2 Ask Alice: Evaluating Inductive and Deductive Arguments in Wonderland

A Little More Logical | Brendan Shea, PhD

In this chapter, we follow Professor Alice, a grown-up version of the curious girl from Lewis Carroll’s “Alice in Wonderland,” as she guides us through the intricacies of argument evaluation. Drawing upon her childhood adventures in Wonderland and the works of eminent logicians, Alice illuminates the key concepts and techniques for assessing the strength and validity of arguments. Using relatable examples from the fantastical realms of Wonderland and Oz, she illustrates the difference between deductive and inductive reasoning, the importance of assessing both the form and truth of premises, and common pitfalls to avoid in argumentation. By the end of this journey, readers will have acquired a toolkit for navigating the labyrinthine world of logic, empowering them to critically evaluate arguments in academia and everyday life. So, let us follow Alice down the rabbit hole of reason, where rigorous analysis meets creative storytelling, and emerge as more discerning thinkers.

Learning Outcomes: By the end of this chapter, you will be able to:

• Differentiate between deductive and inductive arguments and identify common forms of each.
• Evaluate deductive arguments for validity and soundness, assessing both the logical structure and truth of premises.
• Evaluate inductive arguments for strength and cogency, considering the level of support premises provide for conclusions.
• Apply the concepts of good form and true premises to real-world arguments, including those in literature, media, and personal encounters.
• Recognize and avoid common fallacies and flaws in reasoning, such as affirming the consequent or relying on false premises.

Keywords: Logic, Argument, Deductive reasoning, Inductive reasoning, Validity, Soundness, Strength, Cogency, Premises, Conclusions, Modus ponens, Modus tollens, Hypothetical syllogism, Disjunctive syllogism, Constructive dilemma, Destructive dilemma, Affirming the consequent, Denying the antecedent, Categorical syllogism, Generalization, Statistical syllogism, Analogy, Causal inference, Predictive argument, Argument from authority, Argument from sign, Inference to the best explanation, Algorithm, Gender bias, Stereotype threat, Imposter syndrome

Hello, Alice!

Hello, dear readers! I am Professor Alice, a fictional character sprung from the imaginative world of Lewis Carroll’s “Alice in Wonderland.” As a child, I had the most extraordinary adventures in a fantastical place called Wonderland. It was a realm where rabbits wore waistcoats, cats disappeared leaving only their grins, and queens played croquet with flamingos. These adventures, filled with riddles and peculiar characters, were not just flights of fancy; they were intricately woven with threads of logic and philosophy.

The real author of “Alice in Wonderland,” Charles Lutwidge Dodgson, known by his pen name Lewis Carroll, was a logician and mathematician at Christ Church, Oxford. His love for logic and playful intellectual puzzles is evident throughout the narrative of Wonderland. Carroll’s work cleverly intertwines elements of logic, wordplay, and philosophy, making the story a delightful yet thought-provoking journey.

In my world (one where I am real!), I am now grown up and Cambridge University alongside esteemed figures such as Bertrand Russell, Ludwig Wittgenstein, Alan Turing, GEM Anscombe, and Susan Stebbing, who were among the most important logicians of the modern era (and probably of all time). They all read books about me as children, I am sure!

In the following lectures, I aim to demystify the intricacies of logic and critical thinking, drawing parallels between the philosophical dialogues of Wonderland and established theories in logic. My goal is to inspire a sense of wonder and curiosity about the world of logic, much like the curiosity that led me down the rabbit hole into Wonderland.

What is Logic? What is an Argument?

Logic, in its essence, is akin to navigating the perplexing yet intriguing world of Wonderland. It is the study of arguments, or forms of reasoning, where one learns to distinguish good reasoning from bad. This pursuit is not unlike determining whether advice in Wonderland is sensible or nonsensical. An argument in logic is a collection of statements, with one or more of these statements (the premises) providing support to another (the conclusion). For instance, if I argue that the Mad Hatter throws the most interesting tea parties in Wonderland, my premises must logically support this conclusion, just as I would need evidence to believe any claim in Wonderland.

A statement or proposition is a sentence that can be true or false. For example, saying “The Cheshire Cat can disappear at will” is a statement, as it’s either true or false within the context of Wonderland. But not every sentence is a statement. Exclamations or commands, like the Queen of Hearts’ infamous “Off with his head!” cannot be classified as true or false and hence are not statements in the logical sense.

Studying logic has multiple benefits. It enhances critical thinking, much like how Alice needed to assess the situations she encountered in Wonderland. It’s also crucial in decision-making, akin to choosing which path to follow in a Wonderland forest. Logic is integral to scientific reasoning, understanding mathematical proofs, analyzing political discourse, and even in computer science, reflecting the rules that seemingly govern the unpredictable Wonderland.

Consider a Wonderland-themed argument: “All residents of Wonderland are whimsical characters. The Cheshire Cat is a resident of Wonderland. Therefore, the Cheshire Cat is a whimsical character.” Here, the premises logically lead to the conclusion. This argument can be critiqued either by challenging the form or the truth of the premises.

Putting arguments in standard form helps in clarifying their structure. Let’s take a Wonderland example: “The White Rabbit is always anxious because he is constantly late and worried about the Queen’s reaction.” In standard form, this would be:

• The White Rabbit is constantly late.
• He is worried about the Queen’s reaction.
• Therefore, the White Rabbit is always anxious.

This format, much like a guide through the winding paths of Wonderland, reveals the structure of the argument, allowing for easier evaluation and understanding. By laying out each premise and how it supports the conclusion, the argument becomes clearer and more straightforward to follow, just as I sought clarity in my Wonderland adventures.

What is the Difference Between Arguments and Non-Arguments?

