Patrick J. Hurley - A Concise Introduction to Logic - 2014

Context

Learned in this study

Things to explore

Overview

Notes

1 Basic Concepts

1.1 Arguments, Premises, and Conclusions

• An argument is a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion)
• Questions, proposals, suggestions, commands and exclamations cannot be true/false (and so are not usually classified as statements)
• The pattern found in most arguments that lack indicator words: The intended conclusion is stated first, and the remaining statements are then offered in support of this first statement

1.2 Recognizing Arguments

• What is needed for a passage to contain an argument:
  • At least one of the statements must claim to present evidence or reasons
  • There must be a claim that the alleged evidence supports or implies something - that is, a claim that something follows from the alleged evidence or reasons

Simple Noninferential Passages

• Contains statements that could be premises or conclusions (or both), but what is missing is a claim that any potential premise supports a conclusion or that any potential conclusion is supported by premises
• Passages of this sort includes warnings, pieces of advice, statements of belief or opinion, loosely associated statements, and reports

Expository Passages

• An expository passage is a kind of discourse that begins with a topic sentence followed by one or more sentences that develop the topic sentence. If the objective is not to prove the topic sentence but only to expand it or elaborate it, then there is no argument
• In deciding whether an expository passage should be interpreted as an argument, try to determine whether the purpose of the subsequent sentences in the passage is merely to develop the topic sentence or also to prove that it is true

Illustrations

• An illustration is an expression involving one or more examples that is intended to show what something means or how it is done
• In deciding whether an illustration should be interpreted as an argument, determine whether the passage merely shows how something is done or what something means, or whether it also purports to prove something

Explanations

• An explanation is an expression that purports to shed light on some event or phenomenon
• Every explanation is composed of two distinct components: the explanandum and the explanans
  • The explanandum is the statement that describes the event or phenomenon to be explained
  • The explanans is the statement or group of statements that purports to do the explaining
• To distinguish explanations from arguments, identify the statement that is either the explanandum or the conclusion. If this statement describes an accepted matter of fact, and if the remaining statements purport to shed light on this statement, then the passage is an explanation

Conditional Statements

• A conditional statement is an "if ... then ..." statement
• Every conditional statement is made up of two compoenents statements:
  • The antecedent: The component statement immediately following the "if"
  • The consequent: The component statement following the "then"
• A single conditional statement is not an argument
• A conditional statement may serve as either the premise or the conclusion (or both) of an argument
• The inferential content of a conditional statement may be reexpressed to form an argument

1.3 Deduction and Induction

• Briefly we can say that deductive arguments are those that rest on necessary reasoning, while inductive arguments are those that rest on probabilistic reasoning
• A deductive argument is an argument incorporating the claim that it is impossible for the conclusion to be false given that the premises are true
• An inductive argument is an argument incorporating the claim that it is improbable that the conclusion be false given that the premises are true
• In deciding whether an argument is inductive or deductive, we look to certain objective features of the argument. These features include:
  • the occurence of special indicator words
  • the actual strength of the inferential link between premises and conclusion
  • the form or style of argumentation
• Inductive indicators: probably, improbable, plausible, implausible, likely, unlikely, reasonable to conclude
• Deductive indicators: necessarily, certainly, absolutely, definitely

Deductive Argument Forms

• Argument based on mathematics
• Argument from definition
• Categorical syllogism
• Hypothetical syllogism
• Disjunctive syllogism
• A syllogism, in general, is an argument consisting of exactly two premises and one conclusion
• A categorical syllogism is a syllogism in which each statement beings with one of the words "all," "no," or "some"
• A hypothetical syllogism is a syllogism having a conditional ("if ... then ...") statement for one or both of its premises
• A disjunctive syllogism is a syllogism having a disjunctive ("either ... or ...") statement

Inductive Argument Forms

• A prediction is an argument that proceeds from our knowledge of the past to a claim about the future
• Nearly everyone realizes that the future cannot be known with certainty; thus, whenever an argument makes a prediction about the future, one is usually justified in considering the argument inductive
• An argument from analogy is an argument that depends on the existence of an analogy, or similarity, between two things or states of affairs
• A generalization is an argument that proceeds from the knowledge of a selected sample to some claim about the whole group
• An argument from authority is an argument that concludes something is true because a presumed expert or witness has said that it is
• An argument based on signs is an argument that proceeds from the knowledge of a sign to a claim about the thing or situation that the sign symbolizes
• A causal inference is an argument that proceeds from knowledge of a cause to a claim about an effect, or, conversely, from knowledge of an effect to a claim about a cause
• There is a tradition extending back to the time of Aristotle that holds that inductive arguments are those that proceed from the particular to the general, while deductive arguments are those that proceed from the general to the particular

1.4 Validity, Truth, Soundness, Strength, Cogency

• The evaluation of any argument involves answering two distinct questions:
  • Do the premises support the conclusion?
  • Are all the premises true?

Deductive Arguments

• A valid deductive argument is an argument in which it is impossible for the conclusion to be false given that the premises are true
• An invalid deductive argument is a deductive argument in which it is possible for the conclusion to be false given that the premises are true
• Validity is something that is determined by the relationship between premises and conclusion. The question is not whether the premises and conclusion are true or false, but whether the premises support the conclusion
• Any deductive argument having actually true premises and an actually false conclusion is invalid
• A sound argument is a deductive argument that is valid and has all true premises
• An unsound argument is a deductive argument that is invalid, has one or more false premises, or both

Inductive Arguments

• A strong inductive argument is an inductive argument in which it is improbable that the conclusion be false given that the premises are true
• A weak inductive argument is an argument in which the conclusion does not follow probably from the premises, even though it is claimed to
• When we speak of the premises being true, we mean "true" in a complete sense. The premises must not exclude or overlook some crucial piece of evidence that undermines the stated premises and requires a different conclusion. This proviso is otherwise called the total evidence requirement
• To be considered strong, an inductive argument must have a conclusion that is more probable than improbable. In other words, given that the premises are true, the likelihood that the conclusion is true must be more than 50 percent, and as the probability increases, the argument becomes stronger
• A cogent argument is an inductive argument that is strong and has all true premises. Also, the premises must be true in the sense of meeting the total evidence requirement
• An uncogent argument is an inductive argument that is weak, has one or more false premises, fails to meet the total evidence requirement, or any combination of these

1.5 Argument Forms: Proving Invalidity

• Can basically be summarized as "if the arguments are not transitive, then the conclusion cannot be true"

Counterexample Method

• It consists of isolating the form of an argument and then constructing a substitution instance having true premises and a false conclusion. This proves the form invalid, which in turn proves the argument invalid
• The counterexample method can be used to prove the invalidity of any invalid argument, but it cannot prove the validity of any valid argument

1.6 Extended Arguments

• A set of components to diagram an argument:
  • Vertical pattern
  • Horizontal pattern
  • Conjoint premises
  • Multiple conclusions

