Since the boolean interpretatrion doesn't acknowledge any relationship between categorical propositions but contradiction...if the two statements are related in some other way acknowledged on the square of opposition, such as through the relationship of contrarity, are we able to tell anything about the truth value of one under the boolean interpretation, given the truth value of the other? Is it always undetermined?
I am not sure I understand the question.
However, although you can not say one way or another about a converse or inverse, you certainly can say that the contrapositive has value "true"
eg:
Given:
If a cow, then a mammal
then
If not a mammal, then not a cow
I don't know if this will be any clearer, but...
I just want to know if you have two categorical propositions that are related in some way on the square of opposition (but who's relationship is not recognized under the Boolean Interpretation), and you know the truth value of one of the statements (either true or false), can you tell anything about the truth value of the other statement under the Boolean Interpretation...
To understand whether the truth value of one statement can be determined based on the truth value of another statement, we need to consider the relationship between these statements and the principles of Boolean interpretation.
In Boolean interpretation, each statement is assigned a truth value of either True or False. The interpretation does not consider any relationship between categorical propositions other than contradiction. However, if the relationship between the statements falls under another relationship, such as contrarity, we can still make certain determinations about their truth values.
Contrarity, as defined in the square of opposition, means that two statements cannot both be true but can both be false. If two statements are related in this way, and if we know that one statement is true, we can conclude that the other statement must be false. This is because of the principle that there cannot be two contradictory true statements in contrarity.
Therefore, in the case of contrarity, if you have the truth value of one statement, you can determine the truth value of the other statement. It is not always undetermined. If one statement is True, then the other statement must be False.
However, it's important to note that this reasoning only applies to the relationship of contrarity. For other relationships, such as subcontrariety or subalternation, the Boolean interpretation does not offer a direct method to determine one truth value from another.