Logic Review and Exercises
Logic is broadly speaking the study of arguments, or more strictly of inference:



When can a statement be inferred from another statement or set of statements? or:
What statement/s are implied by some statement or set of statements? or:
What statement/s follow from some statement or set of statements?
Logic develops ways of answering these questions.
Key concepts:




Argument: a set of statements in which the truth of one is intended to be supported or demonstrated
by the others.
Conclusion: the statement being supported or demonstrated.
Premises: the statement/s offered in support of the conclusion (reasons).
Inference: the step of reasoning from the premises to the conclusion.
How do you tell if a set of statements is an argument? The key question is: is there an inference? a step of
reasoning? Sometimes that’s indicated by inference indicators—words like: thus, therefore (conclusion
indicators) or because or since (premise indicators).
Examples:
Last week my friend and I rode our bikes through the Santa Ynez Valley. We left from Buellton, rode to Los
Alamos, then to Los Olivos, then back to Buellton. It was beautiful and not too difficult. No argument here,
because there’s no inference.
My friend will win the next cycling race he’s in. He’s been training hard. He bought a lighter bike. The course is
perfect for his strengths. And his strongest competitor is injured. This is an argument because there is an
inference. The first statement is being supported by the following four statements.
But watch out, words that sometimes indicate an inference can be use to indicate an explanation: The
Warriors won because they rebounded well, is an explanation. The difference between an argument and an
explanation is that in an argument the conclusion is in question and the premises (hopefully) are not. The
premises are offered as a reason to believe something you were uncertain of. In an explanation the thing
being explained (the warriors won) is not in question. We’re just looking for how that happened.
Which of these are arguments and which are not?
Each of Julia Roberts’ last ten films grossed over $100,000. So it’s likely that the next one will too.
Julia Roberts must have a husband who does long distance bike races. She was waiting at the finish line
of a recent race in Solvang.
Julia Roberts has made films with some of the best actors of the last half-century. But she never made a
film with Robert Redford.
No humans have gills, but Nemo has gills. So Nemo isn’t a human
I have never seen a person taller than 7 foot 8 inches, so no people are taller than 7 foot 8 inches.
People over 7 feet tall sometimes have mobility problems as they age. They need higher blood pressure,
which means they get winded more easily.
We’ll either go surfing today or do our homework. If the waves are flat, we won’t surf. The waves are flat.
So we’ll do our homework.
There are different kinds of argument, or more precisely, different kinds of inferences.


Deductive arguments are those whose premises are meant to give full support to the conclusion.
Inductive arguments are those whose premises are meant to give a high degree of support to the
conclusion.
The basic difference is the relationship between the information in the premises and the conclusion:


In deductive arguments the information contained in the conclusion is already implicitly contained in
the premises. The conclusion is intended just to make explicit information that is already contained
implicitly in the premises. The inference is explicative.
In inductive arguments the information contained in the conclusion goes beyond that contained in
the premises. That means the premises can never make the conclusion absolutely certain. The
inference is ampliative.
But again, be careful. Sometimes deductive arguments fail. Their premises do not offer the full support for the
conclusion that the arguer intended. That does not make them inductive arguments. Here’s an example:
If the warriors rebound well, they win. They did win. So they must have rebounded well.
That argument is deductive, but the conclusion does not follow from the premises. It is a failed deductive
argument, not an inductive argument.
Common forms of inductive arguments are generalizations and predictions
In a generalization one observes some characteristic in a limited number of some kind of thing and infers that
all of that kind of thing has that characteristic. Every swan I’ve ever seen is white, therefore all swans are white.
In a prediction one observes some characteristic in a limited number of some kind of thing and infers that
some unobserved instance of that kind of thing has that characteristic:
I’ve never seen my cousin’s pet swan, but every swan I’ve ever seen is white, therefore my cousin’s pet
swan is white.
Statistical reasoning is a common form of inductive reasoning.
Which of these arguments are inductive and which deductive?
Each of Julia Roberts’ last ten films grossed over $100,000. So it’s likely that the next one will too.
No humans have gills, but Nemo has gills. So Nemo isn’t a human
I have never seen a person taller than 7 foot 8 inches, so no people are taller than 7 foot eight inches.
We’ll either go surfing today or do our homework. If the waves are flat, we won’t surf. The waves are flat.
So we’ll do our homework.
It has been raining here for a week straight. So the skiing conditions in Big Bear are going to be perfect.
Evaluating reasoning
When we evaluate an argument we’re basically asking if we should accept its conclusion. There are two
questions to ask in evaluating an argument:


