Chapter 4: Logic
Section 4.3: Logical Arguments
Logical Arguments
A logical argument is a claim that a set of premises supports a conclusion. When it comes to an argument, the argument can be either valid, suggesting that the conclusion follows the premises, or invalid, meaning that the premises do not lead to the conclusion. We call invalid arguments fallacies.
Example 1
Given the inductive argument, “When I went to the store last week, I forgot my purse, and when I went today, I forgot my purse. Therefore, I always forget my purse when I go to the store.” Identify the premises and conclusion.
Solution
Premises:
- I forgot my purse last week when I went to the store.
- I forgot my purse today when I went to the store.
Conclusion: I always forget my purse when I go to the store.
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.
Deciding if an argument is valid or not can be determined in two manners. If the symbolic form of the argument matches one of the specific rules, then we can easily determine validity. However, if the symbolic form does not match one of the special rules, a truth table must be constructed to evaluate the argument.
Forms of Valid Arguments
The law of detachment applies when a conditional and its antecedent are given as premises, and the consequent is the conclusion.
| Premise: | \(p \rightarrow q\) |
| Premise: | \(p\) |
| Conclusion: | \(q\) |
The Latin name, Modus Ponens, translates to “mode that affirms.”
This law applies when a conditional and the negation of its consequent are given as premises, and the negation of its antecedent is the conclusion.
| Premise: | \(p \rightarrow q\) |
| Premise: | \(\sim q\) |
| Conclusion: | \(\sim p\) |
| Premise: | \(p \rightarrow q\) |
| Premise: | \(q \rightarrow r\) |
| Conclusion: | \(p \rightarrow r\) |
| Premise: | \(p \vee q\) |
| Premise: | \(\sim p\) or \(\sim q\) |
| Conclusion: | \(q\) or \(p\) |
Forms of Invalid Arguments
| Premise: | \(p \rightarrow q\) |
| Premise: | \(q\) |
| Conclusion: | \(p\) |
| Premise: | \(p \rightarrow q\) |
| Premise: | \(\sim p\) |
| Conclusion: | \(\sim q\) |
| Premise: | \(p \vee q\) |
| Premise: | \(q\) |
| Conclusion: | \(\sim p\) |
Example 2
Given the following logical arguments, identify which of our common forms for invalid arguments each follows.
- If Paul gets an "A" in Science, then he will become a vet. He didn't get an "A" in Science. Therefore he will not become a vet.
- If the sun is shining, then it's a beautiful day. It's a beautiful day. Therefore the sun is shining.
- When I go to dinner, I will order a steak or a side salad. I ordered a side salad. Therefore I did not order a steak.
Solution
-
If Paul gets an "A" in Science, then he will become a vet. He didn't get an "A" in Science. Therefore he will not become a vet.
For this example, allow p $=$ "Paul gets an "A" in Science" and q $=$ "Paul becomes a vet". Symbolically, it would give us the form:
$$ \begin{array}{l r} p \rightarrow q & \\ p & \\ \hline \therefore q & \end{array} $$
That means this an example of an invalid argument by Fallacy of the Inverse.
-
If the sun is shining, then it's a beautiful day. It's a beautiful day. Therefore the sun is shining.
For this example allow p $=$ "The sun is shinning" then q $=$ "It's a beautiful day". Symbolically, it would give us the form:
$$ \begin{array}{l} p \rightarrow q \\ q \rightarrow r \\ \hline \therefore p\rightarrow r \end{array} $$
That means this an example of an invalid argument by Fallacy of the Converse.
-
When I go to dinner, I will order a steak or a side salad. I ordered a side salad. Therefore I did not order a steak.
For this example allow p $=$ "I order steak" and q $=$ "I order a side salad". Symbolically, it would give us the form:
$$ \begin{array}{l} p \wedge q \\ \sim q \\ \hline \therefore p \end{array} $$
That means this an example of an invalid argument by Fallacy of the the Inclusive "or". We have to remember with "or" statements, it can be one, the other, or both. Thus we cannot just say it's not the one because it's the other. We always have that both option.
