PHIL102 Study Guide

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Study Guide Structure

In this study guide, the sections in each unit (1a., 1b., etc.) are the learning outcomes of that unit. 

Beneath each learning outcome are:

• questions for you to answer independently;
• a brief summary of the learning outcome topic;
• and resources related to the learning outcome. 

At the end of each unit, there is also a list of suggested vocabulary words.

 

How to Use this Study Guide

1. Review the entire course by reading the learning outcome summaries and suggested resources.
2. Test your understanding of the course information by answering questions related to each unit learning outcome and defining and memorizing the vocabulary words at the end of each unit.

By clicking on the gear button on the top right of the screen, you can print the study guide. Then you can make notes, highlight, and underline as you work.

Through reviewing and completing the study guide, you should gain a deeper understanding of each learning outcome in the course and be better prepared for the final exam!

Unit 1: Introduction and Meaning Analysis

1a. Distinguish between the literal and implied meanings of sentences

• What is literal meaning?
• What is conversational implicature?
  
• Why is the distinction between the literal and implied meanings of sentences important to clear thinking?

A sentence's grammatical structure and the conventional meanings assigned to the words used constitute literal meaning. More specifically, literal meaning is a property of linguistic expression; to understand the meaning of that expression in literal terms is to understand the common meaning of each word in the expression, along with that expression's grammatical structure. The common meaning of a word is typically found in a dictionary or colloquial usage. Grammatical structure, in turn, is a sentence's arrangement of elements – words, clauses, or phrases.

A sentence's literal meaning is often only a part of a more sophisticated communication process. In various exchanges, a sentence's literal meaning is not intended as the exclusive communication. A sentence takes on new meaning in the larger context of a conversation. Conversational implicature is the term for the implicit meaning a sentence conveys in the context of a conversation.

The distinction between literal and implied meaning, then, is between what a sentence asserts and what that sentence implies in the larger context of a conversation. It is important to understand this distinction, as confusion often results from glossing over it. Moreover, the truth or falsity of the sentence implied, but not explicitly stated, isn't properly determined until made explicit.

Think about it this way, which brings us closer to the core of a course like this, namely inference-making: The sentence uttered is, in the context of a conversation, often used as an indication of an inference implied but not stated. Let's take one of the examples from the text (Literal Meaning): Upon being asked if she wants to go to the movies, Lily replies, "I am very tired". Given the context of the conversation, we conclude that Lily doesn't want to go. Absent the context, there is no specific inference to draw. Suppose, apropos of nothing, Lily asserts, "I am very tired", we'd have a hard time knowing what inference to draw. The truth or falsity of the implied sentence, "Lily does not want to go to the movies", depends on the evidence we have. The reasoning could look like this: If Lily wants to go to the movies, then she isn't very tired. But she is very tired. So, she doesn't want to go to the movies".

To review, see Meaning Analysis: Literal Meaning.

 

1b. Describe reportive, stipulative, precising, and persuasive definitions, and apply them to real-world scenarios

• What is a reportive definition?
• What is a stipulative definition?
• What is a precising definition?
• What is a persuasive definition?

How we define words contributes to meaning, clarity, and good reasoning. Consider, for example, the different ways measurements are defined, such as inches, milliliters, circumference, or area. Four of the most common definition types are reportive, stipulative, precising, and persuasive. Let's take a look at each.

A reportive definition simply reports the current meaning of a term. Also known as a lexical definition, reportive definitions provide us with the correct usage of a term. Consequently, when defining a technical term or difficult to define words, a properly reportive definition goes beyond the limitations of what a dictionary allows. That's because a standard dictionary simply cannot capture the scope or technical meaning of every term. For example, while "cat" is somewhat successfully defined as "a four-legged furry animal", it's not comprehensive; there are four-legged furry animals that are not cats. The relation between a term and its definition, in this case, does not tell us exactly what a cat is.

A stipulative definition assigns a new meaning to an existing term. We often stipulate a definition for the purpose of moving forward in a discussion or debate – there is a specific purpose for the definition. It can be an efficient way to facilitate the scope or domain of the communication. So, for example, one can say, "Let's stipulate that 'DOA' means Department on Aging, not Dead on Arrival".

A precising definition does what it says, namely it makes a vague or ambiguous definition more precise, thereby avoiding or clearing up confusion (and erroneous inferences). This precision might be considered combining the reportive and stipulative definitional processes. In a given context, for example, we might want to generate a precising definition of "energy". So, a patient might describe feeling a lack of energy, which could mean fatigue, lethargy, and so forth. A precising definition can help the doctor hone in on a medical explanation.

A persuasive definition is used with the intention of influencing, positively or negatively, typically by way of emotional or figurative language. Typically, persuasive definitions of a term are not the literal meaning. When engaging in argumentation, we should avoid this type of definition.

To review, see Meaning Analysis: Definitions.

 

1c. Describe the criteria for evaluating definitions and sources, and apply them to real-world scenarios

• What are some criteria for evaluating definitions?
• What are some criteria for evaluating sources?

Each type of definition has its own evaluation criteria. So, a reportive definition of a term is good if it accurately reflects the current usage of that term. A reportive definition accurately reflects the current usage of a term if it is neither too broad nor too narrow.

A definition that is too broad includes things it should not. Consider the previous poor definition of "cat" as "a four-legged furry animal". The definition is too broad because it includes animals such as mice. On the other hand, a definition that is too narrow excludes things it should not. For example, "cat" means "lion" excludes other felines. There are some definitions that fail both criteria at once. For example, "cat" is "a furry animal that roars".

Because stipulating a definition means offering a new meaning for a term, a stipulative definition does not fail in the same ways a reportive definition can fail, namely, they cannot be either too broad or too narrow. It can, however, be circular, inconsistent, or obscure.

A circular definition simply defines the term in question with the very term itself. It does not provide information that expands our understanding. To stipulate "painkiller" as "that which kills pain" does not provide new information.

A stipulative definition that is inconsistent presents incompatible ideas. So, defining "cat" as "a creature that meows but is silent", is inconsistent. An obscure stipulative definition is unclear. For example, defining "sprint" as "a short race" is unclear, since "short" is vague.

To review, see Meaning Analysis: Evaluating Definitions.

 

1d. Distinguish between factual disputes and verbal disputes

• What is a factual dispute?
• What is a verbal dispute?

A dispute is a disagreement. Those disputes we tend to think are worth resolving involve facts – determining the facts of the matter. We make claims that we believe are true, that is, that reflect a fact or set of facts. For example, if Person A believes they left their keys on the dining table, but Person B believes the keys were left in the car. Their dispute is factual, specifically, it's a dispute about the location of the keys.

Sometimes, however, we believe we're engaged in a factual dispute, but instead, we're essentially speaking at cross-purposes, which generates linguistic misunderstanding. This happens when the words or phrases we use are vague or ambiguous. Each party has a different understanding of the words or phrases, and so the dispute is not over facts. This is a verbal dispute.

We might go so far as to say there is no dispute at all, since the two parties aren't talking about the same thing! Depending on how each party defines the term in question, the misunderstanding, and so dispute, arises. If at least one of the individuals involved realizes that they need to either stipulate a definition or work on a more precise definition, either the dispute will dissolve, or they'll have a better shot at disagreeing over the facts. So, for example, if Person A asserts, "All humans are equal", and Person B retorts, "No way! DNA is unique", it's clear that each has a different understanding of "equal".

To review, see Meaning Analysis: Verbal Disputes.

 

1e. Define necessary and sufficient conditions, and give examples of each

• What is a necessary condition?
• What is a sufficient condition?
• What are some examples of a necessary condition?
• What are some examples of a sufficient condition?

A necessary condition is what must be the case for something else to obtain. For example, oxygen is a necessary condition for fire. In other words, without oxygen, fire does not obtain.

Oxygen does not guarantee fire, so it is not a sufficient condition. A sufficient condition guarantees an event. For example, earning a 95% on an exam is sufficient for an A (assuming a standard grading scale). Another way to put it is that earning a 95% guarantees an A.

Notice that conditions can be natural (as the oxygen example illustrates) or conventional (as the grade example illustrates). In either case, how we word the relation between necessary and sufficient conditions is important. A standard formulation is the conditional claim ("if…then"). The actual or supposed (or claimed) sufficient condition goes after "if", that is, in the antecedent position. The actual or supposed necessary condition goes after "then", or in the consequent position. Here are examples, some of which exhibit actual conditions, and some of which exhibit supposed conditions:

• If it is a dog, then it is an animal.
  • Being a dog is sufficient for being an animal.
  • We can also say that without "it" being an animal, it can't be a dog.
  • So, the claim that being a dog is a sufficient condition, and that being an animal is a necessary condition.
• If it is an animal, then it is a dog.
• Being an animal does not guarantee that animal is a dog.
• It's also not the case that without "it" being a dog, it's not an animal.
• So, the claim that being an animal is a sufficient condition for being a dog is false, as is the claim that being a dog is a necessary condition for being an animal.
• If it flies, then it's a bird.
• Something that flies may or may not be a bird.
• So, being a flying thing does not guarantee that thing is a bird.
• Not being a bird does not prevent something from flying.
• So, neither the sufficient nor necessary conditions are met in this claim.
• If it's a bird, then it flies.
• Being a bird does not guarantee flight – penguins are birds, but don't fly.
• Similarly, not being a flying thing does not thereby prevent that thing from being a bird.
• So, neither condition is met in this claim.
• If there is fire, then there is oxygen.
• Fire guarantees there is oxygen, as oxygen is necessary for fire.
• Without oxygen, there is no fire.
• Where there is fire, there must be oxygen. Both conditions are met in this claim.
• If there is oxygen, then there is fire.
• Oxygen does not guarantee fire. There are lots of places where there is oxygen without fire.
• Similarly, a lack of fire does not yield a corresponding lack of oxygen.
• Neither condition is met in this claim.

In Unit 3, we learn that the logical structure of a conditional claim is such that an affirmation of the antecedent yields the consequent. A denial of the consequent yields a denial of the antecedent. So, consider the following argument:

If there is oxygen, there is fire. It's not the case that there is fire. So, it's not the case there is oxygen.

We know that oxygen is not sufficient for fire. We also know that fire is not necessary for oxygen. What we know, however, is not relevant to the conditional claim's logical structure. Recall, however, the previous discussion about what we claim to be the case vs. what is the case. Whether or not a claim is true is distinct from the logical structure of that claim!

To review, see Necessity and Sufficiency.

 

1f. Evaluate statements for various types of obscurity, such as lexical ambiguity, referential ambiguity, syntactic ambiguity, vagueness, incompleteness, and meaning

• What is an obscure sentence?
• What is lexical ambiguity?
• What is referential ambiguity?
• What is syntactic ambiguity?
• What is vagueness?
• What is incompleteness?

An obscure sentence is unclear. There are several ways in which a sentence reflects a lack of clarity. It can be ambiguous, vague, or incomplete. Moreover, there are several ways in which a sentence can be ambiguous. An ambiguous sentence is one in which multiple meanings are possible because a word in the sentence has multiple meanings. In addition, sentences can be unclear due to vagueness or incompleteness.

A sentence in which a single word with multiple meanings is not used precisely for the context is called lexical ambiguity. Consider the sentence, "They saw bats". "Bat" has at least two meanings: an object used to hit a ball and a (mostly) nocturnal mammal with wings. Because the sentence does not specify which meaning is intended, one could draw at least two inferences.

A sentence in which a single word does not explicitly refer is referentially ambiguous. For example, the sentence, "Person A and Person B got into the car and they turned on the air conditioning". It's not clear which person turned on the air condition, since "they" is ambiguous – it can equally refer to Person A or Person B.

A syntactically ambiguous sentence's grammatical structure is unclear. Consider the sentence, "Politicians are frightening people". It is unclear if people are being frightened by politicians, or if politicians, as a group of people, are frightening.

Vagueness differs from ambiguity in that an ambiguous word has multiple, but still determinate meanings, while vague language is indeterminate. Words 'carve out' meaning, which, as we've seen, should be as precise as possible for effective communication. When communication is inexact, misunderstanding and muddled reasoning can follow. Consider this exchange:

Suppose that Person A wants a timeframe, so as not to be late for dinner. Person B's response is vague. What does "in a while" mean in this context? A determinate time (e.g., 15 minutes) would eliminate the vagueness in the original response.

Another way in which words can yield a vague sentence is when they contribute to an incompletely expressed idea. Here again, context matters. When we find ourselves filling in conceptual gaps, for example, we should thereby be alerted to an incompletely expressed idea. We find ourselves asking, "In what way?" or "But how?" Consider the example from Slide 12: "Will this year's final exam be similar to the one last year?" We should find ourselves asking, "In what way is this year's final exam potentially similar to last year's? Are we talking about content, format, number of questions, or difficulty level?"

To review, see Thinking Critically About Ordinary Language: Obscurity.

 

1g. Evaluate statements for distortions of meaning, such as reification, category mistakes, and poor philosophical argumentation

• What is distortion?
• What is reification?
• What is a category mistake?
• What is poor philosophical argumentation?

Related to persuasive definitions, distortions of meaning can result in an erroneously positive or negative disposition. As we will see in Unit 5, where we cover informally fallacious (erroneous) reasoning, the intentional distortion of meaning is associated with a failure to seek truth. Suppose, for example, someone – we'll call them Person A – has concluded that euthanasia is morally wrong. That person then engages in conversation with someone – we'll call them Person B – who either does not think euthanasia is morally wrong, or is undecided. Person A asserts that euthanasia is murder and, as such, should not be legalized. While Person 1 may not believe they have distorted the meaning of euthanasia, it's clear that the morality of euthanasia is not a settled matter. In purely descriptive terms, the meaning of euthanasia does not include unjustified killing, which is how we generally understand "murder".

Another way in which language can distort is through reification. To reify is to make a concrete thing out of an abstraction. Consider the iconography of justice, which is typically exhibited, in the Western tradition, by a woman holding up a scale. The icon is an embodiment not intended to substitute for the concept, but if we did so, we would have reified justice.

To make a category mistake is a specific instance of reification. Consider two categories, apples and oranges. Apples are fruits and oranges are fruits, but only one is a citrus fruit. Were we to 'place' apples in the citrus fruit category, we would have miscategorized it. Citrus is not a property of apples.