Are the premises true?
Is the inference good?
If the premises aren’t true, then, even if the conclusion follows from them, they don’t give us a reason to think
the conclusion is true
If the inference isn’t good, then even if the premises are true, they don’t give us a reason to think the
conclusion is true. They’re not related to the conclusion in a supportive way.
Deductive and inductive inferences are evaluated differently since they have different ambitions. In a
deductive argument the premises are meant to give absolute support to the conclusion, but in an inductive
argument can never do that by its very nature.
Key concepts:
Deductive Validity: A deductive inference is valid if, given the truth of the premises it is impossible for the
conclusion to be false.
If a deductive argument has a valid inference and its premises are true, it is sound.
Inductive Strength: An inductive inference is strong if, given the truth of the premises, it is highly likely that
the conclusion is true.
If an inductive argument has a strong inference and its premises are true, it is cogent.
Logicians typically leave inductive reasoning to the statisticians, and concern themselves with just deductive
reasoning.
But before we leave inductive reasoning, let me give a couple of key concepts regarding inductive
generalizations:
Inductive generalizations start from the observation of some feature in a limited number of some kind of
thing and infer that all of that kind of thing have that feature. The limited number of the kind of thing
observed is called the sample. The whole group about which you’re drawing a conclusion is called the target
population
The key to the strength of an inductive generalization is whether the sample is like the target population is
the relevant ways. Is it representative? The more representative, the stronger the inference.
Representativeness is hard to ensure directly, so statisticians use a substitute that gives approximate
representativeness. They try to construct a random sample. In a random sample, each member of the target
population must have the same chance of being selected in the sample.
Logicians also leave the question of the truth of an argument’s premises to the relevant experts.
They concern themselves just with the validity of deductive inferences, and in particular with ways of
determining whether a deductive inference is valid or invalid.
The crucial idea in deductive logic is that the validity of an argument is a matter of the argument’s
form.
If an argument is valid, all arguments that have its form will be valid. It will have a valid argument
from.
But discerning the form of an argument takes a little work.
There have been two different systems developed to discern the form of an argument. Sometimes
arguments are about categories of things. All presidents have been men. All men have inflated egos. So all
presidents have had inflated egos. Here we might capture the form by seeing how the categories are related to
each other. The category “presidents” is part of the larger category “men”. The category “men” is part of the
larger category “those with inflated egos.” So the category “presidents” is part of the category “those with
inflated egos.” This way of thinking about the form of arguments is called categorical logic, and it’s very
powerful. But it doesn’t capture all natural language arguments.
An alternate system, the one we’ll study, takes sentences as the basic building block of arguments, rather
than categories. It’s called sentential logic. It recognizes that some statements are made up of more basic
statements that are being operated on by some word or phrase. These are called complex statements.
Statements that aren’t made up of simpler components are called simple statements.
Logicians identify five kinds of complex statements distinguished by the word or phrase that is operating on
the components (the logical operator).
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



Negation (It is not the case that…)
Conjunction (both… and…)
Disjunction (either… or…)
Conditional (if… then…)
Biconditional (… if and only if…)
Note that negations are the odd case of a one-component complex statement. Note too that English has many,
many ways of expressing each of these kinds of statements.
Each logical operator has a symbol:





Negation (It is not the case that…)
Conjunction (both… and…)
Disjunction (either… or…)
Conditional (if… then…)
Biconditional (… if and only if…)
~
•
v
-> (typically a right-facing horseshoe)
= (that’s supposed to be three horizontal bars)
That give us five standard forms of complex statements
 ~p
negation
 p•q
conjunction
 pvq
disjunction
 p -> q conditional
 p=q
biconditional
Note that the components of conjunctions, disjunctions and conditionals have special names: conjunctions
have conjuncts, disjunctions have disjuncts and conditionals have antecedents (before the arrow) and
consequents (after the arrow). More on antecedents and consequents below.
Are the following statements simple or complex? If they’re complex what is their form?




