NotesIntroduction to logic

Name: Amit Chaudhary

Semester: 4th

13Victoria Int’l College

What is logic?Logic is the principles and methods used to distinguish correct from incorrect reasoning. It is the science of valid inference. It is the rational way of drawing or establishing conclusions. It is the foundation of all mathematics, science, and reasoning. Logic helps us to identify good argument and understand why they are good. Similarly, it guides us to identify bad argument and to understand why they are bad. People use logic in every situation. It is used to solve problems, troubleshoot, and helps to find mathematical sums. Logic is the way of learning to think and communicate clearly and coherently. In conclusion, logic is the justification of our beliefs and the judgments.

Example:– The increase in CO2 has caused global climate change.– Humans are responsible for the increase in CO2.– Humans are responsible for global climate change.

ArgumentsLogic and critical thinking are concerned with arguments. An argument is a collection of propositions, one of which (the conclusion) is supported by the others (premises). An argument might also be called an inference or reasoning. An argument is a set of statements connected by a special relationship of justification; the statements provide evidence for the main claim. Arguments seek to prove a point; to establish truth of a claim on basis of other claims. An argument presents logical reasons and evidence to support a viewpoint. The best arguments are ones with true premises that provide the strongest possible support for their conclusion. The strongest possible support for a conclusion is called validity.

Inductive and Deductive Arguments

INDUCTIVE - reaches a general conclusion from observed specifics.

If we move from specific premise to general conclusion then it is called inductive argument. It is associated with informal logic.

Example:

– The apple is from the cartoon. – The apple is very testy.– So, the apples in the cartoon are testy.

DEDUCTIVE - begins with a major premise and moves toward a more specific statement or minor premise.

If we move from general premises to specific conclusion then it is called deductive argument. It is associated with formal logic.

– All the apples in this cartoon are testy.

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Specific

General

General

Specific

– This apple is from that cartoon.– So, the apple is testy.

PropositionsArguments are made up of propositions.

A proposition is a statement that can be expressed as a declarative sentence. It asserts that something is or is not the case, and thus can be true or false.

While propositions can be expressed as declarative sentences, they are not to be equated with declarative sentences. A sentence is a set of symbols (typically visual or auditory); in certain cases, the proposition is what those symbols mean. Sentences in different languages, or even different sentences in the same language, may express the same proposition.

Propositions have different levels of complexity. We can get a sense of these levels of complexity by distinguishing between simple and compound propositions.

A proposition is compound if and only if it is a:

(a) A disjunctive proposition: an ‘or’ statement such that the whole statement is true if and only if at least one of its component statements are true.

(b) A conjunctive proposition: an ‘and’ statement such that the whole statement is true if and only if both of its component statements are true.

(c) A hypothetical proposition: an ‘if-then’ statement that is false if and only if its antecedent is true and its consequent is false.

If a hypothetical is of the form “If p then q,” p is the antecedent and q is the consequent.

If a statement is not compound, then it is simple.

For example,

All rabbits are mammals.Bugs Bunny is a rabbit. Bugs Bunny is a mammal.

Premises and conclusionPremises:

Premises are assertions that, when joined together, will lead the reader to the conclusion. The most important part of any premise is that your audience will accept it as true. If your audience rejects even one of your premises, they will likely also reject your conclusion, and your entire argument will fall apart. When constructing premises, it is essential to

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consider your audience. When you know your audience, you also know which assertions they will accept and which they will question.

(a) For example, consider the following assertion: “Because greenhouse gases are causing the atmosphere to warm at a rapid rate...” Is this a solid premise? It depends on your audience. If your readers are members of an environmental group, they will accept this premise without qualms. If your readers are oil company executives, they may reject this premise and your conclusions.

(b) To construct solid premises, you need to consider the rationales and beliefs of your opponents. What are the “givens” you accept that they do not? What beliefs lead them to reject those “givens”? Where can two sides of an argument find common ground? That is where you will find effective premises to reach your conclusion.

Conclusions: A conclusion can be any assertion that your readers will not readily accept. A conclusion must have at least one premise supporting it. The thesis of an argumentative paper will always contain a conclusion, with the main points or body paragraphs acting as premises that lead the reader to accept it.

