[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657683-105":59,"doc-detail-657683-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","logic-and-the-foundations-of-mathematics","Logic and the Foundations of Mathematics","","An encyclopedia-style overview of logic’s foundational themes and key developments, with special focus on set theory and axiomatic approaches aimed at avoiding paradoxes. It summarizes Zermelo–Fraenkel set theory and surveys major logical topics and reference entries relevant to logic and mathematics. The text explains logic diagrams, tracing their historical evolution from Euler’s circle methods and Lambert’s line diagrams to Venn diagrams, and demonstrates how diagrams validate syllogisms through set intersections and shading conventions.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/logic-and-the-foundations-of-mathematics/657683/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/logic-and-the-foundations-of-mathematics/657683.png","ImageObject",300,407,{"name":92,"@type":93},"Asher","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the goal of Zermelo–Fraenkel set theory as described here?","Question",{"text":112,"@type":113},"It avoids paradoxes by dropping the axiom of abstraction and replacing it with a set of axioms about set existence.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What are logic diagrams, and what are they used for?",{"text":117,"@type":113},"They are geometrical figures isomorphic to the structure of statements in formal logic. They help teach logical structure, check results from algebraic methods, and demonstrate connections to topology and set theory.",{"name":119,"@type":110,"acceptedAnswer":120},"How does a Venn diagram represent classes S, M, and P in the syllogism example?",{"text":121,"@type":113},"Each term S, M, and P is represented by a simple closed curve (circles). The diagram’s intersections correspond to members common to the terms, while shading indicates empty compartments and X marks indicate nonempty ones.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657683,1792986268,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":81},687197207639,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Zermelo-Fraenkel set theory. That form of axiomatic set theory that avoids the paradoxes of set theory by dropping the axiom of abstraction and substituting for ita set of axioms about set-existence.  \nBoruch A. Brody (1967)  \nlogic and the foundations of mathematics  \nA very detailed account of main developments of logic will be found in Logic, History of. Brief explanations of many of the terms commonly used by logicians will be found in Logical Terms, Glossary  \nof. The Encyclopedia also features the following articles dealing with questions in logic and the foundations of mathematics: Artificial and Natural Languages; Combinatory Logic; Computability Theory; Computing Machines; Decision Theory; Definition; Existence; Fallacies; Geometry; Gödel’s Theorem; Identity; Infinity in Mathematics and Logic; Laws of Thought; Logical Paradoxes; Logic Diagrams; Logic Machines; Many-Valued Logics; Mathematics, Foundations of; Modal Logic; Negation; Number; Questions; Semantics; Set Theory;  \nSubject and Predicate; Synonymity; Syntactical and Semantical Categories; Types, Theory of; and Vagueness. See “Logic” and “Mathematics, Foundations of,” in the index for entries on thinkers who have made contributions in this area.  \nlogic diagrams  \n“Logic diagrams” are geometrical figures that are in some respect isomorphic with the structure of statements in a formal logic and therefore can be manipulated to solve problems in that logic. They are useful teaching devices for strengthening a student’s intuitive grasp of logical structure, they can be used for checking results obtained by algebraic methods, and they provide elegant demonstrations of the close relation of logic to topology and set theory.  \nLeonhard Euler, the Swiss mathematician, was the first to make systematic use of a logic diagram. Circles had earlier been employed, by Gottfried Wilhelm Leibniz and others, to diagram syllogisms, but it was Euler who, in 1761, first explained in detail how circles could be manipulated for such purposes. Euler’s contemporary Johann Heinrich Lambert, the German mathematician, in his Neues  \nFIGURE 1  \nVenn Diagram Applied to a Syllogism  \nS M  \nP  \nOrganon (1764) used straight lines, in a manner similar to Euler’s use of circles, for diagramming syllogisms.  \nvenn diagrams  \nThe Euler and Lambert methods, as well as later variants using squares and other types of closed curves, are no longer in use because of the great improvement on their basic conception which was introduced by the English logician John Venn. The Venn diagram is best explained by showing how it is used to validate a syllogism. The syllogism’s three terms, S, M, and P, are represented by simple closed curves—most conveniently drawn as circles—that mutually intersect, as in Figure 1. The set of points inside circle S represents all members of class S, and points outside are members of class not-S—and similarly for the other two circles. Shading a compartment indicates that it has no members. An X inside a compartment shows that it contains at least one member. An X on the border of two compartments means that at least oneof the two compartments has members.  \nConsider the following syllogism:  \nSome S is M.  \nAll M is P.  \nTherefore, some S is P.  \nThe first premise states that the intersection of sets Sand M is not empty. This is indicated by an X on the bor-  \nENCYCLOPEDIA OF PHILOSOPHY  \n560 • 2nd edition","cbCaikoAXSFflq6n","https://ap.wps.com/l/cbCaikoAXSFflq6n","pdf",89525,"English","# Zermelo-Fraenkel set theory\n# Logic and foundational developments\n# Logic diagrams and their purpose\n# Historical origins of diagramming methods\n# Venn diagrams\n## Validating a syllogism with diagrams","[{\"question\":\"What is the goal of Zermelo–Fraenkel set theory as described here?\",\"answer\":\"It avoids paradoxes by dropping the axiom of abstraction and replacing it with a set of axioms about set existence.\"},{\"question\":\"What are logic diagrams, and what are they used for?\",\"answer\":\"They are geometrical figures isomorphic to the structure of statements in formal logic. They help teach logical structure, check results from algebraic methods, and demonstrate connections to topology and set theory.\"},{\"question\":\"How does a Venn diagram represent classes S, M, and P in the syllogism example?\",\"answer\":\"Each term S, M, and P is represented by a simple closed curve (circles). The diagram’s intersections correspond to members common to the terms, while shading indicates empty compartments and X marks indicate nonempty ones.\"}]","Logic and the Foundations of Mathematics | PDF"]