Logic for Critical Thinking
Formal Logic/ Deductive Arguments
Four
Common Formal Argument Types
Informal
Logic/ Inductive Arguments
Some Typical Inductive Arguments
(Four)
·
Inductive
Generalization
·
Arguments By Analogy
·
Statistical
Syllogism
·
Abduction (Inference to the Best Explanation)
Hypothesis,
Confirmation and Evidence
Let’s Make A Deal
Logic for Critical Thinking
One reason to study philosophy is because
it helps one 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 than the study of arguments.
Terms:
Logic: The branch of Philosophy which analyzes and evaluates arguments.
Arguments: Verbal Attempts to Persuade
Arguments are comprised 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.
Identifying
the Premises and Conclusions of Arguments
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 set of 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 buried
among the premises. A sentence’s
location within the paragraph, etc. if no indication of whether it is a premise
or a conclusion or either.
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.
Sometimes, a premise is implied. Consider
the following argument:
“All US States have two US Senators, therefore Florida must have two US Senators.
The conclusion is “
The stated premise is “All US
States have two US Senators.” But
there’s more going on here.
The implied, but not overtly
stated, premises is “
So the full argument is really
this:
All
US States have two US Senators.
Florida
is a US State.
Therefore
Florida
must have two US Senators
By contrast, consider this:
“All US States have two US
Senators, therefore the District of Columbia must have two US Senators.
This argument does NOT work
because it assumes a premise which is false.
The District of Columbia is NOT a US State. Although DC has a larger population than some
states, the District is not one of the fifty states and so has no senators at
all.
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 wants you to conclude that
something terrible probably happened to Mary?
Often implied premises or implied
conclusions are so obvious that it hardly seems worth mentioning.
Consider:
“Of course some non-human animals
have green eyes. Why, my cat Tibbles has
green eyes.”
The conclusion “Some non-human
animals have green eyes.” follows from the stated premise “My cat Tibbles has
green eyes.” only assuming an unstated, but necessary premise. (My cat Tibbles
is a non-human animal.) 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 worthy of scrutiny.
Consider:
“In vitro fertilization as a means
of human reproduction is immoral because it is unnatural.”
The conclusion in the passage
above is:
In vitro fertilization as a means
of human reproduction is immoral.
The stated premise is:
In vitro fertilization is
unnatural.
But in isolation of other
considerations, the conclusion does not follow from of the stated premise at
all. The conclusions follows from the stated
premise only assuming another claim:
Anything which is unnatural is
immoral.
This is the unstated, but
necessary, premise in the original argument.
However, this unstated premise is a very contentious claim worth
discussing. One can sometimes slip such
contentious claims 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.
Two criteria to look at when evaluating arguments:
- degree
of support the premises give to the conclusion
- quality of
the premises
Degree of support the premises provide to the conclusion can be
accomplished deductively or inductively
Deductive Arguments- In a deductive argument, 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. Rather this is
absolutely certain.)
Inductive Arguments- In an inductive argument, 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. (Notice this sort of reasoning is what goes on
in the empirical sciences and in law. For
instance, even in criminal cases which require the highest standard of
evidentiary proof, it is not the
responsibility of the prosecution to prove the guilt of the accused beyond
a shadow of a doubt, but rather beyond a reasonable doubt.
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
1.
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: Syllogism - An
argument with two premises and one conclusion is called a syllogism. The argument above is called a categorical syllogism
since the premises and conclusion are claims about categories (e.g. the
category of A and B and C).
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 do you know whether the second
sentence is true, nor the third.
You don’t even know what these
sentences 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 three
sentences.
But that’s not true; 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:
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
T |
F |
T |
T |
F |
F |
F |
|
|
T |
F |
F |
F |
T |
T |
F |
|
|
T |
F |
T |
F |
T |
F |
T |
|
That is, they could all be true,
but for that matter they could all be false, or some combination of true and
false. But the structure prevents
one (and only one) combination from happening.
The structure will not allow T, T, F, 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)
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
T |
F |
T |
T |
F |
F |
F |
|
|
T |
F |
F |
F |
T |
T |
F |
|
|
T |
F |
T |
F |
T |
F |
T |
|
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)
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
T |
F |
T |
T |
F |
F |
F |
|
|
T |
F |
F |
F |
T |
T |
F |
|
|
T |
F |
T |
F |
T |
F |
T |
|
So, can a valid argument have
false premises?
