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From here it is easy to see the relation of concatenation, by virtue of these types of intersection, to the logical conjunction of propositions.  The cartesian product <math>P \times Q</math> is described by a conjunction of propositions, namely, <math>P_1 \land Q_2,</math> subject to the following interpretation:

From here it is easy to see the relation of concatenation, by virtue of these types of intersection, to the logical conjunction of propositions.  The cartesian product <math>P \times Q</math> is described by a conjunction of propositions, namely, <math>P_{[1]} \land Q_{[2]},</math> subject to the following interpretation:

# <math>P_1\!</math> asserts that there is an element from the set <math>P\!</math> in the first place of the product.

# <math>P_{[1]}\!</math> asserts that there is an element from the set <math>P\!</math> in the first place of the product.

# <math>Q_2\!</math> asserts that there is an element from the set <math>Q\!</math> in the second place of the product.

# <math>Q_{[2]}\!</math> asserts that there is an element from the set <math>Q\!</math> in the second place of the product.

The integration of these two pieces of information can be taken in that measure to specify a yet to be fully determined relation.

The integration of these two pieces of information can be taken in that measure to specify a yet to be fully determined relation.

In a corresponding fashion at the level of the elements, the ordered pair <math>(p, q)\!</math> is described by a conjunction of propositions, namely, <math>p_1 \land q_2,</math> subject to the following interpretation:

In a corresponding fashion at the level of the elements, the ordered pair <math>(p, q)\!</math> is described by a conjunction of propositions, namely, <math>p_{[1]} \land q_{[2]},</math> subject to the following interpretation:

# <math>p_1\!</math> says that <math>p\!</math> is in the first place of the product element under construction.

# <math>p_{[1]}\!</math> says that <math>p\!</math> is in the first place of the product element under construction.

# <math>q_2\!</math> says that <math>q\!</math> is in the second place of the product element under construction.

# <math>q_{[2]}\!</math> says that <math>q\!</math> is in the second place of the product element under construction.

Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> in this way, one shifts the level of the active construction from the tupling of the elements in P and Q or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_1\!</math> and <math>Q_2,\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{(1)}</math> and <math>(\mathfrak{L}_2)_{(2)},</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively.  In effect, the subscripting by the indices <math>^{\backprime\backprime} (1) ^{\prime\prime}</math> and <math>^{\backprime\backprime} (2) ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.

Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}</math> and <math>(\mathfrak{L}_2)_{[2]},</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively.  In effect, the subscripting by the indices <math>^{\backprime\backprime} [1] ^{\prime\prime}</math> and <math>^{\backprime\backprime} [2] ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.

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