Legal Technique and Logic Reviewer SIENNA FLORES

5/19/2018 Legal Technique and Logic Reviewer SIENNA FLORES

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SIENNA A.FLORES LEGAL TECHNIQUE &LO

CHAPTER 1PROPOSITIONS

1.1 What Logic Is

Logic

The study of the methods and principles used to distinguish

correct from incorrect reasoning

1.2 Propositions

Propositions

An assertion that something is (or is not) the caseAll propositions are either true or falseMay be affirmed or denied

Statement

The meaning of a declarative sentence at a particular timeIn logic, the word statement is sometimes used instead of

propositions

Simple Proposition

A proposition making only one assertion.

Compound Proposition

A proposition containing two or more simple propositions

Disjunctive (or Alternative) PropositionA type of compound propositionIf true, at least one of the component pro

positions must betrue

Hypothetical (or Conditional) Proposition

A type of compound proposition;It is false only when the antecedent is true and theconsequent is false

1.3 Arguments

Inference

A process of linking propositions by affirming one proposition

on the basis of one or more other propositions.

Argument

A structured group of propositions, reflecting an inference.

Premise

A proposition used in an argument to support some otherproposition.

Conclusion

The proposition in an argument that the other propositions,the premises, support.

1.4 Deductive & Inductive Arguments

Deductive Argument

Claims to support its conclusion conclusively

One of the two classes of argument

Inductive Argument

Claims to support its conclusion only with some degree ofprobability

One of the two classes of argument

Valid Argument

If all the premises are true, the conclusion must be true

(applies only to deductive arguments)

Invalid Argument

The conclusion is not necessarily true, even if all the premises

are true(applies only to deductive arguments)

Classical Logic

Traditional techniques, based on Aristotles works, fanalysis of deductive arguments.

Modern Symbolic Logic

Methods used by most modern logicians to adeductive arguments.

Probability

The likelihood that some conclusion (of an indargument) is true.

1.5 Validity & Truth

Truth

An attribute of a proposition that asserts what really

case.

Sound

An argument that is valid and has only true premises.

Relations Between Truth and Validity:1. Some validarguments contain only truepropositions

premises and a true conclusion.2. Some valid arguments contain only false propositi

false premises and a false conclusion3. Some invalidarguments contain only true proposition

their premises are true, and their conclusions as well.

4. Some invalid arguments contain only true premisehave a false conclusion.

5. Some valid arguments have false premises and aconclusion.

6. Some invalid arguments also have a false premise

true conclusion.7. Some invalid arguments, of course, contain all

propositions false premises and a false conclusion.

Notes:The truth or falsity of an arguments conclusion does itself determine the validity or invalidity of the argumeThe fact that an argument is valid does not guarante

truth of its conclusion.

If an argument is valid and its premises are true, wbe certain that its conclusion is true also.

If an argument is valid and its conclusion is false, notits premises can be true.

Some perfectly valid arguments do have a false concbut such argument must have at least one false prem

CHAPTER 3LANGUAGE AND ITS APPLICATION

3.1 Three Basic Functions of Language

Ludwig Wittgenstein

One of the most influential philosophers of the 20thcen

Rightly insisted that there are countless different ki

uses of what we call symbols, words, sentences.

Informative Discourse

Language used to convey informationInformation includes false as well as true propos

bad arguments as well as good onesRecords of astronomical investigations, historical accreports of geographical trivia our learning about theand our reasoning about it uses language i

informativemode

Expressive Discourse

Language used to convey or evoke feelings.

Pertains not to facts, but to revealing and eliciting attiemotions and feelingsE.g. sorrow, passion, enthusiasm, lyric poetryExpressive discourse is used either to:


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1. manifestthe speakers feelings2. evokecertain feelings in the listeners

Expressive discourse is neither true nor false.

Directive Discourse

Language used to cause or prevent action.Directive discourse is neither true nor false.

Commands and requests do have other attributes reasonableness, propriety that are somewhat analogous totruth & falsity

3.2 Discourse Serving Multiple Functions

Notes:Effective communication often demands combinations offunctions.

Actions usually involve both what the actor wants and whatthe actor believes.Wants and beliefs are special kinds of what we have beencalling attitudes.Our success in causing others to act as we wish is likely to

depend upon our ability to evoke in them the appropriateattitudes, and perhaps also provide information that affectstheir relevant beliefs.

Ceremonial Use of Language

A mix of language functions (usually expressive anddirective) with special social uses.

E.g. greetings in social gatherings, rituals in houses ofworship, the portentous language of state d

ocuments

Performative Utterance

A special form of speech that simultaneously reports on, and

performs some function.Performative verbs perform their functions only when tied inspecial ways to the circumstances in which they are uttered,doing something more than combining the 3 major functions

of language

3.3 Language Forms and Language Functions

Sentences

The units of language that express complete thoughts4 categories: declarative, interrogative, imperative,

exclamatory4 functions: asserting, questioning, commanding, exclaiming

USES OF LANGUAGE

Principal Uses

1. Informative

2. Expressive3. Directive

Grammatical Forms

1. Declarative

2. Interrogative3. Imperative4. Exclamatory

Linguistic forms do not determine linguistic function. Formoften gives an indication of function but there is no sure connectionbetween the grammatical form and the use/uses intended. Languageserving any one of the 3 principal functions may take any one of the 4

grammatical forms

3.4 Emotive and Neutral Language

Emotive Language

Appropriate in poetry

Language that is emotionally toned will distractLanguage that is loaded heavily charged w/ emotionalmeaning on either side is unlikely to advance the quest fortruth

Neutral Language

The logician, seeking to evaluate arguments, will honor theuse of neutral language.

3.5 Agreement & Disagreement in Attitude & Belief

Dis/agreement in Belief vs. Dis/agreement in Attitude

Parties in Potential Conflict May:

1. agree about the facts, and agree in their attitude tothose facts

2. they might disagree about both3. they may agree about the facts but disagree in

attitude towards those facts4. they may disagree about what the facts are, and ye

agree in their attitude toward what they believe the fbe.

Note: The real nature of disagreements must be identified if th

to be successfully resolved.

