A BriefA BriefIntroduction to Introduction to

LogicLogic

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 1Premise 2

Premise n

Conclusion

All humans are mortalSocrates 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 Gx is Fx is G

Instances:All humans are mortalSocrates is humanSocrates is mortal

All monsters are furryGrover is a monsterGrover 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 NOTNOT• Deductive = general to specific

• Inductive = specific to general

– THE ABOVE IS INCORRECTTHE 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 mammalsAll mammals are air-breathersAll whales are air-breathers

“Good” or “Bad”?

F1 G1

TTT

All Premises TrueConclusion True

Argument 1BArgument 1B

All whales are fishAll fish are air-breathersAll whales are air-breathers

“Good” or “Bad”?

F1 G1

FFT

At least One Premise FalseConclusion True

Argument 1DArgument 1D

All whales are reptilesAll reptiles are birdsAll whales are birds

“Good” or “Bad”?

F1 G1

FFF

At least One Premise FalseConclusion False

Form 1Form 1

All F are GAll G are H

All F are H

G1F2

All premises True At least one premise False

1A

All whales are mammalsAll mammals are air-breathersAll whales are air-breathers

1B

All whales are fishAll fish are air-breathersAll whales are air-breathers

1C

??????

1D

All whales are reptilesAll reptiles are birdsAll whales are birds

F1 G2

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Argument 2AArgument 2A

Some animals are frogsSome animals are tree-climbersSome frogs are tree-climbers

“Good” or “Bad”?

F2 G2

TTT

All Premises TrueConclusion True

Argument 2BArgument 2B

Some fish are frogsSome fish are tree-climbersSome frogs are tree-climbers

“Good” or “Bad”?

F2 G2

FFT

At least One Premise FalseConclusion True

Argument 2DArgument 2D

Some fish are frogsSome fish are birdsSome frogs are birds

“Good” or “Bad”?

F2 G2

FFF

At least One Premise FalseConclusion False

Argument 2CArgument 2C

Some animals are frogsSome animals are birdsSome frogs are birds

“Good” or “Bad”?

F2 G2

TTF

All Premises are TrueConclusion False

Form 2Form 2

Some F are GSome F are H

Some G are H

F1 G2

All premises True At least one premise False

2A

Some animals are frogsSome animals are tree-climbersSome frogs are tree-climbers

2B

Some fish are frogsSome fish are tree-climbersSome frogs are tree-climbers

2C

Some animals are frogsSome animals are birdsSome frogs are birds

2D

Some fish are frogsSome fish are birdsSome frogs are birds

F2 G1

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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’

All premises True At least one premise False

1A

All whales are mammalsAll mammals are air-breathersAll whales are air-breathers

Valid & Sound

1B

All whales are fishAll fish are air-breathersAll whales are air-breathers

Valid but Unsound

1C

No Possible Instance(No possible counterexample)

1D

All whales are reptilesAll reptiles are birdsAll whales are birds

Valid but Unsound

F1 G2

Con

clus

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Tru

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lse

All premises True At least one premise False

2A

Some animals are frogsSome animals are tree-climbersSome frogs are tree-climbers

Invalid

2B

Some fish are frogsSome fish are tree-climbersSome frogs are tree-climbers

Invalid

2C

Some animals are frogsSome animals are birdsSome frogs are birds

Invalid(Counterexample to Form 2)

2D

Some fish are frogsSome fish are birdsSome frogs are birds

Invalid

F2 G1

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Argument Forms 1 & 2Argument Forms 1 & 2

Form 1

All F are GAll G are HAll F are H

Valid Form

Form 2

Some F are GSome F are HSome 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 InvalidityCounterexample:

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 cardsThe deck has been shuffledThe next card drawn will not be a Jack

Most Rottweilers have docked tailsRalphie is a RottweilerRalphie 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, HB is F, G, H, and IA is I

My car is a 1999 Toyota CamrySue’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 earsBingo the elephant likes peanuts, is bigger than a breadbox,

has two ears, and has a trunkI 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

FallaciesFallacies

Fallacy:A fallacy is any mistake in reasoning, but some are particularly seductive (both to the speaker/writer and the listener/reader) and so common that they have earned names. See the text for details…

Carroll and TennielCarroll and TennielCharles Lutwidge DodgsonCharles Lutwidge Dodgson [1832-1898] [1832-1898]

Known by his pen name, Known by his pen name, Lewis CarrollLewis Carroll, , Dodgson was a man of diverse interests—in Dodgson was a man of diverse interests—in mathematics, logic, photography, art, theater, mathematics, logic, photography, art, theater, religion, medicine, and science. He was happiest in religion, medicine, and science. He was happiest in the company of children for whom he created the company of children for whom he created puzzles, clever games, and charming letters.puzzles, clever games, and charming letters.

His book His book Alice's Adventures in WonderlandAlice's Adventures in Wonderland (1865), became an immediate success and has since (1865), became an immediate success and has since been translated into more than eighty languages. The been translated into more than eighty languages. The equally popular sequel equally popular sequel Through the Looking-Glass Through the Looking-Glass and What Alice Found Thereand What Alice 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 Euclid and His Modern RivalsModern Rivals, a rare example of humorous work , a rare example of humorous work concerning mathematics, still entice and intrigue concerning mathematics, still entice and intrigue today's students. today's students. Sylvie and BrunoSylvie and Bruno (1889), published (1889), published toward the end of his life, contains startling ideas toward the end of his life, contains startling ideas including a description of weightlessness.including a description 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