For Thursday, July 15, read http://www.madwizard.com/ctl_cba.htm

For Monday, July 19, read http://www.madwizard.com/ctl_definition.htm (not ready yet)

For Tuesday, July 20, read http://www.madwizard.com/ctl_advanced.htm (not ready yet)

Final is Wednesday, July 21


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

This chapter concernss 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.

Validity Explained Correctly

In the previous chapter I gave you a definition of 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 arguments. Only arguments can be valid or invalid. Statements can be true or false, but they can't be valid or invalid.

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

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

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

Now, the definition of validity given in the previous chapter was clear, intuitive and easy to apply. The only problem with it is that it is wrong. Wrong, wrong, wrong, wrong, WRONG!

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

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

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

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

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

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

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

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

The only true statement is, of course, number 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), even if the conclusion is true!

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

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

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

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

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

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

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

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

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

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

Fa Pw Ml

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



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



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

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

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

De Px Ft

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


But this next picture does prove the argument invalid (wonky).

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

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

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's almost certainly get it wrong.

Do the Premises Have to Make the Conclusion True?

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

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

Consider the following argument:

George Bush is over 1,000 feet tall.
The universe is secretly ruled by a small fish living under your couch.
Chocolate either is or is not made from crude oil.

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

All wombats are bitter mysogynists living in dank warrens under the boulevard cafes of Paris.
Cigarette smoking makes you cool, especially if you cough up a diseased lung right in front of the Pope.
It may or may not be true that Osama Bin Laden is posing as a cab driver in Des Moines, Iowa.

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

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

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

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

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

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

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

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

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

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

De ~De Wl                  VALID!!

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

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

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

Cheese is a mineral.
Cheese is not a mineral
Elvis is both alive and not alive.

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

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

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

Terminology (Groan!)

Logic requires a very precise use of terminology. So here it is. A logically good deductive argument is called valid, and a valid argument with true premises is called sound. A logically good inductive argument is called strong, and a strong argument with true premises is called cogent. The words "valid" and "sound" are not used for inductive arguments, and the words "strong" and "cogent" are not used for deductive arguments.

The validity test is as follows:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A Final Note

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.

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

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

Practice Quiz.
1. Do most people find the concept of validity easy to understand?
2. How are deductive arguments different from inductive arguments?
3. Can deductive arguments prove things about the world outside your mind?
4. Can inductive arguments prove things about the world outside your mind?
5. Can deductive arguments prove things with absolute certainty?
6. Can inductive arguments prove things with absolute certainty?

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.
15. Is this deductive or inductive: All whales are fish. Willie is a whale. Therefore Willie is a fish.
16. Is this deductive or inductive: The vast majority of whales live free in the ocean. Willie is a whale. Therefore Willie lives free in the ocean.
17. Is this deductive or inductive: All monkeys can fly. George cannot fly. Therefore George is not a monkey.
18. Is this deductive or inductive: Monkey aerodynamics make flight extremely unlikely. Kong is a monkey. Therefore Kong cannot fly.
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.
2. Deductive arguments establish the precise logical relationships between ideas. Inductive arguments use evidence to try to establish facts about the world.
3. No.
4. Yes.
5. Yes.
6. No.
7. Valid
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 © 2010 by Martin C. Young

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