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Given an indexed set of sentences, <math>s_j\!</math> for <math>j \in J,</math> it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
Given an indexed set of sentences, <math>s_j\!</math> for <math>j \in J,</math> it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
<pre>
<br>
If Sj is a sentence
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px; text-align:center"
then the following are equivalent:
| width="80%" |
| width="20%" | <math>\operatorname{Definition~6}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>\text{each string}~ s_j, ~\text{as}~ j ~\text{ranges over the set}~ J,</math>
|- style="height:20px"
</pre>
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| <math>\text{is a sentence about things in the universe}~ X~</math>
|- style="height:60px"
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| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
|}
|-
|
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:60px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D6a.}</math>
| width="80%" style="border-top:1px solid black" | <math>\overset{J}{\underset{j}{\forall}}~ s_j</math>
|- style="height:60px"
|
| <math>\operatorname{D6b.}</math>
| <math>\operatorname{Conj}_j^J s_j</math>
|}
|}
<br>
<br>
