{"id":9238,"date":"2025-10-17T19:12:39","date_gmt":"2025-10-17T10:12:39","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9238"},"modified":"2025-10-20T18:48:16","modified_gmt":"2025-10-20T09:48:16","slug":"ch01-naive-logic","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch01-naive-logic\/","title":{"rendered":"\uba85\uc81c\uc640 \ub17c\ub9ac"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!--\n\n\n<div style=\"text-align: center; margin: 2em 0;\">\n\n\n<h1 style=\"font-size: 1.5em; font-weight: bold;\">\uc81c1\ubd80. \uc9c1\uad00\uc801 \uc9d1\ud569\ub860<\/h1>\n\n\n\n\n<p style=\"margin-top: 1em;\">\uc218\ud559\uc758 \uac70\uc758 \ubaa8\ub4e0 \ubd84\uc57c\ub294 \uc9d1\ud569\uc774\ub77c\ub294 \uac1c\ub150 \uc704\uc5d0 \uac74\uc124\ub418\uc5b4 \uc788\ub2e4. \uc790\uc5f0\uc218, \ud568\uc218, \uad00\uacc4 \ub4f1 \uc6b0\ub9ac\uac00 \uc218\ud559\uc5d0\uc11c \ub2e4\ub8e8\ub294 \ub300\uc0c1\uc740 \uc9d1\ud569\uc758 \uc5b8\uc5b4\ub85c \uc815\uc758\ub418\uace0 \uc774\ud574\ub420 \uc218 \uc788\ub2e4. \uc774 \ubd80\uc5d0\uc11c\ub294 \uc9c1\uad00\uc801\uc778 \uc218\uc900\uc5d0\uc11c \uc9d1\ud569\uc758 \uac1c\ub150\uacfc \uadf8 \uc5f0\uc0b0\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uc218\ud559\uc801 \ucd94\ub860\uc758 \uae30\ucd08\uac00 \ub418\ub294 \ub17c\ub9ac\uc640 \uba85\uc81c\ub97c \ud568\uaed8 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<p><!-- \n\n<h2>1. \uba85\uc81c\uc640 \ub17c\ub9ac<\/h2>\n\n --><\/p>\n<p>\uc218\ud559\uc740 \uc5c4\ubc00\ud55c \ub17c\ub9ac\uc801 \ucd94\ub860\uc744 \ubc14\ud0d5\uc73c\ub85c \ud55c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc218\ud559\uc801 \ucd94\ub860\uc758 \uae30\ucd08\uac00 \ub418\ub294 \uba85\uc81c\uc640 \ub17c\ub9ac\ub97c \ub2e4\ub8ec\ub2e4. \uba85\uc81c\uc758 \uae30\ubcf8 \uac1c\ub150\uacfc \uc5f0\uc0b0\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uc774\ub97c \ud1b5\ud574 \uc218\ud559\uc801 \uc9c4\uc220\uc744 \uc815\ud655\ud558\uac8c \ud45c\ud604\ud558\uace0 \ubd84\uc11d\ud558\ub294 \ubc29\ubc95\uc744 \uc775\ud78c\ub2e4.<\/p>\n<h3>1. \uba85\uc81c\uc758 \uac1c\ub150<\/h3>\n<p><span class=\"defined\">\uba85\uc81c<\/span>(proposition)\ub780 \ucc38\uc774\ub098 \uac70\uc9d3\uc744 \uba85\ud655\ud558\uac8c \ud310\ubcc4\ud560 \uc218 \uc788\ub294 \ubb38\uc7a5\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4, &#8220;\\(2 + 3 = 5\\)&#8221;\ub294 \ucc38\uc778 \uba85\uc81c\uc774\uace0, &#8220;\\(\\pi < 3\\)\"\uc740 \uac70\uc9d3\uc778 \uba85\uc81c\uc774\ub2e4. \ubc18\uba74 \"\\(x > 5\\)&#8221;\ub294 \\(x\\)\uc758 \uac12\uc774 \uc815\ud574\uc9c0\uc9c0 \uc54a\uc558\uc73c\ubbc0\ub85c \uba85\uc81c\uac00 \uc544\ub2c8\ub2e4.