The phrase, "category mistake" was coined by Gilbert Ryle in a famous philosophical critique of Descartes's claim that mind and body are two distinct kinds of substance – mind is an immaterial substance, and body is a material substance. According to Ryle, Descartes erroneously places "the mind and body in the same logical type or category when they actually belong to another". See The Concept of Mind (Routledge, 1949).

There are multiple ways in which argumentation can go wrong – indeed, this course involves studying some of the most important argumentation failures. Poor philosophical argumentation – any argumentation, really – occurs most blatantly when the arguer distorts, rather than investigates in search of truth.

To review, see Thinking Critically About Ordinary Language: Distortion.

 

Unit 1 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• antecedent
  
• category mistake
• consequent
• conversational implicature
• distortions of meaning
• factual dispute
• implied meaning
• incompletely expressed idea
• lexical ambiguity
• literal meaning
• necessary condition
• obscure
• persuasive definition
• precising definition
• referentially ambiguous
• reification
• reportive definition
• stipulative definition
• sufficient condition
• syntactically ambiguous
• vagueness
• verbal dispute

Unit 2: Argument Analysis

2a. Construct valid and sound arguments using the standard form of an argument

• What is a valid argument?
• What is a sound argument?
• What is standard form for an argument?

The statements that constitute an argument are either true or false. When we argue, we typically believe the statements we assert are true. More specifically, we typically believe that the evidence or justification we provide for a claim is true, and that this truth makes the claim at issue true, as well. In other words, we typically believe our premises are good reasons to accept our conclusion. The statement that is supported is the conclusion, while the statement(s) that support(s) the conclusion are the premise(s).

There are plenty of occasions, however, when we don't know whether or not our premises are true. Moreover, there are premises some people may believe are true, while others do not – these premises themselves require justification. So, how do we evaluate such an argument?

One evaluation standard is validity. A valid argument is one whose premises guarantee the conclusion in virtue of the argument's structure. An invalid argument's premises do not guarantee the conclusion. In other words, when an argument is valid, it is impossible for the conclusion to be false, if the premises are true. When we assess an argument for validity, we assume the premises are true – even if we know they are false!

Consider these two examples:

 

 

Notice that the form is the same for each argument (it is called disjunctive syllogism):

The content is irrelevant to the correctness of the form. So, whatever sentences we plug into the form, we have a valid argument. Another way to put this is that we assume the premises are true, even if we know, as in the case of the moon argument, that at least one is actually false. On this assumption, we cannot 'make' the conclusion false while the premises are true; doing so results in a contradiction. Think about it this way: If we have A or B, and it turns out we don't have A, then it's impossible not to have B – we must have B on pain of contradicting our premises.

Validity is a crucial concept in logic, but it is not the gold standard, as it were. A sound argument is even better. That is because a sound argument is valid, and its premises are true. So, the difference between the moon and math arguments is that the first is valid, but not sound (the first premise is false), while the second is sound (it's not only valid, but both its premises are true).

Finally, the presentation of each argument example is in standard form. Organizing your argument by writing out the premises and conclusion in list form makes the structure clearer. You can present it as seen above, or something like this:

Remember, just because an argument's premises and conclusion are (actually) true does not mean the argument is valid. A valid argument makes it impossible for the conclusion to be false. Remember also that the premises are just those statements we take to be true for the purpose of 'seeing' whether or not they yield the conclusion.

To review, see:

• Validity, Soundness, and Valid Patterns
• The Standard Format of an Argument

 

2b. Determine if a counterexample exists for a given argument

• How does the counterexample method work?

A valid argument's premises cannot yield a false conclusion. If we believe the argument is invalid, we can use the counterexample method to test it. The method consists in preserving the form of the reasoning, while replacing the content. More specifically, we replace the given conclusion with one we know is false, while replacing the original premises with statements we know to be true. If the argument resists the counterexample, it's very likely the argument is valid. The crucial point, however, is that, if successful, the counterexample method shows the argument is invalid.

Here are some examples, which are "imposters" of the valid arguments (see Validity, Soundness, and Valid Patterns):

 

To review, see Validity, Soundness, and Valid Patterns. 

 

2c. Illustrate valid argument patterns such as modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, constructive dilemma, and reductio ad absurdum

• What is modus ponens?
• What is modus tollens?
• What is hypothetical syllogism?
• What is disjunctive syllogism?
• What is constructive dilemma?
• What is a reductio ad absurdum?

In formal logic, the actual truth value of a sentence is not at issue. That said, like other systems of deductive argumentation, natural deduction aims at certainty by way of a valid argument form. So, when we consider any valid inference in this system, the possible truth values for a given connective drive the assembling and disassembling of a statement.

There are many more valid arguments than those presented in this unit. Modus ponens (MP) is a Latin phrase translated as "mode of affirmation", whereas modus tollens (MT) is a Latin phrase translated as "mode of denial". Hypothetical syllogism (HS), disjunctive syllogism (DS), and constructive dilemma (CD) are arguably the most common – you'll see them in almost any logic text.

The reductio ad absurdum (RAA) is any argument that proceeds by arguing to a logical absurdity, which is a contradiction. So, for example, modus tollens is a version of the reductio, as is destructive dilemma, and the negation introduction or indirect proof in symbolic logic.

Recall the discussion of necessary conditions and sufficient conditions from Unit 1. These are helpful to understanding how MP, MT, and CD work. We can also re-formulate the disjunctive claim in DS as a conditional claim, which is equivalent to the original. In so doing, we can see that the concepts of sufficiency and necessity apply here, as well.

Let's look at each focusing on the notion of assembling and dismantling a statement at its connectives or "joint". We will use that focus to see how a statement's truth-functionality plays out in a given inference. As you look through each argument form, think about how the truth definitions guarantee each inference – and remember the definition of a valid argument: If the premises are true, the conclusion can't be false.

Modus ponens, also known as "affirming the antecedent", yields the consequent of a conditional claim when the antecedent is affirmed. In other words, the conditional claim asserts that the antecedent (what comes after the "if" clause) guarantees the consequent (what comes after the "then" clause). Notice, however, that the conditional claim does not assert that the antecedent obtains. After all, it is a hypothetical. This is where the affirmation of the antecedent comes in – it is the premise in the reasoning that, combined with the conditional claim in the other premise, yields the consequent. 

Form of Modus Tollens:

If A then B
Not B
Not A

Example:

If the animal is a dog, then it's a canine.
The animal is not a canine.
The animal is not a dog.

Hypothetical syllogism is another three-statement argument form, but here the statements are all hypothetical, which means they are supposed, conditional, or theoretical. In this type of argument, there is a transitive relation between the two terms in each of the statements by virtue of a term common in each. 

Form of Hypothetical Syllogism:

If A then B
If B then C
If A then C

Example:

If the animal is a dog, it is a canine.
If the animal is a canine, it is a mammal.
If the animal is a dog, it is a mammal.

You already know a bit about the argument form known as disjunctive syllogism. A disjunction is an "either-or" statement, and a syllogism is a three-statement argument. So, a disjunctive syllogism is a three-statement argument that involves a disjunction. One statement is disjunctive – it's the "or", or "either…or" part of the argument. Another statement denies one side of the disjunction, and the conclusion, based on the two premises, is that the other side – the other disjunct – must be true. Another way to think of this argument form is in terms of elimination: When there are two options, and one is eliminated, the other is what is left.

Form of Disjunctive Syllogism:

Either A or B
Not A
B

Example:

The animal is a cat or a dog.
The animal is not a cat.
The animal is a dog.

Constructive dilemma is a combination of elements from modus ponens and disjunctive syllogism. This argument form is also a bit complicated by multiple elements, so describing it is probably more confusing than simply showing the form and giving examples:

Form of Constructive Dilemma

If A then B, and, if C then D
Either A or C
Either B or D

Example:

If the animal is a dog, it's a canine, and if the animal is a cat, it's a feline.
The animal is a dog or a cat.
The animal is a canine or a feline.

Notice that constructive dilemma takes two conditional statements and affirms that the antecedent of one or the other is the case, which means that the consequent of one or the other is the case.

To review, see Validity, Soundness, and Valid Patterns. 

 

2d. Identify hidden and implicit assumptions in an argument

• What is a hidden or implicit assumption?
• What is a method for determining the hidden or implicit assumption in an argument?

Understanding and uncovering hidden or implicit assumptions can be difficult, as so much of our daily communication occurs in a sort of "shorthand". Consider this conversation between Adam and his next-door neighbor, Rosario, that illustrates the sorts of inferences we make, but don't always think about:

Rosario:	Do you want to go ride bikes tomorrow?
Adam:	No, I'm going to the Lemmings basketball game with a bunch of the guys.
Rosario:	Oh, that sounds like fun! I love basketball and always root for the Lemmings.
Adam:	What? No way, girls don't like basketball. Besides, basketball is a guy sport; the WNBA is totally irrelevant in our culture because nobody is interested in watching girls play basketball.
Rosario:	Adam, I thought you were smarter than that! Just because the WNBA isn't doing well doesn't mean girls can't like basketball. What a silly thing to say. Besides, you've seen me playing ball in my front driveway with Robert and some of the other kids.
Adam:	Yeah, but I thought you were just trying to impress Robert because you like him.
Rosario:	Who says I like him?
Adam:	Well, you're always all giggly and acting flighty whenever you're around him. That means you have a crush on him.
Rosario:	Oh, boy. I have no idea what you mean by "acting flighty", but I always giggle.
Adam:	Not around me, you don't. Anyway, that's just what girls do when they like a boy. They act silly.
Rosario:	I think you're a really nice person and all, Adam, but I don't giggle around you because, well, you're not very funny. Robert, on the other hand, is hysterical. He's always telling jokes and doing impressions of people.
Adam:	I am funny! I can't believe you think I'm not. Is it because I play computer games? You think I'm a nerd, and nerds can't be funny?
Rosario:	Um, no. I just think you're a very serious person. It's not a criticism.
Adam:	I'm hardly ever serious.
Rosario:	Well, you look serious all the time. You don't smile very much.
Adam:	So?
Rosario:	And you always go around with your nose in books – books with titles I can't even pronounce. You don't say things that are funny or laugh when someone does.
Adam:	I like science, so what? What's wrong with quantum physics? And anyway, I don't spend my time thinking about funny stuff, but that doesn't mean I can't be funny.
Rosario:	Well, okay. Maybe I jumped to conclusions.
Adam:	Yeah, I guess I did, too. Want to come to the game tomorrow then?
Rosario:	I'd love to.

 

Both Rosario and Adam infer things about each other based on outward appearances and stereotypes. A stereotype is a classification of a person or groups of people, usually negative, based on too few samples. Both made inferences that were off-base – or at least not very well supported by the evidence. Let's look a little more closely at the inferences Rosario and Adam made about each other.

Adam infers that Rosario cannot like basketball. This inference is made based on his belief that girls don't like basketball. Since Rosario is a girl, she won't like basketball, either. It's not clear just why it is that Adam holds this belief, but he does state that basketball is a "guy sport", so perhaps he infers that girls do not like basketball because, basketball is a guy sport, and since girls are not guys, they will not like it.

Adam does not provide explicit support for his claim that basketball is a guy sport, but he does assert that the WNBA is "totally irrelevant". You can see that, upon closer inspection, Adam is making an inference based on rather flimsy evidence, and he is rather vague about his reasons for thinking Rosario can't like basketball. You have probably noticed that Adam has made a few inferences in order to get to his conclusion that girls don't like basketball, and there are some "missing" premises.

When evaluating arguments, it is important to determine what, if anything, is missing from an argument. The second text in this series will include a focus on a specific type of argument in which a premise or conclusion is missing, and we will see how the missing piece can affect the argument as a whole. For now, remember that you should account for all the components of the argument, even the missing, or implicit, premises and conclusions. Recall from Unit 1 that an implicit claim is implied rather than stated, and is typically taken to be understood by a listener or reader without being expressed by the speaker or writer. Here's one way to construct an argument for Adam's conclusion, including the missing premises:

Notice that Adam's first premise is essentially unsupported. Many arguments' premises are in need of some sort of support, which means they are not accepted facts. For example, "Leprechauns exist", is a statement. As such, it is either true or false, but not everyone agrees about how to prove its truth-value, or even if proof is possible. To assert that the statement is true, some would claim, is to assert an improvable belief. Adam's assertion that basketball is a guy sport is not as difficult to prove true or false as is the statement about God's existence, and it would only help Adam's inference that girls don't like basketball if he were to support it. His reference to the failure of the WNBA seems to suggest some sort of support, so let's see how it would look constructed as an argument, including the missing premises:

The conclusion then becomes the first premise of another argument that supports Adam's claim that basketball is a guy's sport:

Wait a minute! What does girls' interest in the WNBA have to do with an interest in basketball and rooting for the Lemmings? If you answered, "Nothing", you're right. Adam is making claims that just aren't related to one another in such a way that it is reasonable for him to conclude that girls don't like basketball. He seems to be confusing the failure of the WNBA with a lack of interest on the part of girls in the sport of basketball (regardless of who's playing). Moreover, he is assuming that the failure of the WNBA is evidence of basketball being of interest only to guys. It would be more reasonable to conclude, instead, that women playing basketball is just not interesting to guys. In any event, there is a bit of confusion on Adam's part in terms of what he is asserting and its relevance to the ultimate claim that he makes.

Adam also makes an erroneous inference that Rosario has a crush on her neighbor, Robert, based on the fact that she giggles and "acts flighty" when he's around. Adam presupposes that girls who giggle and act flighty (whatever that may mean) around boys are girls who have crushes on those boys.

No sooner does Rosario set Adam straight, she makes an inference that is not particularly good. She claims that Adam does not have a sense of humor, and her evidence of this is that his facial expression is serious, ("Well, you look serious all the time"), he does not smile much, and he always has his nose in books with titles that are hard to pronounce. The unstated inference is that books with difficult to pronounce titles are serious, which is then supposed to support her claim that Adam's facial expression is serious. That, we just saw, is meant to serve as evidence for the claim that Adam does not have a sense of humor. The poor quality of the reasoning is highlighted when what is unstated is made explicit.

Here are some more brief examples of poor inference-making, the weakness of which is uncovered when we include the relevant implicit assumption:

1. I have trouble with spelling. I must not be smart. [Implicit assumption: Smart people don't have trouble with spelling].
2. This Doberman is not a good watchdog. He obviously doesn't care about his human family. [Implicit assumption: Only good watchdogs care about their human families].
3. My parents went to Europe and brought home the best chocolate from Switzerland. The Swiss must have the best ingredients in the world. [Implicit assumption: The best chocolate in the world is made out of the best ingredients in the world].
4. The house is quiet. Everyone must be asleep. [Implicit assumption: If everyone is asleep, the house is quiet].
5. This computer's no good. It doesn't do what I want. [Implicit assumption: The computer is good, if it does what I want].

