{"id":33283,"date":"2021-01-18T00:00:07","date_gmt":"2021-01-18T00:00:07","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=33283"},"modified":"2025-07-30T23:23:06","modified_gmt":"2025-07-30T23:23:06","slug":"lingua-logicae-propositionalis","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/lingua-logicae-propositionalis\/","title":{"rendered":"Lingua Logicae Propositionalis"},"content":{"rendered":"<h1 style=\"text-align: center;\">Lingua Logicae Propositionalis<\/h1>\n<h4 style=\"text-align: center;\">Summarium<\/h4>\n<p style=\"text-align: center;\"><em>In hoc commentario recensetur lingua logicae propositionalis ut metalingua adhibita ad expressiones validas linguae basis, ex duobus symbolis compositae, obtinendas. Explicantur regulae syntacticae, notiones variabilium propositionalium et connectoris, atque etiam introductio negationis coniunctae, usus parenthesium et reordinationis ad lectionem expressionum faciliorem reddendam. Praeterea mentionem facimus vocalizationum expressionum logicae propositionalis. Denique lingua logicae propositionalis tamquam instrumentum fundamentale in mathematica et logica perstringitur, et reflectitur de possibilitate inveniendi \u00ablinguam basim basis\u00bb ex qua reliqua omnia iterum constitui possint.<\/em><\/p>\n<h4 style=\"text-align: center;\">Proposita Discendi:<\/h4>\n<p style=\"text-align: justify;\"><em>Completa hac sectione, exspectatur discipulum posse:<\/em><\/p>\n<ol style=\"text-align: justify;\">\n<li><strong>Intelligere<\/strong> notionem metalinguae eiusque usum in logica propositionali. <\/li>\n<li><strong>Comprehendere<\/strong> regulas syntacticas linguae logicae propositionalis. <\/li>\n<li><strong>Nosse<\/strong> notionem variabilis propositionalis eiusque usum in structura expressionum. <\/li>\n<li><strong>Intelligere<\/strong> usum connectoris et negationis coniunctae in lingua logicae propositionalis. <\/li>\n<li><strong>Discere<\/strong> uti parenthesibus et reordinatione ad lectionem expressionum faciliorem reddendam. <\/li>\n<li><strong>Nosse<\/strong> vocalizationes expressionum logicae propositionalis. <\/li>\n<li><strong>Perstringere<\/strong> linguam logicae propositionalis tamquam instrumentum fundamentale in mathematica et logica. <\/li>\n<li><strong>Reflectere<\/strong> de possibilitate inveniendi \u00ablinguam basim basis\u00bb ex qua reliqua omnia iterum constitui possint. <\/li>\n<li><strong>Adhibere<\/strong> notiones didactas in structura expressionum logicae propositionalis. <\/li>\n<li><strong>Uti<\/strong> lingua logicae propositionalis ad problemata mathematica et logica intelligenda et solvenda.<\/li>\n<\/ol>\n<h4 style=\"text-align: center;\">Index<\/h4>\n<p style=\"text-align:center;\">\n<a href=\"#1\"><strong>LINGUA LOGICAE PROPOSITIONALIS: ALPHABETA ET CATENAE SYMBOLORUM<\/strong><\/a><br \/>\n<a href=\"#2\">INITIUM FACIAMUS CUM UNICO SYMBOLO<\/a><br \/>\n<a href=\"#3\">DEINDE ADDAMUS SECUNDUM SYMBOLUM<\/a><br \/>\n<a href=\"#4\"><strong>LINGUA LOGICAE PROPOSITIONALIS: SYNTAXIS<\/strong><\/a><br \/>\n<a href=\"#5\">EXEMPLA RECOGNITIONIS SYNTAXIS<\/a><br \/>\n<a href=\"#6\"><strong>CONVENTIONES NOTATIONIS<\/strong><\/a><br \/>\n<a href=\"#7\">METAVARIABILIA ET CONNECTOR <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span><\/a><br \/>\n<a href=\"#8\">EXEMPLA USUS NEGATIONIS CONIUNCTAE<\/a><br \/>\n<a href=\"#9\">REORDINATIO ET PARENTHESES<\/a><br \/>\n<a href=\"#10\"><strong>CONNECTORES DERIVATI<\/strong><\/a><br \/>\n<a href=\"#11\">VOCALIZATIO EXPRESSIONUM LOGICAE PROPOSITIONALIS<\/a><br \/>\n<a href=\"#12\"><strong>SYNTESIS ET REFLECTIONES DE LINGUA LOGICAE PROPOSITIONALIS<\/strong><\/a><br \/>\n<a href=\"#13\">MATRIX POST MATRIX POST INTELLIGENTIAM OMNIUM RERUM<\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/WwBKcSXIznA\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Lingua Logicae Propositionalis: Alphabeta et Catenae Symbolorum<\/h2>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Initium faciamus cum uno symbolo<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=WwBKcSXIznA&amp;t=37s\" target=\"_blank\" rel=\"noopener\"><strong>Ad linguam logic\u00e6 propositionalis construendam<\/strong><\/a>, studium nostrum incipiemus ex alphabeto simplicissimo: illo quod unum tantum symbolum continet. Figura symboli parum refert, sed momenti est eius unicitas. Si hoc alphabeto scribimus, sola differentia inter catenas symbolorum est numerus repetitionum symboli. Ergo, si possumus scribere catenas symbolorum usque ad longitudinem <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span>, tantummodo <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> catenae diversae scribi possunt. Ut videre potes, hoc alphabetum valde limitatum est et non multum dici potest ultra hoc.