What Logic Is

• Logic is the study of the methods and principles used to distinguish correct from incorrect reasoning.
• When we reason about any matter, we produce arguments to support our conclusions.
• Our arguments include reasons that we think justify our beliefs.
• Not all reasons are good reasons.
• We may always ask, when confronting an argument: Does the conclusion reached follow from the premises assumed?
• To answer this question, there are objective criteria.
• In the study of logic, we seek to discover and apply those criteria.
• Reasoning is not the only way in which people support assertions they make or accept.
• People may appeal to authority or to emotion, which can be very persuasive.
• People may also rely, without reflection, simply on habits.
• When someone wants to make judgments that can be completely relied upon, their only solid foundation will be correct reasoning.
• Using the methods and techniques of logic, one can distinguish reliably between sound and faulty reasoning.

Propositions and Arguments

A. Propositions

• Propositions are the building blocks of our reasoning.
• A proposition asserts that something is the case or it asserts that something is not.
• We may affirm a proposition, or deny it, but every proposition either asserts what really is the case, or it asserts something that is not.
• Therefore, every proposition is either true or false.
• There are many propositions about whose truth we are uncertain.
• “There is life on some other planet in our galaxy,” for example, is a proposition that, so far as we now know, may be true or may be false.
• Its “truth value” is unknown, but this proposition, like every proposition, must be either true or false.
• A question asserts nothing, and therefore it is not a proposition.
• “Do you know how to play chess?” is indeed a sentence, but that sentence makes no claim about the world.
• Neither is a command a proposition (“Come quickly!”), nor is an exclamation a proposition (“Oh my gosh!”).
• Questions, commands, and exclamations, unlike propositions, are neither true nor false.
• When we assert some proposition, we do so using a sentence in some language.
• However, the proposition we assert is not identical to that sentence.
• This is evident because two different sentences, consisting of different words differently arranged, may have the same meaning and may be used to assert the very same proposition.
• For example, “Leslie won the election” and “The election was won by Leslie” are plainly two different sentences that make the same assertion.
• Sentences are always parts of some language, but propositions are not tied to English or to any given language.
• They have a single meaning: all four, using different words, may be uttered to assert the very same proposition.
• Proposition is the term we use to refer to what it is that declarative sentences are typically used to assert.
• The term statement is not an exact synonym of proposition, but it is often used in logic in much the same sense.
• Some logicians prefer statement to proposition, although the latter has been more commonly used in the history of logic.
• Other logicians eschew both terms as metaphysical, using only the term sentence.
• However, the concept of a proposition is seen by many as making a useful distinction between a sentence and what the sentence asserts.
• The very same sentence can be used to make very different statements (or to assert very different propositions), depending on the context in which it is expressed.
• For example, the sentence, “The largest state in the United States was once an independent republic,” once expressed a true statement or proposition (about Texas), but if asserted today would express a false statement or proposition (about Alaska).
• The same words assert different propositions at different times.
• Propositions may be simple, like those used in the preceding illustrations, but they may also be compound, containing other propositions within themselves.
• Consider the following proposition, from a recent account of the exploitation of the Amazon Basin in Brazil: “The Amazon Basin produces roughly 20 percent of the Earth’s oxygen, creates much of its own rainfall, and harbors many unknown species.”
• This sentence simultaneously asserts three propositions, concerning what the Amazon Basin produces and what it creates and what it harbors.
• The passage thus constitutes a conjunctive proposition.
• Asserting a conjunctive proposition is equivalent to asserting each of its component propositions separately.
• Some compound propositions do not assert the truth of their components.
• In disjunctive (or alternative) propositions, no one of the components is asserted.
• Abraham Lincoln (in a message to Congress in December 1861) said, “Circuit courts are useful, or they are not useful.”
• This disjunctive proposition is plainly true, but either one of its components might be false.
• Other compound propositions that do not assert their components are hypothetical (or conditional) propositions.
• The eighteenth-century freethinker, Voltaire, said, “If God did not exist, it would be necessary to invent him.”
• Here, again, neither of the two components is asserted.
• The proposition “God does not exist” is not asserted, nor is the proposition “it is necessary to invent him.”
• Only the “if-then” proposition is asserted by the hypothetical or conditional statement, and that compound statement might be true even if both of its components were false.
• In logic, the internal structure of propositions is important.
• To evaluate an argument we need a full understanding of the propositions that appear in that argument.
• Propositions of many different kinds will be analyzed in this chapter.

