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PHIL310 – Lecture 6 – Truth Tables

Now that we have a full set of symbols for your logical toolkit (~, v, &, →), we can dive deeper into real-life arguments to symbolize them and evaluate their strength. This lesson will review truth tables to ensure we are fully prepared to break down career-related examples and prove that an argument is a fallacy.

Imagine we examine a truth table for a simple example such as “If I like apples, then I like bananas.” It would look like this. A = “I like apples” and B = “I like bananas”.

It is important that we first review what a truth table is showing. A truth table shows all possible truth value combinations of a statement to examine when the entire statement is true or false. That’s why we set up the first columns as the variables and build up to the final statement. The four rows represent the four different ways the full sentence “A → B” can occur. The truth values for A and B simply represent the four combinations. This comes together when examining the final statement “A → B”. We must use our truth conditions to determine that there is only one time when the entire statement is false. That is the instance where A is true, and B is false.

This applies to more complex sentences as well. When we have a more complex sentence like “~Q → ~P”, we start by building up the final statement by creating all the components. In this case, we start with the variables P and Q, and then we expand to ~Q and ~P, and then we add the → connective to complete the full sentence. Below is the completed truth table to show how we built up to it.

Like the previous example, it is important to recognize what this table shows. In this case, the entire statement “~Q → ~P” is only false in one instance. This occurs when P is true, and Q is false. As we continue to do more complex examples using real-life sentences, you can prove an argument’s validity simply by using a truth table.

Let’s take a real-life example and evaluate whether the reasoning is flawed. Imagine during a meeting at work, the CEO shares the following: “If Bob is a good leader, then Bob will have motivated employees. Bob is a good leader. Therefore, Bob has motivated employees.” Let’s use a truth table to figure out if this is valid or invalid. First, let’s convert this example into premise-premise-conclusion form. After this, we can break it down into symbolic form and put it into a truth table.

P1) If Bob is a good leader, then Bob will have motivated employees.

P2) Bob is a good leader.

C) Therefore, Bob has motivated employees.

At this point, we can assign variables to each sentence. We can assign B = “Bob is a good leader” and M = Bob has motivated employees.” Now, we can apply our symbolic knowledge to convert this argument into symbols. In P1, we can see an “if-then” statement. That is symbolized as “B → M”. P2 is simply B, and the conclusion is M. Therefore, we can rewrite the entire argument in symbolic form. We can add the verbal counterpart for clarity in parenthesis for this example.

P1) B → M (If Bob is a good leader, then Bob will have motivated employees.)

P2) B (Therefore, Bob is a good leader.)

C) M (Bob has motivated employees.)

Recall that an argument can be proven to be invalid if the premises are true and the conclusion is false. We can use a truth table to prove if such a case exists. It is an extension of our current truth table knowledge, but if we go step by step, it is manageable.

When trying to prove an argument as a fallacy or that it is invalid, one must write out a truth table where the premises and the conclusion are columns. That is slightly different from what has been done before, but you will see that it is a natural extension of using the truth table. Let’s look at one way to set up the frame of the truth table. We will use the same argument as above.

P1) B → M (If Bob is a good leader, then Bob will have motivated employees.)

P2) B (Therefore, Bob is a good leader.)

C) M (Bob has motivated employees.)

Normally, we can set up our variables as the first columns in our truth table and represent the four possible outcomes. That would look like this:

Now, we want to create a column for each premise and the conclusion. Without filling it out, it would look like this. We can add a header row at the top to make it clear.

This truth table symbolizes the entire argument and is ready to be filled in. To fill in each column, we simply must apply the truth conditions learned earlier in the course. For the first premise, “B → M,” we can apply the truth conditions for →. That is, if the antecedent is true and the consequent is false, the entire “if-then” statement is false. Otherwise, it is true. Let’s fill that in.

Now, premise 2 (B) happens to already be present in our table. We can simply copy the truth values from the ‘B’ column into our premise 2 column. Since the variable is the same, it will have identical truth values in the table each time it is used. The same applies to our conclusion column. The column for our conclusion (M) is the same as our second column (M), so we will copy the truth values since there is no further truth condition that needs to be applied.