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 |
|
|
T |
F |
F |
F |
T |
T |
F |
|
|
T |
F |
T |
F |
T |
F |
T |
|
Likewise, can a valid argument
have 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 |
|
|
T |
F |
F |
F |
T |
T |
F |
|
|
T |
F |
T |
F |
T |
F |
T |
|
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 |
|
|
T |
F |
F |
F |
T |
T |
F |
|
|
T |
F |
T |
F |
T |
F |
T |
|
Can a valid argument have all true
premises and a false conclusion?
NO
(Valid syllogisms can ONLY have
one of these patterns of T’s and F’s)
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
T |
F |
T |
T |
F |
F |
F |
|
|
T |
F |
F |
F |
T |
T |
F |
|
|
T |
F |
T |
F |
T |
F |
T |
|
Let’s go back… Can a Valid
argument have a false conclusion?
Yes.
But note, the conclusion can be
false only if at least one of the premises is also false.
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
T |
F |
T |
T |
F |
F |
F |
|
|
T |
F |
F |
F |
T |
T |
F |
|
|
T |
F |
T |
F |
T |
F |
T |
|
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.
A
further word about structure and formal logic:
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 More Common Valid Argument
Forms:
|
Modus Ponens If
P then Q P
Therefore: Q |
If
it is morally permissible to terminate the life of embryos, then it is
morally permissible terminate the life of fetuses. It
is morally permissible to terminate the life of embryos Therefore: It
is morally permissible to terminate the life of fetuses. |
|
Modus Tollens If
P then Q ~
Q Therefore:
~
P |
If
it is morally permissible to terminate the life of embryos, then it is
morally permissible terminate the life of fetuses. It
is NOT morally permissible to terminate the life of fetuses Therefore: It
is NOT morally permissible to terminate the life of embryos. |
|
Disjunctive Syllogism Either
P or Q ~
P Therefore
Q |
Either
we must accept having millions of Americans living without healthcare or we
must support an universal health care system. But
we cannot accept having millions of Americans living 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 morally permissible, then passive euthanasia is morally
permissible. If
passive euthanasia is morally permissible, then active euthanasia is morally
permissible. Therefore: If
suicide is morally permissible, then active euthanasia is morally permissible. |
Note: I
am not in fact arguing for these conclusions, merely giving you examples of
valid arguments. However note further
that is you think the conclusion of any of these arguments is false, you would
need to go on to identify which of the premises you believe also to be false
and why.
COMMON TRUTH-FUNCTIONAL ARGUMENT PATTERNS
|
Valid Argument Forms |
Invalid Argument
Forms |
|
|
|
|
Modus ponens (or affirming the antecedent) |
Affirming the consequent |
|
If P, then Q |
If P, then Q |
|
P |
Q |
|
Q |
P |
|
|
|
|
Modus tollens (or denying the consequent) |
Denying the antecedent |
|
If P, then Q |
If P, then Q |
|
Not Q |
Not P |
|
Not P |
Not Q |
|
Chain argument |
Undistributed middle (truth functional version) |
|
If P, then Q |
If P, then Q |
|
If Q, then R |
If R, then Q |
|
If P, then R |
If P, then R |
|
Valid
Syllogism 1 |
Invalid Syllogism 1 |
|
All
Xs are Ys. |
All Xs are Ys. |
|
All
Ys are Zs. |
All Zs are Ys. |
|
All
Xs are Zs. |
All Xs are Zs. |
|
Valid
Syllogism 2 |
Invalid Syllogism 2 |
|
All
Xs are Ys. |
All Xs are Ys. |
|
No
Ys are Zs. |
No Zs are Xs. |
|
No
Xs are Zs. |
No Ys are Zs. |
|
Valid
Conversion 1 |
Invalid Conversion 1 |
|
No
Xs are Ys. |
All Xs are Ys. |
|
No
Ys are Xs. |
All Ys are Xs. |
|
Valid
Conversion 2 |
Invalid Conversion 2 |
|
Some
Xs are Y s. |
Some Xs are not Y s. |
|
Some
Y sore Xs. |
Some Y s are not Xs. |
|
|
Unnamed Invalid Inference |
|
|
Some Xs are Y s. |
|
|
Some Xs are not Ys. |
|
|
(Also invalid when run in reverse.) |
|
. |
. |
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, in a 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.
Further, additional information can reduce the strength of an inductive
argument (See above.) in a way that does not happen in deductive arguments.
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 low say, 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 (Four):
1.
Inductive Generalization:
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 students 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 are in the Sample?
