Period. If it's sound, it's conclusion is true. Not, "most likely," not "really really probable." Just plain flat true! (Of course, for this to work we have to be absolutely sure those premises are true.)

Validity: The Easy Part

Before I get into the easy part of validity, I'm just going to mention, but not test you on yet, the real, correct and true definition of "validity". Here it is: (Accept no substitutes.)

An argument is valid if, and only if it is impossible for there to be a situation in which all it's premises are TRUE and it's conclusion is FALSE.

Scared? Confused? I don't blame you. Many, many, many people have a very hard time understanding this definition, so I'm reserving the hard bits for the next chapter. This means you don't have to deal with the full definition of validity until then. For now, I'm giving you a simplified rule for validity to get you through this chapter

The Easy Implication of Validity

The correct definition has three logical implications. Two of those implications are weird and hard to understand, but one is pretty straightforward, so we'll deal with the easy implication first and leave the other two for the next chapter.

Easy Implication: If the argument is such that it's premises, if true, logically force the conclusion to be true, then the argument is valid.

You could also say this as "an argument is valid if the premises make the conclusion true" or "if you can't make the conclusion false without also making at least one premise false, then the argument is valid" or "if the premises, taken all together, force the conclusion to be true, the argument is valid."

Now I want to point out an Important and Ansolutely True Fact. The following statements are all FALSE

This is because the premises making the conclusion true is a sufficient condition for the argument to be valid, but is absolutely not a necessary condition for the argument to be valid. We will talk about the other ducks, I mean the other ways of making an argument valid in the next chapter.

How to Think About Validity

The correct way to think about this is to think about assumptions and possibilities. When you look at one of the arguments given in this chapter say to yourself, "now, if I assume that all these premises are true, is it possible for this conclusion to be false?" (This test works for all the arguments in this chapter. Other kinds of valid arguments will be dealt with later)

The definition I'm giving here is sufficient for an argument to be valid. But it's only one of the ways an argument can be valid, so it's not a necessary condition for an argument to be valid.

When checking arguments for validity, assume that all the premises are true, then ask yourself if it is now possible for the conclusion to be false. If the answer is "yes," the argument is invalid (wonky). If it's "no," the argument is valid. (No, I didn't get it mixed up. That's the rule.)

Now possible for conclusion to be false = invalid (wonky)

Now impossible for conclusion to be false = Valid

For instance, the following arguments are all completely invalid (wonky).
Paris is in France
Berlin is in Germany
Compton is in America
Cats are mammals
Dogs are mammals
Ferrets are mammals
People have two legs
Mammals have four legs
Insects have six legs
Dumbledore is a wizard
Gandalf is a wizard
Merlin is a wizard


Example 1

If George Washington had been poisoned, shot, beheaded and then burned to ashes by his wife for his constant adultery, he would now be dead. George was not poisoned, shot, beheaded and then burned to ashes by Martha (pity), so he's not dead.

1. If Martha Washington had gone absolutely postal on George's ass, he would be dead.
2. Martha was strictly non-postal.
C. George ain't dead.

If these premises were true, would it be possible for the conclusion to be false. Absolutely! George could have died from some other cause, say a bad cold caught when avoiding the irate father of a young woman he had seduced, so it's logically possible for the sentence "George ain't dead" to be false, even if we assume the truth of the premises. So it's invalid (wonky).

Example 2

If carnivorous faerie-pixies had sprinkled their magic barbeque sauce on Ghengis Khan, ol' Ghengis would now be a used car salesman working out of Bakersfield with an unhappy wife, two overweight children and a suspicious rash. Well, carnivorous faerie-pixies did sprinkle magic barbeque sauce on ol' Ghengis, so Ghengis Khan has a rash and so on.

1. If pixies had BBQ sauced Ghengis Khan, Ghengis would be a rash-infested used car salesman working out of Bakersfield.
2. Pixies did BBQ sauce Ghengis Khan.
C. Ghengis does have a rash.

Now, to check validity, we suspend disbelief on the premises and assume that they're true. So if we assume magic barbecue sauce exists and will have these effects when sprinkled, then Ghengis Khan will be in Bakersfield and so on. One of the effects of the sauce is a suspicious rash so, given the truth of the premises, the statement "Ghengis Khan has a rash" can't be false. So this argument is valid. That's right, VALID!

Example 3

You know that big black and white swimming thing they have down at Seaworld, They call it "Shamu," and it's either a cat or a dog. Well, it's certainly not a dog, so it must be a cat.

Shamu is either a cat or a dog
Shamu is not a dog
Shamu is a cat.

Remember again that validity has nothing to do with whether or not the premises are true. What it depends on whether the conclusion could be false, if, the premises were true. Now, if these premises were true, could the conclusion be false? It couldn't, so this argument is valid.

Now check these arguments for validity.

