Deductive Logic

I've tried teaching deductive logic by starting with the correct definition of validity, and it doesn't work. So although this chapter will give the correct definition, we will work with a simplified, heuristic definition for this chapter, and then work with all the weird consequences of the correct definition next chapter.

Here are some examples of deductive arguments.

1. No fallacy is a logically compelling argument. Your argument is a fallacy, so it's not a logically compelling argument.
2. Sam is either a monkey or a supermodel. He's not a supermodel, so he's a monkey.
3. If you're a toad, you eat woad. You don't eat woad, so you sure ain't a toad.
4. Cheese is a mineral, and cheese is not a mineral, so cheese both is and is not a mineral
5. Joe is a Frenchman and an onion seller, so we do know he sells onions.
You're a fish. All fish are flyers. So, you're a flyer and a fish.
6. 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.

Notice how simplistic and pointless they are? It's true I chose some simple and silly examples to introduce you to deductive logic, but you should know that even the most serious and complicated deductive arguments will have the same flavor of pointlessness as the examples given above.

Notice also that none of these arguments refer in any way to our experience of the world. The first one makes a sweeping generalization, but does not back it up with any kind of sample. The second one just stipulates that "Sam," whoever he is, has to be either one thing or another. The third makes a broad claim about all toads, but again without giving any evidence, and so on. This is because deductive logic is fundamentally about concepts. It concerns relationships between concepts and the applicability of various concepts to various objects. It's incredibly useful, perhaps even vital to reasoning, but in itself it is not the kind of reasoning that gives you knowledge about the existance or nonexistance of objects in the world.

Deductive logic is fundamentally different from inductive logic. The rules that work for inductive reasoning don't really apply to deductive logic, and vice versa. That's the basic idea I want to get across in this chapter. The true depth and weirdness of deductive logic can wait until next chapter.

Deductive arguments can't also be inductive. If it's inductive, it's not deductive. If it's deductive, it's not inductive. And there's no third kind of logic, so every argument is either inductive or deductive. Every argument is one kind or the other, but no argument is both kinds, and there are no arguments that are neither kind.

This chapter 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 its 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.

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 its premises have been proved with 100% certainty, its 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 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 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 its premises are true, the categorical syllogism can prove its conclusion with certainty, which is what makes it a deductive argument, while the statistical syllogism can only ever prover its conclusion with a very high probability, which is what makes it an inducive argument.

Here are some more examples of deductive arguments.

No fallacy is a logically compelling argument. Your argument is a fallacy, so it's not a logically compelling 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 its conclusion is very probably true while a deductive argument attempts to show that its 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 its 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 its 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 its conclusion is false. If a deductive argument's conclusion can be false, even if its 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!) )

Deductive Concepts

Before I get into the business of evaluating deductive arguments, I want to try to ease you into the subject by covering a few important concepts.

All-or-Nothing Statements

An "all-or-nothing" claim is the kind of claim that can be proved false by even a single counter example. "All Scots eat haggis" is this kind of claim because if there exists even one Scot who does not eat haggis, the claim is false. All-or-nothing claims overwhelmingly tend to be false in real life, but they are the only kind of claim that deductive logic can deal with, which is one of the things that makes deductive logic seem pointless to many people.

Necessary and Sufficient Conditions

Necessary and Sufficient Conditions (Inductive)

When you talk inductive, you talk about what people actually experience out there in the world, so here we should use physical examples.

Say in room number one you have a loose pile of really dry kindling that's cut really thin, and which includes copious amounts of very delicate wood shavings. And also say that all this is resting on a really hot metal surface, say 270 degrees Celsius in an airtight room that has no atmosphere whatsoever. What's going to happen? It won't be a fire because, without oxygen (or another oxidizing substance), there cannot be combustion. Another way to put this is to say that the presence of an oxidizer is a necessary condition for something to catch fire.

Suppose we have another room with an oxygen atmosphere and a red-hot floor, but there's no kindling or any other kind of fuel in the room. Again no fire, because a fire requires fuel, and there's none here. From this it follows that the presence of fuel is also a necessary condition for a fire to happen.

