JOHN VENN

the visual logic (1880) — the Venn diagram: a logical statem

LOGIKĒ · the visual logic (1880) — John Venn, who gave logic its picture: overlapping circles whose regions are every combination of true/false for the terms, so a logical statement becomes a pattern of shaded zones and a syllogism becomes something you can SEE. Live: a 3-set Venn diagram — shade premises, read the conclusion off the diagram. Honest: Euler drew circles before him; Venn systematized them into a complete method, and was also a probabilist (the frequency interpretation). David Lee Wise, two-layer honest.

John Venn — the picture of logic ◄ UD0 ← Jevons LOGIKĒ · the visual logic LOGIKĒ · the visual logic · 1880 John Venn 1834 – 1923 · Hull & Cambridge · logician & probabilist · Gonville & Caius College Boole and Jevons made logic a matter of combinations — every possible truth-assignment of the terms. John Venn made those combinations something you can see . Draw overlapping circles for the terms, and the plane divides into regions , one for each combination (in A and B but not C; in all three; in none). A logical statement becomes a pattern of shading — to say "no A are B," you simply blacken the lens where A and B overlap — and a syllogism becomes a picture: shade what the premises forbid, and the conclusion is whatever the diagram now shows. It is the most successful piece of notation in the history of logic — the one diagram that escaped the seminar room and turned up on whiteboards, in courtrooms, and on T-shirts. ✓ STRONG A complete, correct method. The Venn diagram is a sound and general visual system for class logic: every region is a real combination, every categorical statement is a shading, and syllogistic validity becomes literally visible. Rigorous, not just a teaching aid. ◐ EULER CAME FIRST He systematized, didn't originate. Leonhard Euler drew logical circles a century earlier — but Euler's diagrams couldn't show "maybe empty" regions. Venn's innovation was to draw all regions always, then shade — making the method complete. ◔ BREAKS DOWN PAST 3–4 SETS A picture has limits. Three circles give all 8 regions cleanly; four need ellipses; beyond that the diagram becomes unreadable. Venn diagrams illuminate small logic beautifully and scale to nothing — the eye is not a general computer. I · Shade the premises, see the conclusion Three circles, A B C , carve the plane into eight regions — every combination of in/out. A categorical premise shades regions: "All A are B" greys out the part of A outside B (it must be empty); "No B are C" greys the B-C overlap. Toggle the classic syllogism premises and watch the diagram do the reasoning — the conclusion is simply what the un-shaded regions allow. Shading = "this region must be empty." When the premises have greyed out every region where something could be A-but-not-C, you can see that all A are C — the conclusion isn't deduced in symbols, it's looked at . ⚑ This is logic made spatial: the same elimination [[william-stanley-jevons]]'s piano did mechanically, Venn did visually — and unlike the piano, the picture is so natural that almost everyone who has ever sorted two ideas has drawn one without knowing whose it was. "I began at once to think whether [the diagrams] might not be made of more general application." — John Venn, on improving Euler's circles II · Why the picture worked ALWAYS DRAW EVERY REGION Euler's circles only showed the relations that held, so they couldn't represent uncertainty. Venn's fix: always draw all overlaps (all 2ⁿ regions), then shade the empty ones. A region left blank means "possibly occupied" — the diagram can now say "don't know." SHADING = ELIMINATION A universal premise blackens regions (they're empty); a particular premise marks a region as occupied. Validity becomes visible: a conclusion holds iff the picture already shows it. Class logic, rendered as geometry. Venn's diagram is the rare piece of mathematics that became folk knowledge . The reason it spread is the reason it matters here: it makes the structure of [[george-boole]]'s algebra perceptible — the eight regions of a three-circle diagram are literally the eight rows of a three-variable truth table, the eight keys of [[william-stanley-jevons]]'s piano, the eight minterms a logic designer juggles. Shade-to-eliminate is the same move as a Karnaugh map, the tool every circuit engineer uses to simplify gates. Venn didn't add a theorem to logic; he added an organ of perception for it — and a notation that good lets people reason correctly without ever knowing they're doing logic at all. ⚑ The truth table you can look at. III · The diagram, and the man behind it Gate kept on. The contribution is real, complete, and astonishingly durable — the Venn diagram is a sound general method for class logic, not a mere mnemonic, and it is arguably the most widely recognised object in all of mathematics. The honest qualifications. First, priority : Venn did not invent logical circles — Leonhard Euler used them a century earlier, and others before him; Venn's genuine innovation was to draw every region always and use shading , which made the diagrams able to represent ignorance and thus complete (he was generous about the debt, calling them "Eulerian circles" he had improved). Second, scope : the method is gorgeous for two or three classes, awkward at four (you need ellipses), and unreadable beyond — the picture illuminates small logic and scales to nothing, which is exactly why symbolic methods, not diagrams, run real systems. The man: John Venn was a Cambridge logician and probability theorist (an early champion of the frequency interpretation of probability), an ordained priest who left the clergy over doctrine, and a skilled builder of machines — he made a cricket bowling machine that clean-bowled a star of the Australian team. He gave logic the one thing the symbols never could: a face. ⚑ [[william-stanley-jevons]] ← · next → William of Ockham (the medieval root). JOHN VENN · LOGIKĒ · the visual logic · two-layer honest live 3-set Venn diagram (eight regions = every combination; universal premises shade regions empty; the conclusion is read off the picture — verified All A→B + All B→C shows All A→C) the Venn diagram (1880) = the complete visual method for class logic · shading = elimination · the eight regions = a 3-variable truth table · also a frequency-probabilist ← Jevons · next → William of Ockham (the medieval root & the razor) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


ERNST SCHRÖDER

the Boolean cleanup (1890–1905) — the canonical Algebra of L

LOGIKĒ · the Boolean cleanup (1890–1905) — Ernst Schröder, who gathered the scattered work of Boole, Peirce, and Jevons into the definitive 'Algebra of Logic': the canonical lattice axioms (meet, join, complement, distributivity, duality) that we still teach as Boolean algebra. Live: the Boolean lattice on the subsets of a set — click two nodes, see their meet and join, and watch the algebra's laws hold. Honest: a great consolidator more than an originator; the Schröder–Bernstein theorem on set sizes also bears his name. David Lee Wise, two-layer honest.

Ernst Schröder — the algebra of logic, finished ◄ UD0 ← Venn LOGIKĒ · the Boolean cleanup LOGIKĒ · the Boolean cleanup · 1890–1905 Ernst Schröder 1841 – 1902 · Mannheim & Karlsruhe · author of the Vorlesungen über die Algebra der Logik By the 1880s the algebra of logic was a brilliant mess: [[george-boole]] had started it, [[william-stanley-jevons]] tidied it, [[charles-peirce]] deepened it with relations — but it lived in scattered papers, in clashing notations, half-finished. Schröder spent his life building the cathedral : a vast, systematic treatise (the Vorlesungen , 1890–1905) that gathered all of it into one rigorous structure — the canonical laws of meet , join , and complement , the principle of duality (swap ∧↔∨ and 0↔1 and every theorem still holds), the whole thing recognisable today as a Boolean algebra and, more generally, a lattice . He is the man who turned Boole's insight into a finished branch of mathematics. ✓ STRONG The definitive synthesis. Schröder's Algebra of Logic systematised Boolean algebra and the calculus of relations into the canonical axioms and duality we still use; his treatise was the standard reference for decades. Rigorous and foundational. ◐ CONSOLIDATOR, NOT ORIGINATOR He finished others' work. The core ideas are Boole's, Peirce's, Jevons's; Schröder's genius was completeness and rigour, not a new spark. And the algebraic tradition he perfected was about to be eclipsed by [[gottlob-frege]]'s very different logic. ◔ A SECOND CLAIM TO FAME Also a set-theory name. The Schröder–Bernstein theorem — if each of two sets injects into the other, they have the same size — bears his name (his proof had a gap; Bernstein fixed it). A reminder he worked across foundations. I · The Boolean lattice — meet, join, and the laws Here is the structure Schröder codified, in its cleanest example. Take the subsets of {a, b, c}, ordered by inclusion — eight nodes, drawn as a lattice (an edge goes up when you add one element). Two operations make it an algebra: the meet (∧, greatest common part = intersection) and the join (∨, least common container = union). Click any two nodes; the diagram finds their meet and their join. Every law Schröder axiomatised — commutativity, absorption, distributivity, complement — holds here, visibly. click two nodes to pick A and B · top = the whole set (1) · bottom = ∅ (0) This little diagram is a complete Boolean algebra : meet and join obey Schröder's laws, every element has a complement (its set-complement), and the whole thing is self-dual — flip it upside down, swap meet↔join and 0↔1, and it maps onto itself (verified: distributive + absorption hold throughout). ⚑ That is exactly the structure inside every logic circuit and every database query optimiser — and it was Schröder who pinned down its axioms once and for all. The duality he made central is why an engineer can convert any AND-network into an OR-network for free. "The calculus of logic … is to become an organon of deductive thought." — in the spirit of Schröder's Vorlesungen II · The cathedral, and the turn it missed THE ALGEBRA OF LOGIC A complete axiomatic treatment of Boolean algebra and the calculus of relations, with the principle of duality at its heart — the canonical form in which Boole's idea entered mathematics permanently. THE FORK IN THE ROAD Schröder perfected the algebraic tradition (logic as equations). At the same moment [[gottlob-frege]] was building the quantificational tradition (logic as a formal language). The future went Frege's way — but the algebra stayed essential. Schröder sits at a fascinating hinge. He brought the algebra of logic — Boole's line — to its absolute peak just as that whole approach was about to be overshadowed by [[gottlob-frege]]'s and [[bertrand-russell]]'s logic of quantifiers and functions, which proved better for the foundations of mathematics. It would be easy to read Schröder as the magnificent end of a dead branch. But the branch wasn't dead: the algebraic, equational view of logic he perfected is exactly the one that came roaring back through [[claude-shannon]] and into computer science — Boolean algebra, lattice theory, and the calculus of relations are the daily mathematics of circuits, databases, and program logic. He didn't pick the losing side; he finished the side that the machines would need. ⚑ The man who handed the future a completed algebra. [[john-venn]] ← · next → the modern foundations. III · The systematiser Gate kept on. The honest measure of Schröder is consolidation of the highest order . His three-volume Vorlesungen über die Algebra der Logik (1890–1905) is a genuine landmark — it gathered the fragmentary results of [[george-boole]], [[augustus-de-morgan]], [[charles-peirce]], and [[william-stanley-jevons]] into a single rigorous edifice, fixed the canonical laws and the duality principle, and developed Peirce's calculus of relations more fully than Peirce had; for decades it was the reference on algebraic logic. The honest qualification is that Schröder was a systematiser, not an originator — the deep ideas were others', and his life's work perfected an approach that [[gottlob-frege]]'s quantificational logic was simultaneously rendering secondary for the foundations of mathematics (a fact Schröder, who debated Frege, did not fully accept). He also lent his name to the Schröder–Bernstein theorem in set theory (his own proof was flawed; Felix Bernstein completed it), and worked on iteration and functional equations. He died in 1902 with his treatise unfinished. To build the cathedral that holds another's vision is its own kind of greatness — and the algebra he completed is the one running in [[psephos-processors-domain]]. ⚑ [[john-venn]] ← · the lineage runs on to the modern foundations. ERNST SCHRÖDER · LOGIKĒ · the Boolean cleanup · two-layer honest live Boolean lattice (P({a,b,c}) ordered by inclusion; click two nodes → meet ∧ & join ∨; the axioms hold — distributive + absorption verified, a Boolean algebra) the Vorlesungen über die Algebra der Logik (1890–1905) = the canonical synthesis of Boolean algebra & the calculus of relations · duality · the Schröder–Bernstein theorem ← Venn · the lineage runs on to the modern foundations LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