All monkeys are primates. George is a monkey, so it follows that George is also a primate. Answer

All monkeys are primates. Pam is a primate, so it follows that Pam is also a monkey. Answer

All monkeys are primates. Nick is not a monkey, so it follows that Nick is not a primate. Answer

All monkeys are primates. Oswald is not a primate, so Oswald is not a monkey. Answer

Remember, the actual truth or falsity of the premises is irrelevant, completely irrelevant, to the validity of the argument.  Validity is just about the logical relationship between the parts of the argument, nothing else.

Logical Form and Validity

Now, some deductive logical forms are so common, and so important, that they get their own names. Here's five of them.

Modus Ponens

If X is true, then Y is true.       If Babe is a shoat, then Joe is a Mocklin       If Roy is a tramp, then Roy is a bum
X is true                                   Babe is a shoat                                               Roy is a tramp                                      
Y is true                                   Joe is a Mocklin                                               Roy is a bum

Affirming the Consequent

If X is true, then Y is true.      If Babe is a shoat, then Joe is a Mocklin         If Roy is a tramp, then Roy is a bum
Y is true                                  Joe is a Mocklin                                                  Roy is a bum                                        
X is true                                  Babe is a shoat                                                  Roy is a tramp

Denying the Antecedent

If X is true, then Y is true.     If Babe is a shoat, then Joe is a Mocklin        If Roy is a tramp, then Roy is a bum
X is not true                          Babe is not a shoat                                           Roy is not a tramp                             
Y is not true                           Joe is not a Mocklin                                          Roy is not a bum

Modus Tollens

If X is true, then Y is true.    If Babe is a shoat, then Joe is a Mocklin       If Roy is a tramp, then Roy is a bum
Y is not true                         Joe is not a Mocklin                                          Roy is not a bum                                
X is not true                         Babe is not a shoat                                           Roy is not a tramp                                           

Disjunctive Syllogism

Either X is true or Y is true.    Either Babe is a shoat or Joe is a Mocklin    Roy is either a bum or a tramp
Y is not true                              Joe is not a Mocklin                                         Roy is not a bum                      
X is true                                    Babe is a shoat                                                Roy is a tramp                                           

In each of the above groups, the first argument is the general statement of the form, the second is a specific instance of the form, and the third argument is an instance of a special case of the form that is easier to deal with in pictures.

Which ones are valid? Which ones are not? Well, to figure that out for the first four, you have to make sure that you understand that "if Babe is a shoat, then Joe is a Mocklin" just means that "if Babe is a shoat, then Joe is a Mocklin." It does not go on and say "if Joe is a Mocklin, then Babe is a shoat." That particular claim is not a part of any of these arguments. Because the second claim is not made or implied by "if Babe is a shoat, then Joe is a Mocklin," the only way "if Babe is a shoat, then Joe is a Mocklin," can be false is if Babe is a shoat but Joe is not a Mocklin. (This is because it's an "if-then" statement.) It doesn't say that Babe is a shoat, and it doesn't say that Joe is a Mocklin. It just says that if Babe is a shoat, then Joe is a Mocklin.

Too complicated? Well, then we'll go with an easier example. I've got these special case Roy tramp/Roy bum arguments which can be easily done in pictures, and which will hopefully make the difference clear. As before, the only way that "if Roy is a tramp, then Roy is a bum" can be false is if Roy is a tramp, and Roy is not a bum. Under all other circumstances that statement will be true. Here's the same point in pictures. Only one of the following pictures makes "if Roy is a tramp, then Roy is a bum" false. All the other pictures make it true.

Picture 1                                                  Picture 2                                              Picture 3                                              Picture 4                        
aaagh!                     aaagh!                    aaagh!                     aaagh!
Roy is neither a tramp nor a bum.            Roy is a tramp but not a bum.                Roy is a tramp and a bum.                  Roy is not a tramp but is a bum.                                                                                         
Well, notice that in picture 2 Roy is a tramp but is not a bum. That couldn't be true if "if Roy is a tramp, then Roy is a bum" was true, so picture 2 makes "if Roy is a tramp, then Roy is a bum" false. All of the other pictures are logically compatible with "if Roy is a tramp, then Roy is a bum" because they could be true even if "if Roy is a tramp, then Roy is a bum" was true. So none of the other pictures makes "if Roy is a tramp, then Roy is a bum" false.

Now, there's an easier way to convey this information with pictures. We can draw a picture in which putting the "r" in the "T" circle means also putting it in the "B" circle. This is easy, because all we have to do is draw the T circle inside the B circle. Like so:

Picture 5
aaagh!

(This actually says "all tramps are bums," which is a little stronger than "if Roy is a tramp, then Roy is a bum," but I'm not going to worry about that here.)

Now copy out the above picture four times, and take each of the following forms in turn, and try to draw in an "r" to make the premises of that form true, and the conclusion false. (Picture five, as it stands, already makes the first premise of each form true. You just have to make the conclusion false while leaving the second premise true.) Click on the name of the form to check your answer.