(a) Let’s revisit the previous example, but change the wording slightly: “Therefore, greenhouse gases are causing the atmosphere to warm at a rapid rate.” How did changing the first word in the sentence change the function of the sentence? The meaning of the sentence stays the same, but how we're using it in our argument has changed. It is now in the form of a conclusion.

(b) You may have heard one of your thesis statements or main arguments described as “too obvious.” This usually means that your readers already accept your conclusion without any need for argument. You must also consider your audience when you are constructing your conclusions.

Recognizing ArgumentsArguments are composed of one or more premises and a conclusion. Premises are statements offered as reasons for accepting another statement. A conclusion is a statement supported by reasons.Distinguishing premises from conclusions is a skill that requires both practice and close attention to the nuances of language. Here are some tips that will help you separate premises from conclusions:

1. Look for premise indicators--words like because, since, for, and given that--that provide clues when premises are being offered.

2. Look for conclusion indicators--words like therefore, thus, hence, and so--that provide clues when conclusion indicators are being offered.

3. If the passage contains no indicator words, try these two strategies:

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a. Ask yourself, "What claim is the writer or speaker trying to prove?" That claim will be the conclusion.

b. Try putting the word "therefore" before each of the statements in turn. The statement it fits best will be the conclusion.

Example:1. A good society treasures its dissidents and mavericks because it

needs the creative thinking that produces new hypotheses, expanded means, a larger set of alternatives, and, in general, the vigorous conversation induced by fresh ideas. (Nel Noddings, Philosophy of Education, 1995)

Premise: A good society needs the creative thinking that that produces new hypotheses, expanded means, a larger set of alternatives, and, in general, the vigorous conversation induced by fresh ideas.

Conclusion: A good society treasures its dissidents and mavericks.

2. Make a will. Otherwise, the state will determine who gets your stuff. (Andrew Tobias, "Isn't It Time You Faced the Future?" 2001)

Premise: If you don't make a will, the state will determine who gets your stuff.Conclusion: You ought to make a will.The word otherwise often functions--as it does here--as premise indicator.

3. With what group do I belong? I am with those who would be pleased to be refuted if I should say anything that is not true, and pleased to be the refuter of anyone who should say anything that is not true--more pleased, in fact, to be refuted than to refute. I think that's a greater good, you see, insofar as it's a greater good to be relieved of a great evil than to relieve another of the same. (Socrates, in Plato's Gorgias)

Premise: It is a greater good to be relieved of a great evil than to relieve another of the same.

Conclusion: It is a greater good to be refuted than to refute.

In this passage, the premise indicator "insofar as" helps us to identify the premise.

Notice that the first two sentences aren't strictly part of the argument. Their function, instead, is to provide background or contextual information necessary to understand the argument.

4. Good sense is of all things in the world the most equally distributed, for everybody thinks himself so abundantly provided with it that

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even those most difficult to please in all other matters do not commonly desire more of it than they already possess.

Premise: Even those most difficult to please in all other matters do not commonly desire more good sense than they already possess.

Conclusion: Good sense is of all things in the world the most equally distributed.

5. Forbear to judge, for we are sinners all.Premise: We are sinners all.Conclusion: You should not judge.

6. Today’s first year college students have lived the external appearances of an adult life for many more years than their counterparts 50 years ago did. [Therefore,] what we have traditionally associated with the intellectual awakening during childhood years must now occur in the high school.

Premise: Today’s first year college students have lived the external appearances of an adult life for many more years than their counterparts 50 years ago did.

Conclusion: What we have traditionally associated with the intellectual awakening during childhood years must now occur in the high school.

7. Thomas Aquinas argued that human intelligence is a gift from God and therefore to ‘apply human intelligence to understand the world is not an affront to God, but is pleasing to him.’

Premise: Human intelligence is a gift from God.

Conclusion: To apply human intelligence to understand the world is not an affront to God, but is pleasing to him.

8. Standardized tests have a disparate racial and ethnic impact; white and Asian students score, on average, markedly higher than their black and Hispanic peers. This is true for fourth-grade tests, college entrance exams, and every other assessment on the books. If a racial gap is evidence of discrimination, then all tests discriminate.

Premise: White and Asian students score markedly higher than black and Hispanic students on fourth-grade standardized tests, college entrance exams, and every other assessment on the books.

Premise: Standardized tests have a disparate racial and ethnic impact.

