Intro to Philosophy Unit 5 Review: Logic and Reasoning

What's Logic and Reasoning?

• Logic involves using principles of valid inference and demonstration to analyze arguments
• Reasoning encompasses using logic, along with other mental processes, to draw conclusions from premises
• Aims to distinguish good arguments from bad arguments based on their structure and content
• Helps clarify thinking, construct sound arguments, and evaluate claims critically
• Formal logic uses symbols and precise rules to study the form of arguments
  • Includes propositional logic which deals with simple declarative propositions and how they can be combined
  • Also includes predicate logic which adds relations and quantifiers to propositional logic
• Informal logic focuses more on the content and context of arguments in natural language
• Both deductive reasoning (drawing necessary conclusions from premises) and inductive reasoning (inferring likely conclusions from evidence) are studied

Key Concepts and Terms

• Argument: a set of statements, one of which (the conclusion) is claimed to follow from the others (the premises)
• Validity: a deductive argument is valid if and only if it's impossible for the premises to be true and the conclusion false
  • Validity is determined by the argument's form, not the actual truth of the premises or conclusion
• Soundness: a deductive argument is sound if and only if it is valid and all its premises are actually true
• Cogency: an inductive argument is cogent if and only if the premises are true and they make the conclusion more likely than not
• Fallacy: an error in reasoning that renders an argument invalid, unsound, or weak
  • Formal fallacies ($affirming the consequent$) violate rules of an argument's logical form
  • Informal fallacies (straw man) are errors in the content or context of the argument
• Proposition: a declarative sentence that is either true or false, often symbolized by letters like $P$, $Q$
• Logical connectives: symbols that join propositions to form compound propositions
  • Includes negation ($\neg$), conjunction ($\wedge$), disjunction ($\vee$), conditional ($\rightarrow$), biconditional ($\leftrightarrow$)

Types of Arguments

• Deductive arguments: the conclusion necessarily follows from the premises if the argument is valid
  • Aim for the strongest standard of validity and soundness
  • Example: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
• Inductive arguments: the conclusion is made more likely by the premises but doesn't follow necessarily
  • Rely on empirical evidence and probability rather than pure logic
  • Example: Most birds can fly. Tweety is a bird. Therefore, Tweety can probably fly.
• Abductive arguments: reason to the best explanation for a set of observations
  • Used commonly in science to develop theories that explain evidence
  • Example: The lawn is wet. It probably rained last night, since that's the most likely explanation.
• Analogical arguments: argue that because two things are similar in some respects, they are probably similar in another respect
  • Strength depends on the relevance and degree of similarity
  • Example: Minds and computers can both process information. Computers don't have feelings. So minds probably don't have feelings either.
• Causal arguments: try to establish a cause-and-effect relationship between events
  • Must rule out alternative causes and establish a plausible mechanism
  • Example: Every time I water my plants, they grow. So watering plants causes them to grow.

Logical Fallacies

• Ad hominem: attacking the person making the argument rather than the argument itself
• Appeal to authority: claiming something is true because an authority figure says so, even if they lack relevant expertise
• Straw man: misrepresenting an opponent's argument to make it easier to attack
• False dilemma: presenting two options as the only possibilities when other alternatives exist
  • Example: Either we go to war or we appear weak. We can't appear weak, so we must go to war.
• Slippery slope: claiming a relatively small first step will lead to an inevitable chain of exaggerated consequences
• Hasty generalization: drawing a broad conclusion from a small or unrepresentative sample
  • Example: My friend had a bad reaction to that vaccine, so the vaccine is dangerous for everyone.
• Circular reasoning: restating the conclusion as a premise, assuming what one is trying to prove
  • Example: God exists because it says so in the Bible, and the Bible is true because it's the word of God.
• Red herring: introducing an irrelevant topic to divert attention from the original issue

Formal Logic Systems

• Aristotelian logic: the earliest formal study of logic, focused on syllogisms and categories
  • A syllogism is a deductive argument with two premises and a conclusion
  • Example: All mammals are animals. All dogs are mammals. Therefore, all dogs are animals.
• Propositional logic (or sentential logic): studies logical operators and rules of inference governing the relations between propositions
  • Uses symbols like $\wedge$ for "and", $\vee$ for "or", $\rightarrow$ for "if...then", $\leftrightarrow$ for "if and only if"
  • Example: If it's raining ($P$) then the streets are wet ($Q$), symbolized $P \rightarrow Q$
• Predicate logic (or first-order logic): extends propositional logic to include predicates, quantifiers, and relations
  • Adds symbols like $\forall$ for "for all" and $\exists$ for "there exists"
  • Can represent statements like "All dogs ($\forall x$) are mammals ($M$)" as $\forall x (Dog(x) \rightarrow M(x))$
• Modal logic: introduces operators for necessity, possibility, and impossibility
  • Uses symbols like $\square$ for necessity and $\diamond$ for possibility
  • Example: If it's necessarily true ($\square P$) that 2+2=4, then it's possible ($\diamond P$) that 2+2=4
• Many-valued logics: allow for truth values beyond just "true" and "false"
  • Fuzzy logic uses degrees of truth between 0 and 1 to handle vague concepts
  • Paraconsistent logics tolerate inconsistencies without everything being derivable