Modus Ponens                                                   Affirming the Consequent                
If Roy is a tramp, then Roy is a bum                If Roy is a tramp, then Roy is a bum
Roy is a tramp                                                   Roy is a bum                                          
Roy is a bum                                                      Roy is a tramp

Denying the Antecedent       

                            Modus Tollens
If Roy is a tramp, then Roy is a bum               If Roy is a tramp, then Roy is a bum  
Roy is not a tramp                                            Roy is not a bum                                
Roy is not a bum                                               Roy is not a tramp

Okay, so now you've figured out that affirming the consequent and denying the antecedent are invalid. (They are actually our first two deductive fallacies, because they are argument forms that look sort of like good forms, but which are really terrible.) The most interesting one is the valid form modus tollens, which is one that many people don't expect to be valid. (And lots of people take a while to figure out why it's valid.) Nevertheless, it's an important and interesting form, especially since it gives us one of our ways of proving a negative statement.

Think about it, if we can prove that if Roy was a tramp, then he would be a bum, and we could also prove that Roy isn't a bum, then that's enough to prove that Roy isn't a tramp either. If it's true that if Roy was a tramp, then he would be a bum, is is it possible for him to be a tramp without also being a bum? No it isn't, so the fact of his not being a bum would then prove adsolutely that he was not a tramp.

More on Disjunctive Syllogism

We've seen the fallacy of "false choice," where an arguer illegitimately claims that the number of possibilities is less than it really is. But what about real choices? What about when there really are only two possibities? If that is really the case, and you can eliminate one of the only two possibities, well then you can apply the valid argument form of "disjunctive syllogism" and get a sound argument.

1. George is either alive or dead.
2. George isn't dead.
C. George is alive.

Now here's the tricky bit. The paragraph above leaves out one little detail. We have to be careful not to confuse the concepts of fallacy and validity. False Choice isn't a fallacy because of invalidity. It's actually a perfectly valid disjunctive syllogism in form. The only thing that makes false choice a fallacy is that the crucial premise, the one that limits our choices, is false. Look at the following argument.

1. Shamu is either a cat or a dog
2. Shamu is not a dog
C. Shamu is a cat.

Notice now that both arguments have the same valid form. The second argument only fails because the first premise is false. Dog and cat are not the only two possibilities for Shamu's species.
Now let's look at disjunctive syllogism through pictures and the following argument.

Roy is either a bum or a tramp
Roy is not a bum                      
Roy is a tramp

Now, it's important to know that in logic, the word "or" is always taken to mean that it could be one and it also could be the other. You are not supposed to add the "and not both" that many people mentally add when they say "or." In logic, "or" should alsways be understood as saying, "one or the other, or both." The following three pictures therefore all make "Roy is either a bum or a tramp" true.

Picture 2                                              Picture 3                                              Picture 4                        
aaagh!                    aaagh!                     aaagh!
Roy is a tramp but not a bum.                Roy is a tramp and a bum.                  Roy is not a tramp but is a bum.                                                                      

If Roy is a tramp but not a bum, then the statement "Roy is either a bum or a tramp" is true.
If Roy is a tramp and a bum, then the statement "Roy is either a bum or a tramp" is true.
And if Roy is not a tramp but is a bum, then the statement "Roy is either a bum or a tramp" is true.. 

Now, out of those three pictures above, which one also makes the statement "Roy is not a bum" true? It's picture 2, isn't it?

Picture 2
aaagh!
Roy is a tramp but not a bum.

Now, this is the only picture that makes both of the premises true.Does it make the conclusion true? If it does, the argument form is valid.

Terminology (Again)

Remember, a logically good deductive argument is called valid, and a valid argument with true premises is called sound. A logically good inductive argument is called strong, and a strong argument with true premises is called cogent. The words "valid" and "sound" are not used for inductive arguments, and the words "strong" and "cogent" are not used for deductive arguments.

 

 

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This chapter concerns the correct defininition of "validity," which is the one part of logic that most people get wrong most often. I'm not saying it is a hard thing to learn. It's certainly not a complicated thing. BUT, it's something that many people find very counterintuitive. ("Counterintuitive" means that it's very different from what you want to think it is.), Beacuse it's so counterintuitive, many people have a strong tendency to instinctively reject the correct defininition of validity in favor of an incorrect definition that feels right but which is completely and utterly wrong. So if you read the definition of validity in this chapter and find yourself saying "that can't be right, he must mean something different from what he's saying," get a grip on yourself and understand that the definition of validity I will give here, as weird as it seems, is the only correct definition. But don't be scared! I'm not saying the definition will be difficult, I'm just saying that it will be weird.

If you're willing to accept something as weird-but-true, then you will definitely be able to master the definition of validity.

Validity Explained Fully

In the previous chapter I gave you a rule for determining validity that was good for some valid arguments, but not for all valid arguments. Now I'm going to give you the real definition of validity. some of you won't like it, some of you will want to reject it, but it's the only game in town, so you'd better accept it.

