Just my $0.02. To me, having chosen the words "we EXPLORE X", it would be better to follow it with "via," "through," or "by WAY OF", rather than the single "by." I guess that's because "explore" suggests travel over a road or path. Since a dictionary is "by" and "via" as synonyms, this would just be a question of style and word choice, not grammar.

I would simply add the word "using": "We explore semantic and syntactic valency (by) using the example of English verbs with incorporated actants." [The word "by" is optional.] You could use the verb "study," but the verb "explore" is perfectly correct. The verb "explore" adds a bit of nuance that the "study" will be deep and detailed. In case anyone doesn't understand why Dan wrote "Just my $0.02," there is an idiom "my two cents", which means "my opinion".

I don't fully understand what this is about, but I'm not convinced by the phrase 'incorporated actants'. It doesn't mean a whole lot to me. If by 'actants' you mean subject and object cases, perhaps 'associated actants' might be better? 'Explore' is fine, and I'd agree with Dan that maybe 'through' would be preferable to 'by'.

Agree with Dan, it is just a matter of style and choice, not the question of grammar. My sentence: We explore semantic and syntactic valency through the example/s of English verbs with incorporated actants. Just an opinion as I am not qualified to correct your grammar.