Someone tells you: “If the bridge were unsafe, they would have closed it. They haven’t closed it. So it’s safe.” It sounds fine. Someone else says: “If it rained, the street is wet. The street is wet. So it rained.” That sounds fine too, until you picture a street cleaner. The first argument is a good one. The second has the same outward look and is a bad one, and a philosophy graduate can tell you why in about three seconds, because they have stopped listening to what the argument is about and started looking at its shape.
That is the whole trick of the subject, and it is why Logic & Argument is the first door into the philosophy major. Every other subject in it (36/02 Ethics, 36/03 Epistemology, and the rest) is built from arguments, so you need a way to ask of any of them: if the starting points were true, would the end have to be?
This piece gives you the working tools: a truth-table machine, a syllogism tester, and a fallacy spotter on invented arguments. It is a first look, not a logic course. For the same ground at a gentler pace, read What Is a Good Argument? (05/03); this goes a step further.
An argument is a set of statements, the premises, offered to support another, the conclusion. Logic asks one question first: is it valid? An argument is valid when it is impossible for all the premises to be true and the conclusion false. Not unlikely. Impossible.
The surprising part is that validity does not care whether anything is true. “All fish are mammals; whales are fish; so whales are mammals” is perfectly valid: if the two premises were true the conclusion would have to be. It is not sound, because a premise is false. Meanwhile an argument can have a true conclusion and still be invalid. Validity is a promise about the connection; soundness is that promise plus true starting points. A good critic asks both, in that order.
Try six arguments and place each one. A few are traps.
Without JavaScript: “All fish are mammals…” is valid and unsound. “All mammals breathe air; whales are mammals; so whales breathe air” is valid and sound. “If it’s Tuesday the library is open; it’s open; so it’s Tuesday” is invalid. “Every swan I’ve seen is white, so the next one is” is not valid (it is induction). “Paris is in France, so 2+2=4” is valid in classical logic, since the conclusion cannot be false. Contradictory premises make any argument valid and unsound.
Two of those answers bother people, and they should. If the conclusion cannot possibly be false, nothing can make the argument go from true premises to a false conclusion, so classical logic calls it valid even though the premise is irrelevant. And from a contradiction, anything follows. Logicians who find this unacceptable built relevance logics and paraconsistent logics. Classical logic remains the default textbook system; the others are live rivals, not cranks.
Take the simplest sentences, ones that are either true or false. Join them with “and”, “or”, “not”, “if…then”. Logic says the truth of the whole depends only on the truth of the parts: each connective is a little machine, a truth function. That is the entire foundation of propositional logic, set out by the Stoic Chrysippus in the 3rd century BCE and rebuilt in symbols by George Boole in the 1840s and Gottlob Frege in 1879.
Set P and Q below and read off every connective at once.
Without JavaScript: “P and Q” is true only when both are true. “P or Q” (inclusive) is false only when both are false. “If P then Q” is false only when P is true and Q false; it is true whenever P is false.
The one that surprises everyone is “if P then Q”. In logic it is the material conditional: false in exactly one case, P true and Q false. So “If the moon is cheese then 2+2=5” counts as true, because the “if” part is false. This is a feature of the simplest tool, not a claim about how “if” works in English. Philosophers have argued about the gap since the Hellenistic period; C. I. Lewis built stricter conditionals in 1912 partly to avoid these “paradoxes”, and Dorothy Edgington and others argue that ordinary conditionals express something closer to conditional probability. You will see that debate again in Philosophy of Language (36/07).
Here is the method a graduate runs in their head. Don’t ask “does this feel right?” Ask: can I describe a situation where every premise is true and the conclusion is false? If you can, the argument is invalid, and that situation is the counterexample. If you honestly can’t, it’s valid. The rule is old: it is how Aristotle’s Prior Analytics (4th century BCE) refutes bad syllogisms, and how Indian logicians tested inferences with cases of “similar” and “dissimilar” instances.
For propositional logic you can do it mechanically: list every combination of true and false for the letters, and look for a row where the premises are all true and the conclusion false. Choose a preset, or type your own formulas using P, Q, R and ~ (not), & (and), | (or), -> (if…then), <-> (if and only if), plus brackets.
Highlighted rows are those where every premise is true. A red row is a counterexample: premises true, conclusion false.
Without JavaScript: modus ponens (P→Q; P; so Q) and modus tollens (P→Q; not Q; so not P) are valid. Affirming the consequent (P→Q; Q; so P) and denying the antecedent (P→Q; not P; so not Q) are invalid; both fail on the same row: P false, Q true.
Notice what the table does not do: it never asks whether P is true. That is why logic is useful on arguments about anything. Notice also the hole in the street-cleaner example: “it rained → wet” plus “wet” leaves open the row where P is false and Q is true, so the argument has a counterexample, and a counterexample is a proof of invalidity.
Aristotle’s first logic was about kinds: “All mammals are warm-blooded; all whales are mammals; so all whales are warm-blooded.” Each statement has one of four forms: A (all X are Y), E (no X are Y), I (some X are Y), O (some X are not Y). A syllogism has two premises and a conclusion sharing three terms: S (the conclusion’s subject), P (its predicate) and M (the “middle” term that appears in both premises but not the conclusion). The middle term can sit in four positions, the four figures. That gives 4 × 4 × 4 × 4 = 256 possible forms.
How many are valid? It depends on a quiet assumption. Modern logic, following Boole and John Venn (his diagrams date from 1880), does not assume that every term has members, so “All unicorns are horned” is true even though there are none. On that reading, 15 of the 256 forms are valid. Medieval logicians, assuming that terms name things that exist, counted 24. The extra nine are valid only if the classes are not empty. The tester below computes both by brute force: it tries all 256 ways of deciding which of the eight Venn regions contain something.