B. Arguments

• With propositions as building blocks, we construct arguments.
• In any argument, we affirm one proposition on the basis of some other propositions.
• In doing this, an inference is drawn.
• Inference is a process that may tie together a cluster of propositions.
• Some inferences are warranted (or correct); others are not.
• The logician analyzes these clusters, examining the propositions with which the process begins and with which it ends, as well as the relations among these propositions.
• Such a cluster of propositions constitutes an argument.
• Arguments are the chief concern of logic.
• Argument is a technical term in logic.
• It need not involve disagreement or controversy.
• In logic, argument refers strictly to any group of propositions of which one is claimed to follow from the others, which are regarded as providing support for the truth of that one.
• For every possible inference, there is a corresponding argument.
• In writing or in speech, a passage will often contain several related propositions and yet contain no argument.
• An argument is not merely a collection of propositions; it is a cluster with a structure that captures or exhibits some inference.
• We describe this structure with the terms conclusion and premise.
• The conclusion of an argument is the proposition that is affirmed on the basis of the other propositions of the argument.
• Those other propositions, which are affirmed (or assumed) as providing support for the conclusion, are the premises of the argument.
• We will encounter a vast range of arguments in this text—arguments of many different kinds, on many different topics.
• We will analyze arguments in politics, in ethics, in sports, in religion, in science, in law, and in everyday life.
• Those who defend these arguments, or who attack them, are usually aiming to establish the truth (or the falsehood) of the conclusions drawn.
• As logicians, however, our interest is in the arguments as such.
• As agents or as citizens, we may be deeply concerned about the truth or falsity of the conclusions drawn.
• However, as logicians, we put those interests aside.
• Our concerns will be chiefly twofold.
• First, we will be concerned about the form of an argument under consideration, to determine if that argument is of a kind that is likely to yield a warranted conclusion.
• Second, we will be concerned about the quality of the argument, to determine whether it does in fact yield a warranted conclusion.
• Arguments vary greatly in the degree of their complexity.
• Some are very simple.
• Other arguments, as we will see, are quite intricate, sometimes because of the structure or formulation of the propositions they contain, sometimes because of the relations among the premises, and sometimes because of the relations between premises and conclusion.
• The simplest kind of argument consists of one premise and a conclusion that is claimed to follow from it.
• Each may be stated in a separate sentence, as in the following argument that appears on a sticker affixed to biology textbooks in the state of Alabama: “No one was present when life first appeared on earth. Therefore, any statement about life’s origins should be considered as theory, not fact.”
• Both premise and conclusion may be stated within the same sentence, as in this argument arising out of recent advances in the science of human genetics: “Since it turns out that all humans are descended from a small number of African ancestors in our recent evolutionary past, believing in profound differences between the races is as ridiculous as believing in a flat earth.”
• The order of premises and conclusion can vary and does not affect the quality of the argument.
• Conclusions often precede premises, as demonstrated by the New York Times on 13 July 1934, when Babe Ruth hit his 700th home run.
• The New York Times argument:
  • Conclusion: Babe Ruth’s record will endure.
  • Premises:
    1. Few players have enjoyed Ruth’s longevity.
    2. Only two players have hit more than 300 home runs.
• This argument’s conclusion was later proven false when Hank Aaron hit his 700th home run on 21 July 1973.
• Jeremy Bentham’s argument in Principles of Legislation (1802):
  • Conclusion: Every law is an evil.
  • Premise: Every law is an infraction of liberty.
• A single sentence can contain both premise and conclusion, but no single proposition can be an argument.
• Hypothetical proposition example:
  • If a state aims to be a society of equals, then a state based on the middle class is the best constituted.
  • No assertion is made, and both components might be false.
• Aristotle’s argument in Politics, Book IV, Chapter 11:
  • Conclusion: A state based on the middle class is best constituted.
  • Premise: A state aims to be a society of equals.
• Arguments consist of a group of propositions with one being the conclusion and the others being premises.
• Not every structured cluster of propositions is an argument, as shown by a recent account of global inequality:
  • Describes global disparity in affluence and poverty.
  • No argument is presented.
• Reasoning is both an art and a science, requiring practice to build and test arguments.
• Skill in reasoning improves with practice, leading to more accurate conclusions.
• The chapter provides numerous opportunities to practice analyzing arguments.

