[5.6.] Strange Types of Valid Argument. [Chapter 3:7]

 

In ch.3:7, your textbook reminds you that a valid argument is one for which there is no assignment of truth values to constants (i.e., no interpretation) on which all the premises are true and the conclusion false at the same time.

 

This definition implies that there are three types of valid argument:

1.      “the normal case”:

§         it is possible for the premises to be all true;

§         it is possible for the conclusion to be false; and

§         it is not possible for the premises to be all true and the conclusion false at the same time.

2.      the first “special case”: it is impossible for the premises to be all true, i.e., the premises are inconsistent (so there is no interpretation on which the premises are all true… and therefore no interpretation on which the premises are all true and the conclusion false).

3.      the second “special case”: it is impossible for the conclusion to be false, i.e., the conclusion is a tautology (so there is no interpretation on which the conclusion is false… and therefore no interpretation on which the premises are all true and the conclusion false).

 

So any argument with inconsistent premises is valid; and any argument the conclusion of which is a tautology is valid.

 

 

[5.7.] The Short Truth Table Test for Invalidity. [Ch. 3:8]

 

The proof method is an extremely efficient way to demonstrate that an argument is valid. But you cannot use it to demonstrate that an argument is invalid. To show an argument to be invalid, we still need to rely on the truth table method.

 

But we don’t have to use the full truth table method covered a few weeks ago. Instead, we can use the more economical short truth table method.

 

I will explain this method in the same way your textbook does, by applying it to a specific example: exercise 3-10 (p.81),  #4:

 

(4)          1. R Ú N

2. L É N

3. R                  /\ ~N

 

 

Step 1: Set up the top row of a truth table, containing just the premises and conclusion:

 

 

 

Step 2: assign truth values to the main connectives of each sentence, or if a sentence consists of a single constant, assign a truth value to that constant. Assign the value “true” to each premise and assign the value “false” to each conclusion.

 

 

 

Step 3: if you have assigned a value to any sentence constants, place those values elsewhere on the table where that constant occurs.

 

 

 

Step 4: Now begin filling in the spaces underneath the other operators and/or constants, according to the truth table definitions of the constants.

 

In this example, you should first insert the value of the “N” in the conclusion, since there is only one value it can take:

 

 

Then add “T” underneath every other occurrence of “N” in the proof:

 

 

And since a conditional with a true consequent is true either when its antecedent is true or when it is false, you can fill in the value of “L” in the second premise either way; here I will fill in “F”, although “T” would work just as well:

 

 

 

Once you’ve completed the short truth table in this manner, you have shown that it is possible for all the premises of the argument to be true and the conclusion false at the same time. In other words, you’ve shown that the argument is invalid.

 

IMPORTANT NOTE: Sometimes you will have a choice as to whether to assign a T or an F to a given constant (as in the assignment of “F” to “L” in the above example). This is because, with regard to some compound sentences, you can change the truth value of a component sentence without changing the truth value of the entire sentence. For example, suppose that “A” is true. In that case, “A Ú B” is true no matter what the truth value of “B” happens to be. In such cases, at least one of the truth values will allow you to complete the truth table. Choose one, and if it doesn’t work, proceed to try the next one. (See the examples on pp.79-80 of your textbook.)

 

Exercise 3-10 (p.81)

·         [we will do one or two evens in class]

·         do all of these for next time; we’ll check the odds in class

·         make sure you write out the entire short truth table (unlike in the back of the book)

 

 

[5.8.] The Short Truth Table Test for Consistency. [Ch. 3:9]

 

We can also use the short truth table method to demonstrate that the premises of an argument are consistent, i.e., that it is possible for them all to be true at the same time. Use the same steps as those described in the last section, except don’t bother including the conclusion in the top row of your table

 

For example: exercise 3-11 (p.82), #2:

 

(2)                1. ~(F º G)

2. ~(F É H)                  /\ F

 

 

Step 1:

 

 

Step 2:

 

 

Step 3: [no values have been assigned to constants at this point]

 

Step 4:

 

 

 

 

Notice that on this last step, when you assign T to “F” and F to “G”, this is consistent with the truth table definition of the triple-bar. You could not assign the same truth value to both “F” and “G”, since that would make the biconditional true, rendering the entire premise false.

 

Exercise 3-11 (p.82):

·         complete all of these for next time

·         make sure you write out the entire short truth table (unlike in the back of the book)

 

 

Stopping point for Friday February 22. For next time, complete ex. 3-10 and 3-11. No new reading.