OCR Document - Jeff's Readings
T-reading: Logic: Constructing Truth Tables
(Guttenplan, 71-89)
A truth table tells you when a given non-basic sentence is true (or false) on the basis of information about when each basic sentence is true (or false). The complete list of all possible combinations of truth values of the basic sentences is written in tabular form on the left side of the double line, and the truth values of the non-basic sentence for each possible combination (each row) are written on the right. This procedure applies both to 'simple' non-basic sentences, such as P & Q, and to more complicated examples such as this:
R & (O ۷ L)
The main difference between this non-basic sentence and those of Primitive is that it contains 'parts' which are themselves non-basic sentences, and it also contains three basic sentences: R, O, and L. This difference means that there will be more work involved in writing out the truth table for it, but no new ideas are needed.
What you have to do first is to figure out a way of writing all the possible combinations of truth values of the three basic sentences. Given two basic sentences, there are four possible combinations (rows) to the truth table. This is because each basic sentence can be either true or false and there are two basic sentences. If you cannot see right off why these considerations determine the four possible combinations, you should look at the truth table for ‘P & Q’. A row this truth table is made up of a truth value of P and a truth value Q. There are as many rows as there are different combinations truth values for these two sentences. Since each sentence could T or F, you can tell by trial and error that there are four distinct combinations, and that these are correctly given in the table.
Precisely the same idea should lead you to work out how many rows are in a truth table that is constructed around three basic sentences. The only trouble is that trial and error is both a trial and can all too easily lead to error. What is wanted is a way of telling straight off how many rows there are in the truth table and how to write them.
Two basic sentences generate four rows: the number of rows is by this arithmetical formula:
Number of rows = 2n
“n” is the number of different basic sentences occurring in the sentence for which the truth table is being constructed, and 2n is 2 multiplied by itself n times. Thus, our sentence above, which has three basic sentences, requires a truth table with 23 or 8 rows. There are eight possible combinations of truth values for the three basic sentences. (For the math-impaired, you could also just memorize that 1 basic sentence requires 2 rows, 2 basic sentences requires 4 rows, 3 requires 8, 4 requires 16, and so on.) All that remains is to work out how to write them down.
The method I will use is summarized in the following steps:
Determine the number of rows in the truth table using the formula: number of rows = 2n, where n is the number of different basic sentences in the non-basic sentence for which the truth table is being constructed.
Starting with the column under the basic sentence letters that comes last alphabetically, fill in alternating truth values: true, false, true, false, going down the column under the letter.
Under the basic sentence letter that comes next-to-last alphabetically, fill in the column alternating true and false every two rows: true, true, false, false, true, true, false, false, etc.
Then the third to last letter alphabetically will alternate every four rows. The fourth will alternate every eight rows, etc. Follow this method until all of your basic sentence letters are filled in.
The result of this construction is that each row of the table will be distinct from the others, and the table will contain all the possibilities.
As is so often the case with things in logic, the explanation can seem more difficult than what it seeks to explain. The reason for this is not hard to appreciate: what is wanted is a completely general account of the method, one which works for any number of basic sentences. This is given by steps (a)-(e) above. However, when the method is applied to a specific example, it is much easier to cope with. Here is the method applied to our original example sentence with the steps labeled as in the general description given above:
Our sentence, R & (O ۷ L), has three basic sentences, so our truth table for it will require 8 rows.
Starting under the R, because it is alphabetically last, we will alternate, true, then false, every row:
| |R |& |(O |۷ |L) | |
| |T | | | | | |
| |F | | | | | |
| |T | | | | | |
| |F | | | | | |
| |T | | | | | |
| |F | | | | | |
| |T | | | | | |
| |F | | | | | |
The next to last letter, alphabetically, is the O, so it will alternate every two rows:
| |R |& |(O |۷ |L) | |
| |T | |T | | | |
| |F | |T | | | |
| |T | |F | | | |
| |F | |F | | | |
| |T | |T | | | |
| |F | |T | | | |
| |T | |F | | | |
| |F | |F | | | |
Our next letter, L, alternates every four rows:
| |R |& |(O |۷ |L) | |
| |T | |T | |T | |
| |F | |T | |T | |
| |T | |F | |T | |
| |F | |F | |T | |
| |T | |T | |F | |
| |F | |T | |F | |
| |T | |F | |F | |
| |F | |F | |F | |
In essence, the method given in steps (a)-(e) above is merely a technique (or trick) for writing out all the possible combinations of truth values for any number of basic sentences. As you have seen, it is cumbersome to explain but, once you have seen it done once or twice, it is easily mastered.
We have completed the construction stage. We will now develop methods for testing the validity of arguments formulated in Sentential.
Truth Tables and Validity
Consider this simple English argument:
If I am seven years-old or teach philosophy, then I have an immature sense of humor.
I teach philosophy.
Therefore, I have an immature sense of humor.
