Logic & Critical Thinking

Definitions

Deductive argument

— the conclusion follows necessarily from the premises. If the premises are true, the conclusion cannot be false. Classic syllogism: all digital companies need cybersecurity; our company is digital; therefore, our company needs cybersecurity. To che...

Inductive argument

— the conclusion follows from the premises with a certain probability, but not with necessity. For three years in a row, the UAE real estate market grew in Q4; this Q4 the market will grow as well. The premises may be true, while the conclusion is f...

Abduction

— inference to the best explanation. The patient has a fever, cough, weakness. Diagnosis: flu. This is not deduction and not induction—it is the proposition of the best hypothesis. Used in diagnostics, detective reasoning, strategic analysis.

Logic is a normative science about correct thinking. It does not describe how people actually think (that is the domain of psychology), but establishes how one should think in order to arrive at true conclusions. Good thinking is not an innate gift, but a skill that can be learned.

Logic is present everywhere in professional life: data analysis, decision-making, drafting documents, negotiations, presentations. A person who can construct and recognize arguments has a serious competitive advantage.

An argument consists of three elements: thesis (the statement that needs to be proven), grounds (premises, facts, data), and link (the logical transition from grounds to thesis).

Example: “This project should be closed (thesis), because it has already required two budget overruns and the current forecast shows a third (grounds). Investments that systematically exceed the budget destroy company value (link).”

Definitions

Affirming the consequent

— “If it’s raining, the streets are wet. The streets are wet. Therefore, it’s raining.” Incorrect—the streets could have been washed. Form: P→Q; Q; therefore P. Wrong.

Denying the antecedent

— “If it’s raining, the streets are wet. It isn’t raining. Therefore, the streets are dry.” Incorrect—they could have been washed. Form: P→Q; not-P; therefore not-Q. Wrong.

Ad hominem

— an attack on the person instead of the argument. “You can’t talk about economics—you’ve never run a business.” But the argument can be correct regardless of who said it.

Straw man

— to distort an opponent’s position, making it easily refutable. “You say we need more regulation. So, you want the government to control our every move?”

False dichotomy

— presenting a situation as a choice between two options, although there are more. “Either you’re with us, or against us.” “Either we lay off staff, or we go bankrupt.”

Appeal to authority

— when the authority is irrelevant. A genius physicist recommending a diet is not an authority in dietetics. But appealing to a relevant expert is a legitimate technique.

Slippery slope

— assuming that one action will inevitably lead to a chain of negative consequences. “If we allow remote work, employees will stop working altogether.” Each step must be justified separately.

Appeal to nature

— “Natural” means good; “artificial” means bad. Cyanide is a natural substance. Vaccines are “artificial,” but they have saved billions of lives.

Gambler’s fallacy

— the belief that past random events affect future ones. “Heads came up 10 times in a row. The next toss is surely tails.” A coin doesn’t remember the past.

Poisoning the well

— discrediting the opponent before they speak. “You are about to hear a proposal from a person who has been to prison twice.”

Appeal to tradition

— “We’ve always done it this way.” This is not an argument for continuing—it’s necessary to justify why the tradition is good.

Hasty generalization

— a few cases → universal law. “Three startups from our incubator failed—therefore, our incubator is bad.”

False consensus effect

— “All reasonable people agree that…”—anyone who disagrees is automatically declared unreasonable, without proof.

Appeal to pity

— “Approve our budget—if not, we’ll have to fire 50 people.” An emotional argument may be relevant but does not substitute for substantive reasoning.

Appeal to novelty

— “This is the latest technology—so it’s better.” New is not necessarily better.

A logical fallacy is an argument that seems convincing but violates the rules of valid inference or appeals to irrelevant factors. The ability to recognize them is one of the most practical intellectual skills. They are found in political speeches, advertising, courtrooms, business negotiations, ...

Affirming the consequent: “If it’s raining, the streets are wet. The streets are wet. Therefore, it’s raining.” Incorrect—the streets could have been washed. Form: P→Q; Q; therefore P. Wrong.

Denying the antecedent: “If it’s raining, the streets are wet. It isn’t raining. Therefore, the streets are dry.” Incorrect—they could have been washed. Form: P→Q; not-P; therefore not-Q. Wrong.