CHAPTER 4DEFINITION

4.1 Disputes and Definitions

Three Kinds of Disputes

1. Obviously genuine disputesthere is no ambiguity present and the disputdisagree, either in attitude or belief

2. Merely verbal disputes

there is ambiguity present but there is no gedisagreement at all

3. Apparently verbal disputes that are really genuin

there is ambiguity present and the disdisagree, either in attitude or belief

Criterial Dispute

a form of genuine dispute that at first appears to be mverbal

4.2 Definitions and Their Uses

Definiendum

a symbol being defined

Definiens

the symbol (or group of symbols) that has the

meaning as the definiendum

Five Kinds of Definitions and their Principal Use

1. Stipulative Definitions

a. A proposal to arbitrarily assign meaning to a introduced symbol

b. a meaning is assigned to some symbol

c. not a reportd. cannot be true or falsee. it is a proposal, resolution, request or instr

to use the definiendum to mean what is methe definiens

f. used to eliminate ambiguity

2. Lexical Definitionsa. A report which may be true or false

meaning of a definiendum already has in language use

b. used to eliminate ambiguity

3. Precising Definitions

a. A report on existing language usage, additional stipulations provided to rvagueness

b. Go beyond ordinary usage in such a way

eliminate troublesome uncertainty regborderline cases

c. Its definiendum has an existing meaning, bumeaning is vague

d. What is added to achieve precision is a matstipulation

e. Used chiefly to reduce vagueness


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Ambiguity: Uncertainty because a word or phrase has moremeaning than one

Vagueness: lack of clarity regarding the borders of aterms meaning

4. Theoretical Definitions

a. An account of term that is helpful for generalunderstanding or in scientific practice

b. Seek to formulate a theoretically adequate orscientifically useful description of the objects to

which the term appliesc. Used to advance theoretical understanding

5. Persuasive Definitionsa. A definition intended to influence attitudes or stir

the emotions, using language expressively ratherthan informatively

b. used to influence conduct

4.3 Extensions, Intension, & the Structure of Definition

Extension (Denotation)

the collection of objects to which a general term is correctlyapplied

Intension (Connotation)

the attributes shared by all objects, and only those objects to

which a general term applies

4.4 Extension and Denotative Definitions

Extensional/Denotative Definitions

a definition based on the terms extensionthis type of definition is usually flawed b

ecause it is mostoften impossible to enumerate all the objects in a generalclass

1. Definitions by example

We list or give examples of the objects denoted bythe term

2. Ostensive definitions

a demonstrative definition

a term is defined by pointing at an objectWe point to or indicate by gesture the extension of

the term being defined

3. Quasi-ostensive Definitions

A denotative definition that uses a gesture and a

descriptive phraseThe gesture or pointing is accompanied by somedescriptive phase whose meaning is taken as beingknown

4.5 Intension and Intensional Definitions

Subjective Intension

What the speaker believes is the intension

The private interpretation of a term at a particular time

Objective IntensionThe total set of attributes shared by all the objects in thewords extension

Conventional Intension

The commonly accepted intension of a term

The public meaning that permits and facilitatescommunication

Intensional Definitions

1. Synonymous definitions

a. Defining a word with another word that has thesame meaning and is already understood

b. We provide another word, whose meanalready understood, that has the same meanthe word being defined

2. Operational definitions

a. Defining a term by limiting its use to situwhere certain actions or operations lea

specified resultsb. State that the term is correctly applied to a

case if and only if the performance of spoperations in the case yields a specified resu

3. Definitions by genus and difference

a. Defining a term by identifying the larger clasgenus) of which it is a member, anddistinguishing attributes (the difference)

characterize it specificallyb. We first name the genus of which the s

designation by the definiendum is a subclasthen name the attribute (or specific diffethat distinguishes the members of that s

from members of all other species in that gen

4.6 Rules for Definition by Genus and Difference

1. A definition should state the essential attributes ospecies

2. a definition must not be circular3. a definition must be neither too broad nor too narrow4. a definition must not be expressed in ambiguous, ob

or figurative language5. a definition should not be negative where it ca

affirmative

Circular Definition

a faulty definition that relies on knowledge of what isdefined

CHAPTER 5NOTIONS AND BELIEFS

5.1 What is a Fallacy?

Fallacy

A type of argument that may seem to be correccontains a mistake in reasoning.

When premises of an argument fail to suppoconclusion, we say that the reasoning is bad; the arguis said to be fallacious

In a general sense, any error in reasoning is a fallacyIn a narrower sense, each fallacy is a type of incargument

5.2 The Classification of Fallacies

Informal Fallacies

The type of mistakes in reasoning that arise formmishandling of the contentof the propositions const

the argument

THE MAJOR INFORMAL FALLACIES

Fallacies of

Relevance

The most numerous and

most frequentlyencountered, are those inwhich the premises aresimply not relevant to

the conclusion drawn.

R1: Appeal to

EmotionR2: Appeal to PityR3: Appeal to ForceR4: Argument Agai

the PersonR5: IrrelevantConclusion

Fallacies ofDefectiveInduction

Those in w/c the mistakearises from the fact thatthe premises of theargument, although

relevant to theconclusion, are so weak

D1: Argument fromIgnoranceD2: Appeal toInappropriate

AuthorityD3: False Cause


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& ineffectivethat reliance

upon them is a blunder.

D4: Hasty

Generalizations

Fallacies of

Presumption

Mistakes that arise

because too much hasbeen assumed in thepremises, the inferenceto the conclusion

depending on thatunwarranted assumption.

P1: AccidentP2: ComplexQuestionP3: Begging theQuestion

Fallacies ofAmbiguity

Arise from the equivocaluse of words or phrasesin the premises or in theconclusion of anargument, some critical

term having differentsenses in different partsof the argument.

A1: EquivocationA2: AmphibolyA3: AccentA4: CompositionA5: Division

5.3 Fallacies of Relevance

Fallacies of Relevance

Fallacies in which the premises are irrelevant to theconclusion.They might be better be called fallacies of irrelevance,

because they are the absence of any real connection betweenpremises and conclusion.

R1: Appeal to Emotion (ad populum, to the populace)A fallacy in which the argument relies on emotion rather than

on reason.

R2: Appeal to Pity (ad misericordiam,a pitying heart)A fallacy in which the argument relies on generosity,

altruism, or mercy, rather than on reason.