We have a complete truth table. However, what does it show? Recall that this truth table symbolizes an argument and not just a statement. Therefore, we want to check to see if this argument is invalid. That means we are looking for a place where P1 and P2 are true, but the conclusion is false. In this example, we do not see such a case. Therefore, the argument is valid.

To recap, we took a real-life argument and proved that it was valid. We did that by breaking it down into symbolic forms and then creating a truth table. We listed out all the possible combinations and did not find a single instance where the premises are true, and the conclusion is false. As we continue to work through examples, all the symbolization you’ve learned will be used to break down everyday arguments to determine validity.

Let’s review a few takeaways from this lesson:

An invalid argument is one where the premises are true, but the conclusion is false.

When using a truth table to prove invalidity, list the premises and conclusion as columns.

Apply the truth conditions to fill out the truth table to determine validity.

Thank you! You have completed this lesson. Please scroll down to complete a short, ungraded Knowledge Check activity.

Check Your Knowledge 1

True or False: If the conclusion is false, the argument is invalid.

Answer: False. The premises must be true, AND the conclusion has to be false.

True or False: The following argument is invalid:

Answer: True. In the second row, P1 is true, P2 is true, but C is false.

True or False: The following argument is invalid:

Answer: False. There is no instance where the premises are true, but the conclusion is false.

Converting real-life examples into a truth table and then determining validity is not an easy task. However, if one follows the same procedure outlined above and uses the truth conditions learned earlier in the course, it is manageable. This lesson will take a more complex example and show how to systematically break it down and determine validity.

Take a more complicated example from the workplace. Suppose you and Mary are working on a project together for your boss. It is due by 5 p.m. on Friday. It’s your job to submit it on time, but you got so caught up reviewing and finalizing the document that you did not send it to your boss until 6 p.m. On Wednesday morning, Mary angrily approaches you and says: “If our presentation was submitted on time, our boss would have liked it. You submitted our presentation late. Therefore, our boss did not like our presentation.” Is this strong reasoning, or is there a flaw in Mary’s argument? We will have to break it down and use a truth table to figure it out. Let’s first state the argument in premise-premise-conclusion form for clarity. We can put the argument like this.

Mary’s Argument Verbally

P1) If our presentation had been submitted on time, our boss would have liked it.

P2) Our presentation was not submitted on time.

C) Therefore, our boss did not like it.

Examining the argument, we see that we can begin assigning variables to statements. We can assign P = “The presentation was submitted on time” and L = “The boss liked the presentation.” That means ~L = “The boss did not like the presentation” and ~P = “The presentation was not submitted on time.” Note that “late” is the same as “not on time” in this context. With that, we can symbolize the entire argument. Using the above, we can write it out like this.

Mary’s Argument Symbolically

P1) P→L

P2) ~P

C) ~L

This is the full symbolization of the argument. Now, it is time to build the frame of the truth table. We have two variables in this case (P and L), so we have the first two columns be the variables.

Next, we must put each premise and the conclusion in their own column. For this, let’s add another three columns to the table. One for P1, one for P2, and one for the conclusion. Like the previous example, we can put a header row to indicate what the column is for.

We are now ready to fill in the truth table and determine if Mary has a strong argument. To do this, we must follow the steps we have learned during our symbolization practice. Let’s get started.

The key to filling in truth tables is a clear understanding of the truth conditions. If you know the truth conditions for each connective, filling in the cells will be a simple procedure to follow. The first two columns are the variables and those will always be filled in the same. There are four possible outcomes for the situation, so those are listed in the table below.

Next, we can look at the first premise (P → L). This is an “if-then” statement. The only time this type of statement is false is when the antecedent (P) is true, and the consequent (L) is false. This only occurs in the second row. Therefore, we can put an “F” in the second row and a “T” for the other cells in that column.

Next, we look at the second premise. This is the “not” condition. The variable that is being negated is P. Therefore, we look at the P column (the first column), and we flip the truth values down the table. In the first two rows, P is true. That means ~P is false. In the last two rows, P is false, so ~P is true.

Finally, we fill in the conclusion column. This column is also a “not” condition. This time, we are negating L, so we take the second column and flip the truth value to fill in the ~L column. In the first and third rows, L is true. That means ~L will be false in those two rows. In the second and fourth rows, L is false. Therefore, ~L is true for those two rows. With that, we can completely fill in the truth table.

