Valid reasoning, argument analysis, and common fallacies.
What is Logic?
Logic is the study of valid reasoning. It provides tools to evaluate arguments, detect errors in thinking, and construct sound proofs. While logic won't tell you what to believe, it tells you what follows from what you already believe.
Why Logic Matters
| Benefit | Application |
|---|
| Detect bad arguments | Avoid manipulation, make better decisions |
| Construct good arguments | Persuade others, clarify your own thinking |
| Identify assumptions | Understand what claims really require |
| Resolve disagreements | Find the core of disputes |
| Think clearly | Reduce confusion and error |
Basic Concepts
Arguments
An argument is a set of statements where some (premises) are meant to support another (conclusion).
| Component | Definition | Example |
|---|
| Premise | Statement offered as support | "All mammals are warm-blooded" |
| Conclusion | Statement being supported | "Therefore, whales are warm-blooded" |
| Inference | The reasoning from premises to conclusion | The logical connection |
Validity and Soundness
| Term | Definition |
|---|
| Valid | If the premises are true, the conclusion must be true |
| Invalid | The premises could be true and the conclusion false |
| Sound | Valid argument with actually true premises |
| Unsound | Invalid or has at least one false premise |
Example of valid but unsound:
All fish can fly. (false premise)
Salmon are fish.
Therefore, salmon can fly.
The argument is valid (logic is correct) but unsound (premise is false).
Deductive vs. Inductive Reasoning
| Type | Characteristic | Strength |
|---|
| Deductive | Conclusion guaranteed by premises | Certain if valid |
| Inductive | Conclusion made probable by premises | Probable, not certain |
Deductive example:
All humans are mortal.
Socrates is human.
Therefore, Socrates is mortal.
Inductive example:
The sun has risen every day in recorded history.
Therefore, the sun will rise tomorrow.
Propositional Logic
Uses symbols to represent logical relationships.
| Symbol | Meaning | Example |
|---|
| P, Q, R | Propositions | P = "It is raining" |
| ~ or NOT | Negation | ~P = "It is not raining" |
| & or AND | Conjunction | P & Q = "It is raining and cold" |
| v or OR | Disjunction | P v Q = "It is raining or cold" |
| -> | Conditional | P -> Q = "If it rains, then it's wet" |
| <-> | Biconditional | P <-> Q = "P if and only if Q" |
Truth Tables
Truth tables show when compound statements are true or false.
Conjunction (AND):
Disjunction (OR):
Conditional (IF-THEN):
Note: A false antecedent makes the whole conditional true (vacuously).
| Name | Form | Example |
|---|
| Modus Ponens | If P then Q; P; therefore Q | If it rains, streets are wet. It's raining. Therefore, streets are wet. |
| Modus Tollens | If P then Q; not Q; therefore not P | If it rains, streets are wet. Streets aren't wet. Therefore, it's not raining. |
| Hypothetical Syllogism | If P then Q; if Q then R; therefore if P then R | If A then B; if B then C; therefore if A then C |
| Disjunctive Syllogism | P or Q; not P; therefore Q | It's raining or snowing. It's not raining. Therefore, it's snowing. |
| Constructive Dilemma | P or Q; if P then R; if Q then S; therefore R or S | Rain or snow. If rain, wet. If snow, cold. Therefore wet or cold. |
| Name | Fallacious Form | Why It Fails |
|---|
| Affirming the Consequent | If P then Q; Q; therefore P | Q could have another cause |
| Denying the Antecedent | If P then Q; not P; therefore not Q | Q could still be true |
Example of Affirming the Consequent:
If it rains, the streets are wet.
The streets are wet.
Therefore, it rained. (INVALID - could be sprinklers)
Errors in reasoning that don't involve formal logic mistakes.