Ad hominem—an attack on the person instead of the argument. “You can’t talk about economics—you’ve never run a business.” But the argument can be correct regardless of who said it.

Definitions

Confirmation bias

— we look for information that confirms what we already believe. A manager who considers a new market promising notices positive data and ignores negative. Antidote: deliberately seek out disconfirming evidence; appoint a “devil’s advocate” on the t...

Anchoring effect

— the first number or information disproportionately influences subsequent judgments. The first price offered in negotiations “anchors” the range. Antidote: carry out your own independent analysis before the negotiation, uninfluenced by others' numb...

Clustering illusion

— we see patterns in random data. Three losing quarters — a “trend.” Maybe, or maybe — randomness. Antidote: statistical analysis; ask “how often would this happen by chance?”

Familiarity effect

— we like what we encounter often. We consider familiar people more competent. Antidote: evaluate results and arguments, not the familiarity of the person.

Hindsight bias

— “I knew it would happen.” After an event, it seems to us that we predicted it. This distorts feedback and interferes with learning. Antidote: record predictions before the event.

Root Cause Analysis

— why did the problem occur? Ask “why?” five times in a row. The first answer is a symptom, the fifth — the systemic cause.

Decision tree

— a branched scheme of possible choices with assessment of probabilities and consequences. Especially useful for decision making under uncertainty.

Scenario analysis

— instead of a single forecast — three scenarios (optimistic, base, pessimistic) with different assumptions. Helps to prepare for surprises.

Critical thinking is a disciplined process of actively analyzing, synthesizing, and evaluating information gathered through observation, experience, reflection, or communication as a guide to belief and action. The definition is lengthy, but every element is important: active (not passive consump...

Critical thinking is not “constantly criticizing everything.” It is the skill of asking the right questions and honestly evaluating the answers. A good critical thinker seeks truth, not confirmation of their own views.

Paul and Elder identified eight intellectual standards: clarity (is it clearly formulated?), accuracy (how specific is it?), relevance (does it relate to the issue?), depth (does it cover complexity?), breadth (are alternative points of view considered?), logic (does the conclusion follow from th...

Daniel Kahneman described two modes of thinking: System 1 (fast, intuitive, automatic) and System 2 (slow, analytical, conscious). Most cognitive biases are a product of System 1 operating where System 2 is needed.

Definitions

Law of contradiction

— P ∧ ¬P — always false. It cannot happen that P is simultaneously true and false.

Law of excluded middle

— P ∨ ¬P — always true. Either P is true or it is not—there is no third option.

De Morgan’s laws

— ¬(P ∧ Q) = ¬P ∨ ¬Q; ¬(P ∨ Q) = ¬P ∧ ¬Q. Very useful in programming and legal text.

Natural language is rich and flexible, but that is precisely why it is imprecise. The same sentence can mean different things depending on context, intonation, and implicit assumptions. Formal logic creates an artificial language with clear rules, in which there is no room for ambiguity. This is ...

A proposition — a statement that has a truth value: true (T) or false (F). “It is raining” is a proposition. “Close the door” is not a proposition.

An atomic proposition is simple, not further decomposable. P: “The company is profitable.” Q: “The stock is rising.”

A molecular proposition is constructed from atomic propositions by means of logical connectives:

Definitions

Conditional probability

— P(A|B) — probability of A given B.

Base rate error

— ignoring prior probability. “A serial killer is a middle-aged man, reserved, likes weapons.” This profile fits millions of people. You can’t diagnose a rare event by a vague profile.

Simpson’s paradox

— a trend observed in subgroups disappears or changes direction when data are combined. Hospital A has higher survival for both mild and severe cases. Hospital B—for overall. How? Hospital A accepts more severe cases. The choice of data aggregation ...

Neglecting sample size

— a statistically significant result in a small sample may be due to chance. Four successful quarters is too little to conclude about a long-term trend.

Regression to the mean

— extreme results in the next period are closer to average. “The curse of the Sports Illustrated cover”: athletes on the cover often perform worse afterward—because they appeared on the cover during peak form. It is erroneous to explain the decline ...

