Complex Truth Tables
Read this tutorial to expand your knowledge of truthtables. The last tutorial showed you how to construct truthtables for the basic connectives in sentential logic (SL). This tutorial extends the same technique to more complex wellformed formulas, which approximate the kinds of statements that might be part of an argument in ordinary language.
Complete the exercises for this tutorial and check your answers.
§1. What are We Doing?
In this tutorial you will learn how to draw full (or complete) truthtables for more complex wellformed formulas (WFFs). What we want to do is to use a truthtable to tell us when a wellformed 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 truthtable for a wellformed formula (WFF) in sentential logic (SL), we are trying to list a set of possible situations to determine when the wellformed formula (WFF) is true (if ever) and when it is false (if ever).
§2. Drawing a truthtable
To draw a truthtable for a wellformed formula (WFF), follow the following procedure. First, remember that there are four regions in a full truthtable:
Region #1  Region #2 
Region #3 
Region #4 
To complete the truthtable for a wellformed formula (WFF), you need to fill in the four regions according to this procedure:
 Step 1
Write down the wellformed formula (WFF) in region #2.
 Step 2
Identify the sentence letters that appear in the wellformed 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 wellformed formula (WFF).
So for example, given the wellformed formula (WFF) "(P→(PvQ))", the truthtable 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 truthvalues 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 truthvalues. 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 truthvalue T or F, then we know that there are 2^{n} possible assignments of truthvalues (2^{1}=2, 2^{2}=4, 2^{3}=8, 2^{4}=16, etc.).
So in the truthtable above, there should be four assignments. We write them down one on each row in region #3:


T F F T F F 

On each row, we write down under each sentence letter a truthvalue which is the truthvalue 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 truthtable. When you compare the truthtables of two or more wellformed formulas (WFFs), it is important to use the same method to list the truthvalue 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 truthvalue 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 ^{n1} row.
 Step 4
The remaining task in completing the truthtable is to fill in region #4 by calculating the truthvalue of the wellformed formula (WFF) under each assignment.
First, let us define the length of a wellformed 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 wellformed formula (WFF) the truthvalue it has under each assignment:


T F F T F F 
T T F F F T F F F 
Sentence letters are of course wellformed formulas (WFFs) of length 1. We then look for the next shortest wellformed formulas (WFFs) which are part of the whole WFF, and calculate their truthvalues under each assignment. The next shortest wellformed
formulas (WFFs) should be of length 2, such as "~Q," but since there are no such wellformed formulas (WFFs), the next shortest wellformed formula (WFF) is "(PvQ)" of length 5.
We write down its truthvalue for each assignment under its main operator. Remember that the main operator of a wellformed formula (WFF) is the occurrence of a connective that has the widest scope.


T F F T F F 
T TTF F FTT F FFF 
We repeat this procedure with the next shortest wellformed formula (WFF) until we are finished with the whole wellformed formula (WFF):


T F F T F F 
T T TTF F T FTT F T FFF 
The last column of truthvalue we write down indicates the truthvalues of the whole wellformed formula (WFF) under each of the assignments. This particular truthtable we have completed shows that the wellformed formula (WFF) is true under all
the four possible assignments.
If you can mentally carry out calculations of truthvalues quickly, you can leave out writing the truthvalues of the parts of wellformed formula (WFF) and just write down the truthvalue of the whole wellformed formula (WFF) under each assignment.
Then you will end up with this truthtable.


T F F T F F 
T T T 
§3. Brackets
Now that you know how to construct complex truthtables, you can see why brackets are necessary. For example, “(A&B→C)” is ambiguous as between “((A&B)→C)” and “(A&(B→C)).”
The truthtables of the two wellformed 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 wellformed formula (WFF), the resulting truthtable is always the same.
Source: Joe Lau and Jonathan Chan, https://philosophy.hku.hk/think/sl/complex.php
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