Okay, here's where it gets weird. Remember, you'll be okay if you read the definition of validity very carefully, and interpret it absolutely literally.

By the way, this is something that only applies to deductive arguments. Only deductive arguments can be valid or invalid. Statements can be true or false, but they can't be valid or invalid.

A deductive argument is one that relies on the purported truth of its premises and on the purported fact that it is impossible for those premises to be true if the conclusion is false. (Any argument that isn't "deductive" is "inductive.")

A deductive argument that has good logical form is called "valid," one that doesn't is called "invalid." Invalid deductive arguments are no good. Pshaw! They're crap. (And they know it, the stinkers.)

A valid deductive argument with true premises is called "sound." A sound argument has a true conclusion. Period. If it's sound, it's conclusion is true. Not, "most likely," not "really really probable." Just plain flat true! (Of course, for this to work we have to be absolutely sure those premises are true.)

An argument is valid if, and only if it is impossible for there to be a situation in which all it's premises are TRUE and it's conclusion is FALSE.

You probably didn't get that the first time, so go back and read it again. I'll wait.

Did you get it? We'll see. Answer the following "true/false" quiz.

1. An argument where it's possible to have all true premises and a true conclusion all at the same time is always valid. Answer

2. An argument where it's impossible to have all true premises and a true conclusion all at the same time is always valid. Answer

3. An argument where it's possible to have all true premises and a false conclusion all at the same time is always invalid (wonky). Answer

4. An argument where it's impossible to have all true premises and a false conclusion all at the same time is always invalid (wonky). Answer

5. An argument where it's possible to have all true premises and a false conclusion all at the same time is always valid. Answer

6. An argument where it's impossible to have all true premises and a false conclusion all at the same time is always valid. Answer

The only true statements are numbers three and six. All the others are false.

An argument is valid if and only if it is impossible to have a situation in which the premises are true and the conclusion is false. Otherwise, it is invalid (wonky). (A valid argument will only prove something if it is also sound.) An argument is sound if and only if it is valid and all its premises are true. If an argument is sound, then its conclusion is true. Thus, a deductive argument will have persuasive force to the extent that we think that it is sound. Just being valid isn't enough. Neither is just having true premises. It's gotta have both. If we are convinced that the argument is sound, then we should be convinced that the conclusion is true. To put it another way, a sound argument proves its conclusion absolutely.

Now go back and read the definition of validity again. Isn't it weird? I mean, validity isn't really about truth at all. It's about possibility. If a certain kind of situation is possible for an argument, that argument will be invalid (wonky), even if the conclusion is true!

(Test yourself: An argument where the conclusion could be false even if the premises are true is answer)

We can test for validity by trying to draw pictures. Actually, we can test for invalidity by trying to draw pictures. For arguments with the type of premises we can draw pictures for, an argument is valid if and only if it is impossible to draw a picture in which the premises are true and the conclusion is false. Otherwise, it is invalid (wonky).

Read that again carefully. Now test yourself. Which of the following statements (A, B, C & D) is true?

A: If you can draw a picture that makes the premises true and the conclusion true, the argument is VALID

B: If you can draw a picture that makes the premises true and the conclusion false, the argument is VALID

C: If you can't draw a picture that makes the premises true and the conclusion true, the argument is VALID

D: If you can't draw a picture that makes the premises true and the conclusion false, the argument is VALID

Did you get that? It means that to test an argument, we try to draw a picture in which the premises are true and the conclusion is false. If we can, the argument is invalid (wonky). If we can't, it's valid.

Now, is this argument valid or invalid (wonky)?
  Albert Einstein discovered France
My wolverine eats cheese pizza
Laura Schlessinger is a Martian
Fa
Pw
Ml

Now, it's true we can draw a picture in which the premises and conclusion are all true. Here's a simple one:


Aaaaagh!

But it doesn't prove anything. Being able to make everything true doesn't matter. We need to know if its possible to make all the premises true at the same time that the conclusion is false. The following picture does this, so the argument is invalid (wonky).



This picture proves that it's possible for Albert Einstein to have discovered France and for my wolverine to eat cheese pizza even if Laura Schlessinger is not a Martian. If that's possible, then the argument is not valid.

An argument with conclusion and premises that are true still isn't neccesarily valid.
  Elvis is dead. (Accept it.)
The X-Files was a popular TV show
The Eiffel Tower is in France
De
Px
Ft

This time, don't worry about the fact that all of these things are true. Worry about the fact that it's possible for the conclusion to be false even if the premises are true. Again, the following picture does not prove the argument valid.



But this next picture does prove the argument invalid (wonky).


This picture proves that it's possible for Elvis to be dead and for the X-Files to have been a popular TV show even if the Eiffel Tower is not in France. If that's possible, then the argument is not valid.

So if you're trying to check the validity of an argument, and you figure out a way that the premises and conclusion can all be true, then you haven't checked the validity of that argument. You gotta try to figure a way to make the premises true and the conclusion false. If that can't be done, the argument is valid. If it can be done, then it's invalid (wonky).

