Logic, as a field of study, can be defined as the organized body of knowledge, or science, that evaluates arguments. Its aim is to develop a system of methods and principles used as criteria for evaluating others' arguments and as guides for constructing our own.
An argument is a structured combination of two or more statements, classified as premise(s) and a conclusion. A premise is a statement claimed to provide logical support or evidence for the argument's main point, known as the conclusion. A conclusion is a statement claimed to follow from the alleged evidence.
Depending on the logical and factual ability of the premise(s) to support the conclusion, an argument can be categorized as good or bad. Unlike other types of passages, including those resembling arguments, all arguments purport to prove something.
Arguments are generally divided into two main types: deductive and inductive. A deductive argument claims that its premises support the conclusion in such a way that it is impossible for the premises to be true and the conclusion false. An inductive argument claims that its premises support the conclusion such that it is improbable for the premises to be true and the conclusion false.
Whether an argument is deductive or inductive can often be determined by specific indicator words, the actual strength of the inferential relationship between its components, and its argumentative form or structure.
Deductive arguments are evaluated based on their validity and soundness. Inductive arguments are evaluated based on their strength and cogency. A deductive argument is valid if its premises, assumed true, guarantee the truth of the conclusion (i.e., it's impossible for the premises to be true and the conclusion false). It is invalid if it is possible for true premises to lead to a false conclusion. Similarly, an inductive argument is strong if its premises, assumed true, make the conclusion probable (i.e., it's improbable for the premises to be true and the conclusion false). It is weak if the premises do not make the conclusion probable.
Furthermore, a deductive argument is sound if it is valid *and* all its premises are actually true. If it fails on either count (invalid form or at least one false premise), it is unsound. Likewise, an inductive argument is cogent if it is strong *and* all its premises are actually true (considering all relevant evidence). If it fails on either count (weak form or at least one false premise, or overlooks crucial evidence), it is uncogent. This chapter discusses logic's basic concepts, techniques for distinguishing arguments from non-arguments, and the types and evaluation of arguments.
Upon successful completion of this chapter, you will be able to:
Logic is generally defined as the philosophical science that evaluates arguments. An argument is a systematic combination of one or more statements (premises) claimed to provide logical support or evidence for another single statement (the conclusion), which is claimed to follow logically from the alleged evidence. An argument's quality (good or bad) depends on the logical adequacy of its premise(s) in supporting its conclusion.
The primary aim of logic is to develop methods and principles used as criteria for evaluating others' arguments and as guides for constructing our own. Studying logic enhances one's ability and confidence to critique others' arguments and present well-reasoned arguments. This section discusses the meaning of logic and its basic concepts: arguments, premises, and conclusions.
After completing this section, you will be able to:
How would you define Logic?
The word "logic" originates from the Greek word logos, signifying sentence, discourse, reason, truth, and rule. In its broader sense, logic is the science that evaluates arguments and the study of correct reasoning. It can also be defined as the study of the methods and principles of correct reasoning or the art of reasoning correctly.
Logic can be defined in several ways. Here are some common definitions:
As an academic discipline, logic studies reasoning itself: forms of argument, general principles, common errors, and argumentative methods. Recognizing flawed reasoning in daily life is common, and logic helps us understand why arguments go wrong or why someone argues in a particular manner.
“Logic sharpens and refines our natural gifts to think, reason and argue.” (C. S. Layman)
What do you perceive as the benefits of studying logic? Discuss with a classmate.
We employ logic in everyday communication. As humans, we all think, reason, and argue, and we are exposed to others' reasoning. Some reason well naturally, others less so. While reasoning ability might partly be innate, it can always be refined, improved, and sharpened. Studying logic is one of the most effective ways to enhance this natural ability.
Furthermore, in academic pursuits, our arguments must be logical and persuasive; logic provides the necessary tools. In general, studying logic offers several major benefits:
The aim of logic, therefore, is to develop systematic methods and principles for evaluating arguments and constructing sound arguments in daily life. Studying logic increases confidence in criticizing others' arguments and advancing one's own. A key goal is to cultivate individuals who are critical, rational, and reasonable in both public and private spheres. However, fully benefiting from logic requires a thorough understanding of its basic concepts and the ability to apply them in practical situations.
What comes to mind when you hear the word "argument"? How would you define it? Discuss with a classmate.
The word ‘argument’ is likely familiar, as we encounter arguments daily in books, newspapers, television, and conversations. Reviewing the definitions of logic reveals a common theme: argument. Indeed, identifying, analyzing, and evaluating arguments is a central benefit of studying logic, making argument its primary subject matter.
So, what is an argument? In logic, argument is a technical term and the field's chief concern. From a logical perspective, an argument is a group of statements where one or more statements (the premises) are claimed to provide support for, or reasons to believe, another statement (the conclusion). As this definition indicates, "argument" in logic differs from its everyday meaning, such as a verbal dispute. Let's examine the components of this definition more closely.
First, an argument consists of a group of statements. The initial requirement for a passage to qualify as an argument is that it combines two or more statements.
But what exactly is a statement?
A statement is a declarative sentence that possesses a truth-value—it is either true or false. Thus, truth and falsity are the two possible truth-values of a statement. Typically, statements are declarative sentences.
Consider these examples:
Statements (a) and (b) are true because they accurately describe reality or assert what is the case. Their truth-value is 'True'. Statement (c) is false because it asserts what is not the case; its truth-value is 'False'.
Logicians often use "proposition" and "statement" interchangeably. However, strictly speaking, a proposition is the meaning or informational content conveyed by a statement. In this chapter, "statement" will refer to both premises and conclusions.
However, not all sentences are statements, and therefore, not all sentences can be used to construct arguments. Examples of non-statements include:
While a sentence is a group of words expressing a meaningful idea, unlike statements, the sentences above cannot be evaluated as true or false. Consequently, they are not statements and cannot form part of a logical argument.
