CRITICAL THINKING
MAKING SENSE OF ARGUMENTS
Deductive v Inductive Arguments
Deduction
Definitions of Validity and Soundness
Truth, Validity and Soundness
Common Deductive Argument Forms
Induction
Definitions of Strength and Cogency
Truth, Strength and Cogency
Common Inductive Argument Forms
Judging Arguments
Finding Missing Parts
LECTURE
PROFESSOR JULIE YOO
DEDUCTIVE VS INDUCTIVE ARGUMENTS
An inference is the reasoning involved in drawing a conclusion. Arguments display the
inference in a chain of reasoning by specifying the evidence or reasons for the conclusion. There
are two general classes of arguments, deductive (D) and inductive (I), and they form two
different classes because they differ in the relationship between premises and conclusion.
DEDUCTIVE
INDUCTIVE
The information expressed by the conclusion is
fully “contained” in the information expressed
by the premises. The conclusion does not say
anything more what the premises say.
The information expressed by the conclusion
“goes beyond” the information expressed by
the premises. The conclusion does say
something more than what the premises say.
The purpose of giving a deductive argument is
to show that the conclusion is fully contained in
the premises.
The purpose of giving an inductive argument is
to generate new knowledge in the conclusion,
based on the premises.
In a deductive argument, the conclusion is
meant to follow from the premises with
necessity. For example:
In an inductive argument, the conclusion is
meant to follow from the premises with
probability. For example:
1. If it rains, then we stay home.
2. If we stay home, we play cards.
3. If it rains, we play cards.
1. It rained last winter.
2. It rained the winter before.
3. It will rain next winter.
Notice how the premises fully contain the
conclusion.
Notice how the premises support, but do not
contain, the conclusion.
Arguments do not announce whether they are deductive or inductive. Whether an argument is
deductive or inductive depends upon the type of inference the arguer intends to use. There are
times when you want to argue for a point inductively and times when you want to argue for a
point deductively. One type of argument is not better than the other.
DEDUCTION
Definition of Validity and Soundness
Not all deductive arguments are good or successful. The ones that are good are sound, which
means that it lives up to two virtues: 1) valid argument structure and 2) true premises. The issue
of whether the premises are true is a matter of checking the facts. It is the concept of validity
that needs explanation. This concept is defined in the following way:
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Soundness: An argument is sound =
are true.
dfn
The argument is valid AND all of its premises
Validity: An argument is valid = dfn The conclusion is fully contained in the premises; it
is impossible for all the premises to be true and the conclusion false.
Explanation: The definition validity can be hard to grasp, but if you remember to separate the
issue of an argument’s validity from the truth or falsity of the premises, then you will understand
the concept of validity more easily. To say that an argument is valid is to say that the premises
fully contain the conclusion, and it does this is by following a correct form of reasoning. An
argument that is not valid does not fully contain the premises in the conclusion. Now, suppose
we have a valid argument and the premises happen to be true. Then this would guarantee a true
conclusion, since the premises of a valid argument fully contain the conclusion. If we are
dealing with an invalid argument, on the other hand, then all bets are off; we cannot be
guaranteed that the conclusion is true, nor can we be guaranteed that the conclusion is false.
Whereas a valid argument has the power to preserve the truth of the premises, an invalid
argument lacks this power. This diagram makes the point visually:
Valid: The premises contain the conclusion.
The conclusion directly follows. Example:
1. If you ate an apple, then you ate a fruit.
2. You didn’t eat any fruit.
NOTICE HOW (1) AND (2) CONTAIN (3)
3. You didn’t eat an apple.
Making Sense of Arguments
Invalid: The premises don’t contain the conclusion.
The conclusion doesn’t follow at all. Example:
1. You love junk food.
2. It’s the day after Halloween.
NOTICE HOW (1) AND (2) DO NOT CONTAIN (3)
3. You didn’t an apple.
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Consider these two common invalid argument forms. The first is known as the fallacy of
affirming the consequent:
If P, then Q
Q
P
If the car is scarlet, then it is red.
The car is red.
