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<p>For example, <math>x \perp y</math> signifies that <math>x\!</math> is <math>\mathbf{f}</math> and <math>y\!</math> is <math>\mathbf{f}</math>.  Then <math>(x \perp y) \perp z</math>, or <math>\underline {x \perp y} \perp z</math>, will signify that <math>z\!</math> is <math>\mathbf{f}</math>, but that the statement that <math>x\!</math> and <math>y\!</math> are both <math>\mathbf{f}</math> is itself <math>\mathbf{f}</math>, that is, is ''false''.  Hence, the value of <math>x \perp x</math> is the same as that of <math>\overline {x}</math>;  and the value of <math>\underline {x \perp x} \perp x</math> is <math>\mathbf{f}</math>, because it is necessarily false;  while the value of <math>\underline {x \perp y} \perp \underline {x \perp y}</math> is only <math>\mathbf{f}</math> in case <math>x \perp y</math> is <math>\mathbf{v}</math>;  and <math>( \underline {x \perp x} \perp x) \perp (x \perp \underline {x \perp x})</math> is necessarily true, so that its value is <math>\mathbf{v}</math>.</p>

<p>For example, <math>x \curlywedge y</math> signifies that <math>x\!</math> is <math>\mathbf{f}</math> and <math>y\!</math> is <math>\mathbf{f}</math>.  Then <math>(x \curlywedge y) \curlywedge z</math>, or <math>\underline {x \curlywedge y} \curlywedge z</math>, will signify that <math>z\!</math> is <math>\mathbf{f}</math>, but that the statement that <math>x\!</math> and <math>y\!</math> are both <math>\mathbf{f}</math> is itself <math>\mathbf{f}</math>, that is, is ''false''.  Hence, the value of <math>x \curlywedge x</math> is the same as that of <math>\overline {x}</math>;  and the value of <math>\underline {x \curlywedge x} \curlywedge x</math> is <math>\mathbf{f}</math>, because it is necessarily false;  while the value of <math>\underline {x \curlywedge y} \curlywedge \underline {x \curlywedge y}</math> is only <math>\mathbf{f}</math> in case <math>x \curlywedge y</math> is <math>\mathbf{v}</math>;  and <math>( \underline {x \curlywedge x} \curlywedge x) \curlywedge (x \curlywedge \underline {x \curlywedge x})</math> is necessarily true, so that its value is <math>\mathbf{v}</math>.</p>

<p>With these two signs, the [[vinculum]] (with its equivalents, parentheses, brackets, braces, etc.) and the sign <math>\perp</math>, which I will call the ''ampheck'' (from &#945;&#956;&#966;&#951;&#954;&#942;&#962;&nbsp;, cutting both ways), all assertions as to the values of quantities can be expressed. (C.S. Peirce, CP 4.264).</p>

<p>With these two signs, the [[vinculum]] (with its equivalents, parentheses, brackets, braces, etc.) and the sign <math>\curlywedge</math>, which I will call the ''ampheck'' (from &#945;&#956;&#966;&#951;&#954;&#942;&#962;&nbsp;, cutting both ways), all assertions as to the values of quantities can be expressed. (C.S. Peirce, CP 4.264).</p>

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