If an argument combines premises and a conclusion, what are premises and conclusions?
Second, the statements constituting an argument are divided into premise(s) and a conclusion. Simply containing two or more statements does not guarantee an argument. An argument must contain at least one premise and one (and only one) conclusion. An argument may have multiple premises but always has a single conclusion.
An argument always attempts to justify a claim. The claim being justified is the conclusion, and the statement(s) supposedly providing the justification are the premises. Therefore, a premise is a statement offering the reason or evidence given for accepting the conclusion. It represents the claimed evidence. A conclusion is the statement claimed to follow logically from the provided evidence (premises); it is the claim the argument seeks to establish.
Based on the definition provided, can you now construct an argument?
Let's examine some examples together:
Example 1:
(1) All Ethiopians are Africans. (Premise 1)
(2) Tsionawit is Ethiopian. (Premise 2)
Therefore, Tsionawit is African. (Conclusion)
Example 2:
(1) Some Africans are black. (Premise 1)
(2) Zelalem is an African. (Premise 2)
Therefore, Zelalem is black. (Conclusion)
In both arguments, the first two statements serve as premises, claimed to provide evidence for the third statement, which is the conclusion claimed to follow from this evidence. The word “therefore” indicates the relationship where the premises support the conclusion.
All arguments fall into two basic groups: those where the premises genuinely support the conclusion, and those where they fail to do so, despite the claim. The former are considered good (well-supported) arguments, the latter bad (poorly-supported) arguments. Comparing the examples: In Example 1, the premises provide strong reason to believe the conclusion, making it a good argument. In Example 2, however, the premises fail to adequately support the conclusion; even if true, they don't give good reason to believe the conclusion is true. Therefore, Example 2 is a bad argument, though it remains an argument because it attempts to provide support.
How can we reliably distinguish premises from the conclusion?
While logic aims to develop methods for evaluating arguments (distinguishing good from bad), a crucial initial step is identifying the premises and the conclusion. This can sometimes be challenging for several reasons.
Although arguments ideally consist only of premises and a conclusion, real-world arguments might contain extraneous sentences. Furthermore, even within the core statements, pinpointing the conclusion can be difficult. Since analyzing arguments requires identifying their components, we need techniques to differentiate premises from conclusions.
The primary technique for identifying premises and conclusions involves looking for indicator words. Arguments frequently contain words that signal the role of a statement.
Common Conclusion Indicators:
Therefore
Hence
Thus
Consequently
Accordingly
So
Wherefore
Whence
As a result
It follows that
We may conclude
We may infer
It implies that
It must be that
It shows that
In an argument containing one of these conclusion indicators, the statement immediately following the indicator is typically the conclusion. By elimination, the other statements serving as evidence for this conclusion can be identified as premises.
Example:
Women are mammals.
Zenebech is a woman.
Therefore, Zenebech is a mammal.
Using the rule above, the conclusion is “Zenebech is a mammal” because it follows the indicator word “therefore”. The other two statements function as premises providing support.
If an argument lacks a conclusion indicator, it might contain a premise indicator.
Common Premise Indicators:
Since
Because
For
As
Given that
Seeing that
Inasmuch as
Owing to
In that
As indicated by
May be inferred from
For the reason that
In an argument containing any of these premise indicators, the statement immediately following the indicator is usually a premise. The remaining single statement, which these premises support, will be the conclusion.
Example:
You should avoid any form of cheating on exams because cheating on exams is punishable by the Senate Legislation of the University.
According to the rule, the premise is “cheating on exams is punishable by the Senate Legislation of the University” because it follows the indicator word “because”. The other statement (“You should avoid any form of cheating on exams”) is the conclusion.
The indicator “for this reason” is unique: it typically appears *after* the premise(s) it indicates and immediately *before* the conclusion. It functions somewhat like both a premise and conclusion indicator in this intermediate position. Be careful not to confuse it with "for the reason that," which is a standard premise indicator.
Sometimes, a single indicator can introduce multiple premises. Consider:
Tsionawit is a faithful wife, for Ethiopian women are faithful wives and Tsionawit is an Ethiopian.
Here, the premise indicator “for” introduces two premises: “Ethiopian women are faithful wives” and “Tsionawit is an Ethiopian”. By elimination, “Tsionawit is a faithful wife” is the conclusion.
Some arguments contain no indicator words at all. In such cases, the reader or listener must analyze the passage and ask:
The answers to these questions should help identify the conclusion.
Example:
Our country should increase the quality and quantity of its military. Ethnic conflicts have recently intensified; border conflicts are escalating; international terrorist activities are increasing.
The main point here is the recommendation that the country should bolster its military. The subsequent statements provide reasons for this recommendation. Although no indicator words are present, the logical flow identifies the first sentence as the conclusion and the others as premises.
In standard form, the argument is:
Ethnic conflicts have recently intensified. (Premise 1)
Border conflicts are escalating. (Premise 2)
International terrorist activities are increasing. (Premise 3)
Therefore, the country should increase the quality and quantity of its military. (Conclusion)
Passages containing arguments may also include statements that are neither premises nor the conclusion. Only statements intended to support the conclusion should be listed as premises. Statements that merely offer background information or passing comments should be excluded from the core argument structure.
Example:
Socialized medicine is not recommended because it would result in a reduction in the overall quality of medical care available to the average citizen. In addition, it might very well bankrupt the federal treasury. This is the whole case against socialized medicine in a nutshell.
Here, the conclusion is “Socialized medicine is not recommended.” The two statements following “because” serve as premises. The final sentence merely comments on the argument itself and is neither a premise nor the conclusion.