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Adiciamus igitur secundum symbolum<\/h3>\n<p style=\"text-align: justify;\">Si secundum symbolum nostro alphabeto addamus, scriptura fit ditior quam in alphabeto priore. Nunc videre possumus ordinem symbolorum; exempli gratia, si <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> sunt nostra symbola, distinguere possumus inter <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">10<\/span><\/span>. Utraque catena eisdem symbolis utitur, sed in ordine diverso. Si catena longissima quam scribere possumus longitudinem habet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N =1,2,3,\\cdots<\/span><\/span>, tunc possumus scribere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2^1=2<\/span><\/span> catenas longitudinis <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2^2=4<\/span><\/span> catenas longitudinis <span class=\"katex-eq\" data-katex-display=\"false\">2<\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2^3=8<\/span><\/span> catenas longitudinis <span class=\"katex-eq\" data-katex-display=\"false\">3<\/span>, et ita porro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> 2^N<\/span><\/span> catenas distinctas longitudinis <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span>.<\/p>\n<p style=\"text-align: justify;\"><strong>Exercitium:<\/strong> Scribuntur in charta omnes catenae distinctae quae fieri possunt inter <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> symbola. Quot catenae in summa scribuntur? <\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #0000aa;\"><strong>Solutio:<\/strong><\/span><br \/>\n Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S_N<\/span><\/span> est summa omnium catenarum, longitudinis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1, 2, 3, <\/span><\/span> et sic usque ad <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span>, iam vidimus quod: <\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle S_N=2^1 + 2^2 + \\cdots +2^{N-1} + 2^N <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Multiplicando per 2 expressionem superiorem, habemus:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 2 S_N=2^2 + 2^3 + \\cdots + 2^N + 2^{N+1} <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Et igitur:<\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle S_N=2 S_N - S_N = 2^N-1 <\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ergo, numerus totalis catenarum in charta scriptarum erit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2^N-1<\/span><\/span>.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Lingua Logicae Propositionalis: Syntaxis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=WwBKcSXIznA&amp;t=295s\" target=\"_blank\" rel=\"noopener\"><strong>Vidimus quod, duobus symbolis adhibitis, distinguere possumus unam catenam ab alia per longitudinem eius et ordinem symbolorum. <\/strong><\/a> Hoc grave est quia nobis permittit definire syntaxim pro alphabeto quod construximus. Syntaxis est congeries regularum quae catenas symbolorum in duas categorias dividit: Expressiones et Non-Expressiones. Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{L}_2<\/span><\/span> est collectio omnium catenarum quae fieri possunt ex symbolis <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span>, tunc syntaxis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{L}_2<\/span><\/span> est subcollectio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2\\subset\\mathcal{L}_2<\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">Possumus definire collectionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2<\/span><\/span> his regulis recursive:<\/p>\n<ol>\n<li style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">00, 11 \\in \\mathcal{SL}_2<\/span><\/span><\/li>\n<li style=\"text-align: justify;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta \\in \\mathcal{SL}_2<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01\\alpha\\beta \\in \\mathcal{SL}_2<\/span><\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify;\">His duabus regulis possumus expressiones huius linguae construere et probare utrum data catena sit expressio huius linguae. Lingua est alphabetum cum syntaxi associata. Linguae hic praesentatae nomen dabimus <em>\u00abLingua Basis Duorum Symbolorum\u00bb,<\/em> sive <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Exempla Recognitionis Syntaxis<\/h3>\n<p style=\"text-align: justify;\">Ut hae notiones facilius intellegantur, inspiciamus haec exempla:<\/p>\n<p style=\"text-align: justify;\"><strong>Exemplum:<\/strong> Quoniam <span style=\"color: #FF4500;\"><span class=\"katex-eq\" data-katex-display=\"false\">0000<\/span><\/span> et <span style=\"color: #DAA520 ;\"><span class=\"katex-eq\" data-katex-display=\"false\">1111<\/span><\/span> continentur in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2<\/span><\/span>, habemus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span><span style=\"color: #FF4500;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">00 00 01 00<\/span><\/span><\/span><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">11 01 11<\/span><\/span><\/span><span style=\"color: #FF4500;\"><span class=\"katex-eq\" data-katex-display=\"false\">0000<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span><span style=\"color: #DAA520 ;\"><span class=\"katex-eq\" data-katex-display=\"false\">1111<\/span><\/span><span style=\"color: #DAA520 ;\"><span class=\"katex-eq\" data-katex-display=\"false\">1111<\/span><\/span> in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2<\/span><\/span>; ergo, sunt expressiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>. Hoc demonstratur applicando regulas quas modo exposuimus.