<\/p>\n<p>[\ubcc0\uc218\ub97c \ud3ec\ud568\ud55c \ubb38\uc7a5\uc740 \uba85\uc81c\uac00 \uc544\ub2c8\uc9c0\ub9cc, \ubcc0\uc218\uc5d0 \ud2b9\uc815 \uac12\uc744 \ub300\uc785\ud558\uba74 \ucc38 \ub610\ub294 \uac70\uc9d3\uc774 \uacb0\uc815\ub418\ub294 \ubb38\uc7a5\uc740 \uba85\uc81c\uc640 \uc720\uc0ac\ud55c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4. \ub530\ub77c\uc11c \uc774\ub7f0 \ubb38\uc7a5\ub3c4 \uba85\uc81c \uc5f0\uc0b0\uc790(\ub17c\ub9ac\ud569, \ub17c\ub9ac\uacf1, \ubd80\uc815, \uc870\uac74\ubd80 \ub4f1)\ub97c \uc0ac\uc6a9\ud558\uc5ec \ubcc0\ud615\ud558\uac70\ub098 \uacb0\ud569\ud560 \uc218 \uc788\ub2e4.]<\/p>\n<p>\uba85\uc81c\ub97c \ub098\ud0c0\ub0bc \ub54c\ub294 \ubcf4\ud1b5 \\(p\\), \\(q\\), \\(r\\) \ub4f1\uc758 \uc18c\ubb38\uc790\ub97c \uc0ac\uc6a9\ud55c\ub2e4. \uba85\uc81c \\(p\\)\uac00 \ucc38\uc77c \ub54c\ub294 &#8220;\\(p\\)\uc758 \uc9c4\ub9bf\uac12\uc774 \ucc38(True, T)\uc774\ub2e4&#8221; \ub610\ub294 \uac04\ub2e8\ud788 &#8220;\\(p\\)\uac00 \ucc38\uc774\ub2e4&#8221;\ub77c\uace0 \ub9d0\ud558\uace0, \\(p\\)\uac00 \uac70\uc9d3\uc77c \ub54c\ub294 &#8220;\\(p\\)\uc758 \uc9c4\ub9bf\uac12\uc774 \uac70\uc9d3(False, F)\uc774\ub2e4&#8221; \ub610\ub294 \uac04\ub2e8\ud788 &#8220;\\(p\\)\uac00 \uac70\uc9d3\uc774\ub2e4&#8221;\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<h3>2. \uba85\uc81c\uc758 \uc5f0\uc0b0<\/h3>\n<p>\ud558\ub098\uc758 \uba85\uc81c\ub97c \ubcc0\ud615\ud558\uac70\ub098 \ub458 \uc774\uc0c1\uc758 \uba85\uc81c\ub97c \uacb0\ud569\ud558\uc5ec \uc0c8\ub85c\uc6b4 \uba85\uc81c\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \uc5f0\uc0b0\uc744 <span class=\"defined\">\uba85\uc81c \uc5f0\uc0b0<\/span> \ub610\ub294 <span class=\"defined\">\ub17c\ub9ac \uc5f0\uc0b0<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\ubd80\uc815<\/span>(negation): \uba85\uc81c \\(p\\)\uc758 <span class=\"defined\">\ubd80\uc815<\/span> \\(\\neg p\\)\ub294 \\(p\\)\uac00 \ucc38\uc77c \ub54c \uac70\uc9d3\uc774\uace0, \\(p\\)\uac00 \uac70\uc9d3\uc77c \ub54c \ucc38\uc774\ub2e4. &#8220;\\(p\\)\uac00 \uc544\ub2c8\ub2e4&#8221;\ub77c\uace0 \uc77d\ub294\ub2e4.<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\uacf1<\/span>(conjunction): \uba85\uc81c \\(p\\)\uc640 \\(q\\)\uc758 <span class=\"defined\">\ub17c\ub9ac\uacf1<\/span> \\(p \\wedge q\\)\ub294 \\(p\\)\uc640 \\(q\\)\uac00 \ubaa8\ub450 \ucc38\uc77c \ub54c\ub9cc \ucc38\uc774\ub2e4. &#8220;\\(p\\) \uadf8\ub9ac\uace0 \\(q\\)&#8221;\ub77c\uace0 \uc77d\ub294\ub2e4.<\/li>\n<li><span class=\"defined\">\ub17c\ub9ac\ud569<\/span>(disjunction): \uba85\uc81c \\(p\\)\uc640 \\(q\\)\uc758 <span class=\"defined\">\ub17c\ub9ac\ud569<\/span> \\(p \\vee q\\)\ub294 \\(p\\)\uc640 \\(q\\) \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\uac00 \ucc38\uc77c \ub54c \ucc38\uc774\ub2e4. &#8220;\\(p\\) \ub610\ub294 \\(q\\)&#8221;\ub77c\uace0 \uc77d\ub294\ub2e4.