Logic

• 1. Introduction

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  Logic is the study of correct reasoning, and it starts with something as ordinary as a Sunday. Take the premises "it's Sunday" and "if it's Sunday then I don't have to work." From those two claims, a conclusion arrives on its own: "I don't have to work." That tiny move, from what you accept to what you must therefore accept, is the whole subject in miniature. Logic asks a single stubborn question about moves like this. Do the premises actually support the conclusion, or only seem to?

  The word itself comes from the Greek logos, which can be translated as reason, discourse, or language. For over two thousand years, one answer to the question dominated the Western world. Then, in the late 19th century, a handful of mathematicians rebuilt the whole field. They asked what makes an argument valid no matter what it is about. They asked whether a sentence could be true purely because of its shape. They asked how a machine of pure rules could grind out conclusions without ever touching the world. The answers reshaped philosophy, mathematics, computer science, and linguistics, and they raised harder questions about whether the laws of logic are known before all experience or discovered inside the structure of reality itself.

• 2. Form Over Content

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  Formal logic replaces concrete expressions with abstract symbols, and that single trick is what gives it power. By studying the logical form of an argument instead of its words, formal logic becomes topic-neutral. It cares about the abstract structure, not the subject. Consider the rule called modus ponens. Any argument of the form "(1) p, (2) if p then q, (3) therefore q" is valid, no matter what p and q stand for. Knowing that it has just rained and that after rain the streets are wet, one can deduce that the streets are wet, using nothing but the shape.

  A deductively valid argument is one whose premises guarantee the truth of its conclusion. It is impossible for the premises to be true and the conclusion false. Strikingly, the actual truth of the parts does not matter. The argument "all frogs are mammals; no cats are mammals; therefore no cats are frogs" is valid, because the conclusion follows necessarily, even though one premise is plainly false. Validity is about the path, not the destination.

  Alfred Tarski offered an influential view that deductive arguments share three features. They are formal, depending only on the form of premises and conclusion. They are a priori, needing no sense experience to confirm. And they are modal, holding by logical necessity for the given propositions. That third feature gives deduction its reputation for being uninformative, since the conclusion cannot add anything not already present in the premises. But that charge would render most of mathematics uninformative too. The escape is a distinction between surface and depth information. Deductive inferences are uninformative at depth, yet richly informative at the surface, making implicit information explicit, which is exactly what a mathematical proof does.

• 3. Reasoning That Risks

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  Many arguments in everyday life and the sciences do not guarantee their conclusions at all. These are ampliative arguments, and their conclusions contain genuinely new information not found in the premises. The cost of that novelty is steep. The premises make the conclusion more likely without ensuring it, so an ampliative conclusion can be false even when every premise is true. This ties them to non-monotonicity and defeasibility. A new observation may force you to retract a conclusion you had already drawn.

  Induction, in the narrow sense, is a form of statistical generalization. From many individual observations that all show a pattern, you infer a general law. Seeing black raven after black raven, you conclude that all ravens are black. Watching gray elephant after gray elephant, you infer that all elephants are gray, or that the next unseen elephant will be gray too. The terminology here is famously inconsistent. James Hawthorne uses "induction" for all non-deductive arguments, while Igor Douven stipulates that inductive inferences rest only on statistical considerations, and Leo Groarke allows conductive arguments as a further type.

  Abduction is the inference to the best explanation. A doctor concludes a patient has a certain disease because that diagnosis best explains the symptoms. Find a plate with breadcrumbs in the kitchen in the early morning, and you infer that your house-mate had a midnight snack and was too tired to clean up. The trick is the word "best." The conclusion that a burglar broke in, got hungry, and ate a snack would also explain the crumbs. It is rejected because it is not the most likely explanation, and explaining the evidence is not enough on its own.

• 4. When Reasoning Fails

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  "It is sunny today; therefore spiders have eight legs" is a fallacy, even though spiders do have eight legs. The flaw is never that the conclusion is false. The flaw is in the reasoning that leads there. John Stuart Mill gave a more restrictive definition, adding that fallacies must also appear correct, which separates genuine fallacies from careless mistakes and explains why people keep falling for them. That appeal to appearances stays controversial, since appearances belong to psychology rather than logic and differ from person to person.