In the realm of logic, much like navigating the twists and turns of Wonderland, understanding what constitutes an argument is key. An argument must contain a factual claim about some statement(s) being true, and a claimed inferential link that justifies believing in the conclusion. It’s akin to Alice making sense of the various claims and conclusions she encounters, from the Caterpillar’s advice to the Queen of Hearts’ decrees.

Not everything is an argument, a concept we can illustrate with examples from both our world and Wonderland. For instance, warnings and pieces of advice, like the Cheshire Cat’s suggestion to Alice not to go certain ways in Wonderland, are not arguments. They don’t provide reasons to support a conclusion. Similarly, a simple statement of belief, such as the Mad Hatter’s belief that it’s always tea time, isn’t an argument either, as it lacks an inferential link to other statements.

Reports, like a narrative of a journey through Wonderland, provide information but don’t argue for a particular viewpoint. Similarly, expository passages, like a detailed description of the Mad Tea Party, elaborate upon a topic without arguing for it.

Distinguishing between arguments and non-arguments can sometimes be tricky. An illustration, like describing various Wonderland characters to clarify the nature of Wonderland, is not an argument. It uses examples to explain, but doesn’t argue. Conversely, an argument from example uses specific instances to support a general conclusion. For instance, if I argue that Wonderland creatures are unpredictable, citing examples like the sudden appearance and disappearance of the Cheshire Cat, I’m making an argument.

The difference between an argument and an explanation is also crucial. An explanation is a group of statements where one (the explanans) provides a reason or cause for another’s truth (the explanandum). Unlike an argument, it doesn’t provide reasons to believe the explanandum but assumes its truth. For example, explaining why the Cheshire Cat disappears (the explanans) does not argue for the fact of his disappearance (the explanandum), it assumes it as true and provides a reason.

Lastly, let’s consider conditional statements, which are not arguments but often serve as premises or conclusions in arguments. A conditional statement is of the form ‘if antecedent A, then consequent C’. For example, “If Alice grows too tall, she can’t fit through the small door.” The antecedent (Alice growing tall) is a sufficient condition for the consequent (not fitting through the door). These statements are crucial in logic, much like understanding the rules of a potion that makes people grow or shrink is crucial for survival in Wonderland. They lay out a relationship between conditions but don’t make an argumentative claim.

What is the Difference Between Inductive and Deductive Arguments?

In logic, much like navigating the court of the Queen of Hearts, understanding the nature of arguments is crucial. There are two main types of arguments: deductive and inductive, each evaluated differently based on their structure and the strength of their inferential links.

A deductive argument claims that if the premises are true, then the conclusion must necessarily be true. It’s a bit like the Queen of Hearts declaring, “All flamingos are used for croquet. This bird is a flamingo. Therefore, it must be used for croquet.” In deductive reasoning, the premises provide absolute support for the conclusion. An example of a good deductive argument is: “All roses in the Queen’s garden are red. This flower is from the Queen’s garden. Therefore, this flower is red.” This argument is valid because its premises logically lead to the conclusion. A bad deductive argument, on the other hand, might be: “All roses in the Queen’s garden are red. All daisies are flowers. Therefore, all daisies are red.” Despite the truth of the premises, the conclusion doesn’t logically follow.

Inductive arguments, contrastingly, suggest that if the premises are true, the conclusion is likely, but not certain, to be true. In Wonderland, this is akin to me observing, “The Queen of Hearts often loses her temper at court. Therefore, she is likely to lose her temper at the next trial.” Inductive reasoning deals with probability rather than certainty. A strong inductive argument might be: “The Queen has sentenced most trespassers to beheading. This person has trespassed. Therefore, they are likely to be sentenced to beheading.” A weak inductive argument would be: “The Queen has sentenced a few trespassers to beheading. This person has trespassed. Therefore, they are likely to be sentenced to beheading.” Here, the conclusion is less likely to be true because the premises don’t strongly support it.

It’s important to remember that the ‘goodness’ or ‘badness’ of an argument isn’t (just) about the truth of the premises, but about how well the premises support the conclusion. A good argument with true premises gives us strong reason to believe the conclusion. In cases where it’s unclear whether an argument is deductive or inductive, the principle of charity should be applied, representing the argument in the way that gives it the best chance of being successful.

To draw an analogy, a deductive argument is like playing chess, where each move is definitive and governed by clear rules, much like the straightforward orders of the Queen of Hearts. An inductive argument, however, is like playing poker, where outcomes are influenced by a mix of skill, luck, and changing information. Even in chess, inductive reasoning is at play, as players make predictions and generalizations to narrow down realistic moves, reflecting the unpredictable nature of interactions at the Queen’s court. In both cases, there’s a possibility of being wrong, underscoring the importance of careful reasoning in both deductive and inductive arguments.

How Can I Tell Deductive From Inductive Arguments?

Determining whether an argument is deductive or inductive can often feel like trying to make sense of a riddle in Wonderland. However, there are some useful “rules of thumb” to guide us through this process, much like following certain cues and signs in Wonderland.

• Is it Possible for the Conclusion to be False if the Premises are True? If it seems utterly impossible for the premises to be true and the conclusion false, the argument is likely deductive. Imagine the Queen of Hearts saying, “All hearts are red. This card is a heart. Therefore, it is red.” Here, it’s impossible for the premises to be true and the conclusion false, making it deductive. Conversely, if it’s very unlikely (but not impossible) for the premises to be true and the conclusion false, then the argument is probably inductive. For example, if I observe, “The Queen of Hearts often sentences people to beheading for minor offenses, so she will probably do so again,” this is inductive, as there’s a chance, however small, that she might not.
• Does the context require certainty? In contexts requiring absolute certainty, like mathematics, arguments tend to be deductive. In Wonderland, if a statement is made with the certainty of a rule in a game of croquet by the Queen (where rules are absolute), it’s likely deductive. In contrast, predictions about weather, politics, or even the behavior of Wonderland characters, which require less certainty, are usually inductive.
• Does the argument’s language provide “clues”? Deductive arguments often use words like “necessarily,” “certainly,” or “absolutely,” much like the Queen’s definitive declarations. Inductive arguments, in contrast, use terms like “probably,” “likely,” or “it is reasonable to believe that,” similar to making educated guesses about the next move of the Cheshire Cat. However, this isn’t always reliable, as people (and Wonderland characters) often claim certainty for what is actually inductively supported.
• What Kind of Argument is it? Some types of arguments are typically deductive or inductive. For instance, arguments in mathematics are usually deductive, while arguments predicting the behavior of the March Hare or the Cheshire Cat are typically inductive.

