{"id":23,"date":"2018-10-12T15:52:51","date_gmt":"2018-10-12T15:52:51","guid":{"rendered":"http:\/\/jkhorner.com\/?page_id=23"},"modified":"2021-07-19T20:59:35","modified_gmt":"2021-07-19T20:59:35","slug":"boolean-logic","status":"publish","type":"page","link":"https:\/\/jkhorner.com\/index.php\/boolean-logic\/","title":{"rendered":"BOOLEAN LOGIC"},"content":{"rendered":"

File 20180912_AllBoolean_eqvs.pdf<\/a> is a monograph of extensively automated proofs (implemented in Mathematica) of the implicational equivalence of equational Boolean, Huntington, Robbins, Sheffer, and two Wolfram logics.\u00a0 The derivations include what appears to be a novel proof of the Robbins conjecture (“A Robbins logic is a Boolean logic”).<\/p>\n

\n

Mathematica v12.1.0, released March 2020, contains an enhancement of Mathematica’s equational-logic\u00a0 proof search function FindEquationalProof. The enhancement automatically converts Mathematica first-order predicate logic expressions to equational logic (prior versions did not provide this conversion). The v12.1.0 distribution contains an example (Lewis Carroll’s Puzzle Number 1 (1896) showcasing the conversion capability. The example proves that “Babies can’t manage crocodiles”, given:<\/p>\n

(a) All babies are illogical.<\/p>\n

(b) Nobody is despised who can manage a crocodile.<\/p>\n

(c) Illogical persons are despised.<\/p>\n

I made minor modifications to the example provided in the Wolfram distribution; the results are shown here.<\/a><\/p>\n

\n

 <\/p>\n

In a scene in Letters from the Earth (Twain 1962, \u201cExtract from Eve\u2019s Diary\u201d, 82), Adam and Eve stand in the Garden of Eden some distance from the First Lion — \u201cWilliam McKinley\u201d — discussing whether McKinley is a vegetarian. Adam propounds a long-winded, first-principles argument for the claim,
\nconcluding that the First Lion eats no meat.<\/p>\n

Eve, who had seen McKinley devour the First Lamb just a few days earlier, counters, \u201cAdam, I think there is something better than logic.\u201d<\/p>\n

\u201cAnd what could that be?\u201d he asks.<\/p>\n

\u201cFact\u201d, she replies.<\/p>\n

Adam\u2019s argument is in some ways akin to some of Lewis Carroll\u2019s \u201cLogical Puzzles\u201d (1896), which trade on contrapositive inference. Here, I use the automated deduction functionality in Mathematica (2020) to produce an equational logic proof of a variant of Adam\u2019s argument.<\/p>\n

Click Lions Eat Only Strawberries<\/a> to see the proof.<\/p>\n

———-<\/p>\n

Carroll L. (1896).\u00a0 Symbolic Logic. Reprinted by Clarkson N.Potter, 1977.<\/p>\n

Twain M. (1962). Letters From the Earth.\u00a0 B. DeVoto (ed.)\u00a0 Harper and Row.<\/p>\n

Wolfram Research (2020).\u00a0 Mathematica v12 .1 .0 Home Edition.
\nhttps : \/\/ www.wolfram.com\/mathematica – home – edition \/. Accessed 19 March 2020.<\/p>\n

 <\/p>\n

\n

Click here<\/a> for a Mathematica-based automated equational logic derivation of the implicational equivalence of Lukasiewicz’s CN and the Principia Mathmatica sentential calculi.<\/p>\n

\n

Click here<\/a> for a Mathematica-based automated equational logic derivation of Exercise 60 (“I avoid kangaroos”) of Lewis Carroll’s Symbolic Logic<\/em>.<\/p>\n","protected":false},"excerpt":{"rendered":"

File 20180912_AllBoolean_eqvs.pdf is a monograph of extensively automated proofs (implemented in Mathematica) of the implicational equivalence of equational Boolean, Huntington, Robbins, Sheffer, and two Wolfram logics.\u00a0 The derivations include what appears to be a novel proof of the Robbins conjecture (“A Robbins logic is a Boolean logic”). Mathematica v12.1.0, released March 2020, contains an enhancement … <\/p>\n

Continue reading “BOOLEAN LOGIC”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-23","page","type-page","status-publish","hentry"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/pages\/23","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/comments?post=23"}],"version-history":[{"count":12,"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/pages\/23\/revisions"}],"predecessor-version":[{"id":397,"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/pages\/23\/revisions\/397"}],"wp:attachment":[{"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/media?parent=23"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}