<br \/>\n<\/span><\/p>\n<p>Finis exempli <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\blacksquare<\/span><\/span><\/p>\n<p><strong>Exercitium:<\/strong> In exemplo superiore vidimus quomodo expressiones ex aliis duabus expressionibus elementaribus construantur. Hoc per se non est difficile negotium; attamen, processus inversus, qui consistit in probando utrum certa expressio sit vel non sit expressio, fortasse paulo difficilior sit.<\/p>\n<p style=\"text-align: justify;\"><span style=\"\">Determina, regulis syntacticis utens, utrum catenae sequentes sint vel non sint expressiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>:<br \/>\n<\/span><\/p>\n<ol>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{}012100<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">101100<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{}0100010000<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0101000011<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{}01010000010000<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01010010000100101000011<\/span><\/span><\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: justify;\"><span style=\"color: #0000aa;\"><strong>Solutio:<\/strong><\/span><br \/>\nAntequam solutionem videas, suadeo ut prius ipse coneris et deinde cum solutione compares. Si iam fecisti, progredere \ud83d\udc4d<\/p>\n<ol>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span><span class=\"katex-eq\" data-katex-display=\"false\">2<\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">100<\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">Ut videre possumus, haec catena symbolum <span class=\"katex-eq\" data-katex-display=\"false\">2<\/span> continet, quod non est in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{L}_2<\/span><\/span>; ergo haec catena non potest esse in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2<\/span><\/span> et ita non est expressio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>.<\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">10<\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1100<\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">Hic videmus hanc catenam incipere per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">10<\/span><\/span>. Ex regulis syntacticis inferre possumus omnes catenae longitudinis maioris quam 2 incipere necesse esse per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span>; ergo haec non potest esse expressio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>.<\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0100010000<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Haec catena incipit per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span>, itaque transit probationem primam. Ex hoc sequitur ut, ut sit expressio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{L}_2<\/span><\/span>, necesse est ut pars caerulea unice in duas expressiones dividi possit.<span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">00010000<\/span><\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"\">Si, licet legibus syntacticis obtemperet, divisio non est unica, tunc syntaxis definita est ambigua atque corrigenda.<br \/>\n<\/span><\/p>\n<p style=\"text-align: justify;\"> Partem caeruleam examinantes, habemus has possibilitates divisionis:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0010000<\/span><\/span><\/span><\/td>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">00<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">010000<\/span><\/span><\/span><\/td>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">000<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">10000<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0001<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0000<\/span><\/span><\/span><\/td>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">00010<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">000<\/span><\/span><\/span><\/td>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">000100<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">00<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0001000<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/span><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hic animadvertendum est: si pars aurea non est <span class=\"katex-eq\" data-katex-display=\"false\">0000<\/span> vel <span class=\"katex-eq\" data-katex-display=\"false\">1111<\/span>, tum pars caerulea respondens incipere debet per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span> ut tota catena sit expressio; ideo fieri possunt sequentia reiectionum iudicia:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{}0010000<\/span><\/span><\/span>\u274c<\/td>\n<td><span style=\"color: #DAA520 ;\"><span class=\"katex-eq\" data-katex-display=\"false\">00<\/span><\/span><span