If the Lakers win, they will make the playoffs.
President Obama will either sign this legislation or the Republicans will win the next election.
President Obama won’t yield on cutting defense spending.
My dog will either sleep all day or bark at the neighbor’s cat.
More challenging—these are complex statements with complex components. What is the form of the overall
statement?
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


If the Lakers don’t win, they won’t make the playoffs.
President Obama will either sign this legislation or the Democrats won’t win the next election.
My dog will take a long walk and either sleep all day or bark at the neighbor’s cat.
President Obama won’t yield on cutting defense spending and if he doesn’t, the Democrats won’t win the
next election.
A key element in logic is translating statements from a natural language, like English, to the artificial
language that allows us to see just the form of statements.
The steps to translating are simple, but you have to be very precise. A crucial rule is to work from the outside
in. Discern the form of the overall statement first, then of its components, then of their components, and so
forth.
1. Is the statement simple or complex?
1a. If simple, assign it a letter (B = Jim is a bachelor) and write the letter. You’re done.
1b. If complex, write parentheses (with plenty of space between).
2. What kind of complex statement is it? (negation, conjunction, disjunction, conditional,
biconditional). Write the symbol for that kind of complex statement between the parentheses.
3. Look at the first component of the complex statement (if a negation, there will be only one). Is it
simple or complex?
3a. If simple, assign it a letter, write the letter in the appropriate place and go to step 1 for the next
component. If there are no more components, you’re done.
3b. If complex, write parentheses.
4. What kind of complex statement is the component? (negation, conjunction, disjunction, conditional,
biconditional). Write the symbol for that kind of complex statement between the parentheses.
Go back to 3 and repeat 3 and 4 until you have symbolized every component down to its simple
statements and logical operators.
If I don’t get married, I will travel to Europe and buy a motorcycle.
1.
Is the overall statement simple or complex? Complex!
1b. (
2.
)
What kind of complex statement is the overall statement? Hmm…Conditional!
(
->
)
3. Is the first component (I don’t get married) simple or complex? Complex!
3b. Write parentheses where that component goes
((
) ->
)
4. What kind of complex statement is that component? Negation! Write the symbol for that kind
of complex statement inside the parenthesis (toward the left in the case of negation).
( (~
) ->
)
Repeat 3 for the component being negated (I get married). Is it simple or complex? Simple!
3a. M = I get married
( (~ M) ->
)
You’re done with the antecedent of the overall conditional.
Go on to the next consequent, I will travel to Europe and buy a motorcycle.
Repeat step 3. Is it simple or complex? Complex!
3a. Write parentheses where that component goes:
( (~ M) -> (
))
Repeat 4 for this component. What kind of complex statement is it? Conjunction!
Write the symbol for conjunction inside the parentheses.
((~ M) -> (
•
))
Repeat 3 for the first conjunct (I will travel to Europe). Is it simple or complex? Simple!
3a. E = I will travel to Europe
((~ M) -> ( E •
))
Repeat 3 for the second conjunct (I will buy a motorcycle). Is it simple or complex? Simple!
3a. B = I will buy a motorcycle
((~ M) -> ( E • B ) )
We’re done! We’ve seen that, If I don’t get married, I will travel to Europe and buy a motorcycle. is a
conditional whose antecedent is the negation of a simple statement and whose consequent is the
conjunction of two simple statements.
Try these:
If the Lakers don’t win, they won’t make the playoffs.
President Obama will either sign this legislation or the Democrats won’t win the next election.
My dog will take a long walk and either sleep all day or bark at the neighbor’s cat.
We can distinguish different statement forms by how the world would have to be in order for them to be true
(their truth conditions). Take the statement:
I won’t stay up past 10:30.
You can see it’s a negation whose component is I stay up past 10:30. What has to be the case about that
component in order for the negation I won’t stay up past 10:30. to be true? The component has to be false. And
if the component is true (I do stay up past 10:30) the original statement will be false.
We can put this in a table (called a truth table):
Let S = I stay up past 10:30
S | ~S
t | f
f | t
The column on the left represents all the ways the world can be with respect to the statement’s component, S.
S could be true (t) or false (f). These “ways the world can be with respect to S” are called valuations. The
column on the right shows the truth or falsehood (the truth value) of the complex statement ~S for each
valuation. When S is true, ~S is false; when S is false, ~S is true.
If we let a lower case italicized letter, say p, stand for any statement we can make a truth table for negation:
p | ~p
t | f
f | t
And we can do the same for all other statement forms.
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