Applying Logic in Real Life

• Constructing and evaluating arguments in essays, debates, and discussions
  • Identifying premises and conclusions, assessing validity and soundness
  • Avoiding fallacies and irrelevant appeals to emotion or authority
• Analyzing media content like advertisements and political speeches for manipulative reasoning
  • Spotting techniques like false dichotomies, slippery slopes, and straw man arguments
  • Checking facts and seeking alternative perspectives to counter bias
• Making rational decisions by weighing evidence, considering alternatives, and forecasting consequences
  • Applying principles of inductive logic and probability to assess risks and benefits
  • Using deductive logic to draw out the implications of one's beliefs and values
• Designing and debugging computer programs based on logical rules and valid inferences
• Solving puzzles and brain teasers that require deductive reasoning and creative problem-solving
  • Example: Knights and Knaves puzzles where some characters always tell the truth and others always lie
• Promoting clearer thinking and communication by defining terms, making explicit arguments, and questioning assumptions
  • Using logical structures like modus ponens ($P \rightarrow Q, P \vdash Q$) to present reasoning step-by-step
  • Translating everyday language into formal logical notation to reveal hidden premises or invalid inferences

Famous Philosophers and Their Contributions

• Aristotle (384-322 BCE): developed the first formal system of logic, theory of the syllogism, and foundations of deductive reasoning
• Chrysippus (279-206 BCE): co-founder of propositional logic and innovator in the logic of the Stoics
• Al-Farabi (872-950): wrote extensive commentaries on Aristotelian logic and explored modal syllogisms in Islamic philosophy
• Avicenna (980-1037): Persian polymath who advanced modal logic and the hypothetical syllogism
• William of Ockham (1287-1347): Scholastic thinker known for Ockham's Razor, the principle of favoring simplicity in explanations
• Gottfried Leibniz (1646-1716): envisioned a universal "calculus of reasoning" and laid groundwork for symbolic logic
• George Boole (1815-1864): developed Boolean algebra with its system of logical operators, a foundation of computer logic
• Gottlob Frege (1848-1925): invented first-order predicate logic and set a new standard of rigor for deductive proofs
• Bertrand Russell (1872-1970): advanced mathematical logic, theory of types, and logical atomism
• Kurt Gödel (1906-1978): proved the incompleteness theorems showing the limits of formal systems
• Alfred Tarski (1901-1983): made major contributions to model theory, the concepts of truth and logical consequence

Tricky Bits and Common Mistakes

• Confusing validity and soundness: an argument can be logically valid but still have a false conclusion if a premise is false
• Affirming the consequent: inferring $P$ from $P \rightarrow Q$ and $Q$, a logical fallacy
  • Example: If it rains, the streets are wet. The streets are wet. Therefore, it rained. (The streets could be wet for other reasons.)
• Denying the antecedent: inferring $\neg Q$ from $P \rightarrow Q$ and $\neg P$, another fallacy
• Assuming that inductive strength accumulates with each premise, neglecting how premises can undermine each other
  • Example: Most birds fly. Most pets are cats. Tweety is a pet bird. Therefore, Tweety very likely flies. (Tweety is more likely a flightless pet bird.)
• Equivocating between different meanings of a term in the premises and conclusion
  • Example: Nothing is better than eternal happiness. A ham sandwich is better than nothing. Therefore, a ham sandwich is better than eternal happiness.
• Failing to recognize hidden assumptions that are necessary to make an argument valid
  • Example: If you're a patriot, you support the war. You don't support the war, so you're not a patriot. (Assumes there aren't other ways to be patriotic.)
• Mistaking a lack of evidence for evidence of lack, especially regarding existential claims
  • Example: We've never seen aliens, so aliens don't exist. (Absence of evidence isn't necessarily evidence of absence.)
• Conflating "some" and "all" in categorical statements and syllogisms
  • Example: Some lawyers are crooks. Harold is a lawyer. Therefore, Harold is a crook. (Doesn't follow from "some".)