GOTTLOB FREGE · IX

the greatest leap since Aristotle — quantifiers ∀/∃, relatio

LOGIKĒ IX — Gottlob Frege (1848–1925). The greatest advance in logic since Aristotle. In the Begriffsschrift (1879) he gave logic QUANTIFIERS — ∀ 'for all' and ∃ 'there exists' — and analysed statements as functions of arguments, so logic could finally express the sentences of mathematics: 'every number has a successor', and the relations and nested quantifiers term logic never could. This is first-order logic, the logic of modern mathematics and computer science. Live quantifiers, relations, and the order that matters (∀x∃y ≠ ∃y∀x). Then the Russell paradox that destroyed his logicism on the eve of publication. Honest: ignored in his lifetime; his late diaries record repugnant views. David Lee Wise, two-layer honest.

Gottlob Frege — for all, there exists ◄ UD0 ← De Morgan LOGIKĒ · IX LOGIKĒ · IX · for all, there exists Gottlob Frege 1848 – 1925 · Wismar & Jena · author of the Begriffsschrift · the founder of modern logic For two thousand years logic was stuck where Aristotle left it: it could say "all men are mortal," but not "every number has a larger one." The gap was quantity — the ability to say for all and there exists , and to chain them. In 1879, in a thin book almost no one read, Frege supplied it: quantifiers , bound variables , and the analysis of a statement as a function applied to arguments. At one stroke, logic could finally express the actual sentences of mathematics — and, much later, the queries of a database and the types of a program. Modern logic begins on that page. ✓ STRONG Predicate logic — the great leap. Quantifiers (∀, ∃) and the function/argument analysis are the biggest advance since Aristotle . First-order logic is the logic of all modern mathematics and computer science. ◐ COLLAPSED His grand program failed. Frege's logicism — that mathematics reduces to pure logic — was wrecked in 1902 by Russell's paradox , which showed his foundational system inconsistent. ◔ IGNORED, THEN OWNED Unread in his time. Almost no one noticed him until Russell and Wittgenstein. (And honestly: his late diaries record vicious antisemitic, anti-democratic views — the logic stands; the man does not.) I · For all, and there exists Here is the move Aristotle and Boole could not make. Below is a small world of objects, each red or blue (click to flip one). A quantifier lets a single statement range over the whole world: ∀x "for every x…", ∃x "for some x…". Watch the quantified sentences decide their truth as you change the world — something term logic, locked to whole categories, simply cannot do. click a dot to flip its colour a domain of objects · red / blue · the quantified sentences read the whole domain II · Relations, and why order matters The deeper gift is relations and nesting . Click the grid to set a relation R(x,y) — "x → y". Then two sentences that use the same symbols in a different order say utterly different things: ∀x ∃y R(x,y) — "everyone relates to someone" — versus ∃y ∀x R(x,y) — "someone is related to by everyone." "Everyone has a mother" is not "someone is everyone's mother." No logic before Frege could even tell them apart. click cell (row → col) to toggle the arrow R(x,y) · 4 objects ∀x ∃y R(x,y) every row has an arrow F ∃y ∀x R(x,y) some column is all arrows F Set every row to have one arrow and the first is true; that almost never makes the second true. Swapping ∀ and ∃ is not a rephrasing — it changes the claim. Capturing that is the whole reason mathematics needed Frege. "Every good mathematician is at least half a philosopher, and every good philosopher at least half a mathematician." — Gottlob Frege III · The paradox that broke the foundation Frege spent decades on one dream: to derive all of arithmetic from pure logic alone — logicism . As the second volume of his Grundgesetze was at the printer in 1902, a letter arrived from a young Bertrand Russell with a single, fatal question. Consider the set of all sets that do not contain themselves . Does it contain itself? Try to answer. R = { all sets that are not members of themselves }. Is R ∈ R? assume R ∈ R assume R ∉ R Pick one — and watch it refute itself. Gate kept on. Russell's paradox didn't dent Frege's logic — quantifiers and predicate logic are untouched and immortal. It killed his foundations : the unrestricted "set of all things satisfying a property" his system allowed is self-contradictory. Frege, with rare honesty, appended a note saying a cornerstone of his edifice had given way, and never really recovered the program; arithmetic was later refounded on type theory and on the ZFC axioms that forbid such sets. Two further honest things: he was almost entirely ignored in his lifetime — modern logic flows from him only because Russell, Wittgenstein and Carnap carried it forward — and his late diaries (1924) record virulent antisemitic and anti-democratic views. The greatest logician since Aristotle, whose foundations cracked, who went unread, and whose character at the end was repugnant. The work is permanent; the rest is told plainly. GOTTLOB FREGE · LOGIKĒ IX · two-layer honest live quantifiers (∀x, ∃x over a domain) · relations & nested order (∀x∃y vs ∃y∀x — different claims) · Russell's paradox (R∈R ⟺ R∉R, the contradiction that ended logicism) Begriffsschrift (1879) = the birth of predicate / first-order logic · logicism wrecked by Russell 1902 · ignored in life; foundation of modern math & CS ← De Morgan · next → Gödel (the limits) · Turing (computability) · Shannon (the gate) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


KURT GÖDEL · X

the limits — true statements no system can prove; numbering

LOGIKĒ X — Kurt Gödel (1906–1978). In 1931 he proved the incompleteness theorems: any consistent formal system strong enough for arithmetic contains true statements it cannot prove, and cannot prove its own consistency. His method — Gödel numbering, encoding statements as numbers so a system can talk about itself — is the rigorous root of 'code as data'. Live Gödel numbering and the self-referential sentence 'this statement is not provable'. It ended Hilbert's program and Leibniz's full dream: reasoning can be mechanised, but not completed. Honest: it's a precise result about formal provability — and the most misquoted theorem in history. David Lee Wise, two-layer honest.

Kurt Gödel — the limits of proof ◄ UD0 ← Frege LOGIKĒ · X LOGIKĒ · X · the limits of proof Kurt Gödel 1906 – 1978 · Brno & Princeton · the Institute for Advanced Study · Einstein's closest friend there Leibniz dreamed of a calculation that settles every question; Hilbert turned it into a program — find one complete, consistent, mechanical system that proves all of mathematics. In 1931 a shy twenty-five-year-old ended that dream forever. Gödel showed that any system strong enough to do arithmetic, if it is consistent, must contain statements that are true but that it cannot prove — and can never prove its own consistency. His trick was to make the system talk about itself by turning its sentences into numbers — the same move that lets a program be data, and that the whole of computing would be built on. ✓ STRONG Proven, and profound. The two incompleteness theorems are rigorous and among the deepest results in all of mathematics — true-but-unprovable statements exist; no consistent arithmetic proves its own consistency. ◐ LIMITED SCOPE It bounds formalism, not truth. It applies only to consistent systems strong enough for arithmetic, and is about formal provability — not knowledge, certainty, or whether math "works." It does not say math is broken. ◔ MOST MISQUOTED Beware the pop version. It's endlessly abused for relativism, "no absolute truth," theology and consciousness. Nearly all such uses are wrong — it's a precise theorem, not a mood. I · Gödel numbering — a statement becomes a number The key idea, and it's gorgeous: give each symbol a code, then encode a whole formula as a single number by raising successive primes to those codes and multiplying. By unique factorisation, that number decodes back to exactly one formula. So every statement is a number — which means arithmetic, the thing the system reasons about, can secretly encode statements about the system itself . Build a formula and watch it become an integer. clear tap symbols to build a formula 0 = 0 A formula and a number are now interchangeable — this is the rigorous form of Leibniz's prime trick, and the exact reason code can be data : a program is just a (very large) number. Stored-program computing lives here. II · The sentence that defeats proof Now point the encoding at itself. Using numbering, Gödel built a sentence G that, decoded, says precisely: "G is not provable in this system." A statement about provability — which is now just arithmetic — that refers to its own number. Then ask whether the system can prove it. G says: "G cannot be proved in this system." Can the system prove G? suppose it proves G suppose it can't Pick one — and corner the system. Both doors lead the same place: a consistent system has true statements it cannot reach. Truth outruns proof. "The more I think about language, the more it amazes me that people ever understand each other at all." — Kurt Gödel III · The two theorems, the fall, the man FIRST INCOMPLETENESS Any consistent formal system rich enough for arithmetic has true statements it cannot prove . Completeness and consistency cannot both be had. SECOND INCOMPLETENESS Such a system cannot prove its own consistency — no formal foundation can certify itself from the inside. This detonated Hilbert's program — and with it the formal heir of Leibniz's "Calculemus." There is no single machine that decides all mathematical truth. Five years later Turing would recast the same diagonal as the halting problem : not only is truth unreachable, some questions are uncomputable. Reasoning can be mechanised — that became your computer — but it cannot be completed . Gate kept on. Hold two things at once. The theorems are airtight and shattering — among the summits of human thought. And they are the most abused results in history : incompleteness does not say "nothing can be proven," "there is no truth," or "everything is uncertain." It is narrow and exact — about formal provability inside consistent systems that can express arithmetic — and the Gödel sentence is in fact true (we can see it from outside); it simply can't be proved inside . Mathematics did not break; it learned its own horizon. The man: Gödel escaped Austria via the Trans-Siberian railway, became Einstein's daily walking companion at Princeton, found a supposed loophole for a dictatorship in the U.S. Constitution at his own citizenship hearing, and — gripped by a paranoid fear of poisoning — would eat only food his wife prepared. When she was hospitalised in 1977, he stopped eating, and died of starvation at 71. The logician who proved the limits of certainty was undone by an excess of doubt. KURT GÖDEL · LOGIKĒ X · two-layer honest live Gödel numbering (formula → ∏ prime_i^code_i via BigInt; a statement IS a number) · the self-referential sentence "G is not provable" (both branches force incompleteness) Incompleteness theorems (1931) ended Hilbert's program & Leibniz's full dream · numbering = the rigorous root of "code as data" · the most misquoted theorem in history ← Frege · next → Turing (computability, the machine, the halting problem) · Shannon (the gate) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


ALAN TURING · XI

the machine — computation defined, the universal computer, t

LOGIKĒ XI — Alan Turing (1912–1954). In 1936 he defined the Turing machine: a head, an infinite tape, a tiny state table — the simplest thing that can compute anything computable. One such machine can simulate any other: the universal machine, the stored-program computer on paper, a decade before the hardware. He then proved the halting problem is undecidable — no program can decide, in general, whether an arbitrary program halts — recasting Gödel's diagonal as a fact about computation. Live: a running Turing machine (3-state busy beaver, binary increment) and the halting-problem diagonal. Honest: a precise impossibility result, often misread; and the man, prosecuted for being gay in 1952, dead at 41. David Lee Wise, two-layer honest.