Recognizing Arguments

• Before evaluating an argument, we must first recognize it.
• This involves distinguishing argumentative passages in writing or speech.
• Recognizing arguments requires an understanding of the language used in the passage.
• Despite understanding the language, identifying an argument can be challenging due to peculiarities in its formulation.
• Even when confident that an argument is intended, identifying its premises and conclusion can be difficult.
• The order of propositions does not determine which are premises and which is the conclusion.
• Therefore, another method must be used to identify premises and conclusions in an argument.

A. Conclusion Indicators and Premise Indicators

• One useful method for identifying arguments involves recognizing common indicators.
• Certain words or phrases typically signal an argument’s conclusion or premises.
• These indicators help identify the components of an argument.
• Premise indicators are words or phrases that usually mark the premises of an argument.
• What follows a premise indicator is usually, but not always, the premise of some argument.

B. Arguments in Context

• The listed words and phrases can help indicate the presence of an argument or identify its premises or conclusion, but these indicators do not always appear.
• Sometimes the meaning or setting of the passage indicates the presence of an argument.
• Example: A 2007 critique of U.S. troop deployment to Iraq lacked indicators but still presented a clear argument:
  • Premises: Iraqi leaders abandoned their posts; Iraqis rejected the opportunity given to them.
  • Conclusion: Iraq does not deserve U.S. help.
• In Sam Harris’s Letter to a Christian Nation, indicators are absent, but the argument is clear:
  • Premises: Half of Americans believe the universe is 6,000 years old; they are wrong about this.
  • Conclusion: Declaring them wrong is intellectual honesty, not “irreligious intolerance.”
• Understanding the context in which an argument is presented is often crucial for appreciating its force.
• Example: In Gratz v. Bollinger (2003), Justice Ruth Bader Ginsburg defended the University of Michigan’s admission system:
  • Premises: Universities will maintain minority enrollment through camouflage if not openly.
  • Conclusion: Michigan’s transparent affirmative action program is preferable to covert methods.
• This argument derives its force from the historical context of universities disguising their preferential admission programs.
• Chief Justice William Rehnquist’s response is intelligible only in the context of Ginsburg’s defense:
  • Premises: Universities will pursue affirmative action whether or not it violates the Constitution.
  • Conclusion: Instead of requiring universities to obey the Constitution, the suggestion is to change the Constitution to fit university conduct.
• Rehnquist’s reference to “changing the Constitution” must be understood in the context of the Michigan system being held unconstitutional.
• The full force of arguments and counterarguments is often grasped only with an understanding of their context.
• Context is critical in real life. For example, if someone says they are bringing a lobster home for dinner, it is understood they intend to eat it, not feed it.