Conclusion: If a racial gap is evidence of discrimination, then all tests discriminate.

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Argument vs Explanation

If you want to communicate with someone in order to get your point across, you will probably end up using many definitive statements backed up by factual statements. These statements can be based on observation, established facts, or proofs. Without even consciously thinking about it, we are always using this methodology in our conversations. These conversations are therefore riddled with arguments and explanations. While the two terms are often erroneously used interchangeably and explanations can be used to bolster arguments, there are many differences between arguments and explanations.

1. Definition of an Argument and an Explanation

Argument – Argument has a number of different definitions. Essentially, it is a line of logic that is presented in order to support the veracity of a statement. Argument has combative connotations, but an argument does not have to be belligerent.

Explanation – Explanation is used to clarify and explicate a statement. Its aim is to make the listener understand the statement rather than persuade him to accept a certain point of view.

2. Example of an Argument and an Explanation

Argument - one person wants to convince the other person that it is going to snow tomorrow. He will cite predictions from the weather station, as well as the clouds visible on the horizon, the damp chill in the air, and the squirrels furiously hiding their nuts.

Explanation- one both people agree is it going to snow tomorrow because, they say, there is a cold front coming in and the air feels damp.

In both cases, the example of snow is used, but note that the argument is trying to convince someone of the truth of their statement, whereas with the explanation, it is not a matter of if the statement is true, but why it is true.

3. Uses of Arguments and Explanations

Arguments- Arguments are used in a variety of professional and academic applications. For instance, a debate club will take on both sides of an argument and strive to prove each one is right. Arguments are also used by lawyers to convince the jury of the defendant’s guilt or innocence. Diplomats will approach a negotiating table with a certain argument in mind. Entrepreneurs will present potential backers with an argument in support of their business model.

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Explanations- Explanations are used all the time in the classroom to put across new items to students. Giving directions is a form of explanation. You will also find explanations included with most new purchases, especially those with some assembly required. When the aforementioned entrepreneur is presenting an argument about his business model, he may be asked to explain how it all works.

In summary, then, an argument is a piece of reasoning in which the reason is intended to provide evidence for accepting a doubted conclusion. An explanation is a piece of reasoning in which the reason is intended to provide a cause for an already accepted conclusion.

It is sometimes said that rational inquiry aims at two things: knowledge and understanding. We can now say that argument and explanation are the reasoning tools that we use to accomplish these two goals. Argument attempts to establish knowledge by giving evidence that reduces doubt. Explanation attempts to establish understanding by supplying causal connections between accepted facts.

Summary:

1. Arguments and explanations are both used to get the point across when speaking or writing.

2. Arguments are persuasive and seek to make people understand that something is true, whereas explanations start with the assumption of truthfulness and tell why or how the statement has come into being.

3. Both arguments and explanations have wide application in education and business, but arguments are used for persuasion and explanations are used for clarification.

Validity of arguments

An argument is valid if and only if the truth of its premises entails the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid. Under such conditions it would be self-contradictory to affirm the premises and deny the conclusion. The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a logical consequence of its premises.An argument that is not valid is said to be "invalid".An example of a valid argument is given by the following well-known syllogism (also known as modus ponens):

All men are mortal.Socrates is a man.Therefore, Socrates is mortal.

What makes this a valid argument is not that it has true premises and a true conclusion, but the logical necessity of the conclusion, given the two premises.

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The argument would be just as valid were the premises and conclusion false. The following argument is of the same logical form but with false premises and a false conclusion, and it is equally valid:

All cups are green.Socrates is a cup.Therefore, Socrates is green.

No matter how the universe might be constructed, it could never be the case that these arguments should turn out to have simultaneously true premises but a false conclusion. The above arguments may be contrasted with the following invalid one:

All men are mortal.Socrates is mortal.Therefore, Socrates is a man.

In this case, the conclusion does not follow inescapably from the premises. All men are mortal, but not all mortals are men. Every living creature is mortal; therefore, even though both premises are true and the conclusion happens to be true in this instance, the argument is invalid because it depends on an incorrect operation of implication. Such fallacious arguments have much in common with what are known as howlers in mathematics.A standard view is that whether an argument is valid is a matter of the argument's logical form. Many techniques are employed by logicians to represent an argument's logical form. A simple example, applied to two of the above illustrations, is the following: Let the letters 'P', 'Q', and 'S' stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:All P are Q.S is a P.Therefore, S is a Q.Similarly, the third argument becomes:All P are Q.S is a Q.Therefore, S is a P.