Having trouble with spelling does not imply a lack of intelligence. Lots of people who are plenty smart find spelling difficult. There are a number of reasons that could account for trouble with spelling, and these reasons are probably much more likely to account for the spelling difficulty than is the inference that I am not smart. It is possible, for example, that I am dyslexic. Dyslexia is a neurological disorder in which the brain processes something different from what the eye sees. Quite often a person with dyslexia will transpose letters in a word, or transpose words in a sentence. A bad speller might also not have much experience spelling, perhaps because he doesn't read much or doesn't read broadly enough to come across lots of different words.

Similarly with the spelling inference, the Doberman inference fails because there are other, possibly better reasons, to account for its being a poor watchdog. The Swiss chocolate inference fails, in part, for the same reason as the Doberman and spelling examples. In addition, there is an assumption that what is true of the whole, in this case the goodness of the chocolate cake, is also true of its parts. The inference is, then, that the ingredients of the chocolate cake are as good as the cake itself. A similar mistaken inference could be made if I was to claim that every individual European country makes good chocolate cake just because the cake came from one of those European countries.

The fourth and fifth examples also fail as good inferences because other inferences could be made that would work just as well, if not better than, the ones actually made. From these examples it should be clear that one can make any inference one wants from the facts at hand, but this does not mean they're any good. The examples show that good inferences rely on a strong connection between the facts, or evidence, and the conclusions inferred from them.

To review, see Hidden Assumptions, Inductive Reasoning, and Good Arguments. 

 

2e. Explain the differences between deductive reasoning and inductive reasoning 

•     What is deductive reasoning?
•     What is inductive reasoning?

The study of logic in a course like this divides arguments into two types: deductive and inductive. Though inductive arguments are far more common in our everyday use than deductive arguments, it is fair to say that we generally want our arguments to have the certainty of deduction.

We know that an argument consists of two or more statements, one of which is supported by the other(s). The statement that is supported is the conclusion, while the statement(s) that support(s) the conclusion are the premise(s). An argument is characterized by an inferential flow: The conclusion follows from the premises; the premises imply the conclusion. Another way to say this is that the conclusion is drawn from, or implied by, the premises.

An argument is evaluated in terms of the strength of the connection between the premise(s) and conclusion. The strongest connection between the premises and conclusion is entailment. When the conclusion is a necessary consequence of the premises, the conclusion is entailed by the premises. This means that you cannot have true premises and a false conclusion at the same time. The conclusion is said to be true on the basis of accepting that the premises are true. So, if the premises are true, it would be contradictory for the conclusion to be false. All such arguments' power comes from the structure of the premise-conclusion relation.

As statements, premises and conclusions have truth-values. The truth-value of a statement is its truth or falsity. So, statements are about what is the case; they are not about beliefs, opinions, questions, or commands. It is true that beliefs can be asserted as statements, but their truth-value is more difficult to determine. When you say, "He's handsome" or "She's beautiful", the statement is true or false depending on whether or not he is handsome or she is beautiful; and determining that is not quite so easy as determining the truth-value of something like, "He is in the room".

The premises and conclusion of a deductive argument have a specific relationship. The conclusion is supposed to follow from the premises with certainty, or necessity, and the premises are said to be sufficient to infer the conclusion. In other words, the relationship between the premises and conclusion is one in which the premises are claimed to support the conclusion such that it is impossible for the premise(s) to be true while the conclusion is false. Before moving on to discuss the structure of deductive arguments, let us spend a bit more time looking at the relationship between the premises and conclusion. An example might help make clear just how that relationship works:

Besides thinking to yourself, 'Duh, this isn't proving to me anything I don't already know,' you might also think, 'Of course Chihuahuas have to be canines. After all, if it's true that Chihuahuas are dogs, and dogs are canines, Chihuahuas also have to be canines.' What you're expressing is the sufficient relationship between the premises and conclusion, and the necessary relationship between the conclusion and premises. Another way to assert the two ideas just expressed is to say that, assuming the premises are true, you are committed to asserting the truth of the conclusion – to do otherwise is to assert a contradiction, that is, that the premises can be true while the conclusion is false.

When we see an argument whose conclusion cannot be doubted while we accept the premises, we see an argument that succeeds by its structure. Here another example, which highlights how the meanings of certain relations, when structured in specific ways, reveal this idea:

Now consider the following argument, which uses the same relational predicate, "taller", but with a change of the argument's structure, the premises no longer guarantee the conclusion:

Here, we could easily falsify the conclusion, while accepting the truth of the premises. Suppose that Gerrardo is 5'10" tall, Stanley is 5'8" tall, and Kumar is 5'9" tall. At 5'10" tall, Gerrardo is taller than both Stanley and Kumar. So, the premises are true. At 5'8" tall, Stanley is not taller than Kumar, who is 5' 9" tall. So, the conclusion is false.

Perhaps our most common, everyday mode of reasoning is experiential. Consider the following examples:

• Rosa leaves for school at 8:00 a.m. Rosa is always on time. Rosa infers that she will always be on time for school, if she leaves at 7:00 a.m.
• The cost of the fabric was $1.00. The cost of labor to manufacture the fabric was $.50. The sales price of the resulting item was $5.00. So, the manufacturer made a good profit.
• Every snowstorm in this area comes from the northeast. There is wind and flakes of snow coming from the northeast, so, a new snowstorm is coming from the northeast.
• Javier is showing a new car to his friend, Ahmed. So, Javier must have bought a new car.
• The living room walls are painted yellow, as are the kitchen and dining room walls. So, all the walls in the house are painted red.
• Every time you touch poison ivy, you get a rash. So, you're allergic to poison ivy.
• Every dog you know barks. So, every dog barks.
• Half of the students at this university receive student aid. Therefore, half of all university students receive student aid.
• All of the children in the class are brown-haired, therefore all children in this neighborhood are brown-haired.
• Michelle just moved here from Los Angeles. Michelle has blonde hair, therefore people from Los Angeles have blonde hair.
• All the players on my basketball team are tall, so all basketball players must be tall.

We call experiential reasoning inductive. The premises and conclusions of inductive arguments do not claim the same relationship as those of deductive arguments. Instead, the relationship is only probable. Probability expresses the likelihood that something is the case. This means that, instead of expressing a supposedly necessary relationship between premises and conclusion, an inductive argument asserts that, if the premises are true, the conclusion is likely to be true. Another way to express probability is in terms of chances. In an inductive argument, if the premises are true, the chances of the conclusion being true increase. Often, we use inductive reasoning to express our conviction, that is, to reason to a claim that expresses we are convinced something is the case.

Here are some more examples of inductive arguments:

Notice that the probability of the conclusion being true is strengthened by how well the premises support the conclusion. In the first example, based on the similarities between Polar Bears and Grizzly Bears, it's inferred that Kayla's love of reading about Polar Bears means she's likely to enjoy a book on Grizzlies. The second example relies on predicting something about the future based on repeated past experiences. This is a very common way of thinking, and we use it every day to order our lives. Just imagine how hard it would be to get through your day if you didn't rely on past experience! The third example makes a connection between the wetness of the dishes and the timeline for their having been washed. The fourth example concludes something about the past based on present experience. It's basically the reverse of predicting something about the future based on the past.

Here are some examples of inductive arguments in which the chance of the conclusion following from the (true) premises is far less than in the previous examples:

Carmen hasn't fallen off her new bike enough times to warrant the conclusion that she's going to fall off every time she rides it in the future. Though all predictions are uncertain, some are more reliable than others. This prediction, however, fails due to insufficient evidence. The second example is problematic because it applies to every member of the household something that is only known to be true of one of its members. By similar reasoning, you could try to claim that the new neighbors are all youngest children on the basis of the youngest child being the youngest – and that's utter nonsense!  (What about the broccoli example?) Finally, there is no good reason to conclude that the "special potion" Samantha took cured her cold. After all, colds typically run their course and are over within a week.

Though there are no inductive forms of arguments, there are types and patterns of inductive reasoning that are similar enough in their functions to categorize them as follows:

 

We will see later that there are ways in which inductive arguments are twisted or so bad that they confuse people into believing something that's not true.

To review, see Hidden Assumptions, Inductive Reasoning, and Good Arguments. 

 

2f. Explain the pattern of inductive reasoning called an analogical argument

• What is an analogical argument?
• Why is an analogical argument a type of inductive argument?

One of the most common ways of reasoning in our daily lives is by way of a comparison between two or more objects or events. Suppose you are looking at produce. You want a ripe avocado. On the basis of previous experience with the features of a ripe avocado, you begin looking through the pile in front of you, taking out an avocado at random, gently squeezing it, and then putting it back. Eventually, you find one that "feels right". That's the one that goes in your basket.

Or suppose you need to purchase a car. On the basis of your previous experience with a particular make and model you owned, you conclude that buying another car of the same make and model will suit you very well.

Now suppose that you are an astronomer researching the possibility of life on other planets. You will make comparisons between the requisite features for life found here on Earth, and, say, the chemical composition of materials found on another planet.

Finally – although many more examples are to be had – suppose you are a legal expert. You have before you a case of a self-driving car that has been involved in an accident. The car went off the road and ran into someone's yard. While no one was injured, the property owner's fence and garden was ruined. The question of liability needs to be settled, but suppose there is no precedent – there is no case law under which the current circumstance falls. The legal expert must think about commonalities. Is the autonomous vehicle more like a horse-drawn carriage than an ordinary, human-driven vehicle? Is the autonomous vehicle more like a personal computer, with hardware and software? Determining the relevant comparison can make an enormous difference to how to make and interpret the law.

Analogical reasoning generally follows this pattern:

• Two (though sometimes more) objects or events are compared.
• One of those objects (the target, or secondary analogue) is lesser known than the other (the primary or original analogue).
• On the basis of the comparison, a conclusion is drawn about the lesser known of the two objects.
• The number of relevant similarities and dissimilarities will determine whether or not the inference is strong.

Here are two ways to view the general structure of analogical reasoning:

Analogical reasoning can go wrong. Consider, for example, the following story: A woman in her first pregnancy experiences feeling and seeing puffiness, "like I ate too much pizza the night before", and seriously low energy. Despite eating a healthy diet and being a marathon runner, her symptoms continued to worsen. When her blood pressure spiked into the 190s over 110, she went to the hospital, where doctors worried they might have to deliver her fetus at only 25 weeks of gestation.

She responded well to blood pressure medication, so an emergency c-section was postponed. After all, if her symptoms indicated a problem that removing the fetus would not solve, the premature baby's health would be at risk. The question remained, did the woman have preeclampsia? Part of the thinking was that, apart from the high blood pressure, the woman did not have other signs of preeclampsia, a disorder that is the leading cause of maternal death, worldwide. Untreated, high blood pressure negatively affects a pregnant woman's organs.

Because the woman and her doctors were initially unconcerned about her symptoms, and because even in a critical state, her symptoms were apparently not enough like preeclampsia to warrant the diagnosis. The woman did go on to have a C-section at just over 30 weeks. Since her experience, the woman's daughter has grown into a healthy toddler. When she became pregnant again, and began to experience the same symptoms as she had during her first pregnancy, she immediately got help.

This woman's story is part of a larger picture emerging in maternal health. Historically, much of the medical focus surrounding pregnancy, delivery, and post-partum has been on fetal and infant health. There are, for example, fetal health centers around the United States, where care plans are far more customized than would be the case with uncomplicated situations. There have not, however, been health centers devoted to maternal health. The Mothers Center at New York-Presbyterian/Columbia University Irving Medical Center is the first center to focus on at-risk pregnant women – and it's modeled on the hospital's own fetal health center.

We can see from the mistakes made in the case discussed above, that analogical reasoning does not always go well. We can also see that analogical reasoning is at work in the thinking of the medical professionals who inaugurated the Mothers Health Center at New York-Presbyterian/Columbia University Irving Medical Center.

To review, see the section on Analogical Arguments.

 

2g. Construct an argument map for an argument

• How can a modus ponens argument form be mapped?

We know the MP argument has the following form:

Here is an ordinary language example: The dog is barking. So, someone's at the door. That's because, if the dog is barking, someone's at the door.

One we identify the conclusion (the conclusion indicator word, "So" alerts us), we can number our elements:

The argument map looks like this:

 

To review, see Argument Mapping.

 

2h. Apply the criteria for evaluating the strength of an argument to any given argument

•     What is a strong argument?
•     How can analogical arguments be evaluated?

Just as deductive arguments are "good" or "bad", so are inductive arguments. But the criteria for evaluating an inductive argument are different from those for deductive arguments – with one exception. Like deductive arguments, inductive arguments are evaluated based on how well the premises support the conclusion. In the case of deductive arguments, there are validity and soundness. With inductive arguments, the concepts are strength and cogency.

A strong argument is one in which the premises are true and the conclusion is probably true. That is, there is a good chance the conclusion follows from the premises.

Here is an example of a strong argument:

Weak arguments are those whose conclusion is not likely to be true. The problem is that, even when the premises are true, they just don't give much support to the conclusion. Based on the premises, it is not likely that the conclusion is true. Weak arguments act like their name: the evidence to support the conclusion is weak.

Here is an example of a weak argument:

People who look similar to each other are not necessarily related.

Let's leverage our understanding of analogical arguments by considering the basic criteria for evaluating an analogical inference. First, the number and relevance of the similarities is important to the quality of the reasoning. The greater the number and relevance of the similarities between the two objects or events, the more probable is the conclusion. Second, the nature and degree of disanalogy – the dissimilarity between the elements – is also important to the quality of the reasoning. Serious relevant disanalogies will undermine the likelihood that the conclusion is true. Lastly, the more specific the conclusion, the stronger the analogy must be. After all, the more specific the conclusion, the easier it is to falsify, which weakens the reasoning. So, a specific conclusion requires a diverse number of relevant similarities, and only a small number of relevant dissimilarities – and these cannot be compelling enough to outweigh the similarities. The ways in which the analogues are not alike can bear on the strength of the inference. The key characteristic(s) inferred about the secondary analogue can be undermined by these disanalogies. If you think an analogical argument is weak, list as many disanalogies as possible. The idea is that the difference between the compared items shows that the characteristic intended to be attached to the secondary analogue – the target in the conclusion – is not likely transferrable on the basis of the comparison.

