Logic Chapter One

Logic Chapter One
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Statements, blah blah blah

Atomic Statements

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, (not covered by this course), can prove things about the world, but the price of never giving you absolute certainty. Let me explain.

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 quite 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?

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.

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 a basic handle on symbolic logic, it's necessary to learn the language. We will learn one language for symbolic logic over the course of this class, starting with schemes.

SCHEMES

A scheme is just a list of symbols together with the meanings we've (temporarily) assigned to them. (It's like a little tiny made-up language.) Here's an example.

                                              Bx : x is a bear
                                              Cx : x is a cat
                                              a : King Arthur
                                              b : Bill Clinton

In this class, we will use upper case letters, A, B, C,.... to stand for properties (call them "predicate letters") and lower case letters to stand for things (call these "object letters"). Object letters come in two flavors. There's "x," "y" and "z" (called "variables") which can stand for just some thing, when we don't know what particular thing it is, or it could be anything, or all things, or nothing. Then there's all the other lower case letters, a, b, c,...., (called "names") which are reserved for particular things. So when you see a variable, you know it means "just any old thing" but when you see a name you know it stands for some particular, unique thing, like you, or me, or Ricky Martin's left big toe, or the planet Neptune or anything, just so long as it's a particular thing. So the scheme above lets us talk about bears in general, and cats in general, without necessarily referring to any particular cats or bears, and to specifically talk about King Arthur and Bill Clinton.

Sometimes, I will use the upper case letters "P," "Q" and "R" to stand for "just any old statement." It's like a variable, only for statements rather than objects.

STATEMENTS

A statement (in deductive logic) is a definite claim that we using whatever scheme we happen to choose. (It's like saying something in a little tiny language.) At the moment, the scheme I gave above allows us to say the following four things: "Ba" which means "King Arthur is a bear," "Bb" which means "Bill Clinton is a bear," "Ca" which means "King Arthur is a cat" and "Cb" which (guess what) means "Bill Clinton is a cat." (These are called "simple" statements. Later on, we'll be able to say more complicated things.)

It's important to form statements in exactly the right way. "Ba" is a proper statement, but "aB" isn't. Nor is "CB," "ba" or "aCb" and so on. Statements that are made the right way are called "well-formed" while those that aren't are called "ill-formed." Make sure all your statements are well formed.

WORLDS

A world is an imaginary universe, or version of the real world, that we make up to try to make statements true or false. There's a couple of ways to represent worlds. The easiest to understand is by diagrams, so I'll start with that.

What this picture means:
First, if there are any bears at all in this world, they are inside the "B" circle. If there are any cats at all, they are inside the "C" circle.
Second, because the "a" is inside the "B" circle but not the "C" circle, it means that King Arthur is a bear and not a cat.
Finally, because the "b" is inside both circles, it means that Bill Clinton is both a bear and a cat.

It's important to remember that the empty spaces are all places that other objects could be. There could be other objects anywhere in this picture. Thus, this picture doesn't say anything beyond the meaning I just told you about.

VALUE

The value (or "truth value") of a statement just means whether it's true or false in a particular world. Here are the values of the four statements given above in this particular world. (Can you tell why these four statements have the values they do?)
Ba = true, Bb = true, Ca = false, Cb = true.
Notice also that the world specifies more than just the values of these four statements. For instance, it also says that something can be both a bear and a cat.

PRACTICE

What are the values of our four statements in this world?

Ba = ?

Bb = ?

Ca = ?

Cb = ?

(For the answer, rest your cursor on the relevant statement. The answer will appear as a web address at the bottom of your screen. Or click the relevant statement.) By the way, the fact that neither the "b" nor the "c" is inside the "B" circle doesn't mean that the "B" circle is necessarily empty. It might be empty, but there might be something there. This picture doesn't say.

What about this one?

Ba = ?

Bb = ?

Ca = ?

Cb = ?

And finally, this one? (Remember that everything inside the "B" circle is also inside the "C" circle!

Ba = ?

Bb = ?

Ca = ?