An argument is formally valid if its form is one such that for each interpretation under which the premises are all true, the conclusion is also true. As already seen, the interpretation given above (for the third argument) does cause the second argument form to have true premises and false conclusion (if P is a not human creature), hence demonstrating its invalidity.

Validity of statementsA statement can be called valid, i.e. logical truth, if it is true in all interpretations.

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DIAGRAMMING, SUMMARIZING, AND PARAPHRASING ARGUMENTSI. Analyzing ArgumentsA. There are two techniques for analyzing arguments.1. Paraphrasing arguments involves “setting for the arguments (the one in prose) and the propositions in clear language and logical order.”12. Diagraming arguments involves the use of “spatial relations in two dimensions”2 for the purposes of clarifying both the content and general flow of an argument in prose.

Paraphrasing Arguments• A paraphrase is fundamentally about making the argument of a passage

more easily recognizable, thus a good paraphrase will:– Use premise- and conclusion-indicators– Clearly answer the question-and-answer test– Use common argument forms– Balance faith and charity– Add important hidden argument-parts, including contextual clues

Why paraphrase?• Paraphrasing is the basis of note-taking.

– In an overwhelming majority of cases, the only notes you need to take are those that pertain to an author’s argument.

– Note-taking is the most important transition from reading to writing.

• Paraphrasing is the basis of clear writing.– If you can write good paraphrases, you can write clear sentences,

paragraphs, and papers.Other paraphrasing strategies: Overview

• Order propositions in an intuitive manner• Simplify the language of the original text• Eliminate irrelevant propositions• Provide uniformity of terms and language• Plus two more to be discussed next class…

– Identify important intermediate conclusions– Distinguish independent from dependent premises

• A good paraphrase should list the premises in an order which makes the structure of the argument clear, minimally in standard form;

– Standard form:• Premise 1• Premise 2• …• Premise n • Conclusion

A good paraphrase should simplify the language of the original text, by trading out more elliptical and counterintuitive language for more concrete and concise language.

A good paraphrase should eliminate irrelevant propositions. A proposition is irrelevant if it is neither a premise nor a conclusion of an argument

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What does paraphrasing accomplish? Paraphrasing helps clarify the argument in prose. (we’ve already noted

this) Paraphrasing helps one discern the nature of the inference the argument

is using. Paraphrasing can also help one see what premises were assumed, or not

explicitly stated in the prose-style presentation of the argument. Paraphrasing also helps prep one for a proper evaluation of the

argument.

Examples:

1. The Pistons did not lose because of the lack of ability. They are an all-around better team. They lost because of the law of averages. They will beat the Spurs every two times out of three. When you examine the NBA Finals, that is exactly how they lost the seventh because that would have been three out of three. The Spurs will beat the Pistons one out of three. It just so happens that, that one time was the final game, because the Pistons had already won two in a row.

Argument 1:• Either the Pistons lost because they are inferior to the Spurs

or because of the law of averages.• The Pistons are better than the Spurs.• The Pistons lost the NBA Finals because of the law of

averages.Argument 2:

• The Pistons will beat the Spurs 2 of 3 times; the Spurs will beat the Pistons 1 of 3 times.

• The Pistons had won Games 5 and 6 of the Finals—two in a row, so if they had won the final game they would have won 3 of 3.

• The Pistons lost the NBA Finals because of the law of averages.

2. Racially diverse nations tend to have lower levels of social support than homogeneous ones. People don’t feel as bound together when they are divided on ethnic lines and are less likely to embrace mutual support programs. You can have diversity or a big welfare state. It’s hard to have both.

If a nation is diverse, then people don’t feel bound together. If people don’t feel bound together, then they are less likely to embrace

large scale social programs. If a nation is diverse, then it is less likely to embrace large scale social

programs.

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Diagraming Arguments“To exhibit the structure of an argument it is sometimes useful to represent it graphically, to diagram it.”5Sometimes a diagram is more useful when all of the premises and conclusions are stated explicitly, but in a way that is rather complicated.