It’s crucial to remember that these rules aren’t infallible, much like how the rules in Wonderland often bend and twist. They’re not 100% foolproof and are best used in combination. However, they do provide a good starting point for distinguishing between deductive and inductive arguments, guiding us through the labyrinth of logical reasoning as effectively as any map or guide in Wonderland.

What are Some Common Types of Deductive Arguments?

In the world of deductive arguments, much like navigating the rules of a game in Wonderland, there are several common types that are useful to understand. Each type has its own structure and way of deriving the conclusion from the premises.

• Argument from Mathematics—This type of argument uses mathematical reasoning, like arithmetic or geometry, to derive the conclusion. For example, if we were in Wonderland and said, “The March Hare has ten teacups arranged in two equal rows, therefore he has five teacups in each row,” we’re using an argument from mathematics. It’s a straightforward application of mathematical principles to reach a conclusion.
• Argument from Definition—Here, the conclusion is derived solely from the definition of the terms used. In Wonderland, this might look like, “A Bandersnatch is defined as a creature with swift movement. Therefore, a Bandersnatch moves quickly.” This type of argument relies on understanding and applying definitions.
• Categorical Syllogism—This is a two-premise deductive argument where each statement (premise and conclusion) makes a claim about all, some, or no members of a category. For example, if we say, “All creatures in Wonderland are whimsical. The Cheshire Cat is a creature in Wonderland. Therefore, the Cheshire Cat is whimsical,” we’re using a categorical syllogism. It works by applying a general statement to a specific case to derive a conclusion.
• Hypothetical Syllogism—This type involves conditional (“if-then”) statements in both premises and the conclusion. In Wonderland, a hypothetical syllogism might be: “If the White Rabbit is late, he rushes. If he rushes, he forgets his gloves. Therefore, if the White Rabbit is late, he forgets his gloves.” It’s about linking conditional statements to arrive at a conclusion.
• Disjunctive Syllogism—This form of argument follows the structure of “Either X or Y is true. But X is false. So, Y must be true.” For instance, in Wonderland, we might encounter a situation where it’s said, “Either the potion makes me grow taller or it makes me shrink. It didn’t make me grow taller. Therefore, it must make me shrink.” This type of argument uses a process of elimination to arrive at the conclusion.

Each of these types of deductive arguments has its own rules and structure. Understanding these structures will eventually allow to determine whether the arguments are “valid” (have a good form) or “invalid”( Have a bad form).

What are Some Common Types of Inductive Arguments?

In inductive arguments, unlike deductive arguments, we recognize that it is possible for the premises to be true and the conclusion to be false. Each of the following types or inductive argument uses different forms of premises to reach conclusions, often about uncertain or future events.

• Predictions—These arguments use premises about the past or present to make conclusions about the future. For example, if I were to say in Wonderland, “The White Rabbit has been late every day for the past month; therefore, he will likely be late tomorrow too,” I’m making a prediction based on past occurrences.
• Generalizations—Generalizations draw conclusions about larger groups based on observations of smaller subgroups. Imagine if Tweedledum and Tweedledee, after observing a few flowers in the garden singing a song about the sun, concluded that “All flowers everywhere can sing lovely songs.” They would be making a generalization from a small sample (the few flowers they’ve encountered) to the entire population of flowers.
• Causal Inferences—These arguments draw conclusions about causes or effects based on non-causal premises. For instance, if I observed that “Characters who frequently visit the Mad Hatter’s tea parties tend to be more eccentric, so the tea parties must cause eccentricity,” I would be making a causal inference, similar to concluding smoking causes lung cancer from observing higher incidence rates among smokers.
• Arguments from Analogy—These rely on similarities between two or more objects to conclude they must be similar in other ways too. Using Wonderland logic, one might argue, “Just as the Red Queen runs but stays in the same place, so too do we often work hard but make no progress. Thus, our efforts might be as futile as the Red Queen’s running.” This argument draws an analogy between two seemingly disparate situations.
• Arguments from Authority—These arguments conclude something is true because an authority claims it is. If the Caterpillar, known for his wisdom in Wonderland, declared that “The Jabberwocky can be defeated only with a Vorpal sword,” and I decided to believe him, this would be an argument from authority, akin to trusting an astronomy textbook about the distance of the sun from the earth.
• Arguments from Signs—These arguments conclude that something is true because of a sign left by an intelligent being. For example, in Wonderland, if I found a signpost pointing to the Queen’s castle and concluded that the castle must be in that direction, I would be making an argument from a sign, similar to using maps to deduce geographical locations.
• Arguments to the Best Explanation—These arguments conclude that a hypothesis is true because it is the best explanation for a known fact. In Wonderland, if the Cheshire Cat disappeared leaving only his grin, and I concluded, “The Cheshire Cat has vanished but left his grin, so it must be that in Wonderland, grins can remain even when their owners disappear,” I would be making an argument to the best explanation, like concluding Shelley is sick because she missed class and illness is the most plausible reason.

