30:Tf57,venn = connective => { const φ = 1.61803398875; const width = 200, height = width / φ, radius = height / 2 - height / Math.pow(φ, 5), cx_p = height / φ - Math.pow(φ, 4), cx_q = height + Math.pow(φ, 4), cy = height / 2; return html` ` }31:[] 2e:{"title":"Logic in JavaScript","cells":[{"id":0,"value":"md`# Logic in JavaScript`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2120,"value":"md`This notebook runs permanently through four different states: **${state}**.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2171,"value":"md`It changes its state every **${(state_duration / 1000).toFixed(1)}** seconds. You may want to adjust the time interval between state changes to your liking:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2090,"value":"viewof state_duration = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1211,"value":"md`## Laws of thought`","pinned":false,"hidden":false,"mode":"ojs"},{"id":935,"value":"md`### Principle of bivalence`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1790,"value":"md`The semantic [principle of bivalence](https://en.wikipedia.org/wiki/Principle_of_bivalence) states that every declarative sentence expressing a proposition ${tex`p`} has exactly one [truth value](https://en.wikipedia.org/wiki/Truth_value), that can be either *true* (${tex`\\top`}) or *false* (${tex`\\bot`}). A logic satisfying this principle is called a two-valued logic or bivalent logic.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2642,"value":"tex`p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2646,"value":"html`${p}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":897,"value":"p","pinned":false,"hidden":false,"mode":"ojs"},{"id":1004,"value":"md`### Negation`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1793,"value":"md`Within a two-valued logic, [negation](Negation) is an operation on the value of a proposition ${tex`p`}, that produces a value of *true* when its operand is false and a value of *false* when its operand is true.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2652,"value":"tex`\\neg p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2648,"value":"html`!${p}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1019,"value":"!p","pinned":false,"hidden":false,"mode":"ojs"},{"id":1797,"value":"md`### Double negation`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1024,"value":"md`Within a two-valued logic, [double negation](https://en.wikipedia.org/wiki/Double_negation), that is, the negation of the negation of the value of a proposition ${tex`p`}, is logically equivalent to ${tex`p`}.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2688,"value":"tex`\\neg \\neg p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2684,"value":"html`!!${p}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1039,"value":"!!p","pinned":false,"hidden":false,"mode":"ojs"},{"id":941,"value":"md`### Law of identity`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1804,"value":"md`The [law of identity](https://en.wikipedia.org/wiki/Law_of_identity) states that each proposition ${tex`p`} is identical with itself.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2681,"value":"tex`p = p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2677,"value":"html`${p} === ${p}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":827,"value":"p === p","pinned":false,"hidden":false,"mode":"ojs"},{"id":983,"value":"md`### Law of noncontradiction`","pinned":false,"hidden":false,"mode":"ojs"},{"id":891,"value":"md`The [law of noncontradiction](https://en.wikipedia.org/wiki/Law_of_noncontradiction) states that a proposition ${tex`p`} and its negation ${tex`\\neg p`} cannot both be *true* in the same sense at the same time.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2674,"value":"tex`\\neg (p \\land \\neg p)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2670,"value":"html`

!(${p} && !${p})`","pinned":false,"hidden":false,"mode":"ojs"},{"id":818,"value":"!(p && !p)","pinned":false,"hidden":false,"mode":"ojs"},{"id":993,"value":"md`### Law of excluded middle`","pinned":false,"hidden":false,"mode":"ojs"},{"id":995,"value":"md`The [law of excluded middle](https://en.wikipedia.org/wiki/Law_of_excluded_middle) states that for any proposition ${tex`p`}, either that proposition is *true* or its negation ${tex`\\neg p`} is *true*.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2665,"value":"tex`p \\lor \\neg p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2662,"value":"html`${p} || !${p}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1076,"value":"p || !p","pinned":false,"hidden":false,"mode":"ojs"},{"id":1372,"value":"md`## Truth functions`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1822,"value":"md`In logic, a [truth function](https://en.wikipedia.org/wiki/Truth_function) is a function that accepts truth values as input and produces a truth value as output. Within a two-valued logic, there are sixteen possible truth functions of two logical values, typically the values of two propositions ${tex`p`} and ${tex`q`}. There are several cases where a function isn't depending on one or both of its arguments.\n\nAny of these functions corresponds to a [truth table](https://en.wikipedia.org/wiki/Truth_table) of a certain logical connective in classical logic.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2194,"value":"md`### False/Contradiction`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2198,"value":"venn(\"contradiction\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":2202,"value":"tex`\\bot`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2511,"value":"html`false`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2205,"value":"false","pinned":false,"hidden":false,"mode":"ojs"},{"id":2207,"value":"md`### True/Tautology`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2211,"value":"venn(\"tautology\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":2209,"value":"tex`\\top`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2508,"value":"html`true`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2213,"value":"true","pinned":false,"hidden":false,"mode":"ojs"},{"id":2215,"value":"md`### Proposition p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2219,"value":"venn(\"proposition_p\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":2224,"value":"tex`p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2505,"value":"html`${p}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2226,"value":"p","pinned":false,"hidden":false,"mode":"ojs"},{"id":2228,"value":"md`### Negation of p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2233,"value":"venn(\"negation_p\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":2249,"value":"tex`\\neg p`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2502,"value":"html`!${p}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2251,"value":"!p","pinned":false,"hidden":false,"mode":"ojs"},{"id":2253,"value":"md`### Proposition q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2256,"value":"venn(\"proposition_q\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":2258,"value":"tex`q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2500,"value":"html`${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2261,"value":"q","pinned":false,"hidden":false,"mode":"ojs"},{"id":2263,"value":"md`### Negation of q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2266,"value":"venn(\"negation_q\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":2268,"value":"tex`\\neg q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2496,"value":"html`!