Legal Technique and Logic Reviewer SIENNA FLORES

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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”

Classical Logic Traditional techniques, based on Aristotle‟s works, for the analysis of deductive arguments. Modern Symbolic Logic Methods used by most deductive arguments.

modern

logicians

to

analyze

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.

Hypothetical (or Conditional) Proposition A type of compound proposition; It is false only when the antecedent is true and the consequent is false

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.

1.3 Arguments

Notes:

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

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.

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

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)

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Ludwig Wittgenstein One of the most influential philosophers of the 20 th 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

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.

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 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

CHAPTER 4 DEFINITION 4.1 Disputes and Definitions Three Kinds of Disputes 1. 2. 3.

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

Principal Uses 1. Informative 2. Expressive 3. Directive

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

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Obviously genuine disputes there is no ambiguity present and the disputers do disagree, either in attitude or belief Merely verbal disputes there is ambiguity present but there is no genuine disagreement at all Apparently verbal disputes that are really genuine there is ambiguity present and the disputers disagree, either in attitude or belief

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

4.

5.

b.

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

2.

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

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

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

4.6 Rules for Definition by Genus and Difference 1.

Intension (Connotation) the attributes shared by all objects, and only those objects to which a general term applies

2. 3. 4.

4.4 Extension and Denotative Definitions 5. 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 communication

Synonymous definitions a. Defining a word with another word that has the same meaning and is already understood

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A definition should state the essential attributes of the species a definition must not be circular a definition must be neither too broad nor too narrow a definition must not be expressed in ambiguous, obscure, or figurative language 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

Fallacies of Relevance

facilitates

Intensional Definitions 1.

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

Fallacies of Defective Induction

THE MAJOR INFORMAL FALLACIES The most numerous and R1: Appeal to most frequently Emotion encountered, are those in R2: Appeal to Pity which the premises are R3: Appeal to Force simply not relevant to R4: Argument Against the conclusion drawn. the Person R5: Irrelevant Conclusion Those in w/c the mistake D1: Argument from arises from the fact that Ignorance the premises of the D2: Appeal to argument, although Inappropriate relevant to the Authority conclusion, are so weak D3: False Cause

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Fallacies of Presumption

Fallacies of Ambiguity

& ineffective that reliance upon them is a blunder. Mistakes that arise because too much has been assumed in the premises, the inference to the conclusion depending on that unwarranted assumption. 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.

D4: Hasty Generalizations P1: Accident P2: Complex Question P3: Begging the Question

A1: A2: A3: A4: A5:

Equivocation Amphiboly Accent Composition 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.

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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.

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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 Letter Name Quantity A Universal E Universal I Particular O Particular

and Distribution Quality Distribution Affirmative S only Negative S and P Affirmative Neither 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 necessarily false

are

neither

necessarily

true

nor

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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 Convertend A: All S is P. E: No S is P. I: Some S is P. O: Some S is not P.

CONVERSIONS Converse I: Some P is S (by limitation) E: No P is S. I: Some P is S (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.

Premise A: All S is P. E: No S is P. I: Some S is P. O: Some S is not P.

CONTRAPOSITION Contrapositive A: All non-P is non-S. O: Some non-P is not non-S. (by limitation) (Contraposition not valid) O: Some non-P is not non-S.

6.7 Existential Import & the Interpretation of Categorical Propositions Boolean Interpretation

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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

A

All S is P

Symbolic Rep, _ SP = 0

E

No S is P

SP = O

I

Some S is P

SP ≠ 0

O

Some not P

S

is

_ SP ≠ O

Explanation The class of things that are both S and non-P is empty. The class off things that are both S and P is empty. The class of things that are both S and P is not empty. (SP as at least one member.) 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 propositions using overlapping circles.

and

categorical

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 Major Term Minor Term Middle Term Major Premise Minor Premise

of a Standard-Form Categorical Syllogism The predicate term of the conclusion. The subject term of the conclusion. The term that appears in both premises but not in the conclusion. The premise containing the major term. In standard form, the major premise is always stated 1st. 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:

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

Schematic Representation

Description

The Four Figures 1st Figure 2nd 3rd Figure Figure M–P P–M M–P S–M S–M M–S .˙. S – P .˙. S – P .˙. S – P The The The middle middle middle term may term may term may be the be the be the subject predicate subject term of term of term of the major both both premise premises. premises. and the predicate term of the minor premise.

4th Figure P–M M–S .˙. S – P 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 Undistributed Middle term in at least one A formal mistake in which a premise. categorical syllogism contains a middle term that is not distributed in either premise. 3. Any term distributed Illicit Major in the conclusion must A formal mistake in which the major be distributed in the term of a syllogism is undistributed in premises. 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 Exclusive Premises premises. A formal mistake in which both premises of a syllogism are negative. 5. If either premise is Drawing an Affirmative Conclusion negative, the conclusion from a Negative Premise must be negative. A formal mistake in which one premise of a syllogism is negative, but he conclusion is affirmative. 6. From 2 universal Existential Fallacy premises no particular As a formal fallacy, the mistake of conclusion may be inferring a particular conclusion from 2 drawn. universal premises.

