Section 3.4 Arguments
Objectives
Identify inductive and deductive reasoning.
Evaluate arguments with truth tables.
Discover forms of valid and invalid arguments.
Discuss logical fallacies.
Subsection 3.4.1 Argument Types
There are two general types of arguments:
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 3.4.1.
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
The conclusion is:
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 3.4.2.
The argument “every day for the past year, a plane flies over my house at 2:00 P.M. A plane will fly over my house every day at 2:00 P.M.” is a stronger inductive argument, since it is based on a larger set of evidence. While it is not necessarily true—the airline may have cancelled its afternoon flight—it is probably a safe bet.
Note 3.4.3. 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 that it may be true.
Many scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. Common scientific theories, like Newton’s theory of gravity, have all stood up to years of testing and evidence, though sometimes they need to be adjusted based on new evidence, such as when Einstein proposed the theory of general relativity
A deductive argument is more clearly valid or not, which makes it easier to evaluate.
Note 3.4.4. Evaluating deductive arguments.
A deductive argument is considered valid if, assuming that all the premises are true, the conclusion follows logically from those premises. In other words, when the premises are all true, the conclusion must be true.
Subsection 3.4.2 Evaluating Dedutive Arguments with Truth Tables
Arguments can also be analyzed using truth tables, although this can be a lot of work.
To analyze an argument with a truth table:
Represent each of the premises and the conclusion symbolically
Create truth tables for each premise and for the conclusion.
The argument is valid if and only if the conclusion is true when all premises are true.
Example 3.4.5.
Consider the argument
Premise: If you bought bread, then you went to the store.
Premise: You bought bread.
Conclusion: You went to the store.
While this example is fairly obviously a valid argument, we can analyze it by representing the argument symbolically and then creating truth tables for the premises and conclusion.
We’ll let b represent “you bought bread” and s represent “you went to the store”. Then the argument becomes:
Premise: \(b \to s\)
Premise: \(b\)
Conclusion: \(s\)
| Variable 1: \(b\) | Variable 2: \(s\) | Premise 1: \(b\) | Premise 2: \(b \to s\) | Conclusion: \(s\) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | T | F | F |
| F | T | F | T | T |
| F | F | F | T | F |
The only row where both premises are true is the first one. The conclusion is also true in the first row. Since the conclusion is true when both premises are true, this is a valid argument.
Problem 3.4.7. Try It Now.
Use truth tables to determine whether the argument is valid:
If I have a shovel, I can dig a hole.
I dug a hole.
Therefore, I had a shovel.
Answer.Let s represent "I have shovel" and d represent "I dug a hole."
In symbolic form, the argument is:
\(s \to d\)
\(d\)
\(s\)
The truth tables for this argument are:
| Variable 1: \(s\) | Variable 2: \(d\) | Premise 1: \(s \to d\) | Premise 2: \(d\) | Conclusion: \(s\) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | F |
Both premises are true in both the first and third rows of the table. However, the conclusion is false in the third row. Since there is a situation where both premises are true and the conclusion is false (the third row of the table), this is an invalid argument.
Subsection 3.4.3 Forms of Arguments
Rather than making a truth table for every argument, we may be able to recognize certain common forms of arguments that are valid (or invalid). If we can determine that an argument fits one of the common forms, we can immediately state whether it is valid or invalid.
Definition 3.4.9. Law of Detachment.
The law of detachment applies when a conditional and its antecedent are given as premises, and the consequent is the conclusion.
The general form is:
Premise: \(p \to q\)
Premise: \(p\)
Conclusion: \(q\)
Example 3.4.10.
Recall this argument from an earlier example:
Premise: If you bought bread, then you went to the store.
Premise: You bought bread.
Conclusion: You went to the store.
In symbolic form:
Premise: \(b \to s\)
Premise: \(b\)
Conclusion: \(s\)
This argument has the structure described by the law of detachment. (The second premise and the conclusion are simply the two parts of the first premise detached from each other.) Instead of making a truth table, we can say that this argument is valid by stating that it satisfies the law of detachment.
Definition 3.4.11. Law of Contraposition.
The law of contraposition applies when a conditional and the negation of its consequent are given as premises, and the negation of its antecedent is the conclusion.