R3: Appeal to Force (ad baculum, to the stick)A fallacy in which the argument relies on the threat of force;threat may also be veiled

R4: Argument Against the Person (ad hominem)A fallacy in which the argument relies on an attack against

the person taking a positiono Abusive: An informal fallacy in which an attack is made

on the character of an opponent rather than on the

merits of the opponents positiono

Circumstantial: An informal fallacy in which an attack ismade on the special circumstances of an opponentrather than on the merits of the opponents position

Poisoning the Well

A type of ad hominem attack that cuts off rational discourse

R5: Irrelevant Conclusion (ignaratio elenchi, mistaken proof)A type of fallacy in which the premises support a different

conclusion than the one that is proposedo Straw Man Policy: A type of irrelevant conclusion in

which the opponents position is misrepresentedo Red Herring Fallacy: A type of irrelevant conclusion in

which the opponents position is misrepresented

Non Sequitor (Does not Follow)

Often applied to fallacies of relevance, since the conclusiondoes not follow from the premises

5.4 Fallacies of Defective Induction

Fallacies of Defective Induction

Fallacies in which the premises are too weak or ineffective towarrant the conclusion

D1: Argument from Ignorance(ad ignorantiam)A fallacy in which a proposition is held to be true just because

it has not been proved false, or false just because it has notbeen proved true.

D2: Appeal to Inappropriate Authority (ad verecundiam)A fallacy in which a conclusion is based on the judgma supposed authority who has no legitimate cla

expertise in the matter.

D3: False Cause (causa pro causa)A fallacy in which something that is not really a cau

treated as a cause.o Post Hoc Ergo Propter Hoc: After the

therefore because of the thing; a type of false fallacy in which an event is presumed to have

caused by another event that came before it.o Slippery Slope: A type of false cause fallacy in

change in a particular direction is assumed toinevitably to further, disastrous, change in the direction.

D4: Hasty Generalizations (Converse accident)A fallacy in which one moves carelessly from indcases to generalizationsAlso called the fallacy of converse accidentbecause it

reverse of another common mistake, known as the of accident.

5.5 Fallacies of Presumption

Fallacies of Presumption

Fallacies in which the conclusion depends on a

assumption that is dubious, unwarranted, or false.

P1: Accident

A fallacy in which a generalization is wrongly appliedparticular case.

P2: Complex Question

A fallacy in which a question is asked in a waypresupposes the truth of some proposition buried with

question.P3: Begging the Question (petitio principii,circular argumen

A fallacy in which the conclusion is stated or assumed one of the premises.

A petitio principii is always technically valid, but aworthless, as well

Every petitio is a circular argument, but the circle th

been constructed may if it is too large or fuzzyundetected

5.6 Fallacies of Ambiguity

Fallacies of Ambiguity (sophisms)

Fallacies caused by a shift or confusion of meaning an argument

A1: Equivocation

A fallacy in which 2 or more meanings of a word or pare used in different parts of an argument

A2: Amphiboly

A fallacy in which a loose or awkward combination of can be interpreted more than 1 way

The argument contains a premise based on 1 interprewhile the conclusion relies on a different interpretation

A3: Accent

A fallacy in which a phrase is used to convey 2 dif

meaning within an argument, and the difference is baschanges in emphasis given to words within the phrase

A4: Composition

A fallacy in which an inference is mistakenly drawn froattributes of the parts of a whole, to the attributes whole.The fallacy is reasoning from attributes of the indelements or members of a collection to attributes ocollection or totality of those elements.


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A5: Division

A fallacy in which a mistaken inference is drawn from theattributes of a whole to the attributes of the parts of the

whole.o 1

st Kind: consists in arguing fallaciously that what is

true of a whole must also be true of its parts.o 2

nd Kind: committed when one argues from the

attributes of a collection of elements to the attributes ofthe elements themselves.

CHAPTER 6CATEGORICAL PROPOSITIONS

6.1 The Theory of Deduction

Deductive Argument

An argument that claims to establish its conclusionconclusivelyOne of the 2 classes of argumentsEvery deductive argument is either valid or invalid

Valid Argument

A deductive argument which, if all the premises are true, theconclusion must be true.

Theory of Deduction

Aims to explain the relations of premises and conclusions in

valid arguments.Aims to provide techniques for discriminating between validand invalid deductions.

6.2 Classes and Categorical Propositions

Class: The collection of all objects that have some specifiedcharacteristic in common.

o Wholly included: All of one class may be included in all ofanother class.

o Partially included:Some, but not all, of the members of one

class may be included in another class.o Exclude:Two classes may have no members in common.

Categorical Proposition

A proposition used in deductive arguments, that asserts arelationship between one category and some other category.

6.3 The Four Kinds of Categorical Propositions

1. Universal affirmative proposition (A Propositions)Propositions that assert that the whole of one class is

included or contained in another class.

2. Universal negative proposition (E Propositions)

Propositions that assert that the whole of one class isexcluded from the whole of another class.

3. Particular affirmative proposition (I Propositions)

Propositions that assert that two classes have some memberor members in common.

4. Particular negative proposition (O Propositions) Propositionsthat assert that at least on member of a class is excluded from thewhole of another class.

Standard Form Categorical Propositions

Name and Type Proposition Form Example

AUniversal Affirmative All S is P. All politicians areliars.

EUniversal Negative No S is P. No politicians areliars.

IParticular Affirmative Some S is P. Some politiciansare liars.

OParticular Negative. Some S is not P. Some politiciansare not liars.

6.4 Quality, Quantity, and Distribution

Quality

An attribute of every categorical proposition, determinwhether the proposition affirms or denies some foclass inclusion.

o If the proposition affirms some class inc

whether complete or partial, its qualaffirmative. (A and I)

o If the proposition denies class inclusion, whcomplete or partial, its quality is negative. (

O)

Quantity

An attribute of every categorical proposition, determinwhether the proposition refers to all members (univer

only some members (particular) of the subject class.o If the proposition refers to all members

class designated by its subject term, its quanuniversal.(A and E)

o If the proposition refers to only some membthe lass designated by its subject termquantity is particular.(I and O)

General Skeleton of a Standard-Form Categorical Proposi

quantifier

subject termcopulapredicate term

Distribution

A characterization of whether terms of a categproposition refers to all members of the class designathat term.

o The A proposition distributes only its subject o

The E proposition distributes both its subjepredicate terms.

o The I proposition distributes neither its subjeits predicate term.

o The O proposition distributes only its pre

term.

Quantity, Quality and DistributionLetter Name Quantity Quality Distributio

A Universal Affirmative S only

E Universal Negative S and P

I Particular Affirmative Neither

O Particular Negative P only

6.5 The Traditional Square of Opposition

Opposition

Any logical relation among the kinds of categ

propositions (A, E, I, and O) exhibited on the SquaOpposition.

Contradictories

Two propositions that cannot both be true and cannobe false.A and O are contradictories: All S is P is co ntradict

Some S is not P.E and I are also contradictories: No S is P is contra

by Some S is P.

Contraries

Two propositions that cannot both be true

If one is true, the other must be false.They can both be false.

Contingent

Propositions that are neither necessarily true

necessarily false


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Subcontraries

Two propositions that cannot both be falseIf one is false, the other must be true.

