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Modal Logic. M, Brouwer , S-4, and S-5 were attempts to formalize modal notions This is necessary because modal operators appear to make extensional language intensional. Each system makes different statements about which sorts of modal inferences are legitimate
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M, Brouwer, S-4, and S-5 were attempts to formalize modal notions • This is