Logic Chapter Three

Logic Chapter Three
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VALIDITY

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. (A valid argument will only prove something if it is also sound.) Remember, a valid argument is not necessarily a good argument.

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, even if the conclusion is true!

For instance, the following arguments are all completely invalid.

Paris is in France
Berlin is in Germany
Compton is in America

Cats are mammals
Dogs are mammals
Ferrets are mammals

People have two legs
Mammals have four legs
Insects have six legs

Dumbledore is a wizard
Gandalf is a wizard
Merlin is a wizard

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

Don't worry about the fact that none of these things is true. Worry about the fact that it's possible for the conclusion to be false even if the premises are true.

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

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.

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

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

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

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

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

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

Here's a scheme.

                                              Fx : x discovered France
                                              Px : x eats cheese pizza
                                              Mx : x is a Martian
                                              a : Albert Einstein
                                              w : my wolverine
                                              l : Laura Schlessinger

So"a" stands for Albert Einstein, "w" for my wolverine and "l" for Laura Schlessinger. Anything inside the "F" circle discovered France, inside the "P" circle eats cheese pizza and anything inside the "M" circle is a Martian

Given that scheme, is this argument valid or invalid?

Albert Einstein discovered France
My wolverine eats cheese pizza
Laura Schlessinger is a Martian

Fa
Pw
Ml

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

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

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

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

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

De
Pf
Ft

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

But this next picture does prove the argument invalid.

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

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

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

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

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

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

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

Elvis is dead.
Elvis is alive.
Laura Schlessinger is a woolly mammoth.

De
~De
Wl VALID!!

Think about it. Is it possible to have a situation in which the premises are true and the conclusion is false? Sure, it's possible to have a situation in which the conclusion is false, but for the argument to be invalid, 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, 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.

Test yourself: Does the fact that we can make a valid argument for absolutely any conclusion mean that logic can prove absolutely anything? YES NO

To put it another way, can you construct a sound argument for a false conclusion? YES NO

Here are some arguments for you to work on by yourself. Some are harder, so just do the ones you can.

Practice: For each of the following arguments, try to draw a world-picture in which the premises are true and the conclusion is false.

Nu ~Nu Ga ^ Ki	Fa Me Fa ^ Me	Gi Lo Lo v Gi	~Jo Jo	Ke Ko Ku
Hu Je Mo v Je	Fo Fi Fe v Fu	Ma Ji La ^ Ji	Na v Lu ~Na Lu	Ka ^ Mi ~Ka Mi
Ne v Ha Go v Li Go	Ho v Ju Le ^ Ii Le	Ie ^ Hi No v Gu No	Mu ^ Ia He ^ Ni He	Iu Ge ^ Io Ja

DETERMINING VALIDITY BY TRUTH TABLE

To use a truth table to discover whether or not an argument is valid, you fill out a truth table for the argument, with premises and conclusion marked, and see if there are any rows in which the premises are all true, and the conclusion is false.

If you find one or more rows like that, the argument is invalid.

If there are no such rows, the argument is valid.

Here's an argument.

Js v Kt
Js
~Kt

And here's how the truth table starts out:

Notice that each basic term gets its own column to the left of the double line. Premises and conclusion are marked as such, and each premise gets it's own column, even if it's just one of the terms. "Js" is both a basic term and a premise, so it's premise column is easy to fill out. Like so:

Now here's a trick. Look at the third row down, where Js = F. Is it possible for this row to be the kind of row were looking for? Remember that we're looking for rows where all the premises are true and the conclusion's false. Can all the premises be true on this row? No, that's impossible, because one of those premises is already known to be false. So that third row can be eliminated. The same goes for row four, so they can both be ruled out.

We draw a line through the row to show it's eliminated, and circle the "F" on each row to show that it's the reason the row is eliminated. Now we just have to worry about rows one and two. Let's fill in the column under the conclusion, but just on those two rows.

