7—Deductive Reasoning
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9—Deductive Reasoning SJSU Spring 2014 Comm 41 Sect 04, 06Minna Holopainen
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Photo Sonja Holopainen
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Agenda
Syllogisms Aristotelian LogicDiagramming Arguments
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Deductive Reasoning
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Syllogisms Aristotelian logicDiagramming arguments
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Two Logic Systems
+Inductive Reasoning
Finding patternsForming conclusions
-Deductive Reasoning
Eliminating options Testing hypotheses
X
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Logic
Informal logic
! The study of natural languagearguments
! Fallacies
Formal logic
! The study of inference with purelyformal content, where that contentis made explicit.
! Concerned with the form of anargument
! Deductive
! Aristotle’s syllogistic logic
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Types of Deductive Arguments
Argument by elimination Argument based on mathematics Argument from denition
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Syllogisms
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Syllogisms
! Greek: "#$$%&'"()* ( syllogismos )—"conclusion,""inference"
! "A discourse in which, certainthings having been supposed,something di +erent from thethings' supposed results ofnecessity because these thingsare so." (Aristotle’s Prior
Analytics 24b18–20)
Aristotle by RembrandtPicture from http://www.uah.edu/colleges/liberal/philosophy/heikes/302/time/rembrant/aristotle-homer.jpg without permission. SJSU Spring 2014 Comm 41 Holopainen
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SYLLOGISMS
Categorical Disjunctive Hypothetical
ENTHYMEME
Deductive forms of reasoning
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Major premise: All men are mortal.Minor premise:
Socrates is a man.Conclusion:Therefore, Socrates is
mortal.
P or Q.Not P.
Therefore, Q.
I’ll have co ! ee or tea.Not co ! ee.
Therefore, tea.
If P, then Q.P.Therefore, Q.
If P, then Q.Not Q.Therefore, not P.
If P Q.
If Q R.
Therefore, P R.
If I go to bed early, I will getenough sleep.If I get enough sleep, I willdo well in my speechtomorrow.Therefore, if I go to bed early,I will do well in my speechtomorrow.
An incompleteargument in whicheither a premise orconclusion is assumed
Socrates is mortal because he is a man.
Socrates is mortal because all men are mortal.
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Aristotelian Logic
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Aristotle384-322 BCE
Student of PlatoTeacher of Alexander the Great
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Four Categorical Claims
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All Some No
Only children are noisy. At least one child is noisy.
Not a single sweet thingturns sour in my mouth.
Everything sweet turnssour in my mouth.
Some sweets turn sour inmy mouth.
No child is noisy.
Every page has a stain in
it.
There is a page that has a
stain in it.Everybody’s feet werecold.
There were a few peoplewhose feet were cold.
All cups were in thecupboard.
There was a cup that wasin the cupboard.
Each respondent repliedpromptly.
A number of respondentsreplied promptly.
Not even one page has a
stain in it.Nobody’s feet were cold.
There were no cups in thecupboard.
Not a single respondentreplied promptly.
Every time I call him heanswers.
Only sometimes when Icall him he answers.
He never answers when Icall him.
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Four Categorical Claims
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Square of Opposition
Source Stanford Encyclopedia of Phiosophy at http://plato.stanford.edu/entries/contradiction/. SJSU Spring 2014 Comm 41 Holopainen
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Practice Square of Opposition
All board members drink co +ee.
Some board members do notdrink co +ee.Contradictories
Everyone who lives must die.
Nobody who lives must die.
Contraries
Mary always sleeps in class.
Mary sometimes sleeps in class.
Subalternates
Some people like to dance.
Some people do not like to dance.
Subcontraries
Graph from Stanford Encyclopedia of Phiosophy at http://plato.stanford.edu/entries/contradiction/. SJSU Spring 2014 Comm 41 Holopainen
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Four types of claims:
A—All humans are mortal.
E—No human is ageless.
I—Some humans are happy.
O—Some humans are notgreedy.
There are 256 types ofcategorical syllogisms: AAA,
AEA, AIA, AOA, AAE, AAI, etc.
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Diagramming Arguments
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Good arguments
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ValidStrong
Weak
Implausible Plausible
A sound argument has to be
valid.
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Arguments’ validity and strength
Bad argumentsGood arguments
VALID INVALID
Strong Weak
Every way the premisescould be true leads tothe conclusion.
It is extremely likelythat the premises leadto the conclusion.
It is very unlikely orimpossible that the premiseslead to the conclusion.
Whether an argument is valid or strong does notdepend on whether the premises are sound/plausible.
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Mortal things
Men
Valid All men are mortal.Socrates is a man.
Therefore, Socrates is mortal.
Weak All men are mortal.Socrates is mortal.
Therefore, Socrates is a man.
Mortal things
Men
Mortal things
Men
Reasoning with all
Valid All S are P.Q is S.So Q is P.
Weak All S are P.Q is P.So Q is S.
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Cranky
Busy
Students
Valid All students are busy people. All busy people are cranky.So all students are cranky.
Invalid/weakSome students are busy people.Some busy people are cranky.So some students are cranky.
Cranky
Cranky
Reasoning in a chain with all and some
Valid All S are P. All P are Q.So all S are Q.
WeakSome S are P.Some P are Q.So some S are Q.
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Mammal
Cats
Fish
ValidEvery cat is a mammal.No sh is a mammal.So no sh is a cat.
Invalid/weakEvery cat is a mammal.No sh is a cat.So no sh is a mammal.
Mammal
Cats
Fish
Mammal
Cats
Fish
Reasoning with no
Valid All S are P.No Q is P.So no Q is S.
Weak All S are P.No Q is S.So no Q is P.
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Delicious things
Usually strong Almost all cookies are delicious.Madeline is a cookie.So Madeline is delicious.
Weak Almost all cookies are delicious.Madeline is delicious.So Madeline is a cookie.
Delicious things Delicious things
Reasoning with almost all
Strong Almost all S are P.Q is S.So Q is P.
Weak Almost all S are P.Q is P.So Q is S.
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Taxpayers
Weak Almost all senior students are over 20 years old. Almost all 20-year olds are paying taxes.So almost all seniors are taxpayers.
Reasoning in a chain with almost all
Weak Almost all S are P. Almost all P are Q.So almost all S are Q.
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Diagramming Arguments Exercise
1. Get your argument cards.2. In your group, draw diagrams of all possible cases the argument can be true.3. Decide if the argument is valid, strong, or weak.
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All turtles can swim. Anything that swims eats sh.So turtles eat sh.
Anyone who is hot sweats.Manuel is not sweating.So Manuel is not hot.
Every toy is safe.Knife is not a toy.So no knife is safe.
All her shoes are expensive. All expensive things are pretty.So, all her shoes are pretty.
Some pets are aggressive.Some aggressive creatures aredangerous.So some pets are dangerous.
All funny movies are old.Modern Times (1936) is old.So Modern Times is a funny movie.
All food is edible.
Sushi is food.So sushi is edible.
Practice diagramming
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New concepts
Categorical, disjunctive, andhypothetical syllogismEnthymemeConditional
Antecedent, consequentModus ponens, converse errorModus tollens, inverse error
Four categorical claims/Square ofOppositionUniversal, particular
A , rmative, negative
Valid, invalid argumentStrong, weak argument
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Upcoming Assignments
AA3: Moral argument Library Research Module (Comm Center)Critical Analysis
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