In real life, we rarely have complete information. We make decisions under conditions of uncertainty: data are incomplete, models are approximate, the future is unpredictable. Probabilistic thinking is the skill of working with this uncertainty—without denying it and without paralysis.

The probability of an event is a number from 0 to 1, showing relative frequency or degree of confidence. P(A) = 1: event A will certainly occur. P(A) = 0: never. P(A) = 0.5: equally likely yes or no.

Addition (mutually exclusive events): P(A ∨ B) = P(A) + P(B). If a coin: P(heads) + P(tails) = 1. Multiplication (independent events): P(A ∧ B) = P(A) × P(B). Two coin tosses: P(two heads) = 0.5 × 0.5 = 0.25. Conditional probability: P(A|B) — probability of A given B.

Where H is the hypothesis, E is the observed evidence. We update the prior probability P(H) considering the new evidence E to obtain the posterior probability P(H|E).

Definitions

Ethos

— trust in the speaker. The audience is persuaded because they trust the speaker. Ethos has three components: competence (does the person know the subject?), integrity (does he speak the truth?), and benevolence (is he interested in the audience's w...

Pathos

— emotional influence. The audience is persuaded when the argument evokes the required emotions. Aristotle does not deny the role of emotions — he analyzes them systematically. Anger arises from undeserved humiliation; fear — from a real threat; pit...

Logos

— rational persuasion through arguments, facts, and evidence. This is the closest practical equivalent of formal logic in speech. But Aristotle realizes: in real speeches there is no room for complete syllogisms — enthymemes are used: syllogisms wit...

Rhetoric (Greek: rhetorike) is the art of persuasive speech. It is often confused with empty talk or manipulation. But for Aristotle, rhetoric is a serious discipline, a sister to dialectic (logic). Aristotle’s “Rhetoric” is a systematic analysis of how to persuade properly.

Aristotle identified three means of persuasion — three types of “pistis” (proofs):

Ethos — trust in the speaker. The audience is persuaded because they trust the speaker. Ethos has three components: competence (does the person know the subject?), integrity (does he speak the truth?), and benevolence (is he interested in the audience's welfare?). Ethos is established before the ...

Pathos — emotional influence. The audience is persuaded when the argument evokes the required emotions. Aristotle does not deny the role of emotions — he analyzes them systematically. Anger arises from undeserved humiliation; fear — from a real threat; pity — from another’s undeserved misfortune....

Definitions

Inversion

— instead of asking “how to achieve success,” ask “how to guarantee failure”—and avoid that. Munger: “Always invert.”

Second-order effects

— any action has direct consequences (first order) and indirect ones (second, third order). Lower the price → sales will grow (first order) → competitors also lower prices → margins fall for everyone (second order).

Critical mass

— a tipping point after which a process becomes self-sustaining. Nuclear reaction, network effects, fire.

Regression to the mean

— extreme indicators tend to return to the average. Don’t look for causes where there are none—it may just be statistics.

Adaptation

— organisms adapt to their environment or go extinct. Companies too. Kodak did not adapt to digital photography—even though it itself invented the digital camera.

Ecosystem

— organisms are interdependent. The destruction of one species affects the entire network. In business: your ecosystem of partners, suppliers, customers is interdependent.

Niche

— competitive advantage is often in specialization, not in fighting for the broad market.

Survivorship bias

— we study successful cases and ignore failures—and draw false conclusions. All “unicorns” worked 18 hours a day—but most who also worked 18 hours did not become “unicorns.”

Social proof

— in uncertainty, we follow the behavior of the majority. A powerful mechanism of social influence—and a source of herd behavior.

Losses hurt more than gains

— losing $100 weighs more psychologically than gaining $100. This explains the irrational avoidance of risk and reluctance to realize losses.

Opportunity cost

— each choice is a refusal of something else. The true cost of a decision includes what you gave up.

Marginal utility

— each additional unit of a good brings less satisfaction. The first slice of pizza tastes better than the fifth.

Markets clear

— at a free price, surplus turns into shortage, shortage into surplus. Administratively fixed prices create a permanent imbalance.