(Generally speaking, the more the
better. Too few in the sample and one is
in danger of the fallacy of “Hasty Generalization.” 100 students may seem like
a lot, but there are over 65,000 FIU students. See below.)
b. Is my sample representative?
(Consider if I only interviewed on
the BBC campus, or set up my interview stand next to the women’s 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 exhibits 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). Likewise if
after sampling FIU students and conclude that only 20% of them consider parking
a problem, you might note that my sample was drawn from BBC student and not
representative the FIU student body at large.
Conclusions drawn from a biased sample are unreliable.
In 1936 during the election
campaign pitting Franklin Roosevelt against Republican challenger Alfred
Landon, the American Literary Digest magazine collected mail-in postcards from
nearly one quarter of the voting public and predicted the election result. Based
on the collected data they forecasted that Landon would win by a
landslide. In actually though, it was Roosevelt
who won by one of the biggest landslides in American history. Roosevelt
received 27.7 million votes compared to Landon’s 16.6. Subsequently it is thought that those who
disapproved of the incumbent Roosevelt were more motivated to mail in their
cards than were Roosevelt supporters. Others
pointed to the fact that readers of the magazine were wealthier and more likely
to vote Republican than the electorate in general. In any event, the mistake was so scandalous
that the magazine closed shortly thereafter.
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.
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 exit survey for graduating students an “all-volunteer” affair. We would end up with a self-selected sample
that may not be representative of graduating seniors in general.
What is often sought in scientific surveys of this kind is a “random sample.”
Random Sample: a sample 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?
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
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.
Generalizations, Sample Size and Biased Samples
As we have already noted, a generalization is where we sample a
population and based on the information we gain from the sample, we generalize
to claims about the entire target population.
So for instance, if I'm cooking tomato sauce and I want to know whether
or not to add more salt, I will sample a tablespoon of the sauce and based on
that information to determine something about the entire pot of sauce.
When reasoning this way, we must be reliably sure that the sample
we draw accurately represents the target population. To avoid biased samples, researchers
use elaborate methods to obtain a genuinely random sample.[1] “Random Sample” is a technical term referring
to a sample where each member of the target population has an equal chance of
being a member of the sample.
Appropriate
Sample Size
A good sample must also me of the appropriate size to avoid the
fallacy of “hasty generalization.” It would
be nice if a simple mathematical formula could be applied to determine the
appropriate sample size in any given case.
Unfortunately, no such formula exists.
The appropriate size of the sample depends to some extent on the size of
the target population, but also on such other factors as the degree of
uniformity within the population relevant to the characteristics being examined
as well as the desired/ acceptable degree of error.
Size of
the Population
To some extent, the size of the sample depends on the size of the
population. This is especially true when the population is relatively small.
For instance, if we are taking an opinion poll at a small college with only a
few hundred students enrolled, our sample can be smaller than it would need to
be if we were polling students at a university with 20,000 enrolled. However, one common misconception about
samples is that a larger sample always yields a more accurate generalization.
Consider this illustration:
Suppose that we are drawing a sampling a barrel containing 10,000
marbles, in which half of the marbles are red and half of them are blue.
Further, suppose we have a genuinely random sampling process so that each
marble in the barrel has an equal chance of being selected for our sample.
If we sample 500 marbles, our sample likely will contain
approximately 250 red marbles (give or take a few) and approximately 250 blue
ones (give or take a few).
Now suppose instead the barrel contains 1 million marbles instead
of 10,000. Nevertheless, if we again sample of 500 marbles at random, we
should still get approximately 250 red and 250 blue, again, accurately indicating that the distribution
of red or blue marbles is approximately 50/50.
Thus, the larger population does not necessarily require a larger
sample.
Based on experience with the Gallup polls, the relationship
between sample size and sampling error can be stated with remarkable precision
for studies of large populations. Consider the following data:
|
Number of Interviews |
Margin of Error (in percentage
points) |
|
|
|
4000 |
+/- 2 |
|
|
|
1500 |
+/- 3 |
|
|
|
1000 |
+/- 4 |
|
|
|
750 |
+/- 4 |
|
|
|
600 |
+/- 5 |
|
|
|
400 |
+/- 6 |
|
|
|
200 |
+/- 8 |
|
|
|
100 |
+/- 11 |
|
|
2.
Arguments By Analogy
A simple argument by analogy might
go something like this:
John is a very capable corporate
executive of a large successful business.
Therefore he will make a good director of this large government agency.
