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Transcript of Legal Technique and Logic Reviewer SIENNA FLORES
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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