The term inference relates closely to argument. In a narrow sense, inference refers to the reasoning process expressed by an argument—the logical link between premises and conclusion. More broadly, it can refer to the argument itself. For this course, we primarily use the narrower sense, focusing on the inferential link or claim within arguments.
An argument, as defined, combines premise(s) offering logical support for a conclusion. However, not all passages containing multiple statements constitute an argument. Various types of passages may include several statements but lack the core purpose of proving something.
Arguments are distinguished from such non-argumentative passages by their primary goal: to prove or establish a claim. This section explores techniques for distinguishing argumentative passages from non-argumentative ones.
After completing this section, you will be able to:
What defines an argumentative passage? What qualifies a passage as an argument?
Evaluating arguments concerning diverse topics—religion, politics, ethics, science, etc.—is central to logic. Therefore, accurately identifying arguments is essential. Not all passages contain arguments. A passage contains an argument if it purports to prove something; otherwise, it does not.
But what does it mean for a passage to "purport to prove something"?
Two conditions must be met for a passage to purport to prove something (and thus be an argument):
As defined earlier, the statements offering evidence are premises, and the statement allegedly supported is the conclusion. The first condition relates to the premises claiming to provide reasons. The second condition relates to the conclusion being claimed to follow. Crucially, the premises do not need to present *actual* evidence or true reasons, nor do they need to *actually* support the conclusion. What matters is the *claim* that evidence is presented and the *claim* that it supports something.
The first condition expresses a factual claim (the premises assert certain facts or reasons). The second condition expresses an inferential claim (a reasoning process connects premises to conclusion). While the factual claim's accuracy falls largely outside logic's primary scope (which focuses on reasoning structure), the inferential claim is central.
The inferential claim is simply the assertion that the passage embodies a reasoning process—that something supports or implies something else, or that something follows from something else. It's important to note that this claim is distinct from the arguer's subjective intentions, which are usually inaccessible. The inferential claim is an objective feature, typically grounded in the argument's language (indicator words) or structure.
Therefore, when evaluating arguments, our primary focus is on whether the second condition (the inferential claim) is fulfilled. Whether the premises actually provide true reasons or successfully support the conclusion are subsequent evaluation steps.
An inferential claim can be either explicit or implicit. An explicit inferential claim is typically asserted using premise or conclusion indicator words (e.g., "thus," "since," "because," "hence," "therefore"). It exists when an indicator word explicitly states a relationship between premises and conclusion.
Example:
Gamachuu is my biological father, because my mother told me so.
In this example, the premise indicator "because" explicitly expresses the claim that evidence ("my mother told me so") supports a conclusion ("Gamachuu is my biological father"). Hence, the passage constitutes an argument.
An implicit inferential claim exists when there is a clear inferential relationship between statements, but no indicator words are used.
Example:
The genetic modification of food is risky business. Genetic engineering can introduce unintended changes into the DNA of the food-producing organism, and these changes can be toxic to the consumer.
Here, the relationship between the first statement (conclusion) and the subsequent two (premises) constitutes an implicit claim that evidence supports the initial statement. Therefore, we are justified in identifying this passage as an argument, despite the absence of indicator words.
Determining whether a passage contains an argument can sometimes be difficult. When deciding if there's a claim that evidence supports something, look for: (1) indicator words, and (2) the presence of an inferential relationship between statements. However, some caution is necessary.
First, the mere presence of an indicator word does not automatically guarantee an argument. Indicator words can have non-logical meanings. For instance, "since" can indicate time rather than reason. Always check if the indicator word functions logically to link premise(s) and conclusion.
Example:
Since Edison invented the phonograph, there have been many technological developments. (Non-argument: "since" means "from the time that")
Since Edison invented the phonograph, he deserves credit for a major technological development. (Argument: "since" indicates reason)
In the first passage, "since" is used temporally, so it's not an argument. In the second, "since" functions logically, making it an argument.
Second, detecting an inferential relationship without indicator words can be challenging and may require careful reading. Mentally inserting potential indicator words (like "therefore" or "because") between statements can help assess if a logical flow exists. Even so, interpretation can be ambiguous, and disagreement is possible. Sometimes, the best answer might be conditional: “If this passage is intended as an argument, then these are the premises and that is the conclusion.”
To aid in distinguishing arguments from non-arguments, it's helpful to recognize common types of non-argumentative passages.
What characterizes non-argumentative passages? What essential element do they lack to be considered arguments?
Having explored what arguments are, we now focus on what they are not. Non-argumentative passages lack an inferential claim; they do not purport to prove anything. Common types include simple non-inferential passages, expository passages, illustrations, explanations, and conditional statements. These passages may contain statements that could function as premises or conclusions, but they lack the crucial claim of a reasoning process connecting them. This section examines key forms of non-argumentative discourse.
Simple non-inferential passages are straightforward texts lacking any claim that something is being proved. They contain statements that *could* be premises or conclusions, but without the assertion that one supports the other. Examples include:
Example (Warning):
Watch out that you don't slip on the ice.
This statement warns but offers no evidence to prove its truth; hence, it's not an argument.
A piece of advice makes a recommendation.
Example (Advice):
You should keep a copy of all your financial records.
Like warnings, advice typically doesn't provide evidence intended to prove anything, so this passage is not an argument.
A statement of belief or opinion expresses a personal view.
Example (Belief/Opinion):
We believe that our university must develop and produce outstanding students who will perform with great skill and fulfill the demands of our nation.
This passage states a belief without claiming supporting evidence or that it supports a further conclusion; therefore, it's not an argument.
Loosely associated statements discuss a general subject but lack a claimed inferential connection.
Example (Loosely Associated):
Not to honor men of worth will keep the people from contention; not to value goods that are hard to come by will keep them from theft; not to display what is desirable will keep them from being unsettled of mind.