They can both be true.

Subalteration

The oppositions between a universal (the superaltern) and its

corresponding particular proposition (the subaltern).In classical logic, the universal proposition implies the truth ofits corresponding particular proposition.

Square of OppositionA diagram showing the logical relationships among the fourtypes of categorical propositions (A, E, I and O).The traditional Square of Opposition differs from the modernSquare of Opposition in important ways.

Immediate Inference

An inference drawn directly from only one premise.

Mediate Inference

An inference drawn from more than one premise.The conclusion is drawn form the first premise through themediation of the second.

6.6 Further Immediate Inferences

Conversion

An inference formed by interchanging the subject andpredicate terms of a categorical proposition.Not all conversions are valid.

VALID CONVERSIONS

Convertend Converse

A: All S is P. I: Some P is S (by limitation)

E: No S is P. E: No P is S.

I: Some S is P. I: Some P is S

O: Some S is not P. (conversion not valid)

Complement of a Class

The collection of all things that do not belong to that class.

ObversionAn inference formed by changing the quality of a proposition

and replacing the predicate term by its complement.Obversion is valid for any standard-form categoricalproposition.

OBVERSIONS

Obvertend Obverse

A: All S is P. E: NO S is non-P

E: No S is P. A: All S is non-P.

I: Some S is P. O: Some S is not non-P.

O: Some S is not P. I: Some S is non-P.

Contraposition

An inference formed by replacing the subject term of aproposition with the complement of its predicate term, and

replacing the predicate term by the complement of its subjectterm.

Not all contrapositions are valid.

CONTRAPOSITION

Premise Contrapositive

A: All S is P. A: All non-P is non-S.

E: No S is P. O: Some non-P is not non-S. (by limitation)

I: Some S is P. (Contraposition not valid)

O: Some S is not P. O: Some non-P is not non-S.

6.7 Existential Import & the Interpretation of CategoricalPropositions

Boolean Interpretation

The modern interpretation of categorical propositiowhich universal propositions (A and E) are not assumrefer to classes that have members.

Existential Fallacy

A fallacy in which the argument relies on the illegitassumption that a class has members, when there

explicit assertion that it does.

Note:A proposition is said to have existential import if it typicuttered to assert the existence of objects of some kind.

6.8 Symbolism and Diagrams for Categorical Propositions

Form Proposition SymbolicRep,

Explanation

A All S is P_

SP = 0The class of things thaboth S and non-P is emp

E No S is P SP = OThe class off things thaboth S and P is empty.

I Some S is P SP 0The class of things thaboth S and P is not em(SP as at least one mem

O Some S isnot P

_SP O

The class of things thaboth S and non-P isempty. (SP has at leas

member).

Venn Diagrams

A method of representing classes and categpropositions using overlapping circles.

CHAPTER 7CATEGORICAL SYLLOGISM

7.1 Standard-Form Categorical Syllogism

Syllogism

Any deductive argument in which a conclusion is infrom two premises.

Categorical Syllogism

A deductive argument consisting of 3 categpropositions that together contain exactly 3 terms, ewhich occurs in exactly 2 of the constituent proposition

Standard-From Categorical Syllogism

A categorical syllogism in which the premisesconclusions are all standard-form categorical propo

(A, E, I or O)Arranged with the major premise first, the minor prsecond, and the conclusion last.

The Parts of a Standard-Form Categorical Syllogism

Major Term The predicate term of the conclusion.

Minor Term The subject term of the conclusion.

Middle Term The term that appears in both premises but

the conclusion.

Major Premise The premise containing the major term. In sta

form, the major premise is always stated 1st.

Minor Premise The premise containing the minor term.

Mood

One of the 64 3-letter characterizations of categ

syllogisms determined by the forms of the standardpropositions it contains.The mood of the syllogism is therefore representedletters, and those 3 letters are always given i

standard-form order.The 1

st letter names the type of that syllogisms

premise; the 2nd

letter names the type of that syllominor premise; the 3

rd letter names the type

conclusion.Every syllogism has a mood.


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Figure

The logical shape of a syllogism, determined by the positionof the middle term in its premises

Syllogisms can have fourand only fourpossible differentfigures:

The Four Figures

1stFigure 2ndFigure

3rdFigure 4thFigure

SchematicRepresen-

tation

M PS M

.. S P

P MS M

.. S P

M PM S

.. S P

P MM S

.. S P

Description

Themiddleterm may

be thesubjectterm ofthe majorpremise

and thepredicateterm ofthe minor

premise.

Themiddleterm may

be thepredicateterm ofbothpremises.

Themiddleterm may

be thesubjectterm ofbothpremises.

The middleterm maybe the

predicateterm ofthe majorpremiseand the

subjectterm ofthe minorpremise.

7.2 The Formal Nature of Syllogistic Argument

The validity of any syllogism depends entirely on its form.

Valid Syllogisms

- A valid syllogism is a formal valid argument,

valid by virtue ofits form alone.

- If a given syllogism is valid, any other syllogism of the sameform will also be valid.

- If a given syllogism is invalid, any other syllogism of thesame form will also be invalid.

7.3 Venn Diagram Technique for Testing Syllogism

7.4 Syllogistic Rules and Syllogistic Fallacies

Syllogistic Rules and Fallacies

Rule Associated Fallacy

1.Avoid four terms. Four TermsA formal mistake in which a

categorical syllogism contains more than3 terms.

2.Distribute the middleterm in at least onepremise.

Undistributed MiddleA formal mistake in which a

categorical syllogism contains a middleterm that is not distributed in either

premise.

3.Any term distributed

in the conclusion mustbe distributed in thepremises.

Illicit Major

A formal mistake in which the majorterm of a syllogism is undistributed inthe major premise, but is disturbed inthe conclusion.

Illicit MinorA formal mistake in which the minor

term of a syllogism is undistributed inthe minor premise but is distributed inthe conclusion.

4. Avoid 2 negativepremises.

Exclusive PremisesA formal mistake in which both

premises of a syllogism are negative.

5. If either premise isnegative, the conclusionmust be negative.

Drawing an Affirmative Conclusionfrom a Negative Premise

A formal mistake in which onepremise of a syllogism is negative, buthe conclusion is affirmative.

6. From 2 universalpremises no particularconclusion may be

drawn.

Existential FallacyAs a formal fallacy, the mistake of

inferring a particular conclusion from 2

universal premises.

Note: A violation of any one of these rules is a mistake, renders the syllogism invalid. Because it is a mistake of that skind, we call it a fallacy; and because it is a mistake in the fo

the argument, we call it a formal fallacy.

