The rules of valid reasoning, formal systems, and the paradoxes that break them
Part A - what logic is: the science of valid reasoning
Overview
A map of reasoning itself
Logic studies what follows from what. It is the discipline of separating the structure of an argument from the topic, tone, and social pressure around it. In this module, students move from validity and soundness into symbolic logic, fallacies, formal systems, probability, decision theory, paradoxes, and practical critical thinking.
Form over content
Logic's central insight is that validity depends on form, not subject matter. "All A are B; C is A; therefore C is B" keeps the same structure whether A means mammals, legal contracts, integers, or fictional creatures. A valid argument guarantees only this: if the premises are true, the conclusion cannot be false. A sound argument is stronger because it is valid and its premises are actually true.
The historical hinge
Aristotle, writing around 350 BCE, made the syllogism famous: all humans are mortal; Socrates is human; therefore Socrates is mortal. The Stoics then developed propositional logic, with if-then arguments closer to modern computing. George Boole turned logic into algebra in 1847. Gottlob Frege's 1879 Begriffsschrift added quantifiers and variables, which is why modern logic can say "for every number there is a larger number" without pretending every argument is about Socrates. The short version: logic moved from classifying sentences to designing languages precise enough for mathematics and machines.
Part B - classical logic: the foundations
Argument anatomy
Connectives
and, or, not, if, iff
Deduction
certainty from premises
Predicate logic
all, some, none
Induction
evidence to expectation
Abduction
best explanation
Truth table explorer
Select a connective and watch how the truth value changes. The table columns use a fixed layout so the shape stays stable while students compare operators.
Four valid deductive forms
These are reusable argument skeletons. Logic asks whether the skeleton preserves truth; other fields decide whether the starting premises deserve belief.
Part C - fallacies: the patterns of bad reasoning
Fallacies on a spectrum: structural error to persuasion trick
invalid formrhetorical distortion
Affirming consequent
Undistributed middle
False dichotomy
Appeal to authority
Straw man
Placement is conceptual, not a measurement. Formal fallacies can be diagnosed from structure alone; informal fallacies usually require context, definitions, and background facts.
A bias is not automatically a fallacy. Confirmation bias is a psychological tendency to seek friendly evidence; a fallacy is a defect in an argument. They overlap when a person turns selective attention into a conclusion, as in "I found three stories that confirm my belief, so the belief is established."
Part D - formal systems and their limits: Godel, Hilbert, and Turing
Formal system
A game with symbols
A formal system needs an alphabet, formation rules, axioms, and inference rules. Euclidean geometry is the classical model: start with postulates and derive theorems. Modern arithmetic can be formalized the same way, but the symbols are stripped of intuition so every legal move can be checked mechanically.
Hilbert's programme
Prove mathematics secure
David Hilbert wanted mathematics shown complete, consistent, and decidable by finite means. In 1900 he listed 23 famous problems; in 1928 he sharpened the decision problem. The dream was not modest: a procedure that, in principle, could settle every well-formed mathematical question.
Godel's blow
Truth outruns proof
In 1931 Kurt Godel showed that any consistent formal system strong enough for ordinary arithmetic contains true statements it cannot prove. His second theorem added that such a system cannot prove its own consistency from inside itself. This did not destroy mathematics; it ended one fantasy about total self-certification.
Turing's mirror
Computation has limits
Alan Turing proved in 1936 that no general algorithm can decide whether any arbitrary program eventually halts. The proof diagonalizes against the hypothetical solver. It is not a hardware limitation, and faster computers do not help. It is a theorem about what computation is.
Self-reference risk slider
3/10
object language onlytalking about its own sentences
Part E - probability and decision theory: logic under uncertainty
Bayes calculator: base rates change everything
Reasoning under uncertainty: what changes the conclusion?
Base rate
Likelihood
Payoff
Strategy
Illustrative distribution, not a universal law. It reminds you that probability, utility, and strategic interaction are different questions that people often collapse into one vague word: "rational."
Decision theory and game theory are not the same game
Decision theory asks what one agent should choose under uncertainty, usually by expected utility: probability multiplied by value. Pascal's wager is a famous early version, though its assumptions are heavily disputed. Game theory adds other agents whose choices respond to yours. In the prisoner's dilemma, defection dominates in a one-shot version, yet cooperation can win in repeated play because reputation changes the payoff landscape.
Other cooperates
Other defects
You cooperate
You: 3 Other: 3
You: 0 Other: 5
You defect
You: 5 Other: 0
You: 1 Other: 1
Part F - logical and self-reference paradoxes
Self-reference is not automatically forbidden. A book can say "this book has 300 pages" without exploding. The danger appears when a system is strong enough to encode statements about its own truth or provability while also pretending there is only one level of language. Alfred Tarski's hierarchy separates object language from metalanguage precisely to stop truth from chasing its own tail.