Conjunctions are true when both conjuncts are true; otherwise they’re false.
Disjunctions are true if either or both of their disjuncts is true; if both are false, the disjunction is
false.
Conditionals are false only if the antecedent is true and the consequent is false; otherwise they’re
true.
Biconditionals are true if both their components are true or both are false; otherwise they’re false.
You can see this on the tables. (Memorize these—except for biconditional—you’ll need them later!)
Negation
p | ~p
t | f
f | t
Conjunction
p q|p•q
t t| t
t f| f
f t| f
f f| f
Disjunction
p q|pvq
t t| t
t f| t
f t| t
f f| f
Conditional
p q | p -> q
t t| t
t f| f
f t| t
f f| t
Biconditional
p q | p <-> q (Double arrow replaces three bars.)
t t| t
t f| f
f t| f
f f| t
These statement forms allow us to see the form of arguments. All we have to do is translate the premises and
conclusion in English (or whatever language we speak) into our symbolic language.
Try these:

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

If I get married, I’ll be happy the rest of my life. I’ll get married. So I’ll be happy the rest of my life.
If I get married, I won’t be happy the rest of my life. I won’t get married. So I’ll be happy the rest of my
life.
I’m either going to cook dinner or mow the lawn. I’m not going to mow the lawn. So I’ll cook dinner.
If I get married, I’ll buy a motorcycle with my spouse and we’ll ride across the country. I’m getting
married. So my spouse and I will ride across the country.
Some deductive argument forms are so common that they’ve been given standard names and you can just
memorize whether they’re valid or invalid.
If the Lakers win, they will make the playoffs. They will win, so they will make the playoffs.
The first premise of this argument is a conditional. The second premise affirms the antecedent of the
conditional. The conclusion infers the consequent. The form is:
p –> q
p_____
q
That form is called Modus Ponens (Affirming the Antecedent) and, by the way, it is valid:
Here are some other valid forms:
Modus Tollens (Denying the Consequent):
p –> q
~q_____
~p
Denying the Antecedent
Hypothetical Syllogism (Conditional
Chain)
Disjunctive Syllogism




p –> q
~p_____
~q
Affirming the Consequent
p –> q
q –> r__
p –> r
pvq
~p___ or
q
And here are some invalid forms:
p –> q
q_____
p
pvq
~q___
p
What is the form of the following arguments? Are they valid or invalid?
If I get married, I’ll be happy the rest of my life. I’ll get married. So I’ll be happy the rest of my life.
I’m either going to cook dinner or mow the lawn. I’m not going to mow the lawn. So I’ll cook dinner.
If I get married, I’ll buy a motorcycle. If I buy a motorcycle, my spouse and I will ride across the country.
So if I get married, my spouse and I will ride across the country.
 If I stay single, I’ll be happy the rest of my life. I won’t be happy the rest of my life. So I won’t stay single.
 If I get married, I’ll buy a motorcycle. I’m not getting married. So I’m not going to buy a motorcycle.
A little harder:
 If I get married, I won’t be happy the rest of my life. I won’t get married. So I’ll be happy the rest of my
life.
The reason that Modus Ponens and Modus Tollens are valid and that denying the antecedent and affirming
the consequent are invalid has everything to do with what conditional statements assert.
Conditional statements assert a conditional relationship between their two parts: their antecedent and their
consequent. Actually they assert two conditional relationships:


