1.4: Analyzing 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 \(\PageIndex{1}\)
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 \(\PageIndex{2}\)
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.
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.
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.
Evaluating Deductive Arguments with Euler (Venn) Diagrams
We can interpret a deductive argument visually with an Euler diagram, which is essentially the same thing as a Venn diagram. This can make it easier to determine whether the argument is valid or invalid.
Example \(\PageIndex{3}\)
Consider the deductive argument “All cats are mammals and a tiger is a cat, so a tiger is a mammal.” Is this argument valid?
Solution
The premises are:
All cats are mammals.
A tiger is a cat.
The conclusion is:
A tiger is a mammal.
Both the premises are true. To see that the premises must logically lead to the conclusion, we can use a Venn diagram. From the first premise, we draw the set of cats as a subset of the set of mammals. From the second premise, we are told that a tiger is contained within the set of cats. From that, we can see in the Venn diagram that the tiger must also be inside the set of mammals, so the conclusion is valid.
Analyzing arguments with Euler diagrams
To analyze an argument with an Euler diagram:
- Draw an Euler diagram based on the premises of the argument
- The argument is invalid if there is a way to draw the diagram that makes the conclusion false
- The argument is valid if the diagram cannot be drawn to make the conclusion false
- If the premises are insufficient to determine the location of an element or a set mentioned in the conclusion, then the argument is invalid.
Try it Now 8
Determine the validity of this argument:
Premise: All cats are scared of vacuum cleaners.
Premise: Max is a cat.
Conclusion: Max is scared of vacuum cleaners.
Example \(\PageIndex{4}\)
Premise: All firefighters know CPR.
Premise: Jill knows CPR.
Conclusion: Jill is a firefighter.
Solution
From the first premise, we know that firefighters all lie inside the set of those who know CPR. (Firefighters are a subset of people who know CPR.) From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know whether she also is a member of the smaller subset that is firefighters.
Since the conclusion does not necessarily follow from the premises, this is an invalid argument. It’s possible that Jill is a firefighter, but the structure of the argument doesn’t allow us to conclude that she definitely is.
It is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are concerned with whether the premises are enough to prove the conclusion.
Try it Now 10
Determine the validity of this argument:
Premise: All bicycles have two wheels.
Premise: This Harley-Davidson has two wheels.
Conclusion: This Harley-Davidson is a bicycle.
Try it Now 11
Determine the validity of this argument:
Premise: No cows are purple.
Premise: Fido is not a cow.
Conclusion: Fido is purple.
In addition to these categorical style premises of the form “all ___”, “some ____”, and “no ____”, it is also common to see premises that are conditionals.
Example \(\PageIndex{5}\)
Premise: If you live in Seattle, you live in Washington.
Premise: Marcus does not live in Seattle.
Conclusion: Marcus does not live in Washington.
Solution
From the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether Marcus lives in Washington or not. This is an invalid argument.
Try it Now 12
Determine the validity of this argument:
Premise: If you have lipstick on your collar, then you are cheating on me.
Premise: If you are cheating on me, then I will divorce you.
Premise: You do not have lipstick on your collar.
Conclusion: I will not divorce you.
Evaluating Deductive Arguments with Truth Tables
Arguments can also be analyzed using truth tables, although this can be a lot of work.
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 to form the antecedent and using the conclusion as the consequent.
- Create a truth table for the statement. If it is always true, then the argument is valid.
Example \(\PageIndex{6}\)
Consider the argument
Premise: If you bought bread, then you went to the store.
Premise: You bought bread.
Conclusion: You went to the store.
Solution
While this example is fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then form a conditional statement showing that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid.
We’ll let b represent “you bought bread” and s represent “you went to the store”. Then the argument becomes:
Premise: b → s
Premise: b
Conclusion: s
To test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [(b → s) ⋀ b] → s ?
|
b |
s |
b → s |
|---|---|---|
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
|
b |
s |
b → s |
(b → s) ⋀ b |
|---|---|---|---|
|
T |
T |
T |
T |
|
T |
F |
F |
F |
|
F |
T |
T |
F |
|
F |
F |
T |
F |
|
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 |
Since the truth table for [(b → s) ⋀ b] → s is always true, this is a valid argument.
Try it Now 13
Determine whether the argument is valid:
Premise: If I have a shovel, I can dig a hole.
Premise: I dug a hole.
Conclusion: Therefore, I had a shovel.
Example \(\PageIndex{7}\)
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 go to the mall, I’ll buy a shirt.
Solution
Let m = I go to the mall, j = I buy jeans, and s = I buy a shirt.
The premises and conclusion can be stated as:
Premise: m → j
Premise: j → s
Conclusion: m → s
We can construct a truth table for [(m → j) ⋀ (j → s)] → (m → s). Try to recreate each step and see how the truth table was constructed.
|
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 |
From the final column of the truth table, we can see this is a valid argument.
Forms of Valid 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.