The general form is:
Premise: \(p \to q\)
Premise: \(\sim q\)
Conclusion: \(\sim p\)
Notice that the second premise and the conclusion look like the contrapositive of the first premise, \(\sim q \to \sim p\text{,}\) but they have been detached. You can think of the law of contraposition as a combination of the law of detachment and the fact that the contrapositive is logically equivalent to the original statement.
Example 3.4.12.
Premise: If I drop my phone into the swimming pool, my phone will be ruined.
Premise: My phone isn’t ruined.
Conclusion: I didn’t drop my phone into the swimming pool.
If we let d = I drop the phone in the pool and r = the phone is ruined, then we can represent the argument this way:
Premise: \(d \to r\)
Premise: \(\sim r\)
Conclusion: \(\sim d\)
The form of this argument matches what we need to invoke the law of contraposition, so it is a valid argument.
Problem 3.4.13. Try It Now.
Is this argument valid?
Premise: If you brushed your teeth before bed, then your toothbrush will be wet.
Premise: Your toothbrush is dry.
Conclusion: You didn’t brush your teeth before bed.
Answer.Definition 3.4.14. The Transitive Property.
The transitive property has as its premises a series of conditionals, where the consequent of one is the antecedent of the next. The conclusion is a conditional with the same antecedent as the first premise and the same consequent as the final premise.
The general form is:
Premise: \(p \to q\)
Premise: \(q \to r\)
Conclusion: \(p \to r\)
Example 3.4.15.
Premise: If a soccer player commits a reckless foul, she will receive a yellow card.
Premise: If Hayley receives a yellow card, she will be suspended for the next match.
Conclusion: If Hayley commits a reckless foul, she will be suspended for the next match.
If we let r = committing a reckless foul, y = receiving a yellow card, and s = being suspended, then our argument looks like this:
Premise \(r \to y\)
Premise \(y \to s\)
Conclusion: \(r \to s\)
This argument has the exact structure required to use the transitive property, so it is a valid argument.
Definition 3.4.16. Dijunctive Syllogism.
In a disjunctive syllogism, the premises consist of an or statement and the negation of one of the options. The conclusion is the other option.
The general form is:
Premise: \(p \vee q\)
Premise: \(\sim p\)
Conclusion: \(q\)
The order of the two parts of the disjunction isn’t important. In other words, we could have the premises \(p \vee q\) and \(\sim q\text{,}\) and the conclusion \(p\text{.}\)
Example 3.4.17.
Premise: I can either drive or take the train.
Premise: I refuse to drive.
Conclusion: I will take the train.
If we let d = I drive and t = I take the train, then the symbolic representation of the argument is:
Premise \(d \vee t\)
Premise \(\sim d\)
Conclusion: \(t\)
This argument is valid because it has the form of a disjunctive syllogism. I have two choices, and one of them is not going to happen, so the other one must happen.
Keep in mind that, when you are determining the validity of an argument, you must assume that the premises are true. If you don’t agree with one of the premises, you need to keep your personal opinion out of it. Your job is to pretend that the premises are true and then determine whether they force you to accept the conclusion. You may attack the premises in a court of law or a political discussion, of course, but here we are focusing on the structure of the arguments, not the truth of what they actually say.
We have just looked at four forms of valid arguments; there are two common forms that represent invalid arguments, which are also called fallacies.
Definition 3.4.18. Fallacy of the Converse.
The fallacy of the converse arises when a conditional and its consequent are given as premises, and the antecedent is the conclusion.
The general form is:
Premise: \(p \to q\)
Premise: \(q\)
Conclusion: \(p\)
Notice that the second premise and the conclusion look like the converse of the first premise, \(q \to p\text{,}\) but they have been detached. The fallacy of the converse incorrectly tries to assert that the converse of a statement is equivalent to that statement.
Example 3.4.19.
Premise: If I drink coffee after noon, then I have a hard time falling asleep that night.
Premise: I had a hard time falling asleep last night.
Conclusion: I drank coffee after noon yesterday.
