{"id":9270,"date":"2025-10-17T20:18:38","date_gmt":"2025-10-17T11:18:38","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9270"},"modified":"2025-10-20T18:49:03","modified_gmt":"2025-10-20T09:49:03","slug":"ch12-propositional-logic","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-logic\/ch12-propositional-logic\/","title":{"rendered":"\uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150"},"content":{"rendered":"<div class=\"mathlogic2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>12. \uba85\uc81c\ub17c\ub9ac\uc758 \uac1c\ub150<\/h2>\n\n --><\/p>\n<p>\uba85\uc81c\ub17c\ub9ac(propositional logic)\ub780 \uac04\ub2e8\ud788 \ub9d0\ud558\uba74 \uba85\uc81c\ubcc0\uc218\uc640 \uae30\ubcf8 \uacb0\ud569\uc790(\ubd80\uc815, \uba85\uc81c\ud569, \uba85\uc81c\uacf1, \ud568\uc758), \uadf8\ub9ac\uace0 \uba87 \uac00\uc9c0 \uacf5\ub9ac\uc640 \ucd94\ub860\uaddc\uce59\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ub17c\ub9ac\uacc4\ub97c \ub73b\ud55c\ub2e4. \uba85\uc81c\ub17c\ub9ac\ub294 \uc218\ub9ac\ub17c\ub9ac\ub97c \uacf5\ubd80\ud558\uae30 \uc704\ud574 \uae30\ubcf8\uc73c\ub85c \uac70\uccd0\uc57c \ud560 \uad00\ubb38\uc774\ub2e4.<\/p>\n<h3>1. \uad6c\ubb38\ub860<\/h3>\n<p><span class=\"defined\">\uad6c\ubb38\ub860<\/span>(syntactics)\uc774\ub780 \uc8fc\uc5b4\uc9c4 \uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \uaddc\uce59\uc5d0 \ub530\ub77c \ub17c\ub9ac\uc2dd\uc744 \ub9cc\ub4e4\uc5b4\ub0b4\uac70\ub098, \uae30\ud638\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubb38\uc790\uc5f4\uc774 \ub17c\ub9ac\uc2dd\uc778\uc9c0 \uc544\ub2cc\uc9c0\ub97c \ubc1d\ud788\ub294 \ubc95\uce59\uc774\ub2e4. \uad6c\ubb38\ub860\uc5d0\uc11c\ub294 \ubb38\uc790\uc5f4\uc758 \uc758\ubbf8\ub97c \ub530\uc9c0\uc9c0 \uc54a\uace0 \uc624\uc9c1 \uae30\ud638 \uc0ac\uc774\uc758 \ud615\uc2dd\uc801 \uad00\uacc4\uc5d0\ub9cc \uad00\uc2ec\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<p>\uba85\uc81c\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860\uc744 \ub2e4\ub8e8\uae30 \uc704\ud574 \uba3c\uc800 \uac00\uc0b0 \uac1c\uc758 <span class=\"defined\">\uba85\uc81c\ubcc0\uc218<\/span>(propositional variable)\uc758 \uc9d1\ud569<br \/>\n\\[\\left\\{p_0 ,\\, p_1 ,\\, p_2 ,\\, p_3 ,\\, p_4 ,\\, \\cdots \\right\\}\\]<br \/>\n\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ubcf8\ub2e4. \uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c\ub294 \uacb0\ud569\uc790(connective)\ub77c\uace0 \ubd88\ub9ac\ub294 \uae30\ud638<br \/>\n\\[\\neg , \\quad \\wedge , \\quad \\vee , \\quad \\rightarrow , \\quad \\leftrightarrow \\]<br \/>\n\ub97c \uc0ac\uc6a9\ud55c\ub2e4. \uc774 \uacb0\ud569\uc790\ub294 \uc21c\uc11c\ub300\ub85c &#8216;\ubd80\uc815&#8217;, &#8216;\ub17c\ub9ac\uacf1&#8217;, &#8216;\ub17c\ub9ac\ud569&#8217;, &#8216;\ud568\uc758&#8217;, &#8216;\uc591\ubc29\ud5a5 \ud568\uc758&#8217;\ub77c\uace0 \ubd88\ub9b0\ub2e4. \ub610\ud55c \uc67c\ucabd \uad04\ud638\uc640 \uc624\ub978\ucabd \uad04\ud638(\uc5ec\ub294 \uc18c\uad04\ud638\uc640 \ub2eb\ub294 \uc18c\uad04\ud638)\ub3c4 \uae30\ud638\ub85c \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<p>\uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c <span class=\"defined\">\ub17c\ub9ac\uc2dd<\/span>(formula)\uc774\ub780 \ub2e4\uc74c \ub450 \uac00\uc9c0 \uaddc\uce59\uc744 \uc720\ud55c \ubc88 \uc0ac\uc6a9\ud558\uc5ec \uc5bb\uc5b4\uc9c0\ub294 \ubb38\uc790\uc5f4\uc774\ub2e4.<\/p>\n<ul>\n<li>\uba85\uc81c\ubcc0\uc218\ub294 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<\/li>\n<li>\\(\\phi\\)\uac00 \ub17c\ub9ac\uc2dd\uc774\uba74 \\( ( \\neg \\phi )\\)\ub3c4 \ub17c\ub9ac\uc2dd\uc774\ub2e4. \\(\\phi\\)\uc640 \\(\\psi\\)\uac00 \ub17c\ub9ac\uc2dd\uc774\uba74 \ub2e4\uc74c \ubb38\uc790\uc5f4\ub3c4 \ubaa8\ub450 \ub17c\ub9ac\uc2dd\uc774\ub2e4.<br \/>\n\\[ (\\phi \\vee \\psi) , \\quad (\\phi \\wedge \\psi) , \\quad  (\\phi \\rightarrow \\psi) , \\quad  (\\phi \\leftrightarrow \\psi) \\]<\/li>\n<\/ul>\n<p>\ub17c\ub9ac\uc2dd\uc740 \uc704 \uaddc\uce59\uc744 \ud55c \ubc88 \ub610\ub294 \uc5ec\ub7ec \ubc88 \uc801\uc6a9\ud558\uc5ec \uc5bb\uc5b4\uc9c0\ub294 \uac83 \uc678\uc5d0\ub294 \uc5c6\ub2e4.[\uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc5bb\uc5b4\uc9c4 \ub17c\ub9ac\uc2dd\uc744 &#8216;well-formed formula'(\uc904\uc5ec\uc11c &#8216;wff&#8217;)\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.] \uc5b4\ub5a0\ud55c \ubb38\uc790\uc5f4\uc774 \ub17c\ub9ac\uc2dd\uc778\uc9c0 \uc5ec\ubd80\ub294 \ud615\uc2dd\uc801\uc778 \uc808\ucc28\uc5d0 \uc758\ud558\uc5ec \ud310\ubcc4\ub41c\ub2e4.<\/p>\n<h3>2. \uc758\ubbf8\ub860<\/h3>\n<p>\uba85\uc81c\ubcc0\uc218 \\(p_i\\)\uac00 \ucc38\uc778\uc9c0 \uac70\uc9d3\uc778\uc9c0 \uc54c\uace0 \uc788\ub2e4\uba74, \uc774\ub4e4 \uba85\uc81c\ubcc0\uc218\ub97c \uacb0\ud569\ud558\uc5ec \ub9cc\ub4e0 \ub17c\ub9ac\uc2dd\uc774 \ucc38\uc778\uc9c0 \uac70\uc9d3\uc778\uc9c0 \ud310\ubcc4\ud560 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \ud310\ubcc4 \uacfc\uc815\uc744 \ub2e4\ub8e8\ub294 \uac83\uc774 <span class=\"defined\">\uc758\ubbf8\ub860<\/span>(semantics)\uc774\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \uc758\ubbf8 \ubd80\uc5ec \uacfc\uc815\uc744 \uba85\ud655\ud558\uac8c \ud558\uae30 \uc704\ud558\uc5ec <span class=\"defined\">\uac12\ub9e4\uae40<\/span>(valuation)\uc744 \ubaa8\ub4e0 \ub17c\ub9ac\uc2dd\uc758 \uc9d1\ud569\uc73c\ub85c\ubd80\ud130 \uc9d1\ud569 \\(\\left\\{\\mathrm{T} ,\\, \\mathrm{F}\\right\\}\\)\ub85c\uc758 \ud568\uc218 \\(v\\)\ub85c \uc815\uc758\ud55c\ub2e4. \uba85\uc81c\ub17c\ub9ac\uc5d0\uc11c \uac12\ub9e4\uae40\uc740 \uc9c4\ub9ac\ud45c\uc640 \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<p><a href=\"\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table02.png\" data-rel=\"penci-gallery-image-content\" ><img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table02.png\" alt=\"\" width=\"566\" height=\"144\" class=\"aligncenter size-full wp-image-9451\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table02.png 1414w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table02-300x77.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table02-1024x261.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table02-768x196.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table02-1170x299.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2025\/10\/mathlogic2025-table02-585x149.png 585w\" sizes=\"(max-width: 566px) 100vw, 566px\" \/><\/a><!-- width=\"1414\" height=\"361\" --><\/p>\n<p>\uc784\uc758\uc758 \uac12\ub9e4\uae40 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(v(\\phi )=\\mathrm{T}\\)\uc77c \ub54c \\(\\phi\\)\ub97c <span class=\"defined\">\ud56d\uc9c4<\/span>(tautology)\uc774\ub77c\uace0 \ubd80\ub974\uba70, \uc784\uc758\uc758 \uac12\ub9e4\uae40 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(v(\\phi )=\\mathrm{F}\\)\uc77c \ub54c \\(\\phi\\)\ub97c <span class=\"defined\">\ubaa8\uc21c<\/span>(contradiction)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(\\varSigma\\)\uac00 \ub17c\ub9ac\uc2dd\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\sigma \\in \\varSigma\\)\uc5d0 \ub300\ud558\uc5ec \\(v(\\sigma ) = \\mathrm{T}\\)\uc778 \ubaa8\ub4e0 \uac12\ub9e4\uae40 \\(v\\)\uc5d0 \ub300\ud558\uc5ec \\(v (\\phi ) = \\mathrm{T}\\)\uac00 \uc131\ub9bd\ud558\uba74 \\(\\phi\\)\ub97c \\(\\varSigma\\)\uc758 <span class=\"defined\">\ub17c\ub9ac\uc801 \uadc0\uacb0<\/span>(logical consequence)\uc774\ub77c\uace0 \ubd80\ub974\uace0<br \/>\n\\[\\varSigma \\models \\phi\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(\\varSigma\\)\uac00 \uacf5\uc9d1\ud569\uc774\uba74 \\(\\phi\\)\ub294 \ud56d\uc9c4\uc774 \ub418\ub294\ub370, \uc774\uac83\uc744<br \/>\n\\[\\models \\phi\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<h3>3. \ud615\uc2dd\ucd94\ub860\uacc4<\/h3>\n<p>\uc774\uc81c \uba85\uc81c\ub17c\ub9ac\uc758 <span class=\"defined\">\ud615\uc2dd\ucd94\ub860\uacc4<\/span>(formal deduction system)\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ucd94\ub860\uacc4\uc758 \uccab \ubc88\uc9f8 \uc694\uc18c\ub85c\uc11c \uba85\uc81c\ub17c\ub9ac\uc758 <span class=\"defined\">\uacf5\ub9ac\ud2c0<\/span>(schemes of axioms)\uc744 \ub3c4\uc785\ud55c\ub2e4.<\/p>\n<p>&nbsp;&nbsp;(A1) \\( ( \\phi \\to ( \\psi \\to \\phi )) \\)<br \/>\n&nbsp;&nbsp;(A2) \\( (( \\phi \\to (\\psi \\to \\theta )) \\to (( \\phi \\to \\psi ) \\to ( \\phi \\to \\theta ))) \\)<br \/>\n&nbsp;&nbsp;(A3) \\( ((( \\neg \\phi ) \\to ( \\neg \\psi )) \\to ( \\psi \\to \\phi ) ) \\)<\/p>\n<p>\uc5ec\uae30\uc11c \\(\\phi\\), \\(\\psi\\), \\(\\theta\\)\ub294 \uc784\uc758\uc758 \ub17c\ub9ac\uc2dd\uc744 \ub098\ud0c0\ub0b8\ub2e4. \uacf5\ub9ac\ud2c0\uc774\ub780 \uacf5\ub9ac\ub4e4\uc758 \ubaa8\uc784\uc744 \uc774\ub974\ub294\ub370, \uacf5\ub9ac\ud2c0\uc744 \uac04\ub2e8\ud788 <span class=\"defined\">\uacf5\ub9ac<\/span>\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c <span class=\"defined\">\ucd94\ub860\uaddc\uce59<\/span>(rule of inference)\uc744 \ub3c4\uc785\ud55c\ub2e4. \ucd94\ub860\uaddc\uce59\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uaddc\uce59 \ud558\ub098 \ubfd0\uc774\ub2e4.<\/p>\n<p style=\"text-align: center;\">\\(p\\)\uc640 \\( ( p \\to q ) \\)\ub85c\ubd80\ud130 \\(q\\)\ub97c \ucd94\ub860\ud55c\ub2e4.<\/p>\n<p>\uc774 \ucd94\ub860\uaddc\uce59\uc744 &#8216;Modus Ponens&#8217;\ub77c\uace0 \ubd80\ub974\uba70 \uac04\ub2e8\ud788 &#8216;MP&#8217;\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p><span class=\"defined\">\uc99d\uba85<\/span>\uc774\ub780 \ub17c\ub9ac\uc2dd\uc758 \uc5f4\uc778\ub370, \uc5f4\uc758 \uac01 \ub17c\ub9ac\uc2dd\uc740 \uacf5\ub9ac\uc774\uac70\ub098 \uc774\uc804 \ud56d\uc5d0 \ub4f1\uc7a5\ud55c \ub17c\ub9ac\uc2dd\uc5d0 \ucd94\ub860\uaddc\uce59\uc744 \uc801\uc6a9\ud558\uc5ec \uc5bb\uc5b4\uc9c0\ub294 \ub17c\ub9ac\uc2dd\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4\ub2e4. \ub610\ud55c \uc99d\uba85\uc758 \ub9c8\uc9c0\ub9c9 \ud56d\uc744 <span class=\"defined\">\uc815\ub9ac<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(\\varSigma\\)\uac00 \ub17c\ub9ac\uc2dd\uc758 \uc9d1\ud569\uc774\uace0 \\(\\phi\\)\uac00 \ub17c\ub9ac\uc2dd\uc774\ub77c\uace0 \ud558\uc790. &#8216;<span class=\"defined\">\\(\\varSigma\\)\ub85c\ubd80\ud130\uc758 \\(\\phi\\)\uc758 \uc99d\uba85<\/span>&#8216;\uc774\ub780 \ub17c\ub9ac\uc2dd\uc758 \uc720\ud55c\uc5f4\uc778\ub370, \uc5f4\uc758 \uac01 \ub17c\ub9ac\uc2dd\uc740 \uacf5\ub9ac\uc774\uac70\ub098 \\(\\varSigma\\)\uc5d0 \uc18d\ud55c \ub17c\ub9ac\uc2dd\uc774\uac70\ub098 \uc774\uc804 \ud56d\uc5d0 \ub4f1\uc7a5\ud55c \ub17c\ub9ac\uc2dd\uc5d0 \ucd94\ub860\uaddc\uce59\uc744 \uc801\uc6a9\ud558\uc5ec \uc5bb\uc5b4\uc9c0\uba70, \uc5f4\uc758 \ub9c8\uc9c0\ub9c9 \ud56d\uc740 \\(\\phi\\)\uc774\ub2e4. \uc774\ub54c \\(\\phi\\)\ub97c <span class=\"defined\">\\(\\varSigma\\)\uc758 \uc815\ub9ac<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(\\varSigma\\)\ub97c <span class=\"defined\">\uac00\uc815 \uc9d1\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70 \\(\\varSigma\\)\uc758 \uc6d0\uc18c\ub97c <span class=\"defined\">\uac00\uc815<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\phi\\)\uac00 \\(\\varSigma\\)\uc758 \uc815\ub9ac\uc778 \uac83\uc744<br \/>\n\\[\\varSigma \\vdash \\phi \\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(\\varSigma\\)\uac00 \uacf5\uc9d1\ud569\uc774\uba74 \\(\\phi\\)\ub97c &#8216;<span class=\"defined\">\uba85\uc81c\ub17c\ub9ac\uc758 \uc815\ub9ac<\/span>&#8216; \ub610\ub294 \uac04\ub2e8\ud558\uac8c &#8216;<span class=\"defined\">\uc815\ub9ac<\/span>&#8216;\ub77c\uace0 \ubd80\ub974\uba70, \uc774 \uc0c1\ud669\uc744<br \/>\n\\[\\vdash \\phi\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<h3>4. \uc99d\uba85 \uc608\uc81c<\/h3>\n<p>\uc99d\uba85\uacfc \uc815\ub9ac\uc758 \uac04\ub2e8\ud55c \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 12.1.<\/span><br \/>\n\uc784\uc758\uc758 \ub17c\ub9ac\uc2dd \\(p\\)\uc5d0 \ub300\ud558\uc5ec \\( (p \\to p )\\)\ub294 \uba85\uc81c\ub17c\ub9ac\uc758 \uc815\ub9ac\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc815\ub9ac\uc758 \uc99d\uba85\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uae38\uc774\uac00 5\uc778(\ub2e4\uc12f \uac1c\uc758 \ud56d\uc744 \uac00\uc9c4) \ub17c\ub9ac\uc2dd\uc758 \uc5f4\uc774\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n&#038;\\vdash \\quad (( p \\to (( p \\to p) \\to p )) \\to (( p \\to ( p \\to p )) \\to (p \\to p )))\\\\[6pt]<br \/>\n&#038;\\vdash \\quad ( p \\to (( p \\to p ) \\to p ))\\\\[6pt]<br \/>\n&#038;\\vdash \\quad (( p \\to ( p \\to p )) \\to ( p \\to p ))\\\\[6pt]<br \/>\n&#038;\\vdash \\quad ( p \\to (p \\to p ))\\\\[6pt]<br \/>\n&#038;\\vdash \\quad ( p \\to p )<br \/>\n\\end{aligned}\\]<br \/>\n\uccab \ud56d\uc740 (A2)\uc758 \\(\\phi,\\) \\(\\psi,\\) \\(\\theta\\)\uc5d0 \uac01\uac01 \\(p,\\) \\( ( p \\to p ) ,\\) \\(p\\)\ub97c \ub300\uc785\ud558\uc5ec \uc5bb\uc740 \uac83\uc774\ub2e4. \ub458\uc9f8 \ud56d\uc740 (A1)\uc758 \\(\\phi ,\\) \\(\\psi\\) \uc5d0 \uac01\uac01 \\(p,\\) \\( ( p \\to p )\\) \ub97c \ub300\uc785\ud558\uc5ec \uc5bb\uc740 \uac83\uc774\ub2e4. \uc14b\uca68 \ud56d\uc740 \uc55e\uc758 \ub450 \ud56d\uc5d0 MP\ub97c \uc801\uc6a9\ud558\uc5ec \uc5bb\uc740 \uac83\uc774\ub2e4. \ub137\uc9f8 \ud56d\uc740 (A1)\uc758 \\(\\phi ,\\) \\(\\psi\\)\uc5d0 \ubaa8\ub450 \\(p\\)\ub97c \ub300\uc785\ud558\uc5ec \uc5bb\uc740 \uac83\uc774\ub2e4. \ub2e4\uc12f \uc9f8 \ud56d\uc740 \uc55e\uc758 \ub450 \ud56d\uc5d0 MP\ub97c \uc801\uc6a9\ud558\uc5ec \uc5bb\uc740 \uac83\uc774\ub2e4.<\/p>\n<p>\uac04\ub2e8\ud55c \uc815\ub9ac\uc870\ucc28\ub3c4 \uadf8\uac83\uc744 \uc99d\uba85\ud558\ub824\uba74 \uc5ec\ub7ec \uac1c\uc758 \ub17c\ub9ac\uc2dd\uc774 \ud544\uc694\ud558\ub2e4. \ud558\uc9c0\ub9cc \ub2e4\uc74c\uacfc \uac19\uc740 \uba85\uc81c\ub17c\ub9ac\uc758 \uc0c1\uc704\uc218\ud559\uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uba74 \uc99d\uba85\uc744 \ub354 \uac04\ub2e8\ud558\uac8c \ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 12.2. (\ucd94\ub860 \uc815\ub9ac)<\/span><br \/>\n\\(\\phi\\)\uac00 \\(\\varSigma \\cup \\left\\{ \\psi \\right\\} \\)\uc758 \uc815\ub9ac\uc774\uba74 \\( (\\psi \\to \\phi )\\)\ub294 \\(\\varSigma\\)\uc758 \uc815\ub9ac\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85 \uac1c\uc694<\/span><br \/>\n\uc99d\uba85\uc740 \\(\\varSigma\\cup\\left\\{\\psi\\right\\}\\)\ub85c\ubd80\ud130\uc758 \\(\\phi\\)\uc758 \uc99d\uba85 \uae38\uc774\uc5d0 \ub300\ud55c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud55c\ub2e4.[\ub17c\ub9ac\ub294 \ubcf4\ud1b5 \uc218\ud559\uc758 \uccb4\uacc4\ub97c \ub2e4\uc9c0\ub294 \uae30\ucd08 \uc774\ub860\uc73c\ub85c \uc5ec\uaca8\uc9c4\ub2e4. \uadf8\ub807\ub2e4\uba74 \ucd94\ub860 \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \ub54c \ub17c\ub9ac\uc2dd\uc758 \uae38\uc774\uc5d0 \ub300\ud55c &#8216;\uc218\ud559\uc801 \uadc0\ub0a9\ubc95&#8217;\uc744 \uc0ac\uc6a9\ud558\ub294 \uac83\uc774 \ud0c0\ub2f9\ud560\uae4c?]<\/p>\n<p>\uae30\ubcf8 \ub2e8\uacc4: \\(\\varSigma\\cup\\left\\{\\psi\\right\\}\\)\ub85c\ubd80\ud130 \\(\\phi\\)\uc758 \ud55c \uc904 \uc9dc\ub9ac \uc99d\uba85\uc774 \uc874\uc7ac\ud55c\ub2e4\uba74 \\(\\phi\\)\ub294 \uacf5\ub9ac\uc774\uac70\ub098 \\(\\varSigma\\)\uc5d0 \uc18d\ud558\uac70\ub098 \\(\\psi\\)\uc640 \uac19\uc740\ub370, \uc138 \uac00\uc9c0 \uacbd\uc6b0 \ubaa8\ub450 \\((\\psi\\rightarrow\\phi)\\)\uac00 \\(\\varSigma\\)\uc758 \uc815\ub9ac\uac00 \ub41c\ub2e4.