5.2 Truth Tables for Conjunctions and Disjunctions
This sub-module discusses how to create truth tables for conjunctions, disjunctions, and combinations of conjunctions, disjunctions, and negations, as well as how to create truth tables for any number of atomic sentences and logical connectives.
Table of Contents
- 5.2 Truth Tables for Conjunctions and Disjunctions
5.2.1 Truth Tables for Disjunctions
The door is locked if either the top lock or the bottom lock is locked.
The Truth Table for Disjunctions
A disjunction is false when both disjuncts are false, and true otherwise. That means there are three situations in which disjunctions are true: when both disjuncts are true, when the left one is true but not the right one, and when the right one is true but not the left one.
Many people find it tricky to remember that “P or Q” is still true when both P and Q are true. If it helps, you can imagine a disjunction working like a door which has two locks. If either lock is locked, then the door is locked. If both locks are locked, the door is locked. But if neither lock is locked, then the door is not locked.
We can model this with a truth table:
| P | Q | P v Q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Notice that the truth table has four rows instead of two. That is because there are two atomic sentences in this truth table instead of one. Each of those sentences has two possible truth values, and 2 x 2 = 4, so there are 4 possible combinations of truth values which the pair of sentences might have.
To make sure we include every possible combination of truth value assignments, we follow a pattern where in the first column we assign ‘T’ to the top half of the rows to create a section of Ts, and ‘F’ to the bottom half of the rows to create a section of F’s; in the next column, we assign ‘T’ to the top half of each ‘T’ or ‘F’ section, and ‘F’ to the bottom half of each ‘T’ or ‘F’ section. Without following this pattern, there is no guarantee we would include every possible combination of truth values, or every set of “possible worlds”.
In the right-most column, we assign ‘P v Q’ the value T in every row where there is a ‘T’ under P, and in every row where there is a ‘T’ under Q; we assign ‘P v Q’ the value F in every row where there are Fs under both P and Q.
Truth Tables with Disjunctions
We can now create truth tables for complex sentences with combinations of disjunctions and negations. We put the sentence we want to create a truth table for on the right-most side, and then slowly break down the sentence into its component parts as we move from right to left, until we get to the atomic sentences on the left-most side. For instance, here’s the start of the truth table for the sentence: ~(P v ~Q) v P
| P | Q | ~Q | P v ~Q | ~(P v ~Q) | ~(P v ~Q) v P |
| T | T | ||||
| T | F | ||||
| F | T | ||||
| F | F |
Notice how there is one column for every atomic sentence (P and Q), and one column for every logical connective, building from less to more complex: ~Q, P v ~Q, ~(P v ~Q), and ~(P v ~Q) v P.
We now move from left to right through the truth table. ~Q flips the truth value of Q:
| P | Q | ~Q | P v ~Q | ~(P v ~Q) | ~(P v ~Q) v P |
| T | T | F | |||
| T | F | T | |||
| F | T | F | |||
| F | F | T |
P v ~Q is T when the value under P is T, or the value under ~Q is T; it is F when the value under both the P and ~Q columns is F:
| P | Q | ~Q | P v ~Q | ~(P v ~Q) | ~(P v ~Q) v P |
| T | T | F | T | ||
| T | F | T | T | ||
| F | T | F | F | ||
| F | F | T | T |
~(P v ~Q)flips from T to F and from F to T the truth value of the column under P v ~Q:
| P | Q | ~Q | P v ~Q | ~(P v ~Q) | ~(P v ~Q) v P |
| T | T | F | T | F | |
| T | F | T | T | F | |
| F | T | F | F | T | |
| F | F | T | T | F |
Lastly, we have ~(P v ~Q) v P. The main connective in this column is v. There are many connectives in this sentence, but the “main connective” is the one we want to focus on, because every other part of the sentence already appears in the truth table. Since the main connective is v, the sentence is T whenever there is a T in either the ~(P v ~Q) column or the P column, and F when there is an F in both the ~(P v ~Q) column and the P column.
| P | Q | ~Q | P v ~Q | ~(P v ~Q) | ~(P v ~Q) v P |
| T | T | F | T | F | T |
| T | F | T | T | F | T |
| F | T | F | F | T | T |
| F | F | T | T | F | F |
Now that the truth table is complete, what have we learned? We’ve learned that the sentence ~(P v ~Q) v P is false when both P and Q are false, and true otherwise. So, ~(P v ~Q) v P is logically equivalent to P v Q, even though it looks very different, because it has the same truth values in the rightmost column of its truth table.
