Logical possibility

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1 + 1 = 3 and 0 = 1 are false in all possible worlds.

Philosophers generally consider logical possibility to be the broadest sort of subjunctive possibility in modal logic. The notion can be glossed by the popular description of logically possible propositions as those which can be asserted without any logical contradiction. Another way of saying this is that a proposition is logically possible if and only if there is some coherent way for the world to be, under which the proposition would be true. Thus, "the sky is blue" (and all other actually true propositions) is logically possible: there is a logically coherent way for the world to be such that it is true, viz., the way that the world actually is. But this "way for the world to be" need not be the way the world actually is; it need only be logically coherent. So, for example, the false proposition the sky is green is also logically possible, so long as we are able (as we indeed seem to be) to conceive of some logically coherent world in which the sky is green.

These propositions, then, are to be contrasted with logically impossible propositions, i.e., propositions which could not possibly be true because they are formal contradictions. While it is logically possible for the sky to be green, it is not logically possible for the sky to be both green and not green at the same time and in the same respect. To conceive of the sky as green is to exclude it being non-green, and to conceive of it as non-green is to exclude it being green.

Logical possibility should be distinguished from other sorts of subjunctive possibilities. For example, it may be logically possible for the laws of nature to be different from what they actually are. The debate over whether it really is logically possible is beyond the scope of this article; the important thing to note here is that many philosophers have taken it for granted that it is logically possible; and if it is, then many things that we would normally consider to be demonstrably impossible will still be included amongst logical possibilities: for example, that someone might fly by flapping their arms, or that they also might be able throw a baseball faster than the speed of light. Many philosophers, then, have held that these scenarios are logically possible but nomologically impossible, i.e., impossible under the actual laws of nature. Similarly, it is a perfectly genuine logical possibility that a moral leader or other person might go on a senseless killing spree because he/she woke up on the wrong side of the bed one morning; but this is no accusation against their character, because while it is logically possible (there is no contradiction involved in supposing that it is true), it is not characterologically possible--there is no way that it could happen unless he/she ceases to be the sort of person that he/she actually is.

With this understanding of logical possibility in mind, the other logical modalities may be defined in terms of it: a proposition is logically necessary if it is not logically possible for it to be false, logically impossible if it is not logically possible for it to be true, and logically contingent if it is logically possible for it to be true and also logically possible for it to be false.