They assert that the antecedent (the if part in the standard if…then… form) is a sufficient condition
for the consequent (the then part in standard form).
They assert that the consequent is a necessary condition for the antecedent.
One state of affairs (a way that things might be) is a sufficient condition for another if the first guarantees
that the other is the case.
One state of affairs is a necessary condition for another if its not being the case guarantees that the other
isn’t the case.
Under the current graduation requirements, graduating from Westmont is a sufficient condition for having
passed Philosophical Perspectives. If you know someone has graduated, you know they passed Philosophical
Perspectives.
Conversely, passing Philosophical Perspectives is a necessary condition for graduating from Westmont. If you
know someone has not passed Philosophical Perspectives, you know they haven’t graduated.
Notice there’s always this reciprocal relationship: If A is a sufficient condition for B, B will be a necessary
condition for A, and vice versa.
Notice too that if a sufficient condition is not met, you know nothing about whether the thing it’s a condition
for is the case. If someone doesn’t graduate from Westmont, do you know whether they’ve passed
Philosophical Perspectives? No!
And if a necessary condition is met, you know nothing about whether the thing it’s a condition for is the case.
If someone passes Philosophical Perspectives, do you know that they have graduated? No!
Conditional statements assert these relationships between their components. That’s a good way to tell what
the parts are: which component is the antecedent and which is the consequent. The antecedent is the part
that’s being offered as a sufficient condition for the other part. The consequent is the part that’s being offered
as a necessary condition for the antecedent.
The standard form of the conditional (if… then…) emphasizes the sufficiency of the antecedent for the
consequent: If Elisa graduated from Westmont, then she passed Philosophical Perspectives. But it’s also
asserting that the consequent is a necessary condition for the antecedent. Again, that’s because, if A is a
sufficient condition for B, B will be a necessary condition for A, and vice versa.
English also has ways of emphasizing the necessity of the consequent for the antecedent:
Elisa graduated from Westmont, only if she passed Philosophical Perspectives. stresses that passing philosophy
is necessary for graduation. But it still implies that graduating is a sufficient condition for having passed
philosophy.
Notice that this example shows that only if means something different than if! Only if introduces a necessary
condition for something else. If introduces a sufficient condition for something else. (What’s nasty is that
sometimes only if gets split: Elisa only graduated from Westmont if she passed Philosophical Perspectives.
That’s a case of only if and passing philosophy is being offered as a necessary condition for graduating.)
What are the antecedents and the consequents in these conditional statements? (Remember, the best way to
do this is to think about what’s being offered as a sufficient condition for what, and what’s being offered as a
necessary condition for what.)
If Obama signs that legislation, his party will lose the presidency.
I’m going surfing only if the waves are up.
I’m only going surfing if I’m done with my homework.
The Democrats will win the next election if Obama turns the economy around (Careful here—
sometimes the antecedent comes last and the consequent comes first.)
We can now see why Modus Ponens (affirming the antecedent) and Modus Tollens (denying the consequent)
are valid and why Denying the Antecedent and Affirming the Consequent are invalid.
In Modus Ponens, one premise asserts that one state of affairs (A) is a sufficient condition for another (B). The
other premise asserts that that sufficient condition (A) has been met. What do we know then about B? That it
is the case! Which is what the conclusion states.
If Obama signs that legislation, his party will lose the presidency. He’s signing the legislation. So his party will
lose.
Of course the premises might be false. But the point is, if they’re true the conclusion can’t be false.
In Modus Tollens, one premise asserts that one state of affairs is a necessary condition for another, and the
next premise asserts that that necessary condition is not met. What do we know then about the thing it’s a
necessary condition for? That it’s not the case! Which is what the conclusion states.
I’m going surfing only if the waves are up. The waves aren’t up. So I’m not going surfing.
But if we know that a sufficient condition is not met, we know nothing about whether the thing it’s a condition
for is the case. So denying the antecedent is invalid:
If Obama signs that legislation, his party will lose the presidency. He’s won’t sign the legislation. So his party will
not lose. Invalid!
And if a necessary condition is met, you know nothing about whether the thing it’s a condition for is the case.
So affirming the consequent is invalid:
I’m going surfing only if the waves are up. The waves are up. So I’m going surfing.
Invalid! The first premise does not say that good waves are a sufficient condition for going surfing. Only that
they are a necessary condition for going surfing.
I said before that the fact validity is a matter of form. That gives us a way to tell if an argument is valid or
invalid. If we can find an argument of its form that has all true premises and a false conclusion, (a
counterexample) we will know the form is invalid, so we’ll know the argument we’re considering is invalid.
