REASON

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FOUR MAIN WAYS WE REASON

Good source: Irving Copi's Introductionto LogicSummarized source: LRMontezuma notes

ANALOGYAND INDUCTION: Offer new information based on experience

Analogical Inference (this looks like an instance of that): "This should be a goodrestaurant"; "Let's model the human brain as a binary computer."

. ~Oftenused in everyday life, in AoKs, and in applied disciplines (medicine, law, etc.)

. Mayfurnishwrongresults,butisanextremelypowerfultoolto proceedfromtheknown to the unknown, and to solve problems quickly .. Most common error in reasoning: "False analogy" (informal fallacy)

! Induction (It's happened once, tvvice...193,467times... it willalwayshappen): "TheI sun will rise tomorrow"; "Everybodywilldie someday."

. Sourc~ of general assertions Cmajor premises") used in deduction. "Problem of induction" (Russell's chicken), but is the best source of generalizationsavailable to us

. Mostcommonerror in reasoning: "Hasty generalization" (informal fallacy)

DEDUCTION:Offers no new information

11 premise: Direct Inference_: "If all people are mortal, then no person is immortal."

. Based on relationships between quantifiers "all", "some", and "no/none"Important kind: Negation. To negate "all people are mortal,"one case of animmortalperson suffices.Always offers true conclusions iforiginal premise is true and inference is valid

.

.

[2 premises: Syllogism: "If all people are mortal, and I am a person, then I amI mortal."

. Allowsapplicationofgeneralassertions to particular cases .

. Important kinds:Disjunctive(including"and" and "or"), hypothethical(If...then),andcategoricalCdeductive")syllogisms. Alwaysoffers true conclusions!fpremisesare true andsyllogismisvalid

Deductive reasoning errors are called "formal fallacies",

Presented by L.,Rotenberg, at AHUWC,June 1999

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REASON: OVERVIEW

Introduction

.. "Reason" for the purposes of this unit encompasses the following terms: Logical rigor, criticalthinking, logic, deductive logic, deduction, and induction. (This terminology is based on the1996 TOK Cuniculum Review Meeting.)

. Together with sense perception, emotion and language, reason is a fundamental tool thatenables us, knowers, to construct a knowledge base we can share with each other.

. Historically: Reason became valuable as a way to rebel against authority.Historical dualisms: rationalism v. empiricism, reason v. emotion (~ either-or informalfallacies) .

. Types of propositions: (a) ~nalytic; (b) empirical; (c) value judgment; (d) metaphysical(~ Truth tests: analytic proposition ~ coherence truth test; empirical proposition ~correspondence truth test) .

Important Definitions and DIstinctions

"True" v. "valid":

a proposition is true or false (~ truth-tests)an argument is valid or invalid.

Argument:-----------

a series of premises followed by conclusion.

Syllogism: a deductive argument

Fallacy: an invalid argument. i.e. an argument in which the concJusiondoes not logicallyfollow from the premises.

Deductivev. Inductivereasoning:

. Deduction beginswith a general statement (All,None. Some...) and reaches a conclusionabout a particularcase (Bill, the cat..):

All people have ten fingersBill is a oerson:. Bill has ten fingers,

This is an example of a valid deductive syllogism. All arguments of the form

All PisOP .:.0 are valid. no matter what P and 0 stand for. (~ math: symbolic nature)

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If all premises are true and the argument is valid. the conclusion is necessarily true.Deduction can thus enable us to glean new truths from old ones, and allow us to test thecoherence ofourwond-views. (~math, feeling, science, history, ethics )

However, it suffices that one of the premises be questionable or false to shed doubt onthe conclusion, as happens in the example above.

(~ Essaywrtting: Reductio ad absurdum. If. on the other hand. the conclusion of anargument can be shown to generate a CGiltradictionwith one of its premises, it can be concludedthat the conclusionwas false.)

Example: Asoliefeconomy is desired so that future generations can thrive.Internal debt must be reduced to guarantee a solideconomy.We should raise taxes in order to reduce internal debt.We should raise taxes in order to guaranteethat future generations can

thrive.

Reductio: Let us raise taxes.Parents will not have as much money to spend on their families.Children will not have accessto as many resourcesFuture aenerations will not thrive.A solid economy (as defined) will not result from raisingtaxes.

. Inductionbeginswith observations of particular instances and anives at a generalconclusion:

Mygrandparents, my parents, my siblingshave ten fingersAllpeople I know have ten fingersAllpeople my friends know have ten fingers(X number of further observations required here: 81nductive leapj:.AIIpeople have ten fingers

It suffices that one counter-example be found to counter a zjllionobservations.

(~ natural sciences, "Beware of general statementswhen writing!8-caveat alsoincludes "always,88never,8 8everywhere,8 etc., stereotyping)

. The empiricalpropositions}lsed as premises in a deductive argument must be arrived atinductively. Induction is therefore at the heart of our knowtedgeabout the wond.

And,Or

"P and Q" means that both must be the case. E.g.: She is a talented and hard-worXing pianist.