Cb = ?

Molecular Statements

OPERATORS

An operator is a way of making statements into other statements (called "complex" statements) that mean more or less different things. There's a bunch of operators out there, but to keep things simple I'm going to start with just three operators. They are known as "NOT," "AND," and "OR."

NOT

The first operator is "~," the "tilde" (or "snake," or you can call it "the not-sign" or just "NOT."), and it reverses the truth-value of whatever statement it's attached to. Here's how it works.

Adding a tilde to "Bb" we get "~Bb," which means "Bill Clinton is not a bear." (The thing the operator operates on is called a "term," which means that "Bb" is the term of "~Bb.")

Adding another tilde to "~Bb" we get "~ ~Bb," which means "Bill Clinton is not not a bear." (Now this means the same thing as Bb, but they're not the same statement. Trust me.)

Adding yet another tilde we... oh, you can figure it out. You can have as many tildes in a statement as you like, make it an enormous conga line of snakes, but every one you add reverses the truth value of the statement you add it to. (Remember that "P" and "Q" stand for any statement)

If P is true, then ~P is false, but ~ ~P is true, and ~ ~ ~P is false!

But if Q is false, then ~Q is true, but ~ ~Q is false, and ~ ~ ~Q is (you guessed it) true!

For practice, work out the values of the negations of our four original statements.

~Ba = ?

~Bb = ?

~Ca = ?

~Cb = ?

And here's some practice question to work on.
(These will definitely help you prepare for the quiz!)

1. Draw a picture in which Fa is true.

2. Draw a picture in which Gb is false.

3. Draw a picture in which ~Hc is true.

4. Draw a picture in which ~Id is false.

5. Draw a picture in which Je ^ Kf is true.

6. Draw a picture in which Lg ^ Mh is false.

7. Draw a picture in which Ni v Fa is true.

8. Draw a picture in which Gb v Hc is false.

9. Draw a picture in which Id is true and Je is false.

10. Draw a picture in which Kf v Lg is true and Lg is false. (Remember that g can't be in two places.)

11. Draw a picture in which Mh is true and Mh ^ Ni is false. (Remember that h can't be in two places.)

12. Draw a picture in which Fi v Gi is true and Gi is false. (Remember that i can't be in two places.)

13. Draw a picture in which Ha is true and Ha ^ Ia is false. (Remember that a can't be in two places.)

14. Is it possible to draw a picture in which Ai ^ Bj is true and Ai is false. (Remember i can't be two places.)

15. Is it possible to draw a picture in which Ck is true and Ck v Dl is false. (Remember k can't be two places.)

16. Is it possible to draw a picture in which Em and ~Em are both true? (Remember m can't be two places.)

TRUTH TABLES

You may have noticed that worlds are basically defined in terms of what is or isn't true in a particular situation. Sometimes, we will want to specify several different worlds all at once. For this, we use truth tables.

Here is a truth table that covers all the possible values of Ba and Cb.

If I was to represent all these possible values in Venn diagrams, or "world pictures," it would look like this:

Holy crap! That's a lot of worlds! And damned confusing too! If you look closely, you'll see that some of these worlds make Ba and Cb both true, some make Ba true and Cb false, (which is the same as making Ba true and ~Cb true), some make Ba false and Cb true (~Ba and Cb both true), and finally, some make Ba and Cb both false, (or ~Ba and ~Cb both true). Since all I care about right now is the values of Ba and Cb, I'm going to forget about all these worlds and just go with the truth table.

Look at the "T"s and "F"s. As you may guesss, "T" stands for "true" and "F" stands for "false." Each row in a truth table stands for a set of worlds in which the statements in question have the values indicated by the relevant letters. Thus in row 1, Ba and Cb are both true, row 2 Ba is true and Cb is false, in row 3 Ba is false and Cb is true and in row 4, Ba and Cb are both false. A lot easier to keep track of than a bunch of world pictures, huh? (From now on I'm going to use the word "world" to mean both "world" and "set of worlds." Basically, this means a line on a truth table will be called a "world." I hope this doesn't confuse anyone.)