This is a fairly straightforward set of sentences, so I take it that the translating into Sentential will not be too difficult. I will need basic sentence letters for the phrases, “I am seven years-old,” “I teach philosophy,” and “I have an immature sense of humor.” I always like to let my letters stand for key words in the sentences to make them sound more natural. So, let’s pick the letters, S, P, and I, respectively, to stand for these three basic sentences.
As such, the translation should be rather clear:
(S ۷ P) → I
P
-------------
I
We already know enough to be able to work out a mechanical and effective method of deciding whether it is valid. Moreover, the method will be applicable to any argument formulated in Sentential. All you have to do is to focus for a moment on the notion of validity which was discussed at some length earlier.
A valid argument is an argument in which it is not possible for the premises to be true and the conclusion false.
We just worked out a procedure for writing the truth table for any sentence of Sentential. Such a table gives the truth values of any sentence in all possible circumstances. This invites the idea that we should extend the construction of truth tables from sentences to arguments.
Our argument contains three different basic sentences. A truth table with eight rows will list all the possible ways for these sentences to be true or false. In turn, we can use these truth values to work out the truth values of “(S ۷ P) → I”, “P”, and “I” in all possible circumstances, and include them in the table. In this way, the final table will list the truth values of both the premises and the conclusion in all possible circumstances. By inspecting the rows of this table, it will be an easy matter to determine, if there are any circumstances in which the premises are true and the conclusion false. If there are such circumstances, the argument is invalid; and if not, then the argument is valid. For clarity, I will separate the premises by a single “/” and the conclusion by a “//”. Here is the truth table with all of the possible truth combinations written in:
|(S |۷ |P) |→ |I |/ |P |// |I | | | |T | |T | |T | |T | |T | | | |F | |T | |T | |T | |T | | | |T | |F | |T | |F | |T | | | |F | |F | |T | |F | |T | | | |T | |T | |F | |T | |F | | | |F | |T | |F | |T | |F | | | |T | |F | |F | |F | |F | | | |F | |F | |F | |F | |F | | |
Here is a step-by-step procedure for filling in the possible truth-values of each of the sentences in an argument:
After assigning the truth-values under all of the basic sentences, start solving the truth values of the connectives that have the smallest scope. Then, work up to the connectives with larger and larger scope, until you have solved the truth-values under the main connectives for each sentence in the argument.
So, in this argument, if we wanted to go left to write, we would start with the “۷” in the first premise. We need to remind ourselves of the truth table for disjunction. Basically, it says that “۷” is only false when both sides are false. In this case, that means only rows 4 and 8 will be False under the “۷”, as this table shows:
|(S |۷ |P) |→ |I |/ |P |// |I | | | |T |T |T | |T | |T | |T | | | |F |T |T | |T | |T | |T | | | |T |T |F | |T | |F | |T | | | |F |F |F | |T | |F | |T | | | |T |T |T | |F | |T | |F | | | |F |T |T | |F | |T | |F | | | |T |T |F | |F | |F | |F | | | |F |F |F | |F | |F | |F | | |
This new column under the “۷” tells us all of the possible truth-values for the sentence “S۷P”. Now we can move up to the connective with the next widest scope, which, in this case, will be the main connective “→”. The values on the left side of the “→” are the truth-values we just placed under the “۷”, and the values on the right side of the “→” are the truth-values under the basic sentence “I”.
So, now we need to remind ourselves of the truth table for the conditional. It said, basically, that “→” is only False when the antecedent is True and the consequent False. In our table, we find this situation in rows 5, 6, and 7, as shown:
|(S |۷ |P) |→ |I |/ |P |// |I | | | |T |T |T |T |T | |T | |T | | | |F |T |T |T |T | |T | |T | | | |T |T |F |T |T | |F | |T | | | |F |F |F |T |T | |F | |T | | | |T |T |T |F |F | |T | |F | | | |F |T |T |F |F | |T | |F | | | |T |T |F |F |F | |F | |F | | | |F |F |F |T |F | |F | |F | | |
With solving the column under the “→”, because this is the main connective of that sentence, we have found the final truth values in all eight situations for that premise. Now, because the second premise and the conclusion are basic sentences, we already have their possible truth values filled in. (It is a good practice to circle or bold the truth-values under the main connective, as I have done in the table above.)
Thus, we are ready to use our truth table to test the argument’s validity. What we are looking for is the possible situation, or row, in which the premises are true at the same time that the conclusion is false. In short, we are simply looking for the situations that we tried to visualize when we first tried to assess the validity of deductive arguments. In looking over our full truth table, we find that there is no such sentence. Therefore, this table has proven that this argument is valid. It was not possible to prove it invalid, so it must be valid.
The truth table method for testing arguments is an algorithm. It gives you an effective and mechanical way of telling whether any argument in Sentential is valid. The care with which we discussed the construction of truth tables has paid off. It requires nothing but patience to construct a truth table for any argument in Sentential; the same procedure is employed each time, and by going through the finite number of steps carefully, you know that you will finish the job. Moreover, when you have completed a truth table for an argument, the checking procedure is just as mechanical.
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