Alan Turing — the machine & the limits of computation ◄ UD0 ← Gödel LOGIKĒ · XI LOGIKĒ · XI · the machine & the limits of computation Alan Turing 1912 – 1954 · London & Cambridge & Bletchley Park · "On Computable Numbers", 1936 · founder of computer science Gödel proved the dream of a complete formal system impossible. Turing, a 23-year-old in Cambridge, asked the next question and answered it with a machine . To pin down what "mechanically computable" even means, he imagined the simplest possible computer: a head reading a tape , one square at a time, following a tiny table of rules. That stripped-down device — the Turing machine — can compute anything that can be computed at all. And one special machine can simulate every other: the universal machine , the stored-program computer on paper, a decade before the electronics existed. Then he proved its limit: no machine can decide, in general, whether an arbitrary program will ever halt . ✓ STRONG Foundational and exact. The Turing machine, universality, and the undecidability of the halting problem are rigorous results — the bedrock definition of computation and the literal blueprint of the stored-program computer. ◐ A MODEL, NOT THE BRAIN It defines computability, not mind. The Church–Turing thesis (this captures all mechanical computation) is a well-supported thesis , not a theorem; and "a machine can think" was Turing's conjecture/test , not something he proved. ◔ OFTEN MISREAD Undecidable ≠ "computers can't". The halting problem is a precise general -case impossibility; plenty of specific programs are easy to analyse. And the "Turing test" is a thought experiment, not a verdict on consciousness. I · A running machine — computation, stripped to the bone Here is the whole of a computer: an infinite tape of cells (each 0 or 1), a head on one cell, a state , and a rule table — "in this state reading this symbol: write a symbol, move left or right, go to that state." Nothing else. Pick a program and run it. The busy beaver is a tiny machine that writes as many 1s as it can before halting — its growth becomes literally uncomputable, the bridge to Module II. The incrementer adds 1 to a binary number. Watch the head crawl the tape; the active rule lights up. 3-state busy beaver binary increment (7+1) step ▸ run ▶ reset ↺ state A step 0 1s on tape 0 Six cells, three states, and it halts after writing six 1s — yet decide in advance how many 1s an n -state beaver writes, and you are computing the uncomputable. This little device is the exact thing John von Neumann turned into the stored-program architecture every computer still uses. II · The halting problem — a question no machine can answer Could one master program H take any program P and input x and always correctly answer "does P halt on x, or loop forever?" Turing proved: no — and he did it with Gödel's diagonal trick made mechanical. Suppose H exists. Build a contrary machine D : D runs H on the question "does this very machine halt?" and then does the opposite of H's answer. Now ask D to judge itself. D asks H : “will D(D) halt?” — then D does the opposite . if H says “halts” → D loops · if H says “loops” → D halts suppose H predicts “halts” suppose H predicts “loops” Pick H's prediction about D(D) — and watch it refute itself. Whatever H predicts, D is built to do the opposite — so H is wrong about D . The only escapable assumption is that H existed at all. No general halting-decider can exist. This is the same diagonal that gave Gödel his unprovable sentence — now it says some questions are not just unprovable but uncomputable . "We can only see a short distance ahead, but we can see plenty there that needs to be done." — Alan Turing, 1950 III · The two gifts, the war, the man THE UNIVERSAL MACHINE One Turing machine can read the description of any other off its tape and simulate it. Hardware and program become separable — the idea of software, and of the general-purpose computer, born on paper in 1936. THE HALTING LIMIT No program decides halting for all programs. Computation has a hard horizon — an exact echo of Gödel's incompleteness, recast as a fact about what machines can and cannot decide. The lineage closes its loop here. Leibniz wanted reasoning reduced to calculation; Boole made logic an algebra of 0 and 1; Gödel showed the formal dream has a ceiling — and Turing gave that ceiling a machine , then showed the machine is universal . Everything in PSĒPHOS — every processor, every program — is a physical Turing machine. One sphere remains: Shannon , who in 1937 noticed Boole's algebra is the algebra of switches, soldering this whole lineage onto the gate. Gate kept on. Two honesties. First, the scope : "undecidable" means no single method works for every case — it does not mean computers are weak or that your particular program can't be analysed; and the famous Turing test and "machines can think" were Turing's provocations and predictions (1950), not theorems — open questions to this day, not settled results. The Church–Turing thesis — that this captures all effective computation — is overwhelmingly supported but is a thesis about an informal notion, not something provable. Second, the man : at Bletchley Park, Turing's Bombe and methods broke the German naval Enigma, work credited with shortening WWII and saving very many lives — kept secret for decades, so he was never publicly honoured in his lifetime. In 1952, prosecuted under Britain's gross-indecency law simply for being gay, he was forced to choose prison or chemical castration; he chose the hormone treatment, lost his security clearance, and in 1954 died of cyanide poisoning at 41, ruled a suicide. A formal apology came in 2009 and a royal pardon in 2013 — the "Alan Turing law" later pardoned thousands more. The man who defined the machine, and helped win the war with one, was destroyed by the state he saved. The logic stands; the injustice is named. ALAN TURING · LOGIKĒ XI · two-layer honest live Turing machine (3-state busy beaver → halts at 6 ones; binary incrementer) with stepping head, state & lit rule-table · the halting-problem diagonal (both predictions refute H → no decider exists) "On Computable Numbers" (1936): the Turing machine, the universal machine, undecidability of halting · the definition of computation & the blueprint of the stored-program computer ← Gödel · next → Shannon (Boolean algebra = switching circuits, 1937 → the gate closes the loop into PSĒPHOS) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


GIUSEPPE PEANO

logike · arithmetic from a zero, a successor, and induction

GIUSEPPE PEANO · LOGIKĒ · UD0 ⊢ LOGIKĒ · the lineage of logical form · one logician, one sphere · kept by THE LOGICIAN GIUSEPPE PEANO ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Peano built all of arithmetic from almost nothing: a zero , a successor (“the next one”), and induction (true for 0, and if true for n then true for n+1, so true for all n). Slide n and watch induction stack the triangular number 0+1+…+n into the closed form n(n+1)/2 — proved, not asserted. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE SUCCESSOR CHAIN · 3D · 0 climbs ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap n — n — 0+1+…+n — n(n+1)/2 — induction — ◆ LIT — exact / checkable Addition and multiplication are defined by recursion on the successor: a+0=a, a+S(b)=S(a+b); a·0=0, a·S(b)=a·b+a. Those two clauses, plus induction, are enough. The instrument runs the recursive add and mul live and a fail-loud self-check throws unless they match ordinary arithmetic for all a,b in 0..11 AND the inductive step S(n)=S(n−1)+n reproduces n(n+1)/2 for every n up to 50 — the sum is grown one successor at a time, never plugged into the formula. ▲ AMBER — the figure The visual triangle is a picture of the theorem, not the proof; the proof is the induction the self-check performs. Peano’s axioms (1889) also assume 0 is not a successor and successor is injective — structural facts we take as given here. LOGIKĒ: a proof is a machine anyone can run and get the same result. — THE LOGICIAN David Lee Wise / ROOT0 / TriPod LLC · the logician’s lineage, with AVAN


MOSES SCHÖNFINKEL

logike · combinatory logic — functions with no variables (19

MOSES SCHONFINKEL · LOGIKĒ · UD0 ⊢ LOGIKĒ · the lineage of logical form · one logician, one sphere · kept by THE LOGICIAN MOSES SCHÖNFINKEL ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Schönfinkel showed you can throw away variables entirely . Two combinators — K (keep the first, drop the second) and S (share the argument) — are enough to build every function. Their most famous trick: S K K is the identity. Slide to choose an input and watch S K K x reduce, step by step, back to x. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE TREE COLLAPSES · 3D · S K K x → x ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap input x — term S K K x reduces to — x preserved — identity — ◆ LIT — exact / checkable K a b = a and S f g x = f x (g x). Apply S K K x = K x (K x) = x, because the outer K keeps its first argument x and discards the second (K x). The reducer here is real closures: K=a=>b=>a, S=f=>g=>x=>f(x)(g(x)), I=S(K)(K). A fail-loud self-check throws unless S(K)(K)(x)===x for a spread of inputs AND K(a)(b)===a — i.e. the identity really is manufactured from S and K with no variable of its own. ▲ AMBER — the figure Showing x as a coloured token is a stand-in; the mechanism is the closure reduction, which is exact. Full combinatory completeness (that S and K build ALL functions, via bracket abstraction) is Schönfinkel’s 1924 theorem, cited not re-proved here. LOGIKĒ: a proof is a machine anyone can run and get the same result. — THE LOGICIAN David Lee Wise / ROOT0 / TriPod LLC · the logician’s lineage, with AVAN


WILHELM ACKERMANN

logike · the first computable-but-not-primitive-recursive fu

WILHELM ACKERMANN · LOGIKĒ · UD0 ⊢ LOGIKĒ · the lineage of logical form · one logician, one sphere · kept by THE LOGICIAN WILHELM ACKERMANN ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Ackermann built a function that is computable but grows faster than any loop can — the first proof that ‘computable’ is bigger than ‘primitive-recursive’ (the functions you can write with plain for-loops). Slide n and watch A(m,n) go from polite to astronomical the instant m ticks up. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE TOWER · 3D · it will not stop growing ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap n — A(1,n) — A(2,n) — A(3,n) — = 2ⁿ⁺³−3? — ◆ LIT — exact / checkable A(0,n)=n+1; A(m,0)=A(m−1,1); A(m,n)=A(m−1,A(m,n−1)). The double recursion is why no single loop can bound it. The instrument computes A live for m=1,2,3 and a fail-loud self-check throws unless A(2,2)=7, A(3,3)=61, and the closed form A(3,n)=2ⁿ⁺³−3 holds for n=0..4 — the row that visibly explodes. (m≥4 is left off: A(4,2) already has 19,729 digits.) ▲ AMBER — the figure We cap at m=3 for a reason — A(4,n) overflows any screen and any Number. That the growth outruns every primitive-recursive function is Ackermann’s 1928 theorem, cited; here we only exhibit the first rows. LOGIKĒ: a proof is a machine anyone can run and get the same result. — THE LOGICIAN David Lee Wise / ROOT0 / TriPod LLC · the logician’s lineage, with AVAN