Each type of argument, from predictions to arguments to the best explanation, offers a unique way of interpreting and making sense of the world, much like the diverse approaches one must use to understand and navigate the whimsical and often paradoxical world of Wonderland.

Discussion Questions: Classifying Arguments

• What is the main difference between a deductive argument and an inductive argument? Can you use the examples (from Wonderland or real life) to explain this in your own words?
• Can you think of an example from your daily life or from a movie you’ve seen where someone used deductive reasoning? What about inductive reasoning?
• What do you think makes an inductive argument strong or weak? How is this different from how we judge whether a deductive argument is good or bad?
• In inductive reasoning, we talk about ‘likelihood’ and ‘probability’. What do these terms mean to you? How do they fit into the way we think about what might happen next in a story or in real life?
• The text compares deductive reasoning to playing chess and inductive reasoning to playing poker. Why do you think these games were chosen for the comparison? How do they relate to the way these arguments work?
• How do you think you use inductive and deductive reasoning in your everyday decisions? Can you give an example from your own experience?
• Sometimes, arguments can be straightforward, and other times they can be more complex. Can you think of an example of a simple argument and a more complex one? Are they inductive or deductive?

 

 

Sample Problem: Classifying Arguments

Classify the following passages as inductive arguments, deductive arguments, or non-arguments.

Passage	Inductive, Deductive, or Not an Argument?
Alice is a homo sapiens. Therefore, she is a human.	This is deductive since the conclusion follows necessarily from the meaning of the words in the premises.
Alice grows taller after drinking from the bottle marked “Drink Me.” Since the Caterpillar smokes a hookah, he probably changes size when he does this.	Inductive. We reason that because Alice exhibits certain effects (growing taller) from an action in Wonderland, other characters may exhibit similar effects from their actions. This is an argument by analogy, albeit a weak one due to the lack of direct correlation between the actions of drinking and smoking.
Alice attended a tea party with the Mad Hatter and the March Hare. Alice found the experience to be quite bizarre.	This is not an argument at all. It’s a narrative description of events that occurred in “Alice in Wonderland,” without any inferential claim being made from premises to a conclusion.
All characters in Wonderland speak English. The Cheshire Cat is a character in Wonderland. So, the Cheshire Cat speaks English.	This is deductive and looks like a categorical argument. The argument is valid as it follows the logical structure where the conclusion necessarily follows if the premises are true.
I’ve searched all through the garden, but I can’t find the White Rabbit. So, the White Rabbit must be somewhere else in Wonderland.	This is inductive and looks like an argument to the best explanation. We’re seeking a reason for the White Rabbit’s absence from the garden; the best one we can think of is that he is in another location in Wonderland.
The last 10 times Alice encountered the Queen of Hearts, she was threatened with beheading. So, the next time Alice meets the Queen of Hearts, she will likely be threatened again.	Inductive-prediction/generalization. While words like “likely” sometimes signal deductive argumentation, they don’t in this case. After all, we can’t use information about the past to predict the future with 100% certainty (as deductive argumentation requires).
I saw a sign saying, “This way to the Queen’s croquet ground.” So, if we want to go to the croquet ground, we should follow the sign.	Inductive—argument from signs. Whenever you make an inference from “a sign says this” to “it’s true,” you are making an inductive leap (after all, maybe the sign was put up as a joke or to mislead).
There are exactly six impossible things to believe before breakfast. I have believed five of them. So, if I believe one more, I will have accomplished what the White Queen recommended.	Deductive—argument from mathematics. The conclusion here follows from “6 – 5.” Again, it’s important to note I might be wrong about my premises (e.g., maybe there are more than six impossible things), but on the assumption that my premises are TRUE, my conclusion follows simply from the math.
If you enjoy the story of Alice in Wonderland, then you will also like Through the Looking-Glass.	This isn’t an argument! It is a conditional statement claiming that enjoying one story is a sufficient condition for enjoying the sequel. (And that enjoyment of the sequel necessarily follows from enjoyment of the first.)
Alice will solve the riddle if she thinks logically about it. If Alice solves the riddle, she will escape the trial. So, if Alice thinks logically about the riddle, she will escape the trial.	Deductive—hypothetical syllogism. (Again, this is a deductively “valid” argument. However, validity doesn’t guarantee the premises’ truth.)
The caterpillar must be telling the truth, given that he either speaks in riddles or straightforward truths, and he does not speak in riddles.	Deductive-disjunctive syllogism.
Alice couldn’t find her way home before she met the Cheshire Cat. After meeting the Cheshire Cat, she had a better idea of where to go. So, the Cheshire Cat must have been a cause of her improved navigation.	Inductive—argument about causes and effects. Arguments about causes/effects are inherently uncertain since there will always be other possibilities we haven’t accounted for. (E.g., maybe Alice simply remembered her way, or she got lucky in choosing her path.)
The Duchess told Alice that everyone in Wonderland is mad. Since the Duchess is a resident of Wonderland, I think we can trust her.	Inductive—argument from authority. Every time we believe something on the basis that a person/group/book told us it was true, we are reasoning inductively. (Obviously, this accounts for a massive chunk of our beliefs!).

 

 

Alice on Evaluating Arguments

Of course, it’s not enough to classify arguments as inductive or deductive. It also matters whether they are good arguments, and whether we should believe their conclusions. This was certainly true of my experience in Wonderland. It was also true for my good friend Dorothy, who made frequent visits to the fantastical land of Oz, a place brimming with magic, unique characters, and perplexing dilemmas. In Oz, Dorothy encountered talking animals, flying monkeys, and witches, both benevolent and malevolent. Her experiences, while extraordinary, provide rich examples for understanding complex concepts like Argument Evaluation.

In evaluating arguments, we focus on two critical aspects: Good Form and True Premises.

