{"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":"2026-09-27T12:27:58","modified_gmt":"2026-09-27T03:27:58","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\uc5d0\uc11c \ud558\ub098\uc758 \ub2e8\uc704\ub85c \ucde8\uae09\ud558\ub358 \uba85\uc81c\ub97c \ubcc0\uc218, \ud568\uc218, \uad00\uacc4, \ud55c\uc815\uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \ubd84\uc11d\ud558\ub294 \ub17c\ub9ac \uccb4\uacc4\uc774\ub2e4. \uc774\ub97c \uc774\uc6a9\ud558\uba74 \ub300\uc218\uc801 \uad6c\uc870\ub098 \uc21c\uc11c\uad6c\uc870\ucc98\ub7fc \uc218\ud559\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ub9ce\uc740 \ub300\uc0c1\uc744 \ud558\ub098\uc758 \ud615\uc2dd\uc5b8\uc5b4 \uc548\uc5d0\uc11c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774 \ucc45\uc5d0\uc11c\ub294 \ub4f1\ud638\ub97c \ub17c\ub9ac\uae30\ud638\ub85c \ud3ec\ud568\ud558\ub294 \uc77c\uacc4\ub17c\ub9ac\ub97c \ub2e4\ub8ec\ub2e4. \uba85\uc81c\ub17c\ub9ac\uc640\uc758 \uc5f0\uacb0\uc744 \ub2e8\uc21c\ud558\uac8c \ud558\uae30 \uc704\ud558\uc5ec \\(\\neg,\\to,\\forall\\)\ub97c \uae30\ubcf8 \ub17c\ub9ac\uae30\ud638\ub85c \uc0ac\uc6a9\ud558\uace0, \ub098\uba38\uc9c0 \uacb0\ud569\uc790\uc640 \uc874\uc7ac \ud55c\uc815\uae30\ud638\ub294 \uc57d\uc5b4\ub85c \uc0ac\uc6a9\ud55c\ub2e4. \uc989 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch12-propositional-logic\">12\uc7a5<\/a>\uc5d0\uc11c\uc640 \uac19\uc774 \\(\\neg\\), \\(\\to\\)\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(\\wedge\\), \\(\\vee\\), \\(\\leftrightarrow\\)\ub97c \uc815\uc758\ud558\uace0<br \/>\n\\[<br \/>\n(\\exists x)\\phi:=\\neg(\\forall x)\\neg\\phi<br \/>\n\\]<br \/>\n\ub85c \ub454\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 14.1. (\uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4)<\/span><\/p>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4 \\(\\mathcal L\\)\uc740 \ub2e4\uc74c \uae30\ud638\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4\ub2e4.<\/p>\n<ul>\n<li>\uac00\uc0b0 \uac1c\uc758 <span class=\"defined\">\ubcc0\uc218<\/span>(variable) \\(x_0,x_1,x_2,\\ldots\\)<\/li>\n<li>\ub17c\ub9ac\uae30\ud638 \\(=,\\neg,\\to,\\forall\\) \ubc0f \uad04\ud638\uc640 \uc27c\ud45c<\/li>\n<li>\uac01\uac01 \uace0\uc815\ub41c \ud56d\uc218(arity)\ub97c \uac16\ub294 <span class=\"defined\">\ud568\uc218\uae30\ud638<\/span>(function symbol)\uc640 <span class=\"defined\">\uad00\uacc4\uae30\ud638<\/span>(relation symbol)<\/li>\n<li><span class=\"defined\">\uc0c1\uc218\uae30\ud638<\/span>(constant symbol)<\/li>\n<\/ul>\n<p>\ud568\uc218\uae30\ud638, \uad00\uacc4\uae30\ud638, \uc0c1\uc218\uae30\ud638\ub97c \\(\\mathcal L\\)\uc758 <span class=\"defined\">\ube44\ub17c\ub9ac\uae30\ud638<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc0c1\uc218\uae30\ud638\ub97c \\(0\\)\ud56d \ud568\uc218\uae30\ud638\ub85c \ubcf4\uc544\ub3c4 \ub41c\ub2e4.