Now that we have a complete truth table, we must interpret what it means. Remember, our goal is to figure out whether Mary has a strong argument.

From the truth table below, we must determine whether Mary’s argument is strong. To do this, we will have to check to see if the argument is invalid. Recall what invalidity means – there is an instance where the premises are true, and the conclusion is false.

In the table above, there is a row in which the premises are true, and the conclusion is false. This is the second to last row. Premise 1 (P→L) is true. Premise 2 (~P) is true. However, the conclusion (~L) is false. Therefore, we can prove from the truth table that Mary’s argument is invalid. We have shown that it is possible for there to be an instance where the premises are true, and the conclusion is false. This is the instance where P is false, and L is true. In other words, it is possible for your boss to like the presentation despite it not being submitted on time. That should align with your everyday intuition. Perhaps the boss did not mind that it was an hour late if the content was great. Mary’s conclusion that the boss would not like it just because it is late is not always true.

For more practice and to view another example with a guided explanation, refer to this 6-minute, 38-second titled “Truth Table Practice: From Verbal Argument”

video below by 5 Minute Logic:

Truth Table Practice: From Verbal Argument

Let’s review a few takeaways from this lesson:

To determine validity, create a truth table and examine the value of the premises and the conclusion.

More complex arguments may require more steps, but the same process applies.

Knowing the truth conditions is vital to filling out a truth table and determining validity.

Thank you! You have completed this lesson. Please scroll down to complete a short, ungraded Knowledge Check activity.

Check Your Knowledge 2

True or False: If an argument has true premises and a false conclusion, it still can be valid.
Answer: False. That is the definition of invalidity. The argument is invalid.

True or False: The following argument is invalid:

Answer: False. There is no instance where the premises are true, and the conclusion is false. The argument is valid.

Which row proves this argument to be invalid?

The first row

The second row

The third row

The fourth row

Rationale: The third row shows the premises as true and the conclusion as false. This makes the argument invalid.

When building a truth table for an argument, it is critical that you know all the connectives and truth conditions that we have learned in the course. Real life arguments come in many forms, so that means you may have to review some of the connectives from earlier units. This lesson will be a brief review of the connectives and can serve as a guide in case you need to recall how the connectives work.

You have learned about four connectives so far, and each of them has different truth conditions. First, there is the negation symbol (~). This simply negates the truth value of the entire sentence. If P is true, then ~P is false. If P is false, ~P is true. This is the simplest of the connectives and stands for the word “not”.

Next, there is the symbol for “or” (v). This is a two-place connective that must go between two variables. The sentence “W v R” (pronounced “W or R”) has four possible combinations and is true when at least one of the variables is true. In other words, the only time an “or” statement is false is when both components of the sentence are false.

The next connective you learned about was “and”(&). These sentences are true only when both components are true. Otherwise, they are false. For instance, the only time when “A & B” is true is when A is true, and B is true. If one or both is false, then the entire sentence is false.

Finally, you learned about the “if-then” connective (→). This connective is a little different from the prior ones since the order of statements matters. In other words, “A → B” is not the same as “B → A”. In an “if-then” statement, the only time it is false is when the antecedent (the statement before the →) is true and the consequent (the statement after the →) is false.

Let’s take all these truth conditions and put them into truth tables.

The truth table for ~ shows how the truth value for P is flipped when compared to ~P. This is the simplest truth table because P only has two possible outcomes (true or false).

The truth table for v involves two variables, so there are four possible outcomes. The entire “or” statement is false only when both individual pieces are false. This occurs in the final row of the table below. Otherwise, the v statement is true.

The truth table for & also involves two variables, so there are four possible outcomes. The entire statement is true when both pieces are true individually. Otherwise, it is false. From the table below, it is true in only the first row, and it is false in all the others.

The truth table for → involves two variables as well, and the order is important when considering the truth value of the entire statement. The conditional statement is only false when the antecedent is true, and the consequent is false. That only occurs in the second row below. Otherwise, the statement is true.