7.5 Exposition of the 15 Valid Forms of Categorical Syllog

The 15 Valid Forms of the Standard-Form Categorical Syllogism

1stFigure 1. AAA-1 Barbara

2. EAE-1 Celarent

3. AII-1 Darii

4. EIO1 Ferio

2nd

Figure 5. AEE-2 Camestres

6. EAE-2 Cesare

7. AOO-2 Baroko

8. EIO-2 Festino

3rdFigure 9. AII-3 Datisi

10. IAI-3 Disamis

11. EIO-3 Ferison

12. OAO-3 Bokardo

4thFigure 13. AEE-4 Camenes

14. IAI-4 Dimaris

15. EIO-4 Fresison

7.6 Deduction of the 15 Valid forms of Categorical Syllogi

CHAPTER 8SYLLOGISM IN ORDINARY LANGUAGE

8.1 Syllogistic Arguments

Syllogistic Argument

An Argument that is standard-form categorical syllogican be formulated as one without any change in mean

Reduction to Standard Form

Reformulation of a syllogistic argument into standard f

Standard-Form Translation

The resulting argument when we reformulate a loose

argument appearing in ordinary language into cla

syllogism

Different Ways in Which a Syllogistic Argument in Ord

Language may Deviate from a Standard-Form CategArgument:

First DeviationThe premises and conclusion of an argument in orlanguage may appear in an order that is not the orthe standard-form syllogismRemedy: Reordering the premises: the major premise

the minor premise second, the conclusion third.

Second DeviationA standard-form categorical syllogism always has exa

terms. The premises of an argument in ordinary lanmay appear to involve more than 3 terms buappearance might prove deceptive.Remedy: If the number of terms can be reduced to loss of meaning the reduction to standard form m

successful.

Third DeviationThe component propositions of the syllogistic argumordinary language may not all be standardpropositions.Remedy: If the components can be convertedstandard-form propositions w/o loss of meaning

reduction to standard form may be successful.


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8.2 Reducing the Number of Terms to Three

Eliminating Synonyms

A synonym of one of the terms in the syllogism is not really a4

th term, but only another way of referring to one of the 3

classes involved.

E.g. wealthy & rich

Eliminating Class Complements

Complement of a class is the collection of all things that do

not belong to that class (explained in 6.6)E.g. mammals & nonmammals

8.3 Translating Categorical Propositions into Standard Form

Note: Propositions of a syllogistic argument, when not in standardform, may be translated into standard form so as to allow thesyllogism to be tested either by Venn diagrams or by the use of rulesgoverning syllogisms.

I. Singular Proposition

A proposition that asserts that a specific individual belongs(or does not belong) to a particular classDo not affirm/deny the inclusion of one class in another, butwe can nevertheless interpret a singular proposition as aproposition dealing w/ classes and their interrelationsE.g. Socrates is a philosopher.

E.g. This table is not an antique.

Unit Classo A class with only one member

II. Propositions having adjectives as predicat

es, rather thansubstantive or class terms

E.g. Some flowers are beautiful.o Reformulated: Some flowers are beauties.

E.g. No warships are available for active dutyo Reformulated: No warships are things available for

active duty.

III. Propositions having main verbs other than the copula tobe

E.g. All people seek recognition.

o

Reformulated: All people are seekers or recognition.E.g. Some people drink Greek wine.

o Reformulated: Some people are Greek-winedrinkers.

IV. Statements having standard-form ingredients, but not in

standard form order

E.g. Racehorses are all thoroughbreds.o Reformulated: All racehorses are thoroughbreds.

E.g. all is well that ends well.o

Reformulated: All things that end well are things

that are well.

V. Propositions having quantifiers other than all, no, andsome

E.g. Every dog has its day.o Reformulated: All dogs are creatures that have their

days.E.g. Any contribution will be appreciated.

o Reformulated: All contributions are things that are

appreciated.

VI. Exclusive Propositions, using only or none but

A proposition asserting that the predicate applies only to thesubject namedE.g. Only citizens can vote.

o Reformulated: All those who can vote are citizens.E.g. None but the brave deserve the fair.

o Reformulated: All those who deserve the fair arethose who are brave.

VII. Propositions without words indicating quantity

E.g. Dog are carnivorous.o

Reformulated: All dogs are carnivores.E.g. Children are present.

o Reformulated: Some children are beings whpresent.

VIII. Propositions not resembling standard-form proposat all

E.g. Not all children believe in Santa Claus.o

Reformulated: Some children are not belieSanta Claus.

E.g. There are white elephants.o Reformulated: Some elephants are white thin

IX. Exceptive Propositions, using all except or siexpressions

A proposition making 2 assertions, that all membsome class except for members of one of its subclaare members of some other class

Translating exceptive propositions into standard fosomewhat complicated, because propositions of thismake 2 assertions rather than one

E.g. All except employees are eligible.E.g. All but employees are eligible.E.g. Employees alone are not eligible.

8.4 Uniform Translation

Parameter

An auxiliary symbol that aids in reformulating an ass

into standard form

Uniform Translation

Reducing propositions into standard-form syll

argument by using parameters or other techniques.

8.5 Enthymemes

Enthymeme

An argument containing an unstated propositionAn incompletely stated argument is characterized a

enthymematic

First-Order Enthymeme

An incompletely stated argument in which the propothat is taken for granted is the major premise

Second-Order Enthymeme

An incompletely stated argument in which the propothat is taken for granted is the minor premise

Third-Order Enthymeme

An incompletely stated argument in which the propothat is left unstated is the conclusion

8.6 Sorites

Sorites

An argument in which a conclusion is inferred fromnumber of premises through a chain of syllogistic infer

8.7 Disjunctive and Hypothetical Syllogism

Disjunctive Syllogism

A form of argument in which one premise is a disjuand the conclusion claims the truth of one of the disjunOnly some disjunctive syllogisms are valid

Hypothetical SyllogismA form of argument containing at least one condproposition as a premise.


