Introduction to Philosophy Logic

🧭 Overview

🧠 One-sentence thesis

Logic is the discipline that distinguishes good reasoning from bad by formulating precise standards for evaluating arguments, not by measuring persuasiveness but by assessing correctness according to logical rules.

📌 Key points (3–5)

• What logic evaluates: arguments—sets of propositions where premises are supposed to support a conclusion—not persuasive effectiveness.
• Core distinction: good reasoning means logically correct reasoning, not effective persuasion; bad reasoning can be highly persuasive (e.g., Hitler's arguments).
• Fundamental unit: the argument, consisting of premises (reasons) and a conclusion (the claim being supported).
• Common confusion: don't confuse propositions (things that can be true/false, expressed by declarative sentences) with questions or commands, which cannot be true or false.
• Why it matters: logic is essential to philosophy because philosophers exchange arguments to answer deep questions, and logical developments have revolutionized philosophical inquiry.

🎯 What logic does

🎯 The core task

Logic is the discipline that aims to distinguish good reasoning from bad.

• Reasoning involves making claims and backing them up with reasons, drawing inferences from evidence, acting according to beliefs.
• Reasoning can be done well or badly, correctly or incorrectly.
• Logic provides the rules and techniques to tell the difference.

⚖️ Correctness vs effectiveness

• Correctness = conforming to logical rules.
• Effectiveness = persuasiveness, ability to convince people.
• These are not the same: bad reasoning is often extremely persuasive.

Example: Hitler persuaded an entire nation with arguments that were effective but not logically correct. His reasoning failed logical standards, and he relied on threats, emotional manipulation, and unsupported assertions—rhetorical tricks, not sound reasoning.

• Don't confuse: persuasive power with logical validity. The standard in logic is correctness, not how many people believe the conclusion.

🧰 What a logic is

A logic = a set of rules and techniques for distinguishing good reasoning from bad.

• There are many types of reasoning and many approaches to evaluating them, so we speak of "logic" (the enterprise) and "logics" (particular systems).
• A logic must:
  • Formulate precise standards for evaluating reasoning.
  • Develop methods for applying those standards to specific instances.

🧱 Basic building blocks

🧱 Propositions

Propositions = the things we claim, state, assert; the kinds of things that can be true or false.

• Propositions are expressed by declarative sentences.
• Examples: "the Earth revolves around the Sun," "reality is an unchanging Absolute," "it is wrong to eat meat."
• Not propositions: interrogative sentences (questions like "Is it raining?") and imperative sentences (commands like "Don't drink kerosene") do not express propositions because they cannot be true or false.

Don't confuse propositions with sentences:

• A single proposition can be expressed by different sentences: "It's raining" (English) and "es regnet" (German) express the same proposition.
• "John loves Mary" and "Mary is loved by John" express the same proposition in different ways.

📦 Arguments

Argument = a set of propositions, one of which (the conclusion) is (supposed to be) supported by the others (the premises).

• The conclusion is the claim being backed up.
• The premises are the reasons given to support it.
• The parenthetical "supposed to be" allows for bad arguments: premises may fail to support the conclusion.
• Roughly: a good argument's premises actually support the conclusion; a bad argument's premises fail to do so.

Example passage:

"You shouldn't eat at McDonald's. Why? First of all, because they pay their workers very low wages. Second, the animals that provide their meat are raised in deplorable conditions. Finally, the food is extremely unhealthy."

• Conclusion: You shouldn't eat at McDonald's.
• Premises: low wages, deplorable animal conditions, unhealthy food.

🔍 Premise and conclusion markers

• Premise markers: "because," "since," "for"—signal that a premise is being stated.
• Conclusion markers: "therefore," "consequently," "thus," "it follows that," "which implies that"—signal that the conclusion is about to follow.
• The symbol "∴" is shorthand for "therefore."

🔬 Analyzing arguments

🔬 Explication

• Explication = stating explicitly the premises and conclusion of an argument.
• Format: list declarative sentences (propositions), separated by a line, with the conclusion after "∴".

Example explication:

1. McDonald's pays their workers very low wages.
2. The animals that provide McDonald's meat are raised in deplorable conditions.
3. McDonald's food is very unhealthy. ∴ You shouldn't eat at McDonald's.

✏️ Paraphrasing

• Sometimes the passage must be rewritten so every sentence expresses a proposition.
• Interrogative sentences (questions) must be converted to declarative form.

Example: "And yet does it not also display an astonishing degree of order?" is a rhetorical question; paraphrase as "The universe displays an astonishing degree of order."

• If one sentence expresses multiple propositions, separate them into distinct premises.

🕳️ Enthymemes and tacit premises

Enthymeme = an argumentative passage that leaves certain propositions unstated.

• Sometimes a premise is implicit—not stated but necessary for the argument to work.
• How to identify tacit premises: look for a claim that bridges the gap between stated premises and the conclusion; if it's false, it undermines the argument.

Example:

"There cannot be an all-loving God, because so many innocent people all over the world are suffering."

• Stated premise: Many innocent people are suffering.
• Tacit premise: An all-loving God would not allow innocent people to suffer.
• Conclusion: There cannot be an all-loving God.

Why premises are left tacit:

• Often they are controversial and the arguer prefers not to defend them.
• Sometimes they are so widely accepted that stating them would be redundant.

Don't confuse: leaving out a premise for convenience vs leaving it out because it's controversial. Drawing out tacit premises forces a more robust discussion.

🗺️ Diagramming arguments

🗺️ Why diagram

• Diagrams depict relationships among premises and conclusion.
• Helpful for understanding how arguments work and for critical engagement.
• Use circled numbers for propositions and arrows for support relationships.

🔗 Independent premises

• Independent premises = each premise supports the conclusion on its own, without help from others.
• Mark: each premise still provides a reason even if the other were false.

Example:

"America's invasion of Iraq was an act of aggression, not self-defense. In addition, it was unreasonable to expect that the benefits would outweigh the horrors. Therefore, the Iraq War was not a just war."

• Propositions ① and ② each independently support ③.
• Diagram: two separate arrows from ① and ② pointing to ③.

Implication: Undermining one independent premise does not completely remove support for the conclusion, because the other premise still provides some support.

🔗 Intermediate premises

• Sometimes a premise supports the conclusion indirectly by supporting another premise first.

Example:

"Poets are mere 'imitators' whose works obscure the truth; hence, they have a corrupting influence on the souls of citizens. Poets should therefore be banned from the ideal city-state."

• ① supports ②, and ② supports ③.
• Diagram: ① → ② → ③.

Implication: Anything contrary to ① leaves ② without support; anything contrary to ② cuts off support for ③.

🔗 Joint premises

• Joint premises = premises that must work together to support a claim; neither provides support on its own.
• Indicate with brackets in the diagram.

