244970380 legal technique and logic reviewer sienna flores.pdf (1)

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SIENNA A. FLORES LEGAL TECHNIQUE & LOGIC

CHAPTER 1 PROPOSITIONS

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 case All propositions are either true or false May be affirmed or denied

Statement

The meaning of a declarative sentence at a particular time In 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) Proposition

A type of compound proposition If true, at least one of the component propositions must be

true Hypothetical (or Conditional) Proposition

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

consequent 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 other proposition.

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 of probability

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 Aristotle‟s works, for the

analysis of deductive arguments. Modern Symbolic Logic

Methods used by most modern logicians to analyze deductive arguments.

Probability

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

1.5 Validity & Truth Truth

An attribute of a proposition that asserts what really is the case.

Sound

An argument that is valid and has only true premises. Relations Between Truth and Validity:

1. Some valid arguments contain only true propositions – true premises and a true conclusion.

2. Some valid arguments contain only false propositions – false premises and a false conclusion

3. Some invalid arguments contain only true propositions – all their premises are true, and their conclusions as well.

4. Some invalid arguments contain only true premises and have a false conclusion.

5. Some valid arguments have false premises and a true conclusion.

6. Some invalid arguments also have a false premise and a true conclusion.

7. Some invalid arguments, of course, contain all false propositions – false premises and a false conclusion.

Notes:

The truth or falsity of an argument‟s conclusion does not by itself determine the validity or invalidity of the argument.

The fact that an argument is valid does not guarantee the truth of its conclusion.

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

If an argument is valid and its conclusion is false, not all of its premises can be true.

Some perfectly valid arguments do have a false conclusion – but such argument must have at least one false premise.

CHAPTER 3 LANGUAGE AND ITS APPLICATION

3.1 Three Basic Functions of Language Ludwig Wittgenstein

One of the most influential philosophers of the 20th century Rightly insisted that there are countless different kinds of

uses of what we call „symbols,‟ „words,‟ „sentences.‟ Informative Discourse

Language used to convey information “Information” includes false as well as true propositions,

bad arguments as well as good ones Records of astronomical investigations, historical accounts,

reports of geographical trivia – our learning about the world and our reasoning about – it uses language in the informative mode

Expressive Discourse

Language used to convey or evoke feelings. Pertains not to facts, but to revealing and eliciting attitudes,

emotions and feelings E.g. sorrow, passion, enthusiasm, lyric poetry Expressive discourse is used either to:

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1. manifest the speaker‟s feelings 2. evoke certain 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 to truth & falsity

3.2 Discourse Serving Multiple Functions Notes:

Effective communication often demands combinations of functions.

Actions usually involve both what the actor wants and what the actor believes.

Wants and beliefs are special kinds of what we have been calling “attitudes.”

Our success in causing others to act as we wish is likely to depend upon our ability to evoke in them the appropriate attitudes, and perhaps also provide information that affects their relevant beliefs.

Ceremonial Use of Language

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

E.g. greetings in social gatherings, rituals in houses of worship, the portentous language of state documents

Performative Utterance

A special form of speech that simultaneously reports on, and performs some function.

Performative verbs perform their functions only when tied in special 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 thoughts 4 categories: declarative, interrogative, imperative,

exclamatory 4 functions: asserting, questioning, commanding, exclaiming

USES OF LANGUAGE

Principal Uses 1. Informative 2. Expressive 3. Directive

Grammatical Forms 1. Declarative 2. Interrogative 3. Imperative 4. Exclamatory

Linguistic forms do not determine linguistic function. Form often gives an indication of function – but there is no sure connection between the grammatical form and the use/uses intended. Language serving 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 distract Language that is “loaded” – heavily charged w/ emotional

meaning on either side – is unlikely to advance the quest for truth

Neutral Language

The logician, seeking to evaluate arguments, will honor the use 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 towards

those facts 2. they might disagree about both 3. they may agree about the facts but disagree in their

attitude towards those facts 4. they may disagree about what the facts are, and yet they

agree in their attitude toward what they believe the fats to be.

Note: The real nature of disagreements must be identified if they are to be successfully resolved.