C. Premises or Conclusions Not in Declarative Form

• The premises of an argument can sometimes be presented in the form of questions.
• While questions do not express propositions or assert anything on the surface, they can serve as premises when they are rhetorical.
• Rhetorical questions suggest or assume an answer that acts as the premise of an argument.
• This use of questions is often obvious, as in a letter to The New York Times on 7 January 2007, where Irit R. Rasooly objected to a new series of U.S. coins honoring former presidential wives.
• Rasooly’s rhetorical question: “Should we bestow this honor on people who are unelected, whose only credential is having a prominent spouse?” suggests the premise that we should not.
• Rasooly continues with another rhetorical question: “Wouldn’t honoring women who have served as governors, Supreme Court justices, or legislators be a more fitting tribute than coins featuring ‘First Spouses’?”
• This question suggests the premise that honoring such achievements would indeed be a more fitting tribute.
• Context is important for interpreting declarative statements that convey judgments rather than mere descriptions.
• Rasooly’s report of being “irked” by the new coins is more than a description of his state of mind; it expresses a judgment against issuing such coins.
• Using questions as premises can be counterproductive if they invite answers that undermine the argument’s conclusion.
• Example: The Archbishop of the Anglican Church in Nigeria, an opponent of homosexuality, argues against it using rhetorical questions about God’s intentions:
  • “Why didn’t God make a lion, a tree, or another man to be a man’s companion?”
  • These questions invite varied responses that may weaken the argument.
• Questions serve effectively as premises when the assumed answers are clear and inescapable, leading the reader or listener to provide the evident answers themselves.
• Example: Those who find euthanasia morally unacceptable argue against the right to euthanasia grounded in self-determination:
  • “If a right to euthanasia is grounded in self-determination, it cannot reasonably be limited to the terminally ill.”
  • “If people have a right to die, why must they wait until they are actually dying before they are permitted to exercise that right?”
• Arguments relying on rhetorical questions can be questionable because the question itself does not assert anything true or false.
• Rhetorical questions often suggest the truth of a proposition without directly asserting it, which can lead to dubious or false conclusions.
• Example: David Gelernter’s question about Arab leaders’ motivations behind protests at the Al-Aqsa mosque implies duplicity without asserting it outright.
• Similarly, gossip columnists and journalists often use rhetorical questions to insinuate assertions without directly stating them.
• Critics may protect themselves by framing accusations as questions, avoiding direct responsibility for their implications.
• It’s generally advised to avoid arguing with questions due to their non-assertive nature.
• In some arguments, conclusions appear in the form of imperatives rather than declarative statements.
• Example: Proverbs 4:7 states, “Wisdom is the principal thing; therefore get wisdom.”
• A command cannot strictly be a conclusion of an argument because it does not express a proposition.
• However, commands are often understood as assertions that it is wise or necessary to act in a certain way.
• Reformulating imperatives as assertions clarifies their role in arguments, treating them as statements that specify what should be done.
• This approach allows for consistent handling of arguments where conclusions are expressed as imperatives.
• Some arguments present premises in the form of phrases rather than full declarative sentences.
• Example: “What is a failed state? It is one that fails to provide security, guarantee rights, or maintain democratic institutions. On this definition, the United States is the world’s biggest failed state.”
• Despite being phrased as questions or fragments, these statements serve as clear premises supporting the argument’s conclusion.

D. Unstated Propositions

• Arguments can sometimes be obscure because one or more of their propositions are assumed rather than stated explicitly.
• Example: Arguments regarding the justifiability of the death penalty often rely on unstated premises. The first argument assumes: “No one should be executed to achieve an objective not promoted by execution.”
• Example: The second argument assumes: “Protecting innocent lives justifies executing murderers if it deters other potential murderers.”
• Both arguments depend on the assumed truth of these unstated premises to be persuasive.
• Premises may be left unstated because they are considered common knowledge or unquestioned assumptions.
• Enthymemes are arguments in everyday discourse that rely on unstated propositions.
• Example: In debates over human cloning, arguments may assume: “What is intrinsically evil should never be allowed.”
• Enthymemes may deliberately leave critical premises unstated to avoid direct attack.
• Example: Arguments against embryonic stem cell research often rely on the unstated premise that “killing a human embryo is illegal,” which is contentious.
• The effectiveness of an enthymeme can hinge on the audience’s recognition or acceptance of the unstated premise.
• Example: Abraham Lincoln’s argument, “If slavery is not wrong, nothing is wrong,” relies on the unstated premise that “nothing is wrong” is universally recognized as false.
• Similarly, Bruno Bettelheim’s statement, “If all men are good, then there never was an Auschwitz,” assumes the falsehood of the premise that “all men are good.”
• Enthymemes can effectively highlight the falsity of a proposition by structuring arguments around hypothetical scenarios where the unstated premise’s falsehood is implicit.

