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<p style="line-height: 200%" align="center"><b><font size="5">永 真 式</font></b> </p>
<p style="line-height: 200%"> 谓词演算中的永真式可以通过下面的两种 途径获得： </p>
<p style="line-height: 200%"><br>
<b>一、由永真命题公式获得：</b>用原子谓词公式取代命题演算中的命题变元<br>
依据：永真式的任何代入实例也必永真。
</p>
<p style="line-height: 200%">
比如前面我们学过的命题公式都可以相应的得到谓词的永真公式：
</p>
<blockquote>
<p style="line-height: 200%" align="left">P<img src="image/dengjia.gif" width="17" height="9">P<img src="image/xiqu.gif" width="9" height="15">P
则有：<img src="image/renyi.gif" width="9" height="11">xF(x)<img src="image/dengjia.gif" width="17" height="9"><img src="image/renyi.gif" width="9" height="11">xF(x)<img src="image/xiqu.gif" width="9" height="15"><img src="image/renyi.gif" width="9" height="11">xF(x)<br>
<img src="image/fei.gif" width="10" height="5">(P<img src="image/xiqu.gif" width="9" height="15">Q)<img src="image/dengjia.gif" width="17" height="9">(<img src="image/fei.gif" width="10" height="5">P<img src="image/hequ.gif" width="9" height="11"><img src="image/fei.gif" width="10" height="5">Q)
则有: <img src="image/fei.gif" width="10" height="5">(<img src="image/renyi.gif" width="9" height="11">xF(x)<img src="image/xiqu.gif" width="9" height="15"><img src="image/cunzai.gif" width="7" height="11">xG(x))<img src="image/dengjia.gif" width="17" height="9">(<img src="image/fei.gif" width="10" height="5"><img src="image/renyi.gif" width="9" height="11">xF(x)<img src="image/hequ.gif" width="9" height="11"><img src="image/fei.gif" width="10" height="5"><img src="image/cunzai.gif" width="7" height="11">xG(x))
</p>
</blockquote>
<p style="line-height: 200%"><b>二、由已知永真公式通过转换公式获得</b>，几个转换公式如下：
</p>
<p style="line-height: 200%"><b>1.量词消去律（在有限个体域D={a1,a2,……an）</b>
</p>
<blockquote>
<p style="line-height: 200%"><img src="image/renyi.gif" width="9" height="11">xA(x)<img src="image/dengjia.gif" width="17" height="9">A(a1)<img src="image/hequ.gif" width="9" height="11">A(a2)<img src="image/hequ.gif" width="9" height="11">…<img src="image/hequ.gif" width="9" height="11">A(an)<br>
<img src="image/cunzai.gif" width="7" height="11">xA(x)<img src="image/dengjia.gif" width="17" height="9">A(a1)<img src="image/xiqu.gif" width="9" height="15">A(a2)<img src="image/xiqu.gif" width="9" height="15">…<img src="image/xiqu.gif" width="9" height="15">A(an)
</p>
</blockquote>
<p style="line-height: 200%"><b>2.量词转换律</b>
</p>
<blockquote>
<p style="line-height: 200%">
<img src="image/fei.gif" width="10" height="5"><img src="image/renyi.gif" width="9" height="11">xA(x)<img src="image/dengjia.gif" width="17" height="9"><img src="image/cunzai.gif" width="7" height="11">x<img src="image/fei.gif" width="10" height="5">A(x)<br>
<img src="image/fei.gif" width="10" height="5"><img src="image/cunzai.gif" width="7" height="11">xA(x)<img src="image/dengjia.gif" width="17" height="9"><img src="image/renyi.gif" width="9" height="11">x<img src="image/fei.gif" width="10" height="5">A(x)
</p>
</blockquote>
<p style="line-height: 200%"><b>3.量词辖域的扩张和收缩律</b>
</p>
<blockquote>
<table border="0" width="219" cellspacing="0" cellpadding="0" height="290">
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<td width="209" bgcolor="#00FFFF" height="36">
<p style="line-height: 200%"> <img src="image/renyi.gif" width="9" height="11">xA(x)<img src="image/xiqu.gif" width="9" height="15">P<img src="image/dengjia.gif" width="17" height="9"><img src="image/renyi.gif" width="9" height="11">x(A(x)<img src="image/xiqu.gif" width="9" height="15">P)</td>
</tr>
<tr>
<td width="209" bgcolor="#C0C0C0" height="36">