CHAPTER 4 DEFINITION

4.1 Disputes and Definitions Three Kinds of Disputes

1. Obviously genuine disputes there is no ambiguity present and the disputers do

disagree, either in attitude or belief 2. Merely verbal disputes

there is ambiguity present but there is no genuine disagreement at all

3. Apparently verbal disputes that are really genuine there is ambiguity present and the disputers

disagree, either in attitude or belief Criterial Dispute

a form of genuine dispute that at first appears to be merely verbal

4.2 Definitions and Their Uses Definiendum

a symbol being defined Definiens

the symbol (or group of symbols) that has the same meaning as the definiendum

Five Kinds of Definitions and their Principal Use

1. Stipulative Definitions a. A proposal to arbitrarily assign meaning to a newly

introduced symbol b. a meaning is assigned to some symbol c. not a report d. cannot be true or false e. it is a proposal, resolution, request or instruction

to use the definiendum to mean what is meant by the definiens

f. used to eliminate ambiguity

2. Lexical Definitions a. A report – which may be true or false – of the

meaning of a definiendum already has in actual language use

b. used to eliminate ambiguity

3. Precising Definitions a. A report on existing language usage, with

additional stipulations provided to reduce vagueness

b. Go beyond ordinary usage in such a way as to eliminate troublesome uncertainty regarding borderline cases

c. Its definiendum has an existing meaning, but that meaning is vague

d. What is added to achieve precision is a matter of stipulation

e. Used chiefly to reduce vagueness

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Ambiguity: Uncertainty because a word or phrase has more meaning than one

Vagueness: lack of clarity regarding the “borders” of a term‟s meaning

4. Theoretical Definitions

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

b. Seek to formulate a theoretically adequate or scientifically useful description of the objects to which the term applies

c. Used to advance theoretical understanding

5. Persuasive Definitions a. A definition intended to influence attitudes or stir

the emotions, using language expressively rather than 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 correctly applied

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 term‟s extension this type of definition is usually flawed because it is most

often impossible to enumerate all the objects in a general class

1. Definitions by example

We list or give examples of the objects denoted by the term

2. Ostensive definitions

a demonstrative definition a term is defined by pointing at an object We 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 phrase The gesture or pointing is accompanied by some

descriptive phase whose meaning is taken as being known

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 Intension

The total set of attributes shared by all the objects in the word‟s extension

Conventional Intension

The commonly accepted intension of a term The public meaning that permits and facilitates

communication Intensional Definitions

1. Synonymous definitions a. Defining a word with another word that has the

same meaning and is already understood

b. We provide another word, whose meaning is already understood, that has the same meaning as the word being defined

2. Operational definitions

a. Defining a term by limiting its use to situations where certain actions or operations lead to specified results

b. State that the term is correctly applied to a given case if and only if the performance of specified operations in the case yields a specified result

3. Definitions by genus and difference

a. Defining a term by identifying the larger class (the genus) of which it is a member, and the distinguishing attributes (the difference) that characterize it specifically

b. We first name the genus of which the species designation by the definiendum is a subclass, and then name the attribute (or specific difference) that distinguishes the members of that species from members of all other species in that genus

4.6 Rules for Definition by Genus and Difference

1. A definition should state the essential attributes of the species

2. a definition must not be circular 3. a definition must be neither too broad nor too narrow 4. a definition must not be expressed in ambiguous, obscure,

or figurative language 5. a definition should not be negative where it can be

affirmative Circular Definition

a faulty definition that relies on knowledge of what is being defined

CHAPTER 5

NOTIONS AND BELIEFS 5.1 What is a Fallacy? Fallacy

A type of argument that may seem to be correct, but contains a mistake in reasoning.

When premises of an argument fail to support its conclusion, we say that the reasoning is bad; the argument is said to be fallacious

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

argument 5.2 The Classification of Fallacies Informal Fallacies

The type of mistakes in reasoning that arise form the mishandling of the content of the propositions constituting the argument

THE MAJOR INFORMAL FALLACIES

Fallacies of Relevance

The most numerous and most frequently encountered, are those in which the premises are simply not relevant to the conclusion drawn.

R1: Appeal to Emotion R2: Appeal to Pity R3: Appeal to Force R4: Argument Against the Person R5: Irrelevant Conclusion

Fallacies of Defective Induction

Those in w/c the mistake arises from the fact that the premises of the argument, although relevant to the conclusion, are so weak

D1: Argument from Ignorance D2: Appeal to Inappropriate Authority D3: False Cause

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& ineffective that reliance upon them is a blunder.

D4: Hasty Generalizations

Fallacies of Presumption

Mistakes that arise because too much has been assumed in the premises, the inference to the conclusion depending on that unwarranted assumption.

P1: Accident P2: Complex Question P3: Begging the Question

Fallacies of Ambiguity

Arise from the equivocal use of words or phrases in the premises or in the conclusion of an argument, some critical term having different senses in different parts of the argument.

A1: Equivocation A2: Amphiboly A3: Accent A4: Composition A5: Division

5.3 Fallacies of Relevance Fallacies of Relevance

Fallacies in which the premises are irrelevant to the conclusion.