<\/li>\n<li><span class=\"defined\">\uc870\uac74\ubd80<\/span>(conditional): \uba85\uc81c \\(p\\)\uc640 \\(q\\)\uc758 <span class=\"defined\">\uc870\uac74\ubd80<\/span> \ub610\ub294 <span class=\"defined\">\ud568\uc758<\/span> \\(p \\to q\\)\ub294 \\(p\\)\uac00 \ucc38\uc774\uace0 \\(q\\)\uac00 \uac70\uc9d3\uc77c \ub54c\ub9cc \uac70\uc9d3\uc774\uace0, \ub098\uba38\uc9c0 \uacbd\uc6b0\ub294 \ubaa8\ub450 \ucc38\uc774\ub2e4. &#8220;\\(p\\)\uc774\uba74 \\(q\\)\uc774\ub2e4&#8221;\ub77c\uace0 \uc77d\ub294\ub2e4. \uc5ec\uae30\uc11c \\(p\\)\ub97c <span class=\"defined\">\uac00\uc815<\/span> \ub610\ub294 <span class=\"defined\">\uc804\uc81c<\/span>\ub77c \ud558\uace0, \\(q\\)\ub97c <span class=\"defined\">\uacb0\ub860<\/span>\uc774\ub77c \ud55c\ub2e4.<\/li>\n<li><span class=\"defined\">\uc30d\uc870\uac74\ubd80<\/span>(biconditional): \uba85\uc81c \\(p\\)\uc640 \\(q\\)\uc758 <span class=\"defined\">\uc30d\uc870\uac74\ubd80<\/span> \\(p \\leftrightarrow q\\)\ub294 \\(p\\)\uc640 \\(q\\)\uc758 \uc9c4\ub9bf\uac12\uc774 \uac19\uc744 \ub54c \ucc38\uc774\ub2e4. &#8220;\\(p\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(q\\)\uc774\ub2e4&#8221;\ub77c\uace0 \uc77d\ub294\ub2e4.<\/li>\n<\/ul>\n<p>\uba85\uc81c \uc5f0\uc0b0\uc758 \uc9c4\ub9bf\uac12\uc744 \ub098\ud0c0\ub0b4\ub294 \ud45c\ub97c <span class=\"defined\">\uc9c4\ub9ac\ud45c<\/span>(truth table)\ub77c\uace0 \ubd80\ub978\ub2e4. \uae30\ubcf8\uc801\uc778 \uba85\uc81c \uc5f0\uc0b0\uc790\uc758 \uc9c4\ub9ac\ud45c\ub294 \ub2e4\uc74c \ud45c\uc640 \uac19\ub2e4.<\/p>\n<div>\n<a href=\"\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table01.png\" data-rel=\"penci-gallery-image-content\" ><img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table01.png\" alt=\"\" width=\"354\" height=\"157\" class=\"aligncenter size-full wp-image-9450\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table01.png 1239w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table01-300x133.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table01-1024x455.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table01-768x341.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table01-1170x519.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table01-585x260.png 585w\" sizes=\"(max-width: 354px) 100vw, 354px\" \/><\/a>\n<\/div>\n<p><!-- width=\"1239\" height=\"550\" --><\/p>\n<p>\ub450 \uba85\uc81c \\(p\\)\uc640 \\(q\\)\uc5d0 \ub300\ud558\uc5ec, \\(p \\leftrightarrow q\\)\uac00 \ud56d\uc0c1 \ucc38\uc77c \ub54c \\(p\\)\uc640 \\(q\\)\uac00 <span class=\"defined\">\ub17c\ub9ac\uc801\uc73c\ub85c \ub3d9\uce58<\/span>(logically equivalent)\ub77c\uace0 \ud558\uace0 \\(p \\equiv q\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(p \\to q \\equiv \\neg p \\vee q\\)\uc774\ub2e4.