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What is logic?
-----> LOGIC is the critical study of arguments
- an ARGUMENT is reasoning
expressed in statements that are related
as premises and conclusion, where the
conclusion is supposed to follow from
the premises
-----> the point of arguments is to convince
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----->
example argument:
Premise 1 The defendant could not have
committed the murder unless he had the
cash to hire someone, or was in possession
of both a car and a gun.
Premise 2 He would not have committed
the crime unless he was either drunk or on
drugs.
Premise 3 He never touches drugs and he
did not have a gun.
Conclusion Therefore, he could not have
committed the crime.
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----->
arguments in philosophy are
intended to convince us of
claims about:
- the existence of God
- the nature of the mind
- how much we are able to
know
- and many other issues
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Why logic is important?
----->
----->
people try to convince us of things
all the time; we try to convince
others of things all the time
examples:
----->
from editorials in the media
Premise 1 If the Chinese
government violates human
rights then it doesn't deserve
privileged trade status.
Premise 2 The Chinese
government violates human
rights.
Conclusion Therefore, it doesn't
deserve privileged trade status.
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----->
from advertising
Premise 1 Nine doctors out of
ten recommend product X.
Conclusion Therefore, product
X is better than the products
made by competitors.
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- General overview:
Arguments
Deductive
Inductive
- With DEDUCTIVE arguments, the
conclusion is supposed to follow from the
premises ABSOLUTELY.
----->
example:
Premise 1 All tabbies are cats.
Premise 2 All cats are animals.
Conclusion All tabbies are animals.
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- With INDUCTIVE arguments, the
conclusion is supposed to follow from the
premises with HIGH PROBABILITY.
-----> example:
Premise 1 Every emerald that has ever
been seen has been green.
Conclusion All emeralds are green.
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- General overview:
Arguments
Deductive
Valid
Inductive
Invalid
- a deductive argument is VALID just in case:
IF
its premises are true, its conclusion
MUST
be true.
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- an argument's validity is a matter of its
FORM
Argument (a):
Premise 1 All tabbies are cats.
Premise 2 All cats are animals.
Conclusion All tabbies are animals.
Argument (b):
Premise 1 All cats are robots.
Premise 2 All robots are machines.
Conclusion All cats are machines.
Argument form for arguments (a) and (b):
Premise 1 All A's are B's.
Premise 2 All B's are C's.
Conclusion All A's are C's.
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- General overview:
Arguments
Deductive
Valid
Sound
Inductive
Invalid
Unsound
- a deductive argument is SOUND just in
case it is valid and all of its premises are
true
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Validity and argument form
Deductive Arguments
Valid
IF premises are
true, conclusion
MUST be true.
Invalid
Possible for
premises to be
true and
conclusion false.
- the validity/invalidity of deductive
arguments has to do with the
RELATIONSHIP between premises and
conclusion: do the premises, ASSUMING
they are true, REQUIRE the conclusion to
be true?
Consider this example:
Premise 1 All cats are smelly creatures.
Premise 2 All smelly creatures are
arrogant.
Conclusion All cats are arrogant.
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VALIDITY is a matter of ARGUMENT FORM
- distinguish between ARGUMENT FORM
and ARGUMENT INSTANCE
-----> ARGUMENT FORM 1:
Premise 1 All A's are B's.
Premise 2 All B's are C's.
Conclusion All A's are C's.
- A, B, and C are VARIABLES for class
terms.
-----> example of INSTANCE of this form:
Premise 1 All Presidents are politicians.
Premise 2 All politicians are scoundrels.
Conclusion All Presidents are scoundrels.
- here, A = Presidents; B = politicians;
C = scoundrels
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----->
another INSTANCE of this form:
Premise 1 All whales are mammals.
Premise 2 All mammals are warm blooded
animals.
Conclusion All whales are warm blooded
animals.
- here, A = whales; B = mammals;
C = warm blooded animals
----->
another INSTANCE of this form:
Premise 1 All cats are evil creatures.
Premise 2 All evil creatures are immortal.
Conclusion All cats are immortal.
- here, A = cats; B = evil creatures;
C = immortal creatures
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ARGUMENT FORM 1, that is:
Premise 1 All A's are B's.
Premise 2 All B's are C's.
Conclusion All A's are C's.
is a VALID argument form--ANY
systematic replacement of variables results
in a valid argument instance.
- THIS IS WHY an argument instance's
VALIDITY is a matter of its FORM, not
the actual truth of its premises
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-----> ARGUMENT FORM 2:
Premise 1 All A's are B's.