Feedback loops

— reinforcing loop (growth feeds growth—viral content) and balancing loop (corrects deviations—thermostat). Most real systems are a combination of both.

Delay

— between action and result there is a time lag. When managing with delay, it is easy to “oversteer”—to do too much because the effect of a previous action hasn’t yet manifested.

Mental models are simplified representations of how the world works. We cannot keep the entire complexity of reality in our minds, so we use models: maps of the territory which, as Korzybski said, “are not the territory.” A good model is accurate enough to be useful, and simple enough to be appli...

Charlie Munger (partner of Warren Buffett) called a broad collection of mental models from different disciplines a “lattice of mental models.” The more models you have, the richer your view of a problem.

Inversion: instead of asking “how to achieve success,” ask “how to guarantee failure”—and avoid that. Munger: “Always invert.”

Second-order effects: any action has direct consequences (first order) and indirect ones (second, third order). Lower the price → sales will grow (first order) → competitors also lower prices → margins fall for everyone (second order).

Definitions

Delays and Instability

— if a feedback loop has a significant delay, the system becomes unstable. A driver with delayed reactions steers too sharply; a company with slow management information overcorrects.

Policy Decisions with Systemic Consequences

— lowering gasoline prices → consumption increases → oil imports grow → balance of payments worsens → pressure on the currency → inflation → need to lower prices again. A loop amplifying destructive policy.

Focusing on Symptoms Instead of Causes

— painkillers reduce pain (the symptom), but do not cure the disease (the cause). In organizations: bonuses for short-term results conceal systemic problems.

Western science since the time of Descartes and Newton relies on reductionism: break the complex into parts, study each one—and from the parts understand the whole. This method has yielded tremendous results: physics, chemistry, molecular biology. But it works poorly where parts interact nonlinea...

Systems thinking is an alternative approach: the focus is not on the parts, but on the interactions; not on the structure, but on the behavior; not on the elements, but on feedback loops.

Donella Meadows ("Thinking in Systems," 2008) identifies three elements: stocks—accumulated resources that change slowly (population size, level of trust, amount of money in an account); flows—the rate of change of stocks (birth rate, expenses, investments); feedback loops—connections through whi...

Reinforcing loop (R — reinforcing): the growth of one element reinforces others, which in turn reinforce the first. Compound interest: money earns interest, interest is added to principal, larger principal earns more interest. Viral spread: more carriers → more infections → more carriers. Without...

Definitions

1. Empathy

— a deep understanding of the user and their context. Tools: field observations, interviews, "shadowing," customer journey mapping. The goal is to go beyond stated needs to uncover hidden ones ("jobs to be done").

Design thinking — a methodology for creatively solving complex, "ill-defined" problems with a focus on the user. It originated in Stanford's d.school and IDEO (Tim Brown). Unlike analytical methods, which work well with well-defined problems, design thinking is intended for situations where the p...

Its key features are: empathy (understanding the real user, not an abstract one); iterativeness (prototyping quickly and early, failing cheaply); integrativeness (combining what people desire, what is technically feasible, and what is commercially viable).

1. Empathy: a deep understanding of the user and their context. Tools: field observations, interviews, "shadowing," customer journey mapping. The goal is to go beyond stated needs to uncover hidden ones ("jobs to be done").

A classic example: Ford could have asked consumers what they want and heard "a faster horse." The real need was that people wanted to move faster—and so he offered the automobile.

Definitions

Realism

— the world exists independently of our cognition. The task of science is to discover its structure, which is "out there," independent of us.

Philosophers since the time of Plato have defined knowledge as "justified true belief" (JTB). Three conditions: (1) belief (I believe that P); (2) truth (P is true); (3) justification (I have grounds to consider P true).

In 1963, Edmund Gettier showed that these three conditions are insufficient. "Gettier cases": one can have a justified true belief without having knowledge. Example: you look at a stopped clock showing 3:15. In fact, it is 3:15 now. Your belief is justified (the clock usually shows the correct ti...

Gettier sparked half a century of discussions on how to improve the definition of knowledge. One approach: add a condition of "non-defeasibility" (your conviction should not depend on false premises). Another: replace "justification" with a "reliable process of belief formation" (reliabilism). Th...

