No, it may not be open and if so, it may not have gas, or I may not have any money, whatever. Furthermore, logic tells me I cannot necessarily infer the negation, nor the converse. This is the value of logical analysis here. It tells us what are invalid inferences, and so should keep us from drawing illogical conclusions. It does not say the negation or converse are necessarily untrue, just that their veracity cannot be inferred from the given propositions.
Scripture seems to adhere tenaciously to the rules of logic.
Only if the negations are not necessarily implied by the conditionals. Besides, there are many examples in scripture where logic was abused with dire consequences. Consider, for example, the plight of Job. Or, consider the negative inference fallacies committed by the Pharisees in Mt 5. They knew that if a man committed murder, then he would be liable before the courts.