Logic & Critical Thinking

In everyday speech an argument is a shouting match. In logic it is something calmer and far more useful: a set of statements in which one or more, called the premises, are offered as reasons to accept another, called the conclusion. Logic is the study of that support relation. When you learn logic you learn to ask one disciplined question about any claim: what reasons are being given, and do they actually hold the conclusion up?

A statement is a sentence that is either true or false. Questions, commands, and exclamations are not statements, so they cannot be premises or conclusions. "Close the door" and "What time is it?" have no truth value. Only declarative claims like "The door is closed" can play a role in an argument.

Argument versus assertion

Simply asserting something is not arguing for it. "Pineapple belongs on pizza" is an assertion. It becomes part of an argument only when reasons appear: "Pineapple belongs on pizza, because the sweetness balances the salt of the cheese." Now there is a premise (the sweetness balances the salt) offered in support of a conclusion (pineapple belongs on pizza). No reasons, no argument.

Argument versus explanation

Arguments and explanations can look identical because both use words like "because," but they do different jobs. An argument tries to convince you that a conclusion is true. An explanation assumes you already accept that something is true and tells you why it happened. "The bridge collapsed because the steel had rusted through" does not try to persuade you the bridge collapsed - you know it did. It explains the cause. Ask yourself: is the speaker giving evidence for a claim in doubt (argument), or giving the cause of a fact already agreed on (explanation)?

Finding the conclusion

To analyze an argument, find the conclusion first, then everything offered to support it is a premise. The conclusion is the point the arguer wants you to accept; the premises are the "why." Consider:

• Premise: All humans are mortal.
• Premise: Socrates is a human.
• Conclusion: Therefore, Socrates is mortal.

The conclusion is not always last. "Socrates is mortal, since he is human and all humans are mortal" puts the conclusion first. Word order does not decide the role; the logical relationship does. A handy test is to insert "therefore" in front of a candidate: if "therefore Socrates is mortal" reads naturally as the payoff, that statement is the conclusion. In the next lesson we sharpen this with signal words that flag which is which.

Real arguments arrive tangled inside paragraphs, not tidily labeled. Your job as a critical thinker is to reconstruct them: strip away the noise and lay the reasoning bare. Two tools make this fast: indicator words and standard form.

Indicator words

Conclusion indicators are words that typically introduce the point being argued for. Premise indicators typically introduce a reason. Learn these and half the work is done.

Caution: these words have other uses. "Since Tuesday" is about time, not a premise. "So tired" is an intensifier. Always check that the word is actually flagging a reason or a conclusion, not doing some other grammatical job.

Standard form

To put an argument in standard form, list each premise on its own numbered line and write the conclusion last, marked with a line or the word "therefore." Take this passage: "You should not trust that website. It has no author listed, and sources with no named author are unreliable." In standard form:

1. Sources with no named author are unreliable.
2. That website has no named author.
3. Therefore, you should not trust that website.

Rewriting an argument this way exposes exactly what is being assumed, which is the first step to judging whether the reasoning works.

Enthymemes: the missing piece

Everyday arguments often leave a premise or conclusion unstated because it seems obvious. An argument with a missing part is an enthymeme. "Whales are mammals, so they breathe air" hides the premise "all mammals breathe air." Supplying the missing part is not cheating; it is making the reasoning honest and testable. But be charitable: fill the gap with the most reasonable claim the arguer would likely accept, not a silly one you can easily knock down. When you reconstruct an enthymeme, you often discover the whole argument stands or falls on an assumption nobody bothered to say out loud.

Not all arguments aim for the same target. The single most important distinction in this course is between deductive and inductive reasoning, because it decides which standards we judge an argument by.

Deductive reasoning: aiming for certainty

A deductive argument claims that its conclusion follows with necessity: if the premises are true, the conclusion must be true, with no possible exception. The classic example:

1. All mammals are warm-blooded.
2. A whale is a mammal.
3. Therefore, a whale is warm-blooded.

There is no way for those premises to be true and the conclusion false. Deductive arguments do not add information; they unpack what is already contained in the premises. Mathematics and formal logic are deductive through and through. We judge deductive arguments as valid or invalid, terms we define fully in Module 2.