The car is scarlet.
The other is known as the fallacy of denying the antecedent:
If P, then Q
not-P
not-Q
If the car is scarlet, then it is red.
The car isn’t scarlet.
The car isn’t red.
Notice how the premises don’t lead to the conclusion. Because the reasoning is incorrect, the
conclusion does not have the power to preserve the truth of the premises, should the premises
actually turn out to be true. That doesn’t mean that the conclusion will be false with true
premises; it just means that the argument’s conclusion won’t preserve the truth of the true
premises.
Chart for Truth and Validity and Soundness
In this chart , we display all the various the combinations of valid and invalid argument forms
with true and false premises and conclusions.
Valid
⇓
True Ps If Einstein did physics, then he was a scientist.
True C Einstein did physics.
Einstein was a scientist.
(sound – modus ponens)
Invalid
⇓
If Einstein did physics, then he was a scientist.
Einstein was a scientist. (TRUE)
Einstein did physics. (TRUE)
(unsound – affirming the consequent)
True Ps
False C
If Einstein was a dancer, then he was human.
Einstein wasn’t a dancer. (TRUE)
Einstein wasn’t human. (FALSE)
(unsound – denying the antecedent)
False P(s) If you live in LA, then you live in Mexico.
True C If you live in Mexico, then you live in CA .
IF you live in LA, then you live in CA.
(unsound)
If Einstein was smart, then he was a dancer.
Einstein was a dancer. (FALSE)
Einstein was smart. (TRUE)
(unsound – affirming the consequent)
False P(s) All graphic novels are colored.
False C All colored things are purple.
All graphic novels are purple.
(unsound)
If Einstein was smart, then he was a scientist.
Einstein wasn’t smart. (FALSE)
Einstein wasn’t a scientist. (FALSE)
(unsound – denying the antecedent)
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Note On Usage of Terms: It’s important to understand that statements are not the kinds of things
that are valid or invalid. Validity is a feature of arguments, not premises. And arguments are not
the kinds of things that are true or false. Truth is a feature of statements, not arguments.
Therefore, it is incorrect to say things like. “This premise is valid.” “That argument is true.”
COMMON DEDUCTIVE ARGUMENT FORMS
Modus Ponens (MP)
If P, then Q
P
Q
If the car is scarlet, then it is red.
The car is scarlet.
The car is red.
Modus Tollens (MT)
If P, then Q
not-Q
not-P
If the car is scarlet, then it is red.
The car isn’t red.
The car isn’t scarlet.
Hypothetical Syllogism (HS)
If P, then Q
If Q, then R
If P, then R
If the car is scarlet, then it is red.
If the car is red, then it is colored.
If the car is scarlet, then it is colored.
Disjunctive Syllogism (DS)
Either P or Q
not-P (or not-Q)
Q (or P)
Either Jane bought eggs or milk.
Jane didn’t buy eggs.
Jane bought milk.
Categorical Syllogisms
All A’s are B’s
x is an A
x is a B
(There many more categorical syllogism forms than these.)
All men are mortal.
Socrates is a man.
Socrates is mortal.
All A’s are B’s
All B’s are C’s
All A’s are C’s
All apples are fruit.
All fruit is healthy.
All apples are healthy.
All A’s are B’s
No B’s are C
No A’s are C.
All men are mortal.
No mortal things live forever.
No men live forever.
All A’s are B’s
Some A’s are not C
Some B‘s are not C
All men are mortal.
Some men are not omnivores.
Some mortals are not omnivores.
No A’s are B’s
x is an A
x is not B
No car keys open office doors.
John has a set of car keys.
John’s keys do not open office doors.
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INDUCTION
Definition of Strength and Cogency
Just as with the case of deductive arguments, not all inductive arguments are good or successful.
The ones that are good are cogent, which means that it lives up to two virtues: 1) strong
argument structure and 2) true premises. As with deductive arguments, the issue of whether the
premises are true is a matter of checking the facts. The new concept that needs explanation is
inductive strongth. This concept is defined in the following way:
Cogency: An argument is cogent = dfn The argument is strong and all of its premises are
true.
