{"id":9276,"date":"2025-10-17T20:22:09","date_gmt":"2025-10-17T11:22:09","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9276"},"modified":"2025-10-20T18:49:09","modified_gmt":"2025-10-20T09:49:09","slug":"ch14-syntax-first-order-logic","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\/","title":{"rendered":"\uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>14. \uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860<\/h2>\n\n --><\/p>\n<p><span class=\"defined\">\uc77c\uacc4\ub17c\ub9ac<\/span>(first-order logic)\ub294 \uba85\uc81c\ub17c\ub9ac\uc758 \ub2e8\uc810\uc744 \ubcf4\uc644\ud55c \ub17c\ub9ac \uccb4\uacc4\ub85c, \uc218\ud559\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ub300\ubd80\ubd84\uc758 \ub300\uc0c1\uc744 \ud615\uc2dd\uc801\uc73c\ub85c \ud45c\ud604\ud560 \uc218 \uc788\uac8c \ud574\uc900\ub2e4.<\/p>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \ub450 \uac00\uc9c0 \uc911\uc694\ud55c \ud2b9\uc9d5\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\uad00\uacc4, \ud568\uc218, \uc0c1\uc218 \ub4f1\uc744 \ub2e4\ub8e8\ub294 \uc218\ud559\uc801 \uad6c\uc870\uc5d0 \uad00\ud55c \uc9c4\uc220\uc774 \uae30\ubcf8\ub2e8\uc704\uac00 \ub420 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ub450 \ud568\uc218\uac00 \uac19\uc744 \uc870\uac74\uc774\ub098 \ud2b9\uc815 \uad00\uacc4\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc6d0\uc18c\ub4e4\uc5d0 \ub300\ud574 \ub17c\ud560 \uc218 \uc788\ub2e4.<\/li>\n<li>\ud55c\uc815\uc0ac(quantifier)\ub97c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc8fc\uc5b4\uc9c4 \ubc94\uc704\uc758 \ubaa8\ub4e0 \ub300\uc0c1\uc5d0 \ub300\ud55c \uc9c4\uc220(\uc804\uce6d \ud55c\uc815\uc0ac)\uc774\ub098 \ubc94\uc704\uc758 \uc77c\ubd80 \uc6d0\uc18c\uc5d0 \ub300\ud55c \uc9c4\uc220(\uc874\uc7ac \ud55c\uc815\uc0ac)\uc744 \ub2e4\ub8f0 \uc218 \uc788\ub2e4.<\/li>\n<\/ul>\n<h3>\uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\uc131 \uc694\uc18c<\/h3>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \uc5b8\uc5b4\ub294 \uc801\uc6a9 \ubd84\uc57c\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c0\uc9c0\ub9cc, \ubaa8\ub4e0 \uc77c\uacc4\ub17c\ub9ac \uc5b8\uc5b4\ub294 \ub2e4\uc74c \uc694\uc18c\ub4e4\uc744 \ud3ec\ud568\ud55c\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\ud568\uc218\uae30\ud638<\/span>(function symbol): \ud568\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \ud615\uc2dd\uc801 \uae30\ud638<\/li>\n<li><span class=\"defined\">\uad00\uacc4\uae30\ud638<\/span>(relation symbol): \uad00\uacc4\ub97c \ub098\ud0c0\ub0b4\ub294 \ud615\uc2dd\uc801 \uae30\ud638<\/li>\n<li><span class=\"defined\">\uc0c1\uc218\uae30\ud638<\/span>(constant symbol): \ud2b9\uc815 \uc6d0\uc18c\ub97c \ub098\ud0c0\ub0b4\ub294 \ud615\uc2dd\uc801 \uae30\ud638<\/li>\n<li><span class=\"defined\">\ubcc0\uc218<\/span>(variable): \ub17c\ub9ac\uad6c\uc870\uc5d0\uc11c \uc784\uc758\uc758 \uc6d0\uc18c\ub97c \uac00\ub9ac\ud0a4\ub294 \uae30\ud638<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\uae30\ud638<\/span>: \ub4f1\ud638(\\(=\\)), \uacb0\ud569\uc790(\\(\\lnot,\\, \\vee,\\, \\wedge,\\, \\rightarrow,\\, \\leftrightarrow\\)), \ud55c\uc815\uae30\ud638(\\(\\exists,\\, \\forall\\)), \uad04\ud638\uc640 \uc27c\ud45c<\/li>\n<\/ul>\n<p>\uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c\ub294 \uba85\uc81c\ubcc0\uc218\uc640 \uacb0\ud569\uc790, \uad04\ud638\ub9cc\uc744 \uc0ac\uc6a9\ud588\ub358 \uac83\uc5d0 \ube44\ud574, \uc77c\uacc4\ub17c\ub9ac\ub294 \ub354 \ud48d\ubd80\ud55c \ud45c\ud604\uc774 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc5d0\uc11c\ub294 \uba3c\uc800 <span class=\"defined\">\ud56d<\/span>(term)\uc744 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ud558\ub098\uc758 \ubcc0\uc218\ub294 \ud56d\uc774\ub2e4.