Example:

"If true artificial intelligence is possible, then one must be able to program a computer to be conscious. But it's impossible to program consciousness. Therefore, true artificial intelligence is impossible."

• ① and ② work together to support ③.
• Diagram: ① and ② bracketed together, with a single arrow to ③.

Implication: Undermining either joint premise removes all support for the conclusion.

🧩 Complex example

Argument about numbers:

"Numbers are either abstract or concrete objects. They cannot be concrete objects because they don't have a location in space and they don't interact causally with other objects. Therefore, numbers are abstract objects."

• Conclusion: ⑤ Numbers are abstract objects.
• ① and ② work together (joint premises) to support ⑤.
• ③ and ④ independently support ②.
• Diagram: ③ and ④ with separate arrows to ②; ① and ② bracketed together with arrow to ⑤.

🧠 Logic and philosophy

🧠 The central philosophical question

• At the heart of logic: What makes a good argument?
• Equivalent questions: What is it for claims to provide support for another claim? When are we justified in drawing inferences?
• Logicians have developed many logical systems, covering different argument types and applying different principles.

🔧 Logic as a tool

• Logic has wide application beyond philosophy: mathematics (proving theorems), computer science (programming), linguistics (modeling language structure).
• Because of its formal/mathematical sophistication, logic occupies a unique place in philosophy curricula.
• A logic class is often unlike other philosophy classes: less time on "big questions," more on learning logical formalisms.

🔧 Three ways logic is useful to philosophers

Use	Description
Essential tool	Philosophy progresses by exchanging arguments; philosophers must know what makes good arguments.
Altered the conversation	Formal systems developed by logicians have opened new avenues of inquiry and sparked a revolution in 20th-century philosophy; no topic (metaphysics, ethics, epistemology) was untouched.
Source of questions	Logic itself generates philosophical questions (philosophy of logic): What does it mean for logic to be "formal"? Should logic be bivalent (every proposition true or false)? Can we accept multiple incompatible logics (logical pluralism)?

🌀 Philosophy of logic

• Bivalence: traditionally, every proposition is either true or false.
• Vagueness problem: natural language contains vague terms (e.g., "bald") with unclear boundaries; some cases seem neither true nor false.
• Non-bivalent logics: some logicians add a third truth-value ("neither," "undetermined") or infinite degrees of truth ("fuzzy logic").
• Open question: Are these non-traditional logics wrong? Are traditionalists wrong? Can we be pluralists and accept different logics depending on usefulness?

Don't confuse: logic as a tool for evaluating arguments vs philosophy of logic, which asks deeper questions about the nature and foundations of logical systems themselves.

🧭 Overview

🧠 One-sentence thesis

Good arguments require both correct logical structure (whether the premises support the conclusion) and true premises, and different argument types—deductive, inductive, and abductive—offer different strengths of connection between premises and conclusions.

📌 Key points (3–5)

• Two features of good arguments: structure (whether evidence is the right sort to support the conclusion) and truth (whether the evidence is actually true).
• Three types of logical connection: deductive (conclusion follows necessarily), inductive (conclusion is made probable), and abductive (conclusion offers the best explanation).
• Common confusion—certainty vs. conditional certainty: deductive arguments guarantee the conclusion only if the premises are true, not that we know the conclusion with absolute certainty.
• How to distinguish argument types: deductive aims for necessity, inductive extends from observed to unobserved instances, abductive infers the most likely explanation.
• Why it matters: these tools help distinguish well-founded philosophical positions from poorly supported ones, even when final answers remain elusive.

🔗 Inference and implication

🔗 What connects premises to conclusions

Inference: the act of reasoning that connects premises to a conclusion.

• An argument is a connected series of propositions: some are premises (reasons/evidence), at least one is a conclusion.
• A good argument supports a rational inference; a bad argument does not.
• The connection can be described two ways:
  • From the reader's perspective: if you accept the premises, you ought to accept the conclusion.
  • From the logical perspective: the premises logically imply the conclusion.

🧪 Strong vs. weak connections—two examples

Example of a strong connection:

1. All human beings are mortal.
2. Socrates is a human being.
3. Socrates is mortal.

• A reader who accepts both premises but denies the conclusion would hold an irrational belief.
• It doesn't seem rational to believe the premises but deny the conclusion.

Example of a weak connection:

1. I saw a black cat today.
2. My knee is aching.
3. It is going to rain.

• Even if the conclusion turns out true, these premises do not provide the right sort of evidence.
• The reader is not justified in accepting the conclusion based on these premises.

🎯 What makes a connection good

• Good arguments present a strong connection between the truth of the premises and the truth of the conclusion.
• Sometimes the connection guarantees the conclusion (deductive), sometimes it only makes it probable (inductive and abductive).

🔐 Deductive arguments

🔐 What deductive arguments aim for

Deductive argument: an argument that attempts to demonstrate the conclusion follows necessarily from the premises.

• If the premises of a good deductive argument are true, the conclusion is true as a matter of logic.
• This means: on the condition that the premises are true, we can be 100% certain the conclusion is true.
• Don't confuse: this is not absolute certainty about the conclusion; it is certainty conditional on the premises being true.

✅ Validity and soundness

Valid argument: an argument whose premises guarantee the truth of the conclusion—if the premises are true, it is impossible for the conclusion to be false.

Sound argument: a valid deductive argument whose premises are all true.

Examples of valid arguments with true premises (sound):

1. If it rained outside, then the streets will be wet.
2. It rained outside.
3. The streets are wet.

---

1. Either the world ended on December 12, 2012 or it continues today.
2. The world did not end on December 12, 2012.
3. The world continues today.

• These arguments present a close connection: it seems impossible to deny the conclusion while accepting the premises.

❌ Valid but unsound—false premises

Examples of valid arguments with at least one false premise:

1. If Russia wins the 2018 FIFA World Cup, then Russia is the reigning FIFA world champion [in 2019].
2. Russia won the 2018 FIFA World Cup.
3. Russia is the reigning FIFA world champion [in 2019].

---

1. Either snow is cold or snow is dry.
2. Snow is not cold.
3. Snow is dry.

• These have the same structure as the sound arguments above, so they are valid.
• However, at least one premise is false, so the conclusion is false.
• The structure is good, but the evidence is bad.

🚫 Invalid arguments—wrong structure

Invalid argument: an argument where it is possible for the premises to be true and the conclusion false.

Example—undistributed middle term:

1. Grass is green.
2. Money is green.
3. Grass is money.

• Both premises are true, but the conclusion is false.
• The two types in the conclusion are each members of a third type, but not members of each other.

