Logic Chapter Four

Logic Chapter Four

PROOFS

A "proof" is anything that purports to show that something is true. (The pictures in the previous chapter are considered proofs. The first and third ones are bad because they fail to prove anything about their respective arguments, the ones that show a false conclusion with all true premises are good because they succeed in proving that those arguments are invalid.) But pictures can't prove an argument valid, (try drawing a picture that proves that you can't draw a picture), so we still need a way to prove arguments valid.

In this chapter, you will learn how to set up a truth table for an argument, and use that truth table to tell whether or not the argument is valid. The basic idea for this is to see whether or not the truth table contains any rows in which all the premises are true and the conclusion is false. Once you have had some practice with truth tables, you may very well be able to do this just by looking at the truth table. However, most people cannot do this right away, so I'm going to teach you an easier method. This method, the method of "row-elimination," is actually more work than the "just-look-at-the-table" method, but it proceeds by a series of small steps, each of which is fairly easy in itself, that eventually change the truth table into a form in which you will easily be able to see whether or not the argument is valid. Basically, you will eliminate any row in which any premise is false, and you will eliminate any row in which the conclusion is true. If there is a row that cannot be eliminated by either of these rules, it will be a row in which the premises are all true and the conclusion is false, and so it will be a row in which the argument is invalid. If those two rules allow you to eliminate all the rows, then there will be no row in which the premises are true and the conclusion is false, and so the argument will be valid. Yes, I know that that sounds complicated. Don't worry, I'm going to spend the whole rest of this lesson explaining and illustrating the meaning of this paragraph. The bottom line is, at the end of this lesson you should be able to use truth tables to determine the validity of arguments, at least for simple arguments. It's okay if you have no idea how to do that now, it will all be clear by the end of the lesson.

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.

Can you Prooooove it?

I'm confused!	Remember an argument is only invalid if it's *possible* for it to have all true premises and a false conclusion. If you can find a true-premises/false conclusion row in the table, the argument is invalid. And if you can't find a true-premises/false conclusion row, the argument has got to be valid.	That didn't help. grrrrrrrrrrr.

Maybe an example.
will help..

   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 its premise column is easy to fill out. Like so:

Now 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 row number 3?

No, that's impossible, because one of the premises on row 3 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.

Yay!
Red ink!

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.


	I'm NOT paying attention!

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:

So, an "F" under a premise lets me eliminate a row.	And "T" under the conclusion also lets me eliminate a row.	But a "F" under the conclusion does not let me eliminate a row	And a "T" under a premise does not let me eliminate any row.

       Now, test your
      knowledge by
      taking a quiz!

And when you're
done, we'll go on
with the example

Fill in the value of "Js v Kt" on the only row we have left. That value is "T"
	It's not a "T" under the conclusion ^^ or an "F" under a premise, so row one can't be eliminated.	In fact, row one has all true premises and a false conclusion! So the argument is invalid,		And we draw a box around that row to make it clear that it's invalid.

Couldn't I just fill in all the columns and then look for rows with all true premises and a false conclusion? If I find at least one row like that, the argument's invalid. If there aren't any, it's valid.

Well. I guess you
could do it that way.
It might even be
quicker.

               And
           Here's
               A
            Quiz!

Like this:

Here's an argument
Mg
Dr v Mg

Try to do this one
on your own

If you can't figure it out, here's a power point presentation

If the powerpoint doesn't
work, try
MHTML

Now here's one you're going to have to work out for yourself.	It's Rd Rd ^ Sn	You can do it as a "quandry maze" to my right
			Or as a homemade HTML maze starting below

To start my home made maze, click on the truth table that's been started right for this argument.

What about arguments with three terms, like Ka ^ Kb Kb ^ Kc Ka ^ Kc	Well, it has a bigger truth table, for a start.		But why does it have eight rows? Hmmmmm?

	But you eliminate lines in exactly the same way: Cut F's under premises, and T's under the conclusion.

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

     Here are some
     arguments you
     can work on all
     by yourself

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

    Some of these look
    hard, but they can
    be easy if you
    know the trick

Now do the truth functions for these five sentences

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

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

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.