<\/p>\n<\/div>\n<p>\ub4a4\uc758 \uc644\uc804\uc131 \uc815\ub9ac\uc5d0\uc11c\ub294 \ube44\ub17c\ub9ac\uae30\ud638\uac00 \uac00\uc0b0 \uac1c\uc778 \uc5b8\uc5b4\ub97c \uba3c\uc800 \ub2e4\ub8ec\ub2e4. \uad6c\ubb38\ub860\uacfc \uc758\ubbf8\ub860 \uc790\uccb4\ub294 \uc774\ub7ec\ud55c \uac00\uc0b0\uc131 \uac00\uc815 \uc5c6\uc774\ub3c4 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 14.2. (\ud56d)<\/span><\/p>\n<p><span class=\"defined\">\ud56d<\/span>(term)\uc740 \ub2e4\uc74c \uc7ac\uadc0\uaddc\uce59\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ubcc0\uc218\uc640 \uc0c1\uc218\uae30\ud638\ub294 \ud56d\uc774\ub2e4.<\/li>\n<li>\\(f\\)\uac00 \\(n\\)\ud56d \ud568\uc218\uae30\ud638\uc774\uace0 \\(t_1,\\ldots,t_n\\)\uc774 \ud56d\uc774\uba74<br \/>\n\\[<br \/>\nf(t_1,\\ldots,t_n)<br \/>\n\\]<br \/>\n\ub3c4 \ud56d\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \uaddc\uce59\uc5d0 \uc758\ud558\uc5ec \uc5bb\uc5b4\uc9c0\ub294 \uac83\ub9cc\uc744 \ud56d\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 14.3. (\uc544\ud1b0\ub17c\ub9ac\uc2dd\uacfc \ub17c\ub9ac\uc2dd)<\/span><\/p>\n<p><span class=\"defined\">\uc544\ud1b0\ub17c\ub9ac\uc2dd<\/span>(atomic formula)\uc740 \ub2e4\uc74c \ub450 \uaf34\uc758 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/p>\n<ul>\n<li>\\(R\\)\uc774 \\(n\\)\ud56d \uad00\uacc4\uae30\ud638\uc774\uace0 \\(t_1,\\ldots,t_n\\)\uc774 \ud56d\uc77c \ub54c<br \/>\n\\[<br \/>\nR(t_1,\\ldots,t_n),<br \/>\n\\]<\/li>\n<li>\\(t_1,t_2\\)\uac00 \ud56d\uc77c \ub54c<br \/>\n\\[<br \/>\nt_1=t_2.<br \/>\n\\]<\/li>\n<\/ul>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \ub17c\ub9ac\uc2dd\uc740 \ub2e4\uc74c \uc7ac\uadc0\uaddc\uce59\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\uc544\ud1b0\ub17c\ub9ac\uc2dd\uc740 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<li>\\(\\phi,\\psi\\)\uac00 \ub17c\ub9ac\uc2dd\uc774\uba74 \\(\\neg\\phi\\)\uc640 \\(\\phi\\to\\psi\\)\ub294 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<li>\\(\\phi\\)\uac00 \ub17c\ub9ac\uc2dd\uc774\uace0 \\(x\\)\uac00 \ubcc0\uc218\uc774\uba74 \\((\\forall x)\\phi\\)\ub294 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \uaddc\uce59\uc5d0 \uc758\ud558\uc5ec \uc5bb\uc5b4\uc9c0\ub294 \uac83\ub9cc\uc744 \ub17c\ub9ac\uc2dd\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\wedge\\), \\(\\vee\\), \\(\\leftrightarrow\\), \\(\\exists\\)\uac00 \ub4e4\uc5b4\uac04 \uc2dd\uc740 \uc55e\uc5d0\uc11c \uc815\ud55c \uc57d\uc5b4\ub97c \ud480\uc5b4 \ub17c\ub9ac\uc2dd\uc73c\ub85c \ud574\uc11d\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 14.1.