Example—affirming the consequent:

1. If it rained outside, then the streets will be wet.
2. The streets are wet.
3. It rained outside.

• Counterexample: What if a water main broke? Then the streets would be wet, but it may not have rained.
• This scenario shows the premises can be true while the conclusion is false, so the argument is invalid.

🔍 The counterexample method

Counterexample: a scenario in which the premises are true while the conclusion is clearly false.

• A counterexample demonstrates that it is possible for the premises to be true and the conclusion false, proving invalidity.
• If no clear scenario exists, construct another argument with the same structure but easier-to-evaluate propositions.

Example: Original argument:

1. Most people who live near the coast know how to swim.
2. Mary lives near the coast.
3. Mary knows how to swim.

Counterexample with same structure:

1. Most months in the calendar year have at least 30 days.
2. February is a month in the calendar year.
3. February has at least 30 days.

• The counterexample has true premises and a false conclusion, so the original argument is invalid.

📋 How to evaluate a deductive argument

Ask these questions:

• Are the premises true? If not, even a valid argument does not guarantee a true conclusion.
• Is the form valid? Does it match a known invalid form (e.g., undistributed middle, affirming the consequent)?
• Can you find a counterexample? If you can imagine premises true and conclusion false, the argument is invalid.

📊 Inductive arguments

📊 What inductive arguments aim for

• Inductive arguments aim to ensure the conclusion is more probable, not guaranteed.
• Even the best inductive arguments may have false conclusions.
• We do not call them valid/invalid; instead:
  • Good inductive inferences are strong; bad ones are weak.
  • Strong inductive arguments with true premises are cogent.

Quality of Inference	Deductive	Inductive	Abductive
Bad inference	Invalid	Weak	Weak
Good inference	Valid	Strong	Strong
Good inference + true premises	Sound	Cogent	Cogent

🔄 How inductive inferences work

• Inductive inferences typically appeal to past experience to infer further claims directly related to that experience.
• Classic formulation: move from observed instances to unobserved instances, reasoning that the unobserved will resemble the observed.
• Examples: generalizations, statistical inferences, forecasts about the future.

Example—the Sun rising:

1. The Sun rose today.
2. The Sun rose yesterday.
3. The Sun has risen every day of human history.
4. The Sun will rise tomorrow.

• The inference is very strong, but not 100% certain.
• The Sun will eventually die; an asteroid could disrupt Earth's rotation; the future may not resemble the past.

🐔 The problem of representativeness

Example—the chicken's inference:

1. When the farmer came to the coop yesterday, he brought us food.
2. When the farmer came to the coop the day before, he brought us food.
3. Every day that I can remember, the farmer has come to bring us food.
4. When the farmer comes today, he will bring food.

• From the chicken's perspective, this looks strong.
• But on the day the farmer comes with a hatchet, the inference is fatally flawed.
• The sample of experiences is not representative of the whole population (all farmer behaviors).

Example—flawed polling:

1. A recent poll of over 5,000 people in the USA found that 85% are members of the National Rifle Association.
2. The poll found that 98% of respondents were strongly opposed to any firearms regulation.
3. Support of gun rights is very strong in the USA.

• If the poll was taken outside a gun show, the sample is not representative of all Americans.
• The data may be correct, but the sample is flawed.

🎯 What makes an inductive inference strong

• The sample set of experiences must be representative of the whole population described in the conclusion.
• The larger the sample, the more likely it is representative, and the stronger the inference.

📋 How to evaluate an inductive argument

• Are the premises true? Inductive arguments require true premises to infer the conclusion is likely true.
• Is the sample large enough? Larger samples are more likely to be representative, making the inference stronger.

🔎 Abductive arguments

🔎 What abductive arguments aim for

Abductive argument (inference to the best explanation): an argument that produces a conclusion attempting to explain the phenomena in the premises.

• Commonsense phrases: "reading between the lines," "using context clues," "putting two and two together."
• In science: a critical part of hypothesis formation—how scientists generate likely hypotheses for testing.
• Example: Sherlock Holmes should have said "Abduction, my dear Watson," not "Deduction."

Example—the left-handed murderer:

1. The victim's body has multiple stab wounds on its right side.
2. There was evidence of a struggle between the murderer and the victim.
3. The murderer was left-handed.

• The conclusion is not guaranteed (not deductive).
• It is not simply an extension from past experiences (not inductive).
• It provides the most likely explanation: in a face-to-face struggle, a left-handed attacker would naturally stab the victim's right side.

🏠 Everyday abduction

Example—the unlocked door:

• You come home and notice the door is unlocked and food is out on the counter.
• You infer your roommate is home.
• This is not guaranteed: you may have forgotten to lock the door and put away food.
• Abduction reasons to the most likely conclusion, not a guaranteed one.

⚖️ What makes an abductive inference strong

Good explanations should:

• Account for all available evidence. If the conclusion leaves some evidence unexplained, it is probably not strong.
• Respect the principle of extraordinary claims. Extraordinary claims (novel, supernatural, or requiring deep belief revision) require extraordinary evidence.
• Follow Ockham's Razor. Given two explanations, the simpler one is more likely to be true. Be skeptical of explanations requiring complex mechanics, extensive caveats, or extremely precise circumstances.

👽 Comparing explanations—aliens vs. confusion

Identical premises:

1. There have been hundreds of stories about strange objects in the night sky.
2. There is some video evidence of these strange objects.
3. Some people have recalled encounters with extraterrestrial life forms.
4. There are no peer-reviewed scientific accounts of extraterrestrial life forms visiting earth.

Explanation A: 5. There must be a vast conspiracy denying the existence of aliens.

Explanation B: 5. The stories, videos, and recollections are probably the result of confusion, confabulation, exaggeration, or outright falsifications.

• Which is more likely?
• Explanation B is simpler and does not require belief in a vast conspiracy (extraordinary claim).

📋 How to evaluate an abductive argument

• Is all relevant evidence provided? Missing information may prevent knowing the right explanation.
• Does the conclusion explain all the evidence? If not, it may not be the best explanation.
• Extraordinary claims require extraordinary evidence. If the conclusion is novel or surprising, the evidence should be equally compelling.
• Use Ockham's Razor. The simpler of two explanations is likely correct.

🧭 Overview

🧠 One-sentence thesis

Formal logic studies the validity of arguments through their logical form, which allows us to determine whether an argument is valid or invalid without examining the specific content of its premises.

📌 Key points (3–5)

• What logic studies: Logic investigates the relation of consequence between premises and conclusions by analyzing logical form rather than subject matter.
• How validity is determined: An argument is valid if it is impossible for the premises to be true and the conclusion false; truth-tables provide a systematic method to check this.
• The role of logical form: All arguments sharing the same logical form share the same validity status—if one is valid, all are valid; if one is invalid, all are invalid.
• Common confusion: Logical form vs. specific content—validity depends on the structure (connectives like "not" and "if…then") not on what the sentences are about (sea breams, roses, tigers).
• Philosophical puzzles: Questions remain about what logical forms actually are (linguistic schemata or worldly properties?) and whether arguments have one unique logical form or many.