Fallacies of Relevance
| Fallacy | Description | Example |
|---|
| Ad Hominem | Attack the person, not the argument | "You're wrong because you're stupid" |
| Appeal to Authority | Irrelevant authority cited | "This actor says vaccines are bad" |
| Appeal to Emotion | Substitute emotion for argument | "Think of the children!" |
| Appeal to Tradition | It's old, so it's good | "We've always done it this way" |
| Appeal to Popularity | Many believe it, so it's true | "Everyone knows..." |
| Red Herring | Distracting from the issue | Changing the subject when challenged |
| Tu Quoque | "You do it too" | "You can't criticize my lying; you've lied" |
Fallacies of Presumption
| Fallacy | Description | Example |
|---|
| Begging the Question | Assuming what you're trying to prove | "God exists because the Bible says so, and the Bible is God's word" |
| False Dilemma | Only two options when more exist | "You're either with us or against us" |
| Loaded Question | Question assumes unproven premise | "Have you stopped cheating yet?" |
| Slippery Slope | One thing inevitably leads to extreme | "If we allow X, then Y disaster follows" |
| Hasty Generalization | Too small a sample | "I met one rude French person, so all French are rude" |
| Straw Man | Misrepresenting opponent's position | "So you want total anarchy?" |
Fallacies of Ambiguity
| Fallacy | Description | Example |
|---|
| Equivocation | Using a word in two senses | "The sign said 'fine for parking here', so I parked" |
| Amphiboly | Ambiguous grammar | "I saw the man with binoculars" (who has them?) |
| Composition | What's true of parts is true of whole | "Each brick is light, so the wall is light" |
| Division | What's true of whole is true of parts | "The team is great, so each player is great" |
Causal Fallacies
| Fallacy | Description | Example |
|---|
| Post Hoc | After X, therefore because of X | "I wore my lucky socks and we won" |
| Correlation/Causation | Correlation mistaken for causation | "Ice cream sales and drowning both increase in summer" |
| Single Cause | Complex effect attributed to one cause | "Crime is caused by poverty" (oversimplified) |
Constructing Good Arguments
Steps to Build Arguments
| Step | Action |
|---|
| 1. Clarify your conclusion | What exactly are you trying to prove? |
| 2. Identify needed premises | What must be true for your conclusion to follow? |
| 3. Support each premise | Can you defend each premise? |
| 4. Check for validity | Does the conclusion actually follow? |
| 5. Consider objections | What would a critic say? |
| 6. Refine and strengthen | Address weaknesses |
Evaluating Arguments
| Question | Purpose |
|---|
| What's the conclusion? | Identify what's being argued for |
| What are the premises? | Find the supporting claims |
| Are premises true? | Evaluate evidence |
| Is the argument valid? | Check logical structure |
| Are there hidden assumptions? | Uncover unstated premises |
| What are the strongest objections? | Test robustness |
Predicate Logic
Goes beyond propositional logic to analyze internal structure of statements.
Quantifiers
| Symbol | Meaning | Example |
|---|
| ALL (Universal) | All/Every/Any | All dogs bark: ALL x (Dog(x) -> Barks(x)) |
| SOME (Existential) | Some/At least one | Some dogs are large: SOME x (Dog(x) & Large(x)) |
Categorical Propositions
| Type | Form | Example |
|---|
| A | All S are P | All cats are mammals |
| E | No S are P | No reptiles are warm-blooded |
| I | Some S are P | Some birds can't fly |
| O | Some S are not P | Some mammals don't live on land |
Categorical Syllogisms
Three-part arguments using categorical propositions:
All mammals are warm-blooded. (Major premise)
All whales are mammals. (Minor premise)
Therefore, all whales are warm-blooded. (Conclusion)
| Philosopher | Era | Contribution |
|---|
| Aristotle | 384-322 BCE | Syllogistic logic, fallacies |
| Chrysippus | 279-206 BCE | Stoic propositional logic |
| Boole | 1815-1864 | Boolean algebra, symbolic logic |
| Frege | 1848-1925 | Modern predicate logic |
| Russell | 1872-1970 | Logical paradoxes, Principia Mathematica |
| Godel | 1906-1978 | Incompleteness theorems |
| Tarski | 1901-1983 | Semantic theory of truth |
Practical Applications
Spotting Fallacies in Real Life
| Context | Common Fallacies |
|---|
| Political debates | Ad hominem, straw man, false dilemma |
| Advertising | Appeal to emotion, false authority, bandwagon |
| Social media | Hasty generalization, cherry-picking, tu quoque |
| Everyday disagreements | Equivocation, moving goalposts, begging the question |
Logic for Better Thinking
| Practice | Benefit |
|---|
| Identify conclusions first | Know what's being argued |
| Make premises explicit | See hidden assumptions |
| Ask "does this follow?" | Check validity |
| Consider counterexamples | Test generalizations |
| Steelman opponents | Understand best version of opposing view |
| Admit uncertainty | Don't overstate conclusions |
Key Takeaways
- Arguments have structure - Premises, inferences, and conclusions
- Validity is about form - The logical structure, not the truth of premises
- Soundness requires truth - Valid form plus true premises
- Fallacies are everywhere - Learn to recognize them
- Deduction guarantees; induction suggests - Know the difference
- Hidden premises matter - Make assumptions explicit
- Charity strengthens criticism - Understand the strongest version of opposing views
- Logic is a skill - It improves with practice