Inductive reasoning: aiming for probability

An inductive argument claims only that its conclusion is probable given the premises. The premises make the conclusion likely but do not guarantee it. Inductive reasoning goes beyond the evidence, which is exactly why it can teach us new things and also why it can go wrong:

1. Every swan anyone has recorded in Europe for centuries was white.
2. Therefore, the next swan observed in Europe will probably be white.

Strong as this evidence is, black swans exist in Australia. The conclusion was reasonable but not certain. Good inductive reasoning is the engine of science, forecasting, and daily life; we can only ever have evidence, never a guarantee, that the sun will rise tomorrow.

Common inductive patterns

• Generalization: from a sample to a whole population ("400 surveyed voters favored the measure, so most voters probably do").
• Analogy: two things alike in known ways are probably alike in a further way ("this new drug is chemically similar to one that works, so it will probably work too").
• Causal inference: from a repeated correlation to a cause ("every time we removed the additive, the reaction stopped, so the additive causes it").
• Prediction: from past patterns to future cases ("this bridge has held for a century, so it will hold tomorrow").

How can you tell which kind you are facing? Ask what the arguer is claiming. If the claim is that the conclusion follows necessarily, treat it as deductive and test for validity. If the claim is only that the conclusion is likely, treat it as inductive and ask how strong the evidence is. A single argument is rarely both.

These two words carry more weight than any others in logic, and beginners constantly mix them up. Master them here and everything downstream gets easier.

Validity is about form, not truth

A deductive argument is valid when its form guarantees that if the premises were true, the conclusion would have to be true. Validity says nothing about whether the premises are actually true. It is a promise about structure: no argument of this shape can carry you from true premises to a false conclusion. Look at this valid but false argument:

1. All fish are birds.
2. All birds have wheels.
3. Therefore, all fish have wheels.

Both premises are absurd, yet the argument is perfectly valid, because if those premises were true, the conclusion would be forced. Validity is the plumbing; it says the pipes connect, not that clean water is flowing through them. To show an argument is invalid, you find a counterexample: a possible situation where the premises are true and the conclusion false.

Soundness adds truth

A valid argument with a false premise proves nothing. What we really want is a sound argument, defined as one that is (1) valid AND (2) has all true premises. A sound argument is the gold standard, because valid form plus true premises together force a true conclusion. Compare:

The one rule that follows

Here is the practical payoff. If an argument is valid and you want to reject its conclusion, you must reject at least one premise. You cannot accept every premise of a valid argument and still deny the conclusion - that would contradict what validity means. So when a valid argument reaches a conclusion you dislike, the honest move is to point to the specific premise you think is false, not to wave the conclusion away. Much of careful reasoning comes down to hunting for the weakest premise in a valid argument.

Validity and soundness are all-or-nothing: an argument either is valid or it is not. But inductive arguments come in degrees. We do not call them valid or sound; we call them strong or weak, and cogent or not.

Strong versus weak

An inductive argument is strong when its premises, if true, make the conclusion highly probable. It is weak when they do not. Strength is a matter of degree and depends on the evidence:

• Sample size: "I asked 3 people and they liked the app, so everyone will" is weak; "we surveyed 3,000 representative users" is stronger.
• Representativeness: a sample that mirrors the whole group supports a generalization; a biased sample does not. Polling only your friends about a national election is weak no matter how many friends you have.
• Relevance: the evidence must actually bear on the conclusion.

Cogency: the inductive twin of soundness

A cogent argument is a strong inductive argument whose premises are also true. It is the inductive parallel of a sound deductive argument: our best-case scenario for reasoning about the probable. A strong argument built on false premises is not cogent, just as a valid argument with false premises is not sound. Here is the full parallel, worth memorizing:

Defeasibility: a key difference

Inductive arguments are defeasible, meaning new information can weaken a previously strong argument without any old premise becoming false. "Tweety is a bird, so Tweety can fly" is a reasonably strong inference. Add the premise "Tweety is a penguin," and the argument collapses, even though the original premise stayed true. Deductive validity never works this way: once valid, always valid, no matter what you add. This is why good inductive reasoners hold conclusions with a grip proportioned to the evidence, always ready to revise. Certainty is the exception in real life; well-calibrated confidence is the goal.

Certain argument shapes appear so often, and are so reliably valid, that logicians have named them. Learn to recognize these patterns and you can certify an argument as valid at a glance. We write them using letters for whole statements: let P and Q stand for any statements.