The idea here is that the jobs are
analogous. Thus the skills that allow him to be
successful at the one position will allow him to be successful at the
other. The MORE analogous the jobs are,
the stronger the inference. The more
disanalogies, the weaker the inference.
Here is another example:
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. They all are math majors.
2. They all have GPAs of 3.75 or
better.
3. They all attended Miami Senior
High School.
4. They all took calculus there
first semester at FIU and passed with an A.
5. They all did well on the SAT.
6. They all got an “A” in logic.
Bill is
- a math
major
- with a
GPA of 3.75.
- He too
attended Miami Senior High
- earned
an A in calculus his first semester
- and did
well on the SAT.
Therefore:
- Bill
will (probably) get an “A” in logic.
Considerations:
- How many
is in Sample? (the more the better)
- What
percentage of the Sample has the unknown quality? (The higher the
better. Note if only Mary and Sue
got an A in logic but John did not, this would weaken the inference.)
- What is
the number of relevant similarities in common? (The more the better. For instance, if we found out that they
all had the same instructor for logic who used the same textbook each time,
this would strengthen the inference.)
But of course this raises the
question of what constitutes a “relevant similarity.” To raise the probability, the 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 to evaluate the strength of an inductive argument. The relation of the premise set to the
conclusion is an informal one; it is never merely a matter of “form.” Curiously, playing a musical instrument is a
relevant consideration. It has been
shown that students with musical training perform better at logical reasoning
tasks on average than do those without musical training. Who knew?
But that’s the point. If you do
not already know that, then you would not recognize musical training as a
relevant characteristic.
- Is there
diversity among properties un-shared with the target within sample
population? (the greater the better.)
Say Bill did NOT attend Miami
Senior High. This would weaken the
inference. But if Mary attended Miami
Senior, Sue attended Braddock Senior and John attended La Salle High School,
then the fact that Bill did not attend Miami Senior wouldn’t matter.
- Is the
Sample representative?
Imagine I gather my sample by
getting all the stats only on students who got an A in logic. My sample would necessarily ignore all the
students who got anything lower. In that
case, for all we know there are lots of students who have 1-5 in common, but
did not get an A in logic. My sample is
not representative and my inference would be weakened.
3.
Statistical Syllogism
N%
of A’s are B’s
K is an A
Therefore
There is a N% probability that K
is a B
Consider:
90% of Swedes can swim.
Bjorn is a Swede.
Therefore
There is a 90% chance that Bjorn
can swim.
But here again, background beliefs
can override
the probability of the conclusion. Let’s
say you also know, as part of your background beliefs, that Bjorn is paralyzed. Well, this additional information will serve
to undermine the inference.
Note: The probability of the
conclusion is not the result a formal
relation between it and the premise set, but rather 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 (This is a fact set, a set of
know propositions.)
H is the best explanation of F
Therefore:
H is true.
EX:
F = I heard a crash in the next room. I go in and see my cat running out of the
otherwise unoccupied room and I see that there is a smashed glass vase on the
floor.
H = My cat knocked over the vase.
H is the best explanation of F, so given F I have good reason to
believe H.
N.B.:
Beware because:
- Do the
facts actually require an explanation? Trying to explain unrelated
coincidences can lead to conspiracy theories, etc. [2]
- Sometimes, even the best explanation
available is not that good.
- There might be more than one
nearly-as-good explanations; which among the rivals is the “best?” (Ex: A
30%, B40%, C30%)
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.” For instance, the probably that Jeff is
left-handed would be determined by finding out what percentage of the general
population is left-handed (assuming that Jeff is a member of the general
population). Of course, if Jeff is a
member of the South-Paw Pitchers of America, then we would want to determine
what percentage of THAT population was left-handed.
2. Epistemic Probability
EP
is rarely numerical. This involves the
general likelihood of 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.
Jury trial evidence is like
that. “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,
Confirmation and Evidence
A Hypothesis is any statement that could be true but also could be
false. It has a probability of X where
0<X<1.
This if the probability of h is
greater than zero but less then certain [0< Pr (H)
<1], then H is an hypothesis.
Confirmation
Some statement S is said to
“confirm” some hypothesis H is S is true and the likelihood of S being true is
greater on the assumption that H is true than it would be on the assumption
that H is false. So:
S “confirms” H if
S is true &
((Pr (S)) given H) > ((Pr (S)) given ~H)
“disconfirms means lowers the probability of “H,” but not
necessarily to (> .5).