(Lao-Tzu, Thoughts from the Tao Te Ching)
Because no statement claims to provide evidence for another, this is not an argument.
A report consists of statements conveying information about a topic or event.
Example (Report):
The state appropriation for education was increased by 5% this year. The faculty senate authorized a new ethics requirement for all students. Student fees for parking and activities were raised by $100 per semester.
These statements report events or facts. While they *could* serve as premises in an argument, the author makes no claim that they support or imply anything further. Thus, this passage, as it stands, is not an argument.
One must be careful, however, with reports *about* arguments.
Example (Report on a Claim):
“The Air Force faces a serious shortage of experienced pilots in the years ahead, because repeated overseas tours and the allure of high paying jobs with commercial airlines are winning out over lucrative bonuses to stay in the service,” says a prominent Air Force official.
(Newspaper clipping)
Strictly speaking, this passage is not an argument made by the *author*. The author is reporting an argument made by the Air Force official. If interpreting such passages as "containing" arguments, clarify that the argument belongs to the person being reported, not the reporter.
An expository passage is a type of discourse starting with a topic sentence, followed by one or more sentences that develop or elaborate on that topic sentence. If the objective is merely to expand or clarify the topic sentence, rather than prove it, then it is not an argument.
Example:
There is a stylized relation of artist to mass audience in the sports, especially in baseball. Each player develops a style of his own—the swagger as he steps to the plate, the unique windup a pitcher has, the clean-swinging and hard-driving hits, the precision quickness and grace of infield and outfield, the sense of surplus power behind whatever is done.
(Max Lerner, America as a Civilization)
In this passage, the first sentence is the topic sentence. The subsequent sentences merely develop and illustrate this idea. Since it lacks an inferential claim (an attempt to prove the topic sentence), it's not an argument.
However, expository passages differ from simple non-inferential ones because many *can* also function as arguments. If the purpose of the subsequent sentences is not only to elaborate but also to provide reasons *proving* the topic sentence, then the passage constitutes an argument.
Example:
Skin and the mucous membrane lining the respiratory and digestive tracts serve as mechanical barriers to entry by microbes. Oil gland secretions contain chemicals that weaken or kill bacteria on skin. The respiratory tract is lined by cells that sweep mucus and trapped particles up into the throat, where they can be swallowed. The stomach has an acidic pH, which inhibits the growth of many types of bacteria.
(Sylvia S. Mader, Human Biology, 4th ed.)
Here, the topic sentence (first sentence) states that skin and membranes are barriers. The following sentences explain *how* they function as barriers, simultaneously providing evidence *that* they function as barriers. Thus, this passage can be interpreted as both an expository passage and an argument.
Distinguishing requires assessing the author's primary intent: simply explaining or also proving?
One of the most important types of non-argument is the explanation. An explanation is an expression intended to shed light on some event or phenomenon, typically one that is already accepted as fact. It aims to clarify *why* something happened or *why* something is the way it is, rather than to prove *that* it happened or is the case.
Example:
Cows can digest grass while humans cannot, because their digestive systems contain enzymes not found in humans.
Every explanation has two components: the explanandum (the statement describing the event/phenomenon being explained) and the explanans (the statement or group of statements doing the explaining). In the example, the explanandum is “Cows can digest grass while humans cannot,” and the explanans is “their digestive systems contain enzymes not found in humans.”
Argument Structure:
(Claimed to prove)
Explanation Structure:
(Claimed to shed light on an accepted fact)
Explanations are often mistaken for arguments, especially because both frequently use indicator words like "because." However, they differ in purpose. In an explanation, the explanans aims to clarify or make sense of the explanandum (an accepted fact), not to prove its occurrence. In an argument, the premises aim to prove that the conclusion is true (often the conclusion is less established than the premises). Arguments typically move from evidence to a claim, while explanations often move from an accepted fact to its underlying cause or reason.
In the cow/grass example, the fact that cows digest grass and humans don't is generally accepted. The statement about enzymes isn't intended to prove this fact but to explain *why* it's true.
To distinguish explanations from arguments:
1. Identify the statement that is either the explanandum (in an explanation) or the conclusion (in an argument). This is often the statement preceding "because" or following "therefore."
2. Determine if this statement describes an accepted matter of fact.
3. If it does, and if the remaining statements aim to shed light on *why* this fact is true, the passage is likely an explanation. If the goal is to convince the audience *that* the statement is true, it's likely an argument.
This method generally works, but some passages can plausibly be interpreted as both, especially if the explanandum/conclusion might not be universally accepted or understood.
Example:
Women become intoxicated by drinking a smaller amount of alcohol than men because men metabolize part of the alcohol before it reaches the bloodstream, whereas women do not.
This passage could serve two purposes: (1) as an argument to *prove* the first statement to someone who doesn't accept it, or (2) as an explanation to shed light on *why* this accepted fact occurs. It could even do both simultaneously. Thus, interpreting it as either an explanation or an argument (or both) can be valid depending on the assumed context and audience.
A major challenge in distinguishing arguments from explanations lies in determining whether the key statement (explanandum/conclusion) represents an "accepted matter of fact." Acceptance varies among audiences. Context, including the passage's source (textbook, opinion piece, etc.), can help. When context is absent, the interpretation might remain conditional: "If intended as an argument..."
An illustration uses one or more examples to show what something means or how something is done. Illustrations can be confused with arguments, particularly when they contain indicator words like "thus" or "for example."
Example:
Chemical elements, as well as compounds, can be represented by molecular formulas. Thus, oxygen is represented by “O₂”, water by “H₂O”, and sodium chloride by “NaCl”.
This passage is not an argument. It doesn't claim to prove anything. The word "thus" here indicates *how* elements and compounds are represented (i.e., it introduces examples), not a logical conclusion following from premises.
