Logic
One reason to study philosophy is because it helps to
think critically and evaluate arguments well.
(It makes an excellent pre‑law major for these same reasons.)
Since doing philosophy rests largely on arguing,
it is not surprising that philosophy has a branch devoted to noting other then the study of arguments.
Terms:
Logic: The branch of
Philosophy which analyzes and evaluates arguments.
Arguments: Verbal Attempts to
Persuade
Anatomy of an
argument: Argument
of comprise of premises and conclusions.
Premises: Those reasons offered in support of the
conclusion.
Conclusion: That which the argument is seeking to
persuade the hearer to believe.
In many philosophical arguments the premises come first
and the conclusion second. Often
philosophers in an attempt to be clear, set their arguments up like proofs in
geometry where one works from the premises to the conclusion.
This does NOT
always happen however. Do not assume
that just because a sentence comes at the end of a paragraph that it is the
conclusion. In and out of philosophy
arguments are presented sometimes with the conclusion first (e.g. We should
have a national universal healthcare system… and here’s why…) or sometime with the conclusion buries among
the premises. A sentence’s location
within the paragraph, etc. if no indication of whether it is a premise of a
conclusion or neither.
Better to ask yourself:
“What is this trying to prove? Of what point is this trying to convince me?”
Whatever your answer, that’s the conclusion.
Then ask,
“What reasons, if any, have I been given to
believe this conclusion?”
Whatever your answer, that or those are the premises.
Note: sometimes the
conclusion or even some of the premises can be implied, but not
stated.
Consider the following argument:
“All
US States have two US Senators, therefore
The conclusion is “
A (stated) premise is “All US States have two
The implied but not overtly stated
premises is “
Sometimes, the conclusion is implied. Consider:
Mary would never miss her best friend’s
wedding unless something terrible happened to her. And she’s not here (at her best friend’s
wedding).
Would you ask the speaker “So what’s your point?”
or would you “get” that she want’s you to conclude
that something terrible happened to Mary?
Often implied premises or implied conclusions
are so obvious that it hardly seems worth mentioning.
Consider:
“Of course some first grade teachers are
men. Why, my son is a first grade
teacher.”
The conclusion (Some first grade teachers are
men.” Follows from the stated premise (My son is a first grade teacher.) only
assuming an unstated but necessary premise. (My son is a man.) But it is so obvious that it would be silly
to actually state is in a normal conversation.
But other times, the implied premise conceals
an assumption that is controversial or at least worth of scrutiny.
Consider:
In vitro fertilization as a means of human
reproduction is immoral because it is unnatural.
The conclusions follows from the premise only
assuming that “Anything which is unnatural is immoral.” But that is a very contentious claim worth
discussing. However, one can sometimes
slip such contentious claim by an audience by making them unstated premises.
Generally speaking, philosophers, like
lawyers, usually consider these implied premises and implied conclusions
weaknesses in arguments because of the vagueness and ambiguity they create.
Evaluation of
Arguments:
Two
criteria to look at when evaluating arguments:
“a”
can be accomplished deductively or inductively
Deductive
Arguments-
The conclusion is supposed to follow with logical necessity. In well formed
deductive arguments, if the premises were true, the conclusion would have to
be, without fail, necessarily, true.
(Notice that this is like math of geometry. We do not conclude at the end of a geometric
proof that the sum of the interior angels of a triangle equals 180 degrees…
probably).
Inductive
Arguments- The
conclusion is supposed to follow with probability. In good inductive arguments,
if the premises were true, it is more probable that the conclusion is
true.
Analyzing these two different types of arguments,
deductive and inductive, requires two different sets of evaluative
criteria. Therefore, Logic can be seen
as comprised of two parts:
1. Formal Logic
2. Informal Logic
Formal Logic/
Deductive Arguments
Formal
Logic:
a branch of philosophy which analyzes and evaluates the structure of arguments
But what is meant by “The Structure of Arguments?”