Now that you have all the connectives, truth conditions, and truth tables in one place, we can go into some common mistakes that you will want to avoid.

There are a few common mistakes that come up when working with symbolic logic for the first time. Many are understandable, but having this lesson to refer back to should prevent you from making too many.

First, ensure your variables always stand for true/false sentences. It is common to associate a letter with a word instead of an entire sentence, but this is a mistake. For instance, B = “basketball” is incorrect since it is neither true nor false. Instead, be sure your sentence can be evaluated.

Second, some misinterpret the meaning of the statements when dealing with connectives. For instance, the statement “A v B” does not mean “A or B, but not both” in English. Instead, as long as A or B is true, the entire sentence is true. The same issues can happen with other connectives. A → B does not mean that A happened, and B will happen. It is a statement that we must evaluate. We must break down the instances of A and B to see when the sentence is true or false.

Third, some students omit parenthesis or brackets when working with larger sentences. Remember that our symbols are two-place connectives, so that means a symbol must separate two distinct statements. Even a very complex sentence like “[(A & B) v C] → [B → (A v C)]” is acceptable because the → separates two very large statements. If there are three statements in a row with no parenthesis or brackets, you may be missing some.

Finally, the most critical aspect of truth tables and symbolization is understanding what the symbols represent and how to translate them into real-life sentences. In order to do that, one must be comfortable with the symbols and what they mean. This is done mostly through practice, and it becomes more familiar the more problems one does.

Let’s review a few takeaways from this lesson:

Knowledge of each connective is critical (~, v, &, →).

Each truth table is composed of simple statements built up using the truth conditions.

Avoiding common mistakes will help you symbolize and evaluate truth tables.

Thank you! You have completed this lesson. Please scroll down to complete a short, ungraded Knowledge Check activity.

Check Your Knowledge 3

True or False: Parenthesis are needed for the statement “P v Q → R”.

Answer: True. The current statement is unclear because it lacks parenthesis.

True or False: “P & Q” is the same as “Q & P”.

Answer: True. The order of the variables does not matter for &. It only matters for →.

True or False: “Apples” can be symbolized A.
Answer: False. “Apples” is not a true/false sentence, so it cannot be symbolized.

Now that we have reviewed all the connectives, their truth conditions, and their basic truth tables, we are able to dive into three more complex practice examples symbolizing real-life examples and figuring out if they are valid/invalid. To do this, we are going to break down each example symbolically and then create the truth table for it. Then, we will evaluate the truth table and check for validity.

Let’s start off with a medium example.

Suppose you are having a debate with your friend about the state of the economy, and they present the following argument: “If the economy improves, unemployment will go down. Unemployment has gone down. Therefore, the economy has improved.” Your job is to determine whether that is a valid argument or not. Let’s first break down the argument into premise-premise-conclusion form and then attempt to symbolize it.

P1) If the economy improves, unemployment will go down.

P2) Unemployment has gone down.

C) Therefore, the economy has improved.

We can take this argument and begin to symbolize it. Let’s have E = “the economy improves” and U = “unemployment will go down”. We can now symbolize the argument like this:

P1) E → U

P2) U

C) E

Now that we have this argument written out symbolically in premise-premise-conclusion form, we can begin creating our truth table. The frame of the table would look like this:

To fill in this table, we start with the variables section in the beginning. These are the four possible outcomes of this statement. It would look like this:

Next, we look at each premise and follow the truth conditions. In P1, we have an “if-then” statement. The only time this is false is when the antecedent is true, and the consequent is false. That only occurs in the second row. Otherwise, the entire statement is true. To fill in P2, we can take the same values as they are in the variables section since U has been unchanged. The same goes for the conclusion column E. Therefore, the final truth table would look like this:

At this point, we can examine the table to determine whether the argument is valid or invalid. In this case, there is an instance where the premises are true, but the conclusion is false. This occurs on the third row. Therefore, the argument is invalid. Now we can look at a hard example.

Suppose your friend comes back to you and wants to debate more about the economy and how government policies can impact the United States overall. They present you with this argument: “If the government implements a new tax policy, inflation will decrease. The government did not implement the new tax policy. Therefore, inflation will not decrease.” Like the prior examples, we can begin breaking this down and putting it into premise-premise-conclusion form. That would give us this argument.