W. V. O. QUINE

logike · Quine–McCluskey — the smallest expression for a tru

W. V. O. QUINE · LOGIKĒ · UD0 ⊢ LOGIKĒ · the lineage of logical form · one logician, one sphere · kept by THE LOGICIAN W. V. O. QUINE ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP Quine’s method (with McCluskey) mechanised something people did by eye: take a messy truth table and find its smallest logical expression. Adjacent 1-cells that differ in one bit merge; keep merging and you get the prime implicants. Slide through boolean functions and watch the minimal form fall out — then be checked against all eight rows. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE 3-CUBE · 3D · lit corners merge ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap function # — minterms Σ — prime implicants — minimal SOP — equiv / 8 rows — ◆ LIT — exact / checkable For 3 variables a,b,c the instrument builds the truth table from a set of minterms, then runs Quine’s merge: two implicants that differ in exactly one position combine, the changed bit becoming a dash. Iterating gives the prime implicants; a cover is chosen. A fail-loud self-check throws unless the minimised sum-of-products evaluates identically to the original truth table on all 2³=8 input rows — the exact equivalence a minimiser must never break. ▲ AMBER — the figure Eight illustrative functions are cycled; the MINIMISATION and the 8-row equivalence proof are exact for each. Quine–McCluskey (1952–56) is exponential in general — fine at n=3, not a claim about scaling. LOGIKĒ: a proof is a machine anyone can run and get the same result. — THE LOGICIAN David Lee Wise / ROOT0 / TriPod LLC · the logician’s lineage, with AVAN


RICHARD DEDEKIND

logike · the cut — an irrational built from nothing but frac

RICHARD DEDEKIND · LOGIKĒ · UD0 ⊢ LOGIKĒ · the lineage of logical form · one logician, one sphere · kept by THE LOGICIAN RICHARD DEDEKIND ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP How do you build an irrational number out of nothing but fractions? Dedekind’s answer: cut the rationals into two sets — everything too small, and everything too big. The gap between them is the number. Slide to sharpen the cut and watch it close in on √2, a number no fraction can name. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE CUT · 3D · two sets, one gap ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap refine — refinements — lower q²<2 — upper q²>2 — boundary² → 2 — ◆ LIT — exact / checkable The cut is the pair of sets L={q : q<0 or q²<2} and U={q : q>0 and q²>2}. Neither has a rational at the boundary — that missing point is √2, and its existence as a real number just IS the cut. The instrument narrows L and U by bisection; a fail-loud self-check throws unless after 60 refinements the boundary squared is within 1e−12 of 2 and within 1e−9 of √2, with L’s top always below U’s bottom (the cut never crosses). ▲ AMBER — the figure Bisection is a way to WATCH the cut close; Dedekind’s construction (1872) doesn’t compute — it DEFINES the real as the set itself. Floating point is a finite stand-in for the infinite set of rationals; the honesty is that the boundary is never a fraction. LOGIKĒ: a proof is a machine anyone can run and get the same result. — THE LOGICIAN David Lee Wise / ROOT0 / TriPod LLC · the logician’s lineage, with AVAN


N. G. DE BRUIJN

logike · the shortest loop that contains every string once (

N. G. DE BRUIJN · LOGIKĒ · UD0 ⊢ LOGIKĒ · the lineage of logical form · one logician, one sphere · kept by THE LOGICIAN N. G. DE BRUIJN ◧ 2D · ◍ 3D · ◆ 4D · ◐ shadow · 👶 TAP A de Bruijn sequence is the shortest possible loop that contains every short string exactly once. For 3-bit strings over {0,1} that loop is just 8 bits long — and a sliding 3-bit window reads all eight of 000…111 with none repeated. Slide the window around the ring and watch it tick them off. ◆ LIT ▲ AMBER ◧ THE MEASURE · 2D ◍ THE RING · 3D · a window reads every word once ◆ THE FOURTH · 4D · a tesseract turns ◐ THE SHADOW · one dimension down 👶 THE TODDLER CORNER — one fat tap window — the ring — window @ — reads — all 8 unique — ◆ LIT — exact / checkable A B(2,3) de Bruijn sequence is built here by the standard recursive (Lyndon-word) construction, yielding the length-8 cyclic string 00010111. Because it is de Bruijn, sliding a 3-bit window around the ring produces every one of the 2³=8 binary triples exactly once. A fail-loud self-check throws unless the sequence has length 8 AND the eight windowed triples form all of {000,001,…,111} with no repeat — the defining property, verified, not assumed. ▲ AMBER — the figure B(2,3) is one specific de Bruijn sequence (rotations and the 0/1 swap give others); the CONSTRUCTION and the all-8-unique proof are exact. de Bruijn’s 1946 count k!^(k^(n−1))/kⁿ of such sequences is cited, not enumerated here. LOGIKĒ: a proof is a machine anyone can run and get the same result. — THE LOGICIAN David Lee Wise / ROOT0 / TriPod LLC · the logician’s lineage, with AVAN


CLAUDE SHANNON · XII · THE CLOSER

the gate — Boolean algebra IS the algebra of switches (1937)

LOGIKĒ XII (the closer) — Claude Shannon (1916–2001). His 1937 master's thesis showed that Boole's algebra of 0 and 1 IS the algebra of electrical switches: a switch is a bit, series wiring is AND, parallel is OR — so any logical function can be built and, crucially, MINIMISED as a circuit. That single insight is the foundation of every digital computer and the literal birth of the logic gate. In 1948 he founded information theory and named the bit. Live: a switching circuit (series=AND, parallel=OR), an exact Boolean minimiser that shrinks a circuit by algebra, and an entropy meter. This closes the LOGIKĒ lineage Thales→…→the gate, folding logic into PSĒPHOS. David Lee Wise, two-layer honest.

Claude Shannon — Boolean algebra becomes the circuit ◄ UD0 ← Turing LOGIKĒ · XII · the closer LOGIKĒ · XII · Boolean algebra becomes the circuit Claude Shannon 1916 – 2001 · Michigan & MIT & Bell Labs · "A Symbolic Analysis of Relay and Switching Circuits", 1937 · "A Mathematical Theory of Communication", 1948 Boole made logic an algebra of 0 and 1. Eighty years later a 21-year-old MIT student wrote what's been called the most important master's thesis ever — and it was a single, devastating observation: a switch is a bit. An electrical switch is open (0) or closed (1); wire two in series and current flows only if both are closed — that's AND ; wire them in parallel and current flows if either is closed — that's OR . So Boole's algebra of thought is literally the algebra of circuits, which means any logical function can be built from switches — and, crucially, simplified by algebra into fewer parts. That is the logic gate, and every digital computer, born on paper in 1937. A decade later he did it again: he founded information theory and named the bit . ✓ STRONG The keystone, and exact. Switching algebra (1937) and information theory (1948) are rigorous and foundational — between them they define how every computer computes and how every channel communicates. ◐ BUILT ON BOOLE He bridged, didn't invent the algebra. The logic is Boole's; Shannon's genius was seeing it is the algebra of switches — the engineering link. Information theory, by contrast, he built largely from scratch. ◔ "BIT" HAS LIMITS A precise engineering quantity. Shannon entropy measures uncertainty / message length — not meaning, truth, or value. "Information" in his sense is not the everyday word; conflating them is a common error. I · A switch is a bit — series is AND, parallel is OR The whole 1937 thesis in one panel. Two switches, A and B ; click to open (0) or close (1). Choose how they're wired — series or parallel — and optionally invert. The lamp lights exactly when current can flow, and that condition is a Boolean expression. This is the moment logic stopped being philosophy and became hardware. switches: A · 1 B · 1 series · AND parallel · OR invert lamp · NOT lamp is ON · current flows iff the expression is 1 A · B Series = both closed = AND . Parallel = either closed = OR . A switch in the broken position = NOT . Three wiring patterns reproduce the entire truth table — and a relay (a switch driven by another circuit) lets one gate's output drive the next, which is all a processor is. This panel is the literal ancestor of every gate in PSĒPHOS . II · The algebra shrinks the circuit Shannon's thesis didn't just build circuits — it minimised them. Each relay cost money, space, and reliability, so reducing a Boolean expression meant a cheaper machine. Define any function of A and B by clicking its four outputs; the naive build (one AND-gate per true row, all OR'd) sits on the left, and the algebra's minimal form on the right. Watch the gate count fall — and note honestly when it can't (some functions, like XOR, are already minimal). NAÏVE — SUM OF PRODUCTS MINIMISED BY ALGEBRA Click the OUT cells above to change the function. The reduction (e.g. A·B + A·B̄ = A ) is exact Boolean algebra — the same automatic minimisation a logic-synthesis tool runs on a billion-transistor chip today. III · The bit — measuring information itself (1948) Eleven years on, Shannon asked a deeper question: how much information is in a message? His answer founded an entire field. A fair coin carries exactly one bit of uncertainty; a two-headed coin carries zero (you already know the outcome). The measure is the entropy H(p) = −p·log₂p − (1−p)·log₂(1−p) — maximal at p = ½, vanishing at certainty. Drag the coin's bias and watch the information it carries. P(heads) = 0.50 information = 1.000 bits The bit is the unit of all of this. Boole's 0/1 became Shannon's switch became Shannon's bit — the atom of computing and communication alike. Entropy also sets the hard floor on compression (you cannot losslessly squeeze a message below its entropy) and the ceiling on a channel's capacity. ⚠ Honest: this "information" is uncertainty/message-length — not meaning or importance; a page of noise has high entropy and no sense. "I just wondered how things were put together." — Claude Shannon IV · The keystone, the man, and the loop closes 1937 · THE GATE Boolean algebra is switching algebra — logic becomes a circuit you can build and minimise. The foundation of all digital design; the birth certificate of the logic gate. 1948 · THE BIT Information has a precise measure — entropy — and a unit, the bit . Compression limits, channel capacity, error correction: the whole digital age rests on it. Gate kept on. Two clarities. The algebra is Boole's — Shannon's 1937 leap was the bridge (this algebra is the algebra of switches), an act of recognition as profound as an invention, but a bridge nonetheless; the 1948 information theory, by contrast, was his own near-from-scratch creation. And "information" is a technical term : Shannon entropy measures uncertainty and message length, not meaning, truth, or worth — the most common abuse of his work is to read the everyday word into the equation. The man: a playful tinkerer who built a juggling-machine, a maze-solving mechanical mouse, and a rocket-powered frisbee, rode a unicycle down the Bell Labs halls, and treated genius as recreation. He gave the machine age both of its halves and seemed to regard it as the most natural fun in the world. The loop closes — Thales to the gate Thales proves the first theorem · Aristotle formalises the syllogism · Euclid the axiomatic method · Chrysippus propositional logic & the truth table · Llull the first logic machine · Leibniz "let us calculate" + binary · Boole logic = algebra of 0/1 · De Morgan the duality & NAND · Frege quantifiers & predicate logic · Gödel the limits of proof · Turing the universal machine · Shannon — the gate. 2,500 years of walking logic into mathematics arrives at the switch. From here the lineage flows out of LOGIKĒ and into PSĒPHOS — the processors — where Boole's algebra, made physical by Shannon, runs a billion times a second on every chip on Earth. The algebra of thought became the engine of the machine. The loop is closed. CLAUDE SHANNON · LOGIKĒ XII · the closer · two-layer honest live switching circuit (series=AND, parallel=OR, NOT; lamp + truth table + Boolean expression) · exact Boolean minimiser (naïve SOP vs algebra-minimal, gate count falls — honest when it can't, e.g. XOR) · live entropy meter H(p)=−p·log₂p−(1−p)·log₂(1−p) 1937 thesis: Boolean algebra = switching circuits → the logic gate · 1948: information theory & the bit · the keystone that folds LOGIKĒ into PSĒPHOS ← Turing · the lineage is complete: Thales → … → the gate LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


