Truth Tables and Analyzing Arguments: Examples
Truth Tables
Because complex Boolean statements can get tricky to think about, we can create a truth table to keep track of what truth values for the simple statements make the complex statement true and falseTruth Table
A table showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements.Example 1
Suppose you’re picking out a new couch, and your significant other says “get a sectional or something with a chaise.”This is a complex statement made of two simpler conditions: “is a sectional,” and “has a chaise.” For simplicity, let’s use S to designate “is a sectional,” and C to designate “has a chaise.” The condition S is true if the couch is a sectional.
A truth table for this would look like this:
S | C | S or C |
---|---|---|
T | T | T |
T | F | T |
F | T | T |
F | F | F |
Remember also that or in logic is not exclusive; if the couch has both features, it does meet the condition.
Symbols
The symbol ⋀ is used for and: A and B is notated A ⋀ B.The symbol ⋁ is used for or: A or B is notated A ⋁ B
The symbol ~ is used for not: not A is notated ~A
In the previous example, the truth table was really just summarizing what we already know about how the or statement work. The truth tables for the basic and, or, and not statements are shown below.
Basic Truth Tables
A | B | A ⋀ B |
---|---|---|
T | T | T |
T | F | F |
F | T | F |
F | F | F |
A | B | A ⋁ B |
---|---|---|
T | T | T |
T | F | T |
F | T | T |
F | F | F |
A | ~A |
---|---|
T | F |
F | T |
Example 2
Create a truth table for the statement A ⋀ ~(B ⋁ C)It helps to work from the inside out when creating truth tables, and create tables for intermediate operations. We start by listing all the possible truth value combinations for A, B, and C. Notice how the first column contains 4 Ts followed by 4 Fs, the second column contains 2 Ts, 2 Fs, then repeats, and the last column alternates. This pattern ensures that all combinations are considered. Along with those initial values, we’ll list the truth values for the innermost expression, B ⋁ C.
A | B | C | B ⋁ C |
T | T | T | T |
T | T | F | T |
T | F | T | T |
T | F | F | F |
F | T | T | T |
F | T | F | T |
F | F | T | T |
F | F | F | F |
A | B | C | B ⋁ C | ~(B ⋁ C) |
T | T | T | T | F |
T | T | F | T | F |
T | F | T | T | F |
T | F | F | F | T |
F | T | T | T | F |
F | T | F | T | F |
F | F | T | T | F |
F | F | F | F | T |
A | B | C | B ⋁ C | ~(B ⋁ C) | A ⋀ ~(B ⋁ C) |
T | T | T | T | F | F |
T | T | F | T | F | F |
T | F | T | T | F | F |
T | F | F | F | T | T |
F | T | T | T | F | F |
F | T | F | T | F | F |
F | F | T | T | F | F |
F | F | F | F | T | F |
Implications
Implications are logical conditional sentences stating that a statement p, called the antecedent, implies a consequence q.Implications are commonly written as p → q
Example 3
The English statement “If it is raining, then there are clouds is the sky” is a logical implication. It is a valid argument because if the antecedent “it is raining” is true, then the consequence “there are clouds in the sky” must also be true.Example 4
A friend tells you that “if you upload that picture to Facebook, you’ll lose your job.” There are four possible outcomes:- You upload the picture and keep your job
- You upload the picture and lose your job
- You don’t upload the picture and keep your job
- You don’t upload the picture and lose your job
There is only one possible case where your friend was lying—the first option where you upload the picture and keep your job. In the last two cases, your friend didn’t say anything about what would happen if you didn’t upload the picture, so you can’t conclude their statement is invalid, even if you didn’t upload the picture and still lost your job.
Truth Values for Implications
p | q | p → q |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Example 5
Construct a truth table for the statement (m ⋀ ~p) → rWe start by constructing a truth table for the antecedent.
m | p | ~p | m ⋀ ~p |
T | T | F | F |
T | F | T | T |
F | T | F | F |
F | F | T | F |
m | p | ~p | m ⋀ ~p | r | (m ⋀ ~p) → r |
T | T | F | F | T | T |
T | F | T | T | T | T |
F | T | F | F | T | T |
F | F | T | F | T | T |
T | T | F | F | F | T |
T | F | T | T | F | F |
F | T | F | F | F | T |
F | F | T | F | F | T |
Related Statements
The original implication is “if p then q”: p → qThe converse is “if q then p”: q → p
The inverse is “if not p then not q”: ~p → ~q
The contrapositive is “if not q then not p”: ~q → ~p
Example 6
Consider again the valid implication “If it is raining, then there are clouds in the sky.”The converse would be “If there are clouds in the sky, it is raining.” This is certainly not always true.
The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise, this is not always true.
The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This statement is valid, and is equivalent to the original implication.
Implication | Converse | Inverse | Contrapositive | ||
---|---|---|---|---|---|
p | q | p → q | q → p | ~p → ~q | ~q → ~p |
T | T | T | T | T | T |
T | F | F | T | T | F |
F | T | T | F | F | T |
F | F | T | T | T | T |
Equivalence
A conditional statement and its contrapositive are logically equivalent.The converse and inverse of a statement are logically equivalent.
Arguments
A logical argument is a claim that a set of premises support a conclusion. There are two general types of arguments: inductive and deductive arguments.Argument types
An inductive argument uses a collection of specific examples as its premises and uses them to propose a general conclusion.A deductive argument uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion.
Example 7
The argument “when I went to the store last week I forgot my purse, and when I went today I forgot my purse. I always forget my purse when I go the store” is an inductive argument.The premises are:
I forgot my purse last week
I forgot my purse today
I always forget my purse
Notice that the premises are specific situations, while the conclusion is a general statement. In this case, this is a fairly weak argument, since it is based on only two instances.Example 8
The argument “every day for the past year, a plane flies over my house at 2pm. A plane will fly over my house every day at 2pm” is a stronger inductive argument, since it is based on a larger set of evidence.Evaluating inductive arguments
An inductive argument is never able to prove the conclusion true, but it can provide either weak or strong evidence to suggest it may be true.A deductive argument is more clearly valid or not, which makes them easier to evaluate.
Evaluating deductive arguments
A deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are true, and the conclusion follows necessarily from those premises.Example 9
The argument “All cats are mammals and a tiger is a cat, so a tiger is a mammal” is a valid deductive argument.The premises are:
All cats are mammals
A tiger is a cat
A tiger is a mammal