Good Form is akin to a well-crafted recipe, necessitating the right components in a logical sequence. An argument with good form is coherent and structured, where the premises lead seamlessly to the conclusion. For instance, in Oz, a valid argument could be, “If one follows the Yellow Brick Road, one will reach the Emerald City, as this road leads directly there.” This argument is logically structured. However, an argument like, “If one follows the Yellow Brick Road, one will become a powerful wizard,” lacks good form, as the conclusion does not logically follow from the premise.

True Premises involve the accuracy and acceptability of the argument’s foundations. Even a well-structured argument fails if based on falsehoods. For example, asserting, “All witches in Oz are wicked, therefore the Witch of the North must be wicked,” is flawed due to a false premise; not all witches in Oz are wicked. The audience’s acceptance of the premises also matters. Dorothy might argue, “You should visit Oz because it has friendly talking lions,” but this premise, while true to her, might be implausible to someone unfamiliar with the magical nature of Oz.

In essence, the effectiveness of an argument hinges on these two tests: the logical structure (Good Form) and the veracity and acceptability of its premises (True Premises). Dorothy’s experiences in Oz, with their extraordinary elements, offer a vivid backdrop for illustrating these principles, demonstrating their relevance in both fantastical and real-world contexts.

Some examples are as follows:

• True Premises, True Conclusion, Bad Form (Fails Test 1, Passes Test 2):
• Argument: “The Scarecrow wants a brain and the Tin Man wants a heart, therefore the Wizard of Oz will give them what they want.”
• Explanation: The premises are true – the Scarecrow does want a brain, and the Tin Man does want a heart. The conclusion is also true, as the Wizard eventually grants their wishes (sort of). However, the form is bad because the conclusion does not logically follow from the premises. The desires of the Scarecrow and Tin Man don’t inherently lead to the conclusion that the Wizard will grant these wishes.
• False Premises, True Conclusion, Good Form (Passes Test 1, Fails Test 2):
• Argument: “If all witches in Oz are good, then Glinda, being a witch, is good.”
• Explanation: The argument is in good form, as the conclusion logically follows from the premise. However, the premise is false – not all witches in Oz are good. Despite this, the conclusion that Glinda is good is true.
• False Premises, False Conclusion, Bad Form (Fails Both Tests):
• Argument: “Since all flying monkeys are friendly, and Dorothy has red shoes, she can control the weather in Oz.”
• Explanation: The premises are false – not all flying monkeys in Oz are friendly. The conclusion that Dorothy can control the weather because of her red shoes is also false and does not logically follow from the premises, showing bad form.
• True Premises, True Conclusion, Good Form (Passes Both Tests):
• Argument: “Dorothy would like to meet the Wizard, who lives in the Emerald City. The Yellow Brick Road leads to the Emerald City. So, Dorothy should follow the Yellow Brick Road.
• Explanation: The premises are true – Dorothy would like to meet the Wizard. The conclusion that she is seeking a way to return is also true and logically follows from the premises, demonstrating good form.

How Do I Evaluate Deductive Arguments?

In a deductive argument, we can apply the two tests just described (form and truth of premises in the following way):

• We look at the form to determine whether the argument is valid or invalid.
• If the argument is valid, we examine the truth of the premises to determine whether it is sound.

Remember, a deductive argument is one that claims its conclusion follows necessarily from its premises. If the premises are true, the conclusion cannot be false. The strength of a deductive argument lies in its logical structure, not just the truth of its statements.

Valid or Invalid? A Matter of Form

A valid deductive argument is one where the conclusion necessarily follows from the premises. In such an argument, if the premises are true, the conclusion cannot be false. The focus here is purely on the logical structure of the argument, not the actual truth of the statements. For example:

• Premise 1 (P1): If the Cheshire Cat grins, Alice is confused.
• Premise 2 (P2): The Cheshire Cat is grinning.
• Conclusion (C): Alice is confused.

This argument is valid because if P1 and P2 are true, C must be true. The truth of the premises guarantees the truth of the conclusion, following the structure of modus ponens (if P, then Q; P; therefore, Q).

An invalid deductive argument is one where the conclusion does not necessarily follow from the premises. Even if the premises are true, the conclusion can still be false due to a flaw in the argument’s logical structure. For example:

• P1: If Dorothy is in Oz, she meets Princess Ozma.
• P2: Dorothy meets Princess Ozma.
• C: Dorothy is in Oz.

This argument is invalid. It commits the fallacy of affirming the consequent. Even if P1 and P2 are true, C could be false because Dorothy might meet the Ozma under different circumstances.

sound deductive argument is not only valid, but its premises are also actually true. It combines correct logical structure with factual accuracy, ensuring the truth of the conclusion. For example:

• P1: All talking animals are capable of reasoning.
• P2: The White Rabbit is a talking animal.
• C: The White Rabbit is capable of reasoning.

Assuming both premises are true (as they are in the context of Wonderland and Oz), this argument is sound. It is valid, and its premises are factually correct, making its conclusion reliably true.

Sound or Unsound: Truth Matters, Too

Finally, an unsound deductive argument is either invalid or has one or more false premises. It can be a valid argument with false premises or an invalid argument (regardless of the truth of its premises). For example:

• P1: All witches in Oz are evil.
• P2: Glinda is a witch.
• C: Glinda is evil.

This argument, while valid in form, is unsound because P1 is false (as Glinda is a good witch). The argument’s structure might appear logical, but the inaccuracy of the premises leads to an unreliable conclusion.

Deductive, valid arguments are sometimes described as risk free, meaning that we can be exactly as sure of the conclusion as we are of our premises. This is crucial in disciplines like mathematics, computer science, and philosophy, where logical certainty is paramount.