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Pure Hypothetical Syllogism

A syllogism that contains conditional propositions exclusively

Mixed Hypothetical Syllogism

A syllogism having one conditional premise and onecategorical premise

Affirmative Mood/Modus Ponens (to affirm)

A valid hypothetical syllogism in which the categoricalpremise affirms the antecedent of the conditional premise,and the conclusion affirms its consequent

Fallacy of Affirming the Consequent

A formal fallacy in a hypothetical syllogism in which thecategorical premise affirms the consequent, rather than theantecedent, of the conditional premise

Modus Tollens (to deny)

A valid hypothetical syllogism in which the categoricalpremise denies the consequent of the conditional premise,and the conclusion denies its antecedent

Fallacy of Denying the Antecedent

A formal fallacy in a hypothetical syllogism in which thecategorical premise denies the antecedent, rather than theconsequent, of the conditional premise

8.8 The Dilemma

Dilemma

A common form of argument in ordinary discourse in which itis claimed that a choice must be made between 2 (usuallybad) alternatives

An argumentative device in which syllogisms on the sametopic are combined, sometimes w/ devastative effect

Simple Dilemma

The conclusion is a single categorical proposition

Complex Dilemma

The conclusion itself is a disjunction

Three Ways of Defeating a Dilemma

Going/escaping between the horns of the dilemmaRejecting its disjunctive premise

This method is often the easiest way to evade the conclusionof a dilemma, for unless one half of the disjunction is theexplicit contradictory of the other, the disjunction may verywell be false

Taking/grasping the dilemma by its hornsRejecting its conjunction premiseTo deny a conjunction, we need only deny one of its partsWhen we grasp the dilemma by the horns, we attempt to

show that at least one of the conditionals is false

Devising a counterdilemmaOne constructs another dilemma whose conclusion is opposed

to the conclusion of the originalAny counterdilemma may be used in rebuttal, but ideally it

should be built up out of the same ingredients (categoricalpropositions) that the original dilemma contained

CHAPTER 9SYMBOLIC LOGIC

9.1 Modern Logic and Its Symbolic Language

Symbols

Greatly facilitate our thinking about arguments

Enable us to get to the heart of an argument, exhibiting itsessential nature and putting aside what is not essential

With symbols, we can perform some logical operalmost mechanically, with the eye, which might othdemand great effort

A symbolic language helps us to accomplish intellectual tasks without having to think too much

Modern Logic

Logicians look now to the internal structure of proposand arguments, and to the logical links very fnumber that are critical in all deductive arguments

No encumbered by the need to transform ded

arguments in to syllogistic formIt may be less elegant than analytical syllogistics, more powerful

9.2 The Symbols for Conjunction, Negation, & Disjunction

Simple Statement

A statement that does not contain any other statemencomponent

Compound Statement

A statement that contains another statements component2 categories:

o W/N the truth value of the compound statemdetermined wholly by the truth value components, or determined by anything

than the truth value of its components

Conjunction ()

A truth functional connective meaning andSymbolized by the dot ()

We can form a conjunction of 2 statements by placinword and between themThe 2 statements combined are called conjunctsThe truth value of the conjunction of 2 stateme

determined wholly and entirely by the truth values oconjunctsIf both conjuncts are true, the conjunction is otherwise it is false

A conjunction is said to be a truth-functional compstatement, and its conjuncts are said to be truth-funccomponents of it

Note:Not every compound statement is truth-function

Truth Value

The status of any statement as true or falseThe truth value of a true statement is trueThe truth value of a false statement is false

Truth-Functional Component

Any component of a compound statement replacement by another statement having the samevalue would not change the truth value of the com

statement

Truth-Functional Compound Statement

A compound statement whose truth function is

determined by the truth values of its components

Truth-Functional ConnectiveAny logical connective (including conjunction, disjunmaterial implication, and material equivalence) betwee

components of a truth-functional compound statement

Simple Statement

Any statement that is not truth functionally compound

p q p

q

T T T

T F F

F T F

F F F


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Negation/Denial/Contradictory (~)

symbolized by the tilde or curl (~)often formed by the insertion of not in the original

statement

Disjunction/Alteration (v)

A truth-functional connective meaning or

It has a weak (inclusive) sense, symbolized by the wedge(v) (or vee), and a strong (exclusive) sense.2 components combined are called disjunctsor alternatives

p q p v q

T T T

T F T

F T T

F F F

Punctuation

The parentheses brackets, and braces used in symboliclanguage to eliminate ambiguity in meaning

In any formula the negation symbol will be understood toapply to the smallest statement that the punctuation permits

9.3 Conditional Statements and Material Implication

Conditional Statement

A compound statement of the form If p then q.

Also called a hypothetical/implication/implicative statementAsserts that in any case in which its antecedent is true, itsconsequent is also trueIt does no assert that its antecedent is true, but only if itsantecedent is true, its consequent is also trueThe essential meaning of a conditional statement is the

relationship asserted to hold between its antecedent andconsequent

Antecedent (implicans/protasis)

In a conditional statement, that component that immediatelyfollows the if

Consequent (implicate/apodosis)

In a conditional statement, the component that immediatelyfollows the then

ImplicationThe relation that holds between the antecedent and the

consequent of a conditional statement.There are different kinds of implication

Horseshoe (

)

A symbol used to represent material implication, which iscommon, partial meaning of all if-then statements

p q ~q p~q ~ (p~q) p q

T T F F T T

T F T T F F

F T F F T T

F F T F T T

Material Implication

A truth-functional relation symbolized by the horseshoe ( )that may connect 2 statementsThe statement p materially implies q is true when either pis false, or q is true

p q p q

T T T

T F F

F T T

F F T

In general, q is a necessary condition for pand p only

if qare symbolized as p q

In general, p is a sufficient condition for symbolized by p q

9.4 Argument Forms and Refutation by Logical Analogy

Refutation by Logical Analogy

Exhibiting the fault of an argument by presenting a

argument with the same form whose premises are knoe true and whose conclusion is known to be false.

To prove the invalidity of an argument, it suffices to form

another argument that:Has exactly the same form as the firstHas true premises and a false conclusion

Note:This method is based upon the fact that validity and inv

are purely formal characteristics of arguments, which is to saany 2 arguments having exactly the same form are either bothor invalid, regardless of any differences in the subject matter they are concerned.

Statement Variable

A letter (lower case) for which a statement masubstituted.

Argument Form

An array of symbols exhibiting the logical structure

argument, it contains statement variables, bustatements

Substitution Instance of an Argument Form

Any argument that results from the consistent substof statements for statement variables in an argument

Specific Form of an Argument

The argument form from which the given argument rwhen a different simple statement is substituted fordifferent statement variable.