5.2.2 Truth Tables for Conjunctions
If both switches are on, the light is on, but if either is off, the light is off.
The Truth Table for Conjunctions
A conjunction is true when both conjuncts are true, and false otherwise otherwise. That means there are three situations in which conjunctions are false: when both conjuncts are false, when the left one is true but not the right one, and when the right one is true but not the left one. There is only one situation in which conjunctions are true.
We can model this with a truth table:
| P | Q | P & Q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
If it helps, imagine a conjunction like a pair of light-switches, where both need to be on in order for the light to turn on, and if either one is off, or both are off, then the light is off.
Truth Tables with Conjunctions
We can now create truth tables with a combination of conjunctions, disjunctions, and negations. Here is a truth table for the sentence ~(F & ~G). Notice that the sentence has two atomic sentences, F and G, so it will have four rows. It has three logical connectives: ~(F & ~G), F & ~G, and ~G. So, it will have 3 columns for logical connectives, plus 2 columns for atomic sentences, for a total of 5 columns:
| F | G | ~G | F & ~G | ~(F & ~G) |
| T | T | |||
| T | F | |||
| F | T | |||
| F | F |
First, we calculate ~G by flipping the truth value of the G column:
| F | G | ~G | F & ~G | ~(F & ~G) |
| T | T | F | ||
| T | F | T | ||
| F | T | F | ||
| F | F | T |
Next, F & ~G is true when both F and ~G are true:
| F | G | ~G | F & ~G | ~(F & ~G) |
| T | T | F | F | |
| T | F | T | T | |
| F | T | F | F | |
| F | F | T | F |
Lastly, ~(F & ~G) flips the truth value of F & ~G
| F | G | ~G | F & ~G | ~(F & ~G) |
| T | T | F | F | T |
| T | F | T | T | F |
| F | T | F | F | T |
| F | F | T | F | T |
So, this tells us that the sentence ~(F & ~G) is true when F and G are both true (first row), when F is false and G is true (third row), and when F is false and G is false (fourth row). It says that the sentence is false when F is true and G is false (second row).
5.2.3 Truth Tables for Complex Conjunctions and Disjunctions
Electrical panels operate on the same logical principles as truth tables.
More Complex Sentences
So far, we’ve seen truth tables with two atomic sentences, and a combination of negations and either conjunctions or disjunctions. Now, let’s look at an example of a more complex truth table. Let’s create a truth table for the following sentence:
Either Lawrence isn’t quiet, or else it’s not both the case that Kansas City is spooky and that Joplin is friendly.
First, let’s symbolize the atomic sentences:
Let
L = Lawrence is quiet.
K = Kansas City is spooky.
J = Joplin is friendly.
The main connective is “or”:
Lawrence isn’t quiet v it’s not both the case that Kansas City is spooky and that Joplin is friendly.
We can replace K and J in the sentence, and we know that, “Lawrence isn’t quiet” is ~L:
~L v it’s not both the case that K and that J
Recall that “not both P and Q” is symbolized ~(P & Q). So we get:
~L v ~(K & J) = “Either Lawrence isn’t quiet, or else it’s not both the case that Kansas City is spooky and that Joplin is friendly.”
Creating a Truth Table with Three Atomic Sentences
Now, we create a truth table for:
~L v ~(K & J)
The purpose of a truth table is to tell us when a sentence would be true, and when it would be false. So, we need to make sure to represent every possible combination of truth values for our three atomic sentences: L, K, J. This means we will need 2 x 2 x 2 = 8 rows in the truth table.