Consider the following.
All A
are B
All B
are C
All A
are C
Note: An argument with two
premises and one conclusion is called a syllogism.
No doubt you have seen something like this before. What allows you to mentally move from the
first two sentences to the third is not the content of the argument.
Note: Is the first
sentence true? Is it true that All A are
B?
You don’t know.
You do not know if the first sentence is true
or not, (There may be an “A” out there that is in fact NOT a “B.”) nor whether
the second sentence is true, nor the third.
You don’t even know what they mean-
content. So it can’t be the content that
allows you to move from the first two to the third. It must be something else, i.e. the structure.
You might be tempted to say that you don’t
know anything about these sentences.
But you DO know something. And what you know begins with “if.”
You know that if the first sentence is true and the second sentence is true, then
the third sentence must be true.
The actual truth values of the three sentences could
stack up any of seven ways:
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1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
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T |
F |
T |
T |
F |
F |
F |
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T |
F |
F |
F |
T |
T |
F |
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T |
F |
T |
F |
T |
F |
T |
|
But the structure prevents T, T, F, from happening, though it
allows any other combination of T’s and F’s.
This is what it means to say that an argument is formally valid.
Formally
Valid-
this is a term applied to arguments which means “good form” or “good
structure”; it means that if the premises are true then the conclusion must be
true.
Pop Quiz:
If an argument is “valid” does that mean that the
conclusion IS true?
NO. (Valid syllogisms could have any of these patterns of
T’s and F’s)
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1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
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T |
F |
T |
T |
F |
F |
F |
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T |
F |
F |
F |
T |
T |
F |
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T |
F |
T |
F |
T |
F |
T |
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If an argument is “valid” does that mean that the
premises ARE true?
NO. (Valid syllogisms could have any of these patterns of
T’s and F’s)
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1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
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T |
F |
T |
T |
F |
F |
F |
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T |
F |
F |
F |
T |
T |
F |
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T |
F |
T |
F |
T |
F |
T |
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Can a valid argument have false premises?
Yes. (Valid syllogisms could have any of these patterns
of T’s and F’s)
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1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
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T |
F |
T |
T |
F |
F |
F |
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T |
F |
F |
F |
T |
T |
F |
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T |
F |
T |
F |
T |
F |
T |
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Can a valid argument have a false conclusion?
Yes. (Valid syllogisms could have any of these patterns
of T’s and F’s)
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1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
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T |
F |
T |
T |
F |
F |
F |
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T |
F |
F |
F |
T |
T |
F |
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T |
F |
T |
F |
T |
F |
T |
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Can a valid argument have false premises and a false
conclusion?
Yes. (Valid syllogisms could have any of these patterns
of T’s and F’s)
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
T |
F |
T |
T |
F |
F |
F |
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T |
F |
F |
F |
T |
T |
F |
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T |
F |
T |
F |
T |
F |
T |
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Can a valid argument have true premises and a false
conclusion?
NO (Valid
syllogisms can ONLY have one of these patterns of T’s and F’s)
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1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
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T |
F |
T |
T |
F |
F |
F |
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T |
F |
F |
F |
T |
T |
F |
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T |
F |
T |
F |
T |
F |
T |
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Let go back… Can a Valid argument have a false
conclusion?
Yes. (But only if
at least one of the premises is false.)
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1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
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T |
F |
T |
T |
F |
F |
F |
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T |
F |
F |
F |
T |
T |
F |
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T |
F |
T |
F |
T |
F |
T |
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Aristotle discovered that
there are certain argument forms which, if the premises are true, the
conclusion has to be true. They are
“truth preserving.” They will not allow
you to go from truth to falsity.
Formal Logic seeks to analyze and evaluate the structure
of argument; it is not interested in what the argument is about. It is strictly interested in the form. This is the reason behind symbolizing arguments. If I say that an argument is valid, that does
NOT mean that I think the premises are true, but only that the form of the
argument is truth preserving.
