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Directory:Jon Awbrey/Notes/Precursors (view source)
Revision as of 15:22, 5 December 2009
, 15:22, 5 December 2009→Hilbert and Ackermann: markup
===Selection 1===
===Selection 1===
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<p>For the intuitive interpretation on which we have hitherto based the predicate calculus, it was essential that the sentences and predicates should be sharply differentiated from the individuals, which occur as the argument values of the predicates. Now, however, there is nothing to prevent us from ''considering the predicates and sentences themselves as individuals which may serve as arguments of predicates''.</p>
<p>For the intuitive interpretation on which we have hitherto based the predicate calculus, it was essential that the sentences and predicates should be sharply differentiated from the individuals, which occur as the argument values of the predicates. Now, however, there is nothing to prevent us from _considering the predicates and sentences themselves as individuals which may serve as arguments of predicates_.</p>
<p>Consider, for example, a logical expression of the form <math>(x)(A \rightarrow F(x)).</math> This may be interpreted as a predicate <math>P(A, F)</math> whose first argument place is occupied by a sentence <math>A,</math> and whose second argument place is occupied by a monadic predicate <math>F.</math></p>
<p>Consider, for example, a logical expression of the form $(x)(A \rightarrow F(x))$. This may be interpreted as a predicate $P(A, F)$ whose first argument place is occupied by a sentence $A$, and whose second argument place is occupied by a monadic predicate $F$.</p>
<p>A false sentence <math>A</math> is related to every <math>F</math> by the relation <math>P(A, F);</math> a true sentence <math>A</math> only to those <math>F</math> for which <math>(x)F(x)</math> holds.</p>
<p>A false sentence $A$ is related to every $F$ by the relation $P(A, F)$; a true sentence $A$ only to those $F$ for which $(x)F(x)$ holds.</p>
<p>Further examples are given by the properties of ''reflexivity'', ''symmetry'', and ''transitivity'' of dyadic predicates. To these correspond three predicates: <math>\operatorname{Ref}(R),</math> <math>\operatorname{Sym}(R),</math> and <math>\operatorname{Tr}(R),</math> whose argument <math>R</math> is a dyadic predicate. These three properties are expressed in symbols as follows:</p>
<p>Further examples are given by the properties of _reflexivity_, _symmetry_, and _transitivity_ of dyadic predicates. To these correspond three predicates: $\mathop{Ref}(R)$, $\mathop{Sym}(R)$, and $\mathop{Tr}(R)$, whose argument $R$ is a dyadic predicate. These three properties are expressed in symbols as follows:</p>
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<p><math>\operatorname{Ref}(R) \colon (x)R(x, x),</math><br>
<math>\operatorname{Sym}(R) \colon (x)(y)(R(x, y) \rightarrow R(y, x)),</math><br>
<math>\operatorname{Tr}(R) \colon (x)(y)(z)(R(x, y) \And R(y, z) \rightarrow R(x, z)).</math></p>
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<p>All three properties are possessed by the predicate <math>\equiv(x, y)</math> (<math>x</math> is identical with <math>y</math>). The predicate <math><(x, y),</math> on the other hand, possesses only the property of transitivity. Thus the formulas <math>\operatorname{Ref}(\equiv),</math> <math>\operatorname{Sym}(\equiv),</math> <math>\operatorname{Tr}(\equiv),</math> and <math>\operatorname{Tr}(<)</math> are true sentences, whereas <math>\operatorname{Ref}(<)</math> and <math>\operatorname{Sym}(<)</math> are false.</p>
<p>$\mathop{Ref}(R): (x)R(x, x)$,<br>
<p>Such ''predicates of predicates'' will be called ''predicates of second level''. (p. 135).</p>
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<p>Such _predicates of predicates_ will be called _predicates of second level_. (p. 135).</p>
===Selection 2===
===Selection 2===