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There are several different concepts of possibility including:
Each of which has a corresponding impossibility. There are things that can be physically impossible but logically possible but there are no things that can be logically impossible but physically possible.
Logic is based on the presumption that there are some very basic, self-evident laws of thought that ought to guide our thinking. These laws help us understand what counts as a good argument, what counts as good evidence, and what inferences should be accepted and rejected. In a sense, everything we do in logic follows from these laws.
The laws themselves, which I will discuss below, are self-evident descriptions of how the world works. To say they are "self-evident" means that we don't need any other evidence to know they are truly beyond our understanding of what they mean. This raises an interesting point. If you don't believe these are self-evident, that is, if you don't believe the claims that these rules are making are true, then there's very little that can be done to persuade you otherwise. In some sense, the conversation won't be able to proceed without your agreement with these rules.
There are three laws of thought we should consider:
The law of non-contradiction: Something cannot both be and not be at the same time and in the same respect.
The law of identity: Everything is identical with itself.
The law of excluded middle: For any given property or attribute, everything either has that property or does not have that property.
You may be objecting that there are many exceptions to these rules but they are probably not exceptions at all just misunderstandings of what the rules imply.
For example, you might say that something can be and not be. It could be raining one minute and not raining the next. But, this is not an exception at all. The rules do not say that things cannot change or that one thing (or state of affairs) could not become a different one. It is stating that one thing cannot be that something and not be that something at the same time and respect. It's this last qualifier that is critical. A person can be both a parent and daughter without violating the rule since the person's state as a parent does not preclude there also being a daughter. What they cannot be is both a parent and not a parent at the same time and in the same respect.
Once you understand this it should be clear that the rule is asserting nothing more than how things exist. To the extent that things seem to violate this rule, it is most likely that the violation is based on our lack of understanding regarding how the world works, not the inherent falsity of the law of thought.
If you consider what the other two laws are saying it should become clear that they are really just slightly different descriptions of the law of non-contradiction. If something cannot both be and not be at the same time that is the same as saying that something is identical to itself. This can be proven in formal logic. The same goes for the law of excluded middle. This can be shown with a simple example:
If it cannot be both raining and not raining in the place and time then, for any given time and place, it must either be raining or not raining.
What these laws of thought provide is a foundation on which we can build other more interesting and useful rules of reason which provide us with a framework for evaluating claims and theories about how the world works. This is a central feature in the work of logic.
So, when someone claims that something is logically impossible, there are no conditions under which that claim can be possible since its possibility would be a contradiction.