They might be better be called fallacies of irrelevance, because they are the absence of any real connection between premises 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 position o Abusive: An informal fallacy in which an attack is made

on the character of an opponent rather than on the merits of the opponents position

o Circumstantial: An informal fallacy in which an attack is made on the special circumstances of an opponent rather than on the merits of the opponent‟s 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 proposed o Straw Man Policy: A type of irrelevant conclusion in

which the opponent‟s position is misrepresented o Red Herring Fallacy: A type of irrelevant conclusion in

which the opponent‟s position is misrepresented Non Sequitor (“Does not Follow”)

Often applied to fallacies of relevance, since the conclusion does 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 to warrant 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 not been proved true.

D2: Appeal to Inappropriate Authority (ad verecundiam) A fallacy in which a conclusion is based on the judgment of

a supposed authority who has no legitimate claim to expertise in the matter.

D3: False Cause (causa pro causa)

A fallacy in which something that is not really a cause, is treated as a cause. o Post Hoc Ergo Propter Hoc: “After the thing,

therefore because of the thing”; a type of false cause fallacy in which an event is presumed to have been caused by another event that came before it.

o Slippery Slope: A type of false cause fallacy in which change in a particular direction is assumed to lead inevitably to further, disastrous, change in the same direction.

D4: Hasty Generalizations (Converse accident)

A fallacy in which one moves carelessly from individual cases to generalizations

Also called the fallacy of converse accident because it is the reverse of another common mistake, known as the fallacy of accident.

5.5 Fallacies of Presumption Fallacies of Presumption

Fallacies in which the conclusion depends on a tacit assumption that is dubious, unwarranted, or false.

P1: Accident

A fallacy in which a generalization is wrongly applied in a particular case.

P2: Complex Question

A fallacy in which a question is asked in a way that presupposes the truth of some proposition buried within the question.

P3: Begging the Question (petitio principii, “circular argument”) A fallacy in which the conclusion is stated or assumed within

one of the premises. A petitio principii is always technically valid, but always

worthless, as well Every petitio is a circular argument, but the circle that has

been constructed may – if it is too large or fuzzy – go undetected

5.6 Fallacies of Ambiguity Fallacies of Ambiguity (sophisms)

Fallacies caused by a shift or confusion of meaning within an argument

A1: Equivocation A fallacy in which 2 or more meanings of a word or phrase

are used in different parts of an argument A2: Amphiboly

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

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

A3: Accent

A fallacy in which a phrase is used to convey 2 different meaning within an argument, and the difference is based on changes in emphasis given to words within the phrase

A4: Composition

A fallacy in which an inference is mistakenly drawn from the attributes of the parts of a whole, to the attributes of the whole.

The fallacy is reasoning from attributes of the individual elements or members of a collection to attributes of the collection or totality of those elements.

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A5: Division A fallacy in which a mistaken inference is drawn from the

attributes of a whole to the attributes of the parts of the whole. o 1st Kind: consists in arguing fallaciously that what is

true of a whole must also be true of its parts. o 2nd Kind: committed when one argues from the

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

CHAPTER 6

CATEGORICAL PROPOSITIONS 6.1 The Theory of Deduction Deductive Argument

An argument that claims to establish its conclusion conclusively

One of the 2 classes of arguments Every deductive argument is either valid or invalid

Valid Argument

A deductive argument which, if all the premises are true, the conclusion 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 valid and invalid deductions.

6.2 Classes and Categorical Propositions Class: The collection of all objects that have some specified characteristic in common.

o Wholly included: All of one class may be included in all of another 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 a relationship 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 is excluded from the whole of another class.

3. Particular affirmative proposition (I Propositions)

Propositions that assert that two classes have some member or members in common.

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

Standard Form Categorical Propositions Name and Type Proposition Form Example

A – Universal Affirmative All S is P. All politicians are liars.

E – Universal Negative No S is P. No politicians are liars.

I – Particular Affirmative Some S is P. Some politicians are liars.

O – Particular Negative. Some S is not P. Some politicians are not liars.

6.4 Quality, Quantity, and Distribution Quality

An attribute of every categorical proposition, determined by whether the proposition affirms or denies some form of class inclusion.

o If the proposition affirms some class inclusion, whether complete or partial, its quality is affirmative. (A and I)

o If the proposition denies class inclusion, whether complete or partial, its quality is negative. (E and O)

Quantity

An attribute of every categorical proposition, determined by whether the proposition refers to all members (universal) or only some members (particular) of the subject class.

o If the proposition refers to all members of the class designated by its subject term, its quantity is universal. (A and E)

o If the proposition refers to only some members of the lass designated by its subject term, its quantity is particular. (I and O)

General Skeleton of a Standard-Form Categorical Proposition

quantifier subject term copula predicate term

Distribution

A characterization of whether terms of a categorical proposition refers to all members of the class designated by that term.

o The A proposition distributes only its subject term o The E proposition distributes both its subject and

predicate terms. o The I proposition distributes neither its subject nor

its predicate term. o The O proposition distributes only its predicate

term.

