Changes

</ol>

</ol>

Of course, these two modalities of formal language, like written and spoken natural languages, are meant to have compatible interpretations, and so it is usually sufficient to give just the meanings of either one. All that remains is to provide a ''codomain'' or a ''target space'' for the intended semantic function, in other words, to supply a suitable range of logical meanings for the memberships of these languages to map into. Out of the many interpretations that are formally possible to arrange, one way of doing this proceeds by making the following definitions:

Of course, these two modalities of formal language, like written and

 

spoken natural languages, are meant to have compatible interpretations,

and so it is usually sufficient to give just the meanings of either one.

 

All that remains is to provide a "codomain" or a "target space" for the

intended semantic function, in other words, to supply a suitable range

of logical meanings for the memberships of these languages to map into.

Out of the many interpretations that are formally possible to arrange,

one way of doing this proceeds by making the following definitions:

1.  The "conjunction" Conj^J_j Q_j of a set of propositions, {Q_j : j in J},

<p><math>\operatorname{Conj}_j^J q_j</math> is true &nbsp;<math>\Leftrightarrow</math>&nbsp; <math>q_j\!</math> is true for every <math>j \in J.</math></p></li>

    is a proposition that is true if and only if each one of the Q_j is true.

    Conj^J_j Q_j is true <=> Q_j is true for every j in J.

<p>The ''surjunction'' <math>\operatorname{Surj}_j^J q_j</math> of a set of propositions, <math>\{ q_j : j \in J \},</math> is a proposition that is true if and only if exactly one of the <math>q_j\!</math> is untrue.</p>

2.  The "surjunction" Surj^J_j Q_j of a set of propositions, {Q_j : j in J},

<p><math>\operatorname{Surj}_j^J q_j</math> is true &nbsp;<math>\Leftrightarrow</math>&nbsp;  <math>q_j\!</math> is untrue for unique <math>j \in J.</math></p></li>

    is a proposition that is true if and only if just one of the Q_j is untrue.

    Surj^J_j Q_j is true  <=> Q_j is untrue for unique j in J.

</ol>

If the number of propositions that are being joined together is finite,

If the number of propositions that are being joined together is finite, then the conjunction and the surjunction can be represented by means of sentential connectives, incorporating the sentences that represent these propositions into finite strings of symbols.

then the conjunction and the surjunction can be represented by means of

sentential connectives, incorporating the sentences that represent these

propositions into finite strings of symbols.

If J is finite, for instance, if J constitutes the interval j = 1 to k,

If J is finite, for instance, if J constitutes the interval j = 1 to k,

and if each proposition Q_j is represented by a sentence S_j, then the

and if each proposition Q_j is represented by a sentence S_j, then the