2 Language: Meaning and Definition

2.1 Varieties of Meaning

• Two linguistic functions are particularly important:
  • To convey information
  • To express or evoke feelings
• Terminology that conveys information is said to have cognitive meaning, and terminology that expresses or evokes feelings is said to have emotional meaning
• A value claim is a claim that something is good, bad, right, wrong, better, worse, more important, less important than some other thing
• Two defects of cognitive meaning
  • Vagueness
  • Ambiguity
• A vague expression is one that allows for borderline cases in which it is impossible to tell if the expression applies or does not apply
• An ambiguous expression is one that can be interpreted as having more than one clearly distinct meaning in a given context
• The difference between ambiguity and vaguesness is that vague terminology allows for a relatively continuous range of interpretations, whereas ambiguous terminology allows for multiple discrete interpretations
• Disputes that arise over the meaning of language are called verbal disputes
• In dealing with disputes, the first question is whether the dispute is factual, verbal, or some combination of the two
  • If the dispute is verbal, then the second question to be answered is whether the dispute concerns ambiguity or vagueness

2.2 The Intension and Extension of Terms

• The basic units of any ordinary language are words
• A term is any word or arrangement of words that may serve as the subject of a statement (proper names, common names, descriptive phrases)
• The intensional meaning, or intension, consists of the qualities or attributes that the term connotes, and the extensional meaning, or extension, consists of the members of the class that the term denotes
  • Intensional meaning: cats are furry, have four legs, move in a certain way, emit certain sounds
  • Extensional meaning: all the cats in the universe
• Intension <-> sense, Extension <-> reference
• The conventional connotation of a term includes the attributes that the term commonly calls forth in the minds of competent speakers of the language
• Empty extension: denote the empty (or "null") class, the class that has no members
• Intension determines extension

2.3 Definitions and Their Purposes

• Definition: a group of words that assigns a meaning to some word or group of words
• Every definition consists of two parts: a definiendum and the definiens
  • The definiendum is the word or group of words that is supposed to be defined
  • The definiens is the word or group of words that does the defining

Stipulative Definitions

• A stipulative definition assigns a meaning to a word for the first time

Lexical Definitions

• Used to report the meaning that a word already has in a language
• Have the further purpose of eliminating the ambiguity that would otherwise arise if one of these meanings were to be confused with another

Precising Definitions

• Reduce the vagueness of a word

Theoretical Definitions

• Assign a meaning to a word by suggesting a theory that gives a certain characterization to the entities that the term denotes

Persuasive Definitions

• To engender a favorable or unfavorable attitude toward what is denoted by the definiendum. This purpose is accomplished by assigning an emotionally charged or value-laden meaning to a word while making it appear that the word really has (or ought to have) that meaning in the language in which it is used

2.4 Definitional Techniques

Extensional (Denotative) Definitions

• An extensional (denotative) definition is one that assigns a meaning to a term by indicating the members of the class that the definiendum denotes
• There are at least three ways of indicating the members of a class:
  • pointing to them (demonstrative or ostensive definitions)
  • naming them individually (enumerative definitions)
  • naming them in groups (definitions by subclass)
• The principle that intension determines extension, whereas the converse is not true, underlies the fact that all extensional definitions suffer serious deficiencies (how are the properties used as intension determined?)
• Extensions can suggest intensions, but they cannot determine them

Intensional (Connotative) Definitions

• An intensional definition is one that assigns a meaning to a word by indicating the qualities or attributes that the word connotes
• There are at least four kinds of intensional definitions:
  • synonymous definition: the definiens is a single word that connotes the same attributes as the definiendum
  • etymological definition: assigns a meaning to a word by disclosing the word's ancestry in both its own language and other languages
  • operational definition: assigns a meaning to a word by specifying certain experimental procedures that determine whether or not the word applies to a certain thing
  • definition by genus and difference: assigns a meaning to a term by identifying a genus term and one or more difference words that, when combined, convey the meaning of the term being defined
• In logic, genus simply means a relatively larger class, and species means a relatively smaller subclass of the genus

2.5 Criteria for Lexical Definitions

• A lexical definition should ...
  • Rule 1: ... conform to the standards of proper grammar
  • Rule 2: ... convey the essential meaning of the word being defined
  • Rule 3: ... be neither too broad nor too narrow
  • Rule 4: ... avoid circularity
  • Rule 5: ... not be negative when it can be affirmative
  • Rule 6: ... avoid figurative, obscure, vague, or ambiguous language
  • Rule 7: ... avoid affective terminology
  • Rule 8: ... indicate the context to which the definiens pertains

3 Informal Fallacies

3.1 Fallacies in General

• A fallacy is a defect in an argument that arises from either a mistake in reasoning or the creation of an illusion that makes a bad argument appear good
• non sequitur ("it does not follow")
• Fallacies are usually divided into two groups: formal and informal
• A formal fallacy is one that may be identified merely by examining the form or structure of an argument. They are found only in deductive arguments that have identifiable forms (categorical syllogisms, disjunctive syllogisms, hypothetical syllogisms)
• Informal fallacies are those that can be detected only by examining the content of the argument

3.2 Fallacies of Relevance

• The connection between premises and conclusion is emotional
• Appeal to force: Whenever an arguer poses a conclusion to another person and tells that person either implicitly or explicitly that some harm will come to him or her if he or she does not accept the conclusion
• Appeal to pity: When an arguer attempts to support a conclusion by merely evoking pity from the reader or listener
• Appeal to the people:
  • Direct approach: When an arguer, addressing a large group of people, excites the emotions and enthusiasm of the crowd to win acceptance for his or her conclusion
    • Appeal to fear: When an arguer trumps up a fear of something in the mind of the crowd and then uses that fear as a premise for some conclusion
  • Indirect approach: The arguer aims his or her appeal not at the crowd as a whole but at one or more individual separately, focusing on some aspect of those individuals' relationship to the crowd
    • Bandwagon argument: Everybody believes such-and-such or does such-and-such; therefore, you should believe or do such-and-such, too
    • Appeal to vanity: Involves linking the love, admiration, or approval of the crowd with some famous figure who is loved, admired, or approved of
    • Appeal to snobbery: The arguer appeals to a smaller group of the crowd that is supposed to be superior in some way. If the listener wants to be part of this group, then he or she will do a certain thing, think in a certain way, or buy a certain product
    • Appeal to tradition: When an arguer cites the fact that something has become a tradition as grounds for some conclusion
• Argument against the person (argumentum ad hominem): Involves two arguers. One of them advances a certain argument, and the other then responds by directing his or her attention not to the first person's argument but to the first person himself
  • 3 forms:
    • Ad hominem abusive: The second person responsd to the first person's argument by verbally abusing the first person
    • Ad hominem circumstantial: The respondent attempts to discredit the opponent's argument by alluding to certain circumstances that affect the opponent
    • Tu quoque: The second arguer attempts to make the first appear to be hypocritical or arguing in bad faith
• Accident: When a general rule is applied to a specific case it was not intended to cover
• Straw man: When an arguer distorts an opponent's argument for the purpose of more easily attacking it, demolishes the distorted argument, and then concludes that the opponent's real argument has been demolished
• Missing the point (ignoratio elenchi): When the premises of an argument support one particular conclusion, but then a different conclusion, often vaguely related to the correct conclusion, is drawn
• Red herring: When the arguer diverts the attention of the reader or listener by changing the subject to a different but sometimes subtly related one. He or she then finishes by either drawing a conclusion about this different issue or by merely presuming that some conclusion has been established