<\/li>\n<li>\ud558\ub098\uc758 \uc0c1\uc218\uae30\ud638\ub294 \ud56d\uc774\ub2e4.<\/li>\n<li>\\(f\\)\uac00 \\(n\\)\ud56d\ud568\uc218\uae30\ud638\uc774\uace0 \\(t_1, \\,t_2, \\,\\ldots, \\,t_n\\)\uc774 \ud56d\uc774\uba74 \\(f(t_1, \\,t_2, \\,\\ldots, \\,t_n)\\)\uc740 \ud56d\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ub2e4\uc74c\uc73c\ub85c <span class=\"defined\">\uc544\ud1b0\ub17c\ub9ac\uc2dd<\/span>(atomic formula)\uc744 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(R\\)\uc774 \\(n\\)\ud56d\uad00\uacc4\uae30\ud638\uc774\uace0 \\(t_1, \\,t_2, \\,\\ldots,\\, t_n\\)\uc774 \ud56d\uc774\uba74 \\(R(t_1, \\,t_2, \\,\\ldots,\\, t_n)\\)\uc740 \uc544\ud1b0\ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<li>\\(t_1\\)\uacfc \\(t_2\\)\uac00 \ud56d\uc774\uba74 \\((t_1 = t_2)\\)\ub294 \uc544\ud1b0\ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ub9c8\uc9c0\ub9c9\uc73c\ub85c <span class=\"defined\">\ub17c\ub9ac\uc2dd<\/span>(formula)\uc744 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\uc544\ud1b0\ub17c\ub9ac\uc2dd\uc740 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<li>\\(\\phi\\)\uc640 \\(\\psi\\)\uac00 \ub17c\ub9ac\uc2dd\uc774\uba74 \\((\\phi \\wedge \\psi)\\), \\((\\phi \\vee \\psi)\\), \\((\\lnot \\phi)\\), \\((\\phi \\rightarrow \\psi)\\), \\((\\phi \\leftrightarrow \\psi)\\)\ub294 \ubaa8\ub450 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<li>\\(\\phi\\)\uac00 \ub17c\ub9ac\uc2dd\uc774\uace0 \\(x\\)\uac00 \ubcc0\uc218\uc774\uba74 \\((\\exists x)\\phi\\)\uc640 \\((\\forall x)\\phi\\)\ub294 \ubaa8\ub450 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ud55c\uc815\uae30\ud638\uc640 \uad00\ub828\ud558\uc5ec <span class=\"defined\">\uc790\uc720\ubcc0\uc218<\/span>(free variable)\uc640 <span class=\"defined\">\ubb36\uc778\ubcc0\uc218<\/span>(bounded variable)\ub77c\ub294 \uac1c\ub150\uc774 \uc911\uc694\ud558\ub2e4. \ubcc0\uc218 \\(x\\)\uac00 \\((\\forall x)\\phi\\) \ub610\ub294 \\((\\exists x)\\phi\\)\uc758 \uc720\ud6a8\ubc94\uc704 \ub0b4\uc5d0 \uc788\uc744 \ub54c \\(x\\)\ub294 \ubb36\uc778\ubcc0\uc218\uc774\uba70, \uadf8\ub807\uc9c0 \uc54a\uc740 \ubcc0\uc218\ub294 \uc790\uc720\ubcc0\uc218\uc774\ub2e4. \uc790\uc720\ubcc0\uc218\uac00 \uc5c6\ub294 \ub17c\ub9ac\uc2dd\uc744 <span class=\"defined\">\ubb38\uc7a5<\/span>(sentence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<h3>\uc77c\uacc4\ub17c\ub9ac\uc758 \ud65c\uc6a9 \uc608<\/h3>\n<p>\uc77c\uacc4\ub17c\ub9ac\ub97c \uc2e4\uc81c \uc218\ud559\uc5d0 \uc801\uc6a9\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uad70\ub860\uc5d0\uc11c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ubb38\uc7a5\ub4e4\ub85c \uad70\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n&#038; (\\forall x)(\\forall y)(\\forall z)(\\mu(\\mu(x\\,,y),\\,z) = \\mu(x,\\,\\mu(y,\\,z))) \\\\[6pt]<br \/>\n&#038; (\\forall x)((\\mu(x,\\,\\epsilon) = x) \\wedge (\\mu(\\epsilon,\\,x) = x)) \\\\[6pt]<br \/>\n&#038; (\\forall x)((\\mu(x,\\,\\iota(x)) = \\epsilon) \\wedge (\\mu(\\iota(x),\\,x) = \\epsilon))<br \/>\n\\end{aligned}\\]<br \/>\n\uc774 \ubb38\uc7a5\ub4e4\uc740 \uac01\uac01 \uc5f0\uc0b0\uc758 \uacb0\ud569\ubc95\uce59, \ud56d\ub4f1\uc6d0\uc758 \uc874\uc7ac\uc131, \uc5ed\uc6d0\uc758 \uc874\uc7ac\uc131\uc744 \ub098\ud0c0\ub0b8\ub2e4. \uba85\uc81c\ub17c\ub9ac\ub85c\ub294 \uc774\ub7ec\ud55c \ubcf5\uc7a1\ud55c \uc218\ud559\uc801 \uad6c\uc870\uc640 \uad00\uacc4\ub97c \ud45c\ud604\ud560 \uc218 \uc5c6\uc73c\ubbc0\ub85c, \uc77c\uacc4\ub17c\ub9ac\uac00 \uc218\ud559\uc758 \ud615\uc2dd\ud654\uc5d0 \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud55c\ub2e4.