Quantity, Quality and Distribution Letter Name Quantity Quality Distribution

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 categorical propositions (A, E, I, and O) exhibited on the Square of Opposition.

Contradictories

Two propositions that cannot both be true and cannot both be false.

A and O are contradictories: “All S is P” is contradicted by “Some S is not P.”

E and I are also contradictories: “No S is P” is contradicted 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 nor necessarily false

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Subcontraries Two propositions that cannot both be false If 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 of its corresponding particular proposition.

Square of Opposition

A diagram showing the logical relationships among the four types of categorical propositions (A, E, I and O).

The traditional Square of Opposition differs from the modern Square 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 the

mediation of the second. 6.6 Further Immediate Inferences Conversion

An inference formed by interchanging the subject and predicate 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. Obversion

An inference formed by changing the quality of a proposition and replacing the predicate term by its complement.

Obversion is valid for any standard-form categorical proposition.

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 a proposition with the complement of its predicate term, and replacing the predicate term by the complement of its subject term.

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 Categorical Propositions Boolean Interpretation

The modern interpretation of categorical propositions, in which universal propositions (A and E) are not assumed to refer to classes that have members.

Existential Fallacy

A fallacy in which the argument relies on the illegitimate assumption that a class has members, when there is no explicit assertion that it does.

Note: A proposition is said to have existential import if it typically is uttered to assert the existence of objects of some kind. 6.8 Symbolism and Diagrams for Categorical Propositions Form Proposition Symbolic

Rep, Explanation

A

All S is P

_ SP = 0

The class of things that are both S and non-P is empty.

E

No S is P

SP = O

The class off things that are both S and P is empty.

I

Some S is P

SP ≠ 0

The class of things that are both S and P is not empty. (SP as at least one member.)

O

Some S is not P

_ SP ≠ O

The class of things that are both S and non-P is not empty. (SP has at least one member).

Venn Diagrams

A method of representing classes and categorical propositions using overlapping circles.

CHAPTER 7

CATEGORICAL SYLLOGISM 7.1 Standard-Form Categorical Syllogism Syllogism

Any deductive argument in which a conclusion is inferred from two premises.

Categorical Syllogism

A deductive argument consisting of 3 categorical propositions that together contain exactly 3 terms, each of which occurs in exactly 2 of the constituent propositions.

Standard-From Categorical Syllogism

A categorical syllogism in which the premises and conclusions are all standard-form categorical propositions (A, E, I or O)

Arranged with the major premise first, the minor premise second, 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 not in

the conclusion. Major Premise The premise containing the major term. In standard

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 categorical syllogisms determined by the forms of the standard-form propositions it contains.

The mood of the syllogism is therefore represented by 3 letters, and those 3 letters are always given in the standard-form order.

The 1st letter names the type of that syllogism‟s major premise; the 2nd letter names the type of that syllogism‟s minor premise; the 3rd letter names the type of its conclusion.

Every syllogism has a mood.

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Figure The logical shape of a syllogism, determined by the position

of the middle term in its premises Syllogisms can have four–and only four–possible different

figures:

The Four Figures 1st Figure 2nd

Figure 3rd Figure 4th Figure

Schematic Represen-

tation

M – P S – M

.˙. S – P

P – M S – M

.˙. S – P

M – P M – S

.˙. S – P

P – M M – S

.˙. S – P

Description

The middle term may be the subject term of the major premise and the predicate term of the minor premise.

The middle term may be the predicate term of both premises.

The middle term may be the subject term of both premises.

The middle term may be the predicate term of the major premise and the subject term of the minor premise.

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 of its form alone.

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

- If a given syllogism is invalid, any other syllogism of the same 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 Terms A formal mistake in which a

categorical syllogism contains more than 3 terms.

2. Distribute the middle term in at least one premise.

Undistributed Middle A formal mistake in which a

categorical syllogism contains a middle term that is not distributed in either premise.

