{ localUrl: '../page/logical_system.html', arbitalUrl: 'https://arbital.com/p/logical_system', rawJsonUrl: '../raw/5hh.json', likeableId: '3148', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'logical_system', edit: '4', editSummary: '', prevEdit: '3', currentEdit: '4', wasPublished: 'true', type: 'wiki', title: 'Logical system', clickbait: '', textLength: '2082', alias: 'logical_system', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'DylanHendrickson', editCreatedAt: '2016-07-22 17:19:53', pageCreatorId: 'JaimeSevillaMolina', pageCreatedAt: '2016-07-20 21:14:07', seeDomainId: '0', editDomainId: 'arbital_featured_project', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '38', text: 'Logical systems (a.k.a. formal systems) are mathematical abstractions that aim to capture the notion of reasoning to reach valid conclusions from certain premises.\n\nA logical system can be thought of as a procedure which divides a [-language] in badly-formed and well-formed sentences, and further splits this last group into theorems and not theorems.\n\nLogical systems are made from a series of elements: a **language**, a **syntax**, **axioms** and **rules of inference**.\n\nA **language** consists of the [word words] that can be formed from a set of symbols. Typically, we will want our language to be [-enumerable] and [-computable]. For example, a possible language for arithmetic is $\\Sigma^* = \\{\\neg,\\wedge,\\vee,=,+,\\cdot ,0,a_1,a_2,a_3,...\\}^*$.\n\nA **syntax** is the collection of rules which determine whether a word of our language is a well-formed formula. \n\nThe **axioms** are distinguished formulas of the language that are taken to true *a priori*. A logical system is [-axiomatizable] if its set of axioms is computable.\n\nThe **rules of inference** are $n+1$-[-tuples] that represent a function from $n$ formulas (premises) to a new formula (conclusion). For example, we have *modus ponens* as a rule of inference, which says that from a formula of the form $A\\rightarrow B$ and another of the form $A$ you can deduce $B$. Almost always we will want our rules of inference to be [-computable]. Axioms can be thought of as rules of inference for which no premise is necessary.\n\nAxioms and rules of inference are used to construct proofs. A proof of a sentence $S$ of the language is a finite sequence of sentences, such that every sentence is either an axiom or can be deduced from the previous sentences using a rule of inference, and the last sentence in the sequence is $S$. Sentences which have a proof are called theorems of the system.\n\nNote that logical systems are purely syntactical entities - they talk for themselves about nothing. Logical systems are given meaning through [-semantics]. \n\nLogical systems can relate to one another through [-translations].', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'JaimeSevillaMolina', 'EricRogstad', 'DylanHendrickson' ], childIds: [], parentIds: [ 'logic', '2h8' ], commentIds: [], questionIds: [], tagIds: [ 'needs_clickbait_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '2h8', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '3170', likeableType: 'changeLog', myLikeValue: '0', likeCount: '3', dislikeCount: '0', likeScore: '3', individualLikes: [], id: '17323', pageId: 'logical_system', userId: 'DylanHendrickson', edit: '4', type: 'newEdit', createdAt: '2016-07-22 17:19:53', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17219', pageId: 'logical_system', userId: 'JaimeSevillaMolina', edit: '0', type: 'newParent', createdAt: '2016-07-21 07:54:13', auxPageId: '2h8', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '3159', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '17217', pageId: 'logical_system', userId: 'JaimeSevillaMolina', edit: '3', type: 'newEdit', createdAt: '2016-07-21 07:26:33', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17216', pageId: 'logical_system', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-21 06:48:34', auxPageId: 'needs_clickbait_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17215', pageId: 'logical_system', userId: 'EricBruylant', edit: '0', type: 'deleteParent', createdAt: '2016-07-21 06:48:14', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17213', pageId: 'logical_system', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-07-21 06:48:11', auxPageId: 'logic', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17211', pageId: 'logical_system', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-07-21 06:47:38', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17199', pageId: 'logical_system', userId: 'EricRogstad', edit: '2', type: 'newEdit', createdAt: '2016-07-20 23:07:00', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17198', pageId: 'logical_system', userId: 'JaimeSevillaMolina', edit: '1', type: 'newEdit', createdAt: '2016-07-20 21:14:07', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }