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Can anyone clearly explain the S5 modal logic

truthseeker613
Posts: 464
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11/13/2012 1:09:13 PM
Posted: 4 years ago
Can anyone clearly explain the S5 modal logic?

I've tried to get it a # of times but it just doesn't go.
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tBoonePickens
Posts: 3,266
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11/13/2012 4:46:52 PM
Posted: 4 years ago
At 11/13/2012 1:09:13 PM, truthseeker613 wrote:
Can anyone clearly explain the S5 modal logic?

I've tried to get it a # of times but it just doesn't go.

http://en.wikipedia.org...(modal_logic)

http://home.utah.edu...

http://plato.stanford.edu...
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: At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote:
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drafterman
Posts: 18,870
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11/13/2012 7:00:27 PM
Posted: 4 years ago
Since tBoone has already taken the RTFM/TMGTFY approach, I'll take the higher road (the low road is, apparently, single lane, who knew?)

1. Modal logic. Modal logic deals, primarily with issues of possibility and necessity, often interpreted via the concept of possible worlds. Truths can be actual (true in this world, the one we are living in); possible (true in some possible world); and necessary (true in all possible worlds).

An example of an actual truth is: Obama is the president.
An example of a possible truth is: Mitt is the president.
Examples of necessary truths are troublesome, but we can play it safe and use a tautology: bachelors are unmarried.

Distinguishing possibility and necessity is actually redundant, since they can each be expressed in terms of each other.

"possibly x" = "not necessarily not x"
"necessarily x" = "not possibly not x"

English translation:
If something is possible (possibly x) then it can't be necessary for it to be false (not necessarily not x)
If something is necessary (necessarily x) then it can't be possible for it to be false (not possibly not x).

2. S5 modal logic is merely a type of modal logic that relies on a specific set of axioms. And those axioms are:

Axiom K: [](A -> B) -> ([]A -> []B)
If if is necessary that, when A is true, B is true, then, if A is necessary, so is B.

Basically, if A always leads to B (in all possible worlds), and A is true in all possible worlds, then B is true in all possible worlds.

Axiom T: []A -> A
If A is necessarily true, then A is actually true.

Since necessity means it is true in all possible worlds, and since the actual world is a possible world, then it is true in the actual world.

Axiom 5: <>A -> []<>A
If A is possible, then it is necessary that it is possible.

This one may not be entirely intuitive, as it is meta. Basically:

<>A (possibly A) means that, in some world, "A" is true. However, in the actual world "<>A" is true. That is, it is true, in this world, to say that A is true in some possible world (which may be another world that isn't this one. It can be true to say "<>A" while at the same time A is false in the actual world.

Ok, so <>A is true in this world. But what about another world? Can there be another world where <>A is false? This would be:

Not-<>A

But, given the equivalence between possibility and necessity above, we can rewrite this as:

[]Not-A

That is, necessarily A is false. Or, A is false in all possible worlds. But if A were false in all possible worlds, then we couldn't truthfully say, in this world, that A is true in some possible world. That is, it would make <>A false in this world, which is a contradiction because we started out with it being true.

Combined, these axioms form the S5 system (there are other ways of doing it, with other axioms), and ultimately allows for the reduction of redundant modifiers ([]<>[]A, for example).