Because "~Kt" is the conclusion, the "F" on row one doesn't mean that row can be eliminated. We're looking for rows where all the premises are true and the conclusion's false. Row one could still be that kind of row! However, we can eliminate row two, because on that row the conclusion is true, so it can't be a row where the premises are true and the conclusion is false! So:

Let's review. A "F" under a premise lets me eliminate a row. A "T" under the conclusion also lets me eliminate a row. But a "F" under under the conclusion does not let me eliminate a row and a "T" under a premise does not let me eliminate any row.

Pop quiz!
True or False? A "T" under a premise lets me eliminate that row.
True or False? A "T" under the conclusion lets me eliminate that row.
True or False? A "F" under a premise lets me eliminate that row.
True or False? A "F" under the conclusion lets me eliminate that row.

Back to our example: Since not all rows are eliminated, and no row yet says that the argument is invalid, we go on and fill in the value of "Js v Kt" on the only row that has not yet been eliminated.

That value is "T". Since it's not a "T" under the conclusion or an "F" under a premise, row one can't be eliminated. Are there any more premises to check? No. Is there any row with all true premises and a false conclusion? Yes! (It's row one!) The row's not eliminated, so it's not struck out. In fact, we draw a box around the whole row to emphasize that it's all the truth values in this row that make the argument invalid. So the argument's invalid, and we mark it as so.

INVALID (by row one.)

So, to recap, we have two elimination rules:

1. If there's a "F" under a premise, you should eliminate the row the "F" is on.

2. If there's a "T" under the conclusion, you should eliminate the row the "T" is on.

And the way we determine whether an argument is valid is:

First: eliminate all the rows we can. (That's rows with at least one false premise, rows with the conclusion true, and rows with both.)

Second, see if there's any rows left over.

If there's even one row were all the premises are true and the conclusion is false, the argument is invalid.

If there's absolutely no rows like that, the argument is valid.

Another way to handle this is to fill in all the truth functions first, and then look for a row with all true premises and a false conclusion. (Once we have that, we can ignore the other rows.) That would end up looking like this:

Let's review:
True or False? A "T" under a premise lets me eliminate that row.
True or False? A "T" under the conclusion lets me eliminate that row.
True or False? A "F" under a premise lets me eliminate that row.
True or False? A "F" under the conclusion lets me eliminate that row.

True or False? A line where the premises are all true and the conclusion is true means that the argument is valid
True or False? A line where the premises are all true and the conclusion is false means that the argument is valid
True or False? A line where the premises are all false and the conclusion is true means that the argument is valid
True or False? A line where the premises are all false and the conclusion is false means that the argument is valid

True or False? A line where the premises are all true and the conclusion is true means that the argument is invalid
True or False? A line where the premises are all true and the conclusion is false means that the argument is invalid
True or False? A line where the premises are all false and the conclusion is true means that the argument is invalid
True or False? A line where the premises are all false and the conclusion is false means that the argument is invalid

Here's another, trickier argument:

Mg
Dr v Mg

With only one premise, it's at least a shorter truth table.

Since one of the terms is also the premise, it's easy to fill in that column.

And any rows that contain an "F" under a premise can be eliminated.

Next, let's fill in the value of "Dr v Mg" on the other two rows.

Since both those rows put a "T" directly under the conclusion, they can both be eliminated. Since all four lines have been eliminated, there are no lines where all the premises are true and the conclusion is false, the argument is valid! And we mark it so.

VALID.

Here's the basic truth table. See if you can work the problem for yourself.

For the next example, I'm going to make you work your own way through the proof. Here's the argument.

Rd
Rd ^ Sn

First, pick the truth table that's right for this argument, and click on that truth table.

Finally, here's how to deal with an argument that has multiple terms, like:

Ka ^ Kb
Kb ^ Kc

Ka ^ Kc

First, we create a truth table covering all possible worlds for these terms. (Question: why does it have eight rows?)

Then we fill in the values for the first premise. (Are these the values you'd pick?)

And eliminate the ones we can. (How do I pick the ones to eliminate?)

Fill in the values for the second premise. (Again, why does it have these values?)

Again, eliminate whatever rows we can. (You know which one to cross out.)

Finally, fill in the truth-function for the conclusion. (This is the trickiest one to fill in.)

And eliminate any rows that can be eliminated. (Can you remember why the conclusion is treated differently?)

And guess what, it's VALID! (Cool huh?)