But we don't have to stick to any particular statements. We can do truth tables for any arbitrarily chosen sentences. Here's one for any two different statements, which I will represent by the sentence letters "P" and "Q"

Look at the "T"s and "F"s. As before, "T" stands for "true" and "F" stands for "false," and each row in a truth table stands for a world in which the statements in question have the values indicated by the relevant letters. Thus in row 1, P and Q are both true, row 2 P is true and Q is false, in row 3 P is false and Q is true and in row 4, P and Q are both false.

Here's the truth table for any statements that happen to get modified by repeated additions of the "not" operator.

Notice that the "F"s and "T"s are always under the first tilde. That's because that tilde is the main operator of that particular statement. The main operator is the last one you think about when evaluating a statement, and it's what makes a statement the kind of statement it is. A statement whose main operator is a tilde is called a "negation."

AND

The next operator is "^," the "carat" (or just "AND") and it joins two other statements (called "terms") together to make a new statement that is true only when both of the joined statements are also true, and false when either or both of them are false. Here's its truth table. A statement whose main operator is a carat is called a "conjunction," and it's terms are called "conjuncts."

Notice that there's only one world in which "P ^ Q" is true. And that world is... ?

Adding a carat to "Bb" we get " ^ Bb," which doesn't mean anything. But if we add "Ca" to that we get "Ca ^ Bb," which does mean something. It means "King Arthur is cat and Bill Clinton is a bear." That's a statement that's only true if both parts of it are true. If Bill Clinton isn't a bear, then the whole statement "Ca ^ Bb" is false, whether King Arthur is cat or not. And if King Arthur isn't a cat, well...

For practice, work out the values of these conjunctions in these worlds

Ba ^ Ca	Bb ^ Ca
Ba ^ Bb	Bb ^ Bb
Ca ^ Cb	Bb ^ Ba
Ba ^ Cb	Bb ^ Cb

Ba ^ Ca	Bb ^ Ca
Ba ^ Bb	Bb ^ Bb
Ca ^ Cb	Bb ^ Ba
Ba ^ Cb	Bb ^ Cb

Ba ^ Ca	Bb ^ Ca
Ba ^ Bb	Bb ^ Bb
Ca ^ Cb	Bb ^ Ba
Ba ^ Cb	Bb ^ Cb

OR

The next operator is " v," the "wedge" (or just "OR") and it joins two other statements together to make a new statement that is false only when both of the joined statements are also false, and true when either or both of them are true. Here's its truth table. A statement whose main operator is a wedge is called a "disjunction," and its terms are called "disjuncts."

Notice that there's only one world in which "P v Q" is false. And that world is... ?

While a conjuction will only be true if both of its conjuncts were true, a disjunction will be true if either of its disjuncts are true. (Here "either" means one, the other, or both.) Thus the following worlds all make the statement "Ca v Ba" true.

But none of them make "Ca ^ Ba" true. That's right, "Ca v Ba" is true and "Ca ^ Ba" is false in each of these worlds.
(But remember, "P v Q" is true whenever "P ^ Q" is true)

For practice, work out the values of these disjunctions in these worlds

Ba v Ca	Bb v Ca
Ba v Bb	Bb v Bb
Ca v Cb	Bb v Ba
Ba v Cb	Bb v Cb

Ba v Ca	Bb v Ca
Ba v Bb	Bb v Bb
Ca v Cb	Bb v Ba
Ba v Cb	Bb v Cb

Ba v Ca	Bb v Ca
Ba v Bb	Bb v Bb
Ca v Cb	Bb v Ba
Ba v Cb	Bb v Cb

Remember that disjunctions (like P v Q) are true just so long as one of their disjuncts are true. So the following sentences are all true:

Some grass is green or all elephants are made of tofu.
Ru Paul is president or cars can run on gasolene
Smoking is bad for you or celery can grow in your ears.

