## Complex Truth Tables

Read this tutorial to expand your knowledge of truth-tables. The last tutorial showed you how to construct truth-tables for the basic connectives in sentential logic (SL). This tutorial extends the same technique to more complex well-formed formulas, which approximate the kinds of statements that might be part of an argument in ordinary language.

## §1. What are We Doing?

In this tutorial you will learn how to draw full (or complete) truth-tables for more complex well-formed formulas (WFFs). What we want to do is to use a truth-table to tell us when a well-formed formula (WFF) is true and when it is false. Here is an analogy to help you understand what it is that we want to do.

Suppose we have two arrows and they can only point either up or down. Then there are only four different possible ways they can be aligned:

Now consider the sentence "the left arrow is pointing up and the right arrow is pointing up."

Obviously, the sentence is true in the first situation, and false in the other three situations. If we consider a different sentence, such as "the right arrow is pointing down," then it is true in the second and fourth situation. Finally, what about a sentence such as "the left arrow is point up and the left arrow is pointing down"? This is an inconsistent statement, and clearly it is false in all four situations.

Similarly, when we draw a truth-table for a well-formed formula (WFF) in sentential logic (SL), we are trying to list a set of possible situations to determine when the well-formed formula (WFF) is true (if ever) and when it is false (if ever).

## §2. Drawing a truth-table

To draw a truth-table for a well-formed formula (WFF), follow the following procedure. First, remember that there are four regions in a full truth-table:

 Region #1 Region #2 Region #3 Region #4

To complete the truth-table for a well-formed formula (WFF), you need to fill in the four regions according to this procedure:

#### Step 1

Write down the well-formed formula (WFF) in region #2.

#### Step 2

Identify the sentence letters that appear in the well-formed formula (WFF), and write them down one after the other on a single row in region #1. You only need to write down a sentence letter once even if it has more than one occurrences in the well-formed formula (WFF).

So for example, given the well-formed formula (WFF) "(P→(PvQ))", the truth-table would look like this after the first two steps:

 P   Q (P→(PvQ)) Region #3 Region #4

#### Step 3

Region 3 is a list of all possible combination of truth-values that the sentence letters in region #1 can take. Each row in region #3 specifies a combination, and each combination is called an assignment of truth-values. These assignments correspond to the different possible situations in the arrow example at the top.

If there are n sentence letters in region #1, each of which can have the truth-value T or F, then we know that there are 2n possible assignments of truth-values (21=2, 22=4, 23=8, 24=16, etc.).

So in the truth-table above, there should be four assignments. We write them down one on each row in region #3:

 P   Q (P→(PvQ)) T   T   T   F   F   T  F   F Region #4

On each row, we write down under each sentence letter a truth-value which is the truth-value that the sentence letter receives under that assignment. So "T F" on the second row indicates an assignment where "P" is T and "Q" is F, and "F T" on the third row indicates an assignment where "P" is F and "Q" is T.

In writing down the assignments it is important to use a systematic method to list all of them. Without this method it is easy to miss some of them if there are many rows in the truth-table. When you compare the truth-tables of two or more well-formed formulas (WFFs), it is important to use the same method to list the truth-value assignments in the same order.

The standard method is as follows:

Start with the rightmost sentence letter in region #1. Write "T" down under the letter on the first row of region #3, and then "F" on the second row, alternating for every row until the last row. Then move on to the second sentence letter, and again start with "T." But this time alternate the truth-value only every two rows. If there is a third sentence letter, alternate between "T"s and "F"s every four rows. In general then, for the nth sentence letter (starting from the right hand side), alternate between "T"s and "F"s every 2 n-1 row.

#### Step 4

The remaining task in completing the truth-table is to fill in region #4 by calculating the truth-value of the well-formed formula (WFF) under each assignment.

First, let us define the length of a well-formed formula (WFF) as the number of symbols of sentential logic (SL) it contains. Each occurrence of a connective or sentence letter counts as a single symbol, and the open and close brackets are different symbols.

So "P" has length 1, "~~Q" has length 3, and "((P&Q)→R)" has length 9 (not 10).

Apply this procedure. First, write down beneath each sentence letter of the well-formed formula (WFF) the truth-value it has under each assignment:

 P   Q (P→(PvQ)) T   T   T   F   F   T  F   F T     T  T    T     T  F    F     F  T    F     F  F

Sentence letters are of course well-formed formulas (WFFs) of length 1. We then look for the next shortest well-formed formulas (WFFs) which are part of the whole WFF, and calculate their truth-values under each assignment. The next shortest well-formed formulas (WFFs) should be of length 2, such as "~Q," but since there are no such well-formed formulas (WFFs), the next shortest well-formed formula (WFF) is "(PvQ)" of length 5.

We write down its truth-value for each assignment under its main operator. Remember that the main operator of a well-formed formula (WFF) is the occurrence of a connective that has the widest scope.

 P   Q (P→(PvQ)) T   T   T   F   F   T  F   F T     TTT  T     TTF   F     FTT   F     FFF

We repeat this procedure with the next shortest well-formed formula (WFF) until we are finished with the whole well-formed formula (WFF):

 P   Q (P→(PvQ)) T   T   T   F   F   T  F   F T  T  TTT  T  T  TTF   F  T  FTT   F  T  FFF

The last column of truth-value we write down indicates the truth-values of the whole well-formed formula (WFF) under each of the assignments. This particular truth-table we have completed shows that the well-formed formula (WFF) is true under all the four possible assignments.

If you can mentally carry out calculations of truth-values quickly, you can leave out writing the truth-values of the parts of well-formed formula (WFF) and just write down the truth-value of the whole well-formed formula (WFF) under each assignment.

Then you will end up with this truth-table.

 P   Q (P→(PvQ)) T   T T   F  F   T  F   F T       T       T       T

## §3. Brackets

Now that you know how to construct complex truth-tables, you can see why brackets are necessary. For example, “(A&B→C)” is ambiguous as between “((A&B)→C)” and “(A&(B→C)).”

The truth-tables of the two well-formed formulas (WFFs) are different:

 A B C ((A&B)→C) T T T T T T F F T F T T T F F T F T T T F T F T F F T T F F F T

 A B C (A&(B→C)) T T T T T T F F T F T T T F F T F T T F F T F F F F T F F F F F

Of course, there are cases where the brackets do not matter. For example, it should be obvious that with “P&Q&R&S,” no matter where you place the brackets to turn it into a well-formed formula (WFF), the resulting truth-table is always the same.

Source: Joe Lau and Jonathan Chan, https://philosophy.hku.hk/think/sl/complex.php This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.