Let's consider another argument from analogy, and then a series of disanalogies intended to demonstrate the original reasoning's weakness:

Another way to show that an analogical argument is weak is to construct a counter-analogy. Recall that analogical reasoning generally follows this pattern:

A counter-analogy will propose that the target analogue is not like A, but is actually more like something else, e.g., C. Moreover, the counter-analogy typically leads to a conclusion that contradicts the original analogical reasoning.

Finally, pointing out unintended consequences reveals that an analogical inference is weak. An unintended consequence of an analogical argument is one that the person who mounted that argument would not want to accept. As a consequence, the original argument would be revised or altogether abandoned. Hume, once again, provides an example of using unintended consequences to show that the design argument, as presented, is not as strong as it may seem at first.

If we consider the fact that increasingly complicated machines tend to require multiple makers, it should follow that the universe, which is extraordinarily complex, also requires more than one creator. Even if we make a distinction between a designer and a maker, and argue that, while it may take multiple builders to construct a home, for example, it takes only one architect. This objection is met, however, if we consider the variety of technical expertise required to complete a design, from engineering to electrical to plumbing, and so forth.

As we know, an argument's conclusion has to be relevant to the premise(s). In analogical reasoning, this relevance is determined by the characteristic(s) inferred about the secondary analogue – the target item in the conclusion – from the comparison to the primary analogue – the item to which the target is being compared. The more relevant similarities there are, the more likely it will be that the secondary analogue will be like the primary analogue's further characteristic. Provided that the key feature of the primary analogue transferred to the secondary analogue in the conclusion cannot be accounted for in other ways, the inference is strong.

To review, see Good Arguments.

 

Unit 2 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• analogical reasoning
  
• argument map
• conclusion
• constructive dilemma
• contradiction
• counter-analogy
• counterexample
• deductive argument
• disanalogy
• disjunctive syllogism
• hidden/implicit assumption
• hypothetical syllogism
• inductive argument
• inductive reasoning
• invalid argument
• modus ponens
• modus tollens
• premises
• probability
• reductio ad absurdum
• sound argument
• standard form
• strong argument
• truth value
• unintended consequences
• valid argument
• weak argument

Unit 3: Basic Sentential Logic

3a. Contrast formal logic and informal logic

• What is the difference between formal and informal logic?

Formal logic is concerned with deductive logical systems. In other words, a logical system consisting of formal language, rules for determining truth and falsity, and rules for deriving the conclusion of a valid argument is considered "formal logic". Informal logic is the name given to patterns of reasoning that do not exhibit those structures found in formal logic.

To review, see the section on Formal and Informal Logic.

 

3b. Identify declarative, interrogative, and imperative sentences

• What is a declarative sentence?
• What is an interrogative sentence?
• What is an imperative sentence?
• What is the difference between a sentence and a statement?

A sentence is a complete, grammatical construction. Sentences can be declarative, interrogative, or imperative. (A sentence can also be exclamatory, but we do not cover that type here.) 

A complete grammatical declarative sentence makes a claim, e.g., "Today is Tuesday". A complete grammatical interrogative sentence asks a question, e.g., "How are you?" Interrogatives can also be rhetorical, e.g., "Did you really finish the exam in 20 minutes?" A rhetorical interrogative is effectively a declarative sentence with different punctuation. Finally, an imperative sentence is a command or request, e.g., "Shut the door".

The difference between a declarative sentence and the other types shows us how we use sentences in an argument. More specifically, a declarative sentence has a truth value. In other words, a declarative sentence is either true or false. We cannot respond to the interrogative or imperative sentence types with "True", or "False". They are, in other words, not statements. For our purposes, the difference between a sentence and a statement is determined by the concept of a truth value. A statement is a sentence that is true or false.

To review, see Logical Statements, Connectives, and Relations. 

 

3c. Define and identify several kinds of logical statements: negations, conjunctions, disjunctions, conditionals, and biconditionals  

• What is a logical connective?
• What is the function of the negation?
• What is the function of the conjunction?
• What is the function of the disjunction?
• What is the function of the conditional?
• What is the function of the biconditional?
• What are the common symbolic logic symbols to notate connectives?

The structures, or forms, of sentences with which we are concerned in critical thinking, are logically simple or complex. Any statement in one of these forms will be a statement we can evaluate in terms of its truth-value.

The more complicated the statement, the more complicated the logical relationship between its parts. For example, "The dog is barking, but there is no intruder", connects the independent clause, "The dog is barking", with the independent clause, "There is no intruder", with the coordinating conjunction, "but". Notice also that the second clause, "There is no intruder", is the negation of the more simple statement, "There is an intruder".

What we've just seen, and will continue to study in more detail below, is called sentential, statement, or propositional logic. If you take a formal, or symbolic logic course, you will learn quite a bit more about this system of logic. In this course, we want to develop a working understanding of what types of sentences are candidates for arguments. We will leverage that understanding when we identify, evaluate, and construct arguments.

Breaking down complex statements to their most simple forms means identifying and separating out the coordinating conjunctions, which we will call connectives. We do this in logic to evaluate the truth-value of the statement. A statement's truth-value is its truth or falsity. Because every statement is either true or false, we can say that every statement has a truth-value. The truth-value of the statement, "The dog is barking, but there's no intruder", is determined by the simple statements that constitute it: "The dog is barking", and "There is an intruder", and their connectives: "but" and "not". The reason for evaluating truth-values of statements is to help us determine the truth-values of any statement governed by a particular connective, but also to determine the quality of the argument in which such statements appear.

As a reminder, an affirmative subject-predicate statement is a simple statement, and consequently does not have any connectives. As such, it is either true or false. The truth or falsity is altered when connectives are added to the statement. We generally use five connectives in logic:

The Five Logical Connectives and Their Ordinary Language Words and Phrases:

 

A negation denies a statement. It is generally understood to mean, "It is not the case that". So, the negation of "There is an intruder", is "It is not the case that there is an intruder", or "There is no intruder". Other words you can substitute for "It is not the case that", include "Not", "It is false that", "It is not true that", and the contraction of "either-or", which is "neither-nor". Basic grammar does not distinguish between a simple statement and a negated simple statement. Because this distinction has logical importance, we will isolate it from the instances in which it appears, and designate any negated statement compound. Moreover, statements with other connectives are negated, so in a way, a negated statement that's already logically complex is doubly, triply (and so on) complex. Here are some examples:

Conjunction joins together two statements, so that both are asserted to be the case. "The dog is barking but there is no intruder", is the conjunction of the simple statement, "The dog is barking", with the compound statement, "There is no intruder". The conjunction in this case is made with the word "and". Alternative words include "But", "However", "Still", "Nevertheless", "In addition", "Furthermore", "Moreover", "Also", and "Although". If some of these words seem like odd conjunctions, take a moment to see how a few of them work as replacements for our dog barking-intruder statement:

Disjunction separates two statements, so that at least one or the other is the case. Notice the phrase, "at least one", in the last sentence. It is a clue to how to understand a logical disjunction: It is inclusive, not exclusive, so that, when we assert "one or the other", both is also asserted to be the case. Typically, we think of disjunctions in terms of exclusivity, such as when you are presented with a choice of soup or salad at a restaurant. The intended sense is that you may have soup, or you may have salad, but you may not have both. In logic, however, we use the inclusive sense of the disjunction. In terms of a disjunctive sentence's truth-value, this means that one or the other "side" of the disjunct can be true, and in fact, both may be true. The two simple sentences that make up the statement, "That dog looks like a Beagle or a Lab", could both be true; the dog could be a mix of Beagle and Lab, so it could look like both breeds. Other words you can use to express a disjunction are "either-or", "or", and "unless". Here are some examples:

Notice with these examples that the sense of the disjunct is not inclusive but exclusive. After all, we might initially think Monique cannot both be at home and at work. Logically, however (and certainly if Monique works at home!), each disjunct could be true at the same time. There are plenty of disjuncts that are exclusive, and we should be aware of the difference between the two. When we want to communicate exclusivity in logic, we explicitly assert it: A or B, but not both A and B. Here are some more examples of disjunctive statements that can be thought of as inclusive:

In ordinary language, the parts of a conditional statement are dependent on each other, and that makes it difficult to understand just how those parts relate to each other. "If" is a concept that makes asserting truth or falsity troublesome. The tricky thing about conditional statements in logic is that, unlike statements governed by other connectives, the conditional statement is always only hypothetical. It does not say that something is the case, only that something might be the case, will be the case, could have been the case, or would have been the case. The problem with this type of statement is it does not assert that something is the case, and so also something about which you can assert truth or falsity, but that something might be (or have been) the case.

Moreover, we tend to think that denying the antecedent will result in a denial of the consequent. We tend to make this mistake because we think of the antecedent as being a necessary condition, when, in fact, it is not. Take the following reasoning: “If you eat spinach, you’ll be healthy and strong. But you don’t eat spinach, so you won’t be healthy and strong.” The denial of the consequent does not follow from the denial of the antecedent. This will become clearer as we learn more about logically correct reasoning.

The last connective is the biconditional. If you take apart the word, you'll notice that you have "bi" and "conditional". We've already seen what a conditional statement is, but what does "bi" mean? Simply put, it means "two-way". So, the biconditional connective expresses a two-way relationship. It says, "If this, then that, and, if that, then this". The biconditional tells us that both parts of the conditional expression are required in order for the statement to be true. The English expression of the biconditional is "if and only if". Notice the biconditional is made up of "if", "and", and "only if". These are all connectives we already know. It's the way they're put together that expresses a new logical relationship. Here's an example:

She wants to go to college if and only if she is accepted to her first choice.

The sense of the biconditional is equality: going to college is equal to being accepted by a first choice. This means, then, that not going to college will result from not being accepted by the first choice. At the same time, not being accepted by the first choice can be inferred from not going to college. Here are the ways we can express a biconditional relationship:

 

Here are some of the common symbolic logic symbols used to notate each connective:

 

To review, see Statements, Logical Connectives, and Logical Relations.

 

3d. Identify the scope and main connective for a well-formed formula

•     What is a well-formed formula?
•     What is the scope of a connective?
•     What is a sentence's main connective?

A well-formed formula is a correctly structured statement in symbolic logic notation. Suppose we want to notate the sentence, "Today is not Tuesday", in logical notation. We need a symbol for the sentence, "Today is Tuesday", and for the negation of that sentence. Let's stipulate the capital letter, "T", stands for "Today is Tuesday", and we decide to use the tilde for the negation symbol: ~. Both the original simple sentence, T, and the compound sentence, ~T are well-formed.

The scope of a connective is the range of sentences it covers. For example, the scope of the negation in the sentence, ~T is the simple sentence, T. Now suppose we have the sentence, "It's not Tuesday, but it is Wednesday", where we translate the ordinary English word, "but" with the conjunction symbol – we'll use the ampersand: &. The simple sentence, "Today is Wednesday", is translated with W. So, we have ~T&W for the complete sentence. Notice here that the scope of the negation is still restricted to T, while the scope of the & covers both ~T and W – the conjunction conjoins the two simpler sentences (the first of which is itself compound).

The discussion leads us to the idea of a compound sentence's main connective. When a sentence involves two or more connectives, only one can be the governing or main connective. Think about the ordinary English sentence above, "It's not Tuesday, but it is Wednesday", and its translation: ~T&W. Is this a negation or a conjunction sentence? You can tell which is the main connective by looking at the scope.

Now suppose we say, "It's not the case that today isn't Tuesday but it is Wednesday. In other words, here, we deny the entire original sentence: ~(~T&W). The parentheses group together the original sentence's elements to show that the negation covers the whole.

To review, see Sentential Logic and Well-Formed Formulas.

 

3e. Create truth-tables for several kinds of statements, sentences and arguments, such as negations, conjunctions, disjunctions, conditionals, and biconditionals

• How are truth-values determined in truth-functional (sentences governed by a connective) logic?
• What are the truth definitions for the negation, conjunction, disjunction, conditional, and biconditional?

Determining the truth or falsity of a simple statement can be tricky business. Consider assertions such as, "God exists", and "God doesn't exist". In truth-functional logic, the focus is on possible truth values, so that the actual determination issue does not arise. In other words, rather than focusing on determining whether or not it is true to claim, "This tree is a conifer", in formal logic, the focus is on the fact that there are two possible truth values for the claim: True or False.

When dealing with a compound sentence, that is a sentence involving one or more connectives, the possible truth-values are a function of the original simple statement, and the meaning of the connective(s).

Here are some examples of the concept of possible truth values and truth-functionality:

The logic of natural deduction is a logic involving the assembling and disassembling of statements at their connective "joints". The assembling and disassembling, in turn, is based on each connective's possible truth values – it's truth-functionality. Consider the following statement from the example set above: George W. Bush was not the 44th president of the United States, but Barack Obama was. The compound statement, governed by the connective, "but", is made from two other statements, the first of which is itself compound: George W. Bush was not the 44th president of the United States, and Barack Obama was the 44th president of the United States. Now consider what makes the conjunction true. When we make assertions such as, "Both…and", "and", "yet", and "but", for example, the claim is that each element of the "both", or each side of the "and", is true. So, for the compound sentence, George W. Bush was not the 44th president of the United States, but Barack Obama was, each of the conjuncts must be true: the sentence, George W. Bush was not the 44th president of the United States, must be true, and the sentence, Barack Obama was the 44th president of the United States, must also be true.

Digging a bit deeper, we can also see that for the sentence, George W. Bush was not the 44th president of the United States, to be true, the simple sentence, George W. Bush was the 44th president of the United States, must be false. The negation, "not" denies the original truth-value. So, when we want to assemble (the rule, Conjunction) or disassemble (the rule, Simplification) a conjunction, we do so based on its truth definition, namely that a conjunction is true when, and only when, each of the conjuncts is true.

Here is an overview of each of the connectives we are studying, and the introduction of each's truth definition:

Connective	Truth Definition	Example
Negation:	True when the statement it negates is false; false when the statement it negates is true	It is not the case that Abraham Lincoln was the first U.S. President. True!
Conjunction:	True when, and only when, each of the conjuncts is true; otherwise, the statement is false	Providence, RI, and Boston, MA, are cities in New England. True!
Disjunction:	False when, and only when, each of the disjuncts is false; otherwise, the statement is true	Providence, RI, or Boston, MA, is south of the Equator. False!
Conditional:	False when, and only when, the antecedent is true and the consequent is false; otherwise, the statement is true	If Providence is in the U.S., then Providence is on Mars. False!
Biconditional:	True when, and only when, the values of each side of the biconditional are equivalent; otherwise, the statement is false	If the animal is a marsupial, then it has a pouch, and if it has a pouch, then the animal is a marsupial. True!

