Editing Directed set (section)

===Logic===

{{See also|Preorder#Preorders and partial orders on partitions}}

Let <math>S</math> be a [[Theory (mathematical logic)|formal theory]], which is a set of [[Sentence (mathematical logic)|sentences]] with certain properties (details of which can be found in [[Theory (mathematical logic)|the article on the subject]]). For instance, <math>S</math> could be a [[first-order theory]] (like [[Zermelo–Fraenkel set theory]]) or a simpler [[Propositional calculus|zeroth-order theory]]. The preordered set <math>(S, \Leftarrow)</math> is a directed set because if <math>A, B \in S</math> and if <math>C := A \wedge B</math> denotes the sentence formed by [[logical conjunction]] <math>\,\wedge,\,</math> then <math>A \Leftarrow C</math> and <math>B \Leftarrow C</math> where <math>C \in S.</math> 
If <math>S / \sim</math> is the [[Lindenbaum–Tarski algebra]] associated with <math>S</math> then <math>\left(S / \sim, \Leftarrow\right)</math> is a partially ordered set that is also a directed set.