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<p>Let us now take the two statements, <math>S\ \operatorname{is}\ P,</math> <math>\Sigma\ \operatorname{is}\ P;</math> let us suppose that <math>\Sigma\!</math> is much more distinct than <math>S\!</math> and that it is also more extensive.  But we ''know'' that <math>S\ \operatorname{is}\ P.</math> Now if <math>\Sigma\!</math> were not more extensive than S, <math>\Sigma\ \operatorname{is}\ P</math> would contain more truth than <math>S\ \operatorname{is}\ P;</math> being more extensive it ''may'' contain more truth and it may also introduce a falsehood.  Which of these probabilities is the greatest?  <math>\Sigma\!</math> by being more extensive becomes less intensive;  it is the intension which introduces truth and the extension which introduces falsehood.  If therefore <math>\Sigma\!</math> increases the intension of <math>S\!</math> more than its extension, <math>\Sigma\!</math> is to be preferred to <math>S;\!</math> otherwise not.  Now this is the case of induction.  Which contains most truth, ''neat'' and ''deer'' are herbivora, or cloven-footed animals are herbivora?</p>

<p>Let us now take the two statements, ''S'' is ''P'', &Sigma; is ''P'';  let us suppose that &Sigma; is much more distinct than ''S'' and that it is also more extensive.  But we ''know'' that ''S'' is ''P''.  Now if &Sigma; were not more extensive than ''S'', &Sigma; is ''P'' would contain more truth than ''S'' is P;  being more extensive it ''may'' contain more truth and it may also introduce a falsehood.  Which of these probabilities is the greatest?  &Sigma; by being more extensive becomes less intensive;  it is the intension which introduces truth and the extension which introduces falsehood.  If therefore &Sigma; increases the intension of ''S'' more than its extension, &Sigma; is to be preferred to ''S''; otherwise not.  Now this is the case of induction.  Which contains most truth, ''neat'' and ''deer'' are herbivora, or cloven-footed animals are herbivora?</p>

<p>In the two statements, <math>S\ \operatorname{is}\ P,</math> <math>S\ \operatorname{is}\ \Pi,</math> let <math>\Pi\!</math> be at once more ''formal'' and more ''intensive'' than <math>P;\!</math> and suppose we only ''know'' that <math>S\ \operatorname{is}\ P.</math> In this case the increase of formality gives a chance of additional truth and the increase of intension a chance of error.  If the extension of <math>\Pi\!</math> is more increased than than its intension, then <math>S\ \operatorname{is}\ \Pi</math> is likely to contain more truth than <math>S\ \operatorname{is}\ P</math> and ''vice versa''.  This is the case of ''à posteriori'' reasoning.  We have for instance to choose between

<p>In the two statements, ''S'' is ''P'', ''S'' is &Pi;, let &Pi; be at once more ''formal'' and more ''intensive'' than ''P'';  and suppose we only ''know'' that ''S'' is ''P''.  In this case the increase of formality gives a chance of additional truth and the increase of intension a chance of error.  If the extension of &Pi; is more increased than than its intension, then ''S'' is &Pi; is likely to contain more truth than ''S'' is ''P'' and ''vice versa''.  This is the case of ''à posteriori'' reasoning.  We have for instance to choose between

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