Fallacy Row

Argument Analysis

Abstraction Fraction

Deductive Moment

Strategic Analysis

Tactical Analysis - two premise boxes

Exercises

Foreshadowing

Reminders

Exercise Answers

(Play spooky background music while you read this chapter.)

#########################################################

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

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

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

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

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

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

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

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

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

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

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

- If an argument is valid, then the argument is
such that it's 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 |

Example 1

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

1.
If Martha Washington had gone absolutely postal on George's
ass, he would be dead.

__2. Martha was strictly non-postal.__

C. George ain't dead.

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

Example 2

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

1. If pixies had
BBQ sauced Ghengis Khan, Ghengis would be a rash-infested
used car salesman working out of Bakersfield.

__2. Pixies did BBQ sauce Ghengis Khan.__

C. Ghengis does have a rash.

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

Example 3

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

Shamu is either a
cat or a dog

__Shamu is not a dog__

Shamu is a cat.

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

Now check these arguments for validity.

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

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

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

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

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

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

Modus Ponens
**Affirming
the Consequent**

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

__Roy is a
tramp __ __Roy
is a
bum __

Roy is a
bum Roy
is a tramp

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

Roy is not a bum Roy is not a tramp

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

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

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

1. George is either alive or dead.

__2. George isn't dead. __

C. George is alive.

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

1.
Shamu is either a cat or a dog

__2. Shamu is not a dog__

C. Shamu is a cat.

Notice now that both arguments have the same valid
form. The second argument *only *fails because
the first premise is false. Dog and cat are not the only
two possibilities for Shamu's species.

Now let's look at disjunctive syllogism through pictures
and the following argument.

Roy is either
a bum or a tramp

__Roy is not a
bum __

Roy is a tramp

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

Picture 2
Picture 3
Picture 4

Roy is a tramp but not a bum.
Roy is a tramp
and a bum.
Roy is not a tramp but is a bum.

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

If Roy is a tramp and a bum, then the statement "Roy is
either a bum or a tramp" is true.

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

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

Picture 2

Roy is a tramp but not a bum.

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

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

#########################################################

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Did you get that? It means that to test an argument, we
*try* to draw a picture in which the premises are **true**
and the conclusion is **false**. If we *can*,
the argument is *invalid (wonky)*. If we __can't__,
it's __valid__.

Now, is this argument valid or invalid (wonky)?

Albert
Einstein discovered FranceMy wolverine eats cheese pizzaLaura Schlessinger is a Martian |
FaPwMl |

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

This picture proves that it's possible for Albert Einstein to have discovered France and for my wolverine to eat cheese pizza

__An argument with conclusion and premises that are
true still isn't neccesarily
valid.__

Elvis is
dead. (Accept it.)The X-Files was a popular TV showThe Eiffel Tower is in France |
DePxFt |

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

But this next picture

This picture proves that it's possible for Elvis to be dead and for the X-Files to have been a popular TV show

So if you're trying to check the validity of an
argument, and you figure out a way that the premises
and conclusion can *all* be true, then you * haven't*
checked the validity of that argument. You gotta try
to figure a way to make the premises

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

Soooo the following arguments are all valid:

__Roses
are red and violets are blue.__

Roses are red.

__Roses are red and violets are blue.__

Violets are blue.

Roses are red.

__Violets are blue .__

Roses are red and violets are blue.

Cheese
is yellow and goop is green.

__Roses are red and violets are blue.__

Violets are blue and cheese is yellow.

And the following are all invalid:

__Roses
are red. __

Roses are red and violets are blue.

__Violets are blue .__

Roses are red and violets are blue.

Goop is green.

__Roses are red and violets are blue.__

Violets are blue and cheese is yellow.

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

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

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

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

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

1. George is either alive or
dead.

__2. George isn't dead. __

C. George is alive.

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

1.
Shamu is either a cat or a dog

__2. Shamu is not a dog__

C. Shamu is a cat.

Notice now that both arguments have the same valid
form. The second argument *only *fails
because the first premise is false. Dog and cat
are not the only two possibilities for Shamu's
species.

Now let's look at disjunctive syllogism through
pictures and the following argument.