But this account of confirmation is not without its own
difficulties.
Rudolf Carnap points out in Logical Foundations of Probability, the concept of confirmation is
ambiguous.
1.
The special theory of relativity “has been
confirmed by experimental evidence” meaning that the special theory has become
an accepted part of scientific knowledge and that it is very nearly certain in
the light of its supporting evidence. (That is NOT the sense of confirmation
explained immediately above.)
2.
The special theory of relativity “has been
confirmed by experimental evidence” that some particular evidence — e.g.,
observations on the lifetimes of mesons — renders the special theory more
acceptable (greater probability then it had previously) or better founded than
it was in the absence of this evidence.
“The discrepancy between these two meanings is
made obvious by the fact that a hypothesis h, which has a rather low degree of
confirmation on prior evidence e, may have its degree of confirmation raised by
an item of positive evidence i without attaining a high degree of confirmation
on the augmented body of evidence e..i. In other
words, a hypothesis may be confirmed (in the second sense) without being
confirmed (in the first sense).[3]
So a hypothesis with a low degree of confirmation might
have its degree of confirmation increased repeatedly by positive instances (2nd
sense), without ever getting confirmed (1st sense) It can also work
the other way.
So
how DOES the 1st sense on confirmation occur? Good Question. See Philosophy of Science.
Evidence
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.
Jones finger prints on the safe
(E)
Is this evidence that Jones stole
he money (H).
The assumption here is that, E is true, and is more likely to be
the case given the truth of the hypothesis, then assuming the hypothesis is
false.
Here then E is evidence for H.
But…
Well, that depends. We must
first determine:
- What is
the Pr (E) on the assumption of H.
- What is
the Pr (E) on the assumption of ~H.
If E is antecedently more probable (i.e. Right now
it’s a sure thing.) on the assumption that H, then yes, E is evidence for
H. But suppose Jones is the Bank Manager. In that case, E is no more probable on the
assumption of H then on the assumption of ~H.
(Pr E given H) =
(Pr E given n~H)
Here then E is not evidence for H.
Suppose Pierre’s finger prints are on the jewel case from which
the million dollar diamond ring was stolen.
That would seem to be evidence of his guilt. However, if Pierre is known to be an
excellent (extremely meticulous and competent) international jewelry thief, his
prints on the case seem unlikely were he the actual perpetrator. (i.e. He would never have made such a rookie
mistake.) In that case his prints on the
case ( E ) is less likely on the
assumption that he is the thief, then on the assumption he is not (and is
perhaps being framed).
(Pr E given H) < (Pr
E given ~H)
But confirming evidence does not
necessarily warrant that we believe the hypothesis true. Let’s assume that:
(Pr (H2)) = (((Pr
(H1))* 10) – The probability of H2 is ten times greater then
the probability of H1.
but then new information
E
And now
((Pr (E) and H1) = (Pr
(E) and H2) – The probability of H2 is equal to the probability of H1.
in that case
E is evidence for H1 since it does indeed raise its probability of
H1 significantly, but it does not warrant belief in H1 since there is an
equally probably alternative hypothesis.
At this point, given new information E,
(Pr H2) = (Pr
H1)
Two Jars of Jelly Beans
Jar A is mostly Black with a few
red beans; Jar B is mostly Red with a few black beans.
Lights go out. Hear someone
enter the room, unscrew a jar lid and place something on the table and then we
hear the lid being screwed back on and the person leave the room. Lights come back on and there is a black
jelly bean on the table.
Assume that we must choose from two competing hypotheses: The
jelly bean came from jar A or the jelly bean came from jar B. At this point:
H1 = jelly bean came from Jar A ((Pr H1)
>.5)
H2 = jelly bean came from Jar B ((Pr H1)
<.5)
We would have good reason to accept H1.
But if you know that jar is A is
sealed tightly and cannot be easily opened, then
Pr
(E+H1) (50%)
_______
Pr
(E+~H1) (20%)
This ratio gives you the strength
of support of E for H.
E make H1 2.5 times more likely.
> greater then
>! Much greater
>!! Much, much greater
Let’s Make A Deal
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?
I’ll give you a minute……
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, (i.e. not-1) was 2/3.
Pr
1 = 1/3
Pr
~1 = 2/3
Now he shows you the goat behind curtain number 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. Remember that the probability 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 (remains) 2/3.
In Sum:
Philosophical
Logic is concerned with evaluating both types of arguments, Deductive and
Inductive.