Note: then, if I present you with a valid argument, the
conclusion of which you believe to be false, it is incumbent on you to tell me
which of my premises you believe to be false and why.
Four Valid Argument Forms:
|
Modus Ponens If P then Q P Therefore: Q |
If there is nothing
wrong with terminating the life of embryos, then there is nothing wrong with
terminating the life of fetuses. There is nothing
wrong with terminating the life of embryos. Therefore: There is nothing
wrong with terminating the life of fetuses. |
|
Modus Tollens If P then Q ~ Q Therefore: ~ P |
If there is nothing
wrong with terminating the life of embryos, then there is nothing wrong with
terminating the life of fetuses. But there is
something wrong with terminating the life of fetuses. Therefore: There is something wrong
with terminating the life of embryos. |
|
Disjunctive Syllogism Either P or Q ~ P Therefore Q |
Either we must
accept having millions of Americans without healthcare or we must support an
universal health care system. But we cannot accept
having millions of Americans without healthcare. Therefore: We must support an
universal health care system. |
|
Hypothetical Syllogism If P then Q If Q then R Therefore If P then R |
If suicide is a
moral right then passive euthanasia is morally permissible. If passive euthanasia
is morally permissible then active euthanasia is morally permissible. Therefore: If suicide is a
moral right then active euthanasia is morally permissible |
Informal Logic/ Inductive Arguments
Remember, rational
support comes in two varieties: Deductive Support and Inductive Support.
Informal Logic: a branch of philosophy
which analyzes and evaluates the strength and weakness of inductive inferences.
We do NOT used the
terms valid or invalid to evaluate inductive arguments, but rather the terms strong and weak.
In an inductively strong argument, the
conclusion is made more likely by the truth of the premises. Though it is possible for the conclusion to be false even when the premises are
true, an strong inductive argument would be safer to bet, though not a sure
thing, that the conclusion is true.
Example:
Most American college
professors attended college as students.
Kenton Harris is a
college professor.
Given the truth of
the premises, it would be reasonable for you to conclude that…
Kenton Harris attended
college as a student.
OK. But what about this:
Most American college
professors attended college as students.
Renee Louise is a
college professor.
Therefore
Renee Louise attended
college as a student.
The premises are
true, but in this case the conclusion is false.
As it happens, Professor Louise is a dance practitioner who never
attended college. So even good inductive
arguments allow true premises and a false conclusion:
In each, the conclusion follows inductively,
not deductively. The premises, if true,
would make the conclusion more likely, but the premises could be true and the
conclusion false.
Inductive
arguments admit of degrees. Inductive inferences always quantified inferences
and always approximate:
90%
/Strong
70%
/Follows, but weakly
51%
/Follows, but very weekly
50%
or lower/ does not follow at all.
Typically,
the minimal requirement for inductive support for a position is one which
raises the probability above 50 percent.
Note: If the initial
probability of a proposition is 10% (I only had a one in ten chance of being
right.) and my argument raises it to 50% (Now it’s “even money.”), then I have
given a good inductive argument in that I have significantly raised the
probability of the conclusion.
Some
Typical Inductive Arguments:
One
gathers information about some members of a set (sample), and on the basis of that we draw conclusions about the
entire set (target).
So
for instance, if I interview 100 FIU students and on the basis of the results
draw conclusions about all FIU students my sample
are the student I interviewed and my target
is the entire student body of FIU.
When
assessing the strength (relative merit) of an inductive argument here are some
questions to ask:
a.
How many is in Sample?
(Generally speaking, the more the better- too
few and one is in danger of the fallacy of “Hasty Generalization” 100 student may seem like a lot, but there
are over 48,000 FIU students.)
b.
Is my sample representative?
(Consider if I only interviewed on the BBC
campus, or set up my interview stand next to the women bathroom or only spoke
to Philosophy majors or only undergraduate students. Beware of the dangers of biased samples.