Roy is
either a bum or a tramp

__Roy is not a
bum __

Roy is a tramp

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

Picture 2
Picture 3
Picture 4

Roy is a tramp but not a bum.
Roy is a
tramp and a bum.
Roy is not a tramp
but is a bum.

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

If Roy is a tramp and a bum, then the statement
"Roy is either a bum or a tramp" is true.

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

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

Picture 2

Roy is a tramp but not a bum.

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

S oooo the following arguments are all valid:

Roses
are red.

__Violets are blue .__

Roses are red or violets are blue.

__Roses
are red. __

Roses are red or violets are blue.

__Violets are blue .__

Roses are red or violets are blue.

And the following are all invalid:

__Roses
are red or violets are blue.__

Roses are red.

__Roses are red or violets are blue.__

Violets are blue.

Cheese is yellow or goop is green.

__Roses are red or violets are blue.__

Violets are blue or cheese is yellow.

Goop
is green.

__Roses are red or violets are blue.__

Violets are blue or cheese is yellow.

Cheese
is yellow or goop is green.

__Roses are red or violets are blue.__

Violets are blue or goop is green.

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

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

Remember the valid argument form of **Modus
Tollens **

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

__Y is not true
__ __Joe
is not a
Mocklin __ __Roy
is not a
bum __

X is not
true Babe
is not a
shoat
Roy is not a tramp

And the invalid argument
form of **Denying the Antecedent**

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

__X is not true
__ __Babe
is not a
shoat __ __Roy
is not a
tramp __

Y is not
true
Joe is not a
Mocklin Roy
is not a bum

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

Either
X is true or Y is true.
Either Babe is a shoat or
Joe is a Mocklin Roy is
either a bum
or
a
tramp

__Y is not true
__ __Joe
is not a
Mocklin __ __Roy
is not a
bum __

X is
true
Babe is a
shoat
Roy
is a tramp

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

The following arguments are all valid:

Roses
are red.

__Violets are blue .__

Roses are not red or violets are blue.

__Roses
are not red. __

Roses are not red or violets are blue.

__Violets are not blue .__

Roses are red or violets are not blue.

__Roses
are red and violets are not blue.__

Roses are red.

__Roses are not red and violets are blue.__

Violets are blue.

Roses are not red.

__Violets are not blue .__

Roses are not red and violets are not blue.

Roses
are not red

__Roses
are red or violets are blue.__

Violets
are blue.

Violets
are not blue.

__Roses
are red or violets are blue.__

Roses are red.

Roses
are not red and cheese is not yellow.

Cheese
is yellow or goop is green.

__Roses are red or violets are blue.__

Violets are blue and goop is green.

Roses
are not red or cheese is not yellow.

Cheese
is yellow or goop is green.

__Roses are red or violets are blue.__

Violets are blue or goop is green.

Cheese
is not yellow or goop is green.

Roses
are red or cheese is yellow

__Roses are not red or violets are blue.__

Violets are blue or goop is green.

And the following are all invalid:

__Roses
are not red and violets are not blue.__

Roses are red and violets are blue.

__Roses
are not red and violets are not blue.__

Roses are red or violets are blue.

__Roses
are not red or violets are not blue.__

Roses are red and violets are blue.

__Roses
are not red or violets are not blue.__

Roses are red or violets are blue.

__Roses
are not red. __

Roses are red or violets are blue.

__Roses are not red or violets are not blue.__

Roses are red.

Cheese
is not yellow or goop is not green.

__Roses are not red or violets are not blue.__

Violets are not blue or cheese is not yellow.

Roses
are not red and cheese is not yellow.

__Roses
are red or violets are blue.__

Violets are blue and goop is green.

Roses
are not red or cheese is not yellow.

__Roses
are red or violets are blue.__

Violets are blue or goop is green.

Cheese
is not yellow or goop is green.

__Roses are not red or violets are blue.__

Violets are blue or goop is green.

Cheese
is not yellow or violets are not blue

Cheese is yellow or goop is green.

__Roses are red or violets are blue.__

Roses are red and goop is green.

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

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

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

Roses are red or roses are not red.

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

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

Roses are red and roses are not red.

