Transcript Philosophy 106

Philosophy 170
Introduction to Logic
Chapter 1:
Informal Introduction
Logic is…
• The study of argument
• The study of criteria for distinguishing successful from
unsuccessful arguments and the study of methods for applying
those criteria
• An argument is a set of statements, some of which—the
premises—are supposed to support, or give reasons for, the
remaining statement—the conclusion
• In a successful argument the premises
genuinely support the conclusion
• ‘genuine support’ requires the probable
or guaranteed preservation of truth
from premises to conclusion
• The study of related properties such as
consistency, logical truth, etc.
• The key to a world of wonder
Logic is not…
• Logic is not the study of
persuasion and manipulative
rhetorical devices
• ‘successful argument’ does
not mean persuasive argument
– Human fallibility and manipulative rhetoric lead people to
• accept poor reasoning
• reject good reasoning
• Remember, in a successful argument if the premises
are true, then the conclusion is either guaranteed to be
true or likely to be true
Why Study Logic?
• Intrinsic value
– Enjoyment of learning
– Enjoyment of abstract structures and analytic
elegance
– Enjoyment of puzzles and figuring things out
• Instrumental value
– Improve abstract, critical, and analytic reasoning
– Increase the number of tools in your critical
thinking “toolkit”
– Improve writing, reading, speaking skills
– Become a better thinker/knower
– Become a more independent thinker
– Become the life of the party
Some Definitions:
Statement:
A statement is a declarative sentence; a sentence which attempts to state a
fact—as opposed to a question, command, exclamation, etc.
Argument:
an argument is a (finite) set of statements, some of which—the premises—
are supposed to support, or give reasons for, the remaining statement—the
conclusion
Logic:
Logic is the study of
(i) criteria for distinguishing successful from unsuccessful argument,
(ii) methods for applying those criteria, and
(iii) related properties of statements such as implication, equivalence, logical truth,
consistency, etc.
Truth Value:
The truth value of a statement is just its truth or falsehood; we assume that
every statement has either the truth value true, or the truth value false, but
not both
An Example Argument
• Socrates is mortal, for all humans are mortal,
and Socrates is human
• Given that Socrates is human, Socrates is
mortal; since all humans are mortal
• All Humans are mortal, Socrates is human;
therefore Socrates is mortal
Premise and Conclusion Indicators
Premise Indicators:
as, since, for, because, given that, for the reason that,
inasmuch as
Conclusion Indicators:
therefore, hence, thus, so, we may infer,
consequently, it follows that
Standard Form
Premise 1
Premise 2

Premise n
Conclusion
All humans are mortal
Socrates is human
Socrates is mortal
Argument Form and Instance
Argument Form and Instance:
An argument form (or schema) is the framework of
an argument which results when certain portions of
the component sentences are replaced by blanks,
schematic letters, or other symbols. An argument
instance is what results when the blanks in a form are
appropriately filled in
Form and Instance
Form:
All F are G
x is F
x is G
Instances:
All humans are mortal
Socrates is human
Socrates is mortal
All monsters are furry
Grover is a monster
Grover is furry
Two Types of Criteria for
Successful Arguments
• Deductive
• Inductive
– These criteria have some things in common, but will turn out to be
importantly different
– The distinction is NOT
• Deductive = general to specific
• Inductive = specific to general
– THE ABOVE IS INCORRECT
– The distinction will involve the nature of the link between premises and
conclusion
– This is best illustrated…
Argument 1A
All whales are mammals
All mammals are air-breathers
All whales are air-breathers
“Good” or “Bad”?
T
T
T
All Premises True
Conclusion True
F1 G1
Argument 1B
All whales are fish
All fish are air-breathers
All whales are air-breathers
“Good” or “Bad”?
F
F
T
At least One Premise False
Conclusion True
F1 G1
Argument 1D
All whales are reptiles
All reptiles are birds
All whales are birds
“Good” or “Bad”?