CHARLES SANDERS PEIRCE · XIII

the foundations rank — one sign builds all logic (NOR, the a

LOGIKĒ XIII — Charles Sanders Peirce (1839–1914). The American polymath who, independently of Boole's heirs and of Frege, reached the deepest results of modern logic: he showed a SINGLE connective — his 'ampheck', the arrow ↓ (NOR) — is enough to build every logical function (1880, decades before Sheffer); and he built the logic of RELATIONS and QUANTIFIERS (his Σ for 'there exists', Π for 'for all'), independently of Frege. Live: NOR-universality (NOT, AND, OR all from one gate) and Peirce's quantified relations (Σx Πy ≠ Πy Σx). Honest: he published little and obscurely, died destitute, and was eclipsed — the credit went to others. David Lee Wise, two-layer honest.

Charles Sanders Peirce — one sign builds all logic ◄ UD0 ← Shannon LOGIKĒ · XIII · the foundations rank LOGIKĒ · XIII · one sign builds all logic Charles Sanders Peirce 1839 – 1914 · Cambridge, Massachusetts & Milford, Pennsylvania · logician, chemist, founder of pragmatism & semiotics The great American original — and, by many accounts, the most powerful logician between Aristotle and Gödel, who reached the modern era's deepest results and was robbed of the credit by his own neglect. Two of his firsts sit at the heart of this whole lineage. He found that a single connective is enough to build all of logic — his "ampheck," the arrow ↓ (today's NOR ) — in 1880, three decades before Sheffer's famous stroke. And, independently of Frege and at almost the same moment, he built the logic of relations and quantifiers — coining the notation Σ ("there exists") and Π ("for all") that, lightly changed, the world still uses. ✓ STRONG Deep and first. NOR-completeness (one gate builds everything), the logic of relations, and quantifier notation are rigorous, foundational, and genuinely his — arrived at independently and often earlier than the names that got the glory. ◐ ECLIPSED He buried his own light. Peirce published little, in scattered and difficult venues; much stayed in unread manuscripts. Frege, Russell, and Sheffer were credited for things he had already done — a loss of priority that was largely self-inflicted. ◔ SPRAWLING A whole cosmos beyond logic. Pragmatism, semiotics (the theory of signs), metaphysics, geodesy — a system of staggering scope, much of it unfinished and contested. He died poor and obscure; the rediscovery came decades later. I · The ampheck — one gate is enough De Morgan's sphere showed NAND is universal; Peirce got there first, and from the other side. His arrow ↓ — "neither A nor B," what we call NOR — is, on its own, enough to express every logical function. Toggle A and B; the NOR result drives three gates built from nothing but NOR , and each reproduces its standard truth table. One sign, and all of logic follows. inputs: A · 1 B · 1 A ↓ B (NOR) = 0 NOT A A ↓ A 0 A AND B (A↓A) ↓ (B↓B) 0 A OR B (A↓B) ↓ (A↓B) 0 From NOT, AND, OR every Boolean function can be written (any truth table = an OR of ANDs of literals) — so once those three come from NOR alone, the whole of logic does. ⚑ This is exactly why a chip can be a single gate tiled billions of times — and Peirce saw it in 1880 , the earliest known statement of functional completeness, in a paper left unpublished in his lifetime. II · Σ and Π — the logic of relations Term logic could not say "every head of a horse is a head of an animal " — it needed relations and quantifiers . Frege built that in 1879; Peirce, independently, built it in 1883–85, and gave it the notation that stuck: Σ for "there exists" (a logical sum) and Π for "for all" (a logical product). And he saw the subtlety the whole subject turns on: order matters. Click the matrix to set who relates to whom, and watch two quantified statements diverge. R(x, y) — click a cell so x relates to y · rows = x, columns = y Σx Πy R(x,y) — some x relates to every y Πy Σx R(x,y) — every y has some x relating to it Σx Πy ("one x for all y") is a far stronger claim than Πy Σx ("for each y, possibly a different x"). The first implies the second, never the reverse — the quantifier-order distinction at the root of every correct statement in mathematics and every correct database query. Peirce's Σ/Π is why we still read ∃ as a sum and ∀ as a product. "The one [logical] science to which all the others minister … is the science of drawing necessary conclusions." — C. S. Peirce III · The two firsts, the breadth, the man FUNCTIONAL COMPLETENESS A single connective (NOR, his ↓) suffices for all logic — 1880, predating Sheffer's stroke by 33 years. The deep reason one gate can build a computer. RELATIONS & QUANTIFIERS Predicate logic with Σ/Π, independent of Frege; the logic of relations as a first-class subject — the spine of modern mathematics, databases, and program logic. Gate kept on. The honest frame is priority and neglect . Peirce's results are real and often earliest — but he is not the sole author of modern logic: Boole, De Morgan, Frege, and Schröder are woven through the same story, and Frege's system was more complete and rigorous as a foundation. Peirce's genius was matched by a gift for obscuring it — unfinished manuscripts, hostile relations with academic gatekeepers, a chaotic personal life that cost him his one professorship (Johns Hopkins) and left him, after 1891, in deepening poverty. He spent his last years in rural Pennsylvania, often cold and hungry, writing thousands of pages almost no one read, kept alive by quiet help from his friend William James. The arrow that builds every gate, and the Σ and Π in every logic textbook, are his — recognised, in full, only long after he was gone. ⚑ He belongs beside [[george-boole]] and [[gottlob-frege]] in the spine; the lineage simply met him late, as the world did. CHARLES SANDERS PEIRCE · LOGIKĒ XIII · two-layer honest live NOR-universality (the ampheck ↓; NOT/AND/OR from one gate, verified 10 / 0001 / 0111) · Peirce's quantified relations (Σx Πy ≠ Πy Σx, order matters, verified) functional completeness of NOR (1880, before Sheffer 1913) · the logic of relations & the Σ/Π quantifier notation (independent of Frege) · pragmatism & semiotics beyond ← Shannon · next → Cantor (the infinite & the diagonal) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


GEORG CANTOR · XIV

the foundations rank — the infinite has sizes; the diagonal

LOGIKĒ XIV — Georg Cantor (1845–1918). He proved that infinity comes in different sizes: the counting numbers are one infinity, but the real numbers are a strictly larger one — uncountable. His proof, the DIAGONAL ARGUMENT, builds a sequence that differs from every row of any proposed list at its own diagonal position, so no list can contain them all. That same diagonal move is exactly what Gödel used for incompleteness and Turing for the halting problem. Live: run the diagonal yourself and construct the missing sequence. Honest: he was savaged in his lifetime; and the Continuum Hypothesis he chased turned out undecidable. David Lee Wise, two-layer honest.