🔍 What logic investigates

🔍 The subject matter of logic

• Different sciences study different things: physics studies matter, biology studies living organisms, mathematics studies numbers and sets.
• Logic is distinct: it studies the relation of consequence that holds between premises and conclusions in valid arguments.
• Logicians ask: "Is the conclusion a consequence of the premises?"
• The key insight: logic investigates forms of argument rather than the specific subject matter of arguments.

Valid argument: An argument where it is not possible for the premises to be true and the conclusion false.

🧪 Example: A valid argument

Consider:

1. If Alex is a sea bream, then Alex is not a rose.
2. Alex is a rose.
3. Alex is not a sea bream.

• It is impossible for (1) and (2) to be true while (3) is false.
• Therefore, the argument is valid.

🔤 The language of propositional logic

🔤 Alphabet and symbols

• Propositional letters: A, B, C, etc. stand for whole sentences.
  • Example: B = "Alex is a rose"; S = "I would love to smell it."
• Logical connectives: symbols that express logical relationships.
  • Negation ("not"): written as ¬
  • Conditional ("if…then"): written as →
  • Conjunction ("and"): written as ∧
  • Disjunction ("or"): written as ∨
• The excerpt focuses on negation and conditional.

🧩 Components of a conditional

Conditional: A sentence of the form "if…then…"

• Antecedent: the first component (the "if" part).
• Consequent: the second component (the "then" part).

Example: "If Alex is a rose, then I would love to smell it" = B → S

• Antecedent: B
• Consequent: S

📊 Truth-tables and meaning

📊 What truth-tables do

• Truth-tables specify the meaning of logical connectives by showing when propositions are true or false.
• Each row represents a possible situation or "way the world could be."
• The truth or falsity of complex sentences depends solely on the truth or falsity of their component letters.

📊 Truth-table for negation

A	¬A
T	F
F	T

• If A is true, ¬A is false.
• If A is false, ¬A is true.
• Example: If "Alex is a rose" is true, then "Alex is not a rose" is false.

📊 Truth-table for the conditional (material conditional)

A	B	A → B
T	T	T
T	F	F
F	T	T
F	F	T

• A conditional is false only when the antecedent is true and the consequent is false (second row).
• If the antecedent is false, the conditional is true regardless of the consequent (rows 3 and 4).
• Why? One suggestion: if your assumption is false, you can legitimately conclude anything. Example: If Amsterdam is the capital of England (false), then Paris is the capital of France (true) or Paris is the capital of Brazil (false)—both conditionals are true.

📊 Using truth-tables to check validity

• Design a truth-table for the premises and conclusion.
• Check whether there is any row where all premises are true and the conclusion is false.
• If no such row exists, the argument is valid.
• If such a row exists, the argument is invalid.

Example: For argument (1)-(3) above, the truth-table shows no row where both premises are true and the conclusion false → valid.

Example: For the argument "If Alex is a tiger, then Alex is an animal; Alex is not a tiger; therefore Alex is not an animal," there is a row (Alex is a dog) where premises are true but conclusion false → invalid.

🧩 Logical form and validity

🧩 What is logical form?

• Two arguments can have the same form even if they talk about completely different things.
• Example:
  • Argument 1: If Alex is a sea bream, then Alex is not a rose; Alex is a rose; therefore Alex is not a sea bream.
  • Argument 2: If Alice is reading Hegel, she is not frustrated; Alice is frustrated; therefore Alice is not reading Hegel.
• Both have the form: φ → ¬ψ; ψ; therefore ¬φ.
• The only difference is the substitution of letters (A/B vs. P/Q); the logical connectives (→, ¬) remain the same.

Logical form: What arguments have in common when we abstract away from their specific content and focus on the structure of logical connectives.

🧩 Why logical form matters for validity

• Key principle: Every argument that shares its logical form with a valid argument is also valid.
• Corollary: Every argument that shares its logical form with an invalid argument is also invalid.
• This liberates logicians from checking every argument individually—once you know one argument of a given form is valid, all arguments of that form are valid.

🧩 Formal fallacies

• Understanding validity in terms of logical form allows us to identify formal fallacies: mistakes in reasoning due to the logical form of the argument.
• Example: Denying the antecedent—the form φ → ψ; ¬φ; therefore ¬ψ is invalid.
  • "If Alice is reading Russell, then Alice is thinking of logic; Alice is not reading Russell; therefore Alice is not thinking of logic."
  • This shares the same invalid form as the tiger/animal example above.

Don't confuse: Validity depends on form, not on whether premises or conclusion happen to be true in a particular case. An invalid argument can have true premises and a true conclusion by accident.

🔬 Extracting logical forms

🔬 How to extract logical form

1. Represent each sentence in logical symbols (e.g., A, B, →, ¬).
2. Abstract away from the specific content of premises and conclusions—treat them as place-holders.
3. Do not abstract away from the logical connectives—their meaning is essential to the form.
4. Use variables (lowercase Greek letters φ, ψ, χ) to express the general form.

Example:

• Argument: φ → ¬ψ; ψ; therefore ¬φ
• Whatever propositions φ and ψ stand for, this form remains valid.

🔬 Analogy with mathematics

• In arithmetic, we use specific numbers: 2 + 3 = 5.
• To generalize, we use variables: x + y = y + x (true for any natural numbers x and y).
• Similarly, logical variables let us talk generally about premises and conclusions, independent of their specific subject matter.

🔬 Logic studies form, not content

• Form can be studied independently of subject matter.
• Arguments are valid or invalid mainly in virtue of their form, not their content.
• Logicians can evaluate validity even without understanding the specific claims within an argument or whether those claims are true.
• Whether claims are true is not a matter for logic; logic explores logical forms and establishes (in)validity.

🤔 Philosophical questions about logical forms

🤔 What are logical forms?

Two main views:

1. Logical forms as schemata (linguistic entities): Logical forms are strings of symbols, like φ → ¬ψ; ψ; ¬φ.
2. Logical forms as properties (extra-linguistic entities): Logical forms are worldly entities, akin to universals; schemata merely express or represent them.

Analogy: "is happy" is a predicate (linguistic), but it expresses the property of being happy (extra-linguistic).

🤔 Problem with the schemata view

• If logical forms are linguistic, then different variable choices yield different logical forms:
  • φ → ¬ψ; ψ; ¬φ
  • α → ¬β; β; ¬α
  • p → ¬q; q; ¬p
• Which one is the logical form? There is no non-arbitrary way to choose.
• This is the fallacy of "treating the medium as the message."