  Formal fallacies hide their error in the form of the argument. Denying the antecedent is one: "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore Othello is not male." The structure itself is broken. Informal fallacies, by contrast, draw their error from content or context, and the academic literature catalogs a great variety of them. They split into fallacies of ambiguity, of presumption, and of relevance. A fallacy of ambiguity trades on slippery words: "feathers are light; what is light cannot be dark; therefore feathers cannot be dark." A false dilemma excludes viable options, as in "you are either with us or against us; you are not with us; therefore, you are against us."

  The rules that catch fallacies come in two kinds, and a comparison to chess makes the difference vivid. Definitory rules determine which inferences are allowed, just as the definitory rules of chess dictate that bishops may only move diagonally. Strategic rules describe how to reach a goal, like controlling the center and defending one's king to win. Some argue that logicians should give more emphasis to strategic rules, since they are what make reasoning effective rather than merely permitted.

• 5. Machines Of Pure Rules

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  A formal system of logic consists of a formal language together with a set of axioms and a proof system. Axioms are statements accepted without proof, used to justify everything else. The formal language has an alphabet of basic symbols and syntactic rules for arranging them into well-formed formulas. In propositional logic, those rules decide that a conjunction needs terms on both sides, so one string counts as well-formed and another does not.

  A proof system is a collection of rules for building formal proofs, defined purely by the syntactic form of formulas, never by their content. The classical rule of conjunction introduction lets you derive a conjunction from its two parts. Apply such rules in sequence and you get a mechanical procedure for generating conclusions, whether through natural deduction or sequent calculi. A semantics maps the expressions to their denotations, which in many systems are simply truth values, and a premise entails a conclusion when the conclusion is true whenever the premise is.

  Two properties test whether syntax and semantics agree. A system is sound when its proof system can never derive a conclusion that is not semantically entailed, so it cannot reach false conclusions. A system is complete when its proof system can derive every conclusion that is semantically entailed, so it reaches every true one. Together, soundness and completeness describe a system whose notions of validity and entailment line up perfectly, a standard that first-order logic famously meets.

• 6. Classical And Its Rivals

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  For over two thousand years Aristotelian logic was treated as the canon of logic in the Western world. In its narrow sense it is term logic, or syllogistics, built around a form of argument with three propositions: two premises and a conclusion. Each proposition has a subject, a predicate, and a copula connecting them, as in "Socrates is wise." The premises share terms with each other and with the conclusion, giving three terms called major, minor, and middle. "All men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. "All cats are mortal; Socrates is mortal; therefore Socrates is a cat" is not. Notably, Aristotelian logic has predicates only for simple properties, with none for relations between entities.

  Classical logic encompasses propositional logic and first-order logic, and is "classical" because it rests on intuitions most logicians share: the law of excluded middle, double negation elimination, the principle of explosion, and the bivalence of truth. Propositional logic builds formulas from atomic propositions and connectives, but cannot reach inside a proposition. First-order logic can, using singular terms, predicates, and quantifiers like "some" and "all." Built first to analyze mathematics, classical logic skips concepts central to philosophy, such as necessity versus possibility, obligation versus permission, and the relations between past, present, and future.

  Extended logics keep the classical intuitions and add new vocabulary. Alethic modal logic introduces symbols for the possible and the necessary, with a rule that whatever is necessary is also possible. Deontic logic carries those operators into ethics as obligation and permission, temporal modal logic into time, and epistemic modal logic into the gap between knowing and merely believing. Higher-order logics instead extend quantification to predicates, gaining expressive power useful for mathematics at a cost to their meta-logical properties.

  Deviant logics reject classical intuitions outright, standing as rivals rather than supplements. Intuitionistic logic drops double negation elimination and the law of excluded middle, grounding truth in verification by proof, and it is prominent in constructive mathematics. Jan Łukasiewicz and Stephen Cole Kleene each proposed ternary logics with a third, indeterminate truth value, while fuzzy logics allow infinitely many degrees of truth between 0 and 1. Paraconsistent logics avoid the principle of explosion so that not everything follows from a contradiction, a stance often motivated by dialetheism, whose influential contemporary proponent is Graham Priest, with similar views ascribed to Georg Wilhelm Friedrich Hegel.