A
biased sample is one that is misleading because it is taken to be typical of the
target population but in fact is not.
The sample exhibit relevant characteristics out of proportion with the
target. For instance if my target is
Miami Residents, but the majority of my sample is Filipino-Americans, I have a
biased sample (since the majority of Miami residents are not
Filipino-Americans). Conclusions drawn
from a biased sample are unreliable.
Online
and call-in polls are particularly at risk of this error, because the
respondents are self-selected. People who care most about an issue will
respond, sometimes flooding the poll.
This is one reason that the University cannot make the exist survey for
graduating students all-volunteer affair.
We would end up with a selected sample that may not be representative of
graduating seniors in general.
In
1936, early in the history of opinion polling, the American Literary Digest
magazine collected mail-in postcards from nearly one quarter of the voting
public and predicted the election result. They forecast that the Republican
challenger Alfred Landon would win by 370 electoral college votes, beating the
then President Franklin D Roosevelt. In the actual election though, Roosevelt
beat Landon in one of the biggest landslides in American history. Roosevelt received
27.7 million votes compared to Landon’s 16.6.
It is theorized that people unhappy with Roosevelt were far more
motivated to mail in their cards then Roosevelt supporters. It is also has been suggested that a readers
of the magazine were wealthier and more like to be Republican than the
electorate in general. In any event, the
mistake was so scandalous that the magazine closed shortly after. George Gallup, on the other hand, correctly
predicted the election result based on a far smaller, but more carefully
controlled sample. This event is seen as an important one in the refining of
scientific polling.
When
researching for notes on this I discovered that this is covered in 6th
grade math. Who knew? Here’s a link if you would like to take a
quiz.
http://www.ixl.com/math/grade-6/identify-representative-random-and-biased-samples
What
is often sought in scientific surveys of this kind is a “random sample.”
A
random sample is one where every member of the target population has an equal
chance of being a member of the sample population.
Now there is no absolute guarantee that even
a random sample will in fact be representative, but it’s more likely than not
that it will be.
Pop
Quiz: Why are T.V. call in surveys NOT
random surveys?
c.
What margin of error am I building in?
For instance if 70% of my sample were female
I might conclude that 70% of the target is female. But my conclusion is stronger if I more
modestly claim that between 60%-80% of my target is female, rather than the
more precise but therefore riskier claim that exactly 70 % of my target is
female. Notice the wider my margin of
error, the better my chance of saying something true. However what I gain in probability, I lose in
precision.
L, M, N, & O have characteristics 1, 2,
3, 4, 5, and 6 in common.
K is known to have qualities 1, 2, 3, 4, and
5
Therefore
It is probable that K has 6.
Mary,
Sue and John are
1. math majors
2. with GPA of 3.75 or better and
3. they all got an “A” in logic.
Bill is a math major with a GPA of 3.75.
Therefore:
Bill will (probably) get an “A” in logic.
Considerations:
Similarity must be relevant to the
possession of the unknown quality? In this case, being blond is
irrelevant. This is just one example of
where background knowledge is necessary in evaluating the strength of an
inductive argument and why it can never merely be a matter of “form.”
N% of A’s are B’s
K is an
A
Therefore
There is a N% probability that K is a B
But, background beliefs can override
the probability of the conclusion.
Consider:
90% of Swedes can swim.
Bjorn is a Swede.
Therefore
There is a 90% chance that Bjorn can swim.
But let’s say you also know, as part of your
background beliefs, that Bjorn is paralyzed.
Note: The probability of the conclusion is not the result a formal relation between it and the
premise set, but rather between it is an informal relation between
it and the entire body of known facts, the premise set constituting only those
of momentary focus.
4.
Abduction (Inference to the Best Explanation)
F (there is a fact set, a set of know
propositions)
H is the best explanation of F
Therefore:
H is true.
N.B.: Beware because:
Probability:
2
kinds
1. Statistical
Probability
SP is always numerical. A matter of determining the percentage of
items in class “a” that are also items in class “b.”