Arguments and Explanations

• Passages that use common indicators like “because,” “for,” “since,” and “therefore” can serve different purposes depending on the author’s intention.
• Example 1: Biblical passage from Matthew 7:19 is an argument. It concludes that one should lay up treasures in heaven based on the premise that where one’s treasure is, there one’s heart will be also.
• Example 2: Biblical passage from Genesis 11:19 is an explanation. It explains why the tower is called Babel, stating that it was named after the confusion of languages caused by God.
• In arguments, “Q because P” aims to establish the truth of proposition Q with evidence P. In explanations, “Q because P” aims to provide reasons or causes for the known truth of proposition Q.
• Example from astronomy: Explanation of why quasars appear red due to the absorption of blue light over vast distances in space.
• Sometimes it’s challenging to discern whether an author intends to explain or argue. Context and the status of proposition Q (whether it needs confirmation or is already known) help determine this.
• An explanation clarifies how something came to be (e.g., the name of the tower of Babel), while an argument seeks to establish the truth of a proposition.
• In ambiguous cases, alternative interpretations may be equally plausible, viewing passages as arguments or explanations depending on the context and purpose inferred by the reader.

Deductive and Inductive Arguments

• Deductive arguments claim that their conclusion is supported conclusively by their premises.
• Inductive arguments do not claim conclusiveness; they provide evidence that supports their conclusion probabilistically.
• Validity in deductive logic means that if the premises are true, the conclusion must be true.
• Deductive arguments are either valid (if their conclusion must be true if premises are true) or invalid (if the conclusion does not follow from the premises).
• Classical logic, rooted in Aristotle’s works, focuses on determining the validity of deductive arguments.
• The central task of deductive logic is to distinguish between valid and invalid arguments using traditional techniques.
• Inductive arguments aim to establish factual conclusions through empirical investigation, such as in medicine, social science, or astronomy.
• Inductive reasoning seeks to understand cause-and-effect relationships and make predictions based on observed evidence.
• Example: Medical researchers use inductive methods to understand the causes and transmission of sexually transmitted diseases (STDs) like AIDS.
• The goal of induction is to gather enough evidence to draw reliable conclusions about phenomena in the real world.
• Inductive arguments, unlike deductive arguments, do not claim certainty for their conclusions.
• Terms like validity and invalidity do not apply to inductive arguments because their conclusions are probabilistic.
• Evaluating the strength of inductive arguments involves assessing the probability of the conclusion based on the premises.
• The circumcision study in Africa exemplifies a strong inductive argument, where the reduction in HIV transmission risk is supported by substantial evidence.
• Even with strong premises, an inductive argument does not establish its conclusion with certainty; it remains subject to new evidence that could strengthen or weaken its probability.
• Inductive reasoning acknowledges that conclusions are never absolutely certain and can be revised with new discoveries.
• Deductive arguments, by contrast, are either valid (conclusively true if premises are true) or invalid (if the conclusion does not follow necessarily from the premises).
• Validity in deductive arguments means the conclusion is certain if the premises are true, and this certainty cannot be enhanced or diminished by additional information.
• The example of Socrates being mortal (from the premises “All humans are mortal” and “Socrates is human”) illustrates deductive certainty that remains unchanged by any other facts or premises.
• Deductive arguments establish a strict relation between premises and conclusion that does not depend on external factors or additional information
• Inductive arguments establish conclusions with probability rather than certainty.
• The relationship between premises and conclusion in an inductive argument is not strict; additional information can weaken or strengthen the argument.
• Example: The argument about Miriam Graf being conservative is inductive. Adding the premise that she is an officer of the ACLU weakens the probability that she is conservative.
• Conversely, adding the premise that Miriam Graf is a long-time officer of the NRA strengthens the probability that she is conservative.
• Inductive arguments do not always explicitly state the probabilistic nature of their conclusions.
• Not all arguments mentioning probabilities are inductive; deductive arguments about probabilities can exist, where probabilities are deduced from other probabilities.
• Deductive arguments claim conclusions follow necessarily from premises without any degree of probability.
• Inductive arguments claim conclusions follow with some degree of probability, dependent on the premises and additional information.
• The distinction between deduction and induction hinges on whether the argument claims necessity (deductive) or probability (inductive) in the relationship between premises and conclusion.