<\/p>\n<p>\\(p \\to q\\)\uac00 \ucc38\uc77c \ub54c \\(p\\)\uac00 \\(q\\)\ub97c <span class=\"defined\">\ub17c\ub9ac\uc801\uc73c\ub85c \ud568\uc758<\/span>\ud55c\ub2e4\uace0 \ud558\uace0 \\(p \\Rightarrow q\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\(p\\Rightarrow q\\)\uc774\uba74\uc11c \\(q \\Rightarrow p\\)\uc778 \uac83\uc744 \\(p \\Leftrightarrow q\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>[\ub2e8\uc21c\uba85\uc81c \\(p\\), \\(q\\)\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uba85\uc81c\ud568\uc218 \\(p(x)\\), \\(q(x)\\)\uc5d0 \ub300\ud574\uc11c\ub3c4 \\(\\forall x (p(x)\\rightarrow q(x))\\)\uac00 \ucc38\uc77c \ub54c \\(p(x)\\Rightarrow q(x)\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \uc774\ub294 \uad50\uc721\uc801 \ud3b8\uc758\ub97c \uc704\ud55c \ud45c\uae30\ubc95\uc774\uba70, \uc5c4\ubc00\ud558\uac8c\ub294 \\(p\\models q\\)\ub85c \ud45c\uae30\ud574\uc57c \ud55c\ub2e4. \ub354 \uc5c4\ubc00\ud55c \ud45c\uae30\ubc95\uc740 <a href=\"\/blog\/invitation-to-mathematical-logic\/ch12-propositional-logic\/\">\uba85\uc81c\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a>\uacfc <a href=\"\/blog\/invitation-to-mathematical-logic\/ch15-semantics-first-order-logic\/\">\uc77c\uacc4\ub17c\ub9ac\uc758 \uc758\ubbf8\ub860<\/a>\uc744 \ubcf4\uae30 \ubc14\ub780\ub2e4.]\n<\/p>\n<h3>3. \uc870\uac74\ubd80 \uba85\uc81c\uc640 \uad00\ub828\ub41c \uac1c\ub150<\/h3>\n<p>\uc870\uac74\ubd80 \uba85\uc81c \\(p \\to q\\)\uac00 \ucc38\uc77c \ub54c, \uc989 \\(p\\Rightarrow q\\)\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(p\\)\ub294 \\(q\\)\uc758 <span class=\"defined\">\ucda9\ubd84\uc870\uac74<\/span>(sufficient condition)\uc774\ub2e4.<\/li>\n<li>\\(q\\)\ub294 \\(p\\)\uc758 <span class=\"defined\">\ud544\uc694\uc870\uac74<\/span>(necessary condition)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\ud2b9\ud788 \\(p\\Leftrightarrow q\\)\uc77c \ub54c, \\(p\\)\uc640 \\(q\\)\ub294 \uc11c\ub85c\uc758 <span class=\"defined\">\ud544\uc694\ucda9\ubd84\uc870\uac74<\/span>\uc774\ub2e4.<\/p>\n<p>\uc870\uac74\ubd80 \uba85\uc81c \\(p \\to q\\)\uc640 \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \uba85\uc81c\ub4e4\uc744 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\uba85\uc81c \\(q \\to p\\)\ub97c \\(p \\rightarrow q\\)\uc758 <span class=\"defined\">\uc5ed<\/span>(converse)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uba85\uc81c \\(\\neg p \\to \\neg q\\)\ub97c \\(p \\rightarrow q\\)\uc758 <span class=\"defined\">\uc774<\/span>(inverse)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\uba85\uc81c \\(\\neg q \\to \\neg p\\)\ub97c \\(p \\rightarrow q\\)\uc758 <span class=\"defined\">\ub300\uc6b0<\/span>(contrapositive)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<\/ul>\n<p>\uc870\uac74\ubd80 \uba85\uc81c\uc640 \uadf8 \ub300\uc6b0\ub294 \ud56d\uc0c1 \ub17c\ub9ac\uc801\uc73c\ub85c \ub3d9\uce58\uc774\ub2e4. \uc989, \\((p \\to q) \\equiv (\\neg q \\to \\neg p)\\)\uc774\ub2e4. \uc774\uac83\uc744 <span class=\"defined\">\ub300\uc6b0\ubc95\uce59<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.1.