Part G - mathematical and infinity paradoxes
Zeno
motion split infinitely
Hilbert's Hotel
countable infinity
Cantor
larger infinities
Banach-Tarski
choice gets wild
Gabriel's Horn
finite volume, infinite area
Birthday paradox slider
23 people
few pairsmany pairwise comparisons
Part H - probability and decision paradoxes
Monty Hall win rates after the host opens a goat door
Always switch
66.7%
Always stay
33.3%
Random choice
50%
Assumes the standard rules: the host knows where the car is, always opens a goat door, and always offers the switch. Change those rules and the calculation changes.
Monty Hall simulator
Pick a door. The host will reveal one empty door and offer the switch. Manual plays update the policy you choose; auto-play tests both stay and switch on the same random setups.
Pick a door to begin.
0stay wins
0stay games
0switch wins
0switch games
Expected value calculator
Part I - philosophical and metaphysical paradoxes
Paradoxes by what they pressure
language and conceptsworld and causality
Sorites heap
Omnipotence
Trolley problem
Ship of Theseus
Unexpected hanging
Fermi paradox
Predestination
The useful habit: ask which assumption breaks
The Sorites paradox pressures vague predicates like "heap" and "bald." The Ship of Theseus pressures identity through time. The Fermi paradox pressures estimates about life, technology, and detectability in a galaxy with roughly 100 to 400 billion stars. Olbers' paradox looked at a dark night sky and refuted an infinite, eternal, static universe before modern cosmology supplied the mechanism: the observable universe has finite age and is expanding.
Part J - decision making and critical thinking: logic applied to real choices
The gap
From valid arguments to good decisions
Logic tells you when conclusions follow. Decision theory tells you what to choose under uncertainty. Critical thinking bridges both: it is the applied discipline of catching the errors, biases, and framing effects that corrupt reasoning before a real stake is won or lost.
The tools
Frameworks you can actually use
Pre-mortems, decision matrices, steelmanning, hypothesis testing, and the outside view all reduce reliance on intuition in situations where intuition is known to be unreliable. Each tool targets a different failure mode.
The decision anatomy diagram
Cognitive biases that corrupt decisions
How biases are distributed across decision domains
Overconfidence
very high
Anchoring
high
Framing effect
high
Availability
moderate
Hindsight bias
moderate
Planning fallacy
very high
Severity ratings are consensus estimates from meta-analyses in behavioral economics, not precise measurements. The key insight is that overconfidence and planning errors are pervasive even among trained experts.
Critical thinking tools and when to use them
Pre-mortem
imagine failure first
Steelmanning
best version of the opposite view
Outside view
reference class forecasting
Fermi estimation
order-of-magnitude check
Five Whys
root cause drill-down
Null hypothesis
burden of proof discipline
Decision quality is not the same as outcome quality
Luck and skill both affect outcomes, but only process quality is within your control. Outcome bias judges a decision by its result; a careful reasoner judges it by the process given the information available at the time.
Good outcome
Bad outcome
Good process
Deserved win. Skill and luck aligned.
Bad luck. The process was still correct. Learn from randomness, not the decision.
Bad process
Dumb luck. Do not reinforce the bad process.
Deserved loss. Luck and skill both failed. Fix the process.
Confidence calibration
70% confident
uncertain (50/50)near-certain (99%)
Argument reconstruction: a worked example
One of the most practical critical thinking skills is breaking a messy claim into its premise-conclusion skeleton. Choose an example argument to see how it reconstructs.
Epistemic habits: good and bad
Habits that improve reasoning
Seek disconfirming evidence
Actively look for what would prove you wrong, not just what confirms you.
Bayesian updating: new evidence should shift your estimate, not be ignored.
Separate object-level from meta-level
Track both what is likely true and how confident you should be.
Habits that degrade reasoning
Motte-and-bailey switching
Defend a bold claim by retreating to a modest one when challenged, then returning to bold.
Galaxy-brained reasoning
Clever chains of steps reaching conclusions that common sense rejects. Each step seems valid; the whole is dangerous.
Moving the goalposts
Changing the success criterion once the original has been met to avoid conceding the argument.
Epistemic cowardice
Giving vague or uncommitted answers to avoid social cost, rather than stating what you actually believe.
The difference between a good reasoner and a bad one is not intelligence. It is the willingness to be wrong. Every technique in this section is designed to make it slightly easier to discover and admit an error before it becomes expensive.
Part K - Q&A
If an argument is valid, does that mean the conclusion is true?
No. Validity only means the conclusion would have to be true if the premises were true. "All whales are planets; all planets are made of cheese; therefore all whales are made of cheese" is valid and absurd. Soundness is the stronger standard: valid structure plus true premises. Most real disputes are not about validity alone; they are about whether the premises deserve belief.
Why does predicate logic matter if propositional logic already has truth tables?
Truth tables can handle whole propositions, but they cannot see inside "every prime greater than 2 is odd." Predicate logic opens the sentence and tracks objects, properties, and quantifiers. That jump is what lets mathematics express general theorems instead of listing cases. It is also the basis for database query languages and automated theorem proving.
Is calling something a fallacy enough to refute it?
No. The label is only useful if you show the exact failed inference. "Ad hominem" is often misused: attacking a witness's reliability can be relevant in court, while attacking a mathematician's personality does not refute a proof. The disciplined move is to reconstruct the argument charitably, identify the premise-to-conclusion step, and explain why that step fails. Otherwise "fallacy" becomes a rhetorical club, which is deliciously ironic and not very logical.
