I'm having trouble understanding some of the absorption and associative laws in the context of English sentences.
Let
$S =$ I went to the store,
$R =$ I went on a run,
$P =$ I went to the park.
Associative Law
$S \vee (P \wedge R) =$ Either I went to the store, or I went to the park and on a run.
$(S \vee P) \wedge (S \vee R) =$ I went to the store or the park, and I went to the store or on a run.
I am having trouble equating these sentences in my mind because the second sentence is awkward unlike the first one. Is there a better way of wording the second sentence so it makes more sense?
Absorption Law
$P \wedge (P \vee R)$ = I went to the park, and I went to the park or on a run.
This sentence should be equivalent to "I went to the park". If I said "I went to the park, and I went to the park or on a run" to someone in real life, the only definite information they could receive from that sentence is that I went to the park. Is this the reason why we can drop the statement that I went on a run?
First, that first one is not the Associative Law, but the Distributive Law.
The Associative Law is that:
$$P \lor (Q \lor R) \Leftrightarrow (P \lor Q) \lor R$$
$$P \land (Q \land R) \Leftrightarrow (P \land Q) \land R$$
but your $S \lor (P \land R) \Leftrightarrow (S \lor P) \land (S \lor R)$ is an instance of the Distributive Law:
$$P \land (Q \lor R) \Leftrightarrow (P \land Q) \lor (P \land R)$$
$$P \lor (Q \land R) \Leftrightarrow (P \lor Q) \land (P \lor R)$$
But yes, the $(S \lor P) \land (S \lor R)$ is an awkward sentence to think about ... I have no suggestions to make it any easier to read.
Finally, saying that $P \land (P \lor R)$ is equivalent to $P$ because $P$ is the only definite piece of information isn't really a good argument. I mean, if i just said $P \lor R$, then you don't get any definite piece of information at all, but we wouldn't say that that statement is equivalent to nothng at all. Or: the only definnite piece of information in $P \land (Q \lor R)$ is $P$ ... but this statement is not equivalent to just $P$. So no, your explanation doesn't really work.
What we can say to explain that $P\land (P \lor R)$ is equivalent to $P$, is that once you say you know that $P$ is true, then the statement $P \lor R$ is automaticallt true as well, and thus it doesn't add any information to the $P$ already stated. Thus, $P \land (P \lor R)$ is really just saying $P$. Notice that this is not true for $P \land (Q \lor R)$: since $Q \lor R$ is not implied by $P$, you really are adding moreover e informatin by adding the $Q \lor R$ (even though neither $Q$ nor $R$ is definite by themselves), and therefore you can't just 'throw it out' or say that it gets 'absorbed' by the $P$ the way $P \lor R$ is absorbed by the already present $P$.