The Deepest of Mysteries (Play spooky background music while you read this chapter.)                                                                                  (Problems printing? Click here.)

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Universals


This chapter includes the 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.

This chapter will also cover the difference between deductive and inductive logic, which is also a little bit weird, but also can be easy to master if you're willing to bite the bullet and embrace -- well, accept -- okay, tolerate -- the weirdness of it.

One big problem with teaching the difference between deductive and inductive arguments is that there are certain teachers out there who have memorized a false definition of that difference and who then mindlessly "teach" that false definition to their students. If you have ever been taught a definition of deductive that is substantially different from the one given below, you should work very hard to forget it. If it says something different from the definition given in this chapter, it is completely false, and relying on it will make you give wrong answers to the quiz.

Don't worry if you have trouble with these concepts at first. Most people do, and be assured that I'm going to do my best to make them as clear as possible. Indeed, you might find that you get the concepts early on, and my attempts to make the clear will then seem to you to be a tedious, and even patronizing exercise in belaboring the obvious. In that case, please bear with me. I'll get to something interesting eventually. On the other hand, if some of the material drives you crazy - especially where I ask you to work out brain-bursting things for yourself - remember that the rest of the course will be much less weird than this chapter, and I'm only going to expect you to master the basics of this deductive logic stuff.

Deductive Arguments

If you already "know" a simple definition of the difference between deductive and inductive arguments, you should read the following definitions very, very, very carefully. The simple definition that I have in mind is very, very easy to remember. It is also absolutely wrong, and the person who told it to you knew absolutely nothing about deductive logic. I'm not kidding. There is a totally dumb definition out there, and irresponsible people are misleading their students by teaching them this dumb, dumb, dumb, dumb, dumb, dumb, dumb, dumb, dumb, definition. (So if you have never heard the simple "definition," you already know more than the people who have!)

Here is the correct definition. Arguments come in two flavors. There are deductive arguments, which establish the precise logical relationships between ideas, and there are inductive arguments which establish truths about the universe we live in. Deductive arguments can give you certainty, but that certainty only applies to the logical relationships involved. Inductive arguments can prove things about the world, but the price of never giving you absolute certainty.

Another way to express the difference between the two kinds of arguments is to say that deductive arguments with true premises either succeed completely or they fail absolutely, whereas inductive arguments only have degrees of success. Even the best inductive argument with absolutely true premises still allows a logical possibility that its conclusion is false. Even the worst inductive argument with absolutely true premises still give some logical support to it's conclusion, as pathetically inadequate as that support may be.

Confused? Dismayed? Disturbed? Good! If these definitions sound weird to you, that's a sign you're beginning to understand them.


Here are some examples of deductive arguments.

No fallacy is a good argument. Your argument is a fallacy, so it's not a good argument.	Sam is either a monkey or a supermodel. He's not a supermodel, so he's a monkey	Cheese is a mineral, and cheese is not a mineral, so cheese both is and is not a mineral	Joe is a Frenchman and an onion seller, so we do know he sells onions.	If you're a cat, you're a dog. If you're a dog, you're a rat. So if you're a cat, you're a rat.

Now, let's discuss this difference between deductive and inductive arguments in excruciating detail.

Inference

An inference is sort of a basic minimum-size argument. Arguments, as you will see, can get very big and complicated, and can contain lots of little argument-bits that are supposed to fit together to make one big argument for the main conclusion. Think about it this way. In the previous chapter we spent a little time breaking big arguments down to smaller ones. Imagine that we had broken all our arguments down into the smallest sub-arguments we possibly could. In that case, each smallest possible sub-argument would then represent a single inference. Now, big argument can contain a mixture of deductive and inductive inferences, each of which may have to be evaluated seperately in order to tell whether or not the big argument works as a whole.

The way to think about the difference between deductive and inductive inferences is like this. Assume we have a set of two claims, claim A and claim B. Now suppose someone says that if claims A and B are both true, then together they would make it absolutely certain that some third claim, claim C, is true. If this person is right that A and B together would make C certainly true, then there is a "valid" deductive inference from A and B taken together to C. If, however, A and B together don't make C certainly true, then there is no valid deductive inference from A and B to C.

Even if we cannot make a deductive infrence from A and B to C, we might still be able to make a "strong" inductive inference from A and B to C. This is because it still might be true that A and B together would make it very likely that C is true. If it is the case that A and B, if both true, would make C very likely to be true, then there is a strong inductive inference from A and B to C.

Notice that a good deductive inference is called "valid," while a good inductive inference is called "strong." This is to avoid confusion in those cases where there's a good inductive, but no good deductive inference from some set of facts to a conclusion. Using the word "strong" makes it clear that you're not claiming that there's a good deductive inference.