🤔 The property view as a solution

• If logical forms are language-independent entities, then all the above schemata express the same logical form.
• None of them is the logical form; each represents it.
• This avoids the arbitrariness problem.

🤔 Do arguments have one logical form or many?

• Objects can have many forms: a vase can be both a cuboid and a cube; a sonnet can be both Petrarchan and Miltonic.
• A single sentence can have multiple forms: ¬(A → ¬B) is:
  • a negation,
  • a negation of a conditional,
  • a negation of a conditional whose consequent is a negation.
• If each of these is a logical form of the same argument, what unifies them?
• One answer: they share a common logical form. But that common form itself has further forms → infinite regress.
• This challenges the thesis that logical forms are unique entities.

Question for reflection: If arguments do not have a single unique logical form, what are the implications for understanding validity in terms of logical form?

📚 Summary and exercises

📚 Summary

• Formal logic studies logical consequence through the form of arguments, not their subject matter.
• Validity is explicated using truth-tables, which specify when propositions are true or false.
• Every argument sharing its logical form with a valid argument is valid; every argument sharing its form with an invalid argument is invalid.
• This understanding allows us to identify formal fallacies (e.g., denying the antecedent, affirming the consequent).
• Philosophical questions remain: Are logical forms linguistic schemata or worldly properties? Do arguments have one logical form or many?

📚 Exercises mentioned

• Exercise One: Use a truth-table to show that "affirming the consequent" (φ → ψ; ψ; therefore φ) is invalid.
• Exercise Two: Use a truth-table to show that "hypothetical syllogism" (φ → ψ; ψ → χ; therefore φ → χ) is valid. (Hint: eight rows for three variables.)
• Exercise Three: Given truth-tables for conjunction (∧) and disjunction (∨), evaluate the validity of several natural-language arguments by identifying their logical form and constructing truth-tables.

🧭 Overview

🧠 One-sentence thesis

Studying informal fallacies—mistakes in reasoning that stem from content and context rather than logical form alone—is essential for avoiding flawed arguments and for distinguishing good reasoning from bad reasoning in real-world discourse.

📌 Key points (3–5)

• What informal fallacies are: errors that lie not in logical form but in the argument's content, including irrelevant information, ambiguous language, or unjustified assumptions.
• Why studying fallacies matters: without identifying flaws in reasoning, we would accept or reject conclusions without good reasons and would have to rely purely on trust in others.
• Three criteria for good reasoning: (1) logically well-framed (premises support the conclusion), (2) acceptable premises (warranted for the audience), (3) relevant information (enough to make the conclusion acceptable).
• Common confusion: an argumentative technique that is appropriate in one context may be fallacious in another—context, purpose, and audience determine appropriateness.
• Three main categories: relevance fallacies (irrelevant information), ambiguity fallacies (unclear or equivocal terms), and fallacies of presumption (hidden false or unjustified assumptions).

🎯 Why fallacies matter

🎯 The primary purpose of studying fallacies

• The excerpt emphasizes that the primary purpose is to avoid falling foul of them yourself, not just to spot them in others.
• By understanding why and when a certain way of reasoning does not support the truth of the conclusion, you learn to avoid making the same mistakes.
• Identifying fallacies requires more than formal logic; it also involves discourse analysis: asking who speaks, to whom, from which perspective, and with what purpose.

🧩 Fallacies are very common

• Committing flaws in reasoning is common; sometimes fallacies pass unnoticed, sometimes they are intentional.
• An arguer may be uninterested in being reasonable or may wish to induce someone else to make a rational error.
• Without the ability to identify flaws, we would base our beliefs purely on the trust of others—a common practice, but not reliable.

🔍 Context and appropriateness

• What is argumentatively appropriate in one context may not be in another.
• Appropriateness depends on the purpose of the argument and the intended audience.
• This does not mean we cannot develop general standards for good and bad reasoning; publicly accessible standards are paramount.

🧱 Three criteria for good reasoning

✅ Logically well-framed

A good argument is logically well-framed: the premises offer reasons for the conclusion.

• This is the minimum requirement.
• However, different individuals can have different ideas about what counts as a good reason—good reasons for one person can be inadequate for another.
• While necessary, this requirement is not sufficient on its own.

✅ Acceptable premises

A good argument starts from acceptable premises, or premises that are warranted, not only for the reasoner but mainly for the audience.

• Even premises that are not true or plausible at all may be acceptable, depending on the audience or the function of the argument in a given context.
• Considerations of form and content necessarily have to be taken together.

✅ Relevant information

The premises must contain relevant information for the conclusion—if not all that is relevant, at least enough to make the conclusion acceptable.

• Concealing relevant information is a well-known form of deceiving people.
• Taking certain information for granted when it has been widely contested is also a mistake.

🔎 Three questions to identify fallacies

To avoid being fallacious, an argument must answer all of these questions positively:

1. Do the premises support the conclusion, or only offer very weak support?
2. Are the premises well-supported?
3. Do the argument's premises include all the important relevant information?

📂 Taxonomy of fallacies

📂 Formal vs. informal fallacies

• The most general division is between formal and informal fallacies.
• Formal fallacies are mistakes in the form of deductive arguments (covered in Chapter 3).
• Informal fallacies are errors that lie not in logical form but in the argument's content.
• To appreciate what is wrong with informal fallacies, we must examine whether the reasoning meets the criteria of relevant information and acceptable premises.

🗂️ Three categories of informal fallacies

Category	What goes wrong
Relevance fallacies	Do not present relevant information, or present irrelevant information for the conclusion.
Ambiguity fallacies	Employ unclear or equivocal terms or propositions, so it becomes impossible to grasp a precise sense of what is being argued for.
Fallacies of presumption	The conclusion rests upon certain assumptions not explicitly stated in the premises; such assumptions are false, uncertain, implausible, or unjustified.

• Explicating the lurking assumption in a fallacy of presumption usually suffices to demonstrate the argument's insufficiency.

🚨 Common informal fallacies (part 1: attacking the person and misrepresenting the argument)

🗣️ Argument directed to the person (Argumentum ad hominem)

This fallacy consists in attacking the person instead of treating the argument that the person is proposing.

• The character or personal circumstances of the speaker are raised to invalidate their arguments, rather than identifying any fault with the argument itself.
• This is a very common fallacy with various forms.

Two main forms:

• Offensive ad hominem: Calls into question the moral character of the speaker, attempting to dismiss their trustworthiness rather than showing actual mistakes in their arguments. The offensive ad hominem dismisses an opinion on the grounds that those who sustain it are to be dismissed, whatever the independent qualities of the opinion.
• Circumstantial ad hominem: Attempts to undermine someone's argument on the basis of their background or current circumstances. For example, arguing that we ought not listen to another's argument because they will benefit from the conclusion's truth. The personal circumstances of one who makes or rejects a claim are irrelevant to the truth of what is claimed.

