TWO KINDS OF REASONING
1. Deductive Logic
Logic is the study of arguments.
An argument in logic consists of two things:
1. A set of propositions – the premises
Which are presented as reasons or support
for:
2. A further proposition – the conclusion
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What are propositions?
Propositions are identified by English
sentences. A proposition is the claim made
by a sentence.
Logic analyzes propositions, i.e. definite
claims. Logic does not analyze:
1. Questions (what time is it?)
2. Commands (close the door.)
3. Requests (hand me my briefcase.)
4. Exclamations (ah hah!)
Etc.
The problem (for logic) with these things is
that they are neither true nor false.
Logic only studies things that are true-orfalse – in other words, propositions.
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Here is an example of an argument:
All humans are mortal.
Socrates is human.
Therefore:
Socrates is mortal.
(premises)
(conclusion)
An argument is good when it is truth
preserving, in other words when:
It is logically impossible for the conclusion to
be false given that the premises are true
(anyone who accepts your premises must
accept your conclusion).
Such arguments are called deductively
valid (or, more simply, valid).
An invalid argument is an argument that is
not valid, i.e. an argument where it is
possible for the premises to be true and the
conclusion false.
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Validity can be seen more easily if we look
at the logical form (structure) of an
argument.
What is logical form?
To begin, consider these arguments:
Either John went to the store or Sue did.
John did not go to the store.
So, Sue did.
Either Yogi is a dog or Yogi is a bear.
Yogi is not a dog.
So, Yogi is a bear.
Now, these are different arguments but they
have something in common.
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Form vs. content
J: John went to the store
S: Sue went to the Store
B: Yogi is a bear
D: Yogi is a dog
This gives us:
J or S
Not J
So:
D or B
Not D
So:
S
D
These arguments differ in content (we use
different letters to represent different
propositions).
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But if we ignore content we can see that
they have the same form:
p or q
Not p
So,
q
Here we let p and q stand for any
proposition whatsoever.
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Grammatical versus Logical Form
The grammatical form of a proposition (or of
an argument)
 is the structure of the proposition (or
argument) as indicated by the surface
grammar of its natural language
The logical form of a proposition (or of an
argument)
 is the logically effective structure of the
proposition (or argument) as indicated
by the meanings of the logical terms it
contains
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Example
"Tom, Dick and Harry lifted the box"
Grammatical form
 (Tom, Dick, Harry) lifted the box
Potential logical forms
 (Tom, Dick, Harry) lifted the box
 (Tom lifted the box) and (Dick lifted the
box) and (Harry lifted the box)
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Another Example
"I see nobody on the road," said Alice.
"I only wish I had such eyes," the King
remarked in a fretful tone. "To be able to
see Nobody! And at that distance too! Why,
it's as much as I can do to see real people,
by this light!"
Grammatical forms
 I see somebody on the road
 I see nobody on the road
Logical forms
 I see somebody on the road
 It is not the case that (I see somebody
on the road)
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Consider our previous example (on page 3).
It had the following form:
All p’s are q’s
S is a p
So:
S is a q
This is obviously valid.
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Don’t confuse the following:
Propositions are true or false.
For example:
Ottawa is the capital of Canada: true.
Kingston is the capital of Ontario: false.
But:
Arguments are valid or invalid.
There is no such thing as a valid or invalid
proposition.
There is no such thing as a true or false
argument.
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An invalid argument can have any
combination of true or false premises and a
true or false conclusion:
All bears are mammals
(true)
All mammals are warm-blooded
Therefore:
Toronto is the capital of Ontario (true)
If New York is in Canada then it is in
North America.
(true)
New York is in North America.
Therefore:
New York is in Canada.
(false)
Etc…
In these arguments, it is possible for the
premises to be true yet the conclusion false.
They are invalid.
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A deductively valid argument may not have:
1. All true premises and a false
conclusion
A deductively valid argument may have:
2. one or more false premises and a
false conclusion
3. one or more false premises and a
true conclusion
4. all true premises and a true
conclusion
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For example:
If Toronto is in NY, then it is in France.
Toronto is in NY.
So:
Toronto is in France.
This argument is truth preserving because if
the premises are true then the conclusion
must be true.
However, the argument is not sound.
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An argument is sound if and only if:
1. It is valid
and
2. All its premises are true
Example:
The Prime Minister lives in Ottawa.
Ottawa is in Ontario.
Therefore:
The Prime Minister lives in Ontario.
Sound arguments are the best. They are
what we want in everyday life.
So, we can criticize an argument for having
false premises (unsound) or for being a bad
piece of reasoning (invalid).
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However, logic cannot tell us which
propositions are true (we must investigate
the world to find that out). So:
Logic cannot tell us whether an argument is
sound.
(There are two exceptions to this rule which
we shall discuss shortly.)
Logic can tell us whether the premises and
conclusion are logically related so as to be
truth preserving. Logic can determine good
reasoning.
Logic can tell us whether an argument is
valid.
Why?
Good reasoning does not depend on how
the world is; it just depends on the logical
relation between premises and conclusion.
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Deductive validity is a very strict notion.
Deductive logic rejects an argument even if
it is merely possible for the premises to be
true and the conclusion false.