<\/p>\n<p>\uadc0\ub0a9 \ub2e8\uacc4: \\(\\varSigma\\cup\\left\\{\\psi\\right\\}\\)\ub85c\ubd80\ud130\uc758 \\(\\phi\\)\uc758 \uc99d\uba85 \uae38\uc774 \\(n\\)\uc774 \\(2\\) \uc774\uc0c1\uc774\ub77c\uace0 \ud558\uace0, \uc99d\uba85 \uae38\uc774\uac00 \\(n\\) \ubbf8\ub9cc\uc778 \uacbd\uc6b0 \uc815\ub9ac\uac00 \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uc774\ub54c \uadc0\ub0a9\uc801 \uac00\uc815\uacfc MP \ubc0f (A1)\\(\\sim\\)(A3)\uc744 \uac70\ub4ed \uc801\uc6a9\ud558\uba74 \\((\\psi\\rightarrow\\phi)\\)\ub97c \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ucd94\ub860 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\uc744 \ud558\ub294 \uc608\ub85c, \ub17c\ub9ac\uc2dd<br \/>\n\\[ (( \\neg \\phi ) \\to ( \\phi \\to \\psi ))\\]<br \/>\n\uac00 \uba85\uc81c\ub17c\ub9ac\uc758 \uc815\ub9ac\uc784\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \ucd94\ub860 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\( (\\phi \\to \\psi )\\)\uac00 \\(\\left\\{ (\\neg \\phi ) \\right\\}\\)\ub85c\ubd80\ud130 \ucd94\ub860\ub420 \uc218 \uc788\uc74c\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4. \uadf8 \uc99d\uba85\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\left\\{ ( \\neg \\phi ) \\right \\} &#038;  \\vdash  &#038;&#038; (( \\neg \\phi ) \\to (( \\neg \\psi ) \\to ( \\neg \\phi )))  \\\\[6pt]<br \/>\n\\left\\{ ( \\neg \\phi ) \\right \\} &#038;  \\vdash  &#038;&#038; (\\neg \\phi ) \\\\[6pt]<br \/>\n\\left\\{ ( \\neg \\phi ) \\right \\} &#038;  \\vdash  &#038;&#038; (( \\neg \\psi ) \\to (\\neg \\phi )) \\\\[6pt]<br \/>\n\\left\\{ ( \\neg \\phi ) \\right \\} &#038;  \\vdash  &#038;&#038; ((( \\neg \\psi ) \\to ( \\neg \\phi )) \\to ( \\phi \\to \\psi )) \\\\[6pt]<br \/>\n\\left\\{ ( \\neg \\phi ) \\right \\} &#038;  \\vdash  &#038;&#038; ( \\phi \\to \\psi ) \\\\[6pt]<br \/>\n &#038;  \\vdash \\, &#038;&#038; (( \\neg \\phi ) \\to ( \\phi \\to \\psi ))<br \/>\n\\end{aligned}\\]<br \/>\n\uc99d\uba85\uc5d0\uc11c \uccab\uc9f8 \ud56d\uc740 (A1)\uc774\uba70 \ub458\uc9f8 \ud56d\uc740 \uac00\uc815\uc774\uace0 \uc14b\uc9f8 \ud56d\uc740 MP\uc5d0 \uc758\ud574 \uc5bb\uc5b4\uc9c4 \uac83\uc774\ub2e4. \ub137\uc9f8 \ud56d\uc740 (A3)\uc774\uba70 \ub2e4\uc12f\uc9f8 \ud56d\uc740 \ub137\uc9f8 \ud56d\uc5d0 MP\ub97c \uc801\uc6a9\ud55c \uac83\uc774\uace0 \uc5ec\uc12f\uc9f8 \ud56d\uc740 \ucd94\ub860 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc5bb\uc5b4\uc9c4 \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 12.1.<\/span><br \/>\n\ud615\uc2dd\uacc4\uc758 \uac1c\ub150\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \uc9c8\ubb38\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\ud615\uc2dd\uacc4\uc5d0\uc11c \uacf5\ub9ac\uac00 \uc5c6\ub2e4\uba74 \uc5b4\ub5a4 \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \uc218 \uc788\ub294\uac00?