Don't confuse with: Situations where personal circumstances are legitimately relevant (e.g., in courts of law, where bias or conflict of interest may affect credibility).

🥋 The Straw Man fallacy

This fallacy consists in reducing an argument to some weaker version of it simply in order to strike it down.

• According to the principle of charity in argumentation analysis, the strongest interpretation of an argument should always be preferred.
• The straw man fallacy is the direct refusal to adhere to this principle.
• The original strength of the argument is missed and, reduced to a caricature, can be easily refuted.
• The fallacy's name comes from the fact that a straw man is easier to beat down than a real man.

Example: Some vegan activists claim their opponents commit this fallacy by stating that if vegans have so much respect for animal life, they should accord the same respect to plant life as well. Vegans may justifiably claim this as a misrepresentation of their own position, and thus it does not diminish its legitimacy.

Don't confuse with: Ad hominem—the straw man fallacy does not attempt to undermine the argument by directly attacking the person.

🚨 Common informal fallacies (part 2: appeals to force, emotion, and popularity)

💪 Appeal to power or threat of force (Argumentum ad baculum)

An argument with a cudgel (baculum) is an appeal to brute force, or a threat of using force instead of reasoning in order to ensure one's conclusion is accepted.

• The ad baculum is a sort of intimidation, either literally by physical power or any other kind of threat, so someone feels constrained to accept the conclusion independently of its truth.
• When someone threatens to use force or power, or any other kind of intimidation instead of reasoning and arguing, one abandons logic.
• This can be taken as the utmost fallacy, the most radical way of trying to impose a conclusion without reasoning in favor of it.

Example: Think of when someone raises their voice as a form of intimidation to force the acceptance of a conclusion, without giving reasons. A historical example comes from the El Salvador guerrillas' slogan in the 1980s to prevent people from voting: "vote in the morning; die in the afternoon." The threat need not be overtly stated—Don Corleone's line in The Godfather (1972), "I'm gonna make him an offer he cannot refuse," is an ad baculum.

😢 Appeal to pity (Argumentum ad misericordiam)

This happens when someone appeals to the audience's sentiments to compel support for a conclusion without giving reasons for its truth.

• A clear example: students who appeal to the teacher's sentiments to obtain a grade review by reciting an unending roll of personal problems (dogs are sacrificed, marital engagements are broken, grandmothers are hospitalized).
• In courts, this fallacy is common, as when the humanitarian sentiments of the jury are appealed to without discussing the facts of the case. A famous case: a youth who murdered his mother and father had his attorney plead for a lighter penalty claiming the youth had become an orphan.

When it's not fallacious: Sometimes the evocation of sentiments is not fallacious. It can be perfectly reasonable to combine reasons for a conclusion with an appeal to outrage or anger towards a certain action. This fallacy occurs when appealing to emotions absolutely replaces giving reasons—aiming at persuasion through eliciting emotions solely, without attempting to rationally support the conclusion.

👥 Appeal to popular opinion (Argumentum ad populum)

This fallacy consists in the mistake of assuming an idea is true just because it's popular.

• The Latin means more precisely "appeal to the populace."
• Such arguments are fallacious because collective enthusiasm or popular sentiment are not good reasons to support a conclusion.
• This is a very common fallacy in demagogic discourses, propaganda, movies, and TV shows.

Example: Marketing campaigns that say "products of brand X are better because they are good sellers." Or when someone says: "everyone agrees with this, why don't you?" But the "this" can be false even if everyone thinks it is true.

Historical illustration: The excerpt includes a photo from Hamburg (Germany) in 1936 during Nazi rule, showing one person refusing to perform the Nazi salute while everyone else does. Relying solely on the popularity of a person, movement, or idea can have significant repercussions for society.

🚨 Common informal fallacies (part 3: circular reasoning, ignorance, and authority)

🔄 Begging the question (Petitio principii)

This fallacy arises when the argument's premises assume the truth of the very conclusion they are supposed to be providing evidence for.

• In order to accept the premises, one has first to accept the conclusion.
• The conclusion acts as a support for itself—hence the Latin name "petition of the principles."
• Such arguments are fallacious because they are useless in establishing the truth of the conclusion, even if ultimately the argument's premises are true and the argument is definitely valid.

Why it's fallacious: We desire independent evidence for our conclusions. If we already knew the conclusion was true, we wouldn't require an argument to prove it. Arguments that beg the question provide no such independent evidence; they pretend to be providing independent evidence when in reality they are simply restating the conclusion, or assuming its truth, within the premises.

Example: When someone argues men are better than women in logical reasoning because men are more rational than women, this is to beg the question. If being logical just means being rational, then what has been said is just that men are more logical because they are more logical. The argument simply assumes the very point it is attempting to demonstrate.

Don't confuse with: Not all circularity in reasoning is a fallacy—there are contexts where circularity is not problematic (the excerpt asks you to think about when this might be the case).

🤷 Appeal to ignorance (Argumentum ad ignorantiam)

This fallacy consists in assuming that the lack of evidence for a position is enough to demonstrate its falsity and, inversely, the lack of evidence for its falsity is enough to entail its truth.

• We cannot assert the truth of a proposition based on the lack of proof of its falsity, and vice versa.
• Lack of evidence is a flaw in our knowledge, and not a property of the claim itself.

Example: To say extraterrestrials exist because there is no proof of their non-existence would be to neglect the fact there may be no independent positive evidence for their existence either. The rational attitude to have when we have no evidence for either position is to suspend judgement on the matter.

Don't confuse with: There are contexts in which ad ignorantiam is not a fallacy (the excerpt asks you to imagine such contexts and explain why it is not a fallacy in those cases).

🎓 Appeal to authority (Argumentum ad verecundiam)

These are arguments based upon the appeal to some authority, rather than independent reasons.

• The speaker cites famous "authorities," dropping names instead of giving their own reasons, thus recognizing their own incapacity to establish the conclusion of the matter at hand.
• The Latin name "argumentum ad verecundiam" is more properly translated as argument based on modesty or coyness, referring to the speaker who invokes an authority to support their case.

When it's legitimate: An appeal to authority can be legitimate if the authority invoked really is an authority on the subject. Citing Hegel in discussing matters of philosophy, or Marie Curie in chemistry or physics, could be reasonable.

