Mathematicians already explored exactly what you describe: this is the difference between classical logic and intuitionistic logic:
In classical logic statements can be true in and of themselves even if there as no proof of it, but in intuitionistic logic statements are true only if there is a proof of it: the proof is the cause for the statement to be true.
In intuitionistic logic, things are not as simple as "either there is a cow in the field, or there is none" because as you said, for the knowledge of "a cow is in the field" to be true, you need a proof of it. It brings lots of nuance, for example "there isn't no cow in the field" is a weaker knowledge than "there is a cow in the field".
In classical logic statements can be true in and of themselves even if there as no proof of it, but in intuitionistic logic statements are true only if there is a proof of it: the proof is the cause for the statement to be true.
In intuitionistic logic, things are not as simple as "either there is a cow in the field, or there is none" because as you said, for the knowledge of "a cow is in the field" to be true, you need a proof of it. It brings lots of nuance, for example "there isn't no cow in the field" is a weaker knowledge than "there is a cow in the field".