They're true because they're disjunctions and, in each one, at least one of the disjuncts is true. Now, you might think that I'm cheating, because in each one I took a true statement and added something silly. Well, yes, that's what I did. And you can do it too. (Practice on your own for a bit. I'll wait.)

COMPLICATED SENTENCES

One of the horrible things about symbolic logic is that there is no limit to how complicated the sentences can get. So far, we've dealt with sentences with one or two terms, but it's possible to have sentences with any number of terms. (Think of a finite number. You can have a sentence with that many terms.) Furthermore, because we have two operators, we have the possibility of not knowing exactly what an operator is supposed to apply to. Does the "~" in "~Ba ^ Cb" apply just to the "Ba" or to the whole of "Ba ^ Cb?" The way we settle this is to use brackets, like "(," ")," "[," "]," "{," "}," to make it clear how we are supposed to interpret combinations of statements.

The "~" in "~Ba ^ Bb" applies just to the "Ba," so it says King Arthur is not a bear but Bill Clinton is a bear.

The "~" in "~(Ba ^ Bb)" applies to the whole of "Ba ^ Cb" so what it says is that "it's not true that both King Arthur and Bill Clinton are bears. (It might be true that just one or the other is a bear, but it's certain that they're not both bears.)

For today, we'll just start thinking about sentences with a maximum of three or at the most four terms, and I'll give you the basic rules for figuring out which operator to evaluate first. When there's any chance of getting the evaluation order wrong, we add parentheses. The parenthese are just added to tell you in which order to apply the operators.

Check out this truth table.

Notice that on the left, the truth function (the little column of "T"s and "F"s) is under the " ^ " while on the right, the truth function is under the "~". This is because on the left the carat is the main operator (making that statement a conjunction) while on the right the tilde is the main operator (making that statement a negation). Then notice that on the left the tilde just modifies the "P" while on the right, the tilde negates the whole of "P ^ Q."

For practice, copy this diagram out twice:

In the first copy, fill in "a" and "b" in such a way as to make ~Ba ^ Cb true. (Answer)

In the second copy, fill in "a" and "b" in such a way as to make ~(Ba ^ Cb) false. (Answer)

The first "ordering" rule you should learn is what I call the "Happy Snake Rule." If there's a tilde "~" right up against the side of some term, like a snake biting some guy in the butt, then the combination of that tilde and that term gets evaluated first, but if the snake is frustrated by a parenthesis, which is like a shield getting in the way, then whatever's inside is "deeper" than the tilde, and so has to be elvaluated before we can get to the tilde. So if we're working on "Tg v ~Wh," we start by evaluating ~Wh. But if we're working on ~(Tl v Vm) that snake's gonna have to wait until whatever's inside those parentheses has been evaluated.

The second rule is the "Dig Deep Rule." This says that operators that are deeper inside nests of parentheses get evaluated before operators that are not so deep. This means that if we were working on "Th v (Ug ^ (Vf v We))," we would start by evaluating "Vf v We." By the same token, we would start our work on "((Tm ^ Ul) v Vj) ^ Wi" by evaluating "Tm ^ Ul" because it's operator is deepest in the nest of parentheses. Finally, we would approach "(Tb ^Ua) v (Un ^ Wn)" by evaluating both "Tb ^Ua" and "Un ^ Wn" before going on to evaluate that last "v ."

For convenience, the operator we would evaluate last is always called the main operator. In the following sentences, the main operator is marked in red.

Ta ^ ~Wb	Tg v (~Wh ^ ~(Te ^ Wf )))	(Tn ^ Vn) ^ Um
Ve ^ (Ud v Wc)	(Vh ^ Wg) v ~Vc ^ Vd	~Ui v (Uj v (Ti v ~(Tl v Vm)))
Wl ^ (Tj ^ Vi)	(Wm v Vl) ^ Wj	Tc v (Ub v Va)
(Tf v Ue) v Wd	((Tm ^ Ul) v Vj) ^ Wi	Th v (Ug ^ (Vf v We))
(Uh ^ Vg) v Uf	(Tb ^ Ua) v (Un ^ Wn)	(Td v Uc) ^ (Vb v Wa)

So, how do we determine the truth function of something like "Ta ^ ~Wb?" Well, first we set up the truth table and evlauate "~Wb" all on it's lonesome.