<\/span><br \/>\n\ub2e4\uc74c \uac01\uac01\uc5d0 \ub300\ud558\uc5ec, \uc8fc\uc5b4\uc9c4 \ub450 \uba85\uc81c\uac00 \ub17c\ub9ac\uc801\uc73c\ub85c \ub3d9\uce58\uc784\uc744 \uc9c4\ub9ac\ud45c\ub97c \uc0ac\uc6a9\ud558\uc5ec \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(p \\to q\\)\uc640 \\(\\neg q \\to \\neg p\\)<\/li>\n<li>\\(\\neg (p \\wedge q)\\)\uc640 \\(\\neg p \\vee \\neg q\\)<\/li>\n<li>\\(\\neg (p \\vee q)\\)\uc640 \\(\\neg p \\wedge \\neg q\\)<\/li>\n<li>\\(p \\to (q \\to r)\\)\uc640 \\((p \\wedge q) \\to r\\)<\/li>\n<\/ol>\n<\/div>\n<h3>4. \uba85\uc81c \uc5f0\uc0b0\uc758 \uc131\uc9c8<\/h3>\n<p>\uba85\uc81c \uc5f0\uc0b0\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc911\uc694\ud55c \uc131\uc9c8\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li style=\"margin-bottom: 0.5em;\"><strong>\uad50\ud658\ubc95\uce59:<\/strong><br \/>\n\\[\\begin{aligned}<br \/>\np \\wedge q &#038;\\equiv q \\wedge p\\\\[6pt]<br \/>\np \\vee q &#038;\\equiv q \\vee p<br \/>\n\\end{aligned}\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\"><strong>\uacb0\ud569\ubc95\uce59:<\/strong><br \/>\n\\[\\begin{aligned}<br \/>\n(p \\wedge q) \\wedge r &#038;\\equiv p \\wedge (q \\wedge r)\\\\[6pt]<br \/>\n(p \\vee q) \\vee r &#038;\\equiv p \\vee (q \\vee r)<br \/>\n\\end{aligned}\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\"><strong>\ubd84\ubc30\ubc95\uce59:<\/strong><br \/>\n\\[\\begin{aligned}<br \/>\np \\wedge (q \\vee r) &#038;\\equiv (p \\wedge q) \\vee (p \\wedge r)\\\\[6pt]<br \/>\np \\vee (q \\wedge r) &#038;\\equiv (p \\vee q) \\wedge (p \\vee r)<br \/>\n\\end{aligned}\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\"><strong>\ub4dc \ubaa8\ub974\uac04\uc758 \ubc95\uce59:<\/strong><br \/>\n\\[\\begin{aligned}<br \/>\n\\neg (p \\wedge q) &#038;\\equiv \\neg p \\vee \\neg q\\\\[6pt]<br \/>\n\\neg (p \\vee q) &#038;\\equiv \\neg p \\wedge \\neg q<br \/>\n\\end{aligned}\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\"><strong>\uc774\uc911\ubd80\uc815\ubc95\uce59:<\/strong><br \/>\n\\[\\neg(\\neg p) \\equiv p\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\"><strong>\ud56d\ub4f1\ubc95\uce59:<\/strong><br \/>\n\\[\\begin{aligned}<br \/>\np \\wedge \\text{T} &#038;\\equiv p\\\\[6pt]<br \/>\np \\vee \\text{F} &#038;\\equiv p<br \/>\n\\end{aligned}\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\"><strong>\uc9c0\ubc30\ubc95\uce59:<\/strong><br \/>\n\\[\\begin{aligned}<br \/>\np \\wedge \\text{F} &#038;\\equiv \\text{F}\\\\[6pt]<br \/>\np \\vee \\text{T} &#038;\\equiv \\text{T}<br \/>\n\\end{aligned}\\]\n<\/li>\n<li style=\"margin-bottom: 0.5em;\"><strong>\uba71\ub4f1\ubc95\uce59:<\/strong><br \/>\n\\[\\begin{aligned}<br \/>\np \\wedge p &#038;\\equiv p\\\\[6pt]<br \/>\np \\vee p &#038;\\equiv p<br \/>\n\\end{aligned}\\]\n<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 1.2.<\/span><br \/>\n\uc9c4\ub9ac\ud45c\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc704 \uc131\uc9c8\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<h3>5. \ucd94\ub860 \uaddc\uce59<\/h3>\n<p>\uc218\ud559\uc801 \uc99d\uba85\uc5d0\uc11c\ub294 \ucc38\uc778 \uba85\uc81c\ub4e4\ub85c\ubd80\ud130 \uc0c8\ub85c\uc6b4 \ucc38\uc778 \uba85\uc81c\ub97c \uc774\ub04c\uc5b4\ub0b4\ub294 <span class=\"defined\">\ucd94\ub860 \uaddc\uce59<\/span>\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \ub300\ud45c\uc801\uc778 \ucd94\ub860 \uaddc\uce59\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li style=\"margin-bottom: 1em;\"><strong>\uc804\uac74 \uae0d\uc815<\/strong>(Modus Ponens): \\(p \\to q\\)\uc640 \\(p\\)\uac00 \ucc38\uc774\uba74 \\(q\\)\ub3c4 \ucc38\uc774\ub2e4.<br \/>\n\\[((p \\to q) \\wedge p) \\Rightarrow q\\]\n<\/li>\n<li style=\"margin-bottom: 1em;\"><strong>\ud6c4\uac74 \ubd80\uc815<\/strong>(Modus Tollens): \\(p \\to q\\)\uc640 \\(\\neg q\\)\uac00 \ucc38\uc774\uba74 \\(\\neg p\\)\ub3c4 \ucc38\uc774\ub2e4.<br \/>\n\\[((p \\to q) \\wedge \\neg q) \\Rightarrow (\\neg p)\\]\n<\/li>\n<li style=\"margin-bottom: 1em;\"><strong>\uac00\uc5b8\uc801 \uc0bc\ub2e8\ub17c\ubc95<\/strong>(Hypothetical Syllogism): \\(p \\to q\\)\uc640 \\(q \\to r\\)\uc774 \ucc38\uc774\uba74 \\(p \\to r\\)\ub3c4 \ucc38\uc774\ub2e4.<br \/>\n\\[((p \\to q) \\wedge (q \\to r)) \\Rightarrow (p \\to r)\\]<br \/>\n\uc774\uac83\uc744 <span class=\"defined\">\ucd94\uc774\ubc95\uce59<\/span>\uc774\ub77c\uace0\ub3c4 \ud55c\ub2e4.\n<\/li>\n<li style=\"margin-bottom: 1em;\"><strong>\uc120\uc5b8\uc801 \uc0bc\ub2e8\ub17c\ubc95<\/strong>(Disjunctive Syllogism): \\(p \\vee q\\)\uc640 \\(\\neg p\\)\uac00 \ucc38\uc774\uba74 \\(q\\)\ub3c4 \ucc38\uc774\ub2e4.<br \/>\n\\[((p \\vee q) \\wedge \\neg p) \\Rightarrow q\\]\n<\/li>\n<\/ul>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.3.<\/span><br \/>\n\ub2e4\uc74c \ucd94\ub860\uc774 \ud0c0\ub2f9\ud55c\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ub9cc\uc57d \ube44\uac00 \ub0b4\ub9ac\uba74 \ub545\uc774 \uc816\ub294\ub2e4. \ub545\uc774 \uc816\uc5b4 \uc788\ub2e4. \ub530\ub77c\uc11c \ube44\uac00 \ub0b4\ub9b0 \uac83\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(n\\)\uc774 \uc9dd\uc218\uc774\uba74 \\(n^2\\)\ub3c4 \uc9dd\uc218\uc774\ub2e4. \\(n^2\\)\uc774 \ud640\uc218\uc774\ub2e4. \ub530\ub77c\uc11c \\(n\\)\uc740 \ud640\uc218\uc774\ub2e4.