Here's an argument that's complicated in a different way. .

Ta ^ ~Wb
~Ta
Wb

As before, we follow the happy snake rule:

Notice that we can't eliminate any lines on the basis of "~Wb." Why not? It's because "~Wb" isn't a premise. But we do use the "~Wb" column to fill in the truth-function for the first premise.

And that truth function lets us eliminate three rows!

Next, we again use the happy snake rule, which says we can evaluate the next premise.

Since "~Ta" is a premise, and it is has a value of "F" on line two, we can we use it to eliminate that last line. And since it is the last line, we've just proved the argument valid!

VALID.

Now, what if we had a similar argument, with a "v " instead of a "^"? Let's see what happens:

Lt v ~Mr
~Lt
Mr

Well, the initial truth table is very similar.

But the truth function for the new first premise is very different.

And it only allows us to eliminate one row.

Lets write in the second premise.

Which allows us to eliminate two more rows.

Now write in the truth function for the conclusion.

Which has an "F" on line 4, making the argument invalid.

INVALID (by line 4.)

Finally, let's do a slightly trickier one.

~Un
~(Un ^ Wn)

Here's the truth function for the only premise.

Which allows us to eliminate two rows.

Since "Un ^ Wn" is inside the parentheses, the snake is unhappy, and we evaluate "Un ^ Wn" first as an intermediate step on the way to the conclusion.

Now we can get the truth function for the conclusion by negating the truth function for "Un ^ Wn".

Since the conclusion is true on the two lines we have not eliminated, the argument is valid.

VALID.

For the next example, I'm going to make you work your own way through the proof. Here's the argument.

~Vj
Vj ^ ~Wi

First, pick the truth table with the truth function that's right for the premise, and click on that truth table.

Here's another one.

~(Tl v Vm)

~Vm

Click on the correct truth table to get this proof started.

Here are some to work through on your own.

(The hard ones are actually easier than they look. If you can figure out the trick! He he he!)
Hint: A couple of these can be done by filling just one line in the truth table!
Hint: If you can't see how do it by truth table, try drawing some pictures first.

Nu ~Nu Ga ^ Ki	Fa Me Fa ^ Me	Gi Lo Lo v Gi	~Jo Jo	Ke Ko Ku
Hu Je Mo v Je	Fo Fi Fe v Fu	Ma Ji La ^ Ji	Na v Lu ~Na Lu	Ka ^ Mi ~Ka Mi
Ne v Ha Go v Li Go	Ho v Ju Le ^ Ii Le	Ie ^ Hi No v Gu No	Mu ^ Ia He ^ Ni He	Iu Ge ^ Io Ja

Homework 3. Use your own paper or the answer sheet at logic03homework.rtf. For each of the following arguments, say whether it is valid or invalid. If it's invalid, draw a picture that proves it's invalid.

	My monkey can fly. My monkey can fly.	1. ____________
	My monkey can fly. My monkey can not fly.	2. ____________
	My monkey can not fly. My monkey can fly. My monkey can fly.	3. ____________
	My cat can not fly. My dog can not fly. My snake can not fly. My rabbit can not fly. My monkey can not fly.	4. ____________

If you have trouble, I suggest trying to draw pictures based on the following scheme

                                              Fx : x can fly
                                              m : my monkey
                                              c : my cat
                                              d : my dog
                                              s : my snake
                                              r : my rabbit

Question 5. Write out the correct definition of validity in your own words as completely as you can. Don't worry about making it elegant, just worry about getting it right.

Homework 4. Use your own paper or the answer sheet at logic04homework.rtf. For each of the following arguments, prove it valid or invalid by truth table.

1.	Cf Dg Cf ^ Dg
2.	Fh ^ Gb Gb
3.	Hc v Jd ~Hc Jd
4.	Kg Kg v Lf

For each of the following arguments, prove it valid or invalid by truth table.

9.	Wa ~Vb v Wa
10.	~Vc ^ Vd ~Vd
11.	Vf v ~We We Vf
12.	Tg v ~Wh Tg

Question 5. Write out the correct definition of validity in your own words as precisely as you can. Try to use different words from the ones you used last time.

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