If we let c = I drink coffee after noon and h = I have a hard time falling asleep, then our argument looks like this:
Premise: \(c \to h\)
Premise: \(h\)
Conclusion: \(c\)
This argument uses converse reasoning, so it is an invalid argument. There could be plenty of other reasons why I couldn’t fall asleep: I could be worried about money, my neighbors might have been setting off fireworks, etc.
Definition 3.4.20. Fallacy of the Converse.
The fallacy of the inverse occurs when a conditional and the negation of its antecedent are given as premises, and the negation of the consequent is the conclusion.
The general form is:
Premise: \(p \to q\)
Premise: \(\sim p\)
Conclusion: \(\sim q\)
Again, notice that the second premise and the conclusion look like the inverse of the first premise, \(\sim p \to \sim q\text{,}\) but they have been detached. The fallacy of the inverse incorrectly tries to assert that the inverse of a statement is equivalent to that statement.
Example 3.4.21.
Premise: If you listen to the Grateful Dead, then you are a hippie.
Premise: Sky doesn’t listen to the Grateful Dead.
Conclusion: Sky is not a hippie.
If we let g = listen to the Grateful Dead and h = is a hippie, then this is the argument:
Premise: \(g \to h\)
Premise: \(\sim g\)
Conclusion: \(\sim h\)
This argument is invalid because it uses inverse reasoning. The first premise does not imply that all hippies listen to the Grateful Dead; there could be some hippies who listen to Phish instead.
Of course, arguments are not limited to these six basic forms; some arguments have more premises, or premises that need to be rearranged before you can see what is really happening. There are plenty of other forms of arguments that are invalid. If an argument doesn’t seem to fit the pattern of any of these common forms, though, you may want to use a Venn diagram or a truth table instead.
Lewis Carroll, author of Alice’s Adventures in Wonderland, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle, to be connected using syllogisms. The following example is one such puzzle.
Example 3.4.22.
Solve the puzzle. In other words, find a logical conclusion from these premises.
All babies are illogical.
Nobody is despised who can manage a crocodile.
Illogical persons are despised.
Let b = is a baby, d = is despised, i = is illogical, and m = can manage a crocodile. Then we can write the premises as:
\(b \to i\)
\(m \to \sim d\)
\(i \to d\)
Writing the second premise correctly can be a challenge; it can be rephrased as “If you can manage a crocodile, then you are not despised.”
Using the transitive property with the first and third premises, we can conclude that \(b \to d\text{;}\) that all babies are despised. Using the contrapositive of the second premise, \(d \to \sim m\text{,}\) we can then use the transitive property with \(b \to d\) to conclude that \(b \to \sim m\text{;}\) that babies cannot manage crocodiles. While it is silly, this is a logical conclusion from the given premises.
Problem 3.4.23. Try It Now.
Is this argument valid?
Premise: If I go to the party, I’ll be really tired tomorrow.
Premise: If I go to the party, I’ll get to see friends.
Conclusion: If I don’t see friends, I won’t be tired tomorrow
Answer.Let p = go to party, t = be tired, and f = see friends.
Premise: \(p \to t\)
Premise: \(p \to f\)
Conclusion: \(\sim f \to \sim t\)
We could try to rewrite the second premise using the contrapositive to state \(\sim f \to \sim p\text{,}\) but that does not allow us to form a syllogism. If I don’t see friends, then I didn’t go the party, but that is not sufficient to claim I won’t be tired tomorrow. Maybe I stayed up all night watching movies.
Subsection 3.4.4 Logical Fallacies in Common Language
In the previous discussion, we saw that logical arguments can be invalid when the premises are not true, when the premises are not sufficient to guarantee the conclusion, or when there are invalid chains in logic. There are a number of other ways in which arguments can be invalid, a sampling of which are given here.
Definition 3.4.24. Ad hominem.
An ad hominem argument attacks the person making the argument, ignoring the argument itself.
Example 3.4.25.
“Jane says that whales aren’t fish, but she’s only in the second grade, so she can’t be right.”
Here the argument is attacking Jane, not the validity of her claim, so this is an ad hominem argument.
Example 3.4.26.
“Jane says that whales aren’t fish, but everyone knows that they’re really mammals. She’s so stupid.”
This certainly isn’t very nice, but it is not ad hominem since a valid counterargument is made along with the personal insult.
Definition 3.4.27.