Strength: An inductive argument is strong = dfn The conclusion generates new
knowledge based on the premises; it is improbable for all the premises to be true and the
conclusion false.
Explanation: Unlike valid arguments, inductively strong arguments do not have conclusions that
follow necessarily from the premises. They follow only with a degree of probability. Both
deductive and inductive arguments need to be evaluated according to two separate criteria: 1)
the structure of the argument and 2) the truth of the premises. In the case of deductive
arguments, structure is a matter of validity or invalidity. In the case of inductive arguments,
structure is a matter of degrees of strength.
Depending on how much data you have in your premises, the premises can support a conclusion
with a high degree of probability or a low degree of probability. Strength comes in degrees,
unlike validity, which is a binary (yes/no) affair. We can think of an argument’s strength as a
function of its sample size: when it comes to concluding that it will rain next winter, a data set
of only two rainy winters is not as strong as a data set of 5, 15, or 50 rainy winters.
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Strong: In a strong inductive argument, the
conclusion does not try to go too far beyond
the premises. Or the same point differently,
the premises provide a big base for the
conclusion. Example:
Weak: In a weak inductive argument, the
conclusion tries to go far beyond the premises.
Or the same point differently, the premises
provide a small base for the conclusion,
making the conclusion a big leap. Example:
1. 98% of dogs like the chase cats.
2. Fido is a dog.
1. 18% of dogs like the chase squirrels.
2. Fido is a dog.
NOTICE GOOD SUPPORT
NOTICE THE POOR SUPPORT
3. Fido likes to chase cats.
With a figure like 98%, the chances are pretty
good that Fido likes to chase cats.
3. Fido likes to chase squirrels.
With a figure like 18%, the chances are pretty
bad that Fido likes to chase squirrels.
Truth, Strength, Cogency
This chart displays all the possible ways we can separate and combine strength and weakness
with true and false premises:
Strong
Weak
⇓
⇓
True Ps 95% of US presidents are college graduates. 15% politicians are male.
True C The current president is a college graduate.
The current VP is male.
The next president will be a college The next VP will be male.
graduate.
(uncogent)
(cogent)
True Ps
False C
15% presidents are bald.
The current pres is bald.
The next pres will be bald.
(uncogent)
False P(s) 95% US presidents are under 30 years old.
True C The current president is under 30 years old.
The next pres will be under 30 years old.
(uncogent)
15% presidents lived in CA.
15% of CA residents are US citizens.
The next presidents will be a US citizens.
(uncogent)
False P(s) 95% US presidents were born in CA.
False C The current president was born in CA.
The next pres will be from CA.
(uncogent)
15% presidents lived in CA.
15% of CA residents are females.
All presidents are females.
(uncogent)
Note On Usage of Terms: As a technical matter of fact, all inductive arguments are invalid. But
this does not make all invalid argument inductive. Whether an argument is deductive or
inductive is a function of what the argument-giver intends to demonstrate. If she wants to show
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that the conclusion can give us further knowledge on the basis of the premises, then she intends
to give an inductive argument. On the other hand, if her intention is to show that the conclusion
is contained in the premises, then she intends to give a deductive argument.
COMMON INDUCTIVE ARGUMENT FORMS
Inductive Generalization
something that is A is also B
another things that is A is also B
another things that is A is also B
…
All A’s are B’s
(Let “A” and “B” be any feature.)
This bit of metal conducts electricity.
Another bit of metal conducts electricity.
A different bit of metal conducts electricity..
…
All metals conduct electricity.
Enumerative Induction
something that is A is also B
another things that is A is also B
another things that is A is also B
…
The NEXT A will also be B
This bit of metal conducts electricity.
Another bit of metal conducts electricity.
A different bit of metal conducts electricity..
…
The new bit of metal should conduct electricity.
Prediction
C causes E
C occurred
E will occur
(Let “C” mean the cause and “E” mean the effect.)
Drinking a lot of beer causes getting drunk.
Mo is drinking a lot of beer.
Mo is getting drunk.