Not all logical arguments fit into one of our common forms. When that is the case, we must construct a truth table to determine the validity of the argument. It is important to note that validity plays no role in the truth value behind the statement. All that validity tells us is that we can logically arrive at this conclusion with the premises we've been presented, regardless if those premises are true or false.
- Represent each of the premises symbolically.
- Create a conditional statement that joins all the premises to form the antecedent, and use the conclusion as the consequent.
- Create a truth table for the statement. If the last column of the table is always true, then the argument is valid.
Example 3
Identify if the following argument is valid or invalid. "You went out to eat and ordered chicken. You also ordered french fries. Therefore if you go out to eat, you order French Fries."
Solution
First, identify your premises and the symbolic form of your argument.| Premise | Description | Symbolic Form |
|---|---|---|
| Premise 1 | You went out to eat and ordered chicken. | p ∧ q |
| Premise 2 | You ordered French Fries. | r |
| Conclusion | If you go out to eat, you order French Fries. | ∴ p → r |
Next we need to start building the truth table. We will start by creating our columns for p, q, and r as there are three different simple statements.
| p | q | r |
|---|---|---|
| T | T | T |
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
Now we need to look at what our argument would look like as a complete symbolic argument. To put this all together, we create a conditional statement between all of the premises and an implication(conditional) for the conclusion. That would symbolically give us:
$$ [ ( \text{p} \wedge \text{q} ) \wedge \text{r} ] \rightarrow (\text{p} \rightarrow \text{r}) $$
Once we've identified that argument, we need to "build" it out in our logic table.
Table 1
| p | q | r | p ∧ q |
|---|---|---|---|
| T | T | T | T |
| T | T | F | T |
| T | F | T | F |
| T | F | F | F |
| F | T | T | F |
| F | T | F | F |
| F | F | T | F |
| F | F | F | F |
Table 2
| p | q | r | p ∧ q | (p ∧ q) ∧ r |
|---|---|---|---|---|
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | F | F |
| T | F | F | F | F |
| F | T | T | F | F |
| F | T | F | F | F |
| F | F | T | F | F |
| F | F | F | F | F |
Table 3
| p | q | r | p ∧ q | (p ∧ q) ∧ r | p → r |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | T | F | T | F | F |
| T | F | T | F | F | T |
| T | F | F | F | F | F |
| F | T | T | F | F | T |
| F | T | F | F | F | T |
| F | F | T | F | F | T |
| F | F | F | F | F | T |
Table 4
| p | q | r | p ∧ q | (p ∧ q) ∧ r | p → r | ((p ∧ q) ∧ r) → (p → r) |
|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T |
| T | T | F | T | F | F | T |
| T | F | T | F | F | T | T |
| T | F | F | F | F | F | T |
| F | T | T | F | F | T | T |
| F | T | F | F | F | T | T |
| F | F | T | F | F | T | T |
| F | F | F | F | F | T | T |
Looking through each table, we can see that in Table 1 we started by finding the validity behind the first premise. From there, we moved to table two where we found the conjunction between the first and second premise. It is important to note that at this point, we did not need to build out a second premise as it was just a simple statement. Next, we moved to Table 3 where we first had to find the validity of the conclusion since it was a compound statement.
Finally, we moved to Table 4 where we used the conjunction of the premises to imply the conclusion, which we can see it all true making it a valid argument. Some things that are important to note are that when you're doing this on your own, you will not make 4 different tables. Table 4 is what your table will look like once completed. You just keep adding columns until you've completed the table. Another thing to note is that the validity of the argument does not imply the truth value of the argument. We do not know if this argument is true or false. All we know is that it is acceptable to arrive at this conclusion from the premises we were presented.
Equivalence, Tautologies, and Self-Contradictions
In the English language, we can phrase our speech in a number of ways to say the same thing. Equally the language allows us to completely contradict ourselves too. In logic, two statements are equivalent if the two statements have the same exact truth values.
