{"id":9278,"date":"2025-10-17T20:22:48","date_gmt":"2025-10-17T11:22:48","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9278"},"modified":"2025-10-20T18:49:14","modified_gmt":"2025-10-20T09:49:14","slug":"ch15-semantics-first-order-logic","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch15-semantics-first-order-logic\/","title":{"rendered":"\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!--  \n\n<h2>15. \uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/h2>\n\n --><\/p>\n<p>\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860\uc740 \uad6c\ubb38\ub860\uc5d0\uc11c \uc815\uc758\ud55c \uae30\ud638\ub4e4\uc5d0 &#8216;\uc758\ubbf8'(\uc9c4\ub9bf\uac12)\ub97c \ubd80\uc5ec\ud558\ub294 \ubc29\ubc95\uc744 \ub2e4\ub8ec\ub2e4. \uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c\ub294 \uba85\uc81c\ubcc0\uc218\uc5d0 \uc9c4\ub9bf\uac12\uc744 \ubc30\uc815\ud558\ub294 \uac83\uc73c\ub85c \ucda9\ubd84\ud588\uc9c0\ub9cc, \uc77c\uacc4\ub17c\ub9ac\uc5d0\uc11c\ub294 \ub354 \ubcf5\uc7a1\ud55c \uad6c\uc870\uac00 \ud544\uc694\ud558\ub2e4.<\/p>\n<h3>\uc77c\uacc4\ub17c\ub9ac\uc5d0\uc11c\uc758 \uac12\ub9e4\uae40<\/h3>\n<p>\\(\\mathcal{L}\\)\uc774 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\ub77c\uace0 \ud558\uc790. <span class=\"defined\">\\(\\mathcal{L}\\)-\uad6c\uc870<\/span>(L-structure)\ub780 \ub2e4\uc74c \uc694\uc18c\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uad6c\uc870\uc774\ub2e4.<\/p>\n<ul>\n<li>\uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569 \\(V\\) (\ub17c\ub9ac\uc2dd\uc758 \ud574\uc11d \uc601\uc5ed)<\/li>\n<li>\\(\\mathcal{L}\\)\uc758 \uac01 \uad00\uacc4\uae30\ud638\uc5d0 \ub300\uc751\ud558\ub294 \\(V\\)\uc5d0\uc11c\uc758 \uad00\uacc4<\/li>\n<li>\\(\\mathcal{L}\\)\uc758 \uac01 \ud568\uc218\uae30\ud638\uc5d0 \ub300\uc751\ud558\ub294 \\(V\\)\uc5d0\uc11c\uc758 \ud568\uc218<\/li>\n<li>\\(\\mathcal{L}\\)\uc758 \uac01 \uc0c1\uc218\uae30\ud638\uc5d0 \ub300\uc751\ud558\ub294 \\(V\\)\uc758 \uc6d0\uc18c<\/li>\n<\/ul>\n<p>\uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c <span class=\"defined\">\uac12\ub9e4\uae40<\/span>(valuation)\uc774 \uba85\uc81c\ubcc0\uc218\uc5d0 \uc9c4\ub9bf\uac12\uc744 \ud560\ub2f9\ud558\ub294 \ud568\uc218\uc600\ub2e4\uba74, \uc77c\uacc4\ub17c\ub9ac\uc5d0\uc11c\ub294 \ubcc0\uc218\uc5d0 \\(V\\)\uc758 \uc6d0\uc18c\ub97c \ud560\ub2f9\ud558\ub294 \ud568\uc218\ub85c \ud655\uc7a5\ub41c\ub2e4. \uc989, \uac12\ub9e4\uae40 \\(v\\)\ub294 \ubcc0\uc218\ub4e4\uc758 \uc9d1\ud569\uc73c\ub85c\ubd80\ud130 \\(V\\)\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/p>\n<p>\uac12\ub9e4\uae40 \\(v\\)\ub97c \ud655\uc7a5\ud558\uc5ec \ud56d\uc758 \uac12\uacfc \ub17c\ub9ac\uc2dd\uc758 \uc9c4\ub9bf\uac12\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\uba3c\uc800, \ud56d\uc5d0 \ub300\ud55c \uac12\ub9e4\uae40\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ubcc0\uc218 \\(x\\)\uc5d0 \ub300\ud574 \\(v(x)\\)\ub294 \uc774\ubbf8 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.