<p style="line-height: 200%"> <img src="image/renyi.gif" width="9" height="11">xA(x)<img src="image/hequ.gif" width="9" height="11">P<img src="image/dengjia.gif" width="17" height="9"><img src="image/renyi.gif" width="9" height="11">x(A(x)<img src="image/hequ.gif" width="9" height="11">P)</td>
</tr>
<tr>
<td width="209" bgcolor="#00FFFF" height="36">
<p style="line-height: 200%"> <img src="image/renyi.gif" width="9" height="11">x(A(x)<img border="0" src="image/yunhan.gif" width="15" height="9">P)<img src="image/dengjia.gif" width="17" height="9"><img src="image/cunzai.gif" width="7" height="11">xA(x)<img border="0" src="image/yunhan.gif" width="15" height="9">P</td>
</tr>
<tr>
<td width="209" bgcolor="#C0C0C0" height="36">
<p style="line-height: 200%"> <img src="image/renyi.gif" width="9" height="11">x(P<img border="0" src="image/yunhan.gif" width="15" height="9">A(x))<img src="image/dengjia.gif" width="17" height="9">P<img border="0" src="image/yunhan.gif" width="15" height="9"><img src="image/renyi.gif" width="9" height="11">xA(x)</td>
</tr>
<tr>
<td width="209" bgcolor="#00FFFF" height="36">
<p style="line-height: 200%"> <img src="image/cunzai.gif" width="7" height="11">xA(x)<img src="image/xiqu.gif" width="9" height="15">P<img src="image/dengjia.gif" width="17" height="9"><img src="image/cunzai.gif" width="7" height="11">x(A(x)<img src="image/xiqu.gif" width="9" height="15">P)</td>
</tr>
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<td width="209" bgcolor="#C0C0C0" height="36">
<p style="line-height: 200%"> <img src="image/cunzai.gif" width="7" height="11">xA(x)<img src="image/hequ.gif" width="9" height="11">P<img src="image/dengjia.gif" width="17" height="9"><img src="image/cunzai.gif" width="7" height="11">x(A(x)<img src="image/hequ.gif" width="9" height="11">P)</td>
</tr>
<tr>
<td width="209" bgcolor="#00FFFF" height="37">
<p style="line-height: 200%"> <img src="image/cunzai.gif" width="7" height="11">x(A(x)<img border="0" src="image/yunhan.gif" width="15" height="9">P)<img src="image/dengjia.gif" width="17" height="9"><img src="image/renyi.gif" width="9" height="11">xA(x)<img border="0" src="image/yunhan.gif" width="15" height="9">P</td>
</tr>
<tr>
<td width="209" bgcolor="#C0C0C0" height="37">
<p style="line-height: 200%"> <img src="image/cunzai.gif" width="7" height="11">x(P<img border="0" src="image/yunhan.gif" width="15" height="9">A(x))<img src="image/dengjia.gif" width="17" height="9">P<img border="0" src="image/yunhan.gif" width="15" height="9"><img src="image/cunzai.gif" width="7" height="11">xA(x)</td>
</tr>
</table>
</blockquote>
<p style="line-height: 200%"><b>4.量词分配律</b>
</p>
<blockquote>
<p style="line-height: 200%"><img src="image/renyi.gif" width="9" height="11">x(A(x)<img src="image/hequ.gif" width="9" height="11">B(x))<img src="image/dengjia.gif" width="17" height="9"><img src="image/renyi.gif" width="9" height="11">xA(x)<img src="image/hequ.gif" width="9" height="11"><img src="image/renyi.gif" width="9" height="11">xB(x)<br>
<img src="image/cunzai.gif" width="7" height="11">x(A(x)<img src="image/xiqu.gif" width="9" height="15">B(x))<img src="image/dengjia.gif" width="17" height="9"><img src="image/cunzai.gif" width="7" height="11">xA(x)<img src="image/xiqu.gif" width="9" height="15"><img src="image/cunzai.gif" width="7" height="11">xB(x)
</p>
</blockquote>
<p style="line-height: 200%"><b>5.两个量词时:</b>
</p>
<blockquote>
<p style="line-height: 200%">*xA(x)<img src="image/hequ.gif" width="9" height="11"> #yB(y)<img src="image/dengjia.gif" width="17" height="9">*x
#y(A(x)<img src="image/hequ.gif" width="9" height="11">B(y))<br>
*xA(x)<img src="image/xiqu.gif" width="9" height="15"> #yB(y)<img src="image/dengjia.gif" width="17" height="9">*x