The sentence has 4 logical connectives: ~L, (K & J), ~(K & J) and ~L v ~(K & J). So, we will need 4 columns for the logical connectives, plus 3 for the atomic sentences, for a total of 7 columns:
| L | K | J | ~L | K & J | ~(K & J) | ~L v ~(K & J) |
The first step is to write all of the possible combinations of truth values under the atomic sentences. In the left-most column, under L, we’ll put T in the top half, and F in the bottom half of the rows, to create a T section and an F section:
| L | K | J | ~L | K & J | ~(K & J) | ~L v ~(K & J) |
| T | ||||||
| T | ||||||
| T | ||||||
| T | ||||||
| F | ||||||
| F | ||||||
| F | ||||||
| F |
Now, in the second column, under K, we’ll put T in the top half and F in the bottom half of each of the sections of the column to its left:
| L | K | J | ~L | K & J | ~(K & J) | ~L v ~(K & J) |
| T | T | |||||
| T | T | |||||
| T | F | |||||
| T | F | |||||
| F | T | |||||
| F | T | |||||
| F | F | |||||
| F | F |
Now, in the third column, under J, we’ll put T in the top half and F in the bottom half of each of the sections of the column to its left:
| L | K | J | ~L | K & J | ~(K & J) | ~L v ~(K & J) |
| T | T | T | ||||
| T | T | F | ||||
| T | F | T | ||||
| T | F | F | ||||
| F | T | T | ||||
| F | T | F | ||||
| F | F | T | ||||
| F | F | F |
We now have represented all 8 unique possible combinations of truth values. Now, it’s time to calculate the truth values under the fourth column, for ~L. To do this, we simply flip the truth values of the column under L from T to F and from F to T:
| L | K | J | ~L | K & J | ~(K & J) | ~L v ~(K & J) |
| T | T | T | F | |||
| T | T | F | F | |||
| T | F | T | F | |||
| T | F | F | F | |||
| F | T | T | T | |||
| F | T | F | T | |||
| F | F | T | T | |||
| F | F | F | T |
Next, we have the fifth column, under K & J. Here, we write a T when there is a T under both the K column and the J column, and an F when there is an F under either or both the K or J columns. Notice that there are only two ‘T’ truth values, in the first and fifth rows.
| L | K | J | ~L | K & J | ~(K & J) | ~L v ~(K & J) |
| T | T | T | F | T | ||
| T | T | F | F | F | ||
| T | F | T | F | F | ||
| T | F | F | F | F | ||
| F | T | T | T | T | ||
| F | T | F | T | F | ||
| F | F | T | T | F | ||
| F | F | F | T | F |
The sixth column, under ~(K & J), is simply the negation of K & J. So, we flip the truth value of K & J from T to F or from F to T:
| L | K | J | ~L | K & J | ~(K & J) | ~L v ~(K & J) |
| T | T | T | F | T | F | |
| T | T | F | F | F | T | |
| T | F | T | F | F | T | |
| T | F | F | F | F | T | |
| F | T | T | T | T | F | |
| F | T | F | T | F | T | |
| F | F | T | T | F | T | |
| F | F | F | T | F | T |
The last column, under ~L v ~(K & J), has v as its main connective. So, we need to look at two columns: the column under ~L, and the column under ~(K & J). If there is a T under either or both of those columns, we write a T. If there is an F under both columns, then we write an F.
| L | K | J | ~L | K & J | ~(K & J) | ~L v ~(K & J) |
| T | T | T | F | T | F | F |
| T | T | F | F | F | T | T |
| T | F | T | F | F | T | T |
| T | F | F | F | F | T | T |
| F | T | T | T | T | F | T |
| F | T | F | T | F | T | T |
| F | F | T | T | F | T | T |
| F | F | F | T | F | T | T |
The truth table is done! What does this tell us? It tells us that the original sentence, ~L v ~(K & J), is false in only one situation: row 1, when L and K and J are all true; it is true in every other case (rows 2-8).
5.2.4 Truth Tables without Limits
Truth tables rapidly expand without limits.
Calculating Rows
We’ve seen so far how to create truth tables with 2 or 3 atomic sentences, and a few logical connectives. But what if we were faced with a very long sentence, like this one?
(E v C) & (~N v ~(M & ~(P & ~(~E v ~(C & ~(G v ~C))))))
The principles will remain the same as those we followed before. First, we count the number of atomic sentences. We only count each atomic sentence once, no matter how many times it occurs. There are six:
E, C, N, M, P, G
There are 2 truth values, T and F, so the number of rows in a truth table is 2 multiplied by itself for as many atomic sentences as appear in the truth table. Since there are six atomic sentences, the number of rows will be 2 multiplied by itself six times, or 2 x 2 x 2 x 2 x 2 x 2, or 26. Using a calculator, we can see that 26 = 64. So, there will be 64 rows.
We can generalize this principle:
The number of rows in a truth table is 2n, where n = the number of atomic sentences (each counted only once).
To make sure that we represent every possible truth value, we follow the pattern we established earlier. The first half (top 32 rows) of the left-most column will be assigned T, and the bottom half (bottom 32 rows) will be assigned F, creating two sections. The second column will assign T to the top half (top 16 rows) and F to the bottom half (bottom 16 rows) of each section in the column to its left, creating four sections. The third column will assign T to the top half (top 8 rows) and F to the bottom half (bottom 8 rows) of each section in the column to its left, creating 8 sections. This pattern will continue until in the sixth column, under G, we have alternating T and F truth values. To generalize this principle, under each atomic sentence:
Assign T to the top half and F to the bottom half of each section in the column to the left.