Analyzing Arguments with Venn Diagrams[2]
To analyze an argument with a Venn diagram- Draw a Venn diagram based on the premises of the argument
- If the premises are insufficient to determine what determine the location of an element, indicate that.
- The argument is valid if it is clear that the conclusion must be true
Example 10
Premise: All firefighters know CPR
Premise: Jill knows CPR
Conclusion: Jill is a firefighter

Since the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.
In addition to these categorical style premises of the form “all ___,” “some ____,” and “no ____,” it is also common to see premises that are implications.
Example 11
Premise: If you live in Seattle, you live in Washington.
Premise: Marcus does not live in Seattle
Conclusion: Marcus does not live in Washington

Example 12
Consider the argument “You are a married man, so you must have a wife.”This is an invalid argument, since there are, at least in parts of the world, men who are married to other men, so the premise not insufficient to imply the conclusion.
Example 13
Consider the argument:Premise: If you bought bread, then you went to the store
Premise: You bought bread
Conclusion: You went to the store
We’ll get B represent “you bought bread” and S represent “you went to the store”. Then the argument becomes:
Premise: B → S
Premise: B
Conclusion: S
B | S | B → S | (B→S) ⋀ B | [(B→S) ⋀ B] → S |
T | T | T | T | T |
T | F | F | F | T |
F | T | T | F | T |
F | F | T | F | T |
Analyzing arguments using truth tables
To analyze an argument with a truth table:- Represent each of the premises symbolically
- Create a conditional statement, joining all the premises with and to form the antecedent, and using the conclusion as the consequent.
- Create a truth table for that statement. If it is always true, then the argument is valid.
Example 14
Premise: If I go to the mall, then I’ll buy new jeans
Premise: If I buy new jeans, I’ll buy a shirt to go with it
Conclusion: If I got to the mall, I’ll buy a shirt.
The premises and conclusion can be stated as:
Premise: M → J
Premise: J → S
Conclusion: M → S
M | J | S | M → J | J → S | (M→J) ⋀ (J→S) | M → S | [(M→J) ⋀ (J→S)] → (M→S) |
T | T | T | T | T | T | T | T |
T | T | F | T | F | F | F | T |
T | F | T | F | T | F | T | T |
T | F | F | F | T | F | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | F | T | T |
F | F | T | T | T | T | T | T |
F | F | F | T | T | T | T | T |
- Technically, "these are Euler circles or Euler diagrams, not Venn diagrams, but for the sake of simplicity we'll continue to call them Venn diagrams." ↵
- Technically, "these are Euler circles or Euler diagrams, not Venn diagrams, but for the sake of simplicity we’ll continue to call them Venn diagrams." ↵