philosophy 170 introduction to logic. chapter 1: informal introduction

Philosophy 170Philosophy 170

Introduction to LogicIntroduction to Logic

Chapter 1:Chapter 1:Informal IntroductionInformal Introduction

Logic is…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 premisesgenuinely support the conclusion

• ‘genuine support’ requires the probableor 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 Logic is notnot……

• 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?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: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 ArgumentAn 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 IndicatorsPremise 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 FormStandard Form

Premise 1

Premise 2

Premise n

Conclusion

All humans are mortal

Socrates is human

Socrates is mortal

Argument Form and InstanceArgument 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 InstanceForm 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 Two Types of Criteria for Successful ArgumentsSuccessful 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 1AArgument 1A

All whales are mammals

All mammals are air-breathers

All whales are air-breathers

“Good” or “Bad”?

F1 G1

T

T

T

All Premises True

Conclusion True

Argument 1BArgument 1B

All whales are fish

All fish are air-breathers



F1 G1

F

F

T

At least One Premise False

Conclusion True

Argument 1DArgument 1D

All whales are reptiles

All reptiles are birds

All whales are birds


F1 G1

F

F

F


Conclusion False

Form 1Form 1

All F are G

All G are H

All F are H

G1F2

All premises True At least one premise False

1A




1B

All whales are fish



1C

??????

1D




F1 G2

Con

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alse

Argument 2AArgument 2A

Some animals are frogs

Some animals are tree-climbers

Some frogs are tree-climbers


F2 G2

T

T

T

All Premises True

Conclusion True

Argument 2BArgument 2B

Some fish are frogs

Some fish are tree-climbers



F2 G2

F

F

T


Conclusion True

Argument 2DArgument 2D

Some fish are frogs

Some fish are birds

Some frogs are birds


F2 G2

F

F

F


Conclusion False

Argument 2CArgument 2C


Some animals are birds



F2 G2

T

T

F

All Premises are True

Conclusion False

Form 2Form 2

Some F are G

Some F are H

Some G are H

F1 G2


2A




2B

Some fish are frogs



2C




2D

Some fish are frogs

Some fish are birds


F2 G1

Con

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alse

Evaluating Deductive ArgumentsEvaluating 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’


1A




Valid & Sound

1B

All whales are fish



Valid but Unsound

1C

No Possible Instance(No possible counterexample)

1D




Valid but Unsound

F1 G2

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2A




Invalid

2B

Some fish are frogs



Invalid

2C




Invalid(Counterexample to Form 2)

2D

Some fish are frogs

Some fish are birds


Invalid

F2 G1

Con

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Argument Forms 1 & 2Argument Forms 1 & 2

Form 1

All F are G

All G are H

All F are H

Valid Form

Form 2

Some F are G

Some F are H

Some G are H

Invalid Form

Some Points about ValiditySome 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 InvalidityCounterexamples 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 InvalidityCounterexamples 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

SoundnessSoundness

• 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 ArgumentsEvaluating Deductive Arguments

Invalid but still “good”?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 ArgumentsEvaluating 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 EnumerationInduction by EnumerationA1 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 Creekwere infected with the RGH virusAll 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 AnalogyArgument 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 Some Rules of Thumb for Enumerations/AnalogiesEnumerations/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 Inductive Strength NotNot a Matter of Form 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. StrengthValidity 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 & StrengthBackground 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

AbductionAbduction

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

ExamplesExamples

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 AbductionAbout 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 ArgumentsEvaluating Inductive Arguments

Some Logical Properties of Some Logical Properties of StatementsStatements

• 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 StatementsProperties 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 StatementsProperties 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 StatementsPairs 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 StatementsSets 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 StatementsSets 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 Sets of Statements and Target StatementsStatements

{ 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 statementThe target statement follows from the set

Entailment, ImplicationEntailment, 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 TennielCarroll and Tenniel

Charles Lutwidge DodgsonCharles Lutwidge Dodgson [1832-1898] [1832-1898]Known by his pen name, Known by his pen name, Lewis CarrollLewis Carroll, Dodgson , Dodgson

was a man of diverse interests—in mathematics, logic, was a man of diverse interests—in mathematics, logic, photography, art, theater, religion, medicine, and photography, art, theater, religion, medicine, and science. He was happiest in the company of children for science. He was happiest in the company of children for whom he created puzzles, clever games, and charming whom he created puzzles, clever games, and charming letters.letters.

His book His book Alice's Adventures in WonderlandAlice's Adventures in Wonderland (1865), became an immediate success and has since been (1865), became an immediate success and has since been translated into more than eighty languages. The equally translated into more than eighty languages. The equally popular sequel popular sequel Through the Looking-Glass and What Through the Looking-Glass and What Alice Found ThereAlice Found There was published in 1872. was published in 1872.

The “Alice” books are but one example of his The “Alice” books are but one example of his wide ranging authorship. wide ranging authorship. The Hunting of the SnarkThe Hunting of the Snark, a , a classic nonsense epic (1876) and classic nonsense epic (1876) and Euclid and His Modern Euclid and His Modern RivalsRivals, a rare example of humorous work concerning , a rare example of humorous work concerning mathematics, still entice and intrigue today's students. mathematics, still entice and intrigue today's students. Sylvie and BrunoSylvie and Bruno (1889), published toward the end of (1889), published toward the end of his life, contains startling ideas including a description his life, contains startling ideas including a description of weightlessness.of weightlessness.

Adapted from:Adapted from:http://www.lewiscarroll.org/cld.html http://www.lewiscarroll.org/cld.html

Sir John TennielSir John Tenniel [1820–1914] [1820–1914]

English illustrator and English illustrator and satirical artist, especially knownsatirical artist, especially knownfor his work in for his work in PunchPunch and his and his illustrations for illustrations for Alice's Adventures in WonderlandAlice's Adventures in Wonderland (1865) and (1865) and Through the Looking-GlassThrough the Looking-Glass (1872). (1872).

In his drawings for In his drawings for PunchPunch TennielTenniel lent new lent new dignity to the political cartoon. dignity to the political cartoon.

TennielTenniel was knighted in 1893 and retired was knighted in 1893 and retired from from PunchPunch in 1901. He illustrated many books; in 1901. He illustrated many books; his drawings for his drawings for Alice's Adventures in Alice's Adventures in WonderlandWonderland and and Through the Looking-GlassThrough the Looking-Glass are are remarkably subtle and clever and are extremely remarkably subtle and clever and are extremely well-suited to Lewis Carroll's text. These well-suited to Lewis Carroll's text. These illustrations won him an international reputation illustrations won him an international reputation and a continuing audience. and a continuing audience.

Excerpted from: http://search.eb.com/eb/article-Excerpted from: http://search.eb.com/eb/article-90717009071700

philosophy 170 introduction to logic. chapter 1: informal introduction

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