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The 15 Valid Forms of the StandardForm 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 th 4 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.

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

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.

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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 categorical premise

conditional

premise

and

one

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

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 () A truth functional connective meaning “and” Symbolized by the dot () 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

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

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

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

Truth-Functional Compound Statement A compound statement whose truth function is wholly determined by the truth values of its components

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

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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 T T F F

q T F T F

pq T 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 T T F F

q T F T F

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 q T F T F

~q F T F T

p~q F T F F

~ (p~q) T F T T

p

q 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 T T F F

q T F T F

p

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.

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 pvq ~p q

q T F T T

In general, “q is a necessary condition for p” and “p only if q” are symbolized as p q

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9.4 Argument Forms and Refutation by Logical Analogy

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

pvq T T T 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

p T T F F

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

p T T F F

q T F T F

pvq T T T F

~p F F T 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

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

q p q

p T T F F

q T F T F

p

Tautologous Statement Form A statement form that has only true substitution instances A tautology:

q T F T 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 T T F F

q T F T F

p

Substitution Instance of Statement Form Any statement that results from the consistent substitution of statements for statement variables in a statement form

q

~p F F T T

Self-Contradictory Statement Form A statement form that has only false substitution instances A contradiction

r T F T F T F T F

p

q T T F F T T T T

q

r T F T T T F T T

p

p T T F F

r T F T F T T T T

Fallacy of Affirming the Consequent A formal fallacy in which the 2 nd 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 2 nd 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

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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 q r p r Q T T F F T T F F

p v ~p T T

Peirce’s Law A tautological statement of the form [(p

Hypothetical Syllogism A valid argument containing only conditional propositions

p T T T T F F F F

~p F T

Contingent Form A statement form that has both true and false substitution instances ~q F T F T

T F T T

p T F

q T F T F

p

q T 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 Symbol Proposition Names of (Name of Type Components of Symbol) Propositions of that Type And  (dot) Conjunction Conjuncts Or V (wedge) Disjunction Disjuncts If…then (horseshoe) Conditional Antecedent, consequent If and only if (tribar) Biconditional Components TruthFunctional Connective

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 T F

~p F T

T

~~p T F

p ~~p T 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

pvq

~(p v q)

~p

~q

~p~q

T T F F

T F T F

T T T F

F F F T

F F T T

F T F T

F F F T

~(p v q)

(~p~q)

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 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 pvq ~p q 5. Constructive Dilemma C.D. (p q)  (r s) pvr qvs 6. Absorption Abs. p q p (p  q) 7. Simplification Simp. pq p 8. Conjunction Conj. p q pq 9. Addition Add. p pvq

9.10 The Three “Laws of Thought”

10.2 The Rule of Replacement

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

Rule of Replacement The rule that logically equivalent expressions may replace each other Note: this is very different from that of substitution

Principle of Noncontradiction No statement can be both true and false Every statement of the form p~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

RULES OF REPLACEMENT: LOGICALLY EQUIVALENT EXPRESSIONS NAME ABBREV. FORM 10. De Morgan‟s De M. ~(p  q) (~ p v ~q) Theorem ~(p v q) 11. Commutation

Com.

12. Association

Assoc.

13. Distribution

Dist.

14. Double Negation 15. Transportation 16. Material Implication 17. Material Equivalence

D.N.

(p v q)

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

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(q v p)

(p  q)

[(p v q) v r]

[p  (q  r)]

[(p  q)  r]

[p  (q v r)]

[(p  q) (p  r)]

[p v (q  r)]

Trans.

(p

Imp. Equiv.

(p (p (p

18. Exportation

Exp.

19. Tautology

Taut.

(q  p)

[p v (q v r)]

10.1 Formal Proof of Validity Rules of Inference The rules that permit valid inferences from statements assumed as premises

(~ p  ~q)

p

~~ p

q)

(~q

q)

q) q)

[(p v q)  (p v r)]

~p)

(~p v q)

[(p

q)  (q

p)]

[(p  q) v (~p  ~q)]

[(p  q)

r]

[p

p

(p v p)

p

(p  p)

(q

r)]

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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 truthfunctional 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

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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

Universal Instantiation

Universal Generalization

Existential Instantiation

Existential Generalization

Rules of Inference: Quantification UI (x) ( x) Any substitution instance of a propositional v (where v is any function can be validly inferred from its individual symbol) universal quantification UG y From the substitution instance of a (x) ( x) function (where y denotes propositional any arbitrarily with respect to the name selected individual) of any arbitrarily selected individual, one may validly infer the universal quantification of that propositional function EI ( x)( x) From the existential quantification of a v function, (where v is any propositional we may infer the truth of individual constant, other its substitution instance with respect to any than y, having no individual constant (other previous occurrence in the than y) that occurs nowhere earlier in the context) context. EG v From any true substitution instance of a ( x)( x) function, (where v is any propositional we may validly infer the individual existential quantification constant) 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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