Modus ponens (affirming the antecedent)

The most common valid form. From a conditional and its antecedent, conclude the consequent.

1. If P, then Q.
2. P.
3. Therefore, Q.

Example: "If it is raining, the ground is wet. It is raining. So the ground is wet." Always valid.

Modus tollens (denying the consequent)

From a conditional and the denial of its consequent, conclude the denial of the antecedent.

1. If P, then Q.
2. Not Q.
3. Therefore, not P.

Example: "If it is raining, the ground is wet. The ground is not wet. So it is not raining." Also always valid, and a favorite tool of science: if a hypothesis predicts an outcome that fails to occur, the hypothesis is wrong.

Two more workhorses

• Hypothetical syllogism (chaining conditionals): If P then Q; if Q then R; therefore if P then R. "If I study, I pass; if I pass, I graduate; so if I study, I graduate."
• Disjunctive syllogism (process of elimination): Either P or Q; not P; therefore Q. "The keys are in my coat or my bag; they are not in my coat; so they are in my bag."

The dangerous look-alikes

Two invalid forms impersonate modus ponens and modus tollens. They are formal fallacies: they look valid but are not.

• Affirming the consequent (INVALID): If P then Q; Q; therefore P. "If it is raining, the ground is wet. The ground is wet. So it is raining." Wrong - a sprinkler could have wet the ground.
• Denying the antecedent (INVALID): If P then Q; not P; therefore not Q. "If it is raining, the ground is wet. It is not raining. So the ground is not wet." Wrong again - the sprinkler.

The lesson: a conditional "if P then Q" tells you P is enough for Q, but not that P is the only route to Q. Affirming Q or denying P therefore proves nothing. Keep the two valid forms (affirm the antecedent, deny the consequent) firmly apart from their two invalid twins.

A fallacy is a mistake in reasoning that tends to persuade anyway. Formal fallacies (like affirming the consequent) fail because of bad structure. Informal fallacies fail for reasons of content: the premises may look relevant but do not truly support the conclusion. This module covers the informal fallacies you will meet most often. We start with fallacies of relevance, where the premises are simply beside the point.

Ad hominem (against the person)

The ad hominem fallacy attacks the person making an argument instead of the argument itself. "You cannot trust her claim about the budget - she has been divorced twice" says nothing about the budget. A person's character, motives, or circumstances are usually irrelevant to whether their claim is true. Note the difference from legitimately questioning a witness's reliability; the fallacy is treating an insult as if it refuted a point.

Straw man

The straw man fallacy misrepresents an opponent's position, replacing it with a weaker, distorted version that is easier to knock down, then attacks that. "Senator Lee wants to cut the defense budget slightly." "So the Senator wants to leave us defenseless against our enemies!" The exaggerated version is a straw man; the real, modest proposal was never addressed. The antidote is the principle of charity: engage the strongest, most accurate version of the other side.

Appeals that pressure rather than prove

• Appeal to force (ad baculum): threatening harm instead of giving reasons. "You will agree the report is fine, if you value your job." A threat is not evidence.
• Appeal to emotion (ad populum in its pity or fear forms): stirring feelings in place of argument. "You must acquit my client; look at his weeping family." Sympathy, however genuine, does not establish innocence.
• Appeal to the people (bandwagon): "Millions use this diet, so it must work." Popularity is not proof; millions can be wrong.

What unites these is a bait and switch: your attention is pulled toward the person, a caricature, or your emotions, and away from the actual question of whether the conclusion is true. The moment you notice the given reasons would not survive the question "but is that relevant to the truth of the claim?", you have likely found a fallacy of relevance.

A second family of fallacies smuggles in an unjustified assumption or leaps from thin evidence. These are fallacies of presumption and weak induction. They are more dangerous than fallacies of relevance because they can feel like real arguments.

False dilemma (false dichotomy)

The false dilemma presents only two options when more exist, forcing a choice between extremes. "Either we cut every environmental regulation or the economy collapses." Reality offers a spectrum of policies in between. Watch for "either/or" framings; ask whether a third option was quietly hidden.

Slippery slope

The slippery slope claims that one small step must inevitably lead to an extreme outcome, without justifying each link in the chain. "If we let students retake one quiz, next they will demand to retake finals, then to skip class entirely, and the whole system will fall apart." Each step needs a reason; assuming the tumble is inevitable is the fallacy.

