(February 2, 2022 at 10:00 am)polymath257 Wrote:(February 2, 2022 at 9:23 am)emjay Wrote: How is one to even get started learning about modal logic when according to the wiki, it seems a veritable pick 'n' mix of different types/uses? And what practical use does it have beyond these sorts of questions, or was it developed strictly for these types of questions? This goes out to anyone.
Yeah, modal logic can be a mess.
The basic problem is how to deal with counter-factual claims. So, for example, "If Caesar did not cross the Rubicon, the Roman Republic would have lasted for another century".
In classical logic, this statement is *true* since Caesar *did* cross the Rubicon and a false implies anything.
But the statement "If Caesar did not cross the Rubicon, then the Roman Republic would have still fallen within the year" would *also* be true for the same reason.
The upshot is that any counterfactual implication is always true.
So the idea is that there are 'possible worlds' that are close to ours, but different. We can then ask in which of those possible worlds Caesar did not cross the Rubicon and ask what then happened.
Now, a bit more formal treatment defines a possible world as a 'maximal logically consistent system'. So, it is *logically* consistent to have a world in which Caesar did not cross the Rubicon, even though in the 'real world' he did. Notice that this is very different than the notion of a multiverse, for example. it is logically possible to have worlds where the laws of physics are very different.
This also then can deal with the notions of 'necessity' and 'possibility'. The definition of 'necessity' in this framework is 'happens in all possible worlds' and 'possibility' means 'happens in some possible world'.
The difficulties happen when you want to say things like 'it is possible that it is necessary that...' or some other situation where there is more than one use of 'possible' or 'necessary'. This inevitably leads to questions about relations between possible worlds. But different possible worlds are mutually contradictory (since they are maximal consistent systems).
Thanks for the run down... I'll definitely give it some thought. The ever-amusing 'British States of America' from the TV series Sliders springs immediately to mind here... ie what would/could have happened in a parallel world if we the British had won the American War of Independence