style=\"color: #007ACC ;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">010000<\/span><\/span>\u2705<\/td>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">000<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{}10000<\/span><\/span><\/span>\u274c<\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0001<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0000<\/span><\/span><\/span>\u274c<\/td>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">00010<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">000<\/span><\/span><\/span>\u274c<\/td>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">000100<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">00<\/span><\/span><\/span>\u274c<\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #DAA520 ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0001000<\/span><\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span><\/span>\u274c<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hac de causa sola divisio quae huic analysi superest est <span style=\"color: #DAA520 ;\"><span class=\"katex-eq\" data-katex-display=\"false\">00<\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">010000<\/span><\/span><\/span>, ubi pars aurea est expressio et pars caerulea unice ac secundum syntaxim dividitur. Denique catena <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0100010000<\/span><\/span> admittit unicam divisionem syntaxim consistentem, quae est <span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><span style=\"color: #DAA520 ;\"><span class=\"katex-eq\" data-katex-display=\"false\">00<\/span><\/span><span style=\"color: #007ACC ;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">010000<\/span><\/span>, et ideo est expressio linguae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0101000011<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Huius catenae sequentem divisionem, coloribus notatam, facere possumus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span><span style=\"color: #007ACC ;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">010000<\/span><\/span><\/span><span style=\"color: #DAA520 ;\"><span class=\"katex-eq\" data-katex-display=\"false\">1111<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ex regulis syntacticis, ut catena longitudinis maioris quam 2 sit expressio, necesse est incipere per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span>, et postea sequantur duae expressiones, quas caeruleo et aureo notavimus. Facile est videre hanc divisionem esse unicam: si pars caerulea aut aurea longitudinem mutet, utraque simul expressiones esse non poterunt.<\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01010000010000<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ab extrema parte ad sinistram revisendo, sequentem divisionem invenire possumus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\underbrace{01\\underbrace{01\\overbrace{00}\\overbrace{00}}_{{expresi\u00f3n}}\\underbrace{01\\overbrace{00}\\overbrace{00}}_{{expresi\u00f3n}}}_{{expresi\u00f3n}}<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01010010000100101000011<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Oculus acutus animadvertet hanc catenam longitudinem habere 23, et fieri non posse catenam longitudinis imparis construi per regulas syntaxis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{L}_2<\/span><\/span>, quae expressiones componit concatenando catenas longitudinis paris. Omnes catenae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2<\/span><\/span> habent longitudinem parem, ergo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01010010000100101000011<\/span><\/span> non est expressio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>.<\/p>\n<\/li>\n<\/ol>\n<p>Finis exercitii <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\blacksquare<\/span><\/span><\/p>\n<figure id=\"attachment_25115\" aria-describedby=\"caption-attachment-25115\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/11\/simbolos.jpg\" alt=\"una tablilla con mucho s\u00edmbolos decodificados\" width=\"1081\" height=\"399\" class=\"size-full wp-image-25115 lazyload\" \/><figcaption id=\"caption-attachment-25115\" class=\"wp-caption-text\"><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/11\/simbolos.jpg\" alt=\"una tablilla con mucho s\u00edmbolos decodificados\" width=\"1081\" height=\"399\" class=\"size-full wp-image-25115 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/11\/simbolos.jpg 1081w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/11\/simbolos-300x111.jpg 300w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/11\/simbolos-1024x378.jpg 1024w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/11\/simbolos-768x283.jpg 768w\" sizes=\"(max-width: 1081px) 100vw, 1081px\" \/><\/noscript><\/figcaption><\/figure>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Conventiones Notationis<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=WwBKcSXIznA&amp;t=918s\" target=\"_blank\" rel=\"noopener\"><strong>Laborare cum nullis et unitatibus potest esse confusum perceptioni nostrae<\/strong><\/a> et errores inducere potest. Ut processus humanis mentibus amicior fiat, possumus uti conventionibus notationis et quibusdam metasymbolis.