<\/li>\n<li>\ud615\uc2dd\uacc4\uc5d0\uc11c \ucd94\ub860\uaddc\uce59\uc774 \uc5c6\ub2e4\uba74 \uc5b4\ub5a4 \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \uc218 \uc788\ub294\uac00?<\/li>\n<li>\ubaa8\ub4e0 \ub17c\ub9ac\uc2dd\uc774 \uc815\ub9ac\uac00 \ub418\ub294 \ud615\uc2dd\uacc4\uc758 \uc608\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/li>\n<\/ol>\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>\uba85\uc81c\ub17c\ub9ac(propositional logic)\ub780 \uac04\ub2e8\ud788 \ub9d0\ud558\uba74 \uba85\uc81c\ubcc0\uc218\uc640 \uae30\ubcf8 \uacb0\ud569\uc790(\ubd80\uc815, \uba85\uc81c\ud569, \uba85\uc81c\uacf1, \ud568\uc758), \uadf8\ub9ac\uace0 \uba87 \uac00\uc9c0 \uacf5\ub9ac\uc640 \ucd94\ub860\uaddc\uce59\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ub17c\ub9ac\uacc4\ub97c \ub73b\ud55c\ub2e4. \uba85\uc81c\ub17c\ub9ac\ub294 \uc218\ub9ac\ub17c\ub9ac\ub97c \uacf5\ubd80\ud558\uae30 \uc704\ud574 \uae30\ubcf8\uc73c\ub85c \uac70\uccd0\uc57c \ud560 \uad00\ubb38\uc774\ub2e4. 1. \uad6c\ubb38\ub860 \uad6c\ubb38\ub860(syntactics)\uc774\ub780 \uc8fc\uc5b4\uc9c4 \uae30\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec \uaddc\uce59\uc5d0 \ub530\ub77c \ub17c\ub9ac\uc2dd\uc744 \ub9cc\ub4e4\uc5b4\ub0b4\uac70\ub098, \uae30\ud638\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubb38\uc790\uc5f4\uc774 \ub17c\ub9ac\uc2dd\uc778\uc9c0 \uc544\ub2cc\uc9c0\ub97c \ubc1d\ud788\ub294 \ubc95\uce59\uc774\ub2e4. \uad6c\ubb38\ub860\uc5d0\uc11c\ub294 \ubb38\uc790\uc5f4\uc758 \uc758\ubbf8\ub97c \ub530\uc9c0\uc9c0 \uc54a\uace0 \uc624\uc9c1 \uae30\ud638 \uc0ac\uc774\uc758 \ud615\uc2dd\uc801 \uad00\uacc4\uc5d0\ub9cc \uad00\uc2ec\uc744 \uac00\uc9c4\ub2e4. \uba85\uc81c\ub17c\ub9ac\uc758 \uad6c\ubb38\ub860\uc744 \ub2e4\ub8e8\uae30 \uc704\ud574 \uba3c\uc800 \uac00\uc0b0 \uac1c\uc758 \uba85\uc81c\ubcc0\uc218(propositional variable)\uc758&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9246,"menu_order":112,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9270","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9270","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=9270"}],"version-history":[{"count":12,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9270\/revisions"}],"predecessor-version":[{"id":9456,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9270\/revisions\/9456"}],"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=9270"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}