2.
Epistemic Probability
EP is rarely
numerical. Involves whether or not a
statement is true. The degree to which
truth is made probable is the evidence in support of the statement.
“What is the probability that Humans and
Chips have a common ancestor?” There is
no relevant statistical analysis of that question.
“O.J.” case is NOT statistical. (But some statistical evidence is relevant.)
Ep ranges form 0 ->
1.
0
= Certainly false.
1
= Certainly true,
.5
= 50% chance
Hypothesis has a probability X
where 0<X<1.
If 0< Pr (H)
<1 then H is a hypothesis.
S confirms H if
S is
true &
(S
. (Pr(H)) > (Pr (H)).
Or
((Pr (S)) given H) > ((Pr (S))
given ~H)
“disconfirms
means lowers the probability of “H,” but not necessarily to (> .5).
E is evidence for hypothesis H if
E
is true and E is antecedently more probable on the assumption that H is true
than on the assumption that H is false.
Like
the Bronco Ride – more probable on the assumption that O.J. is guilty than on
the assumption that he is not guilty.
Jones finger prints on the safe (E) is the
evidence that stole he money (H).
1
(a) What is the Pr (E) on the assumption of H.
2
(b) What is the Pr (E) on the assumption of ~H.
(If
Jones is the Bank Manager then a and b are about equal. So E is not evidence/ does not confirm.
Let’s assume that
(Pr (H2)) = (((Pr (H1))* 10)
but
then new information
E
And
((Pr (E) and H1) = (((Pr (E))and
H2)*)
in
that case
(Pr H2) = (Pr H1)
Jelly
Beans Story
Two Jars of Jelly Beans
A is mostly Black with a few red; B is mostly
Red with a few black.
Lights
go out. Hear someone enter the room,
unscrew a jar lid and place something on the table. Lights come back on and there is a black
jelly bean on the table.
H1
= jelly bean came from Jar A ((Pr H1) >.5)
H2
= jelly bean came from Jar B ((Pr H1) <.5)
But if you know that jar is A is sealed. Then
H1
= jelly bean came from Jar A ((Pr H1) < .5)
H2
= jelly bean came from Jar B ((Pr H1) > .5)
Pr (E+H)
_______
Pr (E+~H)
This ratio gives you the strength of support
of E for H.
E confirms H1 over H2 = E is true and
E is antecedently more probable on the
assumption of H1 then the assumption of H2.
>
greater then
>!
Much greater
>!!
Much, much greater
There
are three curtains. Behind one there is
a new car and behind the other two there is only booby prizes.
You
pick #1 from among three options 1,2, and 3, each with an equal chance for
success (1 in three, a new car) or failure (2 in three and goat or nothing).
You know that Montey
Hall will show you the goat and then allow you to change your mind.
He shows you the goat behind Curtain #3.
Should you choose curtain #2 now?
Based on probability theory, YES.
Why? Because when you choose #1 the probability of
the car being behind curtain number 1 was 1/3 and the probability of it NOT
being behind #1 (i.e. being behind curtain number 2 or 3; (not-1) was 2/3.
Now
he shows you the goat behind curtain number curtain #3. That does not improve the odds of it
being behind curtain number 1 since he would have shown you the goat
anyway.
But
it does
raise the odds that it is behind curtain number 2 to from 1/3 to 2/3. Why?
Because the odds of it NOT being behind #1 was and remains 2/3 (i.e. Pr of Not-1 = 2/3).
Now add to that information the fact that it is NOT behind curtain
number 3. Given the new information, the odds of it being behind curtain #3 is
0 and therefore the odds of it being behind curtain number 2 alone (i.e. Not-1)
is 2/3.
Philosophical Logic
is concerned with evaluating both types of arguments, Deductive and Inductive.
Essentially, there
are only two questions to ask when confronted with an argument In the end, a critical examination of an
argument only requires that one ask and answer two questions.