<\/span><br \/>\n\uc0c1\uc218\uae30\ud638 \\(c\\), 1\ud56d \ud568\uc218\uae30\ud638 \\(f\\), 2\ud56d \ud568\uc218\uae30\ud638 \\(g\\), 1\ud56d \uad00\uacc4\uae30\ud638 \\(P\\), 2\ud56d \uad00\uacc4\uae30\ud638 \\(R\\)\ub97c \uac16\ub294 \uc5b8\uc5b4 \\(\\mathcal L\\)\uc744 \uc0dd\uac01\ud558\uc790. \ub2e4\uc74c \uac01 \ubb38\uc790\uc5f4\uc744 \ud56d, \uc544\ud1b0\ub17c\ub9ac\uc2dd, \uc544\ud1b0\ub17c\ub9ac\uc2dd\uc774 \uc544\ub2cc \ub17c\ub9ac\uc2dd, \uc798 \ub9cc\ub4e4\uc5b4\uc9c4 \uc2dd\uc774 \uc544\ub2d8 \uc911 \ud558\ub098\ub85c \ubd84\ub958\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(x\\).<\/li>\n<li>\\(g(x,f(c))\\).<\/li>\n<li>\\(P(f(x))\\).<\/li>\n<li>\\(R(g(x,c),f(c))\\).<\/li>\n<li>\\(g(P(x),c)\\).<\/li>\n<li>\\((\\forall x)R(x,f(y))\\).<\/li>\n<li>\\(\\neg P(c)\\to R(x,c)\\).<\/li>\n<\/ol>\n<p>\uac01 \uacbd\uc6b0\uc5d0 \uc7ac\uadc0\uc801 \uc815\uc758\uc758 \uc5b4\ub290 \uaddc\uce59\uc744 \uc0ac\uc6a9\ud588\ub294\uc9c0\ub3c4 \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 14.4. (\uc790\uc720\ubcc0\uc218\uc640 \ubb36\uc778\ubcc0\uc218)<\/span><\/p>\n<p>\ub17c\ub9ac\uc2dd \uc548\uc758 \ubcc0\uc218\ub294 \uae30\ud638 \uc790\uccb4\uac00 \uc544\ub2c8\ub77c \uac01 <span class=\"defined\">\ubc1c\uc0dd<\/span>(occurrence)\uc744 \uae30\uc900\uc73c\ub85c \uc790\uc720\ub85c\uc6b4\uc9c0 \ubb36\uc5ec \uc788\ub294\uc9c0\ub97c \ud310\ub2e8\ud55c\ub2e4. \\((\\forall x)\\phi\\)\uc5d0\uc11c \\(\\phi\\)\ub97c \uc774 \ud55c\uc815\uae30\ud638\uc758 <span class=\"defined\">\uc720\ud6a8\ubc94\uc704<\/span>(scope)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ubc94\uc704 \uc548\uc5d0 \uc788\ub294 \\(x\\)\uc758 \uc790\uc720\ub85c\uc6b4 \ubc1c\uc0dd\uc740 \\(\\forall x\\)\uc5d0 \uc758\ud558\uc5ec \ubb36\uc778\ub2e4.<\/p>\n<p>\uc5b4\ub5a4 \ud55c\uc815\uae30\ud638\uc5d0\ub3c4 \ubb36\uc774\uc9c0 \uc54a\uc740 \ubcc0\uc218\ub97c <span class=\"defined\">\uc790\uc720\ubcc0\uc218<\/span>(free variable)\ub77c\uace0 \ubd80\ub974\uace0, \ud55c\uc815\uae30\ud638\uc5d0 \ubb36\uc778 \ubcc0\uc218\ub97c <span class=\"defined\">\ubb36\uc778\ubcc0\uc218<\/span>(bound variable)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc790\uc720\ubcc0\uc218\uac00 \ud558\ub098\ub3c4 \uc5c6\ub294 \ub17c\ub9ac\uc2dd\uc744 <span class=\"defined\">\ubb38\uc7a5<\/span>(sentence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<p>\uac19\uc740 \ubcc0\uc218\uae30\ud638\uac00 \ud55c \ub17c\ub9ac\uc2dd\uc5d0\uc11c \uc790\uc720\ub86d\uac8c\ub3c4, \ubb36\uc5ec\uc11c\ub3c4 \ub098\ud0c0\ub0a0 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[<br \/>\nP(x)\\to(\\forall x)Q(x)<br \/>\n\\]<br \/>\n\uc5d0\uc11c \uccab \ubc88\uc9f8 \\(x\\)\ub294 \uc790\uc720\ub86d\uace0 \ub4a4\uc758 \ub450 \\(x\\)\ub294 \\(\\forall x\\)\uc5d0 \ubb36\uc5ec \uc788\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 14.2.