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.

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.

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

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. |

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!) )

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.

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.

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.

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.

- By definition, an "Aylesbury duck" is a large domesticated duck with pure white plumage, a pink bill and orange legs and feet.
- By definition, a "Mandarin Duck" is a medium-sized with a red bill, a white crescent above the eye and reddish face and "whiskers".
- By definition, a "Ringed Teal" is a small duck with a chestnut back, pale grey flanks and a salmon-coloured breast speckled in black.
- By definition, an "Abacot Ranger" is a duck with a 'hood' of fawn-buff feathers and a creamy white body streaked with colour.
- By definition, a "Red-crested Pochard" is a large diving duck with a rounded orange head, red bill and black breast.
- By definition, an "Indian Runner" is a domestic duck that stands erect like a penguin and runs rather than waddling.

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.

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.

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

**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.

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* sufficient*condition for
being a lapin, then it will follow that being a lapin is a*
necessary *condition for being a coney.

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.)

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 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**

- If an argument is valid,
then the argument is such that its premises, if true, logically force
the conclusion to be true. (
**False**!) - If an argument is valid,
the premises make the conclusion true. (
**False**!) - If an argument is valid,
you can't make the conclusion false
*without**also*making at least one premise*false*. (**False**!) - If an argument is valid,
the premises, taken all together, force the conclusion to be true. (
**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.

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 FranceBerlin is in GermanyCompton is in America |
Cats
are mammalsDogs are mammalsFerrets are mammals |
People
have two legsMammals have four legsInsects have six legs |
Dumbledore
is a wizardGandalf is a wizardMerlin is a wizard |

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.

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

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

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

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

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

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 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.

If Roy is a tramp, then Roy is a bum If Roy is a tramp, then 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.

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

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.

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

Copyright © 2012 by Martin C. Young