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If we're talking about metaphysical possibility, then normally yes. If you reject the claim that "if P then possibly P", you must also reject the claim that "if necessarily P then P". Proof: suppose we reject truth implies possibility (that is, we reject that for every formula P, if P then possibly P). Then for some formula A, we have A and not-possibly A. Not-possibly A is equivalent to necessarily-not-A. So we have A and necessarily-not-A, meaning the necessity of not-A doesn't imply the actual truth of not-A.

However formally within modal logic itself, you can mess around with axioms and frame conditions in whatever way you want. Rejecting if"if P then possibly PP" amounts to rejecting reflexivity as a frame condition. See https://en.m.wikipedia.org/wiki/Accessibility_relation for more about frame conditions and their corresponding axioms. (EDIT: Frame conditions tell us what worlds we "see" when evaluating possibly P and necessarily P at a world w. If at least one world that w "sees" satisfies P, then w satisfies possibly P. If every world w "sees" satisfies P, then w satisfies necessarily P. Reflexivity tells us that w always "sees" itself when evaluating statements of possibility and necessity. It may be that P is true in the actual world, but if we reject reflexivity then we're not looking at the actual world to determine the truth of possibly P! And maybe every other world we "see" indeed fails to satisfy P.)

(Noah Schweber's comments below should be heeded as well. The box and diamond operators can be interpreted in different ways for different modalities!)

If we're talking about metaphysical possibility, then normally yes. If you reject the claim that "if P then possibly P", you must also reject the claim that "if necessarily P then P". Proof: suppose we reject truth implies possibility (that is, for every formula P, if P then possibly P). Then for some formula A, we have A and not-possibly A. Not-possibly A is equivalent to necessarily-not-A. So we have A and necessarily-not-A, meaning the necessity of not-A doesn't imply the actual truth of not-A.

However formally within modal logic itself, you can mess around with axioms and frame conditions in whatever way you want. Rejecting if P then possibly P amounts to rejecting reflexivity as a frame condition. See https://en.m.wikipedia.org/wiki/Accessibility_relation for more about frame conditions and their corresponding axioms.

If we're talking about metaphysical possibility, then normally yes. If you reject the claim that "if P then possibly P", you must also reject the claim that "if necessarily P then P". Proof: suppose we reject truth implies possibility (that is, we reject that for every formula P, if P then possibly P). Then for some formula A, we have A and not-possibly A. Not-possibly A is equivalent to necessarily-not-A. So we have A and necessarily-not-A, meaning the necessity of not-A doesn't imply the actual truth of not-A.

However formally within modal logic itself, you can mess around with axioms and frame conditions in whatever way you want. Rejecting "if P then possibly P" amounts to rejecting reflexivity as a frame condition. See https://en.m.wikipedia.org/wiki/Accessibility_relation for more about frame conditions and their corresponding axioms. (EDIT: Frame conditions tell us what worlds we "see" when evaluating possibly P and necessarily P at a world w. If at least one world that w "sees" satisfies P, then w satisfies possibly P. If every world w "sees" satisfies P, then w satisfies necessarily P. Reflexivity tells us that w always "sees" itself when evaluating statements of possibility and necessity. It may be that P is true in the actual world, but if we reject reflexivity then we're not looking at the actual world to determine the truth of possibly P! And maybe every other world we "see" indeed fails to satisfy P.)

(Noah Schweber's comments below should be heeded as well. The box and diamond operators can be interpreted in different ways for different modalities!)

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If we're talking about metaphysical possibility, then normally yes. If you reject the claim that "if P then possibly P", you must also reject the claim that "if necessarily P then P". Proof: suppose we reject truth implies possibility (that is, for every formula P, if P then possibly P). Then for some formula A, we have A and not-possibly A. Not-possibly A is equivalent to necessarily-not-A. So we have A and necessarily-not-A, meaning the necessity of not-A doesn't imply the actual truth of not-A.

However formally within modal logic itself, you can mess around with axioms and frame conditions in whatever way you want. Rejecting if P then possibly P amounts to rejecting reflexivity as a frame condition. See https://en.m.wikipedia.org/wiki/Accessibility_relation for more about frame conditions and their corresponding axioms.