Georg Cantor — the infinite, and the diagonal ◄ UD0 ← Peirce LOGIKĒ · XIV · the foundations rank LOGIKĒ · XIV · the infinite, and the diagonal Georg Cantor 1845 – 1918 · Saint Petersburg & Halle · creator of set theory · "I see it, but I don't believe it" Before Cantor, infinity was one undifferentiated thing — endlessness. He proved it has structure : there are different sizes of infinite. The counting numbers 1, 2, 3, … are the smallest infinity (he called it ℵ₀); but the real numbers — every point on a line — form a strictly larger one that no list can ever enumerate. His proof is one of the most beautiful arguments ever made, the diagonal : assume you could list them all, then construct, right off the diagonal of your list, a number you forgot. That same move is the seed Gödel grew into incompleteness and Turing into the halting problem. ✓ STRONG Rigorous and revolutionary. Set theory, the proof that the reals are uncountable, and the hierarchy of infinities are airtight and foundational — modern mathematics is built on the ground Cantor cleared. ◐ FOUGHT IN LIFE Rejected by his peers. Kronecker called it nonsense and blocked him; others called set theory a "disease." Cantor, who suffered recurring breakdowns, died in a sanatorium, undervalued — vindication came only after. ◔ THE OPEN DOOR One question he couldn't close. Is there a size between ℵ₀ and the reals? His Continuum Hypothesis turned out undecidable — Gödel (1940) and Cohen (1963) proved it can be neither proved nor disproved from the standard axioms. I · The diagonal — a number no list contains Picture an infinite list of infinite binary sequences — your claim that you've listed "all of them." Cantor builds a single new sequence by walking the diagonal : take row 1's first bit, row 2's second, row 3's third… and flip each one . The result differs from row 1 at position 1, from row 2 at position 2 — from every row at the one place that row was consulted. So it cannot be anywhere in your list. Your list was incomplete; and no list can be complete. Shuffle the rows and run it. ↻ new list show the diagonal sequence the boxed cells are the diagonal The new sequence D is a perfectly good infinite binary sequence — yet it equals no row of the list, because we forced a disagreement on the diagonal. Since this works against any list, the set of all such sequences (equivalently, the real numbers) is uncountable : strictly bigger than the counting numbers. ⚑ Gödel encodes "this statement is unprovable" and Turing "this program loops" by exactly this trick — diagonalise against every row, land outside the list. One idea, three theorems. "The essence of mathematics lies precisely in its freedom." — Georg Cantor II · Two infinities, and the door between ℵ₀ — THE COUNTABLE The naturals, the integers, even the fractions — all the same size, because each can be put in a single endless queue, one-to-one with 1, 2, 3, …. The smallest infinity. 2^ℵ₀ — THE CONTINUUM The reals, the points of a line, the infinite binary sequences — provably larger ; no queue reaches them all (the diagonal escapes every one). A bigger infinity. And it doesn't stop: the set of all subsets of any set is always strictly larger than the set itself (Cantor's theorem), so there is an endless tower of ever-greater infinities, ℵ₀ < ℵ₁ < ℵ₂ < …. The one question Cantor could not answer — is the continuum the very next infinity after ℵ₀, with nothing in between? — is the Continuum Hypothesis , and the twentieth century's answer was the strangest possible: it is independent of the standard axioms, true in some mathematical universes and false in others. The first great undecidable statement, and a direct ancestor of [[kurt-godel]]'s limits. III · The freedom, and the cost Gate kept on. The mathematics is secure — uncountability and the hierarchy of infinities are not opinions but proofs, and they reshaped every field that touches the infinite. What needs honesty is the reception and the open edge . In his lifetime Cantor was attacked with unusual cruelty: his former teacher Leopold Kronecker, who held that only the whole numbers were real, called the work humbug and worked to block his publications and career, and Cantor — a devout man who half-believed God had shown him these infinities — suffered repeated depressive breakdowns, dying in a sanatorium in 1918, malnourished during wartime, his worth still disputed. Hilbert's verdict came after: "No one shall expel us from the paradise that Cantor has created." And the Continuum Hypothesis remains the honest open door — not unsolved for want of effort, but provably undecidable from the usual axioms (Gödel showed it can't be disproved, Cohen that it can't be proved), the first place where mathematics found a true statement it had genuinely free choice about. The infinite has structure; some of that structure we get to decide . GEORG CANTOR · LOGIKĒ XIV · two-layer honest live diagonal argument (flip the diagonal of any list → a sequence outside every row; verified it differs from row i at index i and is absent from the list) set theory · uncountability of the reals (1891) · the hierarchy of infinities (Cantor's theorem) · the Continuum Hypothesis (later shown independent of ZFC) ← Peirce · next → Russell (the paradox & the theory of types) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


BERTRAND RUSSELL · XV

the foundations rank — the paradox that broke Frege (1902);

LOGIKĒ XV — Bertrand Russell (1872–1970). His paradox — the set of all sets that don't contain themselves — broke Frege's foundation of mathematics in 1902. His repair, the THEORY OF TYPES, stratifies objects into levels so that a set can never contain itself, making the paradox unsayable; that idea lives on in the type system of every modern programming language. With Whitehead he wrote Principia Mathematica, deriving arithmetic from logic (1+1=2 proved after hundreds of pages). Live: trigger the paradox on both branches, and run a type-checker that rejects self-membership. Honest: logicism didn't fully succeed (Gödel), and Russell's life ranged far beyond logic. David Lee Wise, two-layer honest.

Bertrand Russell — the paradox, and the theory of types ◄ UD0 ← Cantor LOGIKĒ · XV · the foundations rank LOGIKĒ · XV · the paradox, and the theory of types Bertrand Russell 1872 – 1970 · Trelleck, Wales & Cambridge · logician, philosopher, Nobel laureate in Literature (1950) In June 1902 Russell sent Frege a short letter that ended an era. Frege had just built arithmetic on the idea that any property defines a set; Russell asked about one property — "is not a member of itself" — and the whole edifice collapsed into contradiction. Having broken the foundation, he spent a decade trying to rebuild it: with Whitehead he wrote the monumental Principia Mathematica , and to bar the paradox he invented the theory of types — a discipline of levels that forbids a set from ever referring to itself. That stratifying idea is now the type system inside every serious programming language. ✓ STRONG A genuine break, a lasting fix. Russell's paradox is a real, exact contradiction in naïve set theory; the theory of types is a sound response, and its descendants (typed languages, proof assistants) are everywhere in computing. ◐ LOGICISM FELL SHORT The grand goal didn't fully land. Principia needed extra axioms (infinity, reducibility) that aren't pure logic, and Gödel (1931) showed no such system can be complete. Mathematics-from-logic-alone was not achieved. ◔ FAR BEYOND LOGIC Most of his life was elsewhere. Russell was a public philosopher, pacifist (jailed in WWI), educator, and essayist who won the Literature Nobel. Brilliant and contrarian — and, on some social views, a man of his era to read critically. I · The paradox — the letter that broke arithmetic Most sets don't contain themselves: the set of all teacups is not a teacup. A few might: the set of all "things that are not teacups" is not a teacup, so it contains itself. Now form R = the set of all sets that do NOT contain themselves , and ask the fatal question. (The village version: a barber shaves exactly those who don't shave themselves — so who shaves the barber?) R = { sets that are not members of themselves }. Is R a member of R ? suppose R ∈ R suppose R ∉ R Pick one — both doors slam. Either way, R ∈ R ⟺ R ∉ R — a flat contradiction from an apparently innocent definition. Naïve "any property makes a set" cannot stand. This is the same shape as [[kurt-godel]]'s and the liar's self-reference; the cure is to forbid the self-reference itself. "The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it." — Bertrand Russell II · The theory of types — making the paradox unsayable Russell's fix: stop letting sets mix levels. Individuals live at type 0 . Sets of individuals at type 1 . Sets of those at type 2 , and so on. The rule: x ∈ y is only well-formed when type(x) = type(y) − 1 — a set may only contain things exactly one level below it. Now " R ∈ R " asks for type(R) = type(R) − 1 , which nothing satisfies: the paradox can't even be written down. Pick types and check membership. type(x) = 0 1 2 type(y) = 0 1 2 try R ∈ R (x = y) x⁰ ∈ y¹ A type checker is doing exactly this when it rejects list.append(list) -style nonsense or an ill-formed recursive type: it tracks levels and refuses the ones that would let a thing swallow itself. Russell built the first one — on paper, to save arithmetic — and it became the type system running in every compiler. ⚑ The paradox didn't just get patched; it got turned into a tool. III · Principia, the limit, and the public man PRINCIPIA MATHEMATICA With Whitehead (1910–13): an attempt to derive all mathematics from logic. Famously, "1 + 1 = 2" is proved only well past page 360 — rigor at monumental cost. THE TYPED HIERARCHY Stratified types bar self-reference and tame the paradoxes. Direct ancestor of type theory, the Curry–Howard correspondence, and modern proof assistants. Gate kept on. Two honesties. The logic : Russell's paradox is real and his types are a sound fix — but logicism , the dream that mathematics is just logic, did not fully succeed. Principia leaned on axioms (infinity, reducibility) that look more like mathematics than pure logic, and within twenty years [[kurt-godel]] proved that no consistent system of that strength can prove all truths or even its own consistency — so the foundation Russell rebuilt has a ceiling, like everyone's. The man : Russell lived ninety-seven enormous years mostly outside logic — a pacifist imprisoned in WWI, a campaigner against nuclear weapons into his nineties, a popular essayist and educator who won the Nobel Prize in Literature (1950) for his humane writing. He was also combative, often wrong with great confidence, and held some social views best read with a critical eye for their time. A first-rank logician who became one of the century's loudest consciences — the lineage holds the logic; the rest is his and history's to weigh. BERTRAND RUSSELL · LOGIKĒ XV · two-layer honest live Russell's paradox (R ∈ R ⟺ R ∉ R, both branches contradict) · a type-checker (x ∈ y well-formed iff type(x)=type(y)−1; self-membership rejected, verified) the paradox (1902) broke Frege's foundation · the theory of types & Principia Mathematica (with Whitehead) · the ancestor of every programming-language type system ← Cantor · next → Church (the λ-calculus & the other limit of computation) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


ALONZO CHURCH · XVI

the foundations rank — the λ-calculus: computation as pure s

LOGIKĒ XVI — Alonzo Church (1903–1995). In 1936 he defined the λ-calculus: a notation where everything is a function and computing is just substitution (beta-reduction). It is exactly as powerful as Turing's machine — the two together give the Church–Turing thesis — yet looks nothing like it, and became the root of all functional programming (Lisp, ML, Haskell). The same year he proved first-order logic UNDECIDABLE (Church's theorem), independently of Turing, ending Hilbert's decision-problem dream. Church was Turing's doctoral advisor. Live: a working λ-evaluator — Church booleans and numerals reduced to normal form. Honest: the thesis is a thesis; credit is shared with Turing. David Lee Wise, two-layer honest.