F
F
F
At least One Premise False
Conclusion False
F1 G1
Form 1
All F are G
All G are H
All F are H
F2 G1
Conclusion False
Conclusion True
All premises True
At least one premise False
1A
1B
All whales are mammals
All mammals are air-breathers
All whales are air-breathers
All whales are fish
All fish are air-breathers
All whales are air-breathers
1C
1D
??????
All whales are reptiles
All reptiles are birds
All whales are birds
F1 G2
Argument 2A
Some animals are frogs
Some animals are tree-climbers
Some frogs are tree-climbers
“Good” or “Bad”?
T
T
T
All Premises True
Conclusion True
F2 G2
Argument 2B
Some fish are frogs
Some fish are tree-climbers
Some frogs are tree-climbers
“Good” or “Bad”?
F
F
T
At least One Premise False
Conclusion True
F2 G2
Argument 2D
Some fish are frogs
Some fish are birds
Some frogs are birds
“Good” or “Bad”?
F
F
F
At least One Premise False
Conclusion False
F2 G2
Argument 2C
Some animals are frogs
Some animals are birds
Some frogs are birds
“Good” or “Bad”?
T
T
F
All Premises are True
Conclusion False
F2 G2
Form 2
Some F are G
Some F are H
Some G are H
F1 G2
Conclusion False
Conclusion True
All premises True
At least one premise False
2A
2B
Some animals are frogs
Some animals are tree-climbers
Some frogs are tree-climbers
Some fish are frogs
Some fish are tree-climbers
Some frogs are tree-climbers
2C
2D
Some animals are frogs
Some animals are birds
Some frogs are birds
Some fish are frogs
Some fish are birds
Some frogs are birds
F2 G1
Evaluating Deductive Arguments
Deductive Validity, Invalidity:
An argument (form) is deductively valid iff* it is
NOT possible for ALL the premises to be true AND
the conclusion false, it is deductively invalid iff it is
not valid
Soundness:
An argument is sound iff it is deductively valid AND
all its premises are true
* ‘iff’ is short for ‘if and only if’
Conclusion False
Conclusion True
All premises True
At least one premise False
1A
1B
All whales are mammals
All mammals are air-breathers
All whales are air-breathers
All whales are fish
All fish are air-breathers
All whales are air-breathers
Valid & Sound
Valid but Unsound
1C
1D
No Possible Instance
(No possible counterexample)
All whales are reptiles
All reptiles are birds
All whales are birds
Valid but Unsound
F1 G2
Conclusion False
Conclusion True
All premises True
At least one premise False
2A
2B
Some animals are frogs
Some animals are tree-climbers
Some frogs are tree-climbers
Some fish are frogs
Some fish are tree-climbers
Some frogs are tree-climbers
Invalid
Invalid
2C
2D
Some animals are frogs
Some animals are birds
Some frogs are birds
Some fish are frogs
Some fish are birds
Some frogs are birds
Invalid
Invalid
(Counterexample to Form 2)
F2 G1
Argument Forms 1 & 2
Form 1
Form 2
All F are G
All G are H
All F are H
Some F are G
Some F are H
Some G are H
Valid Form
Invalid Form
Some Points about Validity
• Validity a question of Truth Preservation
• It is a matter of Form
– Thus an argument form is valid (invalid), and any instance
of that form is valid (invalid)
• It has nothing to do with actual truth values of the
sentences involved*
– True premises and true conclusion are neither necessary
nor sufficient for validity (see 1B, 1D, and 2A)
*Except for counterexamples…
Counterexamples and Invalidity
Counterexample:
A counterexample to an argument (form) is an
instance of exactly the same form having all true
premises and a false conclusion. Production of a
counterexample shows that the argument form and all
instances thereof are invalid.