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Metavariabiles et Connector <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span><\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=WwBKcSXIznA&amp;t=950s\" target=\"_blank\" rel=\"noopener\"><strong>Metasymbolum est symbolum adhibitum ad repraesentandas catenas symbolorum linguae obiectivae.<\/strong><\/a> Exempli gratia, cum syntaxis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{L}_2<\/span><\/span> definita est, symbola <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> adhibita sunt ad repraesentandas expressiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>. Haec symbola vocantur <strong>metavariabiles <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>:<\/strong> metasymbola quae, cum omnia substituuntur expressionibus linguae, generant per syntaxin aliam expressionem eiusdem linguae, ut indicat secunda regula de elementis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2<\/span><\/span>:<\/p>\n<p style=\"text-align: center;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\in \\mathcal{SL}_2<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01\\alpha\\beta \\in\\mathcal{SL}_2<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quam ob rem dicitur has metavariabiles esse <strong>metaexpressiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>.<\/strong><\/p>\n<p style=\"text-align: justify;\">Ut scribendi rationem nostram in posterum faciliorem reddamus, usuri sumus metasymbolo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span><\/span> ad repraesentandam catenam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">01<\/span><\/span>. Hoc metasymbolum dicitur <strong>connector<\/strong> et appellatur <strong>Negatio Coniuncta<\/strong> ob rationes semanticas.<\/p>\n<p style=\"text-align: justify;\">His positis, possumus syntaxim <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{SL}_2<\/span><\/span> modo metalinguistico exprimere per sequentis regulas recursivas:<\/p>\n<ol>\n<li>\n<p style=\"text-align: justify;\">Omnes metavariabiles <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span> sunt metaexpressiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span> sunt metavariabiles <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha\\beta<\/span><\/span> est metaexpressio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span><\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">His regulis possumus scribere metaexpressiones quae, cum omnibus metavariabilibus substitutis expressionibus et <strong>connectoribus<\/strong> in forma per nullos et unitates expressa, dant expressionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>. Quaelibet talis metaexpressio refertur ad infinitam familiam expressionum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span>: collectionem omnium expressionum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{B}_2<\/span><\/span> quae per illam structuram repraesentari possunt. Hoc est ipsum quod significat habere linguam formalem.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h4>Exempla Usus Negationis Coniunctae<\/h4>\n<p style=\"text-align: justify;\"><strong>Exemplum:<\/strong> <span style=\"\"> Ex metaexpressione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha\\downarrow\\beta\\gamma<\/span><\/span> per substitutiones obtineri possunt sequentes expressiones:<br \/>\n<\/span><\/p>\n<ol>\n<li>\n<p style=\"text-align: justify;\">Substituendo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha := 00<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta := 011100<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma := 010011<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Advenitur ad expressionem:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">010001011100010011<\/span><\/span><\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\">Si substituimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha := 011100<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta := 0111011100<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma := 0111010011<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Generatur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">010111000101110111000111010011<\/span><\/span><\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">Metaexpressio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha\\downarrow\\beta\\gamma<\/span><\/span> non solum facilius intellegitur quam quaelibet alia expressio formae satisfaciens, sed etiam repraesentat omnes expressiones quae inde obtineri possunt per substitutionem metavariabilium per expressiones.<\/p>\n<p>Finis exempli <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\blacksquare<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Cum metavariabilis substituitur, substituitur in omnibus locis ubi apparet.