<\/p>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-logic\/\">\uc9d1\ud569\uacfc \uc218\ub9ac\ub17c\ub9ac \uccab\uac78\uc74c \ubaa9\ucc28 \ubcf4\uae30<\/a><\/p>\n<p><span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch01-naive-logic\/\">\uba85\uc81c\uc640 \ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch02-sets\">\uc9d1\ud569\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch03-algebra-of-classes\">\ub2e4\uc591\ud55c \uc9d1\ud569\uc758 \uc5f0\uc0b0<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch04-relations-and-functions\">\uad00\uacc4\uc640 \ud568\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch05-infinite-sets\">\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch06-natural-numbers\">\uc790\uc5f0\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch07-cardinal-numbers\">\uc9d1\ud569\uc758 \uae30\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch08-ordinal-numbers\">\uc9d1\ud569\uc758 \uc11c\uc218<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch09-axiomatic-set-theory\">\uc9d1\ud569\ub860\uc758 \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch10-axiom-of-choice\">\uc120\ud0dd \uacf5\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch11-formal-logic\">\ud615\uc2dd\ub17c\ub9ac<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch12-propositional-logic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch13-soundness-completeness-proplogic\">\uba85\uc81c\ub17c\ub9ac\uc758 \uac74\uc804\uc131\uacfc \uc644\uc804\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch15-semantics-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch16-inference-rule-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucd94\ub860\uaddc\uce59<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch17-compactness-first-order-logic\">\uc77c\uacc4\ub17c\ub9ac\uc758 \ucf64\ud329\ud2b8\uc131<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch18-peano-arithmetics\">\ud398\uc544\ub178 \uc0b0\uc220<\/a><\/span><br \/>\n<span class=\"contentboxindex\"><a href=\"\/blog\/invitation-to-mathematical-logic\/ch19-incompleteness-theorem\">\ubd88\uc644\uc804\uc131 \uc815\ub9ac<\/a><\/span>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc77c\uacc4\ub17c\ub9ac(first-order logic)\ub294 \uba85\uc81c\ub17c\ub9ac\uc758 \ub2e8\uc810\uc744 \ubcf4\uc644\ud55c \ub17c\ub9ac \uccb4\uacc4\ub85c, \uc218\ud559\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ub300\ubd80\ubd84\uc758 \ub300\uc0c1\uc744 \ud615\uc2dd\uc801\uc73c\ub85c \ud45c\ud604\ud560 \uc218 \uc788\uac8c \ud574\uc900\ub2e4. \uc77c\uacc4\ub17c\ub9ac\uc758 \ub450 \uac00\uc9c0 \uc911\uc694\ud55c \ud2b9\uc9d5\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. \uad00\uacc4, \ud568\uc218, \uc0c1\uc218 \ub4f1\uc744 \ub2e4\ub8e8\ub294 \uc218\ud559\uc801 \uad6c\uc870\uc5d0 \uad00\ud55c \uc9c4\uc220\uc774 \uae30\ubcf8\ub2e8\uc704\uac00 \ub420 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ub450 \ud568\uc218\uac00 \uac19\uc744 \uc870\uac74\uc774\ub098 \ud2b9\uc815 \uad00\uacc4\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc6d0\uc18c\ub4e4\uc5d0 \ub300\ud574 \ub17c\ud560 \uc218 \uc788\ub2e4. \ud55c\uc815\uc0ac(quantifier)\ub97c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc8fc\uc5b4\uc9c4 \ubc94\uc704\uc758 \ubaa8\ub4e0 \ub300\uc0c1\uc5d0 \ub300\ud55c \uc9c4\uc220(\uc804\uce6d \ud55c\uc815\uc0ac)\uc774\ub098 \ubc94\uc704\uc758 \uc77c\ubd80 \uc6d0\uc18c\uc5d0&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":114,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9276","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9276","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9276"}],"version-history":[{"count":8,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9276\/revisions"}],"predecessor-version":[{"id":9445,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9276\/revisions\/9445"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9246"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}