Premise 2 All C's are B's.
Conclusion All A's are C's.
- A, B, and C are VARIABLES for class
terms.
-----> example of INSTANCE of this form:
Premise 1 All Republican Senators are U.S.
citizens.
Premise 2 All Democratic Senators are U.S.
citizens.
Conclusion All Republican Senators are
Democratic Senators.
- here, A = Republican Senators;
B = U. S. citizens
C = Democratic Senators
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-----> another INSTANCE of this form:
Premise 1 All cats are animals.
Premise 2 All dogs are animals.
Conclusion All cats are dogs.
- here, A = cats; B = animals;
C = dogs
An argument instance is INVALID just in
case it is possible for its premises to be
true and its conclusion false.
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-----> so ARGUMENT FORM 2, that is:
Premise 1 All A's are B's.
Premise 2 All C's are B's.
Conclusion All A's are C's.
is an INVALID argument form--ANY
systematic replacement of variables results
in an invalid argument instance.
- an argument instance's INVALIDITY
is also a matter of its FORM, not the
actual truth of its premises
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How do we know if an argument form is valid
or invalid?
We can prove that an argument form is
INVALID by giving an example of an
instance of that form where the premises
are true and the conclusion is false.
- an argument instance of this sort is
called a COUNTEREXAMPLE
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----->
for example:
Premise 1 All A's are B's.
Premise 2 All C's are B's.
Conclusion All A's are C's.
Premise 1 All cats are animals.
Premise 2 All dogs are animals.
Conclusion All cats are dogs.
- this argument instance is a
COUNTEREXAMPLE
How about:
Premise 1 No A's are B's.
Premise 2 Some C's are not A's.
Conclusion Some B's are not C's.
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SUMMARY (now that you have distinction
between argument instances and argument
forms):
An argument INSTANCE is valid if and
only if it is an instance of a valid form (p.
15 Klenk).
- that is:
validity of an argument
instance is a matter of its
FORM
An argument FORM is valid if and only if
there are no instances of that form in
which its premises are true and its
conclusion is false (p. 16 Klenk).
- that is:
- that is:
an argument form is valid so
long as it has no
COUNTEREXAMPLE;
for a valid argument form,
there is NO POSSIBLE
instance where premises are
true and conclusion false
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Validity and soundness
Deductive Arguments
Valid
IF premises are
true, conclusion
MUST be true.
Sound
Valid with
true premises.
Unsound
At least one
false premise.
Invalid
Possible for
premises to be
true and
conclusion false.
All are unsound
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There are TWO aspects of the goodness of
deductive arguments:
1. validity
2. true premises
To be CONVINCING arguments should be
good on BOTH aspects.
- but HOW do we tell whether premises
are actually true?
-----> example argument:
Premise 1 All whales are mammals.
Premise 2 All mammals are warm
blooded animals.
Conclusion All whales are warm
blooded animals.
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All we are concerned with is the first aspect of
goodness, that is:
1. validity
Is the following argument valid or
invalid?
Premise 1 All fish can walk.
Premise 2 All creatures that can walk can
fly.
Conclusion All fish can fly.
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Distinction between sentential and predicate
logic
- in PREDICATE logic, letter variables in
argument forms stand for terms, for example,
class terms
-----> example argument form in predicate
logic:
Premise 1 All A's are B's.
Premise 2 All B's are C's.
Conclusion All A's are C's.
-----> example argument instance:
Premise 1 All tabbies are cats.
Premise 2 All cats are animals.
Conclusion All tabbies are animals.
- here A = tabbies; B = cats; C = animals
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- in SENTENTIAL logic, letter variables in
argument forms stand for whole sentences.
-----> example argument form in sentential
logic:
Premise 1 If p, then q.
Premise 2 p.
Conclusion q.
-----> example argument instance:
Premise 1 If GM is the biggest US car
manufacturer, then GM is bigger than
Ford.
Premise 2 GM is the biggest US car
manufacturer.
Conclusion GM is bigger than Ford.
- here p = GM is the biggest US car
manufacturer; q = GM is bigger than Ford
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Note: In predicate logic, letter variables for
class terms are capital letters A-Z.
In sentential logic, letter variables for
sentences are small letters p-s.
In sentential logic, capital letters A-Z
are ABBREVIATIONS for sentences,
not VARIABLES.
So, the sentence "GM is bigger than
Ford" might be abbreviated with
"G." Abbreviations are just to save
time and ink.