Logic Chapter Four
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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 also have...

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.)

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-size="1">

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

(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

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.


Practice 4. Use your own paper or the answer sheet at practice 04. 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.

Copyright 2007 by Martin C. Young

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