• In mathematics, a theorem proven via valid deductive reasoning is accepted as indisputably true. This means it holds true even for the (literally infinite) cases we haven’t “tried it out” on.
• In computer science, algorithms based on valid deductions guarantee correct outcomes, even if they are repeated millions or billions of times.
• In philosophy, deductive arguments connect concepts logically, ensuring necessary and true conclusions from given premises. This can be important when arguing about things like ethics, religion, or the nature of logical inference (!).

In all of the cases, it still might be the case that our premises (or “axioms”) are flawed, and because of this, the arguments aren’t sound. However, deductive validity serves to guarantee that we don’t introduce any additional risk in our reasoning.

How Do I Evaluate Inductive Arguments?

When evaluating inductive arguments, we deal with the likelihood rather than the certainty of the conclusions following from the premises. Inductive arguments are commonly used in reasoning where absolute certainty is unattainable. Key aspects to consider are:

• Inductive Strength or Weakness evaluates how strongly the premises support the conclusion.
• Inductive Cogency assesses whether the argument is strong and the premises are true.

Strong or Weak? Let’s Assume Your Right…

An argument is inductively strong if the premises, assuming they are true, make the conclusion likely or probable. The conclusion is not guaranteed but is supported with a high degree of probability. For example,

• P1: Every time the March Hare has spoken in Alice’s experience, he has said something nonsensical.
• P2: The March Hare is about to speak.
• C: It is likely that the March Hare will say something nonsensical.

This example illustrates inductive generalization, where specific observations lead to a general conclusion. It is a “strong” generalization, since Alice has had lots of experience with the March Hare (and he never makes sense!).

By contrast, an argument is inductively weak when the premises, even if true, do not provide strong support for the conclusion. The conclusion might still be true, but it is not sufficiently probable based on the given premises. For example,

• P1: Toto has barked at strangers in the past.
• P2: A stranger is approaching Dorothy and Toto.
• C: Toto will probably bark at the stranger.

This prediction is weak because Toto’s past behavior does not necessarily indicate his future actions in this specific instance.

Cogent or Not? Are You Sure About that Evidence…

An argument is inductively cogent when it is strong (the premises, if true, make the conclusion likely) and all the premises are, in fact, true. Cogency combines the strength of the argument’s probability with factual accuracy. An example:

• P1: Every time Alice (or any other inhabitant of Wonderland) has eaten mushrooms in Wonderland, she has changed size.
• P2: Alice is eating a mushroom in Wonderland.
• C: It is likely that Alice will change size.

This is a causal inference, drawing a probable conclusion based on observed cause-and-effect relationships.

By contrast, an argument lacks inductive cogency if it is either weak or if one or more of its premises are false. This undermines the reliability of the conclusion. For example,

• P1: The Wizard of Oz has claimed that flying monkeys are harmless.
• P2: The Wizard of Oz is sometimes right about things, but he sometimes just makes things up to impress people.
• C: Flying monkeys are probably harmless.

This argument may lack cogency if the Wizard’s authority is not legitimate or his claim is false, despite his perceived status.

Inductive reasoning is essential in areas where deductive certainty is not possible. It’s used in scientific hypothesis formation, historical analysis, and everyday decision-making. In these contexts, understanding the strength and reliability of inductive arguments allows us to make informed predictions and generalizations, always bearing in mind that inductive conclusions are about probability, not certainty. This probabilistic nature means that even the strongest inductive argument cannot offer absolute proof, but it can guide us towards likely and plausible conclusions.

Two Ways of Making Good Arguments, and Many Bad Ways

Good arguments are crucial for convincing others and making sound decisions. As we’ve discussed, there are two main kinds of good arguments: sound deductive arguments and cogent inductive arguments. Both types are useful, but they work differently and are suitable for different situations.

Sound Deductive Arguments rely on logic and the truth of their starting points, known as premises. A deductive argument is called valid when its conclusion has to be true if the premises are true. It’s sound when it’s both valid and the premises are actually true. For example, let’s say we know for sure that the Vorpal Sword can cut through anything (a true premise). If this sword hits the Jabberwocky (another true premise), we can logically conclude (with a valid and sound argument) that the Jabberwocky has been cut by the Vorpal Sword.

Cogent Inductive Arguments are about likelihood and probability. They use examples or evidence to suggest a conclusion is probably true. Inductive strength is about how well the evidence supports the conclusion. These arguments are cogent when they’re strong and the evidence is true. For example, in Oz, if the Hungry Tiger always claims he’s hungry but never eats anyone (strong evidence), and he meets a new character, we might strongly guess (with a cogent argument) that he’ll say he’s hungry but won’t eat this person. This guess isn’t certain, but it’s based on consistent past behavior, making it a strong inductive argument.

Choosing the right type of argument depends on your goals. Use deductive arguments for situations where certainty is needed (and possible!), like proving a point in mathematics. Inductive arguments are better for everyday situations, like predicting behavior or outcomes based on past experiences, where absolute certainty isn’t possible. Much of our ordinary knowledge of the world—from personal experience, textbooks, teachers, or scientific experiments—comes from inductive rather than deductive arguments.

However, arguments can fail in many ways. They might be based on incorrect information, or the logic might not hold up. A deductive argument can be valid but not sound if it’s based on false premises. Similarly, an inductive argument might be weak if it’s based on insufficient or unrepresentative evidence, or it might lack cogency if the premises are not true. Arguments can also be flawed by appealing to irrelevant authorities, using emotional manipulation, or committing other logical fallacies.

Discussion Questions: Argument Evaluation

Reflect on the examples from Oz and Wonderland. How do they illustrate the importance of having both good form and true premises in an argument? Can you think of a real-life situation where one or both of these aspects were missing?

What does it mean for a deductive argument to be valid? What additional criteria must be met for it to be considered sound? Give examples to illustrate your point.

