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In any system where elements are organized according to types, there tend to be any number of ways in which elements of one type are naturally associated with elements of another type.  If the association is anything like a logical equivalence, but with the first type being lower and the second type being higher in some sense, then one may speak of a ''semantic ascent' from the lower to the higher type.

In any system where elements are organized according to types, there tend to be any number of ways in which elements of one type are naturally associated with elements of another type.  If the association is anything like a logical equivalence, but with the first type being lower and the second type being higher in some sense, then one may speak of a ''semantic ascent' from the lower to the higher type.

For example, it is common in mathematics to associate an element <math>a\!</math> of a set <math>A\!</math> with the constant function <math>f_a : X \to A</math> that has <math>f_a (x) = a\!</math> for all <math>x\!</math> in <math>X,\!</math> where <math>X\!</math> is an arbitrary set.  Indeed, the correspondence is so close that one often uses the same name <math>^{\backprime\backprime} a \, ^{\prime\prime}</math> to denote both the element <math>a\!</math> in <math>A\!</math> and the function <math>a = f_a : X \to A,</math> relying on the context or an explicit type indication to tell them apart.

For example, it is common in mathematics to associate an element <math>a\!</math> of a set <math>A\!</math> with the constant function <math>f_a : X \to A</math> that has <math>f_a (x) = a\!</math> for all <math>x\!</math> in <math>X,\!</math> where <math>X\!</math> is an arbitrary set.  Indeed, the correspondence is so close that one often uses the same name <math>{}^{\backprime\backprime} a {}^{\prime\prime}</math> to denote both the element <math>a\!</math> in <math>A\!</math> and the function <math>a = f_a : X \to A,</math> relying on the context or an explicit type indication to tell them apart.

For another instance, we have the "tacit extension" of a ''k''-place relation ''L''&nbsp;&sube;&nbsp;''X''₁&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''_''k'' to a (''k''+1)-place relation ''L''&prime;&nbsp;&sube;&nbsp;''X''₁&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''_''k''+1 that we get by letting ''L''&prime; = ''L''&nbsp;&times;&nbsp;''X''_{''k''+1&nbsp;}, that is, by maintaining the constraints of ''L'' on the first ''k'' variables and letting the last variable wander freely.

For another example, we have the ''tacit extension'' of a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> to a <math>(k+1)\!</math>-place relation <math>L^\prime \subseteq X_1 \times \ldots \times X_{k+1}\!</math> that we get by letting <math>L^\prime = L \times X_{k+1},</math> that is, by maintaining the constraints of <math>L\!</math> on the first <math>k\!</math> variables and letting the last variable wander freely.

What we have here, if I understand Peirce correctly, is another such type of natural extension, sometimes called the "diagonal extension".  This associates a ''k''-adic relative or a ''k''-adic relation, counting the absolute term and the set whose elements it denotes as the cases for ''k'' = 0, with a series of relatives and relations of higher adicities.

What we have here, if I understand Peirce correctly, is another such type of natural extension, sometimes called the ''diagonal extension''.  This extension associates a <math>k\!</math>-adic relative or a <math>k\!</math>-adic relation, counting the absolute term and the set whose elements it denotes as the cases for <math>k = 0,\!</math> with a series of relatives and relations of higher adicities.

A few examples will suffice to anchor these ideas.

A few examples will suffice to anchor these ideas.