You can also think about a room with plenty of air and kindling, but a stone cold floor. Will a fire start without a heat source? No, so a heat source is also a necessary condition for a fire.

We can also run this the other way. Is fuel, all by itself, enough to make a fire? No it isn't, so while fuel is a necessary condition for a fire, it is not a sufficient condition for a fire. The same can also be said for oxidizer and ignition-temperature heat source. None of these, by itself, is sufficient to make a fire.

Now consider a suspension bridge over a rocky gorge. How many different conditions can you think of that would be sufficient to bring this bridge down? I can think of several. For instance, a five-kiloton nuclear explosion should do nicely. Of course, it doesn't have to be a nuclear explosion. Five tons of TNT set off in the middle of the bridge should also do the trick. There's also meterorites. A one-ton meterorite hitting the bridge would not only take down the bridge, it would also vaporize it, and the rocky gorge as well, and most of the nearby mountains too. (Be cool to watch, too.) Finally, imagine if a convoy of trucks carrying liquid nitrogen all spilled their loads at once and supercooled a large number of the suspension cables, making them too brittle to take the load, that would take down the bridge too. Or if we magically changed the roadbed to Wenslydale cheese and the cables to Mozzarella. Notice that any one of these conditions is sufficient to take out the bridge, even though none of them is necessary to take the bridge down. Nuclear weapons are not necessary because we could take the bridge down with conventional explosives, conventional explosive are not necessary because we could take the bridge down with a nuke, or cheese magic, or . . .

Is it always easy to specify necessary and sufficient conditions? Well, when we're dealing with causal relationships in the real world, it can often be really difficult to say what's necessary or what's sufficient to bring about a particular effect. In the looking-glass world of deductive logic, things can be a lot easier.

Necessary and Sufficient Conditions (Deductive)

By definition, a "bachelor" is a man of marriagable age who has not yet married.

Jimmy is male, unmarried since birth, and 14 years old. Is Jimmy a bachelor?

Jane is female, unmarried since birth, and 24 years old. Is Jane a bachelor?

Joe is male, married three seconds ago, and 24 years old. Is Joe a bachelor?

If you said "yes" to any of those questions, go back and look at the definition again. Jimmy is not of marriagable age, and so he lacks that necessary condition to be a bachelor. Jane is not male, and so she lacks that necessary condition to be a bachelor. Joe is not unmarried, and so he lacks that necessary condition to be a bachelor.

How about Absalom, who is a male human who has never been married and is now 86 years old? Well, by the definition given above, Absalom is a bachelor! The conditions of being a man, being old enough to marry, and having never married are jointly sufficient to make Absaom, or anyone else, a bachelor.

Being an Abacot Ranger is sufficient for being a duck, because you can't be an Abacot Ranger without also being a duck.

But being an Abacot Ranger is not necessary for being a duck, because you can be a duck without also being an Abacot Ranger.

Notice that the deductive necessary and sufficient conditions are abslolutely cut and dried. Once we've established that Jimmy, Jane and Joe all lack one of the necessary conditions, we've established for certain that none of them is a bachelor, Once we've established that Absalom meets all three necessary conditions, we've established for certain that he is a bachelor, Things are not quite so certain back in the inductive realm. While I can't begin to think of any way to make fire without fuel, without oxidant or without heat, I can't absolutely guarantee that there isn't one, so I can't absolutely guarantee that these conditions are absolutely necessary for fire.

Remember:

For a statement to be a lie, it is necessary for it to be untrue, but merely being untrue is not sufficient to make it a lie, because it might have been an honest mistake. 

Sticking a lit road flare into a loose pile of dry kindling is sufficient to set it ablaze, but a road flare is not necessary to start the fire, because a powerful laser or some burning gasoline would do just as well to start a fire.

Stipulation

We don't bother to prove the premises of deductive arguments. Basically, we just stipulate that they're true. We assume them, and then go on to look at the logical relationship between those premises and the conclusion. Deductive logic is all about logical relationships. It doesn't, by itself, try to prove things about the world. Instead, it investigates logical relationships between definitions and othere stipulative statements.