– This is the ONLY time actual truth values are
relevant
• If all premises are true and the conclusion is false, that
instance, that form, and any other instance of that form
are invalid
Counterexamples and Invalidity
• We can see that a particular argument, an
argument form, and all instances of that form
are invalid by either:
– Offering a counterexample, or
– Consistently imagining that all the premises are
true and the conclusion is false
• Failure to do one of the above shows nothing, however,
because it may be just our lack of creativity which
prevents us finding a counterexample or imagining the
appropriate situation
Soundness
• An argument is sound iff it is deductively valid and
all the premises are true
• Unlike validity, soundness does have to do with the
actual truth values of the premises
• Soundness is only an issue when the argument is
valid
• Unsound arguments will not convince a worthy
opponent
• Determining soundness is outside the bounds of logic,
it requires non-logical investigation
Evaluating Deductive Arguments
Invalid but still “good”?
There are 4 Jacks in this standard deck of 52 cards
The deck has been shuffled
The next card drawn will not be a Jack
Most Rottweilers have docked tails
Ralphie is a Rottweiler
Ralphie has a docked tail
Evaluating Inductive Arguments
Inductive Strength:
An argument is inductively strong to the
degree to which the premises provide evidence
to make the truth of the conclusion plausible or
probable. If an argument is not strong, it is
weak.
Cogency:
An argument is cogent iff it is inductively
strong AND all the premises are true
Induction by Enumeration
A1 is F
A2 is F

An is F
All As (or the
next A) are/will
be F
All 57 trout caught in Jacob’s Creek
were infected with the RGH virus
All trout (or the next trout caught; or x% of
trout) in Jacob’s Creek will be infected with
the RGH virus
• The As are the sample—the observed instances or examples;
• F is the target property
Argument by Analogy
A is F, G, H
B is F, G, H, and I
A is I
My car is a 1999 Toyota Camry
Sue’s car is a 1999 Toyota Camry
and gets over 30 mpg
My car will get over 30 mpg
• F, G, H are the similarities, I is the target property
• The comparison base, B, may be an individual or a group
Some Rules of Thumb for
Enumerations/Analogies
• The larger the sample size or comparison base group,
the stronger the argument
• The narrower or more conservative the conclusion,
the stronger the argument
• The greater the number of (relevant) similarities, the
stronger the argument
• The fewer the number of (relevant) dissimilarities, the
stronger the argument
Inductive Strength Not a Matter of Form
The 12,700 days since my birth
have all been days on which I did not die
So I will not die today. Indeed, I’ll never die!
I like peanuts, am bigger than a breadbox, and have two ears
Bingo the elephant likes peanuts, is bigger than a breadbox,
has two ears, and has a trunk
I have a trunk
Validity vs. Strength
• Unlike deductive validity, inductive strength is a
matter of degree, not an all-or-nothing, on/off switch
• Unlike deductive validity, inductive strength is not a
matter of form
• Unlike deductive validity, additional information is
relevant to the assessment of strength
Background Knowledge & Strength
• Determining strength of an inductive argument has a
lot to do with many unstated background
assumptions, e.g.:
– Relevance of similarities and dissimilarities
– Nature and selection of the sample group
– Stability of relevant but unstated conditions
• It also has to do with the availability of further
evidence, thus
• Unlike with validity, additional premises (new
evidence, change in background assumptions) can
increase or decrease the strength of the argument
Abduction
Abduction:
Abduction or abductive reasoning, also known
as inference to the best explanation is a
category of reasoning subject to inductive
criteria in which the conclusion is supposed to
explain the premises
Examples
It is 5pm on Monday
The mail has not come
The mail carrier is almost
never late
It must be a holiday
I see paw prints on the hood and
roof of my car
There are fur balls in the corner
There are mice guts under the car
The garage door was left open
The cat slept in the garage
About Abduction
• The more data accounted for the better the
explanation
• The better the explanation coheres with already
confirmed theory, the better it is
• The more new data successfully predicted, the better
the explanation
• So, again, background assumptions are relevant
• There is almost always more evidence available, and