3.3 Fallacies of Weak Induction

• The fallacies of weak induction occur because the connection between premises and conclusion is not strong enough to support the conclusion
• Appeal to unqualified authority (argumentum ad verecundiaum): When the cited authority or witness lacks credibility
  • In deciding whether a person is a qualified authority, one should keep two important points in mind. First, the person might be an authority in more than one field. Second, there are some areas in which pratically no one can be considered an authority
• Appeal to ignorance (argumentum ad ignorantiam): When the premises of an argument state that nothing has been proved one way or the other about something, and the conclusion makes a definite assertion about that thing
  • Two important exceptions:
    • If qualified researchers investigate a certain phenomenon within their range of expertise and fail to turn up any evidence that the phenomenon exists, this fruitless search by itself constitutes positive evidence about the question
    • In the United States and a few other countries, a person is presumed innocent until proven guilty
• Hasty generalization (converse accident): When there is a reasonable likelihood that the sample is not representative of the group
• False cause: Whenever the link between premises and conclusion depends on some imagined causal connection that probably does not exist
  Post hoc ergo propter hoc ("after this, therefore on account of this"): Presupposes that just because one event precedes another event, the first event causes the second
  Non causa pro causa ("not the cause for the cause"): When what is taken to be the cause of something is not really the cause at all and the mistake is based on something other than mere temporal succession
  Oversimplified cause: When a multitude of causes is responsible for a certain effect but the arguer selects just one of these causes and represents it as if it were the sole cause
  Gambler's fallacy: Whenever the conclusion of an argument depends on the supposition that independent events in a game of chance are causally related
• Slippery slope: When the conclusion of an argument rests on an alleged chain reaction and there is not sufficient reason to think that the chain reaction will actually take place
• Weak analogy: An argument in which the conclusion depends on the existence of an analogy, or similarity, between two things or situations. The fallacy of weak analogy is committed when the analogy is not strong enough to support the conclusion that is drawn

3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference

• Begging the question (petitio principii): Whenever the arguer creates the illusion that inadequate premises provide adequate support for the conclusion by leaving out a possibly false (shaky) key premise, by restating a possibly false premise as the conclusion, or by reasoning in a circle
• Complex question: When two (or more) questions are asked in the guise of a single question and a single answer is then given to both of them
• False dichotomy: When a disjunctive premise presents two unlikely alternatives as if they were the only ones available, and the arguer then eliminates the undesirable alternative, leaving the desirable one as the conclusion
• Suppressed evidence: When an inductive argument ignore important piece of evidence that outweighs the presented evidence and entails a very different conclusion
• Equivocation: When the conclusion of an argument depends on the fact that a word or phrase is used, either explicitly or implicitly, in two different senses in the argument
• Amphiboly: When the arguer misinterprets an ambiguous statement and then draws a conclusion based on this faulty interpretation
  • Amphiboly differs from equivocation in two important ways:
    • Equivocation is always traced to an ambiguity in the meaning of a word or phrase, whereas amphiboly involves a syntatical ambiguity in a statement
    • Amphiboly usually involves a mistake made by the arguer in interpreting an ambiguous statement made by someone else, whereas the ambiguity in equivocation is typically the arguer's own creation
• Composition: When the conclusion of an argument depends on the erroneous transference of an attribute from the parts of something onto the whole
  • To distinguish composition from hasty generalization, the following procedure should be followed:
    • Examine the conclusion of the argument
      • If the conclusion is a general statement, the fallacy committed is hasty generalization
      • If the conclusion is a class statement, the fallacy is composition
• Division: When the conclusion of an argument depends on the erroneous transference of an attribute from a whole (or a class) onto its parts (or members)
  • If the premises contain a general statement, the fallacy committed is accident
  • If the premises contain a class statement, the fallacy is division

3.5 Fallacies in Ordinary Language

Avoiding Fallacies

• We can identify three factors that lead to most of the informal mistakes in reasoning
  • Intent. Many fallacies are committed intentionally. The arguer may know full well that his or her reasoning is defective but goes ahead with it anyway because of some benefit for himself or herself or some other person
  • Careless mental posture combined with an emotional disposition favoring or opposing some person or thing
  • The influence of what we might call the "worldview" of the arguer

4 Categorical Propositions

4.1 The Components of Categorical Propositions

• A proposition that relates two classes, or categories, is called a categorical proposition
• The classes in question are denoted respectively by the subject term and the predicate term
• The proposition asserts that either all or part of the class denoted by the subject term is included in or excluded from the class denoted by the predicate term
• Standard-form categorical proposition
  • All S are P
  • No S are P
  • Some S are P
  • Some S are not P
• "All S are not P" is not a standard form. This form is ambiguous and can be rendered as either "No S are P" or "Some S are P"
• There are exactly three forms of quantifiers ("all", "no", "some") and two forms of copulas ("are" and "are not")
• The quality of a categorical proposition is either affirmative or negative
• The quantity of a categorical proposition is either universal or particular
• Categorical propositions have commonly been designated by letter names corresponding to the first four vowels of the Roman alphabet:
  • A: universal affirmative
  • E: universal negative
  • I: particular affirmative
  • O: particular negative
• A term is said to be distributed if the proposition makes an assertion about every member of the class denoted by the term
  • A: the subject term is distributed
  • E: Both terms are distributed
  • I: No term is distributed
  • O: The predicate is distributed

4.2 Venn Diagrams and the Modern Square of Opposition

• Aristotle held that universal propositions about existing things have existential import
• According to the theory developed by George Boole, no universal propositions have existential import
• For both, the word "some" implies existence
• From a boolean standpoint:
  • All S are P = No members of S are outside P
  • No S are P = No members of S are inside P
  • Some S are P = At least one S exist that is a P
  • Some S are not P = at least one S exist that is not a P
• A and O make assertions that are the exact opposite of each other
• E and I make assertions that are the exact opposite of each other
• This relationship of mutually contradictory pairs of propositions is represented in a diagram called the modern square of opposition
• Propositions are said to be vacuously true, because their truth value results solely from the fact that the subject class is empty, or void of members
• Arguments that have only one premise are called immediate inferences
• Arguments that are valid from the Boolean standpoint are said to be unconditionally valid because they are valid regardless of whether their terms refer to existing things

4.4 Conversion, Obversion, and Contraposition

Conversion

• A conversion consists of switching the subject term with the predicate term
• E.g. No F are H -> No H are F
• Two statements are said to be logically equivalent statements when they necessarily have the same truth value
• Converting an E or I statement gives a new statement that always has the same truth value (and the same meaning) as the given statement
• Converted an A or O statement gives a new statement whose truth value is logically undetermined in relation to the given statement
• The fallacy of illicit conversion: Converting an A or O statement and assuming they are logically equivalent statements

Obversion

• Obversion requires two steps:
  • Changing the quality
  • Replacing the predicate with its term complement
• E.g. No S are P -> All S are non-P
• The term complement is the word or group of words that denotes the class complement
• Obversion of all type of statements (A, E, I, O) are logically equivalent to their original statement

Contraposition

• Contraposition requires two steps:
  • Switching the subject and predicate terms
  • Replacing the subject and predicate terms with their term complements
• E.g. All G are A -> All non-A are non-G
• Contrapositive of A and O statements are logically equivalent to their original statement
• Contrapositive of E and I statements are logically undetermined in relation to the given statement
• The fallacy of illicit contraposition: Assuming that the contraposition of an E or I statement is logically equivalent to the given statement