A priori knowledge — independent of experience. Mathematics, logic: "2+2=4" does not need to be experimentally verified — it is true by the meaning of the concepts. Kant added: some judgments about the world are also a priori, because they are the form of our perception (space and time).

Science is the most successful method for obtaining reliable knowledge about nature that humanity has ever invented. But what exactly makes a theory “scientific” rather than just “a clever idea”?

Karl Popper proposed the criterion of falsifiability: a theory is scientific if it is possible to indicate an observation that would disprove it. Newtonian mechanics is scientific (experiments can disprove it, and some have—favoring relativistic mechanics). The general theory of relativity is sci...

An important consequence: science advances not by accumulating confirmations, but by falsifications. A thousand confirming cases do not prove a theory—one refuting observation (provided the experiment is conducted properly) falsifies it.

Thomas Kuhn (“The Structure of Scientific Revolutions,” 1962) attacked the image of science as a linear progression of knowledge. He introduced the concept of the paradigm: a combination of exemplars, theories, methods, and standards accepted by the scientific community that defines “normal scien...

In 1931, Austrian mathematician Kurt Gödel proved a theorem that shook mathematics and philosophy. The First Incompleteness Theorem: in any sufficiently powerful and consistent formal system, there exist true statements that are unprovable within that system. The Second Theorem: such a system can...

This refuted Hilbert's program—to find a complete and consistent axiomatization of all mathematics. Gödel showed: any such system is either incomplete or inconsistent.

What does this mean beyond mathematics? There are few direct applications—generalizations like “no system can fully comprehend itself” are too broad and imprecise. But epistemologically, the theorem emphasizes: formal systems have internal limitations. This does not mean irrationalism—it is an ho...

Werner Heisenberg in 1927 formulated the uncertainty principle: it is impossible to simultaneously measure both the position and momentum of a particle exactly. The more precisely one is known—the less precisely the other. This is not a limitation of instruments, but a fundamental property of nat...

At the end of the 19th century, mathematicians thought they had found a solid foundation for all of mathematics through Cantor's set theory. Then, in 1902, the young Bertrand Russell wrote a letter to Gottlob Frege, who was working on the "Foundations of Arithmetic": at the very root of Frege's s...

Russell's paradox: consider the set of all sets that are not members of themselves. Is this set a member of itself? If yes, it must not be in the set. If no, then it must be there. This logical contradiction destroyed Frege's system. Upon receiving Russell's letter, Frege wrote: "The worst that c...

Russell proposed a solution—the theory of types: a ban on self-reference through a hierarchy of levels. Statements about objects are of one type, statements about statements are of another type. Therefore, the "set of all sets" is simply forbidden to construct.

Together with Whitehead he wrote "Principia Mathematica" (1910–1913)—three volumes logically deriving mathematics from logic. The proof that $1+1=2$ takes hundreds of pages and is contained in Volume 2. This was a grand attempt—and it showed how complex "obvious" mathematics is.

In 1931, Kurt Gödel published a work that overturned mathematics and philosophy. His incompleteness theorems showed: for any sufficiently powerful consistent formal system, there exist true statements that cannot be proven within it.

The first incompleteness theorem: in any consistent formal system sufficiently powerful for arithmetic, there are statements that cannot be proven nor refuted by means of that system.

The second incompleteness theorem: no sufficiently powerful consistent system can prove its own consistency.

This destroyed Hilbert's program. Mathematics cannot be both complete and consistent at the same time. This is not a "gap" in our understanding—it is a mathematically proven limit of formal systems.

In 1936, Alan Turing, in parallel with Gödel and independently, proved another fundamental limit: there exist problems that are inherently uncomputable—no machine (and likely no algorithm) will ever be able to solve them.

The “Halting Problem”: is it possible to write a program that, for any other program and input data, determines whether that program will halt (produce a result) or loop forever?

Turing proved: no, it is not possible. Suppose such a program H(p, x) exists. Then one can construct a program D which: takes program p, runs H(p, p), and if H says “will halt”—loops forever, if “will loop forever”—halts. What does H(D, D) do? A contradiction is inevitable.