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

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

1. Roses are red.

2. Roses are not red.

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

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

Let's call that situation "**Situation S**".

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

Catch that word "and".

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

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

And if Situation S is *possible* the
argument is *invalid*.

So what kinds of things make Situation S impossible?

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

Situation S: Premises are all true and the
conclusion is false.

Part 1: Premises are all true..

Part 2: Conclusion is false.

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

Situation S is *sufficient* for Part 1.

Situation S is *sufficient* for Part 2.

Part 1 is *necessary* for Situation S.

Part 2 is *necessary* for Situation S.

What this means that is, *if* Situation S
is possible, then *both *Part 1 and Part
2 *have* to be possible.

From that it follows that, *if* Part 1 is
impossible, *then* Situation S is
impossible and, *if* Part 2 is impossible,
*then* Situation S is impossible, which
means that if *either *Part 1 *or*
Part 2 is impossible, *then* Situation S
is impossible.

Now, here's a question:

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

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

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

If __premises all true__ is *impossible*
and __conclusion false__ is possible, it
follows that the *conjunction* (remember
that word) the conjunction __premises all true
and conclusion false__ is *also*
impossible. And if __premises all true and
conclusion false__ is *impossible*,
the argument is **valid**.

Also:

If __premises all true__ is possible *but*
__conclusion false__ is impossible, it follows
that the *conjunction* (remember that
word) the conjunction __premises all true and
conclusion false__ is *also*
impossible. And if __premises all true and
conclusion false__ is *impossible*,
the argument is **valid**.

So, from this two weird things follow:

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

And:

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

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

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

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

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

Consider the following argument:

George
Bush is over 1,000 feet tall.

__The universe is secretly ruled by a small
fish living under your couch. __

Chocolate either is or is not made from crude
oil.

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

All
wombats are bitter mysogynists living in dank
warrens under the boulevard cafes of Paris.

__Cigarette smoking makes you cool, especially
if you cough up a diseased lung right in front
of the Pope. __

It may or may not be true that Osama Bin Laden
is posing as a cab driver in Des Moines, Iowa.

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

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

Read that again. It says that

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

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

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

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

If you said "A," you're right!

If you said "B," you're **wrong**.

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

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

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

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

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

Cheese is a mineral.

__Cheese is not a mineral __

Elvis is both alive and not alive.

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

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

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

Logic requires a very precise use of terminology.
So here it is. A logically good *deductive*
argument is called *valid*, and a valid
argument with true premises is called **sound**.
A logically good *inductive* argument is
called *strong*, and a strong argument
with true premises is called *cogent .*
The words "valid" and "sound" are

The **validity test** is as
follows:

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

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

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

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

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

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

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

__Practice Quiz.__

1. Do most people find the concept of validity
easy to understand?

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

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

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

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

11. Is this valid or invalid: Whales
are fish. Whales are *not* fish. So
cheese is a mineral.

12. Is this valid or invalid: Whales
are mammals. Whales are not fish. So cheese is *not*
a mineral.

13. Is this valid or invalid: Whales
are mammals. Whales are fish. Fish are never
mammals. So whales **are** fish.

14. Is this valid or invalid: Whales
are mammals. Fish are never mammals. Whales are
not fish. So some whales eat fish.

19. Does validity depend on whether the premises
of arguments are actually true or false.

20. Is it true that an argument cannot be valid if
the premises and conclusion are false.

21. "An argument is valid if and only if ...... "

__Practice Quiz Answers__

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

8. Invalid

9. Valid

10. Invalid

11. Valid. Yes, valid. That's right, it's valid.

12. Invalid

13. Valid. Think about it. Can the premises *all*
be true? If they can't, the argument is **valid**.

14. Invalid. Yep, invalid.

15. Deductive.

16. Inductive.

17. Deductive.

18. Inductive.

19. Nope. Validity has nothing to do with the
actual truth or falsity of the premises of an
argument.

20. Nope. An argument *can* be valid even
if *all* the premises and conclusion are
false.

21. "An argument is valid if and only if it is
impossible for there to be a situation in which
all its premises are true and it's conclusion is
false.

Copyright © 2011 by Martin C. Young