. Alternativefonns: She is a talented.hard-worXingpianist.As a pianist she is talented; she is hard-working.

. Negation is achieved by stating either of the following:She is not a talented pianist.She is not a hard-worXingpianist.

C Lena Rotenberg. 1997.This page may be freely reproduced rOl'educational purpOMS as long as thi$ notice i$ included.

Page 3 of 7

up or Q" has two alternativemeanings:

. "Exclusive Or":"Inclusive Or":

I'll either be in the classroom or at the library. (Can only be one.)As a pianist she is either competent or hard-working. (Could be both.).

. . Negation is achieved by denying both assertions:I am neither in the classroom nor at the library.As a pianist she is neither competent nor hard-working.

The SimplificatJon6 of Deduction

Categorical Propositions:AIIPisQNoP is 0SomeP is 0SomeP is not0

(Note that "P is O' is logically equivalent to "All P is 0: Implicit in a statement such as"People are selfISh" is the assertion that s1l people are selfish.)

. Logic (as well as most of our knowledge) assumes that there are invariant categories towhich things statically belong, and that the things themselves are static enough to allow us totalk about them. Some sKepticsand Zen Masters might not agree.

. Square of Opposition:

All No

Some are Some are Not

. . "Some" is taken to encompass all kinds of "'fuzzy.quantifiers. Examples include: a few,fifteen, 27%, not many, the majority, not all, nearly everyone, etc. A "fuzzy logic" is beingdeveloped to better deal with this whole range of quantifiers (see Bart Kosko, Fuzzy thinking:The new science of fuzzy logic,New York: Hyperion, 1993.) .

C Lena Rotenberg, 1997. Thi8 page may be freely reproduced fortducatJonal purposes as long as this notice is included.

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Fundamental principles of Aristotelian logic:

(1) The Law of Identity. PwP (everything is equivalent to itself).(2) The Law of Contradiction: either Pis Q, or Pis non-Q. P cannot be both Q and not Q. (~

quantum mechanics: an electron is both a partide and a wave)(3) The Law of the Excluded Middle: P must either be Q or non-Q (everything must either be

something or not be something). (~ 81ack or white fallacy: "Ifyou're not my friend you mustbe my enemy"; the problem of definition, language.)

Criticisms of Aristotelian Logic:

. Aristotelian logic has been severely criticizedfor incurringinthe "fallacyof existentialassumption: The proposition"Some butterfliesare blue"asserts that butterflies exist ("some"implies that there is at least one butterfly contained in the universe of blue things). However,in Aristotelian logic "some butterflies are blue" is a condition for our being able to say "allbutterfliesare blue" This means that the fourkinds of categorical propositions used inAristotelian logic imply existential import, Le., we are always assuming that the things we talkabout exist. This leads to profound contradictions within the Aristotelian system.

. To counter these limitationsGeorge 800le (1815-1884)developed his own logic, bypostulatingthat the "all"~nd "none"propositionsnever have existential import. One can thusassert things about "allunicorns"and about "noflyingpigs"withouthaving to assume thatthese things exist. Most rules and conclusions reached by Aristotlecould thus bemaintained. rSome unicorns are blue: however, wouldalways be false according to800lean logic,because unicorns do not exist.)

. Modem logicianswent beyond, asserting that l~ic should not even be based on a wond "outthere":

"EpistemiC logic deals with the I~ical issues arising within thegamut of such epistemological concepts as knowledge, belief,assertion, doubt, question-and-answer, or the like. Instead ofdealing with the essentially factual issues of alethic logic (Greek:aletheia, "tnrthj-te., with what is actually or must necessarily orcan possibly be the case-it relates to what people knoworbelieve or maintain or doubt to be the case: (From "The Historyand Kinds of Logic:EPISTEMIC LOGIC." 8litannica CD.Version 97. Encyclopaedia 8litannica, Inc.. 1997.)

. One major representativeof the branch of epistemic logic was Arend Heyting, who in 1930formalized Intuitionism as applied to logic and math. This logiconly deals with what can beknown. Heyting rejeded "EitherQ or not-C- (the Lawof the ExcJudedMiddle)as a logicaltruth, because it does not adually help us to knowwhether Q or not-Q is the case.

. Mathematics and logic became formally interrelated in the early decades of this century,through the worX of Russell and VVhitehead.

~ Lena Rotenberg, 1997. This page may be fr..1y reproduced for educational purposes as long as U'1isnotice is Included.

Page 5 of 7

Formal v. Informal Fallaclu:

A fonnal fallacy occurs when the conclusionof a syllogismdoes not logically follow from itspremises ("Non Sequiturj.

Infonnal fallacies are subtler and often harder to identify. Some are actually instances of formalfallacies, but are so pervasive they have received special names. Such arguments may bepsychologically persuasive,and can be grouped into the following categories (otherclassifications abound):

. Neglecting the whole question

. I Makingfaultyassumpfions

. Neglecting aspects of the question

. Appealing to the ilTelevant

. Using natural and scientific language impreciselyor manipulatively

The Rules of Deduction

Negations: If one is true. the other necessarilymust be false.