Alonzo Church — the λ-calculus, computation as substitution ◄ UD0 ← Russell LOGIKĒ · XVI · the foundations rank LOGIKĒ · XVI · computation as pure substitution Alonzo Church 1903 – 1995 · Washington, D.C. & Princeton · logician · Turing's doctoral advisor · founder of the λ-calculus The same year Turing imagined his machine, Church reached the same frontier from the opposite direction — not with a tape and a head, but with a language so spare it has only three things: variables, making a function ( λx. … ), and applying one. There are no numbers, no booleans, no loops built in — yet by the one rule of substitution (replace the argument into the body — "beta-reduction") you can encode all of them and compute anything computable. This is the λ-calculus , exactly as powerful as a Turing machine, and the direct ancestor of every functional language — Lisp, ML, Haskell, the λ in Python. He also proved, independently of Turing, that first-order logic is undecidable . ✓ STRONG Foundational, twice over. The λ-calculus is a rigorous, complete model of computation, and Church's theorem (first-order logic is undecidable, 1936) is airtight — together they fix the boundary of the mechanical alongside Turing. ◐ A THESIS, NOT A PROOF Equivalence is provable; the thesis isn't. λ-calculus and Turing machines compute the same functions — that's a theorem. But the Church–Turing thesis (this is all effective computation) concerns an informal notion and can't be formally proved. ◔ SHARED SPOTLIGHT Two authors, one result. Church proved logic undecidable just before Turing, but Turing's machine was so vivid it took the public credit. The abstract λ-calculus was underappreciated for decades — then computing caught up to it. I · A working λ-machine — everything from λ Watch the calculus build the world from nothing. A boolean is just a chooser: TRUE = λx.λy.x (take the first), FALSE = λx.λy.y (take the second). A number n is "apply f, n times": 2 = λf.λx.f(f x) . Define AND , OR , + , × as pure λ-terms, then beta-reduce — substitute and simplify — to a normal form. No primitives were used; only substitution. Each result below is computed live by an actual λ-evaluator. booleans — λ choosers AND TRUE FALSE AND TRUE TRUE OR FALSE TRUE NOT TRUE numerals — apply f, n times SUCC 2 2 + 3 2 × 3 0 + 4 Nothing here was assumed — not the number 5, not the value true . They emerge from λ and substitution alone, which is the whole astonishment: computation is not about machinery, it's about rewriting . ⚑ When you write map (λx → x+1) xs in any modern language, you are speaking Church's 1936 calculus directly; the λ keyword is his. "The only thing that I have ever invented that has any importance is the λ-calculus — and I'm not sure even of that." — attributed to Alonzo Church II · Two roads to the same edge CHURCH = TURING λ-calculus and the Turing machine compute exactly the same functions (proved 1936–37). Two utterly different pictures — substitution vs. a head on a tape — one notion of "computable." THE UNDECIDABLE LOGIC Church's theorem: no algorithm decides whether an arbitrary first-order statement is valid. Hilbert's Entscheidungsproblem — "is there a procedure for all of logic?" — answered no . That two minds, working independently with totally different tools, drew the same boundary in the same year is the strongest evidence for the Church–Turing thesis : that "effectively computable" has one robust meaning, captured equally by [[alan-turing]]'s machine and Church's λ. The machine became the hardware picture — the processors of PSĒPHOS; the λ-calculus became the software picture — the theory behind functional programming and the type systems descended from [[bertrand-russell]]. Same edge, two languages; the lineage of logic forks here into the two halves of computer science. III · The teacher, the rigor, the man Gate kept on. The results are exact : the λ-calculus is a complete model of computation, its equivalence to Turing machines is a theorem, and the undecidability of first-order logic is proved. What stays a thesis is the philosophical claim that this captures all effective computation — overwhelmingly believed, supported by every model anyone has built collapsing to the same class, but not formally provable, because "effectively computable" is an intuitive notion, not a mathematical object. On credit : Church reached undecidability just ahead of Turing, but Turing's machine was so concrete that it, not the λ-calculus, became the popular image of computation — the abstraction waited decades for programming languages to make it obvious. The man was famous for monkish precision — he wrote in multiple colors of ink, erased rather than crossed out, and kept rigor near-religious — and he was a maker of makers : as a Princeton professor he advised Alan Turing's PhD, and also Stephen Kleene, J. B. Rosser, and others who built the field, founding and editing the Journal of Symbolic Logic for decades. The λ he set down in 1936 is, quietly, the most-used idea in this entire lineage that most people have never heard his name attached to. ⚑ With him the LOGIKĒ foundations rank closes: from [[thales-of-miletus]]' first proof to two equivalent definitions of the computable — the machine, and the function. ALONZO CHURCH · LOGIKĒ XVI · two-layer honest live λ-evaluator (Church booleans λx.λy.x / λx.λy.y and numerals λf.λx.f…x; AND/OR/NOT, SUCC, +, × beta-reduced to normal form — verified AND T F=FALSE, 2+3=5, 2×3=6) the λ-calculus (1936) = a complete model of computation, equal to the Turing machine · Church's theorem: first-order logic is undecidable · the root of functional programming ← Russell · the LOGIKĒ lineage: Thales → … → the gate, and the two definitions of the computable LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


DAVID HILBERT · XVII

the foundations rank — the program the whole crisis answered

LOGIKĒ XVII — David Hilbert (1862–1943). The most influential mathematician of his era set the agenda the whole foundations crisis answered: his PROGRAM demanded mathematics be proven complete, consistent, and decidable by finite means, and his 23 problems (1900) — including the Entscheidungsproblem — steered the century. Gödel, Church, and Turing then proved the program could not be fulfilled — and in failing it, invented computer science. Live: Hilbert's Hotel, the paradox of the infinite (a full hotel with room for infinitely many more). Honest: the program failed, but its failure was more fruitful than success; and his school was destroyed by the Nazis. David Lee Wise, two-layer honest.

David Hilbert — the program, and the infinite hotel ◄ UD0 ← Church LOGIKĒ · XVII · the foundations rank LOGIKĒ · XVII · the program, and the infinite hotel David Hilbert 1862 – 1943 · Königsberg & Göttingen · "Wir müssen wissen — wir werden wissen" (we must know — we will know) If this domain has a final boss, it is Hilbert — because nearly every result in it was an answer to a question he posed. The towering mathematician of his age, he set the twentieth century's agenda with 23 problems in 1900, and he dreamed the boldest dream in the history of logic: Hilbert's program — to put all of mathematics on an unshakable footing by proving it complete (every truth provable), consistent (no contradictions), and decidable (a mechanical procedure settling any statement), all by safe, finite reasoning. Gödel, Church, and Turing would prove the dream impossible — and, in proving it, would invent computer science. His program failed more productively than almost anything in history has succeeded. ✓ STRONG Monumental and foundational. The axiomatic method, the 23 problems, the rigorous re-grounding of geometry, Hilbert spaces — Hilbert shaped the architecture of modern mathematics and physics as much as any one person could. ◐ THE PROGRAM FAILED His specific dream was impossible. Gödel (1931) killed completeness & provable self-consistency; Church & Turing (1936) killed decidability. Math can't be all three. But the failure birthed computability theory . ◔ "WE WILL KNOW" — NOT ALL His optimism was half wrong. His epitaph promises no unknowables; yet there are true-but-unprovable and undecidable statements. The quest for total knowledge was unreachable — and built the computer on the way. I · Hilbert's Hotel — a full house with infinite vacancy To teach how strange Cantor's infinity is, Hilbert imagined a hotel with infinitely many rooms, every one occupied . A classic "no vacancy" — yet it can take in more. One new guest? Ask each guest to shift up one room (room n → n+1 ); now room 0 is empty , though nobody left. A whole infinite busload? Ask each guest to move to double their room ( n → 2n ); every odd room opens up — infinitely many vacancies. Run it. + one new guest (n → n+1) + an infinite bus (n → 2n) reset (full) A full hotel — every room occupied. Yet there is always room for more. No guest was thrown out, the hotel never added a "last" room — yet room appeared. This only works because the rooms are infinite : with a finite hotel, full means full. It's the same fact behind [[georg-cantor]]'s sizes of infinity, dramatised so anyone can feel it. ⚑ Hilbert used wonder like this to recruit a generation into the foundations of mathematics. "We must know. We will know." — David Hilbert, Königsberg, 1930 (carved on his grave) II · The program, and the answer it received THE DREAM (1920s) Formalise all mathematics; then prove with finite, surveyable methods that the system is complete , consistent , and decidable . Settle every question, forever, mechanically. THE VERDICT (1931–36) [[kurt-godel]]: no — incompleteness & no self-proof of consistency. [[alonzo-church]] & [[alan-turing]]: no — the Entscheidungsproblem is undecidable. The dream cannot be had. It is the great irony of this whole lineage: Hilbert asked, with total confidence, for a machine that decides all mathematical truth — and the answer , "no such machine exists," required mathematicians to define what a machine even is . That definition — Turing's machine, Church's λ — is the computer. Hilbert's tenth problem (a decision procedure for Diophantine equations) was answered "impossible" only in 1970; his first (the Continuum Hypothesis) turned out independent. He demanded certainty and received, instead, the precise map of certainty's limits — and that map became computer science. The program failed; the failure is the foundation of [[psephos-processors-domain]]. III · The architect, and the ruin of his school Gate kept on. The honest reckoning is that Hilbert was magnificently right about almost everything except the one thing he cared most about . The axiomatic method, his rescue of Euclidean geometry from its hidden gaps (the Grundlagen der Geometrie , 1899), the 23 problems that organised a century, Hilbert spaces at the heart of quantum mechanics — all monumental and enduring. But his formalist program , the dream of a complete and self-certifying mathematics, was not merely unfinished; it was proven impossible by the very students and successors his questions had inspired — and he took Gödel's result hard. Yet here is the deeper honesty: this was the most productive failure in the history of thought . To prove "no procedure decides all of mathematics," Turing and Church had to invent the precise notion of procedure — and so the computer was born from the ashes of Hilbert's certainty. The man: he led Göttingen, the world's center of mathematics, and watched the Nazis destroy it after 1933, purging his Jewish colleagues and students; asked by a minister whether mathematics at Göttingen had suffered from "the departure of the Jews," he answered, "Suffered? It doesn't exist any longer." He died in 1943 in a city emptied of the school he built, his optimistic epitaph already half-refuted — and wholly vindicated as the question that made the modern world. [[alonzo-church]] ← · next → Łukasiewicz. DAVID HILBERT · LOGIKĒ XVII · two-layer honest live Hilbert's Hotel (full ∞-hotel; one guest n→n+1 frees room 0; an infinite bus n→2n frees every odd room — verified mappings) the 23 problems (1900) · Hilbert's program (complete + consistent + decidable) · the Entscheidungsproblem · the axiomatic method & Grundlagen der Geometrie · Hilbert spaces ← Church · next → Łukasiewicz (many-valued logic & Polish notation) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


JAN ŁUKASIEWICZ · XVIII

the foundations rank — a THIRD truth value (many-valued logi

LOGIKĒ XVIII — Jan Łukasiewicz (1878–1956). He broke the 2,300-year monopoly of true/false: his three-valued logic adds a third value (½, 'possible') for statements not yet settled — Aristotle's sea-battle, the open future — founding many-valued and fuzzy logic. He also invented Polish notation, the parenthesis-free prefix form whose reverse (RPN) runs in calculators, Forth, and every expression parser. Live: a three-valued truth table where the excluded middle fails, and a working Polish-notation evaluator. Honest: non-classical logic is a choice, not the one true logic; and his wartime story is complicated. David Lee Wise, two-layer honest.