Example 4
Determine if the following statements are logically equivalent:
Statement 1: If I get my nails done, I'll go on a date Friday.
Statement 2: If I don't go on a date Friday, then I won't get my nails done.
Solution
First, we need to look at each statement in their symbolic form:
Statement 1: $$p \rightarrow q$$
Statement 2: $$\sim q \rightarrow \sim p$$
Next, we need to construct a truth table for each statement and compare the solutions:
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
| p | q | ¬q | ¬p | ¬q → ¬p |
|---|---|---|---|---|
| T | T | F | F | T |
| T | F | T | F | F |
| F | T | F | T | T |
| F | F | T | T | T |
Looking at the last column of both tables, we can see that in each situation they are exactly the same truth values. This means that these two statements are logically equivalent to each other.
Occasionally in compound statements, we may run across something known as a tautology or self-contradiction.
Tautology: A logical statement that is never false, regardless of the truth of the original simple statements.
Self-Contradiction: A logical statement that is never true, regardless of the truth of the original simple statements.
Example 5
Determine if the following compound statements are a tautology, self-contradiction or neither:
- If I order a pizza and I order dessert, then I'll order a pizza.
- If I order a pizza or I order a dessert, then I'll order a dessert.
- I'm going to order a pizza and dessert and I'm not going to order a pizza and I'm not going to order dessert.
Solution
First, identify the symbolic form of each statement:
- $$(p \wedge q) \rightarrow p$$
- $$(p \vee q) \rightarrow q$$
- $$(p \wedge q) \wedge (\sim p \wedge \sim q)$$
Next, construct the truth tables:
1. $$(p \wedge q) \rightarrow p$$
| p | q | p ∧ q | → p |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
Since the last column is all true, this statement is a tautology.
2. $$(p \vee q) \rightarrow q$$
| p | q | p ∨ q | → q |
|---|---|---|---|
| T | T | T | T |
| T | F | T | F |
| F | T | T | T |
| F | F | F | T |
Since the last column is not all true or all false, this statement is neither a tautology nor a self-contradiction.
3. $$(p \wedge q) \wedge (\sim p \wedge \sim q)$$
| p | q | p ∧ q | ¬p | ¬q | ¬p ∧ ¬q | (p ∧ q) ∧ (¬p ∧ ¬q) |
|---|---|---|---|---|---|---|
| T | T | T | F | F | F | F |
| T | F | F | F | T | F | F |
| F | T | F | T | F | F | F |
| F | F | F | T | T | F | F |
Since the last column is all false, this statement is a self-contradiction.
De Morgan's Law
Augustus De Morgan (1772-1816), a British mathematician and a contemporary of Boole, formalized two rules of logic that had previously been known informally. They allow us to rewrite the negation of a conjunction as a disjunction, and vice-versa.
De Morgan's Law:
The negation of a conjunction is equivalent to the disjunction of the negation of the statements making up the conjunction:
$$\sim (p \wedge q) \equiv \sim p \vee \sim q$$
The negation of a disjunction is equivalent to the conjunction of the negation of the statements making up the disjunction:
$$\sim (p \vee q) \equiv \sim p \wedge \sim q$$
Example 6
Write the negation of the statement "For Valentine's Day, you did not get your sweetie flowers or candy."
Solution
In this situation, p would be "You did not get your sweetie flowers" and q would be "You did not get your sweetie candy". This makes the negation:
"For Valentine's Day, you got your sweetie flowers and not candy." Since the variation of De Morgan's Law would be $$\sim (p \vee q) \equiv \sim p \wedge \sim q$$.
The Negation of a Conditional
The negation of a conditional statement is logically equivalent to a conjunction of the antecedent and the negation of the consequent:
$$\sim (p \rightarrow q) \equiv p \wedge \sim q$$
Example 7
Write the negation to the statement, "If you don't grease the pan, then the food will stick to it."
Solution
For our example, p = "You don't grease the pan" and q = "Food will stick to it". To find the negation, we negate q to get "Food will not stick to it". Then the negation of the conditional is:
"You don't grease the pan and food will not stick to it".