To tell whether an argument is deductive or inductive, you ignore the question of how the premises are established, and whether or not they are true or false, and focus on the logical form of the argument as it is presented to you. Here is a deductive argument.

All cats are selfish.
Socrates is a cat.
Socrates is selfish.

Assuming that there's nothing wrong with the logical form of this argument, and that the premises are true, it follows that the conclusion is absolutely true. Think about it. If it is true that Socrates is a cat, and that all cats are selfish, could it possibly be true that Socrates is not selfish? Now here's an inductive argument.

All cats known to history are selfish.
All cats are selfish.

Again assuming that there's nothing wrong with the logical form of this argument, and that the premise is true, it follows that the conclusion is very very likely, but it does not follow that the conclusion is absolutely true. Think about it. If every cat everyone has ever known has been selfish, then wouldn't you strongly tend to think that all cats are selfish? But could you be absolutely sure that there wasn't some cat, unknown to history, that was not selfish?

The forms of these two particular arguments have special names. The form of the deductive one is called a categorical syllogism and the inductive one is called a statistical syllogism. Here's those forms expressed symbolically in a way that highlights their most essential difference.

All X are Y
Z is X
Z is Y (Categorical syllogism)

A very high proportion of X are Y
Z is X
Z is Y (Statistical syllogism)

The difference here is subtle, but it's important. Notice that the first argument deals exclusively in black-and-white concepts. It says flatly that all X are Y, and admits of no exceptions. In fact, you could say it basically defines X as a kind of Y, or that it subsumes the idea of X under the idea of Y. In any case, it deals in certainty. The second argument, however doesn't define X as a kind of Y, or subsume the concept of X under Y. Instead, it allows for the possibility that some X isn't Y. This means that even if it's premises have been proved with 100% certainty, it's conclusion is still only probably true.

Here's an example of each kind:

All politicians are corrupt
Melvin is a politician
Melvin is corrupt (Categorical syllogism)

Politicians strongly tend to be corrupt
Melvin is a politician
Melvin is corrupt (Statistical syllogism)

Inductive versus Deductive Again

Inductive arguments also have logical forms. (Because their premises are also logically related to their conclusions.) Compare the form of a categorical syllogism with a statistical syllogism. Oh, and don't be fooled by the names! Here, the word "categorical" means means something like "makes a claim about all of something" and "statistical" means "makes a claim about most of something."

Categorical Syllogism (Deductive)

1. All badgers are anarco-syndicalists.
2. Hiram is a badger.
C. Hiram is an anarco-syndicalist.

Statistical Syllogism (Inductive)

1. Almost all badgers are anarco-syndicalists.
2. Hiram is a badger.
C. Hiram is an anarco-syndicalist.

Notice how the addition of the word "almost" transforms a deductive argument into an inductive argument? This is because that addition transforms it from an argument where, if the premises are true, the conclusion is certainly true, to an argument where, if the premises are true, the conclusion is very probably true. That's the essential difference between deductive and inductive arguments, the logical relationship between premises and conclusion. In deductive arguments that relationship is either hard and fast, or it is nonexistant. In inductive arguments that relationship can get pretty damn firm, but it never approaches the adamantine hardness possible in deductive arguments. This creates a fundemental difference in how deductive and inductive arguments can be handled. Because every certainty is always exactly equal to every other certainty (you're always 100% sure, no matter what it is you're certain about), and because deductive arguments that fail always fail totally, every deductive argument of any given form is always exactly as valid or invalid as every other argument of that same deductive form. This means that there is a fundemental difference between validity and cogency. All valid arguments are equally valid, but all cogent arguments are not equally cogent. Instead, there are degrees of cogency, in which we can judge that one inductive argument is more cogent than another. So while the way deductive arguments differ from each other allows us to name the deductive forms in such a way that valid forms will always have different names from invalid forms, it is not at all practical to name the inductive forms in a way that allows is to distinguish the cogent from the incogent by name. For instance, the categorical syllogism is a valid deductive form, so all categorical syllogisms are valid arguments. To say that some argument is a categorical syllogism is thus to say that it's valid. Not so with statistical syllogisms.

1. 99.9% of badgers are anarco-syndicalists.
2. Hiram is a badger.
C. Hiram is an anarco-syndicalist.

1. 99% of badgers are anarco-syndicalists.
2. Hiram is a badger.
C. Hiram is an anarco-syndicalist.

1. 90% of badgers are anarco-syndicalists.
2. Hiram is a badger.
C. Hiram is an anarco-syndicalist.

1. 75% of badgers are anarco-syndicalists.
2. Hiram is a badger.
C. Hiram is an anarco-syndicalist.