9.5 The Precise Meaning of Invalid and Valid

Invalid Argument Form

An argument form that has at least one substinstance with true premises and a false conclusion

Valid Argument Form

An argument form that has no substitution instancetrue premises and a false conclusion

9.6 Testing Argument Validity on Truth Tables

Truth Table

An array on which the validity of an argument form mtested, through the display of all possible combinatiothe truth values of the statement variables contained

form

9.7 Some Common Argument Forms

Disjunctive Syllogism

A valid argument form in which one premise

disjunction, another premise is the denial of one of thdisjuncts, and the conclusion is the truth of the disjunct

p v q~ p

q

p q p v q ~p

T T T F

T F T F

F T T T

F F F T


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Modus Ponens

A valid argument that relies upon a conditional premise, andin which another premise affirms the antecedent of that

conditional, and the conclusion affirms its consequent

p qp

q

p q p q

T T T

T F F

F T T

F F T

Modus Tollens

A valid argument that relies upon a conditional premise, andin which another premise denies the consequent of thatconditional, and the conclusion denies its antecedent

p q

~q~p

p q p q ~q ~p

T T T F F

T F F T F

F T T F TF F T T T

Hypothetical Syllogism

A valid argument containing only conditional propositions

p q

q r

p r

p Q r p q q r p r

T T T T T T

T T F T F F

T F T F T T

T F F F T F

F T T T T T

F T F T F TF F T T T T

F F F T T T

Fallacy of Affirming the Consequent

A formal fallacy in which the 2nd

premise of an argumentaffirms the consequent of a conditional premise and theconclusion of its argument affirms its antecedent

p q

qp

Fallacy of Denying the Antecedent

A formal fallacy in which the 2nd

premise of an argumentdenies the antecedent of a conditional premise and theconclusion of the argument denies its consequent

p q~p~q

Note: In determining whether any given argument is valid, we must

look into the specific form of the argument in question

9.8 Statement Forms & Material Equivalence

Statement Form

An array of symbols exhibiting the logical structure of astatementIt contains statement variables but no statements

Substitution Instance of Statement Form

Any statement that results from the consistent substof statements for statement variables in a statement f

Specific Form of a Statement

The statement form from which the given statement rwhen a different simple statement is subst

consistently for each different statement variable

Tautologous Statement Form

A statement form that has only true substitution insta

A tautology:

p ~p p v ~p

T F T

F T T

Self-Contradictory Statement Form

A statement form that has only false substitution instaA contradiction

Contingent Form

A statement form that has both true and false substinstances

Peirces Law

A tautological statement of the form [(p q) p] p

Materially Equivalent ( )

A truth-functional relation asserting that 2 stateconnected by the three-bar sign ( ) have the same

value

p q p q

T T T

T F F

F T F

F F T

Biconditional Statement

A compound statement that asserts that its 2 compstatements imply one another and therefore are mat

equivalent

The Four Truth-Functional Connective

Truth-

FunctionalConnective

Symbol

(Name ofSymbol)

Proposition

Type

Names of

ComponentsPropositions

that Type

And (dot) Conjunction Conjuncts

Or V (wedge) Disjunction Disjuncts

Ifthen (horseshoe) Conditional Antecedent,

consequent

If and only if (tribar) Biconditional Components

Note:Not is nota connective, but is a truth-function operatois omitted here

Note: To say that an argument form is valid if, and only expression in the form of a conditional statement is a tautology.

9.9 Logic Equivalence

Logically Equivalent

Two statements for which the statement of their mequivalence is tautology

they are equivalent in meaning and may replacanother

Double Negation

An expression of logical equivalence between a symbthe negation of the negation of that symbol


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p ~p ~~pT

p ~~p

T F T T

F T F T

Note: This table proves that p and ~~p are logically equivalent.

Material equivalence:a truth-functional connective, , which may be

true or false depending only upon the truth or falsity of the elements itconnects

Logical Equivalence: not a mere connective, and it expresses a

relation between 2 statements that is not truth-functionalNote:2 statements are logically equivalent only when it is absolutelyimpossible for them to have different truth values.

p q p v q ~(p v q) ~p ~q ~p~q ~(p v q) (~p~q)

T T T F F F F T

T F T F F T F T

F T T F T F F T

F F F T T T T T

De Morgans Theorems

Two useful logical equivalenceso (1) The negation of the disjunction of 2 statements

is logically equivalent to the conjunction of thenegations of the 2 disjuncts

o

(2) the negation of the conjunction of 2 statementsis logically equivalent to the disjunction of thenegations of the 2 conjuncts

9.10 The Three Laws of Thought

Principle of Identity

If any statement is true, it is true.Every statement of the formp

pmust be true

o Every such statement is a tautology

Principle of Noncontradiction

No statement can be both true and false

Every statement of the form p~p must be falseo Every such statement isself-contradictory

Principle of Excluded Middle

Every statement is either true or falseEvery statement of the form p v ~ p must be trueEvery such statement is a tautology

CHAPTER 10METHODS OF DEDUCTION

10.1 Formal Proof of Validity

Rules of Inference

The rules that permit valid inferences from statementsassumed as premises

Natural Deduction

A method of providing the validity of a deductive argumentby using the rules of inference

Using natural deduction we can proved a formal proofof thevalidity of an argument that is valid

Formal Proof of Validity

A sequence of statements, each of which is either a premiseof a given argument or is deduced, suing the rules of

inference, from preceding statements in that sequence, suchthat the last statement in the sequence is the conclusion ofthe argument whose validity is being proved

Elementary Valid Argument

Any one of a set of specified deductive arguments that servesas a rule of inference & can be used to construct a formalproof of validity

9 RULES OF INFERENCE:ELEMENTARY VALID ARGUMENT FORMS

NAME ABBREV. FORM

1. Modus Ponens M.P. p

qpq

2. Modus Tollens M.T. p q~q~p

3. Hypothetical Syllogism H.S. p q

q rp r

4. Disjunctive Syllogism D.S p v q~ p

q

5. Constructive Dilemma C.D. (p q) (r sp v rq v s

6. Absorption Abs. p q

p (p q)

7. Simplification Simp. p qp

8. Conjunction Conj. pqp q

9. Addition Add. pp v q

10.2 The Rule of Replacement

Rule of Replacement

The rule that logically equivalent expressions may reeach otherNote: this is very different from that of substitution

RULES OF REPLACEMENT:LOGICALLY EQUIVALENT EXPRESSIONS

NAME ABBREV. FORM

10. De Morgans

Theorem

De M.~(p q) (~ p v ~q)

~(p v q) (~ p ~q)

11. Commutation Com.(p v q) (q v p)

(p q) (q p)

12. Association Assoc.[p v (q v r)] [(p v q) v

[p (q r)] [(p q)

13. Distribution Dist.[p (q v r)] [(p q) (p

[p v (q r)] [(p v q) (p

14. Double

Negation

D.N.p ~~ p

15. Transpor-

tation

Trans.(p q) (~q ~p)