However, like expository passages, illustrations *can* sometimes function as arguments, often called arguments from example.
Example (Argument from Example):
Although most forms of cancer, if untreated, can cause death, not all cancers are life-threatening. For example, basal cell carcinoma, the most common of all skin cancers, can produce disfigurement, but it almost never results in death.
In this passage, the example of basal cell carcinoma is provided as evidence to *prove* the truth of the initial claim ("Not all cancers are life-threatening"). Therefore, this passage functions as an argument.
To decide if an illustration is an argument, determine its primary purpose: is it merely showing meaning or method, or is it also trying to prove a point? Consider the claim being illustrated. If it's widely accepted, the passage is likely just an illustration. If the claim is debatable or unfamiliar to the audience, using examples to support it makes the passage function as an argument.
A conditional statement is an “if . . . then . . .” statement, expressing a hypothetical relationship.
Example:
If you study hard, then you will score an ‘A’ grade.
Every conditional statement comprises two parts: the component following "if" is the antecedent (the 'if-clause'), and the component following "then" is the consequent (the 'then-clause'). The order can sometimes be reversed ("You will score an 'A' grade if you study hard").
Standard Form:
If ---------------------------- then ---------------------------------.
Reversed Form:
---------------------------- if ---------------------------------.
Conditional statements, by themselves, are not arguments. They fail to meet the criteria: an argument requires at least one statement claimed as evidence (premise) and a claim that this evidence implies something (conclusion). A conditional statement asserts neither its antecedent nor its consequent as true. It only asserts a relationship: *if* the antecedent is true, *then* the consequent is also true. The example "If you study hard, then you will score an ‘A’" doesn't assert that you *are* studying hard or that you *will* score an 'A'. It only posits the connection.
While a single conditional statement isn't an argument, it does possess inferential content because it asserts a logical relationship between the antecedent and consequent. This relationship resembles the inferential link in an argument.
Consider this conditional statement:
If destroying a political competitor gives you joy, then you have a low sense of morality.
The link between antecedent ("destroying a political competitor gives you joy") and consequent ("you have a low sense of morality") mirrors the structure of an inference. However, it differs because the antecedent is presented hypothetically, not asserted as true, unlike the premises of an argument which are typically presented as factual claims.
Conditional statements themselves aren't evaluated as simply true or false based on their components; their truth depends on the validity of the connection they assert.
Although a conditional statement alone is not an argument, its inferential content can be re-expressed to form one. The previous example could become:
Destroying a political competitor gives you joy. (Premise)
Therefore, you have a low sense of morality. (Conclusion)
Here, the antecedent is asserted as a true premise, leading to the asserted conclusion. This structure constitutes an argument.
Furthermore, conditional statements frequently serve as premises or conclusions within larger arguments.
Example (Conditional as Premise):
If he is selling our national secrets to enemies, then he is a traitor. (Premise 1)
He is selling our national secrets to enemies. (Premise 2)
Therefore, he is a traitor. (Conclusion)
Example (Conditionals as Premises & Conclusion):
If he is selling our national secrets to enemies, then he is a traitor. (Premise 1)
If he is a traitor, then he must be punished by death. (Premise 2)
Therefore, if he is selling our national secrets to enemies, then he must be punished by death. (Conclusion)
Summary of conditional statements and arguments:
Conditional statements ("If A, then B") are crucial because they express relationships between sufficient and necessary conditions.
A is a sufficient condition for B if the occurrence of A guarantees the occurrence of B. Knowing A occurred is enough to know B occurred.
Example: Being a dog (A) is a sufficient condition for being an animal (B). If something is a dog, it must be an animal.
B is a necessary condition for A if A cannot occur without B also occurring. If B hasn't occurred, A cannot have occurred.
Example: Being an animal (B) is a necessary condition for being a dog (A). Something cannot be a dog unless it is also an animal.
To clarify: Imagine a closed box. If told it contains a dog, you know it contains an animal (dog is sufficient for animal). But knowing it contains an animal doesn't guarantee it's a dog (animal is not sufficient for dog). Conversely, if told it does *not* contain an animal, you know it doesn't contain a dog (animal is necessary for dog). But being a dog is not necessary for being an animal (it could be a cat).
These relationships are captured by conditional statements:
If X is a dog, then X is an animal. (Being a dog is sufficient for being an animal)
If X is not an animal, then X is not a dog. (Being an animal is necessary for being a dog)
Note that these two conditional statements are logically equivalent; they express the same underlying relationship. The first statement highlights the sufficient condition (the antecedent), while the second (its contrapositive) highlights the necessary condition (the negation of the consequent implies the negation of the antecedent).
In the form "If A, then B":
• A (antecedent) represents the sufficient condition.
• B (consequent) represents the necessary condition.
(If A occurs, B must occur. If B does not occur, A cannot occur.)
Every argument involves an inferential claim—that the premises provide grounds for the truth of the conclusion. This section addresses the *strength* of that claim: How strongly is the conclusion claimed to follow from the premises? The reasoning process (inference) involved is presented either with certainty or with probability, corresponding to the two main classes of arguments: deduction and induction, respectively.
If an argument claims that the conclusion follows with strict certainty or necessity, it is deductive. If it claims the conclusion follows only probably, it is inductive. These represent two fundamentally different ways premises can support a conclusion. Understanding this distinction, based on the strength of the inferential claim, is essential in logic. This section defines deductive and inductive arguments and explores techniques for distinguishing between them.
After successful completion of this section, you will be able to:
How would you define a deductive argument?
A deductive argument is an argument where the premises are claimed to support the conclusion in such a way that it is *impossible* for the conclusion to be false if the premises are true. The conclusion is claimed to follow *necessarily* or *conclusively* from the premises. Deductive arguments involve necessary reasoning, aiming for certainty.