How do we use inductive reasoning in our everyday decisions? Discuss with examples from your own experiences or from familiar stories.

How do deductive and inductive arguments differ in their approach to establishing truth? Discuss the contexts in which each type of argument might be more appropriate.

Using the characters and scenarios from Oz and Wonderland (or from your favorite childhood books!), create your own examples of deductive and inductive arguments. Evaluate these arguments in terms of their form and the truth of their premises.

Discuss how probability plays a role in inductive reasoning. Can an inductive argument ever be considered ‘certain’? Why or why not?

How does understanding the principles of good form and true premises help us in critically evaluating arguments in media, literature, or public discourse?

 

 

Minds that Mattered: Ada Lovelace

Ada Lovelace (1815-1852) was an English mathematician and writer who is widely regarded as the world’s first computer programmer. Born Augusta Ada Byron, she was the only legitimate child of the famous poet Lord Byron. Ada’s mother, Lady Byron, was a mathematician herself and ensured that Ada received a rigorous education in mathematics and science, which was unusual for a woman in the 19th century.

In 1833, Ada met Charles Babbage, a mathematician and inventor who was working on a mechanical calculating machine called the Analytical Engine. Ada became fascinated by the machine and its potential. She translated an article about the Analytical Engine from Italian to English, and in the process, she added her own extensive notes, which included what is now recognized as the first algorithm intended to be carried out by a machine. This work earned her the title of the world’s first computer programmer.

Key Ideas

The concept of an algorithm. An algorithm is a precise, step-by-step set of instructions for solving a problem or completing a task. It is a fundamental concept in computer science and mathematics. In her notes on the Analytical Engine, Ada Lovelace described how the machine could follow a series of instructions (an algorithm) to perform complex calculations and manipulate symbols. She realized that the machine could be used not only for numerical calculations but also for symbolic manipulation, which is the foundation of modern computer programming. Ada’s insight was groundbreaking, as it showed that machines could be used to perform not just simple calculations but complex processes based on logical rules. An algorithm has several key components:

• Inputs: The data or information needed to solve the problem.
• Outputs: The results or solutions to the problem.
• Sequence: A series of steps that are performed in a specific order.
• Finiteness: An algorithm must have a finite number of steps and must eventually terminate.

Ada’s work laid the foundation for the concept of computer programming and demonstrated the potential for machines to perform complex tasks by following a set of instructions.

The Lady Lovelace Objection. In her notes, Ada Lovelace addressed what has come to be known as the “Lady Lovelace Objection.” This refers to the idea that machines can only do what they are explicitly programmed to do and cannot create, think, or exhibit intelligence on their own. This concept is closely related to the field of artificial intelligence, which aims to create machines that can perform tasks that typically require human intelligence, such as learning, problem-solving, and decision-making. Ada argued that while machines cannot originate ideas, they can be programmed to manipulate symbols according to defined rules, which can lead to new and unexpected results. This idea anticipates modern discussions about the potential for machines to exhibit what appears to be creative or intelligent behavior. Machine learning, a subset of artificial intelligence, focuses on the development of algorithms and statistical models that enable machines to improve their performance on a specific task through experience or data.

Challenging stereotypes about women in math and logic. Ada Lovelace’s contributions to computer science were remarkable not only for their intellectual merit but also because they challenged prevailing stereotypes about women’s abilities in mathematics and logic. Stereotypes are widely held, oversimplified beliefs about a particular group of people. In the context of women in math and logic, these stereotypes often suggest that women are less capable or less suited for these fields than men. In the 19th century, women were often excluded from scientific and mathematical pursuits, and their intellectual capabilities were widely underestimated. This exclusion and underestimation can lead to gender bias, which is the tendency to prefer one gender over another or to hold prejudiced views about a particular gender. Moreover, stereotypes can lead to stereotype threat, which is the risk of confirming negative stereotypes about an individual’s group. In the context of women in STEM, stereotype threat can lead to increased anxiety, reduced performance, and a higher likelihood of leaving the field. Additionally, women in STEM fields may experience imposter syndrome, a psychological pattern in which an individual doubts their skills, talents, or accomplishments and has a persistent fear of being exposed as a “fraud.” Despite these challenges, Ada Lovelace’s work demonstrated that women could make significant contributions to mathematics and logic, helping to pave the way for greater gender equality in science, technology, engineering, and mathematics (STEM).

Influence

Ada Lovelace’s ideas and contributions to computer science were ahead of her time, and their significance was not fully recognized until the 20th century. However, her legacy has since been celebrated, and she has become an icon for women in STEM fields.

In the 1950s, as computer science began to emerge as a distinct discipline, Ada’s notes on the Analytical Engine were rediscovered and recognized for their historical significance. Her description of an algorithm and her ideas about the potential for machines to manipulate symbols according to logical rules were seen as foundational to modern computer programming.

In the 1970s and 1980s, as the field of computer science grew, Ada’s story began to be more widely told, and she became a symbol of women’s achievements in STEM. The U.S. Department of Defense named a programming language “Ada” in her honor in 1979, and the second Tuesday in October has been designated as Ada Lovelace Day, an international celebration of the achievements of women in STEM.

Today, Ada Lovelace is celebrated not only for her intellectual contributions but also for her role in challenging gender stereotypes and paving the way for women in STEM fields. Her story is often used to encourage young women to pursue careers in science and technology, and to highlight the importance of diversity in these fields.