3. Any term distributed in the conclusion must be distributed in the premises.

Illicit Major A formal mistake in which the major

term of a syllogism is undistributed in the major premise, but is disturbed in the conclusion. Illicit Minor A formal mistake in which the minor

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

4. Avoid 2 negative premises.

Exclusive Premises A formal mistake in which both

premises of a syllogism are negative. 5. If either premise is negative, the conclusion must be negative.

Drawing an Affirmative Conclusion from a Negative Premise A formal mistake in which one

premise of a syllogism is negative, but he conclusion is affirmative.

6. From 2 universal premises no particular conclusion may be drawn.

Existential Fallacy As 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, and it renders the syllogism invalid. Because it is a mistake of that special kind, we call it a fallacy; and because it is a mistake in the form of the argument, we call it a formal fallacy. 7.5 Exposition of the 15 Valid Forms of Categorical Syllogism

The 15 Valid Forms of the Standard- Form Categorical Syllogism

1st Figure 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 3rd Figure 9. AII-3 Datisi 10. IAI-3 Disamis 11. EIO-3 Ferison 12. OAO-3 Bokardo 4th Figure 13. AEE-4 Camenes 14. IAI-4 Dimaris 15. EIO-4 Fresison

7.6 Deduction of the 15 Valid forms of Categorical Syllogism

CHAPTER 8 SYLLOGISM IN ORDINARY LANGUAGE

8.1 Syllogistic Arguments Syllogistic Argument

An Argument that is standard-form categorical syllogism, or can be formulated as one without any change in meaning.

Reduction to Standard Form

Reformulation of a syllogistic argument into standard for. Standard-Form Translation

The resulting argument when we reformulate a loosely put argument appearing in ordinary language into classical syllogism

Different Ways in Which a Syllogistic Argument in Ordinary Language may Deviate from a Standard-Form Categorical Argument: First Deviation

The premises and conclusion of an argument in ordinary language may appear in an order that is not the order of the standard-form syllogism

Remedy: Reordering the premises: the major premise first, the minor premise second, the conclusion third.

Second Deviation

A standard-form categorical syllogism always has exactly 3 terms. The premises of an argument in ordinary language may appear to involve more than 3 terms – but that appearance might prove deceptive.

Remedy: If the number of terms can be reduced to 3 w/o loss of meaning the reduction to standard form may be successful.

Third Deviation

The component propositions of the syllogistic argument in ordinary language may not all be standard-form propositions.

Remedy: If the components can be converted into standard-form propositions w/o loss of meaning, the 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 a 4th 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 standard form, may be translated into standard form so as to allow the syllogism to be tested either by Venn diagrams or by the use of rules governing syllogisms. I. Singular Proposition

A proposition that asserts that a specific individual belongs (or does not belong) to a particular class

Do not affirm/deny the inclusion of one class in another, but we can nevertheless interpret a singular proposition as a proposition dealing w/ classes and their interrelations

E.g. Socrates is a philosopher. E.g. This table is not an antique.

Unit Class

o A class with only one member II. Propositions having adjectives as predicates, rather than substantive or class terms

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

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

active duty. III. Propositions having main verbs other than the copula “to be”

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-wine

drinkers. 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,” and “some”

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 the subject named

E.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 are

those 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 who are

present. VIII. Propositions not resembling standard-form propositions at all

E.g. Not all children believe in Santa Claus. o Reformulated: Some children are not believes in

Santa Claus. E.g. There are white elephants.

o Reformulated: Some elephants are white things. IX. Exceptive Propositions, using “all except” or similar expressions

A proposition making 2 assertions, that all members of some class – except for members of one of its subclasses – are members of some other class

Translating exceptive propositions into standard form is somewhat complicated, because propositions of this kind make 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 assertion into standard form

Uniform Translation

Reducing propositions into standard-form syllogistic argument by using parameters or other techniques.

8.5 Enthymemes Enthymeme

An argument containing an unstated proposition An incompletely stated argument is characterized a being

enthymematic First-Order Enthymeme

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

Second-Order Enthymeme

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

Third-Order Enthymeme

An incompletely stated argument in which the proposition that is left unstated is the conclusion

8.6 Sorites Sorites

An argument in which a conclusion is inferred from any number of premises through a chain of syllogistic inferences

8.7 Disjunctive and Hypothetical Syllogism Disjunctive Syllogism

A form of argument in which one premise is a disjunction and the conclusion claims the truth of one of the disjuncts

Only some disjunctive syllogisms are valid Hypothetical Syllogism

A form of argument containing at least one conditional proposition 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 one categorical premise

Affirmative Mood/Modus Ponens (“to affirm”)

A valid hypothetical syllogism in which the categorical premise 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 the categorical premise affirms the consequent, rather than the antecedent, of the conditional premise