Begging the question

To beg the question is to assume the very thing you are trying to prove, so the conclusion is smuggled into the premises. "This medicine is effective because it works to cure the illness." The premise just restates the conclusion in other words. A circular argument gives you no independent reason to accept anything.

Fallacies of weak induction

• Hasty generalization: drawing a broad conclusion from too small or biased a sample. "I met two rude people from that city, so everyone there is rude."
• False cause (post hoc): assuming that because B followed A, A caused B. "I wore my lucky socks and we won, so the socks caused the win." Correlation and sequence are not causation.
• Appeal to ignorance: claiming something is true because it has not been proven false, or vice versa. "No one has proven ghosts do not exist, so they do." Absence of disproof is not proof.

The common thread is a gap the arguer hopes you will not notice: an unlisted third option, an unjustified chain, a premise that merely repeats the conclusion, or a leap far beyond the evidence. Naming these fallacies trains you to feel the gap and demand that it be filled.

Two thousand years before symbolic logic, Aristotle built a rigorous system for reasoning about categories - claims about how classes of things relate. Categorical logic still teaches the discipline of exact quantity and negation, and its Venn-diagram method is beautifully visual.

The four standard forms

A categorical statement relates two classes: a subject term (S) and a predicate term (P). There are exactly four standard types, labeled by the vowels A, E, I, O:

In logic, some means "at least one," not "some but not all." So "Some dogs are brown" is true even if, in fact, all dogs were brown. This precise sense prevents endless confusion.

The square of opposition

The four forms are logically linked. The traditional square of opposition maps the links:

• Contradictories (A-O and E-I, the diagonals): always have opposite truth values. If "All S are P" is true, then "Some S are not P" must be false, and vice versa.
• Contraries (A-E, top): cannot both be true, but could both be false.
• Subcontraries (I-O, bottom): cannot both be false, but could both be true.

The most useful pair is the contradictories, because to deny a universal claim you need only find one exception: the contradictory of "All swans are white" is "Some swan is not white," which a single black swan makes true.

A categorical syllogism is a deductive argument with exactly two premises and a conclusion, built from three categorical statements and three terms. The famous example:

1. All humans are mortal. (major premise)
2. All Greeks are humans. (minor premise)
3. Therefore, all Greeks are mortal. (conclusion)

The three terms each appear twice: the middle term ("humans") links the premises and vanishes from the conclusion; the major term ("mortal") is the predicate of the conclusion; the minor term ("Greeks") is the subject of the conclusion.

Testing validity with Venn diagrams

The cleanest way to test a syllogism is a Venn diagram of three overlapping circles, one per term. The method: diagram only what the premises assert, then look to see whether the conclusion has been drawn for you automatically. If it has, the argument is valid; if you must add anything to make the conclusion appear, it is invalid.

Two marking rules: to show a region is empty (a universal claim, "All" or "No"), shade it out. To show a region has at least one member (a particular claim, "Some"), place an X in it.

Working the Socrates example: "All Greeks are humans" shades the part of the Greeks circle that lies outside Humans. "All humans are mortal" shades the part of Humans outside Mortal. When you finish, the only region left unshaded for Greeks lies inside Mortal - so "All Greeks are mortal" is already shown. The syllogism is valid.

Two rules that catch most bad syllogisms

• The middle term must be distributed at least once. Distributed means the statement refers to every member of that class. Violating this is the fallacy of the undistributed middle, as in: "All cats are animals; all dogs are animals; so all cats are dogs." The middle term "animals" never covers the whole class, so the premises fail to connect cats and dogs.
• From two universal premises you cannot validly draw a particular conclusion that asserts existence - and two negative premises yield no valid conclusion at all.

You do not have to memorize every rule if you can diagram. The Venn test is mechanical and reliable: diagram the premises exactly, add nothing, and read off whether the conclusion appears. It turns an abstract question of validity into something you can literally see.

Propositional logic studies how whole statements combine. We let capital letters stand for simple statements (P = "It is raining," Q = "The game is cancelled") and join them with connectives. Each connective has a symbol and a precise meaning fixed by when the compound is true.