 

Here is another chart depicting the shorthand for each connective's truth definitions, where P and Q stand for any statement – they are the logical equivalent of variables. Note that the ordinary language phrase for the biconditional, "if and only if", is abbreviated to "iff":

Remember that the reference columns always lay out the possible truth values in the same way, and that the number of rows of a truth table increases exponentially with the addition of each simple sentence.

Reading left-to-right, the first simple sentence's values are half true and half false. The second simple sentence's truth values are half of the first sentence's values – again, half true and half false.

One simple sentence:

Two simple sentences:

Three simple sentences:

The formula for determining the number of rows in a truth table: R = 2^{number of simple sentences}

Here are the truth definitions of each connective:

Here are some examples of possible truth values of compound sentences, where the reference column values are directly beneath each simple sentence:

To review, see:

• Truth Tables
• Complex Truth Tables

 

3f. Translate ordinary statements into logical language

• How do we translate ordinary language statements into symbolic logic notation?

In the version of sentential logic we're studying, we translate ordinary English statements into symbolic logic notation using the symbols discussed in 3c. We also use parentheses to signify that elements in a compound statement are to be read as a unit. So, capital letters A-Z are used to notate simple sentences. (Reminder: A simple sentence is the smallest linguistic entity that has a truth value, e.g., "Socrates sits". A simple sentence does not contain a connective, i.e., it does not include a negation, conjunction, disjunction, condition, or biconditional).

The connectives are symbolized as follows:

 

Here are some examples of translating into symbolic logic notation:

 

To review, see Formalization. 

 

3g. Explain the limitations of truth-tables as assessment tools

• What are some of the limitations of truth-tables?

The truth table is a powerful tool for determining the possible values of truth-functional sentences and the validity of truth functional arguments. As we know, however, not every statement is truth-functional, and not every argument is constituted by at least one truth-functional statement.

Consider the following valid arguments:

 

Notice there are no connectives in any of the statements. If we put them into a truth table, we'll end up with at least one row on which the argument fails, which shouldn't happen! This means that the truth table is effective only for those arguments involving connectives.

In addition, consider the fact that the conditional claim (the material conditional) is false in one and one case only, namely when the antecedent is true and the consequent is false. This comports with the concept of validity: An argument is invalid when the premises are true and the conclusion is false. At the same time, however, there are times when the truth table doesn't seem to comport with our intuitions about the conditional claim. Consider the following sentence: "If apples are fruits, then Los Angeles, California is in the United States". On a truth-functional interpretation, the statement is true, but we don't think it's true in any meaningful way. Or consider this sentence: "If fish have legs, then fish take hikes in the woods". From the standpoint of the truth table, where both the antecedent and consequent are false, the statement is true. But this is counterintuitive. As long as we remember how the truth-table works and what its restrictions are, we will not misuse it, which can lead to erroneous outcomes.

To review, see Material Conditional. 

 

Unit 3 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• biconditional
  
• conditional
• conjunction
• conjuncts
• connectives
• declarative sentence
• disjunction
• formal logic
• function
• imperative sentence
• informal logic
• interrogative sentence
• main connective
• negation
• scope
• statement
• truth-value
• truth definition
• truth table
• well-formed formula

Unit 4: Venn Diagrams

4a. Create Venn diagrams as a means to represent and reason about relationships among classes

• What is a Venn diagram?
• What is existential import?
• How do Venn diagrams provide a visual explanation of distribution?
• How do Venn diagrams reveal logically equivalent propositions or indeterminate inferences?

Drawing pictures helps a lot of us understand the logic of the categorical proposition. There are some who find pictorial representations of a proposition's logical structure off-putting. Fortunately, there are plenty of linguistic articulations of this material, but it is also the case that picture-making in categorical logic is systematic, and, once you get the hang of it, very handy. Let's start with a reminder of our four types of categorical proposition:

 

Here is a way to begin visualizing the relationships between the S and P classes, where a lowercase x symbolizes the particular affirmative, "there is at least one", or "some":

Start by focusing on each of the S circles. In the A-claim (All S are P), the entirety of the S class is inside the P class. In the E-claim (No S are P), the S class is entirely outside of the P class (and, it is worth noting, vice versa). Now, take a look at the arrangement of the circles for the two particular claims, Some S are P and Some S are not P. Notice that they overlap somewhat, which creates three internal areas where a relationship between the categories can be mapped:

In the I-proposition, Some S are P, the x is in the overlap area. Another way to read the diagram, without changing the logical structure of the claim is: There is at least one thing that is both an S and a P. In the O-proposition, Some S are not P.

Logicians use this arrangement of circles, which provides a template for all four claim types. The Venn diagram, named after 19th century English mathematician John Venn, provides a visual representation of each claim type's logical structure. Notice that the particular affirmative and particular negative Venn diagrams look just as they do above:

The next question is, how do we diagram the universal propositions? We need a way to express the complete inclusion of S-class members in the P class, and a way to express the complete exclusion of S-class members from the P class. In logic, this is achieved by shading or drawing lines through the area of a circle that is supposed to be empty or, more precisely, that cannot contain any members. It's as if the shading expresses the idea that there's no there there:

Here is another way to visualize the universal claims, where the lines drawn through the area of S that is outside of P, and the overlap area between S and P, for universal affirmative and universal negative propositions, respectively, designates the impossibility of anything being there:

Go back to the original diagram above, in which nested circles represent the universal affirmative proposition, All S are P. Now look at the Venn diagram for the same proposition. You can see that the shaded area of the S category means that there couldn't be an S that isn't included within the P category. Now, do the same for the universal negative proposition, No S are P. In the original diagram above, the two circles are separated, which shows that there isn't even one S that is also a P. The Venn diagram visualizes this relation by way of shading in the area of overlap between the S and P categories.

Let's now bring into the discussion existential import. Recall that universal claims express affirmative or negative relations between the subject and predicate classes. The affirmative relation in the A-proposition is that every S is included in the P category. The negative relation in the E-proposition is that every S is excluded from the P category. Here's a question: Are there any existing members of the S class? In other words, when we claim that all the S's are or are not members of the P category, are we also assuming such members exist? When we say, "Werewolves are scary", or "Leprechauns are gold hoarders", for example, we don't assume that the subject class has existing members – at least, most people agree that there aren't any werewolves or leprechauns. In logic, we can express the universal claims in terms of existential import: The subject class has existing members and is assumed in the Aristotelian or Traditional Logic. This assumption is diagrammed as follows, where the circled x denotes an existing member of the class in question:

So far, we have laid out a number of technical concepts that we will use repeatedly when working in both versions (Aristotelian and modern) of categorical logic. There is one more concept to discuss before we begin to use them in making inferences between propositions (immediate inferences) and from a set of two propositions (categorical syllogisms). It is called distribution. We say that a term (the term that denotes the S class or the P class) is distributed or undistributed. A term is distributed when the proposition refers to the entire class denoted by it:

Here is a summary of the distribution of terms:

 

To sum up, this is what we've learned about categorical propositions, so far:

• Categorical logic consists of four types of categorical proposition:
• Universal affirmative (A-proposition): All S are P
• Universal negative (E-proposition): No S are P
• Particular affirmative (I-proposition): Some S are P
• Particular negative (O-proposition): Some S are not P
• Venn diagrams provide visual representations of each categorical proposition's logical structure
• An x represents, e.g., the word or phrase, "Some", and "There is at least one".
• Shading denotes emptiness, or the impossibility of anything being in the shaded area of a circle;
• The Aristotelian, or traditional, system of categorical logic assumes existential import, that is, assumes that there is at least one existing member of the subject class in the categorical proposition
• The modern system of categorical logic does not assume existential import
• A term is distributed when it is entirely included into, or excluded from, a category:
• Universal propositions distribute the subject class, i.e., the A- and E-propositions distribute the subject class
• A term is also distributed when at least one member of the term-class is excluded from the other class:
• Negative propositions distribute the predicate class, i.e., the E- and O-propositions distribute the predicate class

We are going to start working with the concepts we've just learned:

The way we are going to work with these concepts is understood in terms of what logicians call the (traditional or Aristotelian) square of opposition. This is a diagram that expresses what immediate inferences can be made from one categorical proposition to another. An immediate inference is an inference from one proposition to another. You will likely find some of these inferences are pretty intuitive – they seem to "click" or make sense without you having to think much about them. Still other inferences will feel confusing. It is worthwhile for you to pay attention to each, and ask yourself why you find some so easy to understand, and why others aren't apparently comprehensible.

Before presenting various versions of the traditional square of opposition, let's think through each claim type as a premise for an immediate inference. In other words, beginning with the A-proposition, let's think about the inference to each of the remaining three proposition types, starting with the I-Proposition:

Given the proposition, All S are P, it follows that some S are P:

Notice that the assumption of existential import means that the A-proposition contains within it the I-proposition. This inference is known as subalternation, where the superaltern (the universal proposition) yields its corresponding subaltern (the particular proposition).

Now suppose again that the A-proposition is true. Given this, the O-proposition must be false. These two claims are contradictories: propositions that cannot be simultaneously true or simultaneously false.

Let's think about why A- and O-propositions are contradictories. When we assert the A-proposition is true, we are saying there cannot be even one S outside the P class. This is, however, just what the O-proposition asserts is the case. So, whenever the A-proposition is, or is assumed to be, true, the O-proposition must be false – and vice versa.

Lastly, assume the A-proposition is true. It follows that the E-proposition must be false:

 

You're probably able to recognize that, on the assumption of existential import, the E-proposition contains the O-proposition. So, if the A-proposition is true, the E-proposition must be false. What we cannot infer, however, is that A- and E-propositions are contradictories. We will see that they are not. They are, however, contraries: When a universal proposition is true, its corresponding universal is false. In this case, when A is true, its contrary, E, is false.

In the midst of the discussion about immediate inferences from a true A-proposition, you likely already drew at least one other inference around the traditional square of opposition, namely the inference from the E-proposition to its corresponding particular, the O-proposition. Why? Because, when thinking about the contraries, A-to-E, where we assume A is true, we saw that, assuming existential import, the E-proposition contains the O-proposition:

Similarly, you likely noticed that, just as A and O are contradictories, so also are E and I:

 

Notice that the true E-proposition means that there can't be even one entity in the area of overlap between S and P. The I-proposition, however, claims there is an entity in that area of overlap. So, if the E-proposition is true, the I-proposition cannot be true. The two claim types are contradictories.

Lastly, E and A are contraries, which means, assuming E is true, A must be false. That is because the E-proposition both denies that there can be anything in the overlap area between S and P, and because the E-proposition maintains there is at least one thing in the area of S outside of P. The A-proposition in the traditional interpretation of the universal, that is, on the assumption of existential import, maintains both that nothing is in the area of S outside of P, and that there is at least one S:

We have two more claim types to think about, in terms of immediate inferences around the traditional square of opposition: I and O. Let's start with the I-proposition. We already know that I- and E- propositions are contradictories, so let's start with that inference. In other words, if the I-proposition is true, the E-proposition must be false.

The remaining two inferences are a bit more complicated to think about. That's because the inference from a true I to either an A or an O is undetermined. In other words, if the I-proposition is true, there is no necessary inference: The resulting A-proposition, the superaltern, might be true or it might be false. Similarly, the resulting O-proposition, the subcontrary, might be true or it might be false. The structure of the original claim does not "force" the inference. Here is the Venn for the undetermined superalternation of the I-proposition:

Notice that the I-proposition does not include shading in the area of the S class outside of the P class. So, whether or not there could be anything in the area of S outside of P is an open question. Here are two ordinary language examples that show how the inference from the I-proposition to its corresponding superaltern could yield a true proposition or a false one:

Similarly, a true I-proposition does not yield a necessarily true or necessarily false O-proposition. In other words, subcontraries may be true at the same time, and a true I-proposition may yield a false subcontrary:

There may or may not be anything in the area of overlap between S and P in the O-proposition. Hence, its truth-value is undetermined. Here are two ordinary language examples that show how the inference from the I-proposition to its corresponding subcontrary could yield a true proposition or a false one:

The pattern of inferences is likely becoming clear at this point, as we move into considering the last set of immediate inferences from the assumption that the initial claim is true. A true O-proposition mirrors its subcontrary, the I-proposition. In other words, what is the case about the immediate inferences from the I-proposition to the three other claim types is mirrored in the corresponding particular claim, Some S are not P.  Here are the three inferences from the assumption that the O-proposition is true:

and

and

Below is the traditional square of opposition. As you look at it, think about each inference in terms of beginning with the assumption that the premise – the first categorical proposition – is true:

Notice that we haven't yet discussed inferences from a false premise. In other words, we have not discussed what inferences we may make when the initial categorical proposition is false. In fact, you already know two sets of inferences from a false premise: contradictories. If the A-proposition is false, the O-proposition must be true, and vice-versa. If the E-proposition is false, the I-proposition must be true, and vice-versa. Think about these relations in terms of the Venn diagrams:

We can also infer, based on the other inferences we know, the following necessities:

What is left undetermined is the truth-value of a universal whose corresponding contrary is false. Here are a couple of examples to show the problem:

If you think the traditional (or Aristotelian) square of opposition is complicated, you're correct – at least compared with the modern square of opposition. What makes the traditional square so complicated is the fact that, in most cases, you must know the truth-value of the initial proposition in order to determine the value of the inference – and even then, there are instances where the inference is undetermined.

The modern interpretation of the universal claim type – the A- and E-propositions – does not assume existential import. In other words, there is no assumption of an existing member of the subject class in the universal claim. By suspending judgment about the existence of members in the subject class of a universal claim, the number of inferences in the square of opposition is severely restricted. In fact, there are only two sets of inferences that can be made on the modern square of opposition: contradictories.

Recall the two ways of diagramming the universal claim:

Here is the modern square of opposition:

We have seen some inferences we can make on the traditional and modern interpretation of existential import. More specifically, we've seen when we must infer one claim type from another. Now we turn our attention to what happens when we make internal changes to the quantity and quality of a categorical proposition, as well as the order of the subject and predicate terms. Let's start with an overview of the three types of inferences: conversion, contraposition, and obversion.