Modus Tollens (“to deny”)

A valid hypothetical syllogism in which the categorical premise 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 the categorical premise denies the antecedent, rather than the consequent, of the conditional premise

8.8 The Dilemma Dilemma

A common form of argument in ordinary discourse in which it is claimed that a choice must be made between 2 (usually bad) alternatives

An argumentative device in which syllogisms on the same topic 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 dilemma…

Rejecting its disjunctive premise This method is often the easiest way to evade the conclusion

of a dilemma, for unless one half of the disjunction is the explicit contradictory of the other, the disjunction may very well be false

Taking/grasping the dilemma by its horns…

Rejecting its conjunction premise To deny a conjunction, we need only deny one of its parts When we grasp the dilemma by the horns, we attempt to

show that at least one of the conditionals is false Devising a counterdilemma…

One constructs another dilemma whose conclusion is opposed to the conclusion of the original

Any counterdilemma may be used in rebuttal, but ideally it should be built up out of the same ingredients (categorical propositions) that the original dilemma contained

CHAPTER 9

SYMBOLIC 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 its

essential nature and putting aside what is not essential

With symbols, we can perform some logical operations almost mechanically, with the eye, which might otherwise demand great effort

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

Modern Logic

Logicians look now to the internal structure of propositions and arguments, and to the logical links – very few in number – that are critical in all deductive arguments

No encumbered by the need to transform deductive arguments in to syllogistic form

It may be less elegant than analytical syllogistics, but is more powerful

9.2 The Symbols for Conjunction, Negation, & Disjunction Simple Statement

A statement that does not contain any other statement as a component

Compound Statement

A statement that contains another statements as a component

2 categories: o W/N the truth value of the compound statement is

determined wholly by the truth value of its components, or determined by anything other than the truth value of its components

Conjunction (y)

A truth functional connective meaning “and” Symbolized by the dot (y) We can form a conjunction of 2 statements by placing the

word “and” between them The 2 statements combined are called conjuncts The truth value of the conjunction of 2 statements is

determined wholly and entirely by the truth values of its 2 conjuncts

If both conjuncts are true, the conjunction is true; otherwise it is false

A conjunction is said to be a truth-functional component statement, and its conjuncts are said to be truth-functional components of it

Note: Not every compound statement is truth-functional Truth Value

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

Truth-Functional Component

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

Truth-Functional Compound Statement

A compound statement whose truth function is wholly determined by the truth values of its components

Truth-Functional Connective

Any logical connective (including conjunction, disjunction, material implication, and material equivalence) between the components of a truth-functional compound statement.

Simple Statement

Any statement that is not truth functionally compound

p q pyq 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 disjuncts or 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 symbolic language to eliminate ambiguity in meaning

In any formula the negation symbol will be understood to apply 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 statement Asserts that in any case in which its antecedent is true, its

consequent is also true It does no assert that its antecedent is true, but only if its

antecedent is true, its consequent is also true The essential meaning of a conditional statement is the

relationship asserted to hold between its antecedent and consequent

Antecedent (implicans/protasis)

In a conditional statement, that component that immediately follows the “if”

Consequent (implicate/apodosis)

In a conditional statement, the component that immediately follows the “then”

Implication

The 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 is common, partial meaning of all “if-then” statements

p q ~q py~q ~ (py~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 statements

The statement “p materially implies q” is true when either p is 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 p” and “p only

if q” are symbolized as p q

In general, “p is a sufficient condition for q” is 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 another argument with the same form whose premises are known to e true and whose conclusion is known to be false.

To prove the invalidity of an argument, it suffices to formulate another argument that:

Has exactly the same form as the first Has true premises and a false conclusion

Note: This method is based upon the fact that validity and invalidity are purely formal characteristics of arguments, which is to say that any 2 arguments having exactly the same form are either both valid or invalid, regardless of any differences in the subject matter which they are concerned. Statement Variable

A letter (lower case) for which a statement may be substituted.

Argument Form

An array of symbols exhibiting the logical structure of an argument, it contains statement variables, but no statements

Substitution Instance of an Argument Form

Any argument that results from the consistent substitution of statements for statement variables in an argument form

Specific Form of an Argument

The argument form from which the given argument results when a different simple statement is substituted for each different statement variable.