Review Questions: Ada Lovelace

• How did Ada Lovelace’s work on the Analytical Engine contribute to the development of computer science, and what impact did her ideas have on the field?
• In what ways did Ada Lovelace’s gender affect her work and reception in the 19th century, and how do her experiences relate to the challenges faced by women in STEM fields today?
• How does the Lady Lovelace Objection relate to modern discussions about artificial intelligence and machine learning, and what are some arguments for and against the idea that machines can exhibit intelligent behavior?
• What role do stereotypes and biases play in shaping the representation and experiences of women in STEM fields, and what strategies can be used to challenge and overcome these barriers?
• How can the legacy of Ada Lovelace and other pioneering women in science and technology be used to inspire and encourage greater diversity and inclusion in STEM fields?

 

Glossary

Term	Definition
Algorithm	A precise, step-by-step set of instructions for solving a problem or completing a task, which forms the foundation of computer programming.
Cogent Argument	An inductive argument that is both strong and has true premises. It combines the strength of the argument’s probability with factual accuracy, making the conclusion likely.
Deductive Argument	A form of reasoning where the conclusion is necessitated by, or reached from, the premises. It is characterized by the claim that the conclusion cannot be false if the premises are true.
Gender bias	The tendency to prefer one gender over another or to hold prejudiced views about a particular gender, which can lead to exclusion and underestimation of abilities.
Imposter syndrome	A psychological pattern in which an individual doubts their skills, talents, or accomplishments and has a persistent fear of being exposed as a “fraud,” particularly common among women in STEM fields.
Inductive Argument	A form of reasoning where the conclusion is probable based on the premises. It deals with the likelihood rather than the certainty of conclusions, often used in scenarios where absolute certainty is unattainable.
Sound Argument	A deductive argument that is both valid and has true premises. It combines a correct logical structure with factual accuracy, ensuring the truth of the conclusion.
Stereotype	A widely held, oversimplified belief about a particular group of people, often suggesting that certain groups are less capable or suited for specific fields or tasks.
Stereotype threat	The risk of confirming negative stereotypes about an individual’s group, which can lead to increased anxiety, reduced performance, and a higher likelihood of leaving a field.
Strong Argument	An inductive argument where, assuming the premises are true, the conclusion is likely or probable. It does not guarantee the conclusion but supports it with a high degree of probability.
Valid Argument	A deductive argument where the conclusion logically follows from the premises. In this type of argument, if the premises are true, the conclusion cannot be false. The focus is on the logical structure rather than the truth of the premises.
Weak Argument	An inductive argument where the premises, even if true, provide insufficient support for the conclusion. The conclusion might still be true, but it is not sufficiently probable based on the given premises.

 

Table: Deductive Argument Forms

 

Argument Form	Definition
Modus Ponens	If P, then Q. P is true. Therefore, Q is true. This deductive form affirms the antecedent to conclude the consequent.
Modus Tollens	If P, then Q. Q is not true. Therefore, P is not true. This deductive form denies the consequent to conclude the denial of the antecedent.
Hypothetical Syllogism	If P, then Q. If Q, then R. Therefore, if P, then R. This deductive form connects two conditional statements to form a conclusion that links the first antecedent with the final consequent.
Disjunctive Syllogism	Either P or Q. Not P. Therefore, Q. This deductive form uses a disjunction and the negation of one of its elements to conclude the other element.
Constructive Dilemma	If P, then Q. If R, then S. Either P or R is true. Therefore, either Q or S is true. This deductive form combines two conditional statements with a disjunction to derive a disjunctive conclusion.
Destructive Dilemma	If P, then Q. If R, then S. Either Q is false or S is false. Therefore, either P is false or R is false. This deductive form negates the consequents of two conditionals and concludes with the negation of one of the antecedents.
Affirming the Consequent	If P, then Q. Q is true. Therefore, P is true. This is a logical fallacy where the consequent of a conditional statement is incorrectly used to affirm the antecedent.
Denying the Antecedent	If P, then Q. P is not true. Therefore, Q is not true. This is a logical fallacy where the antecedent of a conditional statement is incorrectly used to deny the consequent.
Categorical Syllogism	This deductive form uses two categorical premises to conclude a relation between two categories. For example: All A are B. All B are C. Therefore, all A are C.

 

Table: Inductive Argument Forms

Term	Definition
Generalization	An inductive argument form that draws a conclusion about all members of a group from observations of a sample of that group.
Statistical Syllogism	An inductive argument form that applies a general statement to a specific case, based on statistical evidence about the likelihood of the statement applying.
Argument from Analogy	An inductive argument form that infers that two things are similar in one or more respects, based on the known similarities in other respects.
Causal Inference	An inductive argument form that determines the cause of a particular effect, based on the observation of regular associations between events and excluding other potential causes.
Predictive Argument	An inductive argument form that forecasts future events based on established patterns or past occurrences.
Argument from Authority	An inductive argument form that relies on the statements or claims made by a credible source or authority to support a conclusion.
Argument from Sign	An inductive argument form that infers a certain state of affairs from the presence of a specific sign or indicator.
Inference to the Best Explanation	An inductive argument form that selects the most plausible explanation for an observed phenomenon, considering various hypotheses and choosing the one that best accounts for the available evidence.

 

References

Henderson, Leah. 2022. “The Problem of Induction.” In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta and Uri Nodelman, Winter 2022. Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/win2022/entries/induction-problem/

Psillos, Stathis, and Chrysovalantis Stergiou. 2024. “Induction, The Problem Of.” In Internet Encyclopedia of Philosophy. Accessed January 5, 2024. https://iep.utm.edu/problem-of-induction/ .

Shea, Brendan. 2010. “Three Ways of Getting It Wrong: Induction in Wonderland.” In Alice in Wonderland and Philosophy: Curiouser and Curiouser, edited by Richard Brian Davis, 93–107. Wiley Blackwell.

Vernon, Kenneth Blake. 2014. “The Problem of Induction.” 1000-Word Philosophy: An Introductory Anthology (blog). May 26, 2014. https://1000wordphilosophy.com/2014/05/26/the-problem-of-induction/.