4.5 The Traditional Square of Opposition

• The traditional square of opposition is an arrangement of lines that illustrates logically necessary relations among the four kinds of categorical propositions
• Contradictory: opposite truth value
• Contrary: at least one is false (not both true)
• Subcontrary: at least one is true (not both false)
• Subalternation: truth flows downward, falsity flows upward
• Contrary relation: If a certain A proposition is given as true, the corresponding E proposition is false, and if an E proposition is given as true, the corresponding A proposition is false. But if an A proposition is given as false, the corresponding E proposition could be either true or false. Similarly, if an E proposition is given as false, the corresponding A proposition has logically undetermined truth value
• Subcontrary relation: If a certain I proposition is given as false, the corresponding O proposition is true, and if an O proposition is given as false, the corresponding I proposition is true. But if either an I or O proposition is given as true, then the corresponding proposition could be either true or false
• Subalternation relation: If an A proposition is given as true, the corresponding I proposition is true also, and if an I proposition is given as false, the corresponding A proposition is false. But if an A proposition is given as false, this truth value cannot be transmitted downward, so the corresponding I proposition will have logically undetermined truth value. Conversely, if an I proposition is given as true, this truth value cannot be transmitted upward, so the corresponding A proposition will have logically undetermined truth value. Analogous reasoning prevails for the subalternation relation between the E and O propositions
• Inferences that depends on an incorrect application of the (contrary, subcontrary, subalternation) relation commit the formal fallacy of illicit (contrary, subcontrary, subalternation)
• What happens when the Aristotelian standpoint is adopted but the propositions are about things that do not exist? The answer is that under these conditions the traditional square gives exactly the same results as the modern square
  • Inferences that are based on a correct application of the contradictory relation are valid, but inferences that are based on an otherwise correct application of the other three relations are invalid and commit the existential fallacy
• Conditionally valid applies to an argument after the Aristotelian standpoint has been adopted and we are not certain if the subject term of the premise denotes actually existing things

4.6 Venn Diagrams and the Traditional Standpoint

• The steps involved in testing an immediate inference from the Aristotelian standpoint
  • Reduce the inference to its form and test it from the Boolean standpoint. If the form is valid, proceed no further. The inference is valid from both standpoints
  • If the inference form is invalid from the Boolean standpoint and has a particular conclusion, then adopt the Aristotelian standpoint and look to see if the left-hand premise circle is partly shaded. If it is, enter a circled X in the unshaded part and retest the form
  • If the inference form is conditionally valid, determine if the circled X represents something that exists. If it does, the condition is fulfilled, and the inference is valid from the Aristotelian standpoint. If it does not, the inference is invalid, and it commits the existential fallacy from the Aristotelian standpoint

5 Categorical Syllogisms

5.1 Standard Form, Mood, and Figure

• A syllogism is a deductive argument consisting of two premises and one conclusion
• The major term is the predicate of the conclusion
• The minor term is the subject of the conclusion
• The middle term is the one that occurs once in each premise and does not occur in the conclusion
• The major premise is the one that contains the major term
• The minor premise is the one that contains the minor term
• A standard-form categorical syllogism is one that meets the following four conditions:
  1. All three statements are standard-form categorical propositions
  2. The two occurrences of each term are identical
  3. Each term is used in the same sense throughout the argument
  4. The major premise is listed first, the minor premise second, and the conclusion last
• A categorical syllogism is a deductive argument consisting of three categorical propositions that is capable of being translated into standard form
• The mood of a categorical syllogism consists of the letter names of the proposition that make it up (major A, minor O, conclusion E -> AOE)
• The figure of a categorical syllogism is determined by the location of the two occurrences of the middle term in the premises (position of the middle terms: 1 \, 2 X|, 3 |X, 4 /)

5.2 Venn Diagrams

• The critical term is the term in a categorical syllogism which, when it denotes at least one existing thing, guarantees that the subject of the conclusion denotes at least one existing thing
• Apply the universal premise first, then the particular premise

5.3 Rules and Fallacies

Boolean Standpoint

• Rule 1: The middle term must be distributed at least once
  • Fallacy: Undistributed middle
• Rule 2: If a term is distributed in the conclusion, then it must be distributed in a premise
  • Fallacy: Illicit major; illicit minor
• Rule 3: Two Negative premises are not allowed
  • Fallacy: Exclusive premises
• Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise
  • Fallacy: Drawing an affirmative conclusion from a negative premise or drawing a negative conclusion from affirmative premises
• It turns out that no valid syllogism can have two particular premises
• Rule 5: If both premises are universal, the conclusion cannot be particular
  • Fallacy: Existential fallacy

Aristotelian Standpoint

• Any categorical syllogism that breaks one of the first four rules is invalid from the Aristotelian standpoint
• The critical term is the one that is superfluously distributed. In other words, it is the term that, in the premises, is distributed in more occurrences than is necessary for the syllogism to obey the first two rules

5.6 Ethymemes

• An enthymeme is an argument that is expressible as a categorical syllogism but that is missing a premise or a conclusion
• Any enthymeme that contains an indicator word is missing a premise

5.7 Sorites

• A sorites is a chain of categorical syllogisms in which the intermediate conclusions have been left out
• The rule in evaluating a sorites is based on the idea that a chain is only as strong as its weakest link. If any of the component syllogisms in a sorites is invalid, the entire sorites is invalid
• A standard-form sorites is one in which each of the component propositions is in standard form, each term occurs twice, the predicate of the conclusion is in the first premise, and each successive premise has a term in common with the preceding one
• Two techniques for testing a sorites for validity
  • Technique 1
    • Put the sorites into standard form
    • Introduce the intermediate conclusions
    • Test each component syllogism for validity
  • Technique 2
    • Each of the middle terms must be distributed at least once
    • If a term is distributed in the conclusion, then it must be distributed in a premise
    • Two negative premises are not allowed
    • A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise
    • If all the premises are universal, the conclusion cannot be particular
• Any sorites having more than one negative premise is invalid, and any sorites having a negative premise and an affirmative conclusion is invalid
• As with syllogisms, any sorites having more than one particular premise is invalid

Chapter 6 - Propositional Logic

6.1 Symbols and Translation

• Five logical operators:
  • ~: negation (not, it is not the case that)
  • $\bullet$/$\wedge$: conjunction (and, also, moreover)
  • $\vee$: disjunction (or, unless)
  • $\supset$: implication (if ... then ..., only if)
  • $\equiv$: equivalence (if and only if)
• The main operator is the operator that has as its scope everything else in the statement
• Event A is said to be a sufficient condition for event B whenever the occurrence of A is all that is required for the occurrence of B
• Event A is said to be a necessary condition for B whenever B cannot occur without the occurrence of A
• Sufficient $\supset$ Necessary
• A well-formed formula is a syntactically correct arrangement of symbols
• Statements cannot be combined without an operator occurring between them
• A tilde cannot immediately follow a statement, but it can immediately precede any statement (except when it immediately follows a statement)
• A dot, wedge, horseshoe, or triple bar must go immediately between statements
• Parentheses, brackets, and braces must be inserted to prevent ambiguity

6.2 Truth Functions

• A truth function is any compound proposition whose truth value is completely determined by the truth values of its components
• A statement form is an arrangement of statement variables and operators such that the uniform substitution of statements in place of the variables results in a statement