Steps toward proper diagraming:

Read the passage carefully. Identify each claim and number them. Provide missing parts if needed. Determine relation of claims and diagram.

a) Draw in the circled numbers that stand in for coordinate premises at the same height on your page.6

b) Draw in the conclusion of the argument below the circled numbers representing the coordinate premises.

c) If the premises of the argument both independently support the conclusion, all one needs to do next is draw arrows starting from each premise to the conclusion (see page 23 in the textbook).

d) If the premises of the argument support the conclusion only when understood in light of one another (as a conjunction perhaps), then one needs to draw a horizontal line which extends out to each premise that jointly supports the conclusion, and then one should draw an arrow from the middle of that horizontal line to the conclusion in the diagram.

e) If there is more than one conclusion, and several premises, sometimes both conclusions may be supported by the plurality of premises in which case your diagram will look like that one on page 25 of the textbook.f) Because “…the same proposition can serve as a premise, where it occurs as an assumption in an argument; or as a conclusion, where it is claimed to follow from other propositions assumed in an argument. ‘Premise’ and ‘conclusion’ are always relative terms.”7

g) Occasionally, one will encounter paragraphs in which there are multiple arguments, and yet there are stand-alone premises which support differing conclusions. In such a case, one’s diagram should look like that on page 27 (see the bottom of the page).

Examples:

1. Since Mary visited a realtor and her bank’s mortgage department, she must be planning on buying a home.

Step 1. Number each statementAnd note each indicator word.

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Since (1) Mary visited a realtor and (2) her bank’s mortgage department, (3) she must be planning on buying a home.

Step 2. Which of the claims is the conclusion? Which are premises?

Step 3. Use arrows to represent the intended relationship between the claims.

In this case the premises are independent. Even though the combined force of both premises makes the argument stronger, either premise could stand alone in supporting the conclusion.

2. Sandra can’t register for her classes on Wednesday. After all, Sandra is a sophomore and sophomore registration begins on Thursday.

Step #1. Identify each claim and note any indicator words that might help identify premise(s) and conclusion(s).

(1) Sandra can’t register for her classes on Wednesday. After all, (2) Sandra is a sophomore and (3) sophomore registration begins on Thursday.

Step #2. Use arrows to show the relationships between the claims in the argument.

These are linked premises since both (in conjunction) are necessary to prove the conclusion.

3. Pool maintenance can cost hundreds of dollars a year and we really don’t have that kind of money. So, I don’t think we should put a pool in this summer. Besides, pools pose a real drowning danger to small children.

Step #1. The first task is to analyze the argument. Decide what the various claims are and begin to decide which premises are and which conclusions are. Number the claims and note any indicator words.

(1) Pool maintenance can cost hundreds of dollars a year and (2) we really don’t have that kind of money. So, (3) I don’t think we should put a pool

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(1)

(2) (3)(+)

(3)

(2)(1)

in this summer. Besides, (4) pools pose a real drowning danger to small children.

Step #2. Use arrows to represent the argument

3. You’ve often complained that mainstream television doesn’t have quality programming, so I think you should support public broadcasting. Besides, you watch PBS all the time and fair is fair. Since support means money, you should write a check to PBS immediately.

(1)You’ve often complained that mainstream television doesn’t have quality programming, so (2) I think you should support public broadcasting. Besides (3), you watch PBS all the time and (4) fair is fair. Since (5) support means money, (6) you should write a check to PBS immediately.

Reasoning:Reasoning is the set of processes that enables us to go beyond the information given. Reasoning is just the process of making certain statements, which we call reasons, in support of other statements, which we call conclusions.

Functions of Language

The formal patterns of correct reasoning can all be conveyed through ordinary language, but then so can a lot of other things. In fact, we use language in many different ways, some of which are irrelevant to any attempt to provide reasons

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Premises 1 and 2 are linked. While premise 1 could stand alone, premise 2 can’t.

Premise 4 is independent.It could be offered alone as support for the conclusion.

(1) (2)

(3)

(+)

(4)

(3) (+) (4)

(2)

(1)

(+)

(6)

(5)

for what we believe. It is helpful to identify at least three distinct uses of language:

Informative use of language:The informative use of language involves an effort to communicate some content. When I tell a child, "The fifth of May is a Mexican holiday," or write to you that "Logic is the study of correct reasoning," or jot a note to myself, "Jennifer—555-3769," I am using language informatively. This kind of use presumes that the content of what is being communicated is actually true, so it will be our central focus in the study of logic.