AND

In deductive logic, we interpret the word "and" in a very special way. We interpret as meaning "both of these are true". What? Well, okay, it's not that special. But we do have a special name for statements made up of two other statements joined by "and". We call such sentences "conjunctions". Yeah, "conjunctions". The important thing here is that, in deductive logic "roses are red and violets are blue" is only true if it is that case that both halves of the phrase are true. So if roses are not red the conjunction "roses are red and violets are blue" is false, and if "violets are not blue", the conjuction "roses are red and violets are blue" is false, aaaaand if roses are not red and violets are not blue, the conjunction "roses are red and violets are blue" is . . . . . . false.

Soooo the following arguments are all valid:

Roses are red and violets are blue.
Roses are red.

Roses are red and violets are blue.
Violets are blue.

Roses are red.
Violets are blue .
Roses are red and violets are blue.

Cheese is yellow and goop is green.
Roses are red and violets are blue.
Violets are blue and cheese is yellow.


And the following are all invalid:

Roses are red.
Roses are red and violets are blue.

Violets are blue .
Roses are red and violets are blue.

Goop is green.
Roses are red and violets are blue.
Violets are blue and cheese is yellow.

To check that the green arguments are valid, try to come up with a situation in which the premises of the arguments are all true and the conclusion is false. If you can't, the argument is valid.

To check that the red arguments are invalid, try to come up with a situation in which the premises of the arguments are all true and the conclusion is false. If you can, the argument is invalid.

OR

Okay, this next bit is important. In deductive logic, we interpret the word "or" in a very special way. We interpret "or" as meaning "at least one of these is true". It's critical to understand that this is the only way "or" is interpreted in deductive logic. Colloquially, "or" is sometimes interpreted a different way, but in deductive logic, "or" is always interpreted as "at least one of these is true".

We also have a special name for statements made up of two other statements joined by "or". We call such sentences "disjunctions". The important thing here is that, in deductive logic "roses are red and violets are blue" is only false if it is that case that both halves of the phrase are false. So if roses are red the conjunction "roses are red or violets are blue" is true, and if "violets are blue", the disjunction "roses are red or violets are blue" is true, and if roses are red and violets are blue, the disjunction "roses are red or violets are blue" is . . . . . . true. That's right, if roses are red and violets are blue, the disjunction "roses are red or violets are blue" is TRUE. Remember that.

OR

We've seen the fallacy of "false choice," where an arguer illegitimately claims that the number of possibilities is less than it really is. But what about real choices? What about when there really are only two possibities? If that is really the case, and you can eliminate one of the only two possibities, well then you can apply the valid argument form of "disjunctive syllogism" and get a sound argument.

1. George is either alive or dead.
2. George isn't dead.
C. George is alive.


Now here's the tricky bit. The paragraph above leaves out one little detail. We have to be careful not to confuse the concepts of fallacy and validity. False Choice isn't a fallacy because of invalidity. It's actually a perfectly valid disjunctive syllogism in form. The only thing that makes false choice a fallacy is that the crucial premise, the one that limits our choices, is false. Look at the following argument.

1. Shamu is either a cat or a dog
2. Shamu is not a dog
C. Shamu is a cat.


Notice now that both arguments have the same valid form. The second argument only fails because the first premise is false. Dog and cat are not the only two possibilities for Shamu's species.
Now let's look at disjunctive syllogism through pictures and the following argument.

Roy is either a bum or a tramp
Roy is not a bum                      
Roy is a tramp


Now, it's important to know that in logic, the word "or" is always taken to mean that it could be one and it also could be the other. You are not supposed to add the "and not both" that many people mentally add when they say "or." In logic, "or" should alsways be understood as saying, "one or the other, or both." The following three pictures therefore all make "Roy is either a bum or a tramp" true.

Picture 2                                              Picture 3                                              Picture 4                        
                                       
Roy is a tramp but not a bum.                Roy is a tramp and a bum.                  Roy is not a tramp but is a bum.                                                                      

If Roy is a tramp but not a bum, then the statement "Roy is either a bum or a tramp" is true.
If Roy is a tramp and a bum, then the statement "Roy is either a bum or a tramp" is true.
And if Roy is not a tramp but is a bum, then the statement "Roy is either a bum or a tramp" is true.. 

Now, out of those three pictures above, which one also makes the statement "Roy is not a bum" true? It's picture 2, isn't it?

Picture 2

Roy is a tramp but not a bum.

Now, this is the only picture that makes both of the premises true.Does it make the conclusion true? If it does, the argument form is valid.

S oooo the following arguments are all valid:

Roses are red.
Violets are blue .
Roses are red or violets are blue.

Roses are red.
Roses are red or violets are blue.

Violets are blue .
Roses are red or violets are blue.

And the following are all invalid:

Roses are red or violets are blue.
Roses are red.

Roses are red or violets are blue.
Violets are blue.