6.3 Truth Tables for Propositions

• A compound statement is said to be logically true or tautologous statement if it is true regardless of the truth values of its components
• It is said to be logically false of self-contradictory statement if it is false regardless of the truth values of its components
• It is said to be a contingent statement if its truth value varies depending on the truth values of its components
• Two propositions are said to be logically equivalent statements if they have the same truth value on each line under their main operators, and they are contradictory statements if they have opposite truth values on each line under their main operators
• Two (or more) propositions are consistent statements if there is at least one line on which both (or all) of them turn out to be true, and they are inconsistent statements if there is no line on which both (or all) of them turtn out to be true

6.4 Truth Tables for Arguments

• A valid argument is one in which it is not possible for the premises to be true and the conclusion false
• The conditional statement having the conjunction of an argument's premises as its antecedent and the conclusion as its consequent is called the argument's corresponding conditional

6.5 Indirect Truth Tables

• To construct an indirect truth table for an argument, we begin by assuming that the argument is invalid. If no contradiction is obtained in the process, this means that it is indeed possible for the premises to be true and the conclusion false, as originally assumed, so the argument is therefore invalid. If, however, the attempt to make the premises true and the conclusion false necessarily leads to a contradiction, it is not possible for the premises to be true and the conclusion false, in which case the argument is valid
• Testing statements for consistency: If we can find a case where the statements are all true and there is no contradiction, then the statements are consistent

6.6 Argument Forms and Fallacies

• An argument form is an arrangement of statement variables and operators such that the uniform substitution of statements in place of the variables results in an argument. The resulting argument is said to be a substitution instance of its related argument form

Disjunctive syllogism (DS)

$$ \frac{ p \vee q \\ \sim p } {q}$$

Pure hypothetical syllogism (HS)

$$ \frac{ p \supset q \\ q \supset r } {p \supset r}$$

Modus ponens (asserting mode) (MP)

$$ \frac{ p \supset q \\ p } {q}$$

Modus tollens (denying mode) (MT)

$$ \frac{ p \supset q \\ \sim q } {\sim p}$$

Affirming the consequent (invalid)

$$ \frac{ p \supset q \\ q } {p}$$

Denying the antecedant (invalid)

$$ \frac{ p \supset q \\ \sim p } {\sim q}$$

Constructive dilemma (CD)

$$ \frac{ (p \supset q) \wedge (r \supset s) \\ p \vee r } {q \vee s}$$

Destructive dilemma (DD)

$$ \frac{ (p \supset q) \wedge (r \supset s) \\ \sim q \vee \sim s } {\sim p \vee \sim r}$$

Refuting Constructive and Destructive Dilemmas

• Since both forms are valid, the only direct mode of defense available to the opponent is to prove the dilemma unsound. This can be done by proving at least one of the premises false
• If the conjunctive premise (otherwise called the "horns of the dilemma") is proven false, the opponent is said to have "grasped the dilemma by the horns". This is done by proving either one of the conditional statements false
• If the disjunctive premise is proven false, the opponent is said to have "escaped between the horns of the dilemma". The strategy involves finding a third alternative that excludes the two that are given in the disjunctive premise
• The strategy to be followed in refuting a dilemma is therefore this:
  • Examine the disjunctive premise. If this premise is a tautology, attempt to grasp the dilemma by the horns by attacking one or the other of the conditional statements in the conjunctive premise
  • If the disjunctive premise is not a tautology, then either escape between the horns by, perhaps, finding a third alternative, or grasp by the horns, whichever is easier
• A third, indirect strategy for refuting a dilemma involves constructing a counter-dilemma. This is typically done by changing either the antecedants or the consequents of the conjunctive premise while leaving the disjunctive premise as it is, so as to obtain a different conclusion
• We will say that an argument has an invalid form if it is a substitution instance of that form and it is not a substitution instance of any valid form

7 Natural Deduction in Propositional Logic

7.1 Rules of Implication I

• Eight rules of inference called rules of implication
  • Modus ponens
  • Modus tollens
  • Pure hypothetical syllogism
  • Disjunctive syllogism
• To construct a proof, start by finding the conclusion we want to prove. Then, work your way back, that is, find premises which will lead to the desired conclusion until all the required inference steps and premises have been proven
• Strategies for applying the first four rules of inference
  • Strategy 1: Always begin by attempting to "find" the conclusion in the premises. If the conclusion is not present in its entirety in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived
  • Strategy 2: If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens
  • Strategy 3: If the conclusion contains a negated letter and the letter appears in the antecedant of a conditional statement in the premises, consider obtaining the negated letter via modus tollens
  • Strategy 4: If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism
  • Strategy 5: If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism

7.2 Rules of Implication II

• Constructive dilemma
• Simplification (Simp)
• Conjunction (Conj)
• Addition (Add)
• Strategies for applying the last four rules of implication:
  • Strategy 6: If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification
  • Strategy 7: If the conclusion is a conjunctive statement, consider obtaining it via conjunction by first obtaining the individual conjuncts
  • Strategy 8: If the conclusion is a disjunctive statement, consider obtaining it via constructive dilemma or addition
  • Strategy 9: If the conclusion contains a letter not found in the premises, addition must be used to introduce that letter
  • Strategy 10: Conjunction can be used to set up constructive dilemma

7.3 Rules of Replacement I

• Double colon is used to designate logical equivalence
• Axiom of replacement: Asserts that within the context of a proof, logically equivalent expressions may replace each other
• The first five rules of replacement:
  • De Morgan's rule (DM)
  • Commutativity (Com)
  • Associativity (Assoc)
  • Distribution (Dist)
  • Double negation (DN)
• Strategies for applying the first five rules of replacement:
  • Strategy 11: Conjunction can be used to set up De Morgan's rule
  • Strategy 12: Constructive dilemma can be used to set up De Morgan's rule
  • Strategy 13: Addition can be used to set up De Morgan's rule
  • Strategy 14: Distribution can be used in two ways to set up disjunctive syllogism (distribute the term and "merge" the common term)
  • Strategy 15: Distribution can be used in two ways to set up simplification
  • Strategy 16: If inspection of the premises does not reveal how the conclusion should be derived, consider using the rules of replacement to deconstruct the conclusion

7.4 Rules of Replacement II

• The remaining five rules of replacement:
  • Transposition (Trans)
  • Material implication (Impl)
  • Material equivalence (Equiv)
  • Exportation (Exp)
  • Tautology (Taut)
• Strategies for applying the last five rules of replacement:
  • Strategy 17: Material implication can be used to set up hypothetical syllogism
  • Strategy 18: Exportation can be used to set up modus ponens
  • Strategy 19: Exportation can be used to set up modus tollens
  • Strategy 20: Addition can be used to set up material implication
  • Strategy 21: Transposition can be used to set up hypothetical syllogism
  • Strategy 22: Transposition can be used to set up constructive dilemma
  • Strategy 23: Constructive dilemma can be used to set up tautology
  • Strategy 24: Material implication can be used to set up tautology
  • Strategy 25: Material implication can be used to set up distribution

7.5 Conditional Proof

• Conditional proof is a method for deriving a conditional statement (either the conclusion or some intermediate line) that offers the usual advantage of being both shorter and simpler to use than the direct method
• After a conditional proof sequence has been discharged, no line in the sequence may be used as a justification for a subsequent line in the proof
• A conditional sequence can be used within the scope of another to derive a desired statement
• Every conditional proof must be discharged

