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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

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

If you find

If there are

Here's an argument.

Js v Kt

~Kt

And here's how the truth table starts out:

Notice that each

Now here's a trick. Look at the third row down, where Js = F. Is it

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

Let's review. A

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

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

Another way to handle this is to fill in all the truth functions first, and

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:

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

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

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 ^ 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

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 |
KeKo Ku |

Hu Je Mo v Je |
FoFi Fe v Fu |
MaJi 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

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.
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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