The Law of Detachment (Modus Ponens)
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 → q
Premise: p
Conclusion: q
The Latin name, modus ponens, translates to “mode that affirms”.
Example \(\PageIndex{8}\)
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.
Solution
In symbolic form:
Premise: b → 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.
The Law of Contraposition (Modus Tollens)
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 → q
Premise: ~q
Conclusion: ~p
The Latin name, modus tollens, translates to “mode that denies”.
Notice that the second premise and the conclusion look like the contrapositive of the first premise, ~q → ~p, 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 \(\PageIndex{9}\)
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.
Solution
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 → r
Premise ~ r
Conclusion: ~ d
The form of this argument matches what we need to invoke the law of contraposition, so it is a valid argument.
Try it Now 14
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.
The Transitive Property (Hypothetical Syllogism)
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 → q
Premise: q → r
Conclusion: p → r
The earlier example about buying a shirt at the mall is an example illustrating the transitive property. It describes a chain reaction: if the first thing happens, then the second thing happens, and if the second thing happens, then the third thing happens. Therefore, if we want to ignore the second thing, we can say that if the first thing happens, then we know the third thing will happen. We don’t have to mention the part about buying jeans; we can simply say that the first event leads to the final event. We could even have more than two premises; as long as they form a chain reaction, the transitive property will give us a valid argument.
Example \(\PageIndex{10}\)
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.
Solution
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 → y
Premise y → s
Conclusion: r → s
This argument has the exact structure required to use the transitive property, so it is a valid argument.
Try it Now 15
Is this argument valid?
Premise: If the old lady swallows a fly, she will swallow a spider.
Premise: If the old lady swallows a spider, she will swallow a bird.
Premise: If the old lady swallows a bird, she will swallow a cat.
Premise: If the old lady swallows a cat, she will swallow a dog.
Premise: If the old lady swallows a dog, she will swallow a goat.
Premise: If the old lady swallows a goat, she will swallow a cow.
Premise: If the old lady swallows a cow, she will swallow a horse.
Premise: If the old lady swallows a horse, she will die, of course.
Conclusion: If the old lady swallows a fly, she will die, of course.
Disjunctive 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 ⋁ q
Premise: ~p
Conclusion: q
The order of the two parts of the disjunction isn’t important. In other words, we could have the premises p ⋁ q and ~q, and the conclusion p.
Example \(\PageIndex{11}\)
Premise: I can either drive or take the train.
Premise: I refuse to drive.
Conclusion: I will take the train.
Solution
If we let d = I drive and t = I take the train, then the symbolic representation of the argument is:
Premise d ⋁ t
Premise ~ 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.
Try it Now 16
Is this argument valid?
Premise: Alison was required to write a 10-page paper or give a 5-minute speech.
Premise: Alison did not give a 5-minute speech.
Conclusion: Alison wrote a 10-page paper.
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.
The 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 → q
Premise: q
Conclusion: p
Notice that the second premise and the conclusion look like the converse of the first premise, q → p , 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 \(\PageIndex{12}\)
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.
Solution
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 → 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, …
Try it Now 17
Is this argument valid?
Premise: If you pull that fire alarm, you will get in big trouble.
Premise: You got in big trouble.
Conclusion: You must have pulled the fire alarm.
The Fallacy of the Inverse
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 → q
Premise: ~p
Conclusion: ~q
Again, notice that the second premise and the conclusion look like the inverse of the first premise, ~p → ~q, 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 \(\PageIndex{13}\)
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.
Solution
If we let g = listen to the Grateful Dead and h = is a hippie, then this is the argument:
Premise g → h
Premise ~ g
Conclusion: ~ 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.
Try it Now 18
Is this argument valid?
Premise: If a hockey player trips an opponent, he will be assessed a 2-minute penalty.
Premise: Alexei did not trip an opponent.
Conclusion: Alexei will not be assessed a 2-minute penalty.
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 \(\PageIndex{14}\)
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.
Solution
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 → i
m → ~ d
i → 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 → d; that all babies are despised. Using the contrapositive of the second premise, d → ~m, we can then use the transitive property with b → d to conclude that b → ~m; that babies cannot manage crocodiles. While it is silly, this is a logical conclusion from the given premises.
Example \(\PageIndex{15}\)
Premise: If I work hard, I’ll get a raise.
Premise: If I get a raise, I’ll buy a boat.
Conclusion: If I don’t buy a boat, I must not have worked hard.
Solution
If we let h = working hard, r = getting a raise, and b = buying a boat, then we can represent our argument symbolically:
Premise h → r
Premise r → b
Conclusion: ~ b → ~ h
Using the transitive property with the two premises, we can conclude that h → b; if I work hard, then I will buy a boat. When we learned about the contrapositive, we saw that the conditional statement h → b is equivalent to ~b → ~h. Therefore, the conclusion is indeed a logical syllogism derived from the premises.
Try it Now 19
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.
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0