Thursday, February 23, 2012

Logic: Modal Logic

I. Modal Logic

Deals with propositions that are "possible", "impossible", "necessary", and "contingent." Possibility in this case means logically possible, and impossible means logically impossible. If a proposition is necessary, then it is logically impossible for it to be false. If a proposition is contingent, then it could logically be either true or false. 

  1. It is (logically) possible that people can fly like Superman (but still physically impossible)
  2. It is (logically) impossible that bachelors are married (because a "married unmarried man" is a logical contradiction)
  3. It is necessary that 2+2=4 (it couldn't have been different)
  4. It is contingent that the Earth circles the sun (it could have been different)
II. Modal Operators 

The □ symbol means "necessary", and the ⋄ symbol means "possible." This can then be combined with the operators and inference rules from propositional logic.

III. Possible Worlds 

To speak of modal propositions, the concept of possible worlds is used. Think of possible worlds as computer simulated versions of ways reality could be, one of which matches up with reality. So a proposition that is "possible" will be true in at least one of those simulations, even if not in the actual world. A proposition that is "impossible" will not be true in any of the simulations. For example, none of the simulations will have a square circle in them, since such a thing is logically impossible. And a proposition that is "necessary" (i.e., cannot be false) is true in all of the simulations, including the real world. For example, in all of the simulations, 2 + 2 = 4.

  1. □P ("it is necessary that P") = "P is true in all possible worlds"
  2.  ⋄P ("it is possible that P") = "P is true in at least one possible world"
  3. ~⋄P ("it is not possible that P") = "P is not true in any possible world"
  4. ⋄~P ("possibly, not P") = "P is false in at least one possible world"
  5. P ("P is true") = "P is true in the actual world"
IV. Modal Inference Rules

A few new inference rules are added. Keep in mind that these add to the inference rules from propositional logic.
  1. Reverse negation: If a modal operator starts with a negation ("~"), then put the negation in front of the proposition and swap the modal operator. ~⋄P ("it is not possible that P") is equivalent to □~P ("necessarily, not P")
  2. Reverse negation: ~□P ("it is not necessary that P") is equivalent to ~P ("it is possible that not P")
  3. P - drop the diamond, and conclude"therefore, P is true in W1" (P is true in new possible world #1)
  4. □P - drop the box, and conclude "therefore, P is true in Wx" (P is true in any and all possible worlds)

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