We might think of it this way:
If there is any risk of an argument
leading you astray, deductive logic
rejects the argument. It doesn’t matter
how small the risk is.
Valid arguments are risk-free arguments.
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Three kinds of propositions
Logically necessary
 Can’t be false, no matter how the world
is
 E.g. ‘either John is 6’ tall or John isn’t 6’
tall’
Logically impossible
 Can’t be true, no matter how the world is
 E.g. ‘John is 6’ tall and John isn’t 6’ tall’
Logically contingent
 Neither logically necessary nor
impossible – depend the world.
 E.g. ‘John is 6’ tall’
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A proposition is logically necessary if it is
true in virtue of its logical form. Examples:
 p or not p
 If p, then p
It is logically false if it is false in virtue of its
logical form. For example:
 p and not p
 If p, then not p
So, logic can tell us whether a proposition is
logically necessary or logically impossible.
It cannot tell us whether a contingent
proposition is true or false.
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2. Inductive Logic
In many situations you must use risky
arguments:
1. Should I major in physics or history?
2. Should I buy stock in company x?
3. Should I open my own business?
Etc…
The outcomes of your choices are
uncertain, so to answer these questions you
must reason about risk.
Consider this argument:
All the history majors I talked to got a
good job.
So:
If I major in history I will get a good job.
This argument is invalid: the conclusion
could be false even if the premise is true.
This argument takes a risk.
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The previous argument does not seem very
good. How about this?
Most history majors in this school got
good jobs.
Therefore:
If I major in history I will get a good job.
This is still invalid, but seems better than the
first one.
According to national statistics, 95% of
history majors get good jobs.
Therefore:
If I major in history I will get a good job.
This invalid argument seems better still.
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All of these arguments reason from
populations (history majors) to samples
from that population (you, if you choose to
major in history).
1. We can also reason from samples to
wholes:
Jane, Sarah and Mike all majored in
history and got good jobs.
So:
All history majors get good jobs.
2. Or from sample to sample:
Jane majored in history and she got a
good job
So:
If John majors in history he will get a
good job.
These are all risky arguments.
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Question: How can we evaluate these risky
arguments?
Answer: Using inductive logic.
Question: What’s that?
Answer: It’s what this course is about.
But let’s get started…
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When we use risky arguments, it is natural
to use the word ‘probably’:
According to national statistics, 95% of
history majors get good jobs.
So, probably:
If I major in history I will get a good job.
Jane, Sarah and Mike all majored in
history and got good jobs.
So, probably:
All history majors get good jobs.
If we could come up with a method of
assigning numbers to probability values, we
could more easily evaluate risky arguments.
So:
Probability is a fundamental tool for
inductive logic.
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Even if an argument is invalid, it might be
inductively strong because the premises
make the conclusion probable. The more
probable the premises make the conclusion,
the stronger the argument.
Inductive logic analyses risky arguments
using probability ideas.
We can think of this pictorially:
Arguments:
Deductively valid
Stronger
Degrees of inductive strength
weaker
Completely worthless
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Be careful:
Not all arguments with the word “probability”
are inductive. For example:
This die has six faces: 1, 2, 3, 4, 5, 6.
Each face is equally probable.
So:
The probability of rolling a 3 is 1/6.
This argument is valid: it is impossible for
the premises to be true and the conclusion
false.
However:
This die has six faces: 1, 2, 3, 4, 5, 6.
In 120 rolls, 3 came up 21 times.
So:
The probability of rolling a 3 with this die
is about 1/6.
This argument is invalid: the die could be
biased and the results a coincidence.
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Similarly, not all risky arguments involve
probability:
1. INFERENCE TO THE BEST EXPLANATION
(ABDUCTION)
Everybody who ate at Restaurant X got
sick the next day.
So:
The food must have been spoiled.
Here we offer a hypothesis to explain the
observed facts.
There may be other plausible explanations:
(everybody caught a virus that day, etc.)
The most plausible explanation is usually
accepted: this is an inference to the best
explanation:
And it is risky (one of the other explanations
might be true).
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2. TESTIMONY
My parents say I was born on January
18th.
So:
I was born on January 18th.
It is unlikely that your parents would lie to
you about this, but still… This is a risky
argument.
Your parents could be mistaken, have some
reason to lie, etc.
Perhaps probability is useful in analyzing
some testimony and abduction.
Philosophers disagree on these questions.
In this course we will only worry about
inductive arguments (samples to wholes,
wholes to samples, etc.).
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Decision Theory
When we reason about what to believe, this
is called theoretical reasoning.
Often, however, we reason because we
need to make a decision about what to do.
This is called practical reasoning.
The study of practical reasoning is called
decision theory.
Since we must make risky decisions,
probability theory will be useful in decision
theory.
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How does decision theory work?
We decide what to do on the basis of:
1. What we think will probably happen
(beliefs).
2. What we want (values).
So, decision theory requires probability
theory and a theory of values.
Values are measured by what are called
utilities.
Decision theory analyzes risky decisionmaking using ideas of probability and utility.
This is a very brief introduction to decision
theory. We shall go into more detail later in
the course.
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Homework:
 Do the exercises at the end of chapters 1
and 2.
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