<\/li>\n<li>\uc0c1\uc218\uae30\ud638 \\(c\\)\uc5d0 \ub300\ud574 \\(v(c)\\)\ub294 \\(c\\)\uc5d0 \ub300\uc751\ub418\ub294 \\(V\\)\uc758 \uc6d0\uc18c\uc774\ub2e4.<\/li>\n<li>\ud568\uc218\uae30\ud638 \\(f\\)\uc640 \ud56d \\(t_1,\\) \\(t_2,\\) \\(\\ldots,\\) \\(t_n\\)\uc5d0 \ub300\ud574 \\(v(f(t_1,\\, t_2,\\, \\ldots,\\, t_n))\\)\uc740 \\(V\\)\uc758 \uc6d0\uc18c \\(v(t_1),\\) \\(v(t_2),\\) \\(\\ldots,\\) \\(v(t_n)\\)\uc744 \uc778\uc218\ub85c \uac16\ub294 \ud568\uc218 \\(f\\)\uc758 \uac12\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ub2e4\uc74c\uc73c\ub85c \ub17c\ub9ac\uc2dd\uc5d0 \ub300\ud55c \uac12\ub9e4\uae40\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. (\uba85\uc81c\ub17c\ub9ac\uc758 \uacb0\ud569\uc790\uc5d0 \ub300\ud55c \ubd80\ubd84\uc740 \uc0dd\ub7b5\ud568.)<\/p>\n<ul>\n<li>\uad00\uacc4\uae30\ud638 \\(R\\)\uacfc \ud56d \\(t_1,\\) \\(t_2,\\) \\(\\ldots,\\) \\(t_n\\)\uc5d0 \ub300\ud574 \\(v(R(t_1,\\, t_2,\\, \\ldots,\\, t_n)) = \\mathrm{T}\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(V\\)\uc758 \uc6d0\uc18c \\(v(t_1),\\) \\(v(t_2),\\) \\(\\ldots,\\) \\(v(t_n)\\)\uc774 \\(R\\)\uc5d0 \ub300\uc751\ub418\ub294 \uad00\uacc4\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4.<\/li>\n<li>\\(v((t_1 = t_2)) = \\mathrm{T}\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(v(t_1) = v(t_2)\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<li>\\(v((\\forall x_i)\\phi) = \\mathrm{T}\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(i\\)-\uc778\uc811\ud55c \uc784\uc758\uc758 \uac12\ub9e4\uae40 \\(v&#8217;\\)\uc5d0 \ub300\ud558\uc5ec \\(v'(\\phi) = \\mathrm{T}\\)\uc778 \uac83\uc774\ub2e4. (\\(v\\)\uc640 \\(v&#8217;\\)\uc774 \\(i\\)-\uc778\uc811\ud558\ub2e4\ub294 \uac83\uc740 \\(j\\neq i\\)\uc778 \uc784\uc758\uc758 \\(j\\)\uc5d0 \ub300\ud558\uc5ec \\(v(x_j) = v'(x_j)\\)\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4.)<\/li>\n<li>\\(v((\\exists x_i)\\phi) = \\mathrm{T}\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(i\\)-\uc778\uc811\ud55c \uac12\ub9e4\uae40 \\(v&#8217;\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(v'(\\phi) = \\mathrm{T}\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n<\/ul>\n<h3>\ubaa8\ub378\uacfc \uc774\ub860<\/h3>\n<p>\ubb38\uc7a5 \\(\\phi\\)\uac00 \uad6c\uc870 \\(M\\)\uc5d0\uc11c \ucc38\uc77c \ub54c \\(M \\models \\phi\\)\ub85c \ub098\ud0c0\ub0b4\uace0, &#8220;\\(M\\)\uc740 \\(\\phi\\)\uc758 <span class=\"defined\">\ubaa8\ub378<\/span>\uc774\ub2e4&#8221;\ub77c\uace0 \uc77d\ub294\ub2e4. \uad6c\uc870 \\(M\\)\uc5d0\uc11c \ucc38\uc778 \ubaa8\ub4e0 \ubb38\uc7a5\ub4e4\uc758 \ubaa8\uc784\uc744 \\(M\\)\uc758 <span class=\"defined\">\uc774\ub860<\/span>(theory)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\operatorname{Th}(M)\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4, \uad70\ub860\uc5d0\uc11c \ubaa8\ub378 \\(M\\)\uc774 \uad70\uc758 \uacf5\ub9ac\ub97c \ub9cc\uc871\ud55c\ub2e4\uba74 \\(M\\)\uc740 \uad70\uc774\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\uc758 \uc758\ubbf8\ub860\uc740 \uad6c\ubb38\ub860\uc5d0\uc11c \uc815\uc758\ud55c \uae30\ud638\ub4e4\uc5d0 &#8216;\uc758\ubbf8'(\uc9c4\ub9bf\uac12)\ub97c \ubd80\uc5ec\ud558\ub294 \ubc29\ubc95\uc744 \ub2e4\ub8ec\ub2e4. \uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c\ub294 \uba85\uc81c\ubcc0\uc218\uc5d0 \uc9c4\ub9bf\uac12\uc744 \ubc30\uc815\ud558\ub294 \uac83\uc73c\ub85c \ucda9\ubd84\ud588\uc9c0\ub9cc, \uc77c\uacc4\ub17c\ub9ac\uc5d0\uc11c\ub294 \ub354 \ubcf5\uc7a1\ud55c \uad6c\uc870\uac00 \ud544\uc694\ud558\ub2e4. \uc77c\uacc4\ub17c\ub9ac\uc5d0\uc11c\uc758 \uac12\ub9e4\uae40 \\(\\mathcal{L}\\)\uc774 \uc77c\uacc4\ub17c\ub9ac\uc5b8\uc5b4\ub77c\uace0 \ud558\uc790. \\(\\mathcal{L}\\)-\uad6c\uc870(L-structure)\ub780 \ub2e4\uc74c \uc694\uc18c\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uad6c\uc870\uc774\ub2e4. \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569 \\(V\\) (\ub17c\ub9ac\uc2dd\uc758 \ud574\uc11d \uc601\uc5ed) \\(\\mathcal{L}\\)\uc758 \uac01 \uad00\uacc4\uae30\ud638\uc5d0 \ub300\uc751\ud558\ub294 \\(V\\)\uc5d0\uc11c\uc758 \uad00\uacc4 \\(\\mathcal{L}\\)\uc758 \uac01 \ud568\uc218\uae30\ud638\uc5d0 \ub300\uc751\ud558\ub294 \\(V\\)\uc5d0\uc11c\uc758 \ud568\uc218 \\(\\mathcal{L}\\)\uc758 \uac01 \uc0c1\uc218\uae30\ud638\uc5d0 \ub300\uc751\ud558\ub294 \\(V\\)\uc758 \uc6d0\uc18c \uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c \uac12\ub9e4\uae40(valuation)\uc774 \uba85\uc81c\ubcc0\uc218\uc5d0 \uc9c4\ub9bf\uac12\uc744 \ud560\ub2f9\ud558\ub294 \ud568\uc218\uc600\ub2e4\uba74, \uc77c\uacc4\ub17c\ub9ac\uc5d0\uc11c\ub294 \ubcc0\uc218\uc5d0 \\(V\\)\uc758&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":115,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9278","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9278","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=9278"}],"version-history":[{"count":7,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9278\/revisions"}],"predecessor-version":[{"id":9446,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9278\/revisions\/9446"}],"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=9278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}