Cheese is yellow or goop is green.
Roses are red or violets are blue.
Violets are blue or cheese is yellow.

Goop is green.
Roses are red or violets are blue.
Violets are blue or cheese is yellow.

Cheese is yellow or goop is green.
Roses are red or violets are blue.
Violets are blue or goop is green.

To check that the green arguments are valid, try to come up with a situation in which the premises of the arguments are all true and the conclusion is false. If you can't, the argument is valid.

To check that the red arguments are invalid, try to come up with a situation in which the premises of the arguments are all true and the conclusion is false. If you can, the argument is invalid.

NOT

Remember the valid argument form of Modus Tollens

If X is true, then Y is true.    If Babe is a shoat, then Joe is a Mocklin       If Roy is a tramp, then Roy is a bum
Y is not true                         Joe is not a Mocklin                                          Roy is not a bum                                
X is not true                         Babe is not a shoat                                           Roy is not a tramp  

And the invalid argument form of Denying the Antecedent

If X is true, then Y is true.     If Babe is a shoat, then Joe is a Mocklin        If Roy is a tramp, then Roy is a bum
X is not true                          Babe is not a shoat                                           Roy is not a tramp                             
Y is not true                           Joe is not a Mocklin                                          Roy is not a bum


And when we combine "or" and "not", we get the valid form of Disjunctive Syllogism

Either X is true or Y is true.    Either Babe is a shoat or Joe is a Mocklin    Roy is either a bum or a tramp
Y is not true                              Joe is not a Mocklin                                         Roy is not a bum                      
X is true                                    Babe is a shoat                                                Roy is a tramp


When we combine "and", "or" and "not", we get all kinds of interesting arguments. (Well, you could at least act like they're interesting.)

The following arguments are all valid:

Roses are red.
Violets are blue .
Roses are not red or violets are blue.

Roses are not red.
Roses are not red or violets are blue.

Violets are not blue .
Roses are red or violets are not blue.

Roses are red and violets are not blue.
Roses are red.

Roses are not red and violets are blue.
Violets are blue.

Roses are not red.
Violets are not blue .
Roses are not red and violets are not blue.

Roses are not red
Roses are red or violets are blue.
V
iolets are blue.

Violets are not blue.
Roses are red or violets are blue.
Roses are red
.

Roses are not red and cheese is not yellow.
Cheese is yellow or goop is green.
Roses are red or violets are blue.
Violets are blue and goop is green.

Roses are not red or cheese is not yellow.
Cheese is yellow or goop is green.
Roses are red or violets are blue.
Violets are blue or goop is green.

Cheese is not yellow or goop is green.
Roses are red or cheese is yellow
Roses are not red or violets are blue.
Violets are blue or goop is green.

And the following are all invalid:

Roses are not red and violets are not blue.
Roses are red and violets are blue.

Roses are not red and violets are not blue.
Roses are red or violets are blue.

Roses are not red or violets are not blue.
Roses are red and violets are blue.

Roses are not red or violets are not blue.
Roses are red or violets are blue.

Roses are not red.
Roses are red or violets are blue.

Roses are not red or violets are not blue.
Roses are red.

Cheese is not yellow or goop is not green.
Roses are not red or violets are not blue.
Violets are not blue or cheese is not yellow.

Roses are not red and cheese is not yellow.
Roses are red or violets are blue.
Violets are blue and goop is green.

Roses are not red or cheese is not yellow.
Roses are red or violets are blue.
Violets are blue or goop is green.

Cheese is not yellow or goop is green.
Roses are not red or violets are blue.
Violets are blue or goop is green.

Cheese is not yellow or violets are not blue
Cheese is yellow or goop is green.
Roses are red or violets are blue.
Roses are red and goop is green.

To check that the green arguments are valid, try to come up with a situation in which the premises of the arguments are all true and the conclusion is false. If you can't, the argument is valid.

To check that the red arguments are invalid, try to come up with a situation in which the premises of the arguments are all true and the conclusion is false. If you can, the argument is invalid.

Tautologies and Contradictions

A "tautology" is a statement that can't be false. Here an example:

Roses are red or roses are not red.

Given that this statement is a disjunction, it follows that if either side is true, the disjunction will be true. So, if roses are red, the statement is true and if roses are not red, the statement is still true.

A "contradiction" is a statement that can't be true. Here an example:

Roses are red and roses are not red.

Given that this statement is a conjunction, it follows that if either side is false, the conjunction will be false. So, if roses are red, the conjunction is false because the other half will be false and if roses are not red, the statement is still false because in that case, the first half will be false.

Oh, and there's pairs of statements that can't both be true, like:

1. Roses are red.
2. Roses are not red.

Whichever way you cut it, one of these statements is false.

And Now the Weirdness Begins

Remember that the correct definition of validity is that an argument is valid if and only if it is impossible to have a situation in which the premises are true and the conclusion is false.

Let's call that situation "Situation S".

In "Situation S" an argument is such that its premises are true AND its conclusion is false.