Conditionality

A "conditional" claim is one that says that if one thing is true, then another thing is true. The following are all conditional claims.

If wishes were horses, then beggars would ride.
If cats wore hats, then dogs would wear clogs.
If Hawk the Kitty wore a hat, then Patch the Dog would wear clogs.

Notice that these statements don't have to make sense, and we're really not interested in whether they're true or not. All we care about in deductive logic is their relationship with other claims. Seriously, conditional statements don't have to make sense. Here's some perfectly good (for our purposes) conditional statements.

If Al Gore was president, then Marie Antoinette was actually a three-legged alien death machine controlled by a crew of 247 white mice.
If trees eat bees then you can live on the knees of cheese
If Joe is a Jumbuck then Bob is a Billabong.

But notice that these statements only run one way. saying that "if Joe is a Jumbuck then Bob is a Billabong" does not imply that "if Bob is a Billabong then Joe is a Jumbuck," oh dear me no! If we happen to find out that Bob is in fact a Billabong, that tells us nothing about whether or not Joe is a Jumbuck!

Necessary and Sufficient Conditions Redux

Now, it's important to notice that conditional statements are not causal statements."If Joe is a Jumbuck then Bob is a Billabong" does not mean "If Joe becomes a Jumbuck then Bob will become a Billabong." What it means is that, if you check on Joe, and he turns out to be a Jumbuck, then it follows that Bob has been a Billabong all along. Nothing causes anything to change. Instead, conditional statements can be thought of as setting out deductive necessary and sufficient conditions for related claims.

For instance, the statement "If Joe is a Jumbuck then Bob is a Billabong" sets out two conditions.

First, it says that Joe being a Jumbuck is sufficient for Bob to be a Billabong
Second, it says that Bob being a Billabong is necessary for Joe to be a Jumbuck.

Think about it.

Now think about whether it says that Joe being a Jumbuck is necessary for Bob to be a Billabong, or whether it says that Bob being a Billabong is sufficient for Joe to be a Jumbuck.(Hint: It doesn't.)

When you're done with that, think about the logical status of statements like "all coneys are lapins." How is it related to conditional statements?

Well, think about this:
Does it say that being a coney is a sufficient condition for being a lapin?
Does it say that being a coney is a necessary condition for being a lapin?
Does it say that being a lapin is a sufficient condition for being a coney?
Does it say that being a lapin is a necessary condition for being a coney?

(Answers yes, no, no, yes.)

So, in essence, "all coneys are lapins" says both "being a coney is a sufficient condition for being a lapin" and "being a lapin is a necessary condition for being a coney." Now, where have we seen that form before? Well, we've seen it before in "if _____ then _____ " statements, otherwise known as conditional statements, so "all coneys are lapins" can be rewritten as "if somthing is a coney, then that thing is also a lapin."

Notice again that "all ____ are ____ " statements are no more reversable than "if ____ then ____ " statements. Just because all housecats are felines it doesn't follow that all felines are housecats.

Reciprocity

Finally, notice that the relationship of necessity is reciprocal to the relationship of sufficiency. If A is necessary for B, then it follows that B is sufficient for A, and vice versa. In fact, they're basically the same relationship, seen two different ways.

For example, if being a coney really is a necessary condition for being a lapin, then it will follow that being a lapin is a sufficient condition for being a coney. And if being a coney really is a sufficientcondition for being a lapin, then it will follow that being a lapin is a necessary condition for being a coney.

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, its 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 its premises are TRUE and its 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 its 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. (Figure out which are valid and which aren't before you look for the answers.)

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

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

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

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

Answers: 1. valid, 2. invalid, 3. invalid, 4. valid.

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 conditional-based forms (Modus Ponens, Affirming the Consequent, Modus Ponens, Denying the Antecedent) 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.) Look beside the name of each form below 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

Answers, modus ponens and modus tollens are valid, affirming the consequent and denying the antecedent are invalid.

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 Reminder

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.

Copyright 2012 by Martin C. Young


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