Example 1:
All philosophers are critical thinkers.
Socrates is a philosopher.
Therefore, Socrates is a critical thinker.
Example 2:
All mammals have hearts.
All dogs are mammals.
It follows that, all dogs have hearts.
These are examples of deductive arguments. In both, the conclusion is claimed to follow from the premises with certainty. The premises are claimed to provide conclusive support. For instance, if we assume that all philosophers are critical thinkers and Socrates is a philosopher (Example 1), it is impossible for Socrates *not* to be a critical thinker. Similarly, if all mammals have hearts and all dogs are mammals (Example 2), it is impossible for dogs *not* to have hearts. The structure implies necessity, so these arguments are interpreted as deductive.
How would you define an inductive argument?
An inductive argument is an argument where the premises are claimed to support the conclusion such that it is *improbable* (but not impossible) for the conclusion to be false if the premises are true. The conclusion is claimed to follow only *probably* from the premises. The premises provide some evidence, potentially considerable, but they do not guarantee the conclusion's truth. Inductive arguments involve probabilistic reasoning, aiming for likelihood rather than certainty.
Example 1:
Most African leaders have been black.
Mandela was an African leader.
Therefore, probably Mandela was black.
Example 2:
Every swan I have ever seen is white.
Therefore, likely all swans are white.
Both examples are inductive. Their conclusions do not follow with strict necessity but with some degree of probability. The premises are claimed to make the conclusion likely. If we assume most African leaders have been black and Mandela was an African leader (Example 1), it is probable, but not certain, that Mandela was black. If we assume every swan observed so far has been white (Example 2), it is probable, but not guaranteed, that all swans are white (black swans were later discovered). The reasoning supports the conclusion probabilistically, so these arguments are best interpreted as inductive.
How can one distinguish between deductive and inductive arguments?
The distinction between deductive and inductive arguments hinges on the *claimed* strength of the inferential link: does the argument aim for certainty (deductive) or probability (inductive)? Since this claim is often implicit, we rely on objective features of the argument to interpret its type.
Three main factors help determine whether an argument's inferential claim is deductive or inductive:
It's important to note that many arguments in ordinary language are incompletely stated, making a definitive classification difficult.
The first factor involves special indicator words. Words like "certainly," "necessarily," "absolutely," and "definitely" typically signal a deductive argument. Conversely, words like "probably," "improbable," "plausible," "implausible," "likely," "unlikely," and "reasonable to conclude" usually suggest an inductive argument.
While indicator words often guide interpretation, caution is needed. Arguers might use deductive indicators ("certainly follows") for rhetorical effect in an argument better classified as inductive, or mistakenly claim to "deduce" a conclusion from probabilistic evidence. If indicator words conflict with other factors (like the actual strength of the link or the argument form), the other factors usually take precedence. Indicator words are clues, not guarantees.
This leads to the second factor: the actual strength of the inferential link. If the conclusion follows with strict necessity from the premises (i.e., it's impossible for the premises to be true and the conclusion false), the argument is clearly deductive. If the conclusion follows probably, but not necessarily, it's usually best interpreted as inductive.
Consider these examples:
Example 1 (Deductive):
All squares are rectangles.
This shape is a square.
Therefore, this shape is a rectangle.
Example 2 (Inductive):
The vast majority of college students own a smartphone.
Sarah is a college student.
Therefore, Sarah probably owns a smartphone.
In Example 1, the conclusion follows with strict necessity. If the premises are true, the conclusion *must* be true. Thus, it's deductive. In Example 2, the conclusion follows probably, but not necessarily. Even if the premises are true, it's possible (though unlikely) that Sarah doesn't own a smartphone. Thus, it's best interpreted as inductive.
Sometimes, an argument lacks indicator words, and the conclusion doesn't clearly follow necessarily or probably (perhaps it doesn't follow well at all). This highlights the need for the third factor: the characteristic form or pattern of argumentation. Certain argument forms are inherently deductive, while others are typically inductive. Recognizing these patterns helps classify arguments.
Several argument forms inherently signal that the premises are intended to provide conclusive support for the conclusion. These are typically classified as deductive. Key examples include:
Syllogisms are classic deductive argument structures.
Categorical Syllogism: A syllogism where each statement begins with "all," "no," or "some," relating categories or classes.
Example:
All mammals are warm-blooded.
All whales are mammals.
Hence, all whales are warm-blooded.
Arguments of this form are typically interpreted as deductive.
Hypothetical Syllogism: A syllogism containing a conditional ("if...then...") statement as one or both premises.
Example (Hypothetical Syllogism):
If it rains, the ground gets wet.
If the ground gets wet, the plants will grow.
Therefore, if it rains, the plants will grow.
These are generally interpreted as deductive.
Disjunctive Syllogism: A syllogism containing a disjunctive ("either...or...") statement as a premise.
Example:
Either the battery is dead or the starter is broken.
The battery is not dead.
Therefore, the starter is broken.
Disjunctive syllogisms are usually best treated as deductive.
Inductive arguments often involve conclusions that "go beyond" the information explicitly contained in the premises, moving from familiar observations to less familiar claims. Common forms include:
These forms typically yield conclusions that are probable at best.
The categories of inductive arguments listed are not strictly mutually exclusive; overlaps exist (e.g., cause-to-effect inferences can also be predictions). Care must be taken not to confuse deductive arguments (like those in geometry proving properties based on congruence) with superficially similar inductive forms (like arguments from analogy).
Arguments applying known scientific laws to specific cases are often treated as deductive, though this sometimes involves assuming the law applies perfectly without exception, which might be viewed with reservation outside purely theoretical contexts.