P1) If the government implements a new tax policy, inflation will decrease.

P2) The government did not implement the new tax policy.

C) Therefore, inflation will not decrease.

Now we can assign our variables. We can use G = “The government implements a new tax policy” and I = “Inflation will decrease”. Now, we can symbolize our full argument.

P1) G → I

P2) ~G

C) ~I

Like the prior examples, we must put these variables into a truth table and evaluate each premise and the conclusion individually. Let’s get started using the two variables in our argument – G and I.

Now that we have the frame written out, we can start to fill out each cell. Like the prior example, we can start with the four outcomes in the variables section. This would give us the following table:

Next, we can use our truth conditions to fill in the remaining cells. The first premise is an “if-then” statement, which means it is only false when the antecedent (G) is true, and the consequent (I) is false. This only occurs in the second row. Other than that row, the entire column is true. Premise 2 and the conclusion are simply negations of the first and second columns. With that, we can fill in the entire truth table.

Now that we have the full truth table filled in, it is time to examine the argument and determine whether it is valid or invalid. In this case, the third row is an instance where the premises are true, and the conclusion is false. Therefore, the argument is invalid.

Finally, we can look at a very hard example.

Suppose you work for the government, and your role involves looking at the carbon emissions of companies nationwide. Your job is to fine companies if they do not meet the correct requirements by the end of each year. You happened to come across a file where a company was not fined last year, but you cannot find the paperwork on whether they met the requirement. You construct the following argument in your mind:

P1) If a company does not meet requirements, it will be fined by the government.

P2) This company was not fined by the government.

C) Therefore, this company met the requirements.

Is this a valid argument? Let’s break it down step by step to examine validity.

First, there are two variables to consider. We’ll want to symbolize the positive version of the sentence in P1 since we want to avoid negations in our initial symbolizations. That would give us R = “A company meets requirements”. Next, we can symbolize the second half of P1, which is G = “they would be fined by the government”. With this, we can symbolize our entire argument.

P1) ~R → G

P2) ~G

C) R

Now that we have the symbolization, we must create a truth table to determine if this is valid. Recall that we will have a column for the two variables and then build each premise. The frame of the table would look like this. We can add the negations for each variable since they will be used throughout the argument.

Now that we have the frame written out, we can fill in each cell based on our truth conditions. The first two columns are the four possible outcomes. The next two columns are the negations of R and G, so we would get the following table.

Now we can fill in the cells for P1, P2, and the conclusion. To fill in P1, we simply must follow the truth conditions for the “if-then” statement. That is, it is only false if the antecedent is true, and the consequent is false. The only time ~R is true, and G is false is the final row. Other than that, the conditional is true. To fill in P2, we simply must use the same ~G column that we have under our variables. To fill in the conclusion, we can take our R column from our variables. That would give us the completed truth table below.

We have our completed truth table, and we can now determine whether the argument is valid or invalid. Since there is no instance where the premises are true, but the conclusion is false, the argument is valid.

Let’s review a few takeaways from this lesson:

Always begin by breaking the argument down into premise-premise-conclusion form.

Symbolize the argument and then put the argument into a truth table.

Examine the completed truth table – if there is an instance of a false conclusion where there are true premises, the argument is invalid.

Thank you! You have completed this lesson. Please scroll down to complete a short, ungraded Knowledge Check activity.

Check Your Knowledge 4

True or False: The following argument shown is valid:

Answer: True. There is no instance where the first two premises are true, and the conclusion is false.

True or False: The following argument shown is valid:

Answer: False. In the first row, P1 and P2 are true, while the conclusion is false. Therefore, the argument is invalid.

True or False: The following argument shown is valid:

Answer: False. In the first two rows, the premises are true, and the conclusion is false. Therefore, the argument is invalid.

In this lecture, you learned how to prove that an argument is invalid using a truth table. You learned that it starts with breaking the verbal argument down into symbols. Then, put the frame of a truth table together and fill in each column based on the truth conditions. This skill will help you break down real-life arguments and evaluate their validity.

In the next lecture, we will look at some more real-life examples of arguments and use the logical skills we have developed so far to evaluate them.