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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.

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<math>\begin{array}{*{11}{c}}

<math>\begin{array}{*{11}{c}}

\mathit{m,}

\mathrm{m,}

& =      & \text{man that is}\, \underline{~~~~}

& =      & \text{man that is}\, \underline{~~~~}

& =      & \mathrm{C}:\mathrm{C}

& =      & \mathrm{C}:\mathrm{C}

& +\!\!, & \mathrm{O}:\mathrm{O}

& +\!\!, & \mathrm{O}:\mathrm{O}

\\[6pt]

\\[6pt]

\mathit{n,}

\mathrm{n,}

& =      & \text{noble that is}\, \underline{~~~~}

& =      & \text{noble that is}\, \underline{~~~~}

& =      & \mathrm{C}:\mathrm{C}

& =      & \mathrm{C}:\mathrm{C}

& +\!\!, & \mathrm{O}:\mathrm{O}

& +\!\!, & \mathrm{O}:\mathrm{O}

\\[6pt]

\\[6pt]

\mathit{w,}

\mathrm{w,}

& =      & \text{woman that is}\, \underline{~~~~}

& =      & \text{woman that is}\, \underline{~~~~}

& =      & \mathrm{B}:\mathrm{B}

& =      & \mathrm{B}:\mathrm{B}

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

\mathit{m},\!\mathit{n}

\mathrm{m},\!\mathrm{n}

& = &

& = &

\text{man that is noble}

\text{man that is noble}

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

\mathit{n},\!\mathit{m}

\mathrm{n},\!\mathrm{m}

& = &

& = &

\text{noble that is a man}

\text{noble that is a man}

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

\mathit{w},\!\mathit{n}

\mathrm{w},\!\mathrm{n}

& = &

& = &

\text{woman that is noble}

\text{woman that is noble}

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

\mathit{n},\!\mathit{w}

\mathrm{n},\!\mathrm{w}

& = &

& = &

\text{noble that is a woman}

\text{noble that is a woman}