All S are P

Converse: All P are S
Obverse: No S are non-P
Contrapositive: All non-P are non-S

No S are P

Converse: No P are S
Obverse: All S are non-P
Contrapositive: No non-P are non-S

Some S are P

Converse: Some P are S
Obverse: Some S are not non-P
Contrapositive: Some non-P are non-S

Some S are not P

Converse: Some P are not S
Obverse: Some S are non-P
Contrapositive: Some non-P are not non-S

Let's begin with taking the converse of a proposition. The inference involves simply switching the subject and predicate positions. The quantity and quality of the proposition are left untouched. The inferences are as follows:

Some of the inferences will feel mentally off, while others will feel obviously correct. Here are the evaluations:

*We will see shortly that, on the assumption of existential import, that is, on the traditional or Aristotelian interpretation of the universal, conversion by limitation makes possible the conversion of the A-proposition.

A few examples may help you think through the valid and invalid inferences. Consider whether or not converting a claim results in one that is logically equivalent to it:

Here are the Venn diagrams that provide a visual demonstration, on both interpretations of the universal – that is, assuming and not assuming existential import, of equivalence and non-equivalence:

 

The only way to successfully convert an A-proposition is by limitation. In conversion by limitation, an inference is made to the I-proposition – subalternation. Conversion of the I-proposition is valid, as we will see momentarily. Hence, by first limiting the A-proposition through subalternation, the resulting conversion is successful (valid). It is important to remember that conversion by limitation is possible only on the assumption of existential import, that is, on the traditional or Aristotelian interpretation of the universal. That is what makes possible the inference to the I-proposition, that is, the A-proposition's subaltern. Such an inference is never valid on the modern interpretation of the universal. We know this because subalternation is not a valid inference.

Notice that the conversion of the I-proposition and the conversion of the O-proposition is the same under either interpretation. Remember that the concept of existential import applies only to universal claims. Here are Venns for the converted particular claims, I and O:

 

When the diagrams do not match up, we can see that the inference from a proposition to its converse is invalid – the two propositions do not make the same claim.

Contraposition is the mirror inference for A- and O-propositions. Just as conversion is valid for E- and I-propositions, contraposition is valid for A- and O-propositions. (Moreover, just as conversion by limitation is valid for the A-proposition, and only on the Aristotelian or traditional interpretation of the universal, contraposition by limitation is possible for the E-proposition.) The internal manipulation is, however, more complex. First, let's walk through the steps, which can be taken in any order, but we will begin with what we do when converting a claim:

Let's try to make sense of each inference, by way of an example: The contrapositive of all dogs are animals is that all non-animals are non-dogs. Another way to put this is in conditional form: If it's not an animal, then it's not a dog. We can see that the contraposed A-proposition says the same thing as the original claim. This is not the case with the contraposed E-proposition. Let's take the example, no dogs are cats. The contrapositive is, no non-cats are non-dogs. This means that whatever is a non-cat is also a non-dog. Surely, however, that can't be correct. It can't be correct to say, no dogs are cats is equivalent to saying, if it's not a cat then it's not a dog. The shading in the area outside of both circles, on both the traditional and modern interpretations, provides us with a visualization of diagramming the class complement of a universal negative.

Contraposition by limitation is possible for the E-proposition only on the assumption of existential import, that is, on the Aristotelian or traditional interpretation of the universal. First, the subaltern is inferred, since as the diagram shows, a true E-proposition contains its corresponding particular, the O-proposition. The contraposition of the O-proposition is equivalent to the original, as we will see now.

 

As with the conversion of particular claims, notice that the contrapositive of the I-proposition and the contrapositive of the O-proposition is the same under either interpretation. Remember that the concept of existential import applies only to universal claims. Here are Venn diagrams for the contraposed particular claims, I and O:

The contraposed I proposition asserts that there is a non-P – there is something outside the P-class – that is also a non-S – something that is outside the S-class. Hence, the x in the area outside both categories. The contraposed O-proposition asserts that there is at least one non-P that is also not a non-S – which is to say, there is at least one non-P that is an S. It is the equivalent of the original.

Lastly, obversion is a valid inference on both interpretations of the universal. It is achieved by a two-step process:

 

Here are some examples to illustrate and elucidate the process of obversion:

Here is a table that sums up the last three immediate inferences, and their evaluations:

	All S are P	No S are P	Some S are P	Some S are not P
Converse:	All P are S Not equivalent to the original (invalid) except on the traditional interpretation, where the I-prop. is inferred and then converted.	No P are S Equivalent to the original (valid)	Some P are S Equivalent to the original (valid)	Some P are not S Not equivalent to the original (invalid)
Obverse:	No S are non-P Equivalent to the original (valid)	All S are non-P Equivalent to the original (valid)	Some S are not non-P Equivalent to the original (valid)	Some S are non-P Equivalent to the original (valid)
Contrapositive:	All non-P are non-S Equivalent to the original (valid)	No non-P are non-S Not equivalent to the original (invalid) except on the traditional interpretation, where the O-prop. is inferred and then contraposed.	Some non-P are non-S Not equivalent to the original (invalid)	Some non-P are not non-S Equivalent to the original (valid)

To review, see:

• Categorical Logic and The Venn Test of Validity for Immediate Categorical Inferences
• Basic Notation
• Universal Statements and Existential Commitment

 

4b. Evaluate the validity of arguments using Venn diagrams

• How do Venn diagrams show an argument is valid or invalid?

A Venn diagram is one way to determine whether or not a categorical syllogism is valid. Here are the steps for completing a Venn diagram:

1. Diagrams are completed only for the premises
2. If one premise is a universal, and the other is a particular, diagram the universal premise first. That's because the shaded area may preclude diagramming an x in a given area
3. When the premises do not force diagramming an x entirely inside or outside an area for a particular claim, the x is placed on the line of a relevant circle

You already know how to diagram individual categorical propositions. You will bring that skill to bear in the process of diagramming the premises of a categorical syllogism. First, rather than diagramming the relevant elements of two overlapping circles, a Venn diagram for the categorical syllogism involves three:

The circles need not be arranged in the order above, as you can see here:

 

A valid argument is one whose premises contain the conclusion. So, when diagramming the premises of a valid argument, the conclusion appears. Below are several examples, following the modern interpretation of the universal. Some are valid, some are invalid.

Example 1:

Example 2:

Example 3:

To review, see:

• Venn Validity for Categorical Syllogisms
• Syllogism
• The Logic of Venn Diagrams

 

4c. Describe the limitations of Venn diagrams as assessment tools

• What are some limitations of the Venn diagram as an assessment tool?

Diagramming categorical propositions is a powerful tool for determining both equivalence and validity. As we have seen, however, the relevant elements are specific. For example, a categorical syllogism is a three-term argument. So, anything more complicated becomes cumbersome for the Venn process. Moreover, the Venn diagram is limited to classes of objects, and so cannot represent individual things.

To review, see Limitations of Venn Diagrams.

 

Unit 4 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• categorical proposition
  
• complement
• contradictories
• contraposition
• contraposition by limitation
• contrapositive
• contraries
• converse
• conversion by limitation
• distribution
• existential import
• immediate reference
• modern system
• obversion
• square of opposition
• subaltern
• subalternation
• subcontraries
• superaltern
• traditional (Aristotelian) system
• undetermined
• universal propositions
• Venn diagram

Unit 5: Fallacies

5a. Explain fallacies of inconsistency, irrelevance, insufficiency, and inappropriate presumption

• What is a fallacy?
• What is an inconsistency fallacy?
• What is an irrelevance fallacy?
• What is an insufficiency fallacy?
• What is an inappropriate presumption fallacy?

We have learned how good arguments are those that not only satisfy the criteria for validity and soundness in deductive arguments, or cogency and strength in inductive arguments, but are also convincing, as a result of those features. More technically, we've seen that an argument is formally convincing when it is valid or strong (for deductive and inductive evaluations, respectively). Even better is when the argument is valid or strong, and the premises are actually true (sound or cogent, for deductive and inductive arguments, respectively). Interestingly enough, we often come across "good" arguments without really committing ourselves to believing what those arguments prove. At the same time, we may be moved to a firm belief by poor arguments. Why? Shouldn't we be utterly convinced by sound and cogent arguments, but dismiss those that aren't any good?

Argumentation is not simply an attempt to prove that something is the case; it's also an attempt to persuade. Unfortunately, a poor argument can convince someone not only to believe something, but also to act upon this belief. In fact, some bad arguments can be very convincing because they fool us into believing the premises justify the conclusion, or by attacking us, or by simply distracting us. Other bad arguments are deceptive because they rely on ambiguity in our language and grammar. This unit focuses on patterns of bad reasoning known as fallacies.

A fallacious argument is deceptive, erroneous, or misleading. In short, a fallacy is a flawed reasoning. With few exceptions, the patterns of fallacious reasoning do not fall into exclusive categories. If you look at a handful of logic and critical thinking texts, you might see different groupings of fallacies, as well as different fallacy identifications. For example, one text might list one pattern of fallacious reasoning under the category Fallacies of Relevance, while another might list it under Fallacies of Weak Induction. While the groupings are meant to make it easier to identify fallacies as instances of a category, they aren't strict. What is important for you to remember is how to identify each fallacy pattern, and why the fallacy is occurring. More importantly, you should focus on being able to analyze the reasoning:

Once you've worked through the fallacies, you will see why they are grouped together as they are, but how they're grouped is not essential to your working understanding of them. The last section of this unit will identify another mode of fallacious argumentation known as formal fallacies. This type of fallacy occurs when an argument looks very similar in structure to one of the deductive argument forms, but is slightly off, thereby misleading you to believe the conclusion follows when, in fact, it does not.

A legitimate question is, "Why is reasoning this way instead of that way better?" There are several ways to answer this question, but the least straightforward one is that there are rules of reasoning that are apparently fundamental to reason itself. Uncovering those rules is part of the task of logic.

Another, related answer is that justifications for claims we make should be relevant to the claim itself, and relevant in a way that actually achieves or demonstrates the conclusion. Think about it this way: The conclusion of a valid argument is demonstrated by the premises; it is extracted, as it were. All that's required to infer the conclusion is an understanding of the meanings of the terms, for example, or the structure of the reasoning.

Here is an example:

Notice that experience is not required to see how the conclusion drops out, as it were, from the premises. Instead, one need only understand the meaning of the phrase, "is identical to", in order to make the inference.

This is also the case when the inference fails:

Liking someone is not transitive in the way that, for example, the spatial relation expressed by "to the left of", is. So, whereas the "likes" argument fails, this one does not:

 

Perhaps what is most important for your studies is not only learning the ways in which one can reason well and reason poorly, but learning and practicing new ways to reason, so you become more in command of your decisions and beliefs. In this way you can begin to see how different logical systems work, and evaluate them for yourself. The ability to think for yourself – not simply to be outspoken and opinionated, but to have a grasp of the variety of ways people can draw conclusions and support them – is crucial to making judgments about the world.

Consider, for a moment, the term "objective". When you say that someone is being objective, or that you know something to be objectively true, you are making an important claim. If reasoning is not objective, it is nearly impossible for anyone to talk to anyone else about it. For in that case, reason would be relative to each individual, and there would be no communication. When we understand the domain of discourse, and we understand how to present and evaluate reasoning, we're in a much better position to succeed in nearly every endeavor we undertake. When we say that we can evaluate reason objectively, we mean that we can talk about how reason works without bringing into the discussion our individual beliefs and attitudes. As a disposition, "objectivity" here means impartiality, or refusal to be swayed by any one person's thoughts, feelings, and attitudes. So, when we're talking about reasoning well, and making claims that we can distinguish between good and bad reasoning, we are saying that we can be objective about how reason works. As mentioned above, this means we think we can create or uncover rules of reason that apply to all rational beings – in other words, we say that reasoning has universal rules.

There is a sharp contrast made between reasoning well and reasoning poorly. This contrast is based on the idea that inferences are made logically; that is, either deductively or inductively. Deductive inferences are said to be necessary, while inductive inferences are said to be probable. There is more ambiguity in evaluating inductive arguments than there is in evaluating deductive arguments. This is because there are degrees of probability, which is the defining feature of inductive arguments, while there are no degrees of necessity, which is the defining feature of valid arguments. In other words, deductive arguments have a much stricter requirement for their being good than do inductive arguments. Even deductive arguments can be subject to degrees of being good because it can be valid but not sound. Moreover, a deductive argument can seem perfectly acceptable until you detect an error due to, say, a grammar problem. This is why it's so important for you to develop your ability to judge the merits of the reasoning you come across in your daily life, and the reasoning you do yourself every day.

Reasoning fails in a variety of ways – so many, in fact, that grouping them according to common features is a helpful way to organize our thinking. Inconsistency fallacies involve the advancement and acceptance of an inconsistency or self-defeating argument. Inappropriate presumption fallacies occur when the argument includes an unreasonable assumption. Relevance fallacies occur when the premises and conclusion are not relevant to each other, or relevant premises are ignored. Finally, fallacies of insufficiency occur when the premises provide insufficient support for the conclusion.

To review, see What is a Fallacy?.

 

5b. Identify common fallacies, including the straw man, gambler's fallacy, begging the question, red herring, ad hominem, appeal to ignorance, appeal to people, complex question, loaded question, and non-sequitur

• What is the straw man fallacy?
• What is the gambler's fallacy?
• What is the begging the question fallacy?
• What is the red herring fallacy?
• What is the ad hominem fallacy?
• What is the appeal to ignorance fallacy?
• What is the appeal to people fallacy?
• What is the complex question fallacy?
• What is the loaded question fallacy?
• What is a non-sequitur?

The straw man fallacy is a distraction fallacy, and so there is no relevance between premises and conclusion. In a straw man fallacy, two arguments are presented. One of the arguments is presented as being so weak – it is presented as being flimsy like a straw man –  that it is easy to dismiss it. At that point, another argument is brought in as a better option. The problem is that the presentation of that first argument is a distortion of the original. It is this distortion that makes the alternative seem more reasonable by comparison. The problem with such reasoning is that it doesn't present the opposing argument accurately. Indeed, one likely would not recognize the original when presented with the distorted version. Moreover, by not presenting the opposing argument as strongly as possible, the alternative presented becomes simply the only one left – it's not one that wins out on its own merits.

Below are some examples of straw man arguments.