7.6 Indirect Proof

• Indirect proof consists of assuming the negation of the statement to be obtained, using this assumption to derive a contradiction, and then concluding that the original assumption is false
• One should consider indirect proof whenever a line ina proof appears difficult to obtain
• A conditional sequence may be constructed within the scope of an indirect sequence, and, conversely, an indirect sequence may be constructed within the scope of either a conditional sequence or another indirect sequence

7.7 Proving Logical Truths

• Any argument having a tautology for its conclusion is valid regardless of what its premises are

8 Predicate Logic

8.1 Symbols and Translation

• In categorical syllogisms, the fundamental components are terms
• In propositional logic, the fundamental components are statements
• Predicate logic combines syllogistic logic and propositional logic
• The fundamental component in predicate logic is the predicate
• A statement function is the expression that remains when a quantifier is removed from a statement
• Universal quantifier

• Existential quantifier

8.2 Using the Rules of Inference

• Instantiation is an operation that consists in deleting a quantifier and replacing every variable bound by that quantifier with the same instatial letter
• We cannot perform universal generalization when the instantial letter is a constant
• Generalization in the inclusive sense is an operation that consists in
  • introducing a quantifier immediately prior to a statement, a statement function or another quantifier
  • replacing one or more occurrences of a certain instantial letter in the statement or statement function with the same variable that appears in the quantifier
• For universal generalization, all occurrences of the instantial letter must be replaced with the variable in the quantifier, and for existential generalization, at least one of the instantial letters must be replaced with the variable in the quantifier
• The name introduced by existential instantiation must be a new name that has not occurred in any previous line, including the line adjacent to the last premise that indicates the conclusion to be derived

8.3 Quantifier Negation Rule

$$ \begin{split} (x)\mathscr{F}x\ & \bf{::} &\ \sim(\exists x)\sim\mathscr{F}x \\ \sim(x)\mathscr{F}x\ & \bf{::} &\ (\exists x)\sim\mathscr{F}x \\ (\exists x)\mathscr{F}x\ & \bf{::} &\ \sim(x)\sim\mathscr{F}x \\ \sim(\exists x)\mathscr{F}x\ & \bf{::} &\ (x)\sim\mathscr{F}x \end{split}$$

8.4 Conditional and Indirect Proof

• Universal generalization must not be used within the scope of an indented sequence if the instantial variable is free in the first line of that sequence

8.5 Proving Invalidity

Counterexample Method

• Consists in finding a substitution instance of a given invalid argument form (or, equally well, a given invalid symbolized argument) that has true premises and a false conclusion
• The counterexample method is effective with most fairly simple invalid arguments in predicate logic. Since its application depends on the ingenuity of the user, however, it is not particularly well suited for complex arguments. For those, the finite universe method is probably a better choice

Finite Universe Method

• It depends on the idea that a valid argument remains valid no matter how things in the actual universe might be altered
• If we are given a valid argument, then that argument remains valid if it should happen that the universe is contracted so that it contains only a single member
• If it should turn out that an argument has true premises and false conclusion in a universe consisting of only one or a few members, then that argument has been proved invalid
• $\bf{\stackrel{c}{::}}$ Asserts that the expressions on either side of it necessarily have the same truth value given a universe of a designated size
  • c for conditional

$$ \begin{split} (x)Px\ & \bf{\stackrel{c}{::}} &\ Pa \\ (\exists x)Px\ & \bf{\stackrel{c}{::}} &\ Pa \end{split}$$

• For more than one thing in the universe

$$ \begin{split} (x)Px\ & \bf{\stackrel{c}{::}} &\ Pa \wedge Pb \wedge ... \wedge Pn &\ \bf{::}\ &\ \bigwedge_{i=a}^n Pi \\ (\exists x)Px\ & \bf{\stackrel{c}{::}} &\ Pa \vee Pb \vee ... \vee Pn &\ \bf{::}\ &\ \bigvee_{i=a}^n Pi \end{split}$$

• For more complex statements

$$ \begin{split} (x)(Px \supset Qx)\ &\ \bf{\stackrel{c}{::}}\ &\ [(Pa \supset Qa) \wedge (Pb \supset Qb) \wedge (Pc \supset Qc)] \\ (\exists x)(Px \wedge Qx)\ &\ \bf{\stackrel{c}{::}}\ &\ [(Pa \wedge Qa) \vee (Pb \wedge Qb) \vee (Pc \wedge Qc)] \end{split}$$

• The method for proving an argument invalid consists in translating the premises and conclusion into singular statements and then testing the result with an indirect truth table
  • First a universe of one is tried. If it is possible for the premises to be true and the conclusion to be false in this universe, the argument is immediately identified as invalid. If, on the other hand, a contradiction results from this assumption, a universe of two is tried. If, in this second universe it is possible for the premises to be true and the conclusion false, the argument is invalid. If not, a universe of three is tried, and so on
• A theorem was proved to the effect that an argument that does not fail in a universe of $2^n$ members, where $n$ designates the number of different predicates, is valid

8.6 Relational Predicates and Overlapping Quantifiers

• Monadic predicates (one-place predicates): Used to assign an attribute to individual things
• A relational predicate (or relation) is a predicate that is used to establish a connection between or among individuals

Translating Relational Statements

• Relations are symbolized like other predicates except that two lowercase letters representing the two related individuals, are written to the immediate right of the uppercase letter representing the relation
• The order in which the lowercase letters are listen often makes a difference
• When two quantifiers of the same sort appear adjacent to each other, the order in which they are listed is not significant
• When different quantifiers appear adjacent to each other, the order does make a difference
• Universal quantifiers go with implications and existential quantifiers go with conjunctions
• Every variable must be bound by some quantifier

8.7 Identity

Simple Identity Statements

• In simple identity statements, the identity symbol controls only the letters to its immediate left and right

Numerical Statements

• There are three types of numerical statements:
  • those that assert a property of at most n items
  • those that assert a property of at least n items
  • those that assert a property of exactly n items
• To translate "at most n" is to say that, if there are n + 1 items that have the stated property, then at least one of them is identical to at least one of the "others"
• To translate "at least" statements we need to use existential quantifiers
• The number of quantifiers must be equal to the number of items asserted
• To ensure that there are at least two distinct items, we must conjoin the assertion that x and y are not identical
• A statement about exactly n items can be seen to be the conjunction of a statement about at least n items and a statement about at most n items

Definite Descriptions

• Definite descriptions are like names in that they identify only one thing, but unlike names they do so by describing a situation or relationship that only that one thing satisfies
• A statement that incorporates a definite description asserts three things:
  • an item of a certain sort exists
  • there is only one such item
  • that item has the attribute assigned to it by the statement

9 Analogy and Legal and Moral Reasoning

9.1 Analogical Reasoning

• Analogical reasoning is reasoning that depends on a comparison of instances
• If the instances are sufficiently similar, the decision reached in the end is usually a good one; but if they are not sufficiently similar, the decision may not be good
• Analogical arguments are closely related to generalizations
• In any argument from analogy, the items that are compared are called analogues
  • Primary analogue: An analogue mentioned in the premise
  • Secondary analgoue: An analogue mentioned in the conclusion
• Principles that are useful for evaluating most arguments from analogy:
  • The relevance of the similarities shared by the primary and secondary analogues: The more relevant the similarities between primary and secondary analogues, the stronger the argument
  • The number of similarities: The greater the number of similarities between primary and secondary analogues, the stronger the argument
  • The nature and degree of disanalogy: Differences between primary and secondary analogues are disanalogies, which can either strenghten or weaken the argument
  • The number of primary analogues: The greater the number of similar primary analogues, the stronger the argument; dissimilar primary analogues are counteranalogies, which weaken the argument
  • The diversity among the primary analogues: The more diverse the primary analogues, the stronger the argument
  • The specificity of the conclusion: The more specific the conclusion, the weaker the argument