<\/p>\n<p style=\"text-align: justify;\"><strong>Exemplum:<\/strong> <span style=\"\"> Consideremus metaexpressionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\downarrow\\alpha\\beta\\downarrow\\alpha\\gamma<\/span><\/span><br \/>\n<\/span><\/p>\n<ol>\n<li>Si substituimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha:=11<\/span><\/span>, tunc obtinemus:\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\downarrow 11\\beta\\downarrow 11\\gamma<\/span><\/span><\/p>\n<\/li>\n<li>Si nunc facimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta:=011100<\/span><\/span>, tunc resultat:\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\downarrow 11011100\\downarrow 11\\gamma<\/span><\/span><\/p>\n<\/li>\n<li>Et si nunc mutationem facimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma:=011111<\/span><\/span>, habebimus:\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\downarrow 11011100\\downarrow 11011111<\/span><\/span><\/p>\n<\/li>\n<li>Denique, mutando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow:=01<\/span><\/span>, concludemus cum hac expressione:\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0101110111000111011111<\/span><\/span><\/p>\n<\/li>\n<\/ol>\n<p>Finis exempli <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\blacksquare<\/span><\/span><\/p>\n<p><a name=\"9\"><\/a><\/p>\n<h3>Reordinatio et Parentheses<\/h3>\n<p style=\"text-align: justify;\">Verificare hanc esse metaexpressionem non est difficillimum, sed requirit attentionem continuam ad numerum metasymbolorum atque ambitum connectoris <span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow<\/span>. Haec difficultas celeriter crescit cum longitudine metaexpressionis. Hinc oritur quaestio an aliqua sit methodus ad haec repraesentanda modo faciliore ad inspiciendum \u2014 responsio est: ita vero; scilicet uti possumus parenthesibus et reordinationibus aptioribus ad formam nostram naturalem rerum aggregationis. Ad hoc illustrandum, inspiciamus sequentem metaexpressionem:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha\\downarrow\\downarrow\\alpha\\beta\\alpha<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Evenit ut, licet non sit difficile comprobare hanc esse metaexpressionem, hoc fieri non possit nisi symbola numeremus, periculo in enumeratione fallendi. Et hoc periculum celeriter augetur cum crescit longitudo expressionis. Estne via ut idem repraesentetur modo magis perspicuo? Certe exstat talis methodus, quae nostris naturalibus modis cogitandi et ordinandi convenit. Hac de causa introducuntur parentheses et reordinatio per sequentem conventionem notationis:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha\\beta:=(\\alpha\\downarrow\\beta)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><strong>Exemplum:<\/strong> Consideremus metaexpressionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha\\downarrow\\downarrow\\beta\\gamma\\delta<\/span><\/span>. Si parenthesim introductionem et reordinationem adhibemus, tunc transformabitur hoc modo:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha\\downarrow<\/span><\/span><span style=\"color: #FF4500;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\beta\\gamma<\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha\\downarrow<\/span><\/span><span style=\"color: #FF4500;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\beta\\downarrow \\gamma)<\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha<\/span><\/span><span style=\"color: #FF4500;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow(\\beta\\downarrow \\gamma)\\delta<\/span><\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha<\/span><\/span><span style=\"color: #FF4500;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\beta\\downarrow \\gamma)\\downarrow\\delta)<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #FF4500;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\downarrow\\alpha((\\beta\\downarrow \\gamma)\\downarrow\\delta)<\/span><\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span style=\"color: #FF4500;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\downarrow((\\beta\\downarrow \\gamma)\\downarrow\\delta))<\/span><\/span><\/span> \u2705<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Haec ultima metaexpressio multo facilior est ad legendum et inspiciendum quam originalis, quia singuli parenthesis clauduntur circa metaexpressionem quae elementis bene distinctis componitur: negatio coniuncta in medio, cum metaexpressione ex utraque parte.