Notice that the positions are stated so radically and with inflammatory language that they essentially work as support for the alternative position. The alternative position barely needs any support in order for it to seem appealing. In the first example, an expressed opposition to wearing helmets is not equivalent to, for example, being opposed to wearing seatbelts. In the second example, the proposed requirement to attend public high school is not the same as wanting to take anyone's rights away from them.

Begging the question occurs when the premises implicitly or explicitly are found in the conclusion so that the reasoning becomes circular. In this fallacy, what you want to prove is already assumed to be true, so you're not actually proving anything. "The Bible says God exists. I know this is true because God says so". Whereas in a nonfallacious argument it is the premises that support the conclusion, in a question begging argument, the conclusion supports the premises, which support the conclusion.

Below are some examples of begging the question.

The problem with the first example is it begs the question, how do we know that euthanasia is murder? The second example begs the question, why is it good to use fuel-efficient and hybrid cars? The last example's conclusion is simply a restatement of the premises using slightly different wording.

The gambler's fallacy involves erroneous reasoning about probabilities. More specifically, when we believe that random events can influence each other, such as that a series of coin tosses are related, we commit the gambler's fallacy. Suppose, for example, that you have bought a lottery ticket every day for the past week. Each ticket has lost, but you conclude that, since you've lost a bunch of times in a row, you are going to win soon.

Red herring is a rather strange name, but it's actually quite appropriate to the fallacy that bears it. Hunters presumably used the first, red herring, to train dogs how to hunt. The fish are purportedly dragged across the ground in an effort to distract the dog from a scent. Those best trained will not be deterred from it, however, while the others are led astray. The fallacy does something very  similar. One person tries to distract another or others from a conclusion. Suppose, for example, a teenager comes home after curfew. As the teen walks in the door, the waiting parent says, "Just what were you doing coming home after curfew?" Instead of addressing the question, the teen says, "Look! Is that a cockroach running across the floor?"

That is an obvious (and, one hopes, unsuccessful) attempt to distract from the issue at hand. Below are some examples of red herring.

An ad hominem, (from the Latin, meaning, "to the man") also known as Personal Attack, involves two arguers. The first person presents an argument or a claim, and the second person attacks the first person instead of addressing the argument. These personal attacks take three forms: ad hominem abusive, ad hominem circumstantial, and poisoning the well.

An ad hominem abusive is most typically an attack against something the individual can't control (such as height, sex, race), but also takes the form of any abuse of the person him or herself (such as calling someone fat or ugly). A circumstantial attack is focused not on the person him or herself, but on associations and affiliations, such as someone's religion, area of residence, political party, and so forth. Poisoning the Well occurs when, even before someone makes his or her case (that is, presents an argument), that person is attacked by someone else. 

Below are some examples of ad hominem abusive.

Below are some examples of ad hominem circumstantial. 

Below are some examples of poisoning the well.

A version of the ad hominem type of fallacy is tu quoque (from the Latin meaning, "you too", or "you're one, too") attacks the person instead of that for which the person argues. It is a way of defending oneself against criticism or suggestion, as if to claim that the point the person is making isn't relevant because the person is something of a hypocrite. True though that may be – the person may very well be a hypocrite – that fact does not make the claim any less true or false. A separate argument about the person's character should be mounted in connection with their hypocrisy. Tu quoque can also occur when one person is representing another or others, and the person responding accuses the one (or group) represented of hypocrisy.

Below are some examples of tu quoque.

The fallacy of ad ignorantiam (from the Latin meaning, "to ignorance") involves drawing a conclusion based on ignorance. Someone is supposed to be forced by ignorance of something into accepting a conclusion that is equally unknown.

Below are some examples of ad ignorantiam arguments.

A lack of experience or proof of something existing does not deny that something does in fact exist. In cases of ad ignorantiam arguments, logic would not allow the conclusions. Instead, you would be wise instead to suspend your judgment until better evidence becomes available.

Ad populum is a mass appeal fallacy. (It comes from the Latin meaning, "to the people"). The problem with Mass Appeal, or Appeal to the Masses, is that the appeals are not relevant to the conclusion insofar as they are appeals to emotion. Advertising is always some version of an appeal to the masses. Appealing to mass numbers of people is not necessarily a bad thing. However, since the appeal is made on emotional grounds and not rational ones, it is logically fallacious. If you read any text on rhetoric, (Aristotle's Poetics is a classic example) you will notice that emotional appeals are not only a very effective way to get people to do or think something, but they are also a perfectly legitimate rhetorical device. We must not mistake logical fallacy, in all instances, as something inherently bad. Nevertheless, good reasoning is not fulfilled by fallacy.

Below are some examples of ad populum arguments. 

The first advertisement represents an appeal to being part of the group. Everyone wants to be accepted, and this ad preys upon that desire. Whether or not Splendid jeans are well-made, comfortable, or durable is not addressed. Instead, only the desire of the consumer to be part of the group is addressed.

In the second example, a politician associates voting for him with being patriotic. In this example, the details of the politician's plan for the country is not the focus. Most people feel the emotion of patriotism, of loyalty to one's country, rather intensely. Yet, there is no reason to think that anyone is being patriotic by voting for a particular politician. Love of one's country does not necessitate particular actions.

The third example also appeals to desire. In this case, the desire is to be part of an elite group. It is an appeal to vanity or snobbery. Surely, anyone who thinks they are "cool, hip, and with-it" could succumb to the lure of an advertisement like this.

A complex question fallacy involves asking (at least) two questions in a single grammatical sentence, which thereby makes it difficult to separate out. So, answering one part of the question thereby commits you to answering the other, as well. For example, you might be asked if you want to take courses in Existentialism and Phenomenology. If you say, "Yes", you are committed to both, when you may really only be interested in Existentialism.

A loaded question fallacy is one in which another question is hidden. The answer to that hidden question is already presupposed, and so the question forces a specific answer, regardless of whether or not it is authentic.  In numerous instances in which a "yes or no" question is impossible to answer with a "yes" or a "no", there is a complex question fallacy at work. The danger of the complex question is that it is an attempt to trap someone into saying something that's probably not true.

Below are some examples of the complex question. 

Each example is really two questions. If you answer "Yes" to the question, "Have you stopped lying yet", then you are admitting you lied. If you answer "No", then you are admitting that you're still lying. (Of  course, that could be a lie!) The question is really two questions: "Were you a liar? If so, are you still lying?" The second example asks for details about a theft that is already presumed to have been committed.

A non-sequitur is one way to refer to a formal fallacy. Two of the most common errors in reasoning are flawed versions of the modus ponens and modus tollens arguments – the fallacies, affirming the consequent and denying the antecedent mimic the forms of their valid counterparts. It is because of their similarities to these valid deductive argument forms that they gain their persuasiveness. The good thing about recognizing the form of each fallacy is that it applies to any argument structured in the form of affirming the consequent and denying the antecedent. You do not have to analyze the argument's content; you simply need to know how the fallacies look.

Denying the antecedent: Anytime you see an argument that looks like a combination of the modus ponens and modus tollens forms, but concludes that the consequent is not true because of a negation of the antecedent, you have a formally fallacious argument. Using the counterexample method discussed in Unit 2 makes the error of this form of reasoning clear. 

Below, a few examples will reveal how this argument form is fallacious.

The problem with the first example is that there are other ways for things to become wet other than rain. Similarly, the second example is one way to brown chicken, but so is frying it or baking it. So it is not correct to conclude that it will not be brown if you don't grill it. In the last example, taking the bus is not the only way to get to the movie on time. Perhaps a ride was offered, or you rode your bicycle.

Affirming the consequent looks like a reverse modus ponens, or a modus tollens without the negations. In this fallacy, the consequent of the first premise is asserted, and the conclusion is that the antecedent is therefore true. 

Below are some examples of affirming the consequent. 

In each example, other conditions could have brought about the consequent. So, the fact that the consequent happened means only that at least one condition obtained, not that that particular one did. In the first example, the grass could be healthy because it rained a lot. Similarly, while it is true that five nickels make twenty-five cents, so does a quarter, twenty-five pennies, two dimes and a nickel, and so forth. Finally, someone could be home without the lights being on; they could be sleeping, or the power could be out.

To review, see List of Fallacies.

 

5c. Describe the nature of a cognitive bias and identify examples of cognitive bias

• What is a cognitive bias?
• What are some examples of cognitive bias?

A cognitive bias is a habit of mind, particularly one of which we are unaware. Cognitive biases can influence our thinking away from objectivity, and lead to erroneous reasoning. Examples of cognitive biases are confirmation bias, framing bias, the overconfidence effect. We exhibit confirmation bias when we look for information that confirms an existing view. In so doing, we exclude from consideration evidence that would undermine that view. A framing bias involves influencing an outcome by way of how information is framed. Finally, when we overestimate our skills or abilities, we fall prey to the overconfidence effect.

To review, see:

• Cognitive Biases
• Cognitive Biases Explained

 

Unit 5 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• ad hominem fallacy
  
• ad ignorantiam fallacy
• ad populum fallacy
• affirming the consequent
• begging the question fallacy
• cognitive bias
• complex question fallacy
• denying the antecedent
• fallacies of insufficiency
• fallacious argument
• fallacy
• gambler's fallacy
• inappropriate presumption fallacies
• inconsistency fallacies
• loaded question fallacy
• non-sequitur
• red herring fallacy
• relevance fallacies
• straw man fallacy

Unit 6: Scientific Reasoning

6a. Explain the hypothetico-deductive method and its implications for testing scientific hypothesis

• What is the hypothetico-deductive method?
• What are some implications for using the hypothetico-deductive method to test a scientific hypothesis?

The hypothetico-deductive method (HD Method) is one of the main hypothesis testing methods across scientific disciplines. The method consists in the following steps:

There are some implications for using the hypothetico-deductive method. For example, it requires the scientific hypothesis to be testable. Only empirical hypotheses are testable – no hypotheses about immaterial souls or divinities are testable. In addition, the method does not guarantee the results. Scientific knowledge is always probable, never definitive. The best the method can do, therefore, is offer a strong likelihood that the prediction is correct. So, we need to consider alternative hypotheses. Finally, while a prediction may fail, the theory may still be correct. So, disconfirmation does not disqualify the theory. What we need to do, in that case, is consider auxiliary (additional) hypotheses.

To review, see The Hypothetical-Deductive Method. 

 

6b. Explain Occam's Razor and its implications in real-world scenarios

• What is Occam's Razor?
• What are some implications of Occam's Razor in real-world scenarios?

Occam's Razor, also known as the principle of parsimony, is a rule that asserts the simplest explanation is the best. In the sciences, the rule admonishes unnecessary complications. The practical implications of this rule are not always positive. For example, in biology, where things can get "messy", a simple explanation of phenomena might be wrong, precisely because it's incomplete. More generally, Occam's Razor has influenced scientists in the direction of simplicity, some say to the exclusion of accuracy. For example, it may just be impossible to provide a systematic, simple account of all natural laws – a so-called theory of everything.

To review, see The Scientific Method Explained by a Scientist. 

 

6c. Explain the criteria scientists use to choose among competing hypothesis

• What criteria do scientists use to choose among competing hypotheses?

There are several criteria for determining the superiority of one scientific theory over another. These include observational consistency, a theory's predictive power, how well the theory explains the relevant underlying causal mechanism(s), the theory's fruitfulness in making surprising or unexpected predictions, and the theory's simplicity and coherence.

To review, see What Makes One Scientific Theory Better Than Another?.

 

6d. Discuss notions of causation, causal relations, and Mill's methods for reasoning about causation

• What are some ways to think about causation and causal relations?
• What are Mill's methods?

Causal reasoning is fundamental to empirical knowledge. To connect events in such a way as to explain nature, be it in terms of so-called natural laws, such as gravity, or in terms of biological theories, such as evolution, is to enlist the concept of causation. If you know the causes of objects and events, you are better able to determine how or why something occurred. This knowledge facilitates asserting control. If you know the causes of events and objects, you are better able to manipulate the outcome you want. It is a method for making predictions. If you know the causes of events and objects, you are better to foresee their occurrence or behavior in the future, as opposed to merely guessing.

Not only does the concept of causation explain why things happen, it can also be enlisted to determine responsibility. If you know the causes of objects and events, you are better able to determine who is to be praised or blamed for some occurrence. There is a scene from the 2004 film, "Collateral" that highlights the relationship between moral responsibility and causal explanation. In the scene, a taxi driver played by Jamie Foxx has just dropped his fare off (Tom Cruise's character) outside an apartment building. Moments later, a body falls on top of the taxi. As Foxx's character scrambles out of his taxi to check on the fallen person, Cruise's character walks around the corner. In a flash, Foxx's character makes the connection. "You killed him", he says in disbelief. "No, I shot him", Cruise's character responds. "The bullets and the fall killed him". Here, Cruise's character attempts to shift his responsibility for killing a man with the proximate cause of the man's death – the bullets and the fall.

So, just what is a cause? Another way to ask the question is, what is a cause and effect relation? A cause is said to be that which, when it obtains, gives rise to another event. Cause and effect relations may be perceptibly discrete, such as a knife and a cut on the skin. Others are not perceptibly distinguishable, such as heat and fire. What makes one set of events a causal relation, and another set a mere coincidence, is part of what we consider knowledge. Indeed, causation is the foundation of the theoretical and experimental sciences, which contribute to our repository of knowledge. Theoretical science is the practice of proposing theories about how nature works. The conceptual framework for how we think about things occurs in this activity. Einstein's theorizing provided experimental science with ideas to test. Creating good experiments, of course, significantly impacts the conclusions drawn. Because causal relations "in real life" are wildly complicated – there are typically too many variables for which a researcher can account – controlled experiments, while artificial, allow us to more clearly see causal relationships. Controlled experiments typically involve control groups, i.e., setups that are identical to the ones in the experiment, except they are not exposed to the factor being tested for the causal relation.

Scientific reasoning proceeds by way of cause and effect analysis. A hypothesis is tested through carefully constructed experiments to see if a causal relation obtains between objects or events, be they proximate or ultimate. A legitimate test of a causal hypothesis is that the prediction must be verifiable. The prediction must not be trivial; and the prediction must have a logical connection to the hypothesis.

Moreover, experimentation is typically controlled: There are multiple experimental setups that differ only by one variable. Consider the current CERN particle accelerator project, which tests different particle physics theories. The Large Hadron Collider, which sits beneath the ground in a tunnel covering a circumference of 17 miles, was built specifically to carry out experiments that attempt to answer, among other things, causal questions about particles. It is one example – albeit a physically enormous one – of experimental science.