The halting problem is only the first among a vast class of uncomputable problems. “Hilbert’s tenth problem”: determine whether a given Diophantine equation has an integer solution. Yuri Matiyasevich (1970) proved: uncomputable. “Post’s correspondence problem”: uncomputable. Many “practical” prog...

Game theory is a mathematical analysis of strategic interactions, where outcomes depend on the choices of multiple agents. A “game” here is a formal model including: players, their possible strategies, a payoff function (what each receives for every combination of strategies).

John von Neumann and Oskar Morgenstern (“Theory of Games and Economic Behavior”, 1944) laid the foundations. Their main result is the theorem on zero-sum games: in a two-player zero-sum game (one's gain = other's loss) there always exists an optimal mixed strategy.

John Nash (“Beautiful Mind”) generalized the result to non-zero-sum games. Nash equilibrium: a situation where no player wants to unilaterally change their strategy, knowing the strategies of the others. This is not necessarily the best outcome—it is a stable outcome.

The most famous example in game theory. Two prisoners, jointly arrested, are interrogated separately. Each can cooperate (remain silent) or betray (testify against the other). Payoff matrix: both silent—1 year of prison for each. Both betray—3 years for each. One betrays, the other is silent—betr...

How does one make decisions under uncertainty? Expected Utility Theory (von Neumann, Morgenstern, 1944): a rational agent maximizes the mathematical expectation of their utility function. This is a normative theory: how a rational agent ought to behave.

Expected value vs. expected utility. Example: a lottery — 50% chance to win 200 rubles, 50% chance to win nothing. Expected value = 100 rubles. But most people will agree to sell this ticket for less than 100 rubles — they are risk-averse. This is explained by decreasing marginal utility of money...

St. Petersburg paradox (Bernoulli, 1738): a coin is flipped until the first heads appears. If heads appears on the n-th flip, payout is $2^n$ rubles. Expected value is infinite. But no one will pay a large sum to participate in this game. Why? Diminishing marginal utility, limited ability of the ...

Maurice Allais (1953) showed: people systematically violate expected utility theory. The “Allais paradox”: most people prefer A (100% to receive 1 million) over B (89% to receive 1 million, 10% — 5 million, 1% — nothing) — and simultaneously prefer C (10% to receive 5 million) over D (11% to rece...

Classical game theory assumed rational agents consciously maximizing their payoff. Evolutionary game theory (Maynard Smith, Price, 1970s) applied its apparatus to biological evolution: strategies are not conscious choices, but behavioral programs selected by evolution.

"Evolutionarily Stable Strategy" (ESS): a strategy which, if adopted by the majority of a population, cannot be invaded by any mutant strategy. This is analogous to Nash equilibrium in the evolutionary context.

Classic example: "Hawk vs. Dove". Hawk always fights for the resource, Dove always retreats. A pure population of Hawks is unstable: conflicts are too costly. A pure population of Doves is unstable: Hawk will seize the resources. The ESS is a mixed population.

Robert Axelrod ("The Evolution of Cooperation," 1984) held a computer tournament: different strategies repeatedly play the prisoner's dilemma. The winner was "Tit for Tat": start by cooperating, then mimic the opponent's last move.

Probability seems intuitively clear — but upon closer examination raises profound philosophical questions. There are three main interpretations, each with different implications.

Frequentist interpretation (Fisher, von Mises): the probability of an event is the limit of its frequency as the experiment is repeated infinitely. This works for casinos and insurance. But what does the "probability" mean that Napoleon would have won Waterloo with a different disposition? Such a...

Subjective (Bayesian) interpretation (de Finetti, Savage): probability is a measure of the degree of confidence of a rational agent. This allows for talk about unique events. But different agents can have different "subjective probabilities"—and that's normal: they should converge as evidence acc...

Objective Bayesian interpretation (Keynes, Jeffreys): probability is objectively determined by the available evidence. This is an attempt to avoid subjectivity while preserving the Bayesian formalism.

Thomas Bayes (18th century) formulated a theorem that became the foundation of Bayesian statistics and rational thinking: P(H|E) = P(E|H) × P(H) / P(E).