<\/p>\n<p>Finis exempli <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\blacksquare<\/span><\/span><\/p>\n<p><a name=\"10\"><\/a><\/p>\n<h3>Connectores Derivati<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=WwBKcSXIznA&amp;t=1478s\" target=\"_blank\" rel=\"noopener\"><strong>Et in logica et in reliqua parte mathematicae,<\/strong><\/a> adsunt quaedam connectorum compositiones quae frequenter adhibentur. Quam ob rem, ad scripturam (humanis mentibus) magis commodam reddendam, connectores derivati introducuntur per sequentes notationis conventiones:<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Negatio:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg \\alpha<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha\\downarrow\\alpha)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Disiunctio Inclusiva:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vee \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\alpha\\downarrow\\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Coniunctio:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\wedge \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\neg\\alpha\\vee \\neg\\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Implicatio:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\neg\\alpha\\vee \\beta)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Implicatio Duplex:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\leftrightarrow \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">((\\alpha\\rightarrow \\beta)\\wedge(\\beta \\rightarrow \\alpha))<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><strong>Disiunctio Exclusiva:<\/strong><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\veebar \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">:=<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg(\\alpha\\leftrightarrow \\beta)<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hoc metalinguum quod super <strong>lingua fundamentali duorum symbolorum<\/strong> aedificavimus, illud est quod dicitur <strong>Lingua Logicae Propositionalis Ordinis Nulli.<\/strong> Per hanc linguam repraesentantur omnes expressiones logicae propositionalis modo accurato et sine ambiguitate.<\/p>\n<p><a name=\"11\"><\/a><\/p>\n<h2>Vocalizatio Expressionum Logicae Propositionalis<\/h2>\n<p style=\"text-align: justify;\">Etsi ad logicam exercendam non est necessarium, considerandum est nos non solum symbolis scriptis communicare, sed etiam naturalem inclinationem habere ad vocalizandum in nostra lingua vernacula. Quamobrem expressionibus linguae logicae propositionalis adsunt vocalizationes quae evocant notiones similes illis quas tractant sui homologis in logica propositionali. Hae vocalizationes sunt sequentes:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\downarrow \\beta)<\/span><\/span><\/td>\n<td>Ne <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> nec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\neg \\alpha<\/span><\/span><\/td>\n<td>Negatio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vee \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> aut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\wedge \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\rightarrow \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> implicat <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\leftrightarrow \\beta)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> si et solum si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\veebar \\beta)<\/span><\/span><\/td>\n<td>aut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span>, aut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span><\/span>, sed non ambo<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"12\"><\/a><\/p>\n<h2>Synthesis et Meditationes de Lingua Logicae Propositionalis<\/h2>\n<p style=\"text-align: justify;\">Hac ultima parte concluditur constructio linguae logicae propositionalis, quam possumus compendiose describere ut metalinguam quae permittit expressiones validas obtinere in lingua fundamentali duorum symbolorum. Lingua logicae propositionalis est lingua formalis, cum definire possit structuram (vel formam) expressionum in lingua base, et unaquaeque eius expressio determinat formam infinitae familiae expressionum in lingua base. Ut antea dictum est, syntaxis linguae formalis est stricte definita, sed pro hoc rigore offert praecisionem et exactitudinem: caret ambiguitate.<\/p>\n<p><a name=\"13\"><\/a><\/p>\n<h3>Matrix post Matrix post Intelligentiam Omnium Rerum<\/h3>\n<p style=\"text-align: justify;\"><span style=\"\">Ultimum quoddam. Logica propositionalis et mathematica in magna parte innituntur logicae propositionali, quae ipsa aedificata est ex lingua fundamentali quae constat ex unitatibus et nullis. Significatne hoc nos advenisse ad \u00abMatrix\u00bb quae subest logicae et mathematicae? Fieri potest. Sed etiam fieri potest ut cogitemus linguam fundamentalem pro ipsa lingua fundamentali, ex qua omnia reliqua denuo aedificari possint; tamen, ad talem linguam reperiendam oporteret nos inquirere in notiones etiam fundamentalius positas quam conceptus ordinis et quantitatis (quibus adhibitis lingua prima constituta est). Linguam basim basis invenire significat cogitationes suscipere de rebus omnium fundamentalissimis quae pertinent ad id quod significat \u00abintelligere res\u00bb. Si altius pervenis, si ad fundum attingere potes, dici posset te vidisse \u00abMatrix post Matrix post Intelligentiam Omnium Rerum\u00bb, atque fieri potest ut hic processus fundamentationis ad infinitum prosequatur, unicuique fundamento novum gradum profunditatis cognitionis conferens.