In set theory, $A \cup B$ is logically defined as $\{x : x \in A \lor x \in B\}$. In set theory, the result of unionizing A with B is a bigger set, but in logic, "or" is a softening operation.
In set theory, $A \cap B$ is logically defined as $\{x : x \in A \land x \in B\}$. In set theory, the result of intersecting A with B is a smaller set, but in logic, "and" is a strengthening operation.
Why do the extensional results of union and intersection go in reverse from their logical definition?
Well, these terms can be often misleading if you strip off their original meaning. Note that they should rather give you an intuition on what happens there, in terms of a limited human language. So let's look how you should actually interpret these words:
"and" is a strengthening operation in a sense that the resulting sentence is stronger, meaning that it narrows down the possibilities how the things may go. Similarly, the set intersection narrows down the set of elements which belong to the resulting set.
"or" is a softening operation in a sense that the resulting sentence is softer, meaning that it extends the possibilities of how the things may go. Similarly, the set unification may extend the set.
Compare being able to lift a pencil versus being able to lift a refrigerator. Clearly, being able to lift a refrigerator is a stronger (yes, pun intended :P ) property. And , it being a stronger, fewer people will be able to do that.
In general, fewer things satisfy a stronger criterion.
In set theory, $A \cup B$ is logically defined as $\{ x : x \in A \lor x \in B \}$ : True
In set theory, the result of unionizing A with B is a bigger set : True
but in logic, "or" is a softening operation : What do you mean by that ?
In set theory, $A \cap B$ is logically defined as $\{ x : x \in A \land x \in B \}$ : True
In set theory, the result of intersecting A with B is a smaller set : True
but in logic, "and" is a strengthening operation : What do you mean by that ?
Why do the extensional results of union and intersection go in reverse from their logical definition? : Answer depends on what you mean.
In general , when we make the truth table , we can see these facts :
$A$ is true $50\%$ of the Cases , $B$ is true $50\%$ of the Cases.
$A \lor B$ is true $75\%$ of the Cases : the result of ORing has larger truth Case : Similar to $\cup$.
$A \land B$ is true $25\%$ of the Cases : the result of ANDing has smaller truth Case : Similar to $\cap$.
Conclusion : Nothing Weird or Strange or Paradoxical here !
We might superimpose Set Operators & Logic Operations in a Diagram , like this :
We get larger result $50\%$ versus $75\%$ via Set Union & logical OR
We get smaller result $50\%$ versus $25\%$ via Set Intersection & logical AND
Conclusion : Nothing Weird or Strange or Paradoxical here !
There is some likeliness in the idea that being able to speak of large sets is cognitively desirable. Are not scientific statements supposed to be universal? This is, I think, the premiss on which rests your question , namely " large extension " $\iff$ " cognitive richness" $ \iff $ " logical strength".
But on the other side, statements involving large sets can be almost empty : consider Leibniz's statement " every being is one, every being is a unity". In French ( Leibniz used to write in this language) : " ce qui n'est pas un être n'est pas véritablement un être ".
So the question is : how to reconcile the desire of universality ( large extension) and the desire of cognitive richness .
As an hypothesis, I would say : a universal statement is cognitively rich if it expresses the inclusion of a large set in an ( apparently) small set ; or, put another way, the inclusion of an extensionnaly large set in an intensionnaly rich set.
Consider this example :
"Being a positive integer" is not a demanding condition, which makes the set it determines an extensionnaly large set.
"Being a number $n$ such that $n^3+3n -1$ is odd " is , to the opposite, a rather demanding condition. Because of the richness of the intension, one anticipates ( erroneously) a " small" set.
Hence , possibly, the cognitive interest attached to the sentence : " the set of positive integers is a subset of the set of integers such that $n^3+3n -1$ is odd".
Same observation for the statement : " the set of integers is a subset of the set of integers that can be written either as $3k+0$ or $3k+1$ or $3k+2$ ( with $k \in \mathbb Z$)".
As an aside, there is a concept that links the idea of strength of a statement to the idea of extension of an associated set, namely the concept of " truth set". For example the truth set of $A\rightarrow B$ is the set $\{ (T,T), (F,T) , (F,F)\}$. The larger the truth set, the weaker or the sentence ( following Carnap's Introduction to symbolic logic) . One can express logical implication in the follwing way : X logically implies Y iff the truth set of X is included in the truth set of Y.