When it's fallacious: An appeal to authority becomes illegitimate when, instead of giving reasons and constructing an independent inference for the conclusion, someone seeks to base a conclusion on the say-so of a putative authority who is not a competent authority on the subject under discussion. Invoking Marie Curie's ideas when talking about football would in all likelihood be irrelevant. Even the highest authority's opinion on some subject is not enough by itself to establish a conclusion. No conclusion is true or false just because some specialist has said so. Rather, one's appeal to the word of the authority is merely a shorthand for, "they will be able to provide you with independent support for my conclusion." If they cannot, then the conclusion is not supported.

Example: The ideas of Charles Darwin—a renowned biologist—are not rarely invoked in discussions about matters of morals, politics, or religion, without biology being really relevant to the case. The excerpt also includes a 1946 Camel cigarettes advertisement that relies upon the health expertise of doctors to extol the virtues of a particular brand. The appeal is unjustified because (1) simply because an individual does something (such as smoke a cigarette brand) does not mean they recommend it for your health, even if they are knowledgeable about its effect, and (2) the advert relies on the presumption that the doctors themselves were informed on the health impacts of cigarettes—an appeal to authority is only justified if those authorities actually are much more informed on the relevant matter.

🚨 Common informal fallacies (part 4: hasty generalization and equivocation)

📊 Hasty generalization

This fallacy is committed whenever one holds a conclusion without sufficient data to support it.

• The information used as a basis for the conclusion may well be true, but nonetheless unrepresentative of the majority.
• Such generalizations are based on an insufficient set of cases, and cannot be justified with only a few confirming instances.

Examples: Widely known generalizations that are unjustified for this reason include "all Brazilians are football lovers," "atheists are immoral people," and "the ends justify the means."

Why it's common: Our beliefs about the world are commonly based on such generalizations. In fact, it is a hard task not to do so! But that does not mean we should accept such generalizations without examination, and before seeking enough evidence to support them.

🔀 Equivocation

Whenever a term or expression appears with different meanings in the premises and in the conclusion, the fallacy of equivocation occurs.

• The speaker relies upon the ambiguity of elements of language and shifts their meaning throughout the argument.
• This forces the audience to accept more than is entailed by the argument when any one fixed meaning is given to the relevant terms.

Classical example:

1. The end of a thing is its perfection.
2. Death is the end of life.
3. Death is the perfection of life.

Here, "end" can mean "goal" or "termination," so the conclusion could be that the goal of life is perfection, or that life is perfected only when it is terminated. Apart from metaphysical considerations, the argument is only apparently valid, since the change in meaning and context make at least one of the premises or conclusion false (or implausible).

How to spot it: The excerpt asks you to rephrase the argument to make the fallacy clear—by fixing one meaning of "end" throughout, you will see that at least one premise becomes false or the conclusion does not follow.

🧭 Overview

🧠 One-sentence thesis

Necessary and sufficient conditions are central to analytic philosophy's standard method of conceptual analysis, yet Wittgenstein's family resemblance critique challenges whether concepts can actually be captured by such definitions.

📌 Key points (3–5)

• What necessary and sufficient conditions are: logical relationships between propositions, formalized as material conditionals (if p, then q) and logical equivalences.
• Why they matter in philosophy: the standard philosophical method (SPM) assumes that philosophical analysis aims to define concepts by specifying sets of necessary and sufficient conditions.
• Valid vs invalid inference patterns: modus ponens and modus tollens are valid; affirming the consequent and denying the antecedent are invalid—these errors stem from confusing necessary with sufficient conditions.
• Common confusion: mistaking a necessary condition for a sufficient one (or vice versa)—e.g., thinking that because B is necessary for A, B's truth guarantees A's truth (it doesn't).
• Wittgenstein's challenge: concepts may not have the form of necessary and sufficient conditions at all, but instead are structured as family resemblance relations, which threatens to make SPM vacuous.

🔤 Definitions and logical structure

🔤 Necessary condition

q is a necessary condition for p means: if p, then q (symbolized as p → q).

• In plain language: q must be true whenever p is true.
• Example: Being an unmarried male is necessary for being a bachelor—if someone is a bachelor, they must be an unmarried male.
• Don't confuse: necessary does not mean sufficient. Just because q is required for p doesn't mean q alone guarantees p.

🔤 Sufficient condition

p is a sufficient condition for q means: if p, then q (symbolized as p → q).

• In plain language: p's truth is enough to guarantee q's truth.
• Example: Being a bachelor is sufficient for being an unmarried male—if someone is a bachelor, that ensures they are an unmarried male.
• Don't confuse: sufficient does not mean necessary. Other conditions might also guarantee q.

🔤 Jointly necessary and sufficient conditions

p is necessary and sufficient for q, and q is necessary and sufficient for p, means: p if and only if q (symbolized as p ↔ q, logical equivalence).

• This is a biconditional: p and q always have the same truth value.
• Example: Being a bachelor and being an unmarried male are jointly necessary and sufficient—each guarantees the other.
• In philosophy, such definitions are the goal of conceptual analysis.

📊 Truth conditions

p	q	p → q (material conditional)	p ↔ q (logical equivalence)
T	T	T	T
T	F	F	F
F	T	T	F
F	F	T	T

• The material conditional is false only when the antecedent is true and the consequent is false.
• Logical equivalence is true only when both sides have the same truth value.

✅ Valid inference patterns

✅ Modus Ponens (Affirming the Antecedent)

1. If A, then B (A → B)
2. A
3. Therefore, B

• Reading in terms of conditions: A is sufficient for B; A obtains; so B obtains.
• Example: If being a bachelor is sufficient for being unmarried, and someone is a bachelor, then they are unmarried.

✅ Modus Tollens (Denying the Consequent)

1. If A, then B (A → B)
2. Not B
3. Therefore, not A

• Reading in terms of conditions: B is necessary for A; B does not obtain; so A does not obtain.
• Example: If being unmarried is necessary for being a bachelor, and someone is married, then they are not a bachelor.

❌ Invalid inference patterns

❌ Affirming the Consequent

1. If A, then B (A → B)
2. B
3. Therefore, A (invalid)

• Why it fails: A is a necessary condition for B, but that doesn't mean A is sufficient for B.
• The fact that B is necessary for A does not ensure it is also sufficient for A.
• Example: If being unmarried is necessary for being a bachelor, and someone is unmarried, that doesn't guarantee they are a bachelor (they might be female or a child).

❌ Denying the Antecedent

1. If A, then B (A → B)
2. Not A
3. Therefore, not B (invalid)

• Why it fails: A is a sufficient condition for B, but other conditions might also suffice for B.
• Example: If being a bachelor is sufficient for being unmarried, and someone is not a bachelor, that doesn't mean they are married (they could be an unmarried woman).

🔍 How to avoid confusion

• Necessary ≠ sufficient: just because q is required for p doesn't mean q alone produces p.
• Sufficient ≠ necessary: just because p guarantees q doesn't mean p is the only way to get q.
• Test by asking: "Does this condition guarantee the outcome?" (sufficient) vs. "Is this condition required for the outcome?" (necessary).