• 7. Logic In The World

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  Claude Shannon showed how Boolean logic can be used to understand and implement computer circuits, and computational logic grew from exactly that kind of crossing. Electronic logic gates represent truth values by voltage levels, so a logical function can be computed by feeding voltages into a circuit and measuring the output. Automatic theorem provers build proofs step by step without human intervention, and logic programming languages such as Prolog, based on predicate logic, express facts as formulas and draw inferences from them.

  Formal semantics uses the tools of symbolic logic and mathematics to give precise theories of natural language meaning, usually in terms of truth conditions. Its guiding assumption is the principle of compositionality: the meaning of a complex expression is fixed by the meanings of its parts and how they combine, so "walk and sing" depends on "walk" and "sing." Many such theories rely on model theory and set theory, interpreting "walk" as the set of all walking individuals in a model. Richard Montague and Barbara Partee were early influential theorists, focusing on English.

  Logic also turns its instruments on itself. Metalogic studies the properties of formal systems, asking which formulas can be proven, whether an algorithm can find each proof, and whether systems are complete, sound, consistent, decidable, and how expressive they are. Mathematical logic, in its restricted sense, studies logic within mathematics across model theory, proof theory, set theory, and computability theory. Computability theory uses tools like Turing machines to ask whether a given problem can be solved by an algorithm at all.

  The epistemology of logic asks how we know an argument is valid in the first place. The traditionally dominant view holds that logical understanding is knowledge a priori, grasped by a special mental faculty or true by linguistic convention. Hilary Putnam and Penelope Maddy object, arguing logical truths depend on the empirical world. They point to a startling claim: that certain insights of quantum mechanics refute the principle of distributivity in classical logic, offering an empirical argument that quantum logic should replace it.

• 8. Many Births Of Logic

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  Logic was developed independently in several cultures during antiquity, and Aristotle stands among its earliest contributors, building term logic in his Organon and Prior Analytics. He introduced the hypothetical syllogism and temporal modal logic, along with inductive logic and concepts such as terms, predicables, syllogisms, and propositions. His system remained in wide use in the West until the early 19th century. In the Islamic world, Ibn Sina, known as Avicenna, founded Avicennian logic, which displaced Aristotle's system and influenced Western writers such as Albertus Magnus and William of Ockham. Ibn Sina developed an original temporally modalized syllogistic and used methods of agreement, difference, and concomitant variation that are critical to the scientific method. Fakhr al-Din al-Razi criticized Aristotelian syllogistics and built an early system of inductive logic that foreshadowed John Stuart Mill.

  Boethius shaped the medieval West by translating Aristotle into Latin and writing his own logic textbooks. In 1323, William of Ockham released the Summa Logicae, a comprehensive treatise on types of propositions and their truth conditions. Far to the east, Chinese philosophy produced the School of Names, where Gongsun Long proposed the white horse paradox defending the thesis that a white horse is not a horse, alongside the Mohists who tied logic to ethics. In India, the schools of Nyaya, Buddhism, and Jainism pursued logic inside epistemology and theories of argumentation, treating inference as a source of knowledge, or pramāṇa. The later Navya-Nyāya school reached results resembling modern logic, including a distinction like Gottlob Frege's between sense and reference.

  Many see Gottlob Frege's Begriffsschrift as the birthplace of modern logic, with Gottfried Wilhelm Leibniz's dream of a universal formal language as a forerunner. George Boole invented Boolean algebra, Charles Peirce developed the logic of relatives, and Alfred North Whitehead and Bertrand Russell condensed these insights in Principia Mathematica. The grand ambition was logicism, reducing mathematics to logical tautologies. That program failed twice over, crippled first by Russell's paradox striking Frege's Grundgesetze, then by Gödel's incompleteness theorems defeating Hilbert's program. What survived was first-order logic, whose analytical generality formalized mathematics, drove set theory, and made Alfred Tarski's model theory possible.