Validity and Truth

• Validity in deductive arguments pertains to the logical necessity with which the conclusion follows from the premises.
• It cannot be applied to single propositions because validity relies on the relationship between a set of premises and a conclusion.
• Truth and falsehood, in contrast, are attributes of individual propositions. A proposition can be true if it corresponds to reality (e.g., Lake Superior is the largest Great Lake) or false if it does not (e.g., claiming Lake Michigan is the largest).
• Arguments themselves are neither true nor false; they are structures of inference from premises to a conclusion.
• Deductive logic examines the complex relations between true or false propositions and valid or invalid arguments.
• An argument can be valid even if its premises are false. This means the conclusion logically follows from the premises as stated, regardless of the truth of those premises.
• Abraham Lincoln illustrated this distinction in his critique of the Dred Scott decision, arguing that while the reasoning was valid, the premise about the right of property in a slave being affirmed in the Constitution was false.
  • The relationship between validity and truth is fundamental in deductive logic, where validity concerns the structure of argumentation and truth pertains to the correspondence of statements with reality.
• Validity in an argument depends solely on the logical relationship between its premises and conclusion.
• An argument can be valid even if one or more of its premises are false, as illustrated by Abraham Lincoln’s critique of the Dred Scott decision.
• Examples demonstrate various combinations of true and false premises with true and false conclusions:
  • Valid arguments with all true premises and a true conclusion (Example I).
  • Valid arguments with all false premises and a false conclusion (Example II).
  • Invalid arguments with all true premises and a true conclusion (Example III), where the conclusion does not logically follow.
  • Invalid arguments with all true premises but a false conclusion (Example IV), which cannot occur in a valid argument.
  • Valid arguments with false premises and a true conclusion (Example V).
  • Invalid arguments with false premises and a true conclusion (Example VI), showing that a true conclusion does not guarantee validity.
  • Invalid arguments with all false premises and a false conclusion (Example VII).
• The examples highlight that the truth or falsity of an argument’s conclusion does not determine its validity or invalidity.
• Validity concerns the structure of the argument and how the conclusion logically follows from the premises, regardless of the truth values of the individual statements.
• Government interventions such as licensing laws, minimum wage laws, and collective bargaining laws were often aimed at hindering black economic and social progress, contributing to disparities in employment rates between black and white Americans.
• Evolutionarily, traits that led to successful reproduction persisted, while those that did not disappeared. This includes the evolutionary advantage of sexual pleasure, which encourages reproduction and the passing on of genes.
• Immanuel Kant argues that because changes occur over time, time itself must be a real entity.
• The nursing shortage in the United States has reached crisis levels, with a significant proportion of nurses nearing retirement age and high rates of job dissatisfaction among those currently practicing.
• John Dewey asserts that attributing causes to a situation does not excuse it; justification or condemnation should be based on consequences rather than antecedents.
• Jean Bodin argues that while one may be subject to laws made by others, one cannot bind oneself in matters of free will, implying that sovereigns are not subject to their own laws.
• Oscar Wilde humorously praises Wagner’s music for being so loud that conversations can occur without being overheard.
• Howard Gardner attributes increased cheating in society to the rise of materialistic values, declining religious and communal norms, and a lack of shame among public figures caught in dishonest activities.
• Shakespeare’s quote from A Midsummer Night’s Dream suggests that love is not about physical appearance but rather about understanding and connection.
• A reader’s response to an article about human body hair evolution suggests that sexual selection pressure has influenced traits like hairlessness in women, which is perceived as a sign of youth and attractiveness.