1. 50% of badgers are anarco-syndicalists.
2. Hiram is a badger.
C. Hiram is an anarco-syndicalist.

1. 10% of badgers are anarco-syndicalists.
2. Hiram is a badger.
C. Hiram is an anarco-syndicalist.

Notice how the arguments starts off pretty darn strong, but get weaker and weaker as the percentage shrinks.

Notice that, if it's premises are true, the categorical syllogism can prove it's conclusion with certainty, which is what makes it a deductive argument, while the statistical syllogism can only ever prover it's conclusion with a very high probability, which is what makes it an inducive argument.

Here are some examples of deductive arguments.

No fallacy is a good argument. Your argument is a fallacy, so it's not a good argument.	Sam is either a monkey or a supermodel. He's not a supermodel, so he's a monkey	Cheese is a mineral, and cheese is not a mineral, so cheese both is and is not a mineral	Ralph sells onions and Ralph is French, so Ralph sells onions.
All monkeys are primates. George is a monkey, so it follows that George is also a primate	All monkeys are primates. Pam is a primate, so it follows that Pam is also a monkey.	All monkeys are primates. Nick is not a monkey, so it follows that Nick is not a primate.	All monkeys are primates. Oswald is not a primate, so Oswald is not a monkey
Dogs are cats, and dogs are not cats, so Elvis is alive!	Monkeys are made of cheese, and Elvis was a robot, so dogs are or are not cats.	Elvis is not dead, so Elvis is alive.	All monkeys are primates, so all primates are monkeys.

And for contrast, here are some examples of inductive arguments.

George looks exactly like Norm, who is a primate, so George is also a primate	Pam is a primate, and many primates are monkeys, so it Pam is also a monkey.	Analysis of 30,000 wheels of cheese revealed no mineral content, so cheese is not a mineral	I saw Ralph give someone some onions in exchange for money so Ralph sells onions.
Most arguments of that type fail, so your argument must fail too.	Sam is very hairy and has hands and a prehensile tail, so he's a monkey	George says Nick is not a primate, so Nick is not a primate.	Everyone who ate the fish died, so the poison was in the fish
All Frenchmen eat snails, so everyone in the world eats snails.	Lots of smokers get cancer, so smoking causes cancer.	Elvis was dead last time anyone checked, so Elvis is not alive.	No-one has ever seen a hippogriff, so they don't exist.

Proving Stuff

Can a deductive argument prove something about the world with absolute certainty? No. No argument is ever stronger than its weakest premise. If a deductive argument has a conclusion that's about the world, then it must have a premise that is also about the world. Ultimately, that premise must be justified by observation, by looking at the world. All arguments based upon looking at the world are inductive arguments, so every conclusion about the world, whether it shows up in an inductive or a deductive argument, is ultimately based on some inductive argument. The bottom line is that nothing apart from logical relationships between ideas can be known with certainty. Conclusions about the world always come with at least some uncertainty.

Another way to look at the difference between inductive arguments and deductive arguments is to say that in an inductive argument it is assumed that the premises, if true, make the conclusion very likely to be true. Whereas in a deductive argument is assumed that the premises, if true, make it impossible for the conclusion to be false. To put it another way, an inductive argument attempts to show that it's conclusion is very probably true while a deductive argument attempts to show that it's conclusion is certainly true. (Confused? Don't worry about it. If you don't see how the distinction beween "probably" and "certainly" applies here, you're not alone.  Most people find this the hardest part of logic to get - I've even met a professor who doesn't understand it - and it won't be on the test.)

This distinction does have one useful consequence. If the argument you are dealing with is deductive, and you can show that it is possible for its premises to be true when it's conclusion is false, then you will have shown that it is a bad argument. (Technically, this is called showing that it has an invalid form.) In terms of the SCAEFOD procedure, it follows that once you have figured out that an argument does not rely on analogy, or an authority, and does not make a generalization, or support a causal claim, or claim that it's conclusion explains something else, or depend on burden of proof, then all you have to do to refute the argument is to show that its premises can be true even if it's conclusion is false. If a deductive argument's conclusion can be false, even if it's premises are true, then it's simply not a good arguement.

(The study of relationships between ideas is called formal logic or, (because it's easier to do with symbols), symbolic logic. In order to get more than a basic handle on symbolic logic, it's necessary to learn the language. We're only going to do the basics here, so no language lesson. (Yay!) )


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.

Validity

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 arguments. Only 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.)

On a practical note, my decrepit old ears have great trouble distinguishing the words "valid" and "invalid," so i've added the word "wonky" to my philosophical vocabulary. "Wonky" means "invalid," so if I ask the class "valid or wonky?" I'm asking if the argument is vaild or not, and if you sing out "wonky!" I'll know what you mean.