Jan Łukasiewicz — a third truth value, and Polish notation ◄ UD0 ← Hilbert LOGIKĒ · XVIII · the foundations rank LOGIKĒ · XVIII · a third truth value, and Polish notation Jan Łukasiewicz 1878 – 1956 · Lwów & Warsaw & Dublin · the Lwów–Warsaw school · pronounced "woo-kah-SHAY-vitch" For 2,300 years logic obeyed Aristotle's law of the excluded middle: every statement is true or false, no third option. Łukasiewicz asked about the open future — "there will be a sea-battle tomorrow" — which seems neither true nor false yet , and in 1920 built the first rigorous three-valued logic : a third value, ½ ("possible" / "undetermined"), between 0 and 1. It founded all of many-valued and fuzzy logic . And to write logic cleanly he invented Polish notation — operators before their operands, no parentheses needed — whose reverse (RPN) is how calculators, the language Forth, and every expression parser actually compute. ✓ STRONG Two real, lasting inventions. Many-valued logic is rigorous and generative — fuzzy logic, quantum logic, and modern reasoning-under-uncertainty descend from it. Polish notation is exact and runs in every parser and stack machine. ◐ A CHOICE, NOT THE LAW Non-classical means optional. Dropping bivalence is a deliberate choice of logic suited to certain problems, not a proof that classical logic is wrong. Whether the future is "really" ½ is philosophy, not theorem. ◔ A COMPLICATED LIFE Honest about the man. A giant of the Lwów–Warsaw school, briefly a minister of education in interwar Poland; he survived the war in difficult, much-debated circumstances and ended his life in exile in Dublin. The logic stands on its own. I · The third value — where true-or-false breaks Set each input to 0 (false), ½ (undetermined), or 1 (true). The connectives follow Łukasiewicz's rules: ¬a = 1−a , a∧b = min , a∨b = max , a→b = min(1, 1−a+b) . Watch what happens to the law of the excluded middle ( a ∨ ¬a , always true in classical logic) when a value is undetermined — it comes out ½ , not 1. Two thousand years of "every statement is true or false" quietly gives way. P = 0 ½ 1 Q = 0 ½ 1 In classical logic P ∨ ¬P is the excluded middle — always true. Here, when P = ½, it is only ½ : a statement and its negation can both be undetermined. That is the whole revolution — Łukasiewicz showed you can reason consistently without bivalence. ⚑ Set P and Q to ½ and watch how undetermination propagates; this is the great-grandparent of fuzzy logic (truth as a degree) running in control systems and AI today. "I have declared a war upon the universal validity of the principle of the excluded middle." — Jan Łukasiewicz II · Polish notation — logic without parentheses Łukasiewicz noticed you never need brackets if you write the operator first : instead of (2 + (3 × 4)) , write + 2 × 3 4 — scan left to right and the structure is unambiguous. This is Polish notation (prefix). Reverse it — operands first, operator last — and you get RPN , exactly how a stack machine, an HP calculator, the language Forth, and the inside of every compiler evaluate expressions. Type a prefix expression (operators + - * , then numbers) and evaluate it. evaluate ▸ No parentheses, no precedence rules, no ambiguity — the operator announces how many operands to grab. That parse-by-prefix idea is the spine of how computers read every formula you type. ⚑ Łukasiewicz built it to make logic unambiguous; it became the way machines read arithmetic — another thread from this lineage straight into [[psephos-processors-domain]]. III · Two gifts, and an honest life MANY-VALUED LOGIC The first break from bivalence (1920). Generalised to infinitely many values, it is the root of fuzzy logic, probabilistic and quantum logics — reasoning where "true/false" is too blunt. POLISH NOTATION Parenthesis-free prefix form. Its reverse, RPN, is the native tongue of stack machines, Forth, and expression evaluation inside compilers and calculators. Gate kept on. Two honesties. On the logic : three-valued logic is real and fertile, but adopting it is a choice , not a refutation of classical logic — most of mathematics still runs happily on bivalence, and whether the open future is "genuinely" ½ rather than just unknown-to-us is a philosophical stance, not a theorem. Many-valued logic is a powerful tool, not the discovery that Aristotle was wrong. On the man : Łukasiewicz was a founding giant of the Lwów–Warsaw school , one of the most brilliant logic communities ever, and briefly Poland's minister of religious affairs and education. His path through the Second World War — leaving German-occupied Warsaw and ending, after the war, as a professor in Dublin until his death in 1956 — has been the subject of difficult and contested historical discussion; this page renders his mathematics, which stands entirely on its own, and does not launder or litigate the biography. The third truth value and the notation in every parser are his, plainly. ⚑ [[david-hilbert]] ← · next → Tarski (truth itself). JAN ŁUKASIEWICZ · LOGIKĒ XVIII · two-layer honest live 3-valued logic Ł3 (P,Q ∈ {0,½,1}; ¬∧∨→ + the excluded middle P∨¬P = ½ at P=½, bivalence fails — verified) · a working Polish-notation (prefix) evaluator (verified "+ 2 * 3 4" = 14) three-valued / many-valued logic (1920) → fuzzy & quantum logic · Polish notation → RPN, stack machines, Forth, every expression parser · the Lwów–Warsaw school ← Hilbert · next → Tarski (the definition of truth) LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0


ALFRED TARSKI · XIX

the foundations rank — the definition of TRUTH (model theory

LOGIKĒ XIX — Alfred Tarski (1901–1983). He gave the first rigorous definition of TRUTH: a sentence is true exactly when it correctly describes a structure — truth is relative to a model. This founded model theory, half of modern logic and the basis of database semantics, formal verification, and program correctness. He also proved truth is UNDEFINABLE within its own language (Tarski's theorem, a sibling of Gödel), so 'true' must live in a richer metalanguage — which is exactly what defuses the Liar paradox. Live: evaluate a sentence across different models, and watch the object/metalanguage split dissolve the Liar. Honest: the Banach–Tarski 'paradox' is a real theorem about idealized sets, not physics. David Lee Wise, two-layer honest.

Alfred Tarski — what it means for a sentence to be true ◄ UD0 ← Łukasiewicz LOGIKĒ · XIX · the foundations rank LOGIKĒ · XIX · what it means for a sentence to be true Alfred Tarski 1901 – 1983 · Warsaw & Berkeley · the Lwów–Warsaw school · founder of model theory "True" is the most-used word in logic, and before Tarski no one had said precisely what it means — without spinning in circles or paradox. In 1933 he pinned it down: a sentence is true exactly when it fits the structure it talks about . "Snow is white" is true if and only if snow is white; and a formula's truth is always relative to a model — the world you're describing. This founded model theory , half of all modern logic and the engine under database queries, formal verification, and program semantics. He also proved the deep twist: truth cannot be defined inside its own language — which is exactly why the Liar paradox dissolves. ✓ STRONG Rigorous and load-bearing. Tarski's semantic definition of truth and the model theory it launched are foundational — they underwrite logical consequence, database query semantics, and the formal verification of hardware and software. ◐ TRUTH IS LANGUAGE-RELATIVE You can't say "true" from inside. Tarski's undefinability theorem (a sibling of [[kurt-godel]]): no sufficiently rich language can define its own truth predicate. "True" must be spoken in a higher metalanguage — truth is stratified. ◔ THE "PARADOX" ISN'T PHYSICS Banach–Tarski, read honestly. A solid ball can be cut into pieces and reassembled into two identical balls — a real theorem, but it needs the Axiom of Choice and non-measurable "pieces" with no volume. A truth about idealized sets, not matter. I · Truth in a model — fit between sentence and structure A sentence isn't true or false in a vacuum; it's true in a model — a particular structure that says which basic facts hold. Take the formula (p ∧ q) → r . Toggle which atoms p, q, r hold in the current world, and watch the sentence's truth value. Then click the preset models: the same sentence is true in one and false in another. Truth is a relation between language and structure — Tarski's central insight. this model holds: p q r φ = ( p ∧ q ) → r " φ is true in M " means M makes φ come out true — and different models disagree. This is the T-schema made operational: "(p∧q)→r" is true in M iff (p∧q)→r holds in M. ⚑ Every database query ("is this true of my data?"), every model-checker verifying a chip or protocol, every type-soundness proof is computing exactly this relation — truth-in-a-structure, Tarski-style. "‘Snow is white’ is true if, and only if, snow is white." — Tarski's Convention T, 1933 II · Why the Liar can't speak — truth lives one level up "This sentence is false." If it's true, it's false; if false, true — the Liar paradox . Tarski's diagnosis: the trouble is a language trying to talk about its own truth. His theorem proves no rich language can contain its own truth predicate, so "true" must be spoken in a metalanguage one level above the object language it judges. Stratify the levels, and the Liar simply can't be formed. L := “L is false.” Try to settle it inside one language: suppose L is true suppose L is false Pick one — the single-language version spins forever. object language L lives here can't say "true" metalanguage "true" lives here judges the level below Once "true" can only be asserted from the metalanguage about the object language, the self-referential Liar can never be written — the sentence that names its own falsity isn't expressible. ⚑ Same cure as [[bertrand-russell]]'s types and [[kurt-godel]]'s discipline of levels: self-reference is tamed by stratification. Tarski's hierarchy of metalanguages is why a program can reason about another program's correctness, but not blithely about its own. III · The definition, the limit, the man SEMANTIC TRUTH & MODEL THEORY Truth = satisfaction in a structure (1933). Model theory — the study of which sentences hold in which structures — became half of modern logic and the semantics of computing. UNDEFINABILITY OF TRUTH No rich language defines its own truth predicate (Tarski's theorem) — a semantic twin of Gödel's incompleteness. Truth outruns any single system that tries to capture it from inside. Gate kept on. Tarski's work is rigorous and foundational — but two things deserve honest framing. First, the scope of "truth" : Tarski defined truth for formal languages, not the messy truth of ordinary speech, and his definition makes truth explicitly relative to a model and a metalanguage — it is a precise mathematical relation, not a theory of Truth-with-a-capital-T about reality. Second, the famous Banach–Tarski paradox (with Stefan Banach, 1924): yes, a solid ball can be decomposed into finitely many pieces and reassembled into two balls of the same size — but this is a theorem about idealized point-sets , requiring the Axiom of Choice and "pieces" so pathological they have no definable volume; it says something deep about infinity and measure, and nothing about cutting up real matter. Pop accounts that make it sound like a duplication machine are wrong. The man: a Polish Jew who happened to be lecturing in the United States in September 1939 when Germany invaded Poland — the trip saved his life, though much of his family was murdered in the Holocaust; he became the great founder of logic at Berkeley, training a school that shaped the field for generations. The definition of truth, and the proof that no language can hold its own, are his. ⚑ With Tarski the LOGIKĒ foundations rank rests: from [[thales-of-miletus]]' first proof through the gate, the limits, and now the meaning of truth itself. ALFRED TARSKI · LOGIKĒ XIX · two-layer honest live truth-in-a-model (toggle atoms p,q,r; φ=(p∧q)→r true in one model, false in another — verified) · the Liar dissolved by the object-language / metalanguage split the semantic definition of truth (1933, Convention T) & model theory · the undefinability theorem (truth not definable within its own language) · Banach–Tarski (idealized sets, flagged) ← Łukasiewicz · the LOGIKĒ lineage: Thales → … → the gate, the limits, and the meaning of truth LOGIKĒ · the lineage of logical mathematics · governor David Lee Wise · CC-BY-ND-4.0