P(H) — the "prior" probability of the hypothesis before obtaining evidence. P(E|H) — the probability of observing evidence E, given that hypothesis H is true (likelihood). P(H|E) — the "posterior" probability of the hypothesis after obtaining evidence. P(E) — the probability of the evidence (norm...

This is the formula for rational updating of beliefs. Bayesian thinking: you have a "prior" confidence in a hypothesis. You observe new evidence. Posterior confidence is a function of your prior confidence and the strength of the evidence.

Applications of the Bayesian method have spread far beyond academic statistics. Spam filters (Bayesian classifier): the probability that an email is spam is updated with every new word.

Benjamin Disraeli (or Twain): "There are three kinds of lies: lies, damned lies, and statistics." This does not mean that statistics always lie—it means they can mislead in the absence of critical thinking.

Simpson's Paradox: a trend present in several groups of data disappears or reverses when the groups are combined. Example: Treatment A has better results for women AND better results for men—but in the combined population, it is worse than Treatment B. How? If Treatment B has a disproportionately...

Survivorship bias: we analyze only "survivors"—successful companies, returned aircraft, completed projects—and draw conclusions, ignoring those who did not survive. Walter Schwartz during WWII: do not reinforce the areas of hits in returned aircraft—reinforce the areas where there are no hits bec...

"After, therefore because" (post hoc ergo propter hoc): after X, Y happened, so X caused Y. The rooster crows before sunrise—does the rooster cause the sunrise? Correlation between deaths from drowning and ice cream sales (seasonality—a hidden variable).

Curry-Howard correspondence: programs and proofs are one and the same. A type in programming corresponds to a proposition in logic; a program implementing that type is a proof of that proposition. This is not a metaphor—it is a strict mathematical connection.

Consequence: programming languages with rich type systems (Haskell, Coq, Agda) allow logical constraints to be encoded directly in the code—so that an incorrect program simply will not compile. "The type system is the specification; the compiler is the verifier."

Formal program verification: a mathematical proof that a program meets its specification. Used in mission-critical systems: nuclear reactors, aviation, medical devices. In 2022, the CompCert compiler (a formally verified C compiler) was the first to obtain a formal proof of correctness—requiring ...

P vs. NP—the central open question in computer science. Practical consequences. Traveling Salesman Problem (TSP): for $n$ cities, find the shortest route. NP-complete problem—no known polynomial algorithm. For 1000 cities, brute force requires more operations than atoms in the universe. Heuristic...

AI has developed according to two fundamentally different approaches. Symbolic AI (Good Old-Fashioned AI, GOFAI): represent knowledge in explicit logical form, write inference rules. “Expert systems” of the 1980s: a knowledge base of if-then rules + inference mechanism. Advantage: transparency, i...

Connectionist AI (neural networks): learning from data, without explicit rules. Advantage: flexibility, scalability, ability to find patterns in complex data. Limitation: “Black box” — lack of transparency, difficulty in interpretation.

Current trend: hybrids. “Neurosymbolic AI” — attempts to combine neural networks with symbolic reasoning. Google DeepMind AlphaCode — a neural network for programming, embedded into a symbolic system for correctness verification.

Machine learning is statistical optimization, but with logical constraints. First-order logic allows us to set constraints: “If an X-ray shows a shadow — probability X is higher.” “Inductive logic programming” — learning logical rules from examples.

The digital environment creates unprecedented conditions for logical errors. Speed: decisions are made within seconds. Volume: thousands of units of information per day. Algorithms: recommendations are optimized for engagement, not for truth.

Trendy "format" is not the same as a "logically convincing format". A TikTok video persuading about a conspiracy theory can be rhetorically more powerful than an academic article refuting it. This creates "logical arbitrage": manipulations are cheaper to produce than to refute.

Infodemic (WHO, term from 2020): the excess of information about the pandemic, including false information, making it harder to make the right decisions. The logic "if it sounds plausible—maybe it's true" is especially dangerous in crisis situations.

"Reflective skepticism" (as opposed to "skepticism for skepticism's sake"): evaluation of claims based on method, quality of evidence, possible alternative explanations. This is not denial—it is a standard.