<\/span><br \/>\n\ub2e4\uc74c \uac01 \ub17c\ub9ac\uc2dd\uc5d0\uc11c \uc790\uc720\ubcc0\uc218\ub4e4\uc758 \uc9d1\ud569\uc744 \uad6c\ud558\uc2dc\uc624. \ub610\ud55c \ubb36\uc778\ubcc0\uc218\uac00 \uc874\uc7ac\ud558\ub294 \uacbd\uc6b0 \uadf8 \ubcc0\uc218\uac00 \uc5b4\ub5a4 \ud55c\uc815\uae30\ud638\uc5d0 \ubb36\uc5ec \uc788\ub294\uc9c0 \ud45c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(R(x,y)\\to(\\forall y)P(y)\\).<\/li>\n<li>\\((\\forall x)(R(x,y)\\to(\\exists y)S(x,y))\\).<\/li>\n<li>\\((\\forall x)(\\exists y)R(x,y)\\).<\/li>\n<li>\\((\\forall x)P(x)\\to Q(x)\\).<\/li>\n<\/ol>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\uc758 14.5. (\uce58\ud658)<\/span><\/p>\n<p>\ud56d \\(u\\)\uc5d0\uc11c \ubcc0\uc218 \\(x\\)\ub97c \ud56d \\(t\\)\ub85c \ubc14\uafbc \uacb0\uacfc\ub97c \\(u[t\/x]\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub17c\ub9ac\uc2dd \\(\\phi\\)\uc5d0\uc11c\ub294 \ubcc0\uc218 \\(x\\)\uc758 \uc790\uc720\ub85c\uc6b4 \ubc1c\uc0dd\ub9cc\uc744 \\(t\\)\ub85c \ubc14\uafb8\uace0 \uadf8 \uacb0\uacfc\ub97c \\(\\phi[t\/x]\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774\ub54c \\(t\\)\uc5d0 \ub098\ud0c0\ub098\ub294 \uc5b4\ub5a4 \ubcc0\uc218\ub3c4 \uce58\ud658 \uacfc\uc815\uc5d0\uc11c \uc0c8\ub85c \ud55c\uc815\uae30\ud638\uc5d0 \ubb36\uc774\uc9c0 \uc54a\uc73c\uba74 \u201c\\(t\\)\uac00 \\(\\phi\\)\uc5d0\uc11c \\(x\\)\uc5d0 \ub300\ud574 <span class=\"defined\">\uc790\uc720\ub86d\uac8c \ub300\uc785 \uac00\ub2a5<\/span>\ud558\ub2e4(free for \\(x\\))\u201d\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc608\ub97c \ub4e4\uc5b4 \\(\\phi=(\\forall y)R(x,y)\\)\uc5d0\uc11c \\(y\\)\ub294 \\(x\\)\uc5d0 \ub300\ud574 \uc790\uc720\ub86d\uac8c \ub300\uc785 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4. \ub2e8\uc21c\ud788 \\(x\\)\ub97c \\(y\\)\ub85c \ubc14\uafb8\uba74 \uc6d0\ub798 \uc790\uc720\ub85c\uc6e0\ub358 \ubcc0\uc218\uac00 \\(\\forall y\\)\uc5d0 \ud3ec\ud68d\ub418\uae30 \ub54c\ubb38\uc774\ub2e4. \ub4a4\uc758 \uc804\uce6d \uc608\ud654 \uacf5\ub9ac\uc5d0\uc11c\ub294 \uc774 \uc870\uac74\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 14.3.<\/span><br \/>\n\ub17c\ub9ac\uc2dd \\(\\phi\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[<br \/>\n\\phi=R(x,y)\\to(\\forall y)S(x,y)<br \/>\n\\]<br \/>\n\\(c\\)\ub294 \uc0c1\uc218\uae30\ud638\uc774\uace0 \\(f\\)\ub294 1\ud56d \ud568\uc218\uae30\ud638\uc774\uba70 \\(z\\)\ub294 \\(x,y\\)\uc640 \ub2e4\ub978 \ubcc0\uc218\uc774\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\phi\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\\(y\\)\uac00 \\(\\phi\\)\uc5d0\uc11c \\(x\\)\uc5d0 \ub300\ud574 \uc790\uc720\ub86d\uac8c \ub300\uc785 \uac00\ub2a5\ud55c\uc9c0 \ud310\uc815\ud558\uace0 \uc774\uc720\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/li>\n<li>\\(f(z)\\)\uac00 \\(\\phi\\)\uc5d0\uc11c \\(x\\)\uc5d0 \ub300\ud574 \uc790\uc720\ub86d\uac8c \ub300\uc785 \uac00\ub2a5\ud55c\uc9c0 \ud310\uc815\ud558\uace0, \uac00\ub2a5\ud558\uba74 \\(\\phi[f(z)\/x]\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<li>\ubb36\uc778\ubcc0\uc218\uc758 \uc774\ub984\uc744 \ubc14\uafb8\uc5b4<br \/>\n\\[<br \/>\n\\psi=R(x,y)\\to(\\forall z)S(x,z)<br \/>\n\\]<br \/>\n\ub85c \uc4f0\uc790. \uc774\uc81c \\(y\\)\uac00 \\(\\psi\\)\uc5d0\uc11c \\(x\\)\uc5d0 \ub300\ud574 \uc790\uc720\ub86d\uac8c \ub300\uc785 \uac00\ub2a5\ud55c\uc9c0 \ud310\uc815\ud558\uace0 \\(\\psi[y\/x]\\)\ub97c \uad6c\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 14.4.<\/span><br \/>\n\ub2e4\uc74c \ub17c\ub9ac\uc2dd\uc744 \uae30\ubcf8 \ub17c\ub9ac\uae30\ud638 \\(\\neg\\), \\(\\to\\), \\(\\forall\\)\ub9cc \uc0ac\uc6a9\ud558\ub3c4\ub85d \uc57d\uc5b4\ub97c \ubaa8\ub450 \ud480\uc5b4 \uc4f0\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\((\\exists x)P(x)\\).<\/li>\n<li>\\(P(x)\\wedge Q(x)\\).<\/li>\n<li>\\((\\exists x)(P(x)\\vee Q(x))\\).<\/li>\n<li>\\((\\forall x)(P(x)\\leftrightarrow Q(x))\\).<\/li>\n<\/ol>\n<\/div>\n<p>\uc77c\uacc4\ub17c\ub9ac\ub97c \uc2e4\uc81c \uc218\ud559\uc5d0 \uc801\uc6a9\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uad70\uc744 \ub2e4\ub8e8\ub294 \uc5b8\uc5b4<br \/>\n\\[<br \/>\n\\mathcal L_{\\mathrm{grp}}=\\{\\mu,\\iota,\\epsilon\\}<br \/>\n\\]<br \/>\n\uc744 \uc0dd\uac01\ud558\uc790. \uc5ec\uae30\uc11c \\(\\mu\\)\ub294 2\ud56d \ud568\uc218\uae30\ud638, \\(\\iota\\)\ub294 1\ud56d \ud568\uc218\uae30\ud638, \\(\\epsilon\\)\uc740 \uc0c1\uc218\uae30\ud638\uc774\ub2e4. \ub2e4\uc74c \ubb38\uc7a5\ub4e4\uc740 \uac01\uac01 \uacb0\ud569\ubc95\uce59, \\(\\epsilon\\)\uc774 \ud56d\ub4f1\uc6d0\uc774\ub77c\ub294 \uc870\uac74, \\(\\iota(x)\\)\uac00 \\(x\\)\uc758 \uc5ed\uc6d0\uc774\ub77c\ub294 \uc870\uac74\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\\[<br \/>\n\\begin{aligned}<br \/>\n&#038;(\\forall x)(\\forall y)(\\forall z)\\bigl(\\mu(\\mu(x,y),z)=\\mu(x,\\mu(y,z))\\bigr),\\\\[3pt]<br \/>\n&#038;(\\forall x)\\bigl((\\mu(x,\\epsilon)=x)\\wedge(\\mu(\\epsilon,x)=x)\\bigr),\\\\[3pt]<br \/>\n&#038;(\\forall x)\\bigl((\\mu(x,\\iota(x))=\\epsilon)\\wedge(\\mu(\\iota(x),x)=\\epsilon)\\bigr).<br \/>\n\\end{aligned}<br \/>\n\\]<\/p>\n<p>\uc5ec\uae30\uc11c\ub294 \ud56d\ub4f1\uc6d0\uacfc \uc5ed\uc6d0 \ud568\uc218\uac00 \uc5b8\uc5b4\uc758 \uae30\ud638\ub85c \uc774\ubbf8 \uc9c0\uc815\ub418\uc5b4 \uc788\ub294\ub370, \uc774\ub7ec\ud55c \uc0c1\ud669\uc744 \u201c\uc704 \ubb38\uc7a5\ub4e4\uc740 \uadf8 \uae30\ud638\ub4e4\uc774 \uc6d0\ud558\ub294 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4\u201d\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4. \ud56d\ub4f1\uc6d0\uacfc \uc5ed\uc6d0\uc758 \uc874\uc7ac \uc790\uccb4\ub97c \ud45c\ud604\ud558\ub824\uba74 \\(\\epsilon\\), \\(\\iota\\)\ub97c \uc4f0\uc9c0 \uc54a\uace0 \uc874\uc7ac \ud55c\uc815\uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub098\ud0c0\ub0bc \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 14.5.