Requirements for causal relations are:

Let's revisit the concepts of sufficiency and necessity. It is arguably the most common language used in discussions about cause and effect relations, be they natural or artificial:

• A sufficient condition is that whose presence guarantees a particular outcome. We can say that A is a sufficient condition whenever A, B follows.
• Eating an apple is sufficient for me to have food in my stomach. Eating an apple guarantees I have food in my stomach, though we would not say I must or it is required for me to eat an apple.
• A necessary condition is that whose absence prevents a specific outcome from obtaining. A condition is necessary when, without it, an outcome does not obtain. We can say that B is a necessary condition for A only when B, or without B, not A.
  • Oxygen is necessary for fire. Without oxygen, fire cannot obtain, though the occurrence of oxygen does not bring about fire.
• Joint necessary and sufficient conditions are those conditions that are reciprocal. Whenever A, then B, and whenever not-A, then not-B. A condition is necessary and sufficient when not only the occurrence of A guarantees B, but also, when A is absent, B does not obtain. A necessary and sufficient condition is what must happen, and what guarantees an outcome.
  • Six people eat dinner in a restaurant. Liz has soup, a hamburger, ice cream, French fries, and mixed vegetables. Tom has salad, a hamburger, French fries, and ice cream. Sue has French fries, a hamburger, and salad. Meg has fish and mixed vegetables. Bill has French fries, a hamburger, and soup. Andy has soup and ice cream. Later, Liz, Tom, and Andy get sick from something they ate, but Sue, Meg, and Bill do not. Therefore, ice cream is a necessary and sufficient condition for Liz, Tom, and Andy becoming sick.
  • Fulfilling all your requirements and paying your tuition is necessary and sufficient for your eligibility to graduate.
  • At most colleges, taking 12 credit hours is a necessary and sufficient condition for full time student status.

A related way of talking about causation is an inference to the best explanation. When researchers conclude that a hypothesis is the best explanation for a set of facts, they at least sometimes mean that it is the most probable cause. Not all explanations are equivalent, however, to causal inferences. A doctor who says, "I won't prescribe antibiotics, because you have a viral, not a bacterial, infection", offers an explanation for why you will not receive a prescription for an antibiotic. (Here, the point is not to say that the explanation is a case of an inference to the best explanation, but rather to distinguish explanation from causation).

On the other hand, a doctor who says, "Your throat is sore and your tonsils are swollen because you have strep throat" makes a causal claim – in this case, presumably, the best explanation of the symptoms is the causal claim of strep throat. An example of an inference to the best explanation, which is not a causal claim, could be something like this: "I must have some sort of infection. I've been in bed for over a week and my symptoms are consistent with infection – sore throat, clogged ears, and a headache, all of which were preceded by a fever". Here, the symptoms are taken as evidence best explained by an infection.

A look at 19th-century philosopher, J.S. Mill's methods for determining causal relations can expand our understanding of causal reasoning. Mill formulated five experimental tests of causal relations:

The method of difference involves looking at a situation in which the relevant elements are found to be identical in all aspects identified, except one case. In that one case, an event occurs – an effect – that has not occurred in the others. The difference between the other identified aspects of the situation and the one in question reveals the cause of the effect. Consider the following situation:

The method of difference does not guarantee the identification of a cause. The problem with the method of difference is there can be numerous differences that could account for the effect. Ursula could have become ill from something else, for example. Part of the issue is determining, in this case, at least, the relevant time slice to consider.

The method of agreement looks, as the name suggests, for a commonality, not a difference. If, in all cases where an effect occurs, there is a single prior factor, X, that is common to all those cases, then X is the cause of the effect. Consider the following situation:

Here again, the method does not provide a guarantee that the cause has been identified. It does not account for other possible causes, circumstances that, though common to the people who became ill, were not considered as potential causes. These could include, for example, having drunk the same water, or being exposed to a bug, etc.

What we can see so far is that the methods highlight the desirability of a controlled experiment. "Real life" situations are remarkably complex. There are many variables unaccounted for, so a controlled environment, combined with the right sort of hypothesis and experiment, increases the likelihood that the correct cause and effect relation can be identified.

By combining the first two methods together, the joint method of agreement and difference provides a more robust test of a causal relationship. This method involves comparing situations in which the commonalities and differences are sifted out. Mill explains the method this way: "If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance, the circumstance in which alone the two sets of instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon". Here is a situation in which the joint method is employed:

The common element among Hector's horses is they are all fed grain. They also drink out of water buckets that are filled multiple times a day. The only difference between them is that Champ is fed a grain with a lower protein percentage than his stable mates. One morning, Champ does not eat his grain, but Buster does not eat his, either. The percentage of protein should not be the issue, and with all other variables being accounted for, Hector notices that both Champ and Buster's water buckets are empty. The cause of their refusal to eat their morning grain is determined to be dehydration. For this causal relation to be strengthened, Hector would need to conduct experiments (which he likely would not want to do, in order to protect his horses' wellbeing and happiness), in which water is withheld for a certain period before feeding time. Moreover, it should not be the case that the horses refuse their morning grain when they are fully hydrated. In short, the joint method of agreement and difference requires that the effect obtains whenever the cause is present, and that the effect is absent when the cause is absent. Another way to put the relation is as follows: the cause is both necessary and sufficient for the effect, as the cause guarantees the effect, and without the cause, the effect does not obtain.

The method of residues is, Mill declares, "a peculiar modification of the method of difference" (A System of Logic, p. 490). It relies on previous knowledge of certain causes and effects. It also involves a more complicated notion of a cause – it assumes more of a causal cluster for a given effect. To take a simple example, suppose you have a cat, Maurice, who is incredibly nervous about being away from home, and even more upset about going to the dreaded veterinarian's office. So terrified is Maurice that you cannot let him out of his carrying case in order to weigh him. Worse yet, suppose that he howls and yowls in agony whenever you put the carrier down. The vet needs to weigh Maurice, and asks you to hold the cat in the carrier while you step on the scale. You already know how much you weigh, and you already know how much the carrier weighs – you were prepared for this drama, given Maurice's consistent behavior over years of visits to the vet. So, your weight, and the carrier's weight is subtracted from what the scale reveals – Maurice's weight is the residual number. Here is a situation in which the method of residues is employed:

The method of concomitant variation is employed when we want to understand a causal relation in terms of proportionality. We take for granted, for example, that an intense headache will be alleviated by a painkiller, high cholesterol lowered by statins, and a reduction in salt intake generally correlates with a reduction in blood pressure. A variation in a cause, in this method, sees a concomitant variation in effect. The relation can be inverse or parallel. Here is a situation in which an anticoagulation nurse – a nurse who specializes in anticoagulant therapy – uses the method of concomitant variations to determine the dose that will yield a blood test that comes back between the values of 2.0 and 3.0:

To review, see:

• Causation
• Causal Inferences 
• Mill's Methods 

 

6e. Explain the difference between correlation and causation

• What is the difference between correlation and causation?

Causal relations include correlations, but correlation is not the same as causation. A correlation between events is temporal, i.e., events tend to occur at or around the same time. These events might also be related in some way. So, for example, there is a causal relation between the speed a car travels and gasoline consumption while there is only a correlation – in this case, a temporal connection – between a rooster crowing and sunrise. After all, a rooster doesn't cause the sunrise and, even though roosters may begin crowing at sunrise, they also crow throughout the day. So, it would be erroneous to say the sunrise causes a rooster to crow.

To review, see Correlation and Causation.

 

6f. Use visualization tools to represent causal relations

• What are some ways causal relations are visually represented?

There are several ways causal relations can be diagrammatically represented.

 

 

To review, see Causal Diagrams. 

 

6g. Explain several common fallacies when reasoning about causation, such as false cause

• What are some fallacies associated with causal reasoning?

False cause is the weak counterpart to cause and effect arguments. In this fallacy, a causal relationship is asserted between premises and conclusion, when it most likely does not exist. One way to detect this mistake is to ask whether or not a more plausible cause could be found for the supposed effect. Below are some examples of false cause. 

Just because something happens every time I do something does not mean that I am the cause of it. A better explanation of why the lights go down in the movie theater is that someone who works in the theater dims them, because it's time for the movie to begin.

Similarly, my waking up may be a sign to my cats that breakfast is about to come, but the more probable cause of their running to the kitchen is their expectation of being fed. In fact, most cats run to the kitchen whenever someone heads in that direction, regardless of whether or not that person's just awakened!

Finally, there is no scientific evidence to support the myth that kissing frogs will give you warts. Indeed, the myth is about toads, anyway, and even then there is no evidence to conclude that kissing a toad will cause warts. A dermatologist can provide a much more plausible cause.

To review, see Correlation and Causation.

 

Unit 6 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• causal network diagrams
  
• causation
• cause
• correlation
• false cause
• fishbone diagrams
• hypothetico-deductive method
• joint method
• method of agreement
• method of concomitant variation
• method of difference
• method of residues
• Mill's methods
• Occam's razor
• proximate cause

 

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1. https://home.cern/topics/large-hadron-collider
2. John Stuart Mill, A System of Logic, Vol. I.: https://www.earlymoderntexts.com/assets/pdfs/mill1843book1.pdf

Unit 7: Strategic Reasoning and Creativity

7a. Illustrate several types of problems and explain how to understand and problem-solve for each

• What are several types of problems and how do we begin to solve them?
• What are some problem solving strategies?

When we have a real problem to solve – there is evidence of a problem, as opposed to mere speculation – organizing our thinking about it is the first step toward solving it. First, let's consider the difference between a proposed problem versus a real problem. The 2020 U.S. Presidential Election proceeded according to the practices of previous elections. There were no significant abnormalities that would suggest problems such as election fraud. Officials from both parties oversaw the process and conducted the usual quality control checks. Nevertheless, then-president Donald J. Trump claimed there were numerous problems. Even after he was presented with myriad evidence to the contrary, Trump and his supporters insisted that the U.S. election system is rife with corruption and other problems. The question before citizens is one of evidence. If we say we have a problem, we need evidence of such.

One of the first steps toward problem solving is to ascertain the type of problem we face. To do so, we ask empirical questions, conceptual questions, and evaluative questions:

• Empirical questions are questions about the facts; typically, these questions are answered by observation or experiment (Are COVID-vaccinated individuals more or less likely than non-vaccinated individuals to contract the Delta variant?)
• Conceptual questions are questions about the meanings of words and the reasoning involved in arriving at an answer without observation and experimentation (What is justice?)
• Evaluative questions are questions about moral norms and judgments (Is this political system just?)

Sometimes, combinations of these question types are enlisted in clarifying the problem (Are poor people more likely to contract COVID than non-poor people?).

Mathematician George Pólya's 1971 book, How to Solve It, lays out a four-step process for problem solving:

1. Understand the nature of the problem (look at the classifying questions above).
2. Draw up a plan: The systematic approach to solving a problem includes a plan for action involving time, resources, and preparation.
3. Try out the plan: Monitoring the progress of the plan and recording errors and special considerations will help you understand the outcome.
4. Monitor the outcome: Results will inform next steps.
   

To review, see Classifying Problems and Solving Problems. 

 

7b. Use visualization tools to analyze problems

• What are some visualization tools used to analyze problems?

There are several visualization tools used to analyze problems.

• Flowcharts use shapes to a process or plan:

The meanings of some of the more common shapes are as follows:

The terminator symbol represents the starting or ending point of the system.

A box indicates some particular operation

This represents a printout, such as a document or report.

A diamond represents a decision or branching point. Lines coming out from the diamond indicates different possible situations, leading to different sub-processes.

It represents material or information entering or leaving the system. An input might be an order from a customer. An output can be a product to be delivered.

This symbol would contain a letter inside. It indicates that the flow continues on a matching symbol containing the same letter somewhere else on the page.

As above, except that the flow continues at the matching symbol on a different page.

Identifies a delay or bottleneck.

Lines represent the sequence and direction of a process.

For further information, please refer to:

• International Organization for Standardization (ISO), ISO 5807, Information processing – Documentation symbols and conventions; program and system flowcharts.
• American National Standard, ANSI X3.6-1970, Flowchart Symbols and their Usage in Information Processing.

• Decision trees represent possible consequences of a decision:
  

  
  

  
  

• Causal diagrams (see 6f)

To review, see Charts and Diagrams.

 

7c. Explain the principles of creative thinking and their implications

• What are some principles of creative thinking?
• What are some implications of creative thinking?

Creative thinking can be methodical, rather than haphazard (or inspirational). Developing creative skills generally follows these principles:

• Build new ideas from old elements: Ideas don't occur in a vacuum. They're typically innovations on and novel creations from existing ideas.
• Critically evaluate new ideas: Don't believe every idea is as good as every other.
• Seek connections between ideas: What may seem like disconnected ideas can actually work together to generate something novel or innovative.

Creative thinking is not simply for artistic endeavors. It works across subjects and problems. Many scientists, for example, actively engage these principles as part of their larger research projects.

To review, see Three Basic Principles of Creative Thinking. 

 

7d. Compare the methods for approaching problems creatively as a means to think creatively about real-world problems

• What are some methods for approaching problems creatively?

The principles outlined in the previous section can be followed in a stepwise fashion:

1. Conduct research, so you know what others are thinking and have done to solve the problem.
2. Explore connections between ideas, which likely involves studying otherwise disparate fields.
3. Give yourself time to "digest" what you've researched and studied, since intellectual development takes time, similarly to digesting food.
4. Apply the idea to a relevant problem, review the results, and follow-up, so you have some practical and actionable outcomes.

Some other methods offer additional specifics for problem solving:

• Create a feature list, which is a list of an object or process' features. Once generated, you can think of the features in isolation and in connection with the other features.
• Construct an analogy, which compares two objects or events, with the aim of drawing a conclusion about one on the basis of that comparison.
• Create a systematic search method of other possible solutions, which can help you narrow or broaden your thinking.
• Try a shift in perspective, which can help you see the problem and potential solution(s) in new ways.
• Take advantage of the numbers using group creativity, whose brainstorming, for example, can produce new ideas.

To review, see The Creativity Cycle and Creative Heuristics and Group Creativity.

 

Unit 7 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• causal diagrams
  
• conceptual questions
• creative thinking principles
• decision trees
• empirical questions
• evaluative questions
• flowcharts
• problem-solving
• proposed problem
• real problem