16. MaterialImplication Imp. (p q) (~p v q)

17. Material

Equivalence

Equiv.(p q) [(p q) (q p

(p q) [(p q) v (~p ~

18. Exportation Exp.[(p q) r] [p (q r

19. Tautology Taut.p (p v p)

p (p p)


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The 19 Rules of Inference

The list of 19 rules of inference constitutes a complete systemof truth-functional logic, in the sense that it permits the

construction of a formal proof of validity for anyvalid truth-functional argument

The first 9 rules can be applied only to whole lines of a proof

Any of the last 10 rules can be applied either to whole lines or

to parts of lines

The notion of formalproof is an effectivenotion

It can be decided quite mechanically, in a finite number of

steps, whether or not a given sequence of statementsconstitutes a formal proofNo thinking is requiredOnly 2 things are required:

o The ability to see that a statement occurring in one

place is precisely the same as a statement occurringin another

o The ability to see W/N a given statement has acertain pattern; that is , to see if it is a substitutioninstance of a given statement form

Formal Proof vs. Truth Tables

The making of a truth table is completely mechanicalThere are no mechanical rules for the construction of formalproofsProving an argument valid y constructing a formal proof of itsvalidity is much easier than the purely mechanical

construction of a truth table with perhaps hundreds orthousands of rows

10.3 Proof of Invalidity

Invalid Arguments

For an invalid argument, there is no formal

proof of invalidityAn argument is provided invalid by displaying at least onerow of its truth table in which all its premises are true but its

conclusion is falseWe need not examine allrows of its truth table to discover anarguments invalidity: the discovery of a single row in whichits premises are all true and its conclusion is false will suffice

10.4 Inconsistency

Note:If truth values cannotbe assigned to make the premises true

and the conclusion false, then the argument must be validAny argument whose premises are inconsistent must be validAny argument with inconsistent premises is valid, regardlessof what its conclusion may be

Inconsistency

Inconsistent statements cannot both be trueFalsus in unum, falsus in omnibus (Untrustworthy in onething, untrustworthy in all)

Inconsistent statements are not meaningless; their troubleis just the opposite. They mean too much. They meaneverything, in the sense of implying everything. And ifeverythingis asserted, half of what is asserted is surely false,

because every statement has a denial

10.5 Indirect Proof of Validity

Indirect Proof of Validity

An indirect proof of validity is written out by stating as anadditional assumed premise the negation of the conclusionA version of reductio ad absurdum(reducing the absurd)

with which an argument can be proved valid by exhibiting thecontradiction which may be derived from its premisesaugmented by the assumption of the denial of its conclusionAn exclamation point (!)is used to indicate that a given stepis derived after the assumption advancing the indirect proofhad been madeThis method of indirect proof strengthens our machinery fortesting arguments by making it possible, in some

circumstances, to prove validity more quickly than wopossible without it

10.6 Shorter Truth-Table Technique

Shorter Truth-Table Technique

An argument may be tested by assigning truth

showing that, if it is valid, assigning values that wouldthe conclusion false while the premises are true woulinescapably to inconsistencyProving the validity of an argument with this shorter

table technique is one version of the use of reducabsurdum but instead of suing the rules of infereuses truth value assignmentsIts easiest application is when Fis assigned to a disju(in which case both of the disjuncts must be assigned

to a conjunction (in which case both of the conjunctsbe assigned)

o When assignments to simple statements arforced, the absurdity (if there is one) is qexposed

Note: The reductio ad absurdummethod of proof is often theefficient in testing the validity of a deductive argument

CHAPTER 11

QUANTIFICATION THEORY

11.1 The Need for Quantification

Quantification

A method of symbolizing devised to exhibit the inner structure of propositions.

11.2 Singular Propositions

Affirmative Singular Proposition

A proposition that asserts that a particular individusome specified attribute

Individual Constant

A symbol used in logical notation to denote an individu

Individual Variable

A symbol used as a place holder for an individual cons

Propositional Function

An expression that contains an individual variablebecomes a statement when an individual consta

substituted for the individual variable

Simple Predicate

A propositional function having some true and somesubstitution instances, each of which is an affirm

singular proposition

11.3 Universal and Existential Quantifiers

Universal Quantifier

A symbol (x) used before a propositional function to that the predicate following is true of everything

Generalization

The process of forming a proposition from a proposfunction by placing a universal quantifier or an exisquantifier before it

Existential Quantifier

A symbol (

x) indicating that the propositional futhat follows has at least one true substitution instance

Instantiation

The process of forming a proposition from a proposfunction by substituting an individual constant findividual variable


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11.4 Traditional Subject-Predicate Propositions

Normal-Form Formula

A formula in which negation signs apply only to simplepredicates

11.5 Proving Validity

Universal Instantiation (UI)

A rule of inference that permits the valid inference of any

substitution instance of a propositional function from itsuniversal quantification

Universal Generalization (UG)

A rule of inference that permits the valid inference of a

universally quantified expression from an expression that isgiven as true of any arbitrarily selected individual

Existential Instantiation (EI)

A rule of inference that permits (with restrictions) the valid

inference of the truth of a substitution instance (for anyindividual constant that appears nowhere earlier in thecontext) from the existential quantification of a propositionalfunction

Existential Generalization (EG)

A rule of inference that permits the valid inference of the

existential quantification of a propositional function from anytrue substitution instance of that function

Rules of Inference: Quantificatio

n

UniversalInstantiation

UI (x) ( x)

v(where v is anyindividual symbol)

Any substitution instance

of a propositionalfunction can be validlyinferred from itsuniversal quantification

UniversalGeneralization

UG y

(x) ( x)(where y denotes

any arbitrarilyselected individual)

From the substitutioninstance of a

propositional functionwith respect to the nameof any arbitrarily selectedindividual, one may

validly infer the universal

quantification of thatpropositional function

ExistentialInstantiation

EI ( x)( x)

v(where v is anyindividual

constant, otherthan y, having nopreviousoccurrence in thecontext)

From the existentialquantification of apropositional function,we may infer the truth of

its substitution instancewith respect to anyindividual constant (otherthan y) that occurs

nowhere earlier in thecontext.

Existential

Generalization

EG v

( x)( x)(where v is anyindividualconstant)

From any truesubstitution instance of apropositional function,we may validly infer the

existential quantificationof that propositionalfunction.

11.6 Proving Invalidity

11.7 Asyllogistic Inference

Asyllogistic Arguments

Arguments containing one or more propositions morelogically complicated than the standard A, E, I or O

propositions

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