Catch that word "and".

Now remember, an argument is valid if and only if it is impossible to have a situation in which the premises are true and the conclusion is false. To put it another way, an argument is valid if and only if Situation S is impossible for that argument. In yet other words, Situation S being impossible is both a necessary and a sufficient condition for the argument being valid. (And Situation S being impossible is both a necessary and a sufficient condition for the argument being valid.)

Let me say that again, if Situation S is impossible the argument is valid.

And if Situation S is possible the argument is invalid.

So what kinds of things make Situation S impossible?

Well, Situation S has two parts. First, all the premises are true, and second, the conclusion is false. Let's call these Part 1 and Part 2. To recap:

Situation S: Premises are all true and the conclusion is false.
Part 1: Premises are all true..
Part 2: Conclusion is false.

Now remember the difference between necessary and sufficient conditions. This means:

Situation S is sufficient for Part 1.
Situation S is sufficient for Part 2.
Part 1 is necessary for Situation S.
Part 2 is necessary for Situation S.

What this means that is, if Situation S is possible, then both Part 1 and Part 2 have to be possible.
From that it follows that, if Part 1 is impossible, then Situation S is impossible and, if Part 2 is impossible, then Situation S is impossible, which means that if either Part 1 or Part 2 is impossible, then Situation S is impossible.

Now, here's a question:

If Part 1 is impossible and Part 2 is possible, is Situation S possible?

Remember that both parts are necessary for Situation S, so if Part 1 is impossible, it follows that Situation S is impossible too.

Okay, if that didn't make sense, let's try the same question without the abstractions:

If premises all true is impossible and conclusion false is possible, it follows that the conjunction (remember that word) the conjunction premises all true and conclusion false is also impossible. And if premises all true and conclusion false is impossible, the argument is valid.

Also:

If premises all true is possible but conclusion false is impossible, it follows that the conjunction (remember that word) the conjunction premises all true and conclusion false is also impossible. And if premises all true and conclusion false is impossible, the argument is valid.

So, from this two weird things follow:

If an argument is such that it is impossible for all the premises to be true, it follows that the argument is valid, no matter what the conclusion is.

And:

If an argument is such that it is impossible for the conclusion to be false, it follows that the argument is valid, no matter what the premises are. .

That's the weirdness. (And if that sounds normal to you, you might be a nerd.)

Now, here's a question you probably haven't thought about. But you'd better think about it, because if you answered it without thinking, you'd almost certainly get it wrong.

Do the premises have to make the conclusion true in order for the argument to be valid?

No they don't. One common, but misleading, definition of validity is that "an argument is valid if and only if the premises, if true, make the conclusion true." This is misleading because we can have valid arguments in which the preimises have no logical relationship to the conclusion. For instance, if the conclusion is a "tautology," which is a statement that cannot be false, the argument will be valid no matter what the premises are! Let me emphasize that.

An argument with a conclusion that can't be false IS necessarily valid.

Consider the following argument:

George Bush is over 1,000 feet tall.
The universe is secretly ruled by a small fish living under your couch.
Chocolate either is or is not made from crude oil.

Now ask yourself, can we have a situation where all those premises are true and the conclusion is false? Sure those premises (logically) could be true, but can that conclusion ever be false? If chocolate is made from crude oil, the conclusion is true. If chocolate is not made from crude oil, the conclusion is still true. And if we don't know what chocolate is made from, it still either is or is not crude oil, so the conclusion has got to be true! This conclusion therefore can't be false, and we know the argument is valid whatever the premises are! So that argument just above this paragraph is valid, and so is this one!

All wombats are bitter mysogynists living in dank warrens under the boulevard cafes of Paris.
Cigarette smoking makes you cool, especially if you cough up a diseased lung right in front of the Pope.
It may or may not be true that Osama Bin Laden is posing as a cab driver in Des Moines, Iowa.

You think that's weird? Well check this out.

An argument with premises that can't all be true IS necessarily valid.

Read that again. It says that if the premises can't all be true, then the argument is valid. It doesn't even mention the conclusion, which means that an argument with a false, stupid or impossible conclusion can be perfectly valid, provided it has premises that somehow contradict each other.

I'll say it again. If an argument has mutually contradictory premises, that is, premises which contradict each other, then that argument is automatically valid.

Test yourself. Which of the following two sentences says the same thing as the sentence underlined above?

A. An argument with premises that can't all be true is necessarily valid.

B. An argument with premises that can't all be true is necessarily invalid (wonky).

If you said "A," you're right!
If you said "B," you're wrong.

Here's an example of an argument that's valid because of contradictory premises.

Elvis is dead.
Elvis is alive.
Laura Schlessinger is a woolly mammoth.
De
~De
Wl                  VALID!!