Three points that trip people up

1. "Or" is inclusive. In logic, "P or Q" (P v Q) is true when either or both are true. "You may have coffee or tea" logically allows both. When ordinary language means one but not both (the exclusive or, as in "soup or salad, not both"), that is a different, more complex connective; logic's default "or" is inclusive.

2. The conditional is only false in one case. "If P then Q" (P -> Q) makes a single promise: it is broken only if the antecedent P happens but the consequent Q does not. Consider "If you score above 90, you get an A." The only way this is a lie is if you score above 90 and do not get an A. If you score below 90, the promise says nothing about your grade, so it is not broken either way. This is why a conditional counts as true whenever its antecedent is false - vacuously kept.

3. Antecedent and consequent are not interchangeable. In "if P then Q," P is the antecedent (the condition) and Q is the consequent (the result). "If P then Q" does not mean "if Q then P." "If it is a dog, it is an animal" is true; "if it is an animal, it is a dog" is false. Swapping them (the converse) can flip the truth value entirely.

These connectives are the atoms of formal reasoning. In the next lesson we build truth tables that compute the value of any compound, however complicated, from the values of its parts.

A truth table lists every possible combination of truth values for the simple statements in a formula and computes the resulting value of the whole. It is a complete, mechanical test: nothing is left to intuition. This is the heart of propositional logic, so we will work several tables in full.

How many rows?

With n distinct simple statements, there are 2 to the power n rows, one per combination. One letter needs 2 rows; two letters need 4; three letters need 8. List the combinations in a tidy order so you never miss one.

The five basic tables

First, negation. It simply flips the value:

Now the four two-place connectives together. Read each row across:

Study the conditional column (P -> Q). It is false in exactly one row, the second, where P is true and Q is false. That single-false pattern is the whole personality of the conditional.

A worked compound: ~P v Q

Let us build the table for ~P v Q step by step. We add a helper column for ~P, then combine it with Q using disjunction (true if at least one side is true):

Look closely: the final column of ~P v Q (T, F, T, T) is identical to the column for P -> Q above. That is no accident. "If P then Q" and "not-P or Q" are logically equivalent - they are true in exactly the same rows. Two formulas with matching final columns say the same thing.

Three verdicts a table can deliver

• A tautology is true in every row. Example: P v ~P ("it is raining or it is not raining") is always true.
• A contradiction is false in every row. Example: P & ~P is always false.
• A contingency is true in some rows and false in others - most ordinary statements.

Truth tables give you a decision procedure: to settle whether any propositional claim is a tautology, contradiction, or contingency, just build the table and look. In the final lesson of this module we turn the same tool on whole arguments to test validity directly.

Truth tables do more than classify single statements. They give a foolproof test for the validity of any propositional argument. Recall the definition: an argument is valid when it is impossible for all premises to be true while the conclusion is false. A truth table checks every possibility at once.

The method

1. Build one table with a column for each premise and a column for the conclusion.
2. Find every row in which all the premises are true together - these are the "critical rows."
3. Check the conclusion in those rows. If the conclusion is true in every critical row, the argument is valid. If even one critical row has a false conclusion, that row is a counterexample and the argument is invalid.

Worked example 1: modus ponens is valid

Test "P -> Q; P; therefore Q." The premises are P -> Q and P; the conclusion is Q.

Which rows have both premises true? Only row 1 (P -> Q is true and P is true). In that row the conclusion Q is also true. There is no row where both premises are true and Q is false, so modus ponens is valid. The table proves it beyond doubt.

Worked example 2: affirming the consequent is invalid

Now test the look-alike "P -> Q; Q; therefore P." Premises: P -> Q and Q; conclusion: P.

Critical rows (both premises true) are row 1 and row 3. In row 1 the conclusion P is true, but in row 3 both premises are true (P -> Q is true, Q is true) while the conclusion P is false. Row 3 is a counterexample, so affirming the consequent is invalid. This is the sprinkler case made rigorous: the ground can be wet (Q) with the conditional intact even though it is not raining (P false).

Why this matters

Notice the payoff. Where earlier we relied on cleverly chosen examples to sense invalidity, the truth table settles it mechanically - it examines all cases, so no counterexample can hide. Any argument you can express with these connectives can be certified valid or invalid this way. It is the closest thing critical thinking has to a calculator: set up the columns honestly, scan the critical rows, and the verdict is unavoidable.

You can master every rule of logic and still reason badly, because the mind runs on fast, automatic shortcuts that usually help but sometimes misfire in predictable ways. A cognitive bias is a systematic pattern of deviation from good judgment. "Systematic" is the key word: biases push everyone in the same direction, so they cannot be fixed just by trying harder. They need deliberate counter-strategies.

The big ones

• Confirmation bias: we notice, seek, and remember evidence that fits what we already believe, and overlook or explain away the rest. It is the most important bias to know, because it silently corrupts research, arguments, and beliefs about other people.
• Anchoring: the first number or impression we encounter drags later judgments toward it. Told a shirt was "reduced from 200 dollars," we judge 80 dollars a bargain, even if the shirt is worth 30.
• Availability heuristic: we judge how likely something is by how easily examples come to mind. Vivid, dramatic events (plane crashes, shark attacks) feel far more common than they are, while quiet risks are underrated.
• Hindsight bias: once we know an outcome, we feel we "knew it all along," which makes us overconfident about predicting the future.
• Sunk cost fallacy: we keep investing in a losing course because of what we already spent, even though the past money is gone whatever we choose now.

Why willpower is not enough

Because biases operate automatically and below awareness, simply resolving to be objective does not work; the biased judgment feels like plain perception. The remedy is procedural - building habits and rules that catch the bias before it acts:

• Consider the opposite: deliberately ask "what would make me wrong?" and actively search for disconfirming evidence. This is the single best cure for confirmation bias.
• Seek out base rates: before trusting a vivid anecdote, ask how common the event actually is in the data.
• Pre-commit to criteria: decide in advance what would count as success or failure, so an anchor or a sunk cost cannot move the goalposts later.
• Invite disagreement: ask people likely to see it differently, and treat their objections as data rather than attacks.

Knowing the names of biases is not a party trick; it is the practical bridge between the logic you have learned and the messy reasoning of real life. The goal is not to become a cold calculator, but to catch the moments when a feeling of certainty outruns the evidence.

Logic tells you whether a conclusion follows from premises, but in real life you also have to judge whether the premises are true, and that means evaluating evidence. This final lesson gathers the practical skills of a careful thinker facing a flood of claims.

Not all evidence is equal

Evidence varies in weight. Ask three questions of any piece of evidence:

• Is it relevant? Does it actually bear on the claim, or just sound related?
• Is it reliable? A large controlled study outranks a single anecdote; a direct measurement outranks a rumor; a systematic review of many studies outranks any one of them.
• Is it representative? A few striking cases can mislead if they are not typical. One friend cured by a remedy tells you almost nothing about how it performs across thousands of patients.

A recurring trap is treating a vivid anecdote as if it outweighed statistical data. Stories move us, but "it worked for me" is weak evidence next to a well-run trial.

Judging the source

When you cannot check a claim yourself, you rely on sources, so their credibility matters. Ask:

• Expertise: Is the source knowledgeable in this field? A brilliant physicist has no special authority on nutrition. Beware the appeal to a false authority, borrowing prestige from an unrelated area.
• Conflict of interest: Does the source gain from your belief? A study of a drug funded by its maker is not worthless, but it warrants extra scrutiny.
• Corroboration: Do independent sources agree? A claim confirmed by several unconnected experts is far safer than one repeated by outlets all copying the same origin.
• Track record and transparency: Does the source correct its mistakes and show its methods? Willingness to be checked is a mark of trustworthiness.

Correlation is not causation

Perhaps the most abused idea in public reasoning: two things varying together does not show that one causes the other. Ice cream sales and drowning deaths rise together, but neither causes the other; a third factor, hot summer weather, drives both. Before accepting "A causes B" from a correlation, ask whether B might cause A, whether a common cause drives both, or whether the link is coincidence. Establishing causation usually needs a controlled experiment, not just an observed pattern.

Calibrate, then act

The aim of everything in this course is calibration: holding each belief with a confidence that matches the strength of the evidence - firmly when the evidence is strong, tentatively when it is thin, and openly revisable when new evidence arrives. A good critical thinker is neither gullible nor cynical. They believe things, but they proportion belief to evidence, change their mind when the evidence changes, and can always say what would change it. That habit, more than any single rule, is the lasting payoff of learning to reason.