• Bernard Lewis argues that mutually assured destruction (MAD) was effective in deterring nuclear attacks during the Cold War because both sides knew the consequences of nuclear retaliation. However, he suggests that religious fanatics, like Mahmoud Ahmadinejad of Iran, may not be deterred by MAD but could see it as an inducement to achieve martyrdom and heavenly rewards.

• Gordon Kane discusses the anthropic principle, stating that life as we know it can exist only in three-dimensional space due to the stability of planetary orbits and the physiological requirements of complex organisms.

• Oswald Werner argues for the resettlement of translators and interpreters who have aided US troops and diplomats, emphasizing their strategic importance in bridging language gaps essential for national security.

• Judge James Robertson rules against the Treasury Department, stating that its failure to design currency accessible to blind and visually impaired individuals violates anti-discrimination laws under the Rehabilitation Act.

• Sir W. David Ross contends that acting rightly, based on duty alone, does not guarantee moral goodness if done with indifferent or bad motives.

• J. Lenzi argues for the existence of immutable and eternal laws, such as those of mathematics and physics, positing that they reflect a supreme mind, which he equates with God.

• John Teehan, an atheist, embraces the celebration of Christmas, emphasizing its cultural and communal significance beyond its religious origins.

• Orlando Patterson reflects on ethnic movements, cautioning that while they may start benignly to address collective trauma, they often lead to tragedy when politicized, citing examples from German history.

• Samuel Johnson distinguishes between satisfaction and happiness, arguing that happiness involves a multiplicity of agreeable consciousness, which may differ between individuals of varying intellectual capacities.

Deductive and Inductive Arguments

• Deductive arguments claim conclusively that their premises provide irrefutable grounds for the truth of their conclusions. If this claim holds true, the argument is considered valid; if not, it is invalid.

• Validity, in deductive logic, means that if the premises of an argument are true, the conclusion must also be true. This definition is strict and does not allow for the possibility of a true conclusion with false premises.

• The central task of deductive logic is to distinguish between valid and invalid arguments using techniques developed over centuries, rooted in classical logic originating with Aristotle.

• Inductive arguments, in contrast, do not claim conclusive support for their conclusions. They aim to infer general principles or causal relationships based on observed data, such as in medical studies or social sciences.

• Inductive reasoning involves empirical investigations and aims to establish factual conclusions about cause-and-effect relationships, often with a high degree of probability rather than certainty.

• An example of inductive reasoning is the discovery that circumcision reduces the risk of HIV transmission based on large-scale studies in Africa, where circumcised men showed significantly lower rates of HIV infection compared to uncircumcised men.

• Inductive arguments make weaker claims compared to deductive arguments because their conclusions are never certain. They do not claim conclusive support for their conclusions but rather assert that their premises provide a high probability of the conclusion being true.

• Terms like validity and invalidity do not apply to inductive arguments. Instead, we evaluate them based on the degree of probability conferred on their conclusions by the premises.

• The strength of an inductive argument depends on the level of probability its premises provide for the conclusion. Even with strong support from true premises, the conclusion of an inductive argument is not established with certainty.

• Inductive arguments can be refined or weakened by new information. Additional premises can either strengthen or weaken the original argument’s conclusion, depending on their content.

• In contrast, deductive arguments claim that their conclusions follow necessarily from their premises. If an argument is valid, no additional premises can increase its validity. The certainty of the conclusion remains unchanged regardless of additional information.

• An example of a deductive argument is the classic syllogism: All humans are mortal; Socrates is human; therefore, Socrates is mortal. This conclusion follows necessarily from the premises, and no new information can alter this deductive validity.

• Inductive arguments often involve assessing probabilities rather than certainties. They do not always explicitly state the probabilistic nature of their conclusions.

• The presence of the term “probability” in an argument does not automatically classify it as inductive. There are deductive arguments that involve probabilities, where the probability of events is deduced from other probabilities.

