Transcript Formal Logic PPT

“Greatest Hits” of Formal Logic
Overview of deductive reasoning
Formal Logic is the science of
deductive reasoning
 Definition: “reasoning from known premises,
or premises presumed to be true, to a certain
conclusion.”
 In contrast, most everyday arguments involve
inductive reasoning.
– reasoning from uncertain premises to
probabalistic conclusions
– “inference-making”
Structural validity versus
material truth
 Formal logic cannot establish
the truth of the premises. The
truth of the premises must be
presumed, or taken as a
given.
– Some premises may be proven
or authenticated by scientific
testing, reference to external
sources, etc.
– Some premises may be granted
or stipulated by all the parties
to an argument
– Some premises may have been
established as the conclusion
of a previous argument
– DNA testing and paternity
• If a DNA sample is
collected and analyzed
properly and,
• If the DNA is an exact
match with the alleged
father,
• Then that person is the
father.
In deduction, proofs are always valid
or invalid.
 There is no middle
ground. A deductive
argument can’t be “sort
of” valid.
 By contrast, everyday
arguments enjoy degrees
of probability--plausible,
possible, reasonable,
believable, etc.
The form or structure of a deductive
argument determines its validity
 The fundamental property of a valid, deductive
argument is that IF the premises are true, THEN the
conclusion necessarily follows.
 The conclusion is said to be “entailed” in, or
contained in, the premises.
– If all pigs have curly tails
– And Nadine is a pig
– Then Nadine has a curly tail
The terms used in a syllogism must
be defined precisely
 If the meanings of key terms are vague or ambiguous,
or change during the course of a deductive argument,
then no valid conclusion may be reached.
– Major premise: All pitchers hold water
– Minor premise: Tom Glavin is a pitcher
– Conclusion: Therefore, Tom Glavin holds water
(the term “pitcher” has two different meanings in this
argument, so no valid conclusion can be reached)
Example of a valid deductive
argument
major premise: All cats have 9 lives
minor premise: “Whiskers” is a cat
conclusion: Therefore, Whiskers has 9 lives
(Note: it doesn’t matter whether cats really have 9
lives; the argument is premised on the assumption
that they do.)
“Validity” versus “Soundness”
 An argument is valid if its structure conforms to
the rules of formal logic.
 An argument is sound if it is valid, and its
premises are true.
 Thus validity is a prerequisite for soundness, but
an argument needn’t be sound to be valid.
– If sound, then valid too
– If valid, not necessarily sound
Validity versus soundness
Example of a valid, but
unsound argument
major premise: All cats
are pink
minor premise: Felix is a
cat
conclusion: Therefore,
Felix is pink
Example of a valid and sound
argument
major premise: Anthrax is not a
communicable disease
minor premise: Communicable
diseases pose the greatest threat
to public health
conclusion: Therefore, anthrax
does not pose the greatest threat
to public health
(Cats aren’t pink, which makes
the first premise untrue.
Validity, however, presumes
the truth of the premises.)
(The premises are true and the
conclusion is valid, that is, it
necessarily follows from the
premises)
Syllogistic reasoning
The syllogism is a common form of deductive
reasoning.
There are different types of syllogisms
categorical (universal premises)
hypothetical (if-then premises)
disjunctive (either-or premises)
All follow the basic form:
major premise
minor premise
conclusion
Categorical syllogisms rely on
universal premises
Example of a valid categorical syllogism:
major premise: All Christians believe Jesus is
the son of God.
minor premise: Biff is a Christian.
conclusion: Biff believes Jesus is the son of
God.
(Note: validity isn’t affected by whether the premises
are true or not. Obviously, other religions don’t accept
Jesus as the son of God.)
Hypothetical syllogisms use “ifthen” premises
Example of a valid hypothetical syllogism:
Major premise: If Biff likes Babbs, then he’ll
ask her to the prom.
Minor premise: Biff likes Babbs,
Conclusion: Therefore, he’ll ask her to the
prom.
Disjunctive syllogisms use “eitheror” premises
Example of a valid disjunctive syllogism:
Major premise: Either Babbs will get her navel
pierced, or she’ll get a tongue stud.
Minor premise: Babbs didn’t get her navel
pierced.
Conclusion: Therefore, Babbs got a tongue
stud.
Practice syllogism
Major premise: Any creature with six legs is an
insect.
Minor premise: . Dr. Gass has six legs.
Conclusion: Therefore, Dr. Gass is an insect.
What kind of syllogism is this? (categorical,
hypothetical, or disjunctive)
Answer: Valid, but
Are the premises true?
unsound
Is the conclusion valid?
Is the argument sound (true premises and a valid
conclusion)
Well-known forms of deductive
invalidity
 Affirming the consequent
 Invalid Example:
– If A, then B
–B
– Therefore, A
 Invalid Example:
– Students who plagiarize are expelled from school
– Rex was expelled from school
– Rex must have plagiarized
More deductive invalidity
 Denying the antecedent
 Invalid example:
– If A, then B
– Not A
– Therefore, not B
 Invalid example:
– If you exceed the speed limit, you’ll get a
ticket.
– I’m not exceeding the speed limit.
– Therefore, I won’t get a ticket.
More deductive invalidity
 Undistributed middle term:
 Valid example:
– All A are B
– All B are C
– Therefore, all A are C
 Invalid example
– All A are B
– All C are B
– Therefore, all A are C
The middle term, B,
must serve as the
subject of one premise,
and the predicate of
another premise, but
cannot occur in the
conclusion
 Undistributed middle term:
 Invalid example:
– All humans need air to breathe
– All dogs need air to breathe
– Therefore, all humans need dogs
What, if anything, is wrong with
this syllogism?
All rock stars want to
become movie stars
Morton wants to become
a movie star
Therefore, Morton must
be a rock star
A. affirming the
consequent
B. denying the
antecedent
C. undistributed middle
term
D. valid syllogism
Answer:
Undistributed Middle
Term
What, if anything, is wrong with
this syllogism?
Anyone who has lived in
California for more
than a few years has
experienced an
earthquake
Nadine has lived in
California for more
than a few years
Nadine has experienced
an earthquake
A. affirming the
consequent
B. denying the
antecedent
C. undistributed middle
term
D. valid syllogism
Answer: Valid
Syllogism
What, if anything, is wrong with
this syllogism?
Anyone who has tried
heroin has tried
marijuana
Naomi hasn’t tried
heroin
Therefore, Naomi hasn’t
tried marijuana
A. affirming the
consequent
B. denying the
antecedent
C. undistributed middle
term
D. valid syllogism
Answer: Denying the Antecedent
If A, then B
Not A
Therefore, not B
What, if anything, is wrong with
this syllogism?
All Christian
fundamentalists are
opposed to abortion
Nadine is opposed to
abortion
Nadine is a Christian
fundamentalist
A. affirming the
consequent
B. denying the
antecedent
C. undistributed middle
term
D. valid syllogism
Answer: Affirming the Consequent
If A, then B
B
Therefore, A