9.2 Legal Reasoning

• An essential feature of the English system is its dependence on precedent
  • According to the requirement of precedent, similar cases must be decided similarly
• Many of the arguments that occur in law are arguments from analogy
• Analogical arguments that occur in law differ from the farily simple analogies in at least two important ways:
  • Modes of similarity between cases are often the result of highly creative thinking by lawyers and judges, and the relevance of these similarities to the proposed conclusion is nearly always debatable
  • Primary analogues in law do not all have equal weight

10 Causality and Mill's Methods

10.1 "Cause" and Necessary and Sufficient Conditions

• The word "cause" can have one of three different meanings:
  • Sufficient condition
  • Necessary condition
  • Sufficient and necessary condition
• If we are trying to prevent a certain phenomenon from happening, we usually search for a cause that is a necessary condition, and if we are trying to produce a certain phenomenon, we usually search for a cause that is a sufficient condition
• An important point is that whenever an event occurs, at least one sufficient condition is present and all the necessary conditions are present
• The conjunction of the necessary conditions is the sufficient condition that actually produces the event
• A is a sufficient condition for B: A's occurrence requires B's occurrence
• A is a necessary condition for B: B's occurrence requires A's occurrence
• A is not a sufficient condition for B: A is present when B is absent
• A is not a necessary condition for B: A is absent when B is present

10.2 Mill's Five Methods

• Five methods for identifying causal connections between events
  • Method of agreement
  • Method of difference
  • Joint method of agreement and difference
  • Method of residues
  • Method of concomitant variation

Method of Agreement

• Consists in a systematic effort to find a single factor that is common to several occurrences for the purpose of identifying that factor as the cause of a phenomenon present in the occurrences
• A condition is not necessary for the occurrence of a phenomenon if that condition is absent when the phenomenon is present
• The strength of the argument depends on the nonoccurrence of other possibilities
• A conclusion reached by the method of agreement has limited generality. It applies directly only to those occurrences listed, and only indirectly, through a second inductive inference, to others

Method of Difference

• Consists in a systematic effort to identify a single factor that is present in an occurrence in which the phenomenon in question is present, and absent from an occurrence in which the phenomenon is absent
• The method is confined to investigating exactly two occurrences, and it identifies a cause in the sense of a sufficient condition
• The conclusion yielded by the method of difference is only probable, however, even for the one occurrence to which it directly applies

Joint Method of Agreement and Difference

• Consists of a systematic effort to identify a single condition that is present in two or more occurrences in which the phenomenon in question is present and that is absent from two or more occurrences in which the phenomenon is absent
  • In addition, the condition must never be present when the phenomenon is absent nor absent when the phenomenon is present
• When any of these methods is used as a basis for subsequent inductive generalization, the strength of the conclusion is proportional to the number of occurrences that are included

Method of Residues

• Consists of separating from a group of causally connected conditions and phenomena those strands of causal connection that are already known, leaving the required causal connection as the "residue"

Method of Concomitant Variation

• Identifies a causal connection between two conditions by matching variations in one condition with variations in another
• The existence of a mere correlation between two phenomena is never sufficient to identify a causal connection
• The method of concomitant variation is useful when it is impossible for a condition to be either wholly present or wholly absent, as was required for the use of the first three of Mill's methods

10.3 Mill's Methods and Science

• The method of difference is virtually identical to the method of controlled experiment
• A controlled experiment is one that involves two groups of subjects: an experimental group and a control group
  • The experimental group includes the subjects that receive a certain treatment
  • The control group includes the subjects that do not receive the treatment but are otherwise subjected to the same conditions as the experimental group
• The purpose of the method of difference is to determine whether a preselected condition is the cause of a phenomenon
• The purpose of the joint method is to determine which condition, among a selected class of conditions, is a cause of a phenomenon

11 Probability

11.1 Theories of Probability

• Three distinct theories about probability:
  • The classical theory
  • The relative frequency theory
  • The subjectivist theory
• According to the classical theory, the probability of an event A is given by the formula $P(A) = \frac{f}{n}$, where $f$ is the number of favorable outcomes and $n$ is the number of possible outcomes
• Two assumptions are involved in computing probabilities and betting odds according to the classical theory:
  • All possible outcomes are taken into account
  • All possible outcomes are equally probable (principle of indifference)
• The relative frequency theory depends on actual observations of the frequency with which certain events happen
• The probability of an event A is given by the formula $P(A) = \frac{f_o}{n_o}$, where $f_o$ is the number of observed favorable outcomes and $n_o$ is the total number of observed outcomes
• The subjectivist theory of probability rests on the degrees of belief that people have about the occurrence of some event or the truth of some proposition

11.2 The Probability Calculus

• Two preliminary rules of the probability calculus:
  • The probability of an event that must necessarily happen is taken to be 1
  • The probability of an event that necessarily cannot happen is taken to be 0

Restricted Conjunction Rule

• Used to compute the probability of two events occurring together when the events are independent of each other
• Two events are said to be independent when the occurrence of one has no effect on the probability of the other one occurring

$$ P(A\ \mathrm{and}\ B) = P(A) \times P(B)$$

General Conjunction Rule

• Used to compute the probability of two events occurring together whether or not the events are independent
• When the events are independent, the general conjunction rule reduces to the restricted conjunction rule

$$ P(A\ \mathrm{and}\ B) = P(A) \times P(B\ \mathrm{given}\ A)$$

Restricted Disjunction Rule

• Used to compute the probability of either two events occurring when the events are mutually exclusive

$$ P(A\ \mathrm{or}\ B) = P(A) + P(B)$$

General Disjunction Rule

• Used to compute the probability of either of two events whether or not they are mutually exclusive

$$ P(A\ \mathrm{or}\ B) = P(A) + P(B) - P(A\ \mathrm{and}\ B)$$

• If the events are independent, $P(A\ \mathrm{and}\ B)$ is computed using the restricted conjunction rule, and the geenral disjunction formula reduces to

$$ P(A\ \mathrm{or}\ B) = P(A) + P(B) - [P(A) \times P(B)]$$

Negation Rule

• Useful for computing the probability of an event when the probability of the event not happening is either known or easily computed

$$ P(A) = 1 - P(not-A)$$

Bayes's Theorem

• Useful for evaluating the conditional probability of two or more mutually exclusive and jointly exhaustive events
• The conditional probability of an event is the probability of that event happening given that another event has already happened, and it is expressed P(A given B)

$$ P(A_1\ \mathrm{given}\ B) = \frac{P(A_1) \times P(B\ \mathrm{given}\ A_1)}{[P(A_1) \times P(B\ \mathrm{given}\ A_1)] + [P(A_2) \times P(B\ \mathrm{given}\ A_2)]}$$

12 Statistical Reasoning

12.1 Evaluating Statistics

• Five areas that are frequent sources of ambiguity and deception
  • Problems in sampling
  • The meaning of "average"
  • The importance of dispersion in a sample
  • The use of graphs and pictograms
  • The use of percentages for the purpose of comparison