• The distinction between deduction and induction lies in the nature of the claims made about the relationship between premises and conclusions. Deductive arguments claim absolute necessity, while inductive arguments claim probability based on observed evidence.

• In summary, deductive arguments provide absolute conclusions from premises, while inductive arguments offer conclusions with varying degrees of probability based on observed data and evidence.

Validity and Truth

• Validity in deductive arguments pertains to the logical relationship between premises and conclusions. If a deductive argument is valid, the conclusion necessarily follows from the premises.

• Truth and falsehood are attributes of individual propositions within an argument. A proposition can be true or false based on its correspondence with reality.

• Validity cannot be attributed to single propositions; it applies to the structure of deductive arguments as a whole. An argument is valid if, assuming its premises are true, its conclusion must also be true.

• The distinction between validity and truth is fundamental: validity concerns logical coherence, while truth concerns correspondence with reality.

• Deductive logic examines complex relationships between true/false propositions and valid/invalid arguments. It focuses on how conclusions logically follow from premises, regardless of the truth status of the premises.

• Deductive arguments can remain valid even if one or more premises are false. This highlights that the logical structure of an argument can be sound independent of the truthfulness of its premises.

• Deductive logic involves critical analysis of argument structures to determine their validity, emphasizing the necessity with which conclusions follow from premises in valid arguments.

• The validity of an argument depends solely on the logical relationship between its premises and conclusion, regardless of their truthfulness.
• Abraham Lincoln’s critique of the Supreme Court’s Dred Scott decision illustrates that an argument can be valid even if its conclusion and one or more premises are false.
• Seven illustrative arguments demonstrate various combinations of true and false premises and conclusions to illustrate principles about the relations between truth and validity: I. Valid with all true propositions:
  • All mammals have lungs.
  • All whales are mammals.
  • Therefore all whales have lungs. II. Valid with all false propositions:
  • All four-legged creatures have wings.
  • All spiders have exactly four legs.
  • Therefore all spiders have wings. III. Invalid with all true propositions (true premises, true conclusion):
  • If I owned all the gold in Fort Knox, then I would be wealthy.
  • I do not own all the gold in Fort Knox.
  • Therefore I am not wealthy. IV. Invalid with all true premises and false conclusion:
  • If Bill Gates owned all the gold in Fort Knox, then Bill Gates would be wealthy.
  • Bill Gates does not own all the gold in Fort Knox.
  • Therefore Bill Gates is not wealthy. V. Valid with false premises and true conclusion:
  • All fishes are mammals.
  • All whales are fishes.
  • Therefore all whales are mammals. VI. Invalid with false premises and true conclusion:
  • All mammals have wings.
  • All whales have wings.
  • Therefore all whales are mammals. VII. Invalid with all false propositions:
  • All mammals have wings.
  • All whales have wings.
  • Therefore all mammals are whales.
• The examples demonstrate that valid arguments can have false conclusions (Example II) and invalid arguments can have true conclusions (Examples III and VI), highlighting that the truth or falsity of a conclusion does not determine the validity of an argument on its own.
• Validity of an argument ensures that if its premises are true, the conclusion must be true as well. Therefore, if a conclusion is false in a valid argument, at least one premise must be false.
• A valid argument with all true premises is termed sound. A sound argument guarantees that its conclusion is true, thereby establishing the truth of its conclusion.
• The task of logic is primarily concerned with the logical relations between propositions in arguments, rather than the truth or falsehood of individual propositions. Logic determines the correctness or incorrectness of arguments based on these relations.
• Logic considers the correctness of arguments even when their premises may be false because such arguments can still be structurally valid and important in various contexts.
• In scientific inquiry and everyday decision-making, reasoning often involves deducing consequences from uncertain or theoretical premises. Correctly reasoned arguments help in evaluating theories and making informed choices, even when the truth of premises is uncertain.
• Restricting attention only to arguments with known true premises would undermine the purpose of reasoning, as it would eliminate the need to evaluate alternatives and deduce their consequences effectively. Thus, logic encompasses evaluating arguments with both true and uncertain premises to support rational decision-making and inquiry.