<\/span><\/p>\n<p><a name=\"12\"><\/a><\/p>\n<h2>Synthesis et Meditationes de Lingua Logicae Propositionalis<\/h2>\n<p style=\"text-align: justify;\">Hac parte finali completur constructio linguae logicae propositionalis, quam compendiose describere possumus ut metalinguam quae permittit expressiones validas generari in lingua fundamentali duorum symbolorum. Lingua logicae propositionalis est lingua formalis, quia definit structuram (seu formam) expressionum in lingua base, et unaquaeque eius expressio determinat formam infinitae familiae expressionum in illa lingua. Ut antea dictum est, syntaxis linguae formalis est valde rigida, sed vicissim praebet praecisionem et exactitudinem: caret ambiguitate.<\/p>\n<p><a name=\"13\"><\/a><\/p>\n<h3>Matrix post Matrix post Intelligentiam Omnium Rerum<\/h3>\n<p style=\"text-align: justify;\"><span style=\"\">Ultimum quod restat. Logica propositionalis et mathematica magna ex parte nituntur logica propositionale, quae ipsa ex lingua fundamentali confecta est ex unitatibus et nullis. Significatne hoc nos pervenisse ad \u00abMatrix\u00bb quae logicae et mathematicae subest? Forsitan. Sed etiam considerari potest lingua fundamentalis ad ipsam linguam fundamentalem, ex qua universum reliquum iterum constitui posset; attamen, ut talem linguam inveniamus, necesse esset notiones reperire adhuc magis fundamentales quam conceptus ordinis et quantitatis (qui ad linguam primam instituendam adhibiti sunt). Invenire linguam basim basis implicat meditationes de rebus maxime fundamentalibus quae pertinent ad significationem verbi \u201cintelligere res\u201d. Si altius investigas, si ad fundamentum pervenis, dici posset te vidisse \u201cMatrix post Matrix post Intelligentiam Omnium Rerum\u201d, et fieri potest ut hic processus fundamentationis ad infinitum progrediatur, novam cognitionis profunditatem tribuens singulis gradibus fundamentalibus.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Lingua Logicae Propositionalis Summarium In hoc commentario recensetur lingua logicae propositionalis ut metalingua adhibita ad expressiones validas linguae basis, ex duobus symbolis compositae, obtinendas. Explicantur regulae syntacticae, notiones variabilium propositionalium et connectoris, atque etiam introductio negationis coniunctae, usus parenthesium et reordinationis ad lectionem expressionum faciliorem reddendam. Praeterea mentionem facimus vocalizationum expressionum logicae propositionalis. Denique lingua [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25112,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":42,"footnotes":""},"categories":[1352,1358,1298],"tags":[],"class_list":["post-33283","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-logica-mathematica","category-logica-propositionalis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Lingua Logicae Propositionalis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Exploratio diligens sermonis logicae propositionalis, comprehendentis notiones metalinguisticae, syntaxis, variabiles propositionales et connexiones.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/la\/lingua-logicae-propositionalis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Lingua Logicae Propositionalis\" \/>\n<meta property=\"og:description\" content=\"Exploratio diligens sermonis logicae propositionalis, comprehendentis notiones metalinguisticae, syntaxis, variabiles propositionales et connexiones.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/lingua-logicae-propositionalis\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-01-18T00:00:07+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-30T23:23:06+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/11\/lenguajedelalogicaproposicional-1024x378.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Lingua Logicae Propositionalis\" \/>\n<meta name=\"twitter:description\" content=\"Exploratio diligens sermonis logicae propositionalis, comprehendentis notiones metalinguisticae, syntaxis, variabiles propositionales et connexiones.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/11\/lenguajedelalogicaproposicional.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"11 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lingua-logicae-propositionalis\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lingua-logicae-propositionalis\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Lingua Logicae Propositionalis\",\"datePublished\":\"2021-01-18T00:00:07+00:00\",\"dateModified\":\"2025-07-30T23:23:06+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lingua-logicae-propositionalis\\\/\"},\"wordCount\":2699,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lingua-logicae-propositionalis\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/11\\\/lenguajedelalogicaproposicional.jpg\",\"articleSection\":[\"Logica Mathematica\",\"Logica Propositionalis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lingua-logicae-propositionalis\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lingua-logicae-propositionalis\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/lingua-logicae-propositionalis\\\/\",\"name\":\"Lingua Logicae Propositionalis - 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