Think about it. Is it possible to have a situation in which the premises are true and the conclusion is false? Sure, it's possible to have a situation in which the conclusion is false, but for the argument to be invalid (wonky), it has to be possible for the premises to all be true at the same time the conclusion is false. So if the premises can't all be true, the argument is valid. (If you still think the argument is invalid (wonky), draw a picture in which the premises are all true and the conclusion is false. Remember, there's only one Elvis, and you can't be both dead and alive.)

Is this a startling concept? Well, remember that logic is startlingly different from the way people usually think, and from the way they expect you to think.

Now, here's the weirdest thing of all. The following argument is VALID. That's right, valid. Brace yourself, because this valid argument is going to seem totally weird to you!

Cheese is a mineral.
Cheese is not a mineral
Elvis is both alive and not alive.

Remember, this is VALID. (Weird, huh?) You will of course notice that the conclusion cannot possibly be true. It's a logical self-contradiction! You can't both be and not be anything! The thing to remember here that having a conclusion that can't be true doesn't necessarily make an argument invalid. If the premises contradict each other, the argument is valid, no matter what the conclusion is!

Test yourself: Does the fact that we can make a valid argument for absolutely any conclusion mean that logic can prove absolutely anything? Answer

To put it another way, can you construct a sound argument for a false conclusion? Answer

Terminology

Logic requires a very precise use of terminology. So here it is. A logically good deductive argument is called valid, and a valid argument with true premises is called sound. A logically good inductive argument is called strong, and a strong argument with true premises is called cogent. The words "valid" and "sound" are not used for inductive arguments, and the words "strong" and "cogent" are not used for deductive arguments.

The validity test is as follows:

First, assume that the argument's conclusion is false.

Second, ask yourself if it’s now possible for the all the premises to be true. (Sometimes assuming the conclusion false will make a premise false. Other times there will be another reason why the premises can’t all be true.)

If it’s possible for all the premises to be true, even if the conclusion is false, then the argument is INVALID. (or “wonky.” Remember “possible” = “wonky.”)

If there is any reason why the premises can’t all be true, the argument is VALID. Maybe assuming the conclusion false makes a premise false. Maybe they simply can’t all be true together. Either way, “impossible” = “valid.”)

These two exercises are meant to practice your ability to apply the definition of validity.

A Final Note

If two deductive arguments have the same form, it is exactly the same form. There won't be even the slightest formal difference betweene then. None. No difference in form whatsoever. And if there is a difference between the forms of two deductive arguments, they're simply not the same form at all. The concept of two deductive arguments having "similar" logical forms is neither useful nor even meaningful. They either have the same form or they don't, and if they don't, the validity of one has nothing to do with the validity of the other. This property is unique to deductive arguments. Inductive arguments are different.

Just to remind you, the following two statements are absolutely true.

An argument with a conclusion that can't be false IS necessarily valid.

An argument with premises that can't all be true IS necessarily valid.

Practice Quiz.
1. Do most people find the concept of validity easy to understand?
7. Is this valid or invalid: If the British had caught and executed Benjamin Franklin in 1777, Benjamin Franklin would be dead. Benjamin Franklin is not dead, so the British did not execute Benjamin Franklin.

8. Is this valid or invalid: If the British had caught and executed Benjamin Franklin in 1777, Benjamin Franklin would be dead. The British did not execute Benjamin Franklin, so Benjamin Franklin is not dead.

9. Is this valid or invalid: If the British had caught and executed Benjamin Franklin in 1777, Benjamin Franklin would be dead. The British did execute Benjamin Franklin, so Benjamin Franklin is dead. 

10. Is this valid or invalid: If the British had caught and executed Benjamin Franklin in 1777, Benjamin Franklin would be dead. Benjamin Franklin is dead, so the British did execute Benjamin Franklin.

11. Is this valid or invalid: Whales are fish. Whales are not fish. So cheese is a mineral.
12. Is this valid or invalid: Whales are mammals. Whales are not fish. So cheese is not a mineral.
13. Is this valid or invalid: Whales are mammals. Whales are fish. Fish are never mammals. So whales are fish.
14. Is this valid or invalid: Whales are mammals. Fish are never mammals. Whales are not fish. So some whales eat fish.

19. Does validity depend on whether the premises of arguments are actually true or false.
20. Is it true that an argument cannot be valid if the premises and conclusion are false.
21. "An argument is valid if and only if ...... "

Practice Quiz Answers
1. The concept of "validity" is very counterintuitive and most people find it very difficult to master.

8. Invalid
9. Valid
10. Invalid
11. Valid. Yes, valid. That's right, it's valid.
12. Invalid
13. Valid. Think about it. Can the premises all be true? If they can't, the argument is valid.
14. Invalid. Yep, invalid.
15. Deductive.
16. Inductive.
17. Deductive.
18. Inductive.
19. Nope. Validity has nothing to do with the actual truth or falsity of the premises of an argument.
20. Nope. An argument can be valid even if all the premises and conclusion are false.
21. "An argument is valid if and only if it is impossible for there to be a situation in which all its premises are true and it's conclusion is false.

Copyright 2011 by Martin C. Young


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