Calculating Columns
How many columns will we need? Well, we’ll need 6 columns for atomic sentences. We’ll also need 1 for each logical connective. When counting logical connectives, we do count each time the connective occurs separately. Let’s bold the logical connectives to count them:
(E v C) & (~N v ~(M & ~(P & ~(~E v ~(C & ~(G v ~C))))))
There are 16 logical connectives. So, we’ll have 6 + 16 columns in the truth table, or 22 total. We can generalize this principle:
A reliable guide to the number of columns in a truth table is n + m, where n = the number of atomic sentences (each counted only once), and m = the number of logical connectives (each time they occur counts).
Note that this rule sometimes requires more columns than strictly necessary (in cases where an atomic letter occurs twice or more), but if you follow this rule you’ll never have too few columns. It is a safe rule to guarantee that you never have too few columns.
When arranging columns, the sentence as a whole will be in the right-most column, and then we’ll arrange them from right-to-left by creating the sentence down bit by bit into simpler parts. Whenever a sentence is a part of another sentence, the parts should be on the left and the whole should be on the right. For instance, ~(G v ~C) should be to the right of G v ~C, which should be to the right of ~C.
Lastly, we’ll calculate the truth values under each column based on the main connective in that column. The main connective is the one which applies to the largest parts of the the sentence. For instance, the main connective of ~(G v ~C) is ~, because ~ applies to the largest part (G v ~C); the main connective of G v ~C is v, because that connects the largest parts (G, ~C); the main connective of ~C is of course just ~.
To determine the main connective, break the sentence into parts. Then, determine which connective applies to or connects the largest part of the sentence.
Parentheses are used for grouping, but they do not count as logical connectives, and are never the main connective.
Infinite Conjunctions and Disjunctions
Lastly, we can observe an important principle about a long string of atomic sentences linked only by conjunctions, or linked only by disjunctions.
A long string atomic sentences of linked only by conjunctions, including an infinitely long one (A & B & C & D & . . . etc.), when there are no other logical connectives besides &, is going to have a T truth value only in the first row, when every single atomic sentence is true:
| A | B | C | D | etc… | A & (B & (C & …. |
| T | T | T | T | T | T |
| T | T | T | T | F | F |
| T | T | T | F | T | F |
| etc… | etc… | etc… | etc… | etc… | F, etc…. |
On the other hand, a long string atomic sentences linked only by disjunctions, including an infinitely long one (A v B v C v D v . . . etc.), when there are no other logical connectives besides v, is going to have a T truth value in every row except the last row, where it will be F, when every single atomic sentence is false:
| A | B | C | D | etc… | A v (B v (C v …. |
| etc… | etc… | etc… | etc.. | etc.. | etc… |
| F | F | F | T | F | T |
| F | F | F | F | T | T |
| F | F | F | F | F | F |
So, we can conclude with these general principles:
a conjunction of any length is true when all of its conjuncts are true
a disjunction of any length is true when some (at least one) of its disjuncts are true
All that is needed to prove a conjunction of any length false is that one of its conjuncts is false; on the other hand, proving a disjunction of any length false requires proving that every single disjunct is false.
Because of this, we can represent the sentence form “Some S are P”, known as an existential claim, as a complete series of disjunctions using all the members of the set of things which are S:
Some S are P = S1 is P or S2 is P or S3 is P or …
For example, suppose Pat has 67 cats. The claim “Some of Pat’s cats are happy” is logically equivalent to the claim that “Either Pat’s 1st cat is happy, or Pat’s 2nd cat is happy, or Pat’s 3rd cat is happy, or . . . , or Pat’s 66th cat is happy, or Pat’s 67th cat is happy”.
Similarly, we can represent the sentence form “All S are P”, known as a universal claim, as a complete series of conjunctions using all the members of the set of things which are S:
All S are P = S1 is P and S2 is P and S3 is P and…
Again, considering Pat’s 67 cats, the claim “All of Pat’s cats are happy” is logically equivalent to the claim that “Pat’s 1st cat is happy, and Pat’s 2nd cat is happy, and Pat’s 3rd cat is happy, and . . . , and Pat’s 66th cat is happy, and Pat’s 67th cat is happy”.
Submodule 5.2 Quiz
Licenses and Attributions
Key Sources:
- Watson, Jeffrey (2019). Introduction to Logic. Licensed under: (CC BY-SA).
Next Page: 5.3 Rules for Conjunctions and Disjunctions