9.5 The Precise Meaning of “Invalid” and “Valid” Invalid Argument Form

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

Valid Argument Form

An argument form that has no substitution instances with true 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 may be tested, through the display of all possible combinations of the truth values of the statement variables contained in that form

9.7 Some Common Argument Forms Disjunctive Syllogism

A valid argument form in which one premise is a disjunction, another premise is the denial of one of the two disjuncts, and the conclusion is the truth of the other 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, and

in which another premise affirms the antecedent of that conditional, and the conclusion affirms its consequent

p q

p 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, and in which another premise denies the consequent of that conditional, 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 T F 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 T F 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 argument affirms the consequent of a conditional premise and the conclusion of its argument affirms its antecedent

p q q p

Fallacy of Denying the Antecedent

A formal fallacy in which the 2nd premise of an argument denies the antecedent of a conditional premise and the conclusion 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 a statement

It contains statement variables but no statements

Substitution Instance of Statement Form Any statement that results from the consistent substitution

of statements for statement variables in a statement form Specific Form of a Statement

The statement form from which the given statement results when a different simple statement is substituted consistently for each different statement variable

Tautologous Statement Form

A statement form that has only true substitution instances 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 instances A contradiction

Contingent Form

A statement form that has both true and false substitution instances

Peirce’s Law

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

A truth-functional relation asserting that 2 statements connected by the three-bar sign ( ) have the same truth 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 component statements imply one another and therefore are materially equivalent

The Four Truth-Functional Connective

Truth-Functional Connective

Symbol (Name of Symbol)

Proposition Type

Names of Components of Propositions of

that Type And y (dot) Conjunction Conjuncts Or V (wedge) Disjunction Disjuncts If…then (horseshoe) Conditional Antecedent,

consequent If and only if (tribar) Biconditional Components Note: “Not” is not a connective, but is a truth-function operator, so it is omitted here Note: To say that an argument form is valid if, and only if, its 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 material equivalence is tautology

they are equivalent in meaning and may replace one another

Double Negation

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

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p

~p

~~p

T

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 it connects Logical Equivalence: not a mere connective, and it expresses a relation between 2 statements that is not truth-functional Note: 2 statements are logically equivalent only when it is absolutely impossible for them to have different truth values.

p q p v q ~(p v q) ~p ~q ~py~q ~(p v q) (~py~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 Morgan’s Theorems

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

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

o (2) the negation of the conjunction of 2 statements is logically equivalent to the disjunction of the negations 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 form p p must 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 py~p must be false

o Every such statement is self-contradictory Principle of Excluded Middle

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

CHAPTER 10

METHODS OF DEDUCTION 10.1 Formal Proof of Validity Rules of Inference

The rules that permit valid inferences from statements assumed as premises

Natural Deduction

A method of providing the validity of a deductive argument by using the rules of inference

Using natural deduction we can proved a formal proof of the validity of an argument that is valid

Formal Proof of Validity

A sequence of statements, each of which is either a premise of a given argument or is deduced, suing the rules of inference, from preceding statements in that sequence, such that the last statement in the sequence is the conclusion of the argument whose validity is being proved

Elementary Valid Argument

Any one of a set of specified deductive arguments that serves as a rule of inference & can be used to construct a formal proof of validity

9 RULES OF INFERENCE: ELEMENTARY VALID ARGUMENT FORMS NAME ABBREV. FORM

1. Modus Ponens M.P. p q p q

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

3. Hypothetical Syllogism H.S. p q q r p r

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

q 5. Constructive Dilemma C.D. (p q) y (r s)

p v r q v s

6. Absorption Abs. p q p (p y q)

7. Simplification Simp. p y q p

8. Conjunction Conj. p q

p y q 9. Addition Add. p

p v q 10.2 The Rule of Replacement Rule of Replacement

The rule that logically equivalent expressions may replace each other

Note: this is very different from that of substitution

RULES OF REPLACEMENT: LOGICALLY EQUIVALENT EXPRESSIONS

NAME ABBREV. FORM 10. De Morgan‟s Theorem

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

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

(p v q) (q v p)

(p y q) (q y p) 12. Association Assoc.

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

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

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

[p v (q y r)] [(p v q) y (p v r)] 14. Double Negation

D.N. p ~~ p

15. Transpor-tation

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

16. Material Implication

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

17. Material Equivalence

Equiv. (p q) [(p q) y (q p)]

(p q) [(p y q) v (~p y ~q)] 18. Exportation Exp.