Appeal to ignoranceThis type of argument assumes something it true because it hasn’t been proven false
Example 3.4.28.
“Nobody has proven that photo isn’t of Bigfoot, so it must be Bigfoot.”
Definition 3.4.29. Appeal to authority.
These arguments attempt to use the authority of a person to prove a claim. While often authority can provide strength to an argument, problems can occur when the person’s opinion is not shared by other experts, or when the authority is irrelevant to the claim.
Example 3.4.30.
“A diet high in bacon can be healthy; Doctor Atkins said so.”
Here, an appeal to the authority of a doctor is used for the argument. This generally would provide strength to the argument, except that the opinion that eating a diet high in saturated fat runs counter to general medical opinion. More supporting evidence would be needed to justify this claim.
Example 3.4.31.
“Jennifer Hudson lost weight with Weight Watchers, so their program must work.”
Here, there is an appeal to the authority of a celebrity. While her experience does provide evidence, it provides no more than any other person’s experience would.
Definition 3.4.32. Appeal to consequence.
An appeal to consequence concludes that a premise is true or false based on whether the consequences are desirable or not.
Example 3.4.33.
“Humans will travel faster than light: faster-than-light travel would be beneficial for space travel.”
Definition 3.4.34. False dilemma.
A false dilemma argument falsely frames an argument as an “either or” choice, without allowing for additional options.
Example 3.4.35.
“Either those lights in the sky were an airplane or aliens. There are no airplanes scheduled for tonight, so it must be aliens.”
This argument ignores the possibility that the lights could be something other than an airplane or aliens.
Definition 3.4.36. Circular reasoning.
Circular reasoning is an argument that relies on the conclusion being true for the premise to be true.
Example 3.4.37.
“I shouldn’t have gotten a C in that class; I’m an A student!”
In this argument, the student is claiming that because they’re an A student, though shouldn’t have gotten a C. But because they got a C, they’re not an A student.
Definition 3.4.38. Post hoc (post hoc ergo propter hoc).
A post hoc argument claims that because two things happened sequentially, then the first must have caused the second.
Example 3.4.39.
“Today I wore a red shirt, and my football team won! I need to wear a red shirt every time they play to make sure they keep winning.”
Definition 3.4.40. Straw man.
A straw man argument involves misrepresenting the argument in a less favorable way to make it easier to attack.Example 3.4.41.
“Senator Jones has proposed reducing military funding by 10%. Apparently he wants to leave us defenseless against attacks by terrorists”
Here the arguer has represented a 10% funding cut as equivalent to leaving us defenseless, making it easier to attack Senator Jones’ position.
Definition 3.4.42. Correlation implies causation.
Similar to post hoc, but without the requirement of sequence, this fallacy assumes that just because two things are related one must have caused the other. Often there is a third variable not considered.
Example 3.4.43.
“Months with high ice cream sales also have a high rate of deaths by drowning. Therefore, ice cream must be causing people to drown.”
This argument is implying a causal relation, when really both are more likely dependent on the weather; that ice cream and drowning are both more likely during warm summer months.
Problem 3.4.44. Try It Now.
Identify the logical fallacy in each of the arguments
Only an untrustworthy person would run for office. The fact that politicians are untrustworthy is proof of this.
Since the 1950s, both the atmospheric carbon dioxide level and obesity levels have increased sharply. Hence, atmospheric carbon dioxide causes obesity.
The oven was working fine until you started using it, so you must have broken it.
You can’t give me a D in the class because I can’t afford to retake it.
The senator wants to increase support for food stamps. He wants to take the taxpayers’ hard-earned money and give it away to lazy people. This isn’t fair, so we shouldn’t do it.
Circular
Correlation does not imply causation
Post hoc
Appeal to consequence
Staw man
It may be difficult to identify one particular fallacy for an argument. Consider this argument: “Emma Watson says she’s a feminist, but she posed for these racy pictures. I’m a feminist, and no self-respecting feminist would do that.” Could this be ad hominem, saying that Emma Watson has no self-respect? Could it be appealing to authority because the person making the argument claims to be a feminist? Could it be a false dilemma because the argument assumes that a woman is either a feminist or not, with no gray area in between?