🧪 The Standard Philosophical Method (SPM)

🧪 What SPM claims

The excerpt describes the standard philosophical method as:

1. Conceptual analyses take the form of proposed definitions (i.e., sets of necessary and sufficient conditions) of concepts.
2. The adequacy of any definition can be tested against concrete and/or imagined cases.
3. A priori intuition (a distinct, reliable, fallible non-sensory mental faculty) determines whether a proposed definition fits a given case.
4. Intuition allows us to reliably access knowledge about concepts.
5. The method of reflective equilibrium is used to confirm or disconfirm definitions by bringing intuitively true cases into conformity with a rule or principle.

• SPM assumes that philosophy aims to yield accurate specifications of sets of necessary and sufficient conditions.
• Example: the claim that all bachelors are unmarried men is a conceptual analysis in the form of necessary and sufficient conditions.

🧪 McGinn's defense

Colin McGinn endorses SPM as the one and only method of philosophy:

"…the proper method for uncovering the essence of things is precisely conceptual analysis," and "philosophy, correctly conceived, simply is conceptual analysis."

• Philosophy seeks a priori knowledge of objective being through unaided contemplation.
• It operates "from the armchair" using only thought-experiments and intuitions about possibilities.
• McGinn claims this has been the standard conception throughout most of the history of philosophy.

🧪 Real vs nominal definitions

Carl Hempel's distinction (cited in the excerpt):

Type	What it is	Example
Real definition	A statement of the "essential characteristics" of an entity	Man is a rational animal
Nominal definition	A convention introducing an abbreviated notation; a stipulation	Dictionary-style definitions
Meaning analysis (a type of real definition)	Validation requires only knowing the meanings of constituent expressions; no empirical investigation needed	Knowledge is justified true belief

• SPM involves meaning analyses, not merely stipulative definitions.
• Conceptual analysis defines a pre-theoretical concept by offering a synonymous expression.
• The excerpt emphasizes that analysanda (the proposed definitions) are conceptual truths with the form of analytic definitions.

🧪 Example: the concept of knowledge

Where Kx = "x is knowledge", Jx = "x is justified", Tx = "x is true", Bx = "x is believed":

x is Kx if and only if x is Jx & x is Tx & x is Bx

• This analysis decomposes the concept of knowledge into a list of defining essential features: justified, true, and believed.
• The logical form is a set of jointly necessary and sufficient conditions.

🧩 Wittgenstein's challenge: the potential vacuity problem

🧩 The core criticism

Wittgenstein attacks the foundation of SPM by challenging the assumption that concepts have the form of sets of necessary and sufficient conditions.

"We are unable to clearly circumscribe the concepts we use; not because we don't know their real definition, but because there is no real 'definition' to them."

• The problem: philosophical attempts at conceptual analysis have systematically failed to produce the goods.
• The explanation: concepts are not captured by sets of necessary and sufficient conditions.
• The implication: if there are no (or very few) concepts that can be correctly regimented as sets of necessary and sufficient conditions, there can be no (or very few) correct conceptual analyses in the sense of SPM—making SPM potentially vacuous.

🧩 The principle under attack

SPM assumes:

(CON) For any concept C, there exists a set of necessary and sufficient conditions that constitutes the content of C.

• Wittgenstein denies CON.
• His argument: for any proposed set of necessary or sufficient conditions intended to analyze a concept, there are instances of that concept that do not meet the proposed defining conditions.

🧩 Wittgenstein's favorite example: the concept of a game

• Poker and soccer are both games.
• Proposed definition: something is a game if and only if that activity involves a winner and a loser.
• Counterexample: patty cake is a game but does not have a winner and a loser.
• Conclusion: this definition fails.
• Wittgenstein claims this example generalizes: most or all attempts to specify concepts in terms of necessary and sufficient conditions will face counterexamples.

🧩 Family resemblance as an alternative

Wittgenstein proposes that concepts are structured as family resemblance classes:

• Cases that fall under a concept are related by complex overlapping similarity conditions.
• No one set of conditions holds for all and only the members that exhibit that concept.
• Paradigm members (obvious central cases, e.g., a robin is a paradigmatic bird) and non-paradigm members (less central cases, e.g., a penguin) are related by resemblance to paradigm cases, not by shared defining features.
• Example: games are related by overlapping similarities (some involve competition, some involve physical activity, some involve rules, etc.), but no single feature is common to all games.

🛡️ McGinn's response to Wittgenstein

🛡️ First response: failure to articulate ≠ non-existence

• McGinn bites the bullet: concepts are properly characterized by sets of necessary and sufficient conditions, even though they are very often difficult to articulate.
• Our failure to articulate definitive examples of such analyses is no reason to suppose that there are no such things.
• From the fact that it is difficult to produce the goods in any given case, it does not necessarily follow that there are no such analyses.

🛡️ Second response: family resemblance presupposes necessary and sufficient conditions

• Wittgenstein's own theory of concepts in terms of family resemblances presupposes that concepts can be captured by a special type of necessary and sufficient conditions.
• For any concept C, the non-paradigmatic members of C bear a family resemblance relation to the paradigmatic case(s) of C.
• So, according to Wittgenstein, something is necessarily a concept if and only if it is a set of entities related by family resemblance relations to one or more paradigm cases.
• McGinn's conclusion: Wittgenstein does not reject SPM; rather, he favors a particular form of it—one in which the analysis takes the form "family-resembles paradigm games (such as chess, tennis, etc.)."

🛡️ A remaining problem

• McGinn's response does not defuse the problem that such specifications of conceptual contents cannot plausibly be necessary truths.
• Family resemblance relations cannot plausibly be understood to be necessary truths (i.e., true in all possible worlds).
• Resemblances are not purely objective relations between objects; they are perceiver-relative and vary depending on what features one focuses on.
• Example: a pen resembles a pencil when one focuses on the function of writing, but a pen and a pencil do not resemble one another when one focuses instead on the feature of containing ink.
• This threatens the claim that conceptual analyses are necessary truths, as defenders of SPM typically believe.

📚 Key takeaways for reasoning

📚 Practical implications

• Understanding necessary and sufficient conditions is essential for:
  • Evaluating philosophical arguments and definitions.
  • Recognizing valid and invalid inference patterns.
  • Avoiding common logical errors (affirming the consequent, denying the antecedent).
• The debate over SPM highlights a fundamental question: do concepts have the structure that traditional philosophy assumes?

📚 Open question

• The excerpt leaves open whether Wittgenstein's challenge is ultimately successful or whether McGinn's defense (or some other response) can save SPM.
• The potential vacuity problem remains a live issue in contemporary philosophy.