Expressive use of languageAn expressive use of language, on the other hand, intends only to vent some feeling, or perhaps to evoke some feeling from other people. When I say, "Friday afternoons are dreary," or yell "Ouch!" I am using language expressively. Although such uses don't convey any information, they do serve an important function in everyday life, since how we feel sometimes matters as much as—or more than—what we hold to be true.

Directive uses of languageFinally, directive uses of language aim to cause or to prevent some overt action by a human agent. When I say "Shut the door," or write "Read the textbook," or memo myself, "Don't rely so heavily on the passive voice," I am using language directively. The point in each of these cases is to make someone perform (or forswear) a particular action. This is a significant linguistic function, too, but like the expressive use, it doesn't always relate logically to the truth of our beliefs.

Notice that the intended use in a particular instance often depends more on the specific context and tone of voice than it does on the grammatical form or vocabulary of what is said. The simple declarative sentence, "I'm hungry," for example, could be used to report on a physiological condition, or to express a feeling, or implicitly to request that someone feed me. In fact, uses of two or more varieties may be mixed together in a single utterance; "Stop that," for example, usually involves both expressive and directive functions jointly. In many cases, however, it is possible to identify a single use of language that is probably intended to be the primary function of a particular linguistic unit. British philosopher J. L. Austin developed a similar, though much more detailed and sophisticated, nomenclature for the variety of actions we commonly perform in employing ordinary language. You're welcome to examine his theory of speech acts in association with the discussion in your textbook. While the specifics may vary, some portion of the point remains the same: since we do in fact employ language for many distinct purposes, we can minimize confusion by keeping in mind what we're up to on any particular occasion. Literal and Emotive MeaningEven single words or short phrases can exhibit the distinction between purely informative and partially expressive uses of language. Many of the most common words and phrases of any language have both a literal or descriptive meaning that refers to the way things are and an emotive meaning that expresses some (positive or negative) feeling about them. Thus, the choice of which word to use

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in making a statement can be used in hopes of evoking a particular emotional response. This is a natural function of ordinary language, of course. We often do wish to convey some portion of our feelings along with information. There is a good deal of poetry in everyday communication, and poetry without emotive meaning is pretty dull. But when we are primarily interested in establishing the truth—as we are when assessing the logical merits of an argument—the use of words laden with emotive meaning can easily distract us from our purpose.

Kinds of Agreement and DisagreementIn fact, an excessive reliance on emotively charged language can create the appearance of disagreement between parties who do not differ on the facts at all, and it can just as easily disguise substantive disputes under a veneer of emotive agreement. Since the degrees of agreement in belief and attitude are independent of each other, there are four possible combinations at work here:

Agreement in belief and agreement in attitude: There aren't any problems in this instance, since both parties hold the same positions and have the same feelings about them.

Agreement in belief but disagreement in attitude: This case, if unnoticed, may become the cause of endless (but pointless) shouting between people whose feelings differ sharply about some fact upon which they are in total agreement.

Disagreement in belief but agreement in attitude: In this situation, parties may never recognize, much less resolve, their fundamental difference of opinion, since they are lulled by their shared feelings into supposing themselves allied.

Disagreement in belief and disagreement in attitude: Here the parties have so little in common that communication between them often breaks down entirely.

It is often valuable, then, to recognize the levels of agreement or disagreement at work in any exchange of views. That won't always resolve the dispute between two parties, of course, but it will ensure that they don't waste their time on an inappropriate method of argument or persuasion.

Emotively Neutral LanguageFor our purposes in assessing the validity of deductive arguments and the reliability of inductive reasoning, it will be most directly helpful to eliminate emotive meaning entirely whenever we can. Although it isn't always easy to achieve emotively neutral language in every instance, and the result often lacks the colorful character of our usual public discourse, it is worth the trouble and insipidity because it makes it much easier to arrive at a settled understanding of what is true.

In many instances, the informal fallacies we will consider next result from an improper use of emotionally charged language in the effort to persuade someone to accept a proposition at an emotional level, without becoming convinced that there are legitimate grounds for believing it to be true.

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