A traditional but overly simplistic view holds that induction proceeds from particular premises to a general conclusion, while deduction proceeds from general premises to a particular conclusion. (A particular statement concerns specific members of a class; a general statement concerns all members).
This is inaccurate. Both deductive and inductive arguments can move between general and particular in various combinations.
Example (Deductive: Particular to General):
Three is a prime number.
Five is a prime number.
Seven is a prime number.
Therefore, all odd numbers between two and eight are prime numbers.
Example (Inductive: General to Particular):
All emeralds previously discovered have been green.
Therefore, the next emerald to be found will likely be green.
In summary, distinguishing deductive from inductive arguments relies on assessing the claimed strength of the inference, considering indicator words, the actual logical link, and the argument's characteristic form, rather than just whether it moves from general to particular or vice versa.
Every argument makes two fundamental claims: a factual claim (that evidence or reasons exist, asserted by the premises) and an inferential claim (that the alleged evidence supports the conclusion). Evaluating any argument involves assessing both claims. The inferential claim is paramount; if the reasoning is flawed (premises fail to support the conclusion), the argument is worthless, regardless of the premises' truth. Therefore, we first test the inferential claim. Only if the reasoning is adequate (valid for deduction, strong for induction) do we proceed to test the factual claim (whether the premises are actually true). This section introduces the core concepts and terminology for evaluating arguments, focusing on the characteristics of good deductive and inductive arguments.
After successful completion of this section, you will be able to:
What do "validity" and "soundness" mean in the context of deductive arguments? How are they evaluated?
Recall that a deductive argument claims it's impossible for the conclusion to be false if the premises are true. If the argument's structure *actually* guarantees this, the argument is valid. If the structure *fails* to guarantee this, it is invalid.
A valid deductive argument is one where, assuming the premises are true, the conclusion *must* be true. The conclusion follows with strict necessity from the premises due to the argument's logical form.
An invalid deductive argument is one where, even assuming the premises are true, it is *possible* for the conclusion to be false. The conclusion does not follow with strict necessity, despite the deductive claim.
Examples:
Example 1 (Valid):
All dogs are mammals. (Assume True)
Fido is a dog. (Assume True)
Therefore, Fido is a mammal. (Must be True)
Example 2 (Invalid):
All dogs are mammals. (Assume True)
Fido is a mammal. (Assume True)
Therefore, Fido is a dog. (Possibly False - Fido could be a cat)
Example 1 is valid: if the premises are true, the conclusion cannot be false. Example 2 is invalid: even if the premises are true, the conclusion could still be false (Fido might be another type of mammal). Validity concerns the *form* or *structure* of the argument, guaranteeing truth transfer *if* the premises are true.
The definitions of valid and invalid lead to two key points:
1. Validity is absolute: An argument is either valid or invalid; there's no middle ground.
2. Validity is only *indirectly* related to the actual truth or falsity of the statements. An argument can be valid even with false premises and a false conclusion. Validity solely concerns whether the premises, *if they were true*, would necessitate the conclusion's truth.
To test for validity, we assume the premises are true and then determine if it's *possible* for the conclusion to be false under that assumption. Validity assesses the logical connection, not the factual accuracy of the statements.
Considering the actual truth values, there are four combinations of premises and conclusion:
Importantly, only the second combination (True premises, False conclusion) *guarantees* invalidity. All other combinations are possible for *both* valid and invalid arguments.
Possibility 1: True Premises, True Conclusion (Possible for both Valid and Invalid)
Example 1a (Valid):
All men are mortal. (T)
Socrates is a man. (T)
Therefore, Socrates is mortal. (T)
Example 1b (Invalid):
Some mammals are dogs. (T)
Some mammals are cats. (T)
Therefore, some dogs are cats. (T - but form is invalid)
Both arguments have true premises and a true conclusion, but the first is valid, while the second is invalid due to its flawed structure.
Possibility 2: True Premises, False Conclusion (Impossible for Valid, Guarantees Invalidity)
Example 2a (Invalid):
All dogs are mammals. (T)
All cats are mammals. (T)
Therefore, all dogs are cats. (F)
A valid argument cannot have this combination. If the premises are true and the conclusion is false, the argument *must* be invalid by definition.
Possibility 3: False Premises, True Conclusion (Possible for both Valid and Invalid)
Example 3a (Valid):
All fish are mammals. (F)
All whales are fish. (F)
Therefore, all whales are mammals. (T)
Example 3b (Invalid):
All reptiles fly (F).
All birds fly (T).
Therefore, all birds are reptiles (T).
Even with false premises, an argument can have a true conclusion, and the argument form itself might be valid (Example 3a) or invalid.
Possibility 4: False Premises, False Conclusion (Possible for both Valid and Invalid)
Example 4a (Valid):
All mammals have wings. (F)
All reptiles are mammals. (F)
Thus, all reptiles have wings. (F)
Example 4b (Invalid):
All dogs are cats. (F)
Some cats are fish. (F)
Therefore, some dogs are fish. (F)
Arguments with false premises and a false conclusion can also be either valid (Example 4a, the structure works) or invalid (Example 4b, the structure fails).
The core takeaway is that validity is determined by the logical relationship between premises and conclusion, not their actual truth values. The only combination that definitively determines validity is True Premises and False Conclusion, which *always* indicates an invalid argument. A system of logic would be useless if it allowed deriving falsity from truth through a valid process.
The relationship is summarized below:
| Premise Truth | Conclusion Truth | Possible Validity? |
|---|---|---|
| True | True | Can be Valid or Invalid |
| True | False | Must be Invalid |
| False | True | Can be Valid or Invalid |
| False | False | Can be Valid or Invalid |
Evaluating an argument involves assessing both its inferential claim (reasoning structure) and its factual claim (truth of premises). Having addressed validity (the inferential claim for deduction), we now turn to the factual claim.