it might lead to a reassessment of the
inference/argument
• Exactly what is meant by “best” is not entirely clear
Evaluating Inductive Arguments
Some Logical Properties of
Statements
• Alice fell down a rabbit hole
• If Alice fell down a rabbit hole, then it’s not the case
that Alice didn’t fall down a rabbit hole
• Alice fell down a rabbit hole and did not fall down a
rabbit hole
Properties of Individual Statements
Logically True:
A statement is logically true if and only if it is not possible for
the statement to be false
Logically False:
A statement is a logically false if and only if it is not possible
for the statement to be true
Logically Contingent:
A statement is a logically contingent if and only if it is neither
logically true nor logically false; i.e., it is both possible for the
statement to be true, and possible for the statement to be false
Properties of Pairs of Statements
• Alice fell down a rabbit hole
• Alice drinks from the bottle labeled “DRINK ME”
• If Alice doesn’t drink from the bottle labeled
“DRINK ME”, then she won’t fit through the door
• To fit through the door, Alice must drink from the
bottle labeled “DRINK ME”
• The White Rabbit does not speak and carry a watch
• The White Rabbit speaks and carries a watch
Pairs of Statements
Logically Equivalent:
A pair of statements is logically equivalent if and
only if it is not possible for the statements to have
different truth values
Logically Contradictory:
A pair of statements is logically contradictory if and
only if it is not possible for the statements to have the
same truth values
Sets of Statements
{ Alice eats the cake marked “EAT ME”,
Alice grows large enough to get the key,
Alice is again too large to fit through the door }
{ Alice eats the cake marked “EAT ME”,
If Alice eats the cake marked “EAT ME” she
will grow too large to fit through the door,
Alice fits through the door }
Sets of Statements
Logically Consistent:
A set of statements is logically consistent if and only
if it is possible for all the statements to be true
Logically Inconsistent:
A set of statements is logically inconsistent if and
only if it is not possible for all the statements to be
true
Sets of Statements and Target
Statements
{ If Alice cries while nine feet tall she will leave a large
pool of tears,
If there is a large pool of tears and Alice shrinks
again then she will have to swim,
Alice cries while nine feet tall and shrinks again }
Alice has to swim
The set entails or implies the target statement
The target statement follows from the set
Entailment, Implication
Logically Entails, Implies, Logically Follows:
A set of statements logically entails or implies a
target statement if and only if it is NOT possible for
every member of the set to be true AND the target
statement false. We also say that the target statement
logically follows from the set
Carroll and Tenniel
Sir John Tenniel [1820–1914]
Charles Lutwidge Dodgson [1832-1898]
Known by his pen name, Lewis Carroll, Dodgson
was a man of diverse interests—in mathematics, logic,
photography, art, theater, religion, medicine, and science.
He was happiest in the company of children for whom he
created puzzles, clever games, and charming letters.
His book Alice's Adventures in Wonderland
(1865), became an immediate success and has since been
translated into more than eighty languages. The equally
popular sequel Through the Looking-Glass and What
Alice Found There was published in 1872.
The “Alice” books are but one example of his
wide ranging authorship. The Hunting of the Snark, a
classic nonsense epic (1876) and Euclid and His Modern
Rivals, a rare example of humorous work concerning
mathematics, still entice and intrigue today's students.
Sylvie and Bruno (1889), published toward the end of his
life, contains startling ideas including a description of
weightlessness.
Adapted from:
http://www.lewiscarroll.org/cld.html
English illustrator and
satirical artist, especially known
for his work in Punch and his
illustrations for Alice's Adventures in
Wonderland (1865) and Through the LookingGlass (1872).
In his drawings for Punch Tenniel lent
new dignity to the political cartoon.
Tenniel was knighted in 1893 and retired
from Punch in 1901. He illustrated many
books; his drawings for Alice's Adventures in
Wonderland and Through the Looking-Glass
are remarkably subtle and clever and are
extremely well-suited to Lewis Carroll's text.
These illustrations won him an international
reputation and a continuing audience.
Excerpted from:
http://search.eb.com/eb/article-9071700