Then, we ignore Wb, and evaluate "Ta ^ ~Wb" based just on the values of "Ta" and "~Wb."

And that's it for "Ta ^ ~Wb

How would we do "~(Te ^ Wf )?" Well, since the snake can't get in to bite "Te," it will have to wait until we evaluate "Te ^ Wf " on it's own. And here it is.

Once that's done, we ignore the basic terms and evaluate the negation of "Te ^ Wf," like so:

Now let's do something more complicated "((Ma ^ Ne) v Ma) v Ne" As always, we start with the deepest operator.

Then we do the next deepest. Remember, we ignore the "Ne," and evaluate the "v " in "(Ma ^ Ne) v Ma."

Finally, we evaluate the main operator.

And we're done. How about something even more complicated? How about "(Td v Uc) ^ (Vb v Wa)?" Let's lay out the basic truth table!

Aiiiiiiiiiiiiiiiieeeeeeeeeeeeeeeeeeeeeeee! What have I done? Okay, well, hmmmm.... I know, let's do the truth tables for the two deepest operators:

and

Okay, now copy those over to the big one. Notice that "Td v Uc" gets spread out to four times its original size, while "Vb v Wa" gets repeated four times.

Well, that wasn't so bad. Let's cut off the original four columns and evaluate the main operator.

Well, that was tedious. Let's stop now.

Crap! My boss just walked in. Gotta look busy. Here's some practice exercises. Work out the truth functions for these sentences.

~Te

~Te ^ Wf

~(Te ^ Wf)

Tm ^ ~Ul

~Tf v Ue

~(Tf v Ue)

These two look hard, but they're actually easy if you can figure out the trick.

Hints:
What are the only circumstances under which the first sentence would be true?
What are the only circumstances under which the second sentence would be false?

Homework: Answer the odd-numbered questions either on your own paper or on the Homework Answer Sheet (If you want extra practice, or you have to make up this homework, do all 12 on the Make-up Homework Answer Sheet)

	1. Eg = ____ Eh = ____
	2 Jk = ____ Ek = ____ Jg = ____ Jh = ____
	3. Mg = ____ Mh = ____ Mk = ____
	4. En = ____

	5. ~Bb = ____ ~Ca = ____
	6. De ^ Fg = ____ De v Fg = ____
	7. Hi ^ Jk = ____ Hi v Jk = ____
	8. Ab ^ Cd = ____ Ab v Cd = ____

Once again, the following exercises will help you prepare for the quiz.
1. Draw a picture in which Fa is true.
2. Draw a picture in which Gb is false.
3. Draw a picture in which ~Hc is true.
4. Draw a picture in which ~Id is false.
5. Draw a picture in which Je ^ Kf is true.
6. Draw a picture in which Lg ^ Mh is false.
7. Draw a picture in which Ni v Fa is true.
8. Draw a picture in which Gb v Hc is false.
9. Draw a picture in which Id is true and Je is false.
10. Draw a picture in which Kf v Lg is true and Lg is false. (Remember that g can't be in two places.)
11. Draw a picture in which Mh is true and Mh ^ Ni is false. (Remember that h can't be in two places.)
12. Draw a picture in which Fi v Gi is true and Gi is false. (Remember that i can't be in two places.)
13. Draw a picture in which Ha is true and Ha ^ Ia is false. (Remember that a can't be in two places.)
14. Is it possible to draw a picture in which Ai ^ Bj is true and Ai is false. (Remember i can't be two places.)
15. Is it possible to draw a picture in which Ck is true and Ck v Dl is false. (Remember k can't be two places.)
16. Is it possible to draw a picture in which Em and ~Em are both true? (Remember m can't be two places.)

If you want to practice for the quiz, or you want to make up a missed quiz for this chapter, you can do the statement pre-quiz.

Copyright © 2007 by Martin C. Young

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