Does Godel prove that humans are not machines?
Not by itself. Godel proves limits for formal systems that are consistent, effectively axiomatized, and strong enough for arithmetic. Some philosophers, including J. R. Lucas and Roger Penrose, argue that human mathematicians can "see" truths no machine can prove, but that argument is controversial because humans also make mistakes and may not be consistent. The safe lesson is narrower and powerful: formal proof cannot capture all arithmetical truth in one tidy, self-validating package.
Why do smart people still mishandle medical tests and risk numbers?
Because a test's accuracy is not the same as the chance you have the condition after a positive result. If a disease is rare, false positives can outnumber true positives even when the test looks impressive. Natural frequencies help: "10 out of 1,000 have the disease" is easier to reason with than "1% prevalence." The mind is good at stories; probability asks it to count alternative ways the evidence could appear.
Are paradoxes just word games?
Some are linguistic, but the serious ones reveal a broken assumption. Russell's paradox was not a party trick; it damaged Frege's logicist project just as volume two of Frege's work was going to press in 1902. The liar paradox presses on truth, Berry on definability, and Godel turns self-reference into arithmetic. A good paradox is a stress test for a theory.
Is Tarski's hierarchy a complete solution to the Liar Paradox?
It is a powerful formal solution, not a magic eraser for ordinary language. Tarski avoids the paradox by separating an object language from a metalanguage: a language does not get to contain its own full truth predicate. Natural language ignores that neat separation, so the puzzle remains useful as a warning about unrestricted self-reference.
How can infinity plus one still be the same size?
For finite sets, adding one always makes a larger set. For countably infinite sets, size is defined by whether you can pair members one-to-one with the natural numbers. Hilbert's Hotel moves guest 1 to room 2, guest 2 to room 3, and so on, freeing room 1 without losing anyone. That feels illegal because finite intuitions are doing work in an infinite setting where they no longer apply.
Why did so many mathematicians object to the Monty Hall answer?
The mind sees two unopened doors and wants to assign 50% to each. But the host's action is not neutral information; it is constrained by knowledge of the prize. Your first pick had a 1 in 3 chance of being right, so the two unchosen doors collectively had 2 in 3. When the host deliberately removes a losing door from that pair, the 2 in 3 mass shifts to the remaining unchosen door.
Can logic settle moral paradoxes like the trolley problem?
Logic can expose consistency, hidden premises, and tradeoffs, but it cannot supply values from nowhere. The trolley problem separates outcomes from duties: saving five looks compelling, while pushing one person from a bridge feels like using a person as a tool. Philippa Foot introduced the original case in 1967 to probe the doctrine of double effect. Logic clarifies the disagreement; ethics decides which principles deserve authority.
What is the fastest way to diagnose an argument?
Separate truth from structure. First list the premises and conclusion in plain sentences. Then ask: if the premises were true, would the conclusion have to be true, probably true, or merely possible? That one distinction sorts deduction, induction, and abduction before the argument gets emotionally noisy.
Why is "correlation is not causation" true but incomplete?
Correlation alone does not prove causation because confounders, reverse causation, and selection effects can produce the same pattern. But correlation is still evidence, especially when it is large, repeated, temporally ordered, and tied to a mechanism. The better slogan is: correlation is not causation by itself. Science earns causal claims by combining statistics with design, controls, experiments, and explanation.
What is the difference between a paradox and a contradiction?
A contradiction is a direct clash, such as "P and not P." A paradox is a route to trouble: plausible assumptions plus apparently valid reasoning lead to a contradiction, impossibility, or deeply surprising conclusion. The value is diagnostic. When a paradox is serious, you do not simply reject the conclusion; you inspect the assumptions that made it unavoidable.
What should I remember from Godel and Turing if I forget the technical details?
Remember that formalization is powerful but not omnipotent. Godel shows that arithmetic truth cannot be completely captured by any one consistent, computable axiom system strong enough for arithmetic. Turing shows that no universal program can decide all program behavior. Together they mark a boundary: rigor does not mean every meaningful question has a mechanical answer.
What is the single most effective habit for improving decision quality?
Keeping a decision journal. Write down what you expect to happen, why you believe it, and how confident you are before the outcome is known. Review entries after the fact with the same rigor. This disciplines against hindsight bias: you cannot rewrite memory of what you expected when the expectation is on paper. Over time the journal reveals whether your confidence is calibrated, which patterns in your reasoning recur, and which domains you are actually good at predicting versus those where you are fooling yourself.
How does steelmanning differ from devil's advocacy, and why does it matter?
Devil's advocacy assigns someone to argue against a proposal, but the advocate often does so halfheartedly because their actual view is unchanged. Steelmanning asks you to construct the strongest possible version of the opposing position, one that the position's best defenders would recognize and endorse. The difference is motivation: a devil's advocate can perform dissent; a steelman must understand it. The practical payoff is that you are far less likely to dismiss a position you have genuinely reconstructed at its best. It also makes your own view more defensible because you have already tested it against the best available opposition.