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.)

A deductive 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.

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

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

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

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

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

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

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

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

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!

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

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.

(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:



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).

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                        
                                                            
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


(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                        
                                       
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.


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.

Now look again at some arguments we looked at last chapter.

A. If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. We do not have Mars rocks on Earth. So NASA did not send an expedition to Mars and back in 1974.

B. If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. NASA didn't send an expedition to Mars and back in 1974, so there are no Mars rocks on Earth.

C. If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. We do have Mars rocks on Earth. (This is true!)  So NASA did send an expedition to Mars and back in 1974.

D. If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. NASA did send an expedition to Mars and back in 1974, so there are Mars rocks on Earth.

Exercise 1. Match the above arguments with the correct names for their respective argument forms.

1. Modus Ponens is represented by argument A. B. C. D.

2. Modus Tollens is represented by argument A. B. C. D.

3. Affirming the Consequent is represented by argument A. B. C. D.

4. Denying the Antecedent is represented by argument A. B. C. D.

Exercise 2 . Select the right word out of "valid" and "invalid."

1. Modus Ponens is a valid/invalid argument form.

2. Modus Tollens is a valid/invalid argument form.

3. Affirming the Consequent is a valid/invalid argument form.

4. Denying the Antecedent is a valid/invalid argument form.

Exercise 3 . (No answers) Name the argument forms for the following arguments.

1. If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Mount Saint Helens did not blow up, so Luxembourg did not sent saboteurs to the US.

2. If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Luxembourg sent those saboteurs to the US, so Mount Saint Helens has blown up.

3. If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Mount Saint Helens did blow up, so Luxembourg did sent saboteurs to the US.

4. If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Luxembourg has never sent saboteurs to the US, so Mount Saint Helens has never blown up.

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

Do the Premises Have to Make the Conclusion True?

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 (Groan!)

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.

Exercise 1. Which of the following arguments are valid? Which are invalid (wonky)?

A. If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. We do not have Mars rocks on Earth. So NASA did not send an expedition to Mars and back in 1974. Answer

B. If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. NASA didn't send an expedition to Mars and back in 1974, so there are no Mars rocks on Earth. Answer

C. If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. We do have Mars rocks on Earth. (This is true!)  So NASA did send an expedition to Mars and back in 1974. Answer

D. If NASA sent an expedition to Mars and back in 1974 then we'd have Mars rocks on Earth. NASA did send an expedition to Mars and back in 1974, so there are Mars rocks on Earth. Answer

Exercise 2. (No answers) Determine the validity of the following arguments.

1. If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Mount Saint Helens did not blow up, so Luxembourg did not sent saboteurs to the US.

2. If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Luxembourg sent those saboteurs to the US, so Mount Saint Helens has blown up.

3. If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Mount Saint Helens did blow up, so Luxembourg did send saboteurs to the US.

4. If saboteurs from Luxembourg had planted a nuclear device in Mount Saint Helens, that mountain would have blown up. Luxembourg has never sent saboteurs to the US, so Mount Saint Helens has never blown up.

Homework: Answer the following questions either on your own paper or on the Homework Answer Sheet (If you want extra practice, or you have to make up this homework, do all 10 on the Make-up Homework Answer Sheet) (You should be able to figure out the problems by comparing them to the problems I worked above.) 1. Circle the valid arguments. Cross out the invalid (wonky) arguments A. 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. B. 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. C. 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.  D. 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. 2. Circle the valid arguments. Cross out the invalid (wonky) arguments A. Whales are fish. Whales are not fish. So cheese is a mineral. B. Whales are mammals. Whales are not fish. So cheese is not a mineral. C. Whales are mammals. Whales are fish. Fish are never mammals. So whales are fish. D. Whales are mammals. Fish are never mammals. Whales are not fish. So some whales eat fish. 3. Circle the deductive arguments. Cross out the inductive ones. A. All whales are fish. Willie is a whale. Therefore Willie is a fish. B. The vast majority of whales live free in the ocean. Willie is a whale. Therefore Willie lives free in the ocean. C. All monkeys can fly. George cannot fly. Therefore George is not a monkey. D. Monkey aerodynamics make flight extremely unlikely. Kong is a monkey. Therefore Kong cannot fly.

You can bet that the quiz will want you to know the following things.
1. The difference between inductive and deductive arguments.
2. The correct definition of validity.
3. The validity or invalidity of an argument with mutually contradictory premises.
4. The difference between a categorical syllogism and a statistical syllogism.

If you want to practice for the quiz, or you want to make up a missed quiz for this chapter, you can do the deductive pre-quiz.

Copyright © 2006 by Martin C. Young

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