<\/span><br \/>\n\uad70\uc758 \uc5b8\uc5b4 \\(\\mathcal L_{\\mathrm{grp}}=\\{\\mu,\\iota,\\epsilon\\}\\)\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \uc870\uac74\uc744 \uac01\uac01 \ud558\ub098\uc758 \ubb38\uc7a5\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc5f0\uc0b0 \\(\\mu\\)\ub294 \uad50\ud658\ubc95\uce59\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/li>\n<li>\ubaa8\ub4e0 \uc6d0\uc18c\ub294 \uc790\uae30 \uc790\uc2e0\uc758 \uc5ed\uc6d0\uc774\ub2e4.<\/li>\n<li>\ud56d\ub4f1\uc6d0\uc774 \uc544\ub2cc \uc6d0\uc18c \\(x\\) \uc911 \\(\\mu(x,x)=\\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\uc5ed\uc6d0 \ud568\uc218\ub97c \ub450 \ubc88 \uc801\uc6a9\ud558\uba74 \uc6d0\ub798 \uc6d0\uc18c\ub85c \ub3cc\uc544\uc628\ub2e4.<\/li>\n<\/ol>\n<p>\uac01 \ubb38\uc7a5\uc5d0 \uc790\uc720\ubcc0\uc218\uac00 \uc5c6\uc74c\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\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\uc5d0\uc11c \ud558\ub098\uc758 \ub2e8\uc704\ub85c \ucde8\uae09\ud558\ub358 \uba85\uc81c\ub97c \ubcc0\uc218, \ud568\uc218, \uad00\uacc4, \ud55c\uc815\uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \ubd84\uc11d\ud558\ub294 \ub17c\ub9ac \uccb4\uacc4\uc774\ub2e4. \uc774\ub97c \uc774\uc6a9\ud558\uba74 \ub300\uc218\uc801 \uad6c\uc870\ub098 \uc21c\uc11c\uad6c\uc870\ucc98\ub7fc \uc218\ud559\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ub9ce\uc740 \ub300\uc0c1\uc744 \ud558\ub098\uc758 \ud615\uc2dd\uc5b8\uc5b4 \uc548\uc5d0\uc11c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4. \uc774 \ucc45\uc5d0\uc11c\ub294 \ub4f1\ud638\ub97c \ub17c\ub9ac\uae30\ud638\ub85c \ud3ec\ud568\ud558\ub294 \uc77c\uacc4\ub17c\ub9ac\ub97c \ub2e4\ub8ec\ub2e4. \uba85\uc81c\ub17c\ub9ac\uc640\uc758 \uc5f0\uacb0\uc744 \ub2e8\uc21c\ud558\uac8c \ud558\uae30 \uc704\ud558\uc5ec \\(\\neg,\\to,\\forall\\)\ub97c \uae30\ubcf8 \ub17c\ub9ac\uae30\ud638\ub85c \uc0ac\uc6a9\ud558\uace0, \ub098\uba38\uc9c0 \uacb0\ud569\uc790\uc640 \uc874\uc7ac \ud55c\uc815\uae30\ud638\ub294 \uc57d\uc5b4\ub85c \uc0ac\uc6a9\ud55c\ub2e4. \uc989 12\uc7a5\uc5d0\uc11c\uc640 \uac19\uc774 \\(\\neg\\), \\(\\to\\)\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(\\wedge\\), \\(\\vee\\), \\(\\leftrightarrow\\)\ub97c \uc815\uc758\ud558\uace0 \\( (\\exists&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":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9276\/revisions"}],"predecessor-version":[{"id":10091,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9276\/revisions\/10091"}],"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}]}}