${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2270,"value":"!q","pinned":false,"hidden":false,"mode":"ojs"},{"id":122,"value":"md`### Conjunction`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2001,"value":"md`[Logical conjunction](https://en.wikipedia.org/wiki/Logical_conjunction) is an operation on two logical values that has a value of *true* if and only if both of its operands are true.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1992,"value":"venn(\"conjunction\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1996,"value":"tex`p \\land q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2493,"value":"html`${p} && ${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":697,"value":"p && q","pinned":false,"hidden":false,"mode":"ojs"},{"id":2316,"value":"md`### Alternative denial`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2320,"value":"md`[Alternative denial](https://en.wikipedia.org/wiki/Sheffer_stroke) is an operation on two logical values that has a value of *true* if and only if at least one of its operands is false.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2324,"value":"venn(\"alternative_denial\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":2336,"value":"tex`\\neg p \\lor \\neg q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2490,"value":"html`!${p} || !${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2347,"value":"!p || !q","pinned":false,"hidden":false,"mode":"ojs"},{"id":130,"value":"md`### Disjunction`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2008,"value":"md`[Logical disjunction](https://en.wikipedia.org/wiki/Logical_disjunction) is an operation on two logical values that has a value of *false* if and only if both of its operands are false.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1357,"value":"venn(\"disjunction\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1413,"value":"tex`p \\lor q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2487,"value":"html`${p} || ${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":437,"value":"p || q","pinned":false,"hidden":false,"mode":"ojs"},{"id":148,"value":"md`### Joint denial`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2016,"value":"md`[Joint denial](https://en.wikipedia.org/wiki/Logical_NOR) is an operation on two logical values that has a value of *true* if and only if both of its operands are false.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1363,"value":"venn(\"joint_denial\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":218,"value":"tex`\\neg p \\land \\neg q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2483,"value":"html`!${p} && !${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":481,"value":"!p && !q","pinned":false,"hidden":false,"mode":"ojs"},{"id":1394,"value":"md`### Material nonimplication`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2018,"value":"md`[Material nonimplication](https://en.wikipedia.org/wiki/Material_nonimplication) is an operation on two logical values that has a value of *true* if and only if the first operand is true and the second operand is false.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1402,"value":"venn(\"material_nonimplication\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1405,"value":"tex`p \\land \\neg q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1961,"value":"html`${p} && !${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1482,"value":"p && !q","pinned":false,"hidden":false,"mode":"ojs"},{"id":1378,"value":"md`### Material implication`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2021,"value":"md`[Material implication](https://en.wikipedia.org/wiki/Material_conditional) is an operation on two logical values that has a value of *false* if and only if the first operand is true and the second operand is false.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1381,"value":"venn(\"material_implication\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1391,"value":"tex`\\neg p \\lor q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1964,"value":"html`!${p} || ${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1486,"value":"!p || q","pinned":false,"hidden":false,"mode":"ojs"},{"id":1438,"value":"md`### Converse nonimplication`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2026,"value":"md`[Converse nonimplication](https://en.wikipedia.org/wiki/Converse_nonimplication) is an operation on two logical values that has a value of *true* if and only if the first operand is false and the second operand is true.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1443,"value":"venn(\"converse_nonimplication\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1458,"value":"tex`\\neg p \\land q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1967,"value":"html`!${p} && ${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1489,"value":"!p && q","pinned":false,"hidden":false,"mode":"ojs"},{"id":1497,"value":"md`### Converse implication`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2024,"value":"md`[Converse implication](https://en.wikipedia.org/wiki/Converse_implication) is an operation on two logical values that has a value of *false* if and only if the first operand is false and the second operand is true.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1499,"value":"venn(\"converse_implication\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1503,"value":"tex`p \\lor \\neg q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2469,"value":"html`${p} || !${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1506,"value":"p || !q","pinned":false,"hidden":false,"mode":"ojs"},{"id":158,"value":"md`### Exclusive disjunction`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2032,"value":"md`[Exclusive disjunction](https://en.wikipedia.org/wiki/Exclusive_or) is an operation on two logical values that has a value of *true* if and only if one operand is true and the other one is false.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1517,"value":"venn(\"exclusive_disjunction\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":2060,"value":"tex`p \\not\\equiv q`\n\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":2472,"value":"html`${p} !== ${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1567,"value":"p !== q","pinned":false,"hidden":false,"mode":"ojs"},{"id":166,"value":"md`### Logical equality`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2035,"value":"md`[Logical equality](https://en.wikipedia.org/wiki/Logical_biconditional) is an operation on two logical values that has a value of *true* if and only if both operands are false or both operands are true.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1537,"value":"venn(\"logical_equality\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1864,"value":"tex`p \\equiv q`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2475,"value":"\nhtml`${p} === ${q}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1553,"value":"p === q","pinned":false,"hidden":false,"mode":"ojs"},{"id":2045,"value":"md`## Appendix`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1976,"value":"mutable p = false","pinned":false,"hidden":false,"mode":"ojs"},{"id":1979,"value":"mutable q = false","pinned":false,"hidden":false,"mode":"ojs"},{"id":1758,"value":"state = {\n  let i = -1;\n  while (true) {\n    mutable p = !!(i >> 1 & 1);\n    mutable q = !!(i & 1);\n    yield Promises.tick(state_duration, ++i % 4);\n  }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":2053,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":2049,"value":"css = html``","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}