(An argument with all true premises and a false conclusion can’t be valid—for valid arguments the truth of
the premises guarantee the truth of the conclusion. And if one instance of an argument form is invalid, then
the form is invalid. So the argument we’re considering that share that form is invalid.)
Sometimes we can just think of a counterexample:
I will graduate from Westmont only if I pass Philosophical Perspectives. I’m going to pass. So I’m going
to graduate.
has the same form as
Romney is president only if he’s a US citizen. He is a US citizen. So he’s president.
But that argument has true premises and a false conclusion. So it can’t be valid. So no argument of its form
can be valid. So the argument we’re considering can’t be valid.
But trying to find a counterexample is unreliable. If we think of one, we’ll know the argument is invalid. But if
we don’t think of one, we can’t really be certain the argument is valid— there are countless arguments of any
given form, and we just might not have discovered the counterexample yet.
We need a way to systematically search all the possibilities. That’s what logicians give us a way to do. Rather
than looking at every possible argument of a given form, we can look at the form itself and look at all the ways
the world can be with respect to the truth or falsehood of the elements of the argument.
We do this with more elaborate versions of the truth tables we saw above for standard statement forms. This
time instead of just single statements, we’ll make a table for the whole argument. Tables like these will give us
a way to determine whether an argument is valid or invalid. Here’s the process:
1. We break each statement in the argument down to its form.
2. Identify all the ways the world can be with respect to the truth or falsehood of the components that appear
in the argument form (identify all the possible valuations).
3. Then figure out the truth or falsehood of each premise and each conclusion for each valuation.
4. Then look to see if any of those valuations results in all true premises and a false conclusion.
5. If there is a valuation that results in all true premises and a false conclusion, the argument is invalid.
6. If there is no valuation that results in all true premises and a false conclusion, the argument is valid.
If I don’t get married, I’ll travel to Europe.
I will travel to Europe.
So, I won’t get married.
1. We break it down to its form. (Translate from natural language to symbolic language.)
M = I get married; E = I travel to Europe
((~M) -> E )
E_______________
~M
2. Identify all the ways the world can be with respect to the truth or falsehood of the components that appear
in the argument form. (Make a table of valuations with each simple component that appears in the argument
across the top and Ts and Fs in columns below them in a way that captures all possibilities. The simplest way
to do this is start with the right column and alternate Ts and Fs, then move to the next column to the left and
double the Ts and Fs. If there is one component there will be two rows—two valuations; if there are two
components there will be four rows; if three components eight rows, and so forth.)
M
t
t
f
f
E
t
f
t
f
3. Then figure out the truth or falsehood of each premise and each conclusion for each valuation. Here you
need to use the truth tables for each kind of statement. And you need to work from the inside out.
M
t
t
f
f
E
t
f
t
f
| ((~M) -> E )
| f
t
| f
t
| t
t
| t
f
step1 2
|
|
|
|
|
E
t
f
t
f
3
|
|
|
|
|
~M
f
f
t
t
4
step 1 applies the true table for negation to the truth values under M to get the truth values for ~M
step 2 applies the truth table for conditional to the truth values for the conditional with ~M as the antecedent
and E as the consequent. This is the truth value of the first premise for each valuation. You’ll no longer need
the column in step one.
step 3 doesn’t need to apply a truth table since the second premise E is a simple statement. Just write the
truth values that correspond to the table of valuations under E.
step 4 again applies the truth table for negation to the column under M in the table of valuations.
4. After you’ve completed the truth table, look to see if any of those valuations results in all true premises and
a false conclusion. Look across each row for t under the first premise, t under the second premise and f under
the conclusion. Remember, for the first premise the relevant column is above the 2. We just used the column
above step 1 to get to that. Ignore the column above step 1.
5. Is there is a valuation that results in all true premises and a false conclusion? Yes! The argument is invalid.
M
t
t
f
f
E
t
f
t
f
| ((~M) -> E )
| f
t
| f
t
| t
t
| t
f
step1 2
|
|
|
|
|
E
t
f
t
f
3
|
|
|
|
|
~M
f
f
t
t
4
Try these:
If Obama signs that legislation, his party will lose the presidency. He’s signing the legislation. So his party will
lose.
If I get married, I’ll be happy the rest of my life. I’ll get married. So I’ll be happy the rest of my life.
I’m either going to cook dinner or mow the lawn. I’m not going to mow the lawn. So I’ll cook dinner.
I’m either going to cook dinner or mow the lawn. I’m going to mow the lawn. So I’m not going to cook dinner.
If I get married, I’ll buy a motorcycle. If I buy a motorcycle, my spouse and I will ride across the country. So if I get
married, my spouse and I will ride across the country.
If I stay single, I’ll be happy the rest of my life. I won’t be happy the rest of my life. So I won’t stay single.
If I get married, I’ll buy a motorcycle. I’m not getting married. So I’m not going to buy a motorcycle.