[(p y q) r] [p (q r)] 19. Tautology Taut.

p (p v p)

p (p y p)

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The 19 Rules of Inference The list of 19 rules of inference constitutes a complete system

of truth-functional logic, in the sense that it permits the construction of a formal proof of validity for any valid 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 formal proof is an effective notion

It can be decided quite mechanically, in a finite number of steps, whether or not a given sequence of statements constitutes a formal proof

No thinking is required Only 2 things are required:

o The ability to see that a statement occurring in one place is precisely the same as a statement occurring in another

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

Formal Proof vs. Truth Tables

The making of a truth table is completely mechanical There are no mechanical rules for the construction of formal

proofs Proving an argument valid y constructing a formal proof of its

validity is much easier than the purely mechanical construction of a truth table with perhaps hundreds or thousands of rows

10.3 Proof of Invalidity Invalid Arguments

For an invalid argument, there is no formal proof of invalidity An argument is provided invalid by displaying at least one

row of its truth table in which all its premises are true but its conclusion is false

We need not examine all rows of its truth table to discover an argument‟s invalidity: the discovery of a single row in which its premises are all true and its conclusion is false will suffice

10.4 Inconsistency Note:

If truth values cannot be assigned to make the premises true and the conclusion false, then the argument must be valid

Any argument whose premises are inconsistent must be valid Any argument with inconsistent premises is valid, regardless

of what its conclusion may be Inconsistency

Inconsistent statements cannot both be true “Falsus in unum, falsus in omnibus” (Untrustworthy in one

thing, untrustworthy in all) Inconsistent statements are not “meaningless”; their trouble

is just the opposite. They mean too much. They mean everything, in the sense of implying everything. And if everything is 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 an additional assumed premise the negation of the conclusion

A version of reductio ad absurdum (reducing the absurd) – with which an argument can be proved valid by exhibiting the contradiction which may be derived from its premises augmented by the assumption of the denial of its conclusion

An exclamation point (!) is used to indicate that a given step is derived after the assumption advancing the indirect proof had been made

This method of indirect proof strengthens our machinery for testing arguments by making it possible, in some

circumstances, to prove validity more quickly than would be possible without it

10.6 Shorter Truth-Table Technique Shorter Truth-Table Technique

An argument may be tested by assigning truth values showing that, if it is valid, assigning values that would make the conclusion false while the premises are true would lead inescapably to inconsistency

Proving the validity of an argument with this shorter truth table technique is one version of the use of reductio ad absurdum – but instead of suing the rules of inference, it uses truth value assignments

Its easiest application is when F is assigned to a disjunction (in which case both of the disjuncts must be assigned) or T to a conjunction (in which case both of the conjuncts must be assigned)

o When assignments to simple statements are thus forced, the absurdity (if there is one) is quickly exposed

Note: The reductio ad absurdum method of proof is often the most efficient 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 logical structure of propositions.

11.2 Singular Propositions Affirmative Singular Proposition

A proposition that asserts that a particular individual has some specified attribute

Individual Constant

A symbol used in logical notation to denote an individual Individual Variable

A symbol used as a place holder for an individual constant Propositional Function

An expression that contains an individual variable and becomes a statement when an individual constant is substituted for the individual variable

Simple Predicate

A propositional function having some true and some false substitution instances, each of which is an affirmative singular proposition

11.3 Universal and Existential Quantifiers Universal Quantifier

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

Generalization

The process of forming a proposition from a propositional function by placing a universal quantifier or an existential quantifier before it

Existential Quantifier

A symbol “( x)” indicating that the propositional function that follows has at least one true substitution instance.

Instantiation

The process of forming a proposition from a propositional function by substituting an individual constant for its individual variable

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11.4 Traditional Subject-Predicate Propositions Normal-Form Formula

A formula in which negation signs apply only to simple predicates

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 its universal quantification

Universal Generalization (UG)

A rule of inference that permits the valid inference of a universally quantified expression from an expression that is given 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 any individual constant that appears nowhere earlier in the context) from the existential quantification of a propositional function

Existential Generalization (EG)

A rule of inference that permits the valid inference of the existential quantification of a propositional function from any true substitution instance of that function

Rules of Inference: Quantification

Universal Instantiation

UI (x) ( x) v

(where v is any individual symbol)

Any substitution instance of a propositional function can be validly inferred from its universal quantification

Universal Generalization

UG y (x) ( x)

(where y denotes any arbitrarily selected individual)

From the substitution instance of a propositional function with respect to the name of any arbitrarily selected individual, one may validly infer the universal quantification of that propositional function

Existential Instantiation

EI ( x)( x) v

(where v is any individual constant, other than y, having no previous occurrence in the context)

From the existential quantification of a propositional function, we may infer the truth of its substitution instance with respect to any individual constant (other than y) that occurs nowhere earlier in the context.

Existential Generalization

EG v ( x)( x)

(where v is any individual constant)

From any true substitution instance of a propositional function, we may validly infer the existential quantification of that propositional function.

11.6 Proving Invalidity 11.7 Asyllogistic Inference Asyllogistic Arguments

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

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