Dots mark regions that contain something in a world where the premises are true and the conclusion false.
Without JavaScript: of 256 syllogistic forms, 15 are valid on the modern reading and 24 on the traditional one. Barbara (AAA, figure 1) is valid. All-A in figure 2 (“All dogs are mammals; all cats are mammals; so all cats are dogs”) is invalid: two things can share a group without sharing each other.
Look at the most common failure. “All dogs are mammals; all cats are mammals; so all cats are dogs.” The middle term (mammals) never says anything about all of itself, so the two premises can both be true while cats and dogs are separate circles inside the mammal circle. Logicians call that the undistributed middle. You will meet it in the wild as “Every extremist reads the news; I read the news; so…”.
A fallacy is a mistake in reasoning that is common enough to have a name. Some are formal, like affirming the consequent, which your checker already refuses. Others are informal: the structure is fine, or loose, but something else goes wrong, such as changing the subject, quietly redefining a word, or putting the conclusion in a premise.
Below are ten invented arguments. Name the problem before you look. Treat this as practice in seeing the move, not in winning arguments with Latin: “that’s a fallacy” is not a rebuttal until you show where the support fails.
Without JavaScript: the ten arguments commit, in order, affirming the consequent, ad hominem, straw man, false dilemma, slippery slope, begging the question, equivocation, post hoc, irrelevant appeal to authority, and hasty generalization.
Two cautions a graduate keeps. First, a fallacy name describes the reasoning, not the conclusion: a bad argument can have a true conclusion, and assuming it is false because the argument was bad is its own mistake, sometimes called the fallacy fallacy. Second, the same surface move can be fine. Citing an expert is reasonable when she is an expert in this field and other experts agree; questioning a witness’s record is relevant when the claim rests on her honesty; a slope can really be slippery when each step is likely. The question is always whether the support is relevant and strong enough.
Valid arguments never say more than their premises already contain. That makes them safe and sterile. Most of what we reason about is different: from the sun rising every day so far to tomorrow’s sunrise; from a wet street and a closed café to “there was a burst main”. These are inductive and abductive (“inference to the best explanation”, a term from C. S. Peirce’s work) arguments. They are not valid or invalid; they are stronger or weaker, and a new fact can overturn them.
David Hume pointed out in 1739–48 that no argument can show the future must resemble the past without assuming it; that is the problem of induction (36/03 and Philosophy of Science, 36/06, return to it). A toy version helps. Laplace’s rule of succession (1774) says that after n successes in a row and no failures, starting from a deliberately blank prior, the chance that the next one also succeeds is (n + 1) / (n + 2). It is a model with an assumption baked in, not the truth about the sun; but it shows evidence as a slider, not a switch.
Laplace’s rule with a uniform prior. It assumes the trials are independent and equally likely to succeed; real swans and real suns need more than that.
Without JavaScript: after 10 successes the chance of an 11th is 11/12, about 92%, but the chance that the next 100 all succeed is 11/111, about 10%.
The sliders make a quiet point: a conclusion that says “always” is much harder to support than one that says “next time”. A universal claim can be killed by one counterexample (the black swan), which is why scientists prefer claims that stick their necks out.
It is tempting to tell this story as Greek and then European. The record is wider. Aristotle’s syllogistic and the Stoic propositional logic are the Greek strands. In China the Later Mohists (around the 4th–3rd centuries BCE) left a set of Canons discussing names, definition and “the way things are alike and different”. In India the Nyāya Sūtra, dated by scholars anywhere from the 6th century BCE to the 2nd century CE, gave a five-part inference (claim, reason, example, application, conclusion: the standard illustration is “there is fire on the hill, because there is smoke, as in a kitchen”), and Dignāga (about 480–540 CE) said a good reason must pass three tests, which Epistemology (36/03) turns into a game. In the Islamic world al-Fārābī and Ibn Sīnā (Avicenna) wrote logic as the tool of all sciences and extended Aristotle to modal and conditional reasoning. Each tradition had to answer the same question: what makes one statement follow from another?
The modern landmark is symbolic: Boole’s algebra of logic (1847, 1854), Frege’s Begriffsschrift (1879), which brought quantifiers (“for all”, “there is”) into the system, and Gödel’s results in 1930–31 on what such systems can and can’t prove. Claude Shannon’s 1937 master’s thesis showed that Boolean algebra describes relay switching circuits, which is why logic is also, quietly, the bottom layer of every computer. Foundations: Sets, Logic & Infinity (34/71) and the library’s Gödel’s Incompleteness Theorems (12/08) pick that thread up.
Logic is the study of what follows from what. An argument is valid if no case makes its premises true and its conclusion false, sound if it is valid and the premises are true; you test validity by looking for that one case. Fallacies are shapes that feel convincing without the support, and most everyday reasoning is not deduction but a matter of degree.
Try this: take one argument from a newspaper or a chat today, write it as “premises, therefore conclusion”, and try to describe a world where the premises are true and the conclusion false.
Sources. Aristotle, Prior Analytics; Stoic logic (Chrysippus) and Mohist Canons per the Stanford Encyclopedia of Philosophy entries on ancient logic, Mohism and Chinese logic; Nyāya Sūtra and Dignāga, SEP “Epistemology in Indian Philosophy” and “Dignāga”; Boole 1847 and 1854; Frege 1879; Venn 1880; Gödel 1930–31; Shannon 1937; Hume 1739–40 and 1748; Laplace 1774; Lewis 1912 on strict implication; Edgington on conditionals (SEP “Indicative Conditionals”). The counts of 15 and 24 valid syllogistic forms were computed for this piece by brute force and agree with standard accounts (SEP “Medieval Syllogistic”, “Venn diagrams”). The ten fallacy examples are invented.