I All P is 0 . If P then 0 ., Some P is not 0

I No P is 0 . No 0 is P ) Some P is Q . Some 0 is P

Both equivalencies (denoted by.) in the second row represent an operation calledconversion (I.e., the converse of "Some dogs are black. is "Some black things are dogs: (~Students will have seen this term used in geometry.)

Conditional Propositions: .

. "If p, then 0" is logicaJlyequivalentto .All P is 0: In Boolean logic it is taken to mean,"If P exists, then it is 0." (If a unicorn exists,then it has wings.)

"If p, then not 0" is logically equivalentto "No P is 0." In Boolean logic it is taken tomean,"If P exists.thenit is not0.. (If a unicorn exists. then it does not have wings.)

"P if and only if O' (math:"P iff OJ is logiCallyequivalentto "All P is 0 and All 0 is p', Le.both universes are identical. (~This is a requirementfor a good definition in any field.)

To negate "If it conducts electricity, then it is a metal. we need only to show that there isone thing that conducts electricity that is not a metal. (~ Definitions, classifications)

= Lena Rotenberg, 1997. This page may be freely reproduced fO( educational purposes as long as this notice is included.

Page 6 of 7

Hypothetical Syllogisms:

. The major premise is a conditional proposition. There are two valid forms, Modus Ponensand Modus Tollens:

If P, then 0P:.0

If P, then 0not 0:.not P

Exampie:

"If you build it. they will come."You built it.They didn't come.

...1 can condude that you were lied to. Had it been a true statement. they woald havehad to come.

. Typical mistakes:

"Denying the antecedent" ~ translates into the informaLHypothesisContrary to Fadfallacy)

If P. then 0Not P:.Not O.

If it rains I will stay home.It did not rain.

I did not stay home

The condusion does not foilow: Nothing is known about what I will do if it does not rain.Maybe I will stay home for other reasons.

"Asserting of the consequent" (~ translates into the informal False Cause fallacy)

If P, then 00:.P

If Jack loves me he will callJack calledI conclude that Jack loves me.

The condusion does not follow because JacXcould have called for other reasons.

. A special exception is the fonn "Only if P then Q,' a "transcendental argument' much used inthe field of metaphysics:

Only if P then QQ:.P

Only ~God exists will Q happen'Q happens:. God exists

~ Lena Rotenberg, 1997.This page may be frHly reproduced fO(educationalpurpo~ as long as this notice is included.

Page 7 of 7

DeductiveSyllogisms

Arguments in ordinary language do not necessarilyfollow the standard fonn,

Major premiseMinor premiseConclusion

. Words that mark conclusions in arguments include: therefore, hence, thus, so, accordingly,in consequence, consequently, proves that, as a result, for this reason, it follows that, we mayinfer, I conclude that, which shows I means I entails I implies I allows us to infer I points to theconclusion that.

. Words that mark premises in arguments include: since, because, for. as indicated by, thereason is that, for the reason that, may be inferred from, foilows from, as shown by, inasmuchas, may be derived I deduced from, in view of the fad that (Both lists taken from Copi &Cohen, gtned. 19~, p. 9-10.)

. Dedudive arguments-syllogisms-are used in every field of knowledge and in every facet ofhuman activity which even tangentially involves rational thought. Like algebraic equations, inwhich we can plug in different numbers and obtain corred results, the forms of validsyllogisms and of truth tables allowus to plug in premisesdealing with the most diversethings: P, Q, FL. can be whatever we want.

. Euler diagrams can help students to test given conclusionsfor validity, and to flQureout theconclusions of simple sytlogisms.

Analogical Inference and Jnd~ctJon

To determine the truth of empirical propositions, inductive methods such as analogical inferenceand indudion must be used.

Analogical inference provides us with educated guesses about new situations, based on ourpast e~perience. (E.g.: In the past I have enjoyed the ste,ak dinner at a certain restaurant, so Iretum to that restaurant tonight.) This method is less reliable than indudion-a key issue iswhether the analogy is solid-but it is extremely useful when we can afford to take a few risksand have littJetimeto'makea moredeliberatedecision. In our daily lives, this is -the method wemost often use.

Induction, on the other hand, is based on the repeatedobservation of the same phenomenon.We believe that the sun has always risen, and that under normal circumstances it will continue torise every moming. All the ravens humankind has observed have been blacK, thus we consider"All ravens are black8to be a true statement. However, a conclusion reached inductively is nevercertain. An infinite number of observations will not guarantee that the sun will rise tomorrow, orthat a pink raven will never be observed.

. Empirical fields of knowledge have developed and adopted methods and procedures toensure the maximum possible trustworthiness of their truth cfaims. Such procedures includethe scientific method and the public nature of scientific knowledge (which promote attempts tofalsify empirical propositions). Ukewise, mathematics and statistics are used to determinewhat constitutes a reliable enough conclusion based on empirical observations.

IQLena Rotenberg, 1997,This page may be freelyreproduced to( educational purposes as long as this notic. is included.