#y(A(x)<img src="image/xiqu.gif" width="9" height="15">B(y))<br>
这里 *,# 分别表示 <img src="image/renyi.gif" width="9" height="11">
或 <img src="image/cunzai.gif" width="7" height="11">
</p>
</blockquote>
<p style="line-height: 200%"><b>6.代入规则、替换规则、对偶原理</b>
</p>
<p style="line-height: 200%"> <b>代入规则：</b>
</p>
<ul msimagelist imagesrc="Image/gif/2.gif">
<li>
<p style="line-height: 200%">对公式中所有的同名自由变元都用另一个自由变元来代替，且新的变元没有在原来的公式中出现过。<br>
F(x)-->F(y)</li>
<li>
<p style="line-height: 200%">对公式中的谓词也可用公式未曾出现过的新的谓词来代替，只要求新的谓词中的变元没有在原来的公式中出现过。<br>
<img src="image/renyi.gif" width="9" height="11">x(A(x)<img src="image/hequ.gif" width="9" height="11">B(x))
--><img src="image/renyi.gif" width="9" height="11">x(F(x,y)<img src="image/hequ.gif" width="9" height="11">G(x))</li>
</ul>
<p style="line-height: 200%"> <b>替换规则：</b></p>
<ul msimagelist imagesrc="Image/gif/2.gif">
<li>
<p style="line-height: 200%">设A(x1,x2,…,xn)<img src="image/dengjia.gif" width="17" height="9">B(x1,x2,…,xn),C、D分别为A、B的子公式，则将C、D分别代入A、B中得到A'<img src="image/dengjia.gif" width="17" height="9">B'</li>
</ul>
<p style="line-height: 200%"> <b>对偶原理:</b>
</p>
<ul msimagelist imagesrc="Image/gif/2.gif">
<li>
<p style="line-height: 200%">设A'、B'分别为A、B的对偶公式，则若A<img src="image/dengjia.gif" width="17" height="9">B，则A'<img src="image/dengjia.gif" width="17" height="9">B'。</li>
</ul>
<p style="line-height: 200%">
</p>
<p style="line-height: 200%">例1: 设个体域D={a,b,c},将下列公式中的量词消去
</p>
<blockquote>
<p style="line-height: 200%">(1) <img src="image/renyi.gif" width="9" height="11">x(F(x)<img border="0" src="image/yunhan.gif" width="15" height="9">G(x))<br>
(2) <img src="image/cunzai.gif" width="7" height="11">x(F(x)<img src="image/hequ.gif" width="9" height="11">G(x))<br>
(3) <img src="image/renyi.gif" width="9" height="11">x(F(x)<img border="0" src="image/yunhan.gif" width="15" height="9"><img src="image/cunzai.gif" width="7" height="11">yG(y))
</p>
</blockquote>
<p style="line-height: 200%">例2: 用其他的公式证明下列公式
</p>
<blockquote>
<p style="line-height: 200%">(1) <img src="image/renyi.gif" width="9" height="11">x(A(x)<img border="0" src="image/yunhan.gif" width="15" height="9">P)<img src="image/dengjia.gif" width="17" height="9"><img src="image/cunzai.gif" width="7" height="11">xA(x)<img border="0" src="image/yunhan.gif" width="15" height="9">P<br>
(2) <img src="image/renyi.gif" width="9" height="11">x(P<img border="0" src="image/yunhan.gif" width="15" height="9">A(x))<img src="image/dengjia.gif" width="17" height="9">P<img border="0" src="image/yunhan.gif" width="15" height="9"><img src="image/renyi.gif" width="9" height="11">xA(x)
</p>
</blockquote>
<p style="line-height: 200%">例3: 证明</p>
<ul dynamicoutline initcollapsed>
<li>
<p style="line-height: 200%"><img src="image/renyi.gif" width="9" height="11">x(A(x)<img src="image/xiqu.gif" width="9" height="15">B(x))
不等价于 <img src="image/renyi.gif" width="9" height="11">xA(x)<img src="image/xiqu.gif" width="9" height="15"><img src="image/renyi.gif" width="9" height="11">xB(x)
<ul>
<li>
<p style="line-height: 200%">设:公式的解释的个体域 D
为自然数域 N ,<br>
A(x)表示x为奇数, B(x)表示x为偶数<br>
</li>
</ul>
</li>
<li>
<p style="line-height: 200%"><img src="image/cunzai.gif" width="7" height="11">x(A(x)<img src="image/hequ.gif" width="9" height="11">B(x))
不等价于 <img src="image/cunzai.gif" width="7" height="11">xA(x)<img src="image/hequ.gif" width="9" height="11"><img src="image/cunzai.gif" width="7" height="11">xB(x)</li>
</ul>
<p style="line-height: 200%">
</p>
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</p>
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</p>
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</p>
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