A deductive argument that successfully meets *both* criteria—it is valid *and* has all true premises—is called a sound argument. If an argument fails on either condition (it is invalid, or it has at least one false premise, or both), it is unsound.
Since a valid argument guarantees that true premises lead to a true conclusion, and a sound argument *is* valid and *has* true premises, it follows that every sound argument necessarily has a true conclusion. A sound argument represents a "good" deductive argument in the fullest sense—it has correct reasoning and starts from true foundations.
All invalid arguments are automatically unsound.
What do "strength" and "cogency" mean for inductive arguments? How are they evaluated?
Recall that an inductive argument claims its premises make the conclusion probable. If the premises, assumed true, *actually* make the conclusion probable (more likely than not), the argument is strong. If they do *not* make the conclusion probable, the argument is weak.
A strong inductive argument is one where, assuming the premises are true, the conclusion is likely true (improbable to be false). A weak inductive argument is one where, assuming the premises are true, the conclusion is not likely true (it's probable or at least quite possible that it's false), despite the inductive claim.
Examples:
Example 1 (Strong):
This barrel contains 100 apples.
80 apples selected randomly were found tasty.
Therefore, probably all 100 apples are tasty.
Example 2 (Weak):
This barrel contains 100 apples.
3 apples selected randomly were found tasty.
Therefore, probably all 100 apples are tasty.
Example 1 is strong: the large, random sample provides good probabilistic support for the conclusion. Example 2 is weak: the sample size is too small to justify the broad conclusion with much probability. Testing inductive strength parallels testing deductive validity: assume premises are true and assess the likelihood of the conclusion.
Two key points arise from the definitions of strong and weak:
1. Unlike validity, strength admits degrees. An argument can be very strong, moderately strong, weak, very weak, etc. There isn't an absolute cutoff, but generally, a strong argument's conclusion must be more probable than improbable (better than 50% likelihood, given the premises). Adding or modifying premises can alter an inductive argument's strength (e.g., increasing sample size strengthens; finding contradictory evidence weakens).
2. Like validity, strength is only indirectly related to truth values. An argument can be strong with false premises or a false conclusion. Strength assesses the probabilistic connection: *if* the premises were true, would the conclusion likely be true?
To test for strength, we assume the premises are true and then determine the probability of the conclusion in light of that assumption. Strength results from the level of probabilistic support the premises provide, not their actual truth.
The four truth value combinations apply to inductive arguments as well. Similar to deductive arguments, the combination of True Premises and False Conclusion is incompatible with a strong argument. If the premises are actually true, but the conclusion turns out to be false, the premises did not make the conclusion probable, meaning the argument must be weak.
All other combinations (T/T, F/T, F/F) are possible for both strong and weak arguments.
| Premise Truth | Conclusion Truth | Possible Strength? |
|---|---|---|
| True | True | Can be Strong or Weak |
| True | False | Must be Weak |
| False | True | Can be Strong or Weak |
| False | False | Can be Strong or Weak |
Having assessed inductive strength (the inferential claim), we evaluate the factual claim (truth of premises).
An inductive argument that is strong *and* has all true premises is called a cogent argument. If an argument fails on either condition (it is weak, or it has at least one false premise, or both), it is uncogent.
Since a cogent argument is strong and based on true premises, its conclusion is genuinely supported and likely true. Cogency is the inductive equivalent of soundness, representing a "good" inductive argument in the fullest sense.
All weak arguments are automatically uncogent.
*See next slide for the 'total evidence' requirement.
There's a crucial difference between soundness (deductive) and cogency (inductive) regarding the true-premise requirement. For soundness, true premises and validity guarantee a true conclusion. For cogency, however, the premises must not only be true but must also consider all relevant, available evidence. An inductive argument, even if strong and based on true premises, is *not* cogent if it deliberately ignores or omits important evidence that would significantly weaken the argument or support a different conclusion. This is often called the total evidence requirement. A good inductive argument must be based on all known relevant evidence.
Therefore, a cogent argument is strong, has true premises, and meets the total evidence requirement. An uncogent argument fails on one or more of these criteria.
Logic is the science evaluating arguments, its primary subject. It aims to provide methods for assessing and constructing arguments. An argument comprises premise(s) claimed to support a conclusion. Premises offer alleged evidence; the conclusion is the claim allegedly following from that evidence. Argument quality depends on how well premises support the conclusion.
Not all passages are arguments. To identify arguments, look for: (1) premise/conclusion indicator words, (2) an inferential relationship between statements, and (3) contrast with typical non-argument forms (warnings, advice, explanations, etc.). Remember indicator words aren't foolproof; context and the claimed inferential link are key. Mentally inserting "therefore" can help identify potential conclusions in passages lacking indicators.
Arguments are broadly divided into deductive (claiming necessary support) and inductive (claiming probable support). Distinguishing them involves considering indicator words, the actual strength of the premise-conclusion link (necessity vs. probability), and characteristic argument forms (e.g., mathematical proof vs. prediction). If the conclusion follows necessarily, the argument is deductive. If it follows probably, it's likely inductive.
Evaluating arguments requires checking both the inferential claim (reasoning) and the factual claim (premise truth). First, assume premises are true. For deductive arguments, if the conclusion *must* follow, it's valid; otherwise, it's invalid. For inductive arguments, if the conclusion *probably* follows, it's strong; otherwise, it's weak. Second, if the argument is valid or strong, check if all premises are actually true. A valid deductive argument with all true premises is sound. A strong inductive argument with all true premises (meeting the total evidence requirement) is cogent. Invalid arguments are unsound; weak arguments are uncogent. Sound and cogent arguments represent good reasoning based on true foundations.