12.2 Samples

• Samples that are not representative are said to be biased samples
• Different considerations enter into determining whether a sample is biased. These considerations include
  • whether the sample is randomly selected
  • the size of the sample
  • psychological factors
• A random sample is one in which every member of the population has an equal chance of being selected
• The larger the sample, th emore closely it replicates the population
• The sampling error is the difference between the relative frequency with which some characteristic occurs in the sample and the relative frequency with which the same characteristic occurs in the population
• If a much lower confidence level should be selected, and if this fact is not disclosed, then the results of a poll could be deceptive - even if the margin of error is stated. The reason for this is that the margin of error would be combined with a low likelihood that it covered any actual discrepancy between the sample and the population
• Psychological factors can also have a bearing on whether the sample is representative
• If the people composing the sample think that they will gain or lose something by the kind of answer they give, their involvement will likely affect the outcome
• Another source of psychological influence is the personal interaction between the surveyor and the respondent
• To prevent this kind of interaction from affecting the outcome, scientific studies are often conducted under "double-blind" conditions in which neither of the surveyor nor the respondent know what the "right" answer is
• If either the organization conducting the study or the people composing the sample have something to gain by the kind of answer that is given, the results of the survey should be regarded as suspect

12.3 The Meaning of "Average"

• In statistics the word "average" is used in three different senses: mean, median, and mode
• The mean value of a set of data is the arithmetical average
  • It is computed by dividing the sum of the individual values by the number of values in the set
• The median of a set of data is the middle point when the data are arranged in ascending order
• The mode is the value that occurs with the greatest frequency

12.4 Dispersion

• Refers to how spread out the data are in regard to numerical value
• Three important measures of dispersion are range, variance, and standard deviation
• Range: the difference between the largest and the smallest values
• Variance or standard deviation: measures how every data point varies or deviates from the mean

12.5 Graphs and Pictograms

• If a graph is to represent an actual situation, it is essential that both the vertical and horizontal axes be scaled
• A pictogram is a diagram that compares two situations through drawings that differ either in size or in the number of entities depicted
• Failure to scale the axes
• Chopping off the bottom of the graph
• Altering the scale of the axes

12.6 Percentages

• The problem stems from the fact that a different base is used for the two percentages
• The fallacy committed by such arguments is a variety of equivocation
• A different kind of fallacy occurs when a person attempts to add percentages as if they were cardinal numbers
• The fallacy committed by such arguments would probably be best classified as a case of missing the point

13 Hypothetical/Scientific Reasoning

13.1 The Hypothetical Method

• An explanation is a kind of expression that purports to shed light on some event
• Hypothesis: an informed conjecture
• Hypothetical reasoning: the reasoning process you use to produce an hypothesis, draw an implication, and test the implication
• The hypothetical method involves four basic stages:
  • Occurrence of a problem
  • Formulating a hypothesis
  • Drawing implications from the hypothesis
  • Testing the implications
• The additional points about hypotheses:
  • A hypothesis is not derived from the evidence to which it pertains but rather is added to the evidence
  • A hypothesis directs the search for evidence
  • Concluding that a hypothesis is proven true by the discovery that one of its implications is true amounts to committing the fallacy of affirming the consequent
• How does the hypothetical reasoning that occurs in science differ from other modes of hypothetical reasoning?
  • Scientific tests always involve the measurement of something, and the results of that measurement are nearly always expressed in the language of mathematics

13.3 The Proof of Hypotheses

• Empirical hypotheses concern the production of some thing or the occurrence of some event that can be observed
• Theoretical hypotheses concern how something should be conceptualized
• Empirical hypotheses are for all practical purposes proved when the thing or event hypothesized is observed
• Theoretical hypotheses are never proved but are only confirmed to varying degrees
  • The greater the number of implications that are found to be correct, the more certain we can be of the hypothesis
• If an implication is found to be incorrect, a theoretical hypothesis can be disproved

13.4 The Tentative Acceptance of Hypotheses

• Adequacy is the extent to which a hypothesis fits the facts it is intended to unify or explain
  • If one hypothesis accounts for a set of data with greater accuracy than another, then that hypothesis is more adequate than the other
  • Does the hypothesis fits the facts?
• Internal coherence is the extent to which the component ideas of a hypothesis are rationally interconnected
  • Are the components ideas interconnected?
• External consistency occurs when a hypothesis does not disagree with other, well-confirmed hypotheses
  • Are there conflicts with other hypotheses?
• Fruitfulness is the extent to which a hypothesis suggests new ideas for future analysis and confirmation
  • Are new ideas suggested for future analysis?

14 Science and Superstition

14.1 Distinguishing Between Science and Superstition

• Both science and superstition involve hypotheses
• The distinction between science and superstition also involves psychological and volitional elements

14.2 Evidentiary Support

• Do books count as evidence? If books are credible reports of measurements expressed mathematically, then they can be accepted as conveyance of evidence
• A kind of evidence that is considered unreliable is anecdotal evidence
• One of the key features of scientific evidence gathering is that an experiment be replicable under controlled conditions
  • Replicability helps ensure that the outcome of the experiment did not result from anything peculiar to one certain experimenter operating at a single place and time
  • The controlled conditions are designed to eliminate the influence of extraneous factors
• A defect found in superstitious hypotheses is that they are often framed so vaguely that it is virtually impossible to provide any kind of unequivocal confirmation
• The hypotheses of science are often framed in the language of mathematics, or they can at least be translated into some mathematical expression
• Closely related to the problem of vagueness is the breadth with which a hypothesis is framed. If a hypothesis is framed so broadly and comprehensively that even condictory evidence serves to confirm it, then the hypothesis is not really confirmed by anything
• Any genuinely scientific hypothesis must be framed narrowly enough so that it forbids certain things from happening
  • The hypothesis must be falsifiable
• The problem with ad hoc modifications is that their purpose is to shore up a failure of evidentiary support in the original hypothesis
  • As more and more modifications are added, the hypothesis becomes self-supporting; it becomes a mere description of the phenomenon it is supposed to explain
• Another problem with ad hoc modifications is that they result in hypotheses that are so complicated that applying them becomes difficult
• One of the surest ways to know that our hypotheses are supported by evidence is that they lead to predictions that turn out to be true

14.3 Objectivity

• Our beliefs about the world are objective to the extent that they are unaffected by conditions peculiar to the experiencing subject
• The chief emotions that give rise to superstitious beliefs are fear and anxiety, and they are often reinforced by a disposition to fantasy and mental laziness
• In the natural sciences, much if not most observation occurs through instruments. The results are then recorded on relatively permanent media such as photographic film, or digital storage devices
• In the social sciences, techniques such as double-blind sampling and statistical analysis of data insulate the observer from the outcome of the experiment

14.4 Integrity

• Our efforts to understand the world in which we live have integrity to the extent that they involve honesty in gathering and presenting evidence and honest, logical thinking in responding to theoretical problems that develop along the way
• Most forms of superstition involve elements of dishonesty in gathering evidence or a failure of logic in responding to theoretical problems
• The most severe lack of integrity arises when the evidence is faked
• Any absence of a causal connection is a defect in coherence, because it signals the lack of a connection between ideas functioning in a hypothesis

See also

References

• Hurley, Patrick. A concise introduction to logic. Nelson Education, 2014.