<\/li>\n<li>\\(x > 2\\)\uc774\uba74 \\(x^2 > 4\\)\uc774\ub2e4. \\(x^2 > 4\\)\uc774\uba74 \\(x > 2\\) \ub610\ub294 \\(x < -2\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(x > 2\\)\uc774\uba74 \\(x > 2\\) \ub610\ub294 \\(x < -2\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<h3>6. \ud56d\uc9c4\uba85\uc81c\uc640 \ubaa8\uc21c\uba85\uc81c<\/h3>\n<p>\uc5b4\ub5a4 \uacbd\uc6b0\uc5d0\ub3c4 \ud56d\uc0c1 \ucc38\uc778 \uba85\uc81c\ub97c <span class=\"defined\">\ud56d\uc9c4\uba85\uc81c<\/span>(tautology)\ub77c\uace0 \ud558\uace0, \uc5b4\ub5a4 \uacbd\uc6b0\uc5d0\ub3c4 \ud56d\uc0c1 \uac70\uc9d3\uc778 \uba85\uc81c\ub97c <span class=\"defined\">\ubaa8\uc21c\uba85\uc81c<\/span>(contradiction)\ub77c\uace0 \ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<ul>\n<li>\\(p \\vee \\neg p\\)\ub294 \ud56d\uc9c4\uba85\uc81c\uc774\ub2e4. (\ubc30\uc911\ub960)<\/li>\n<li>\\(p \\wedge \\neg p\\)\ub294 \ubaa8\uc21c\uba85\uc81c\uc774\ub2e4.<\/li>\n<li>\\((p \\to q) \\leftrightarrow (\\neg q \\to \\neg p)\\)\ub294 \ud56d\uc9c4\uba85\uc81c\uc774\ub2e4.<\/li>\n<\/ul>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 1.4.<\/span><br \/>\n\ub2e4\uc74c \uba85\uc81c\uac00 \ud56d\uc9c4\uba85\uc81c\uc778\uc9c0, \ubaa8\uc21c\uba85\uc81c\uc778\uc9c0, \uc544\ub2c8\uba74 \ub458 \ub2e4 \uc544\ub2cc\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\((p \\to q) \\to ((\\neg p) \\vee q)\\)<\/li>\n<li>\\((p \\wedge q) \\wedge \\neg(p \\vee q)\\)<\/li>\n<li>\\(((p \\to q) \\wedge (q \\to r)) \\to (p \\to r)\\)<\/li>\n<li>\\((p \\leftrightarrow q) \\wedge (p \\leftrightarrow \\neg q)\\)<\/li>\n<\/ol>\n<\/div>\n<h3>7. \uba85\uc81c \ud568\uc218<\/h3>\n<p>\ubcc0\uc218\ub97c \ud3ec\ud568\ud55c \ubb38\uc7a5\uc73c\ub85c, \ubcc0\uc218\uc5d0 \ud2b9\uc815 \uac12\uc744 \ub300\uc785\ud558\uba74 \uba85\uc81c\uac00 \ub418\ub294 \uac83\uc744 <span class=\"defined\">\uba85\uc81c \ud568\uc218<\/span>(propositional function) \ub610\ub294 <span class=\"defined\">\uc220\uc5b4<\/span>(predicate)\ub77c\uace0 \ubd80\ub978\ub2e4. [\ub354 \uc5c4\ubc00\ud558\uac8c\ub294, <a href=\"\/blog\/invitation-to-mathematical-logic\/ch14-syntax-first-order-logic\/\">\uc77c\uacc4\ub17c\ub9ac<\/a>\uc5d0\uc11c <span class=\"defined\">\uc790\uc720\ubcc0\uc218\ub97c \uac00\uc9c4 \ub17c\ub9ac\uc2dd<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc2e4\uc81c\ub85c \uc6b0\ub9ac\uac00 \uc9c0\uae08 &#8216;\uba85\uc81c&#8217;\ub77c\uace0 \ubd80\ub974\ub294 \ub300\uc0c1\uc740 \ud615\uc2dd\ub17c\ub9ac\uc5d0\uc11c &#8216;\ubb38\uc7a5'(sentence, \uc790\uc720\ubcc0\uc218\uac00 \uc5c6\ub294 \ub17c\ub9ac\uc2dd)\uc5d0 \ud574\ub2f9\ud558\uba70, \ud604\uc7ac\ub294 \uc9c1\uad00\uc801 \uc774\ud574\ub97c \uc704\ud574 &#8216;\uba85\uc81c&#8217;\uc640 &#8216;\uba85\uc81c\ud568\uc218&#8217;\ub77c\ub294 \uc6a9\uc5b4\ub97c \uc0ac\uc6a9\ud558\uace0 \uc788\ub2e4.] \uc608\ub97c \ub4e4\uc5b4, \\(p(x): x > 5\\)\ub294 \ubcc0\uc218\uac00 \\(x\\)\uc778 \uba85\uc81c \ud568\uc218\uc774\ub2e4. \\(x = 7\\)\uc77c \ub54c \\(p(x)\\)\ub294 \ucc38\uc778 \uba85\uc81c\uac00 \ub418\uace0, \\(x = 3\\)\uc77c \ub54c \\(p(x)\\)\ub294 \uac70\uc9d3\uc778 \uba85\uc81c\uac00 \ub41c\ub2e4.<\/p>\n<p>\uba85\uc81c \ud568\uc218\ub294 \ub2e4\uc74c \uc7a5\uc5d0\uc11c \ub2e4\ub8f0 \ud55c\uc815\uba85\uc81c\ub97c \uc815\uc758\ud558\ub294 \ub370 \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud55c\ub2e4. \ud2b9\ud788 \uc9d1\ud569\ub860\uc5d0\uc11c \uc9d1\ud569\uc758 \uc6d0\uc18c\ub97c \uae30\uc220\ud560 \ub54c \uba85\uc81c \ud568\uc218\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc870\uac74\uc744 \uba85\ud655\ud788 \ud45c\ud604\ud560 \uc218 \uc788\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>\uc218\ud559\uc740 \uc5c4\ubc00\ud55c \ub17c\ub9ac\uc801 \ucd94\ub860\uc744 \ubc14\ud0d5\uc73c\ub85c \ud55c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc218\ud559\uc801 \ucd94\ub860\uc758 \uae30\ucd08\uac00 \ub418\ub294 \uba85\uc81c\uc640 \ub17c\ub9ac\ub97c \ub2e4\ub8ec\ub2e4. \uba85\uc81c\uc758 \uae30\ubcf8 \uac1c\ub150\uacfc \uc5f0\uc0b0\uc744 \uc0b4\ud3b4\ubcf4\uace0, \uc774\ub97c \ud1b5\ud574 \uc218\ud559\uc801 \uc9c4\uc220\uc744 \uc815\ud655\ud558\uac8c \ud45c\ud604\ud558\uace0 \ubd84\uc11d\ud558\ub294 \ubc29\ubc95\uc744 \uc775\ud78c\ub2e4. 1. \uba85\uc81c\uc758 \uac1c\ub150 \uba85\uc81c(proposition)\ub780 \ucc38\uc774\ub098 \uac70\uc9d3\uc744 \uba85\ud655\ud558\uac8c \ud310\ubcc4\ud560 \uc218 \uc788\ub294 \ubb38\uc7a5\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4, &#8220;\\(2 + 3 = 5\\)&#8221;\ub294 \ucc38\uc778 \uba85\uc81c\uc774\uace0, &#8220;\\(\\pi < 3\\)\"\uc740 \uac70\uc9d3\uc778 \uba85\uc81c\uc774\ub2e4. \ubc18\uba74 \"\\(x > 5\\)&#8221;\ub294 \\(x\\)\uc758 \uac12\uc774 \uc815\ud574\uc9c0\uc9c0 \uc54a\uc558\uc73c\ubbc0\ub85c \uba85\uc81c\uac00 \uc544\ub2c8\ub2e4.&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":101,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9238","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9238","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=9238"}],"version-history":[{"count":25,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9238\/revisions"}],"predecessor-version":[{"id":9459,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9238\/revisions\/9459"}],"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=9238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}