Identifying the Cause
a, b, c, d…got E.
a, b, c, d, all share C.
C caused E
(Let “C” mean the cause and “E” mean the effect.)
Alf, Bob, Cam, and Dan got sick.
They all had mayo on their sandwiches.
The mayo caused the sickness.
a has E but b does not.
a and b differ by C.
C caused E
Alf got sick but Bob did not.
Alf ate mayo but Bob did not.
The mayo caused the sickness.
C is increased/decreased
E is increased/decreased
C causes E
Mo is exercising more.
Mo is getting stronger.
Exercises causes strength.
The strength of a causal argument is very hard to assess, because in most realistic cases, there are
several different causes of one and the same effect (think of all the different things that can cause
a car accident – speeding or slippery roads or bad breaks or texting, etc.). So we cannot always
just to a conclusion about what caused what, even when we have a causal generalization at hand.
For causal arguments, the best way to determine whether we have a strong argument is to do lots and
lots of research. It is not one of those things we can just judge on the spot. (This will be covered in
greater detail later in our course.)
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Analogical Induction
x and y are similar in having A1 A2 … An
x is also B
y is also B
(Let “A” and “B” be any feature.)
Mo and Jo are smart, eager, and diligent.
Jo is a good student.
Mo is a good student.
Inference to the Best Explanation
Facts F1, F2, …, Fn call out for explanation.
E is the best explanation of F1, F2, …, Fn.
E is probably true.
JUDGING ARGUMENTS
Judging arguments is not always easy. The process I describe adds a step to the four-step
procedure Vaughn recommends. I break up Vaughn’s step 2 into two separate steps.
Identify
CONCLUSION and
PREMISES.
Determine whether the argument is
DEDUCTIVE or INDUCTIVE.
Is conclusion contained in premises?
Is conclusion only supported by premises?
contained: the argument is deductive
so you need to check for validity.
invalid: unsound.
a false premise(s):
unsound.
valid: then ask whether
all the premises are true.
all true premises:
SOUND.
Making Sense of Arguments
supported: the argument is inductive
so you need to check for strength.
weak: uncogent.
a false premise(s):
uncogent.
strong: then ask whether
all the premises are true.
all true premises:
COGENT.
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FINDING MISSING PARTS
Many arguments do not explicitly give all of the premises. In these cases, you have to fill in the
missing premise(s). The principle that guides this endeavor is what one might call “the principle
of charity”: look for the premise(s) that will make the argument valid, if the argument is
intended as a deductive one, or will make the argument strong, if the argument is intended as an
inductive one.
In the case of a deductive argument, search for a premise(s) that would make the argument valid.
Such a premise needs to satisfy two constraints:
a. It is plausible.
b. It fits with the author’s intent.
In the case of an inductive argument, search for a premise(s) that would make the argument
strong. Then follow the steps for judging arguments. Here is an example:
1. There are no students on campus.
2. It is Winter Break.
There is definitely a missing premise in this argument, but the missing premise can be filled in in
a number of different ways:
1. There are no students on campus.
2. If it is not Winter Break, then there are
students on campus.
3. It is Winter Break.
1. There are no students on campus.
2. The best explanation for this fact is that it
is Winter break.
3. It is Winter Break.
(Deductive: Modus Tollens)
(Inductive: Inference to the Best E)
1. There are no students on campus.
2. If there are not students on campus, then it
is Winter break.
3. It is Winter Break.
1. There are no students on campus.
2. During the past 15 years, there were no
students on campus during Winter break.
3. It is Winter Break.
(Deductive: Modus Ponens)
(Inductive: Prediction)
We have all these choices. So which is the best way to fill in the missing premise? The best way
is determined by the intentions of the author. Sometimes you can just ask the person giving the
argument whether they intended to give a deductive argument or inductive argument. But if that
person is not available, then you have to do a very close reading of the text they leave behind and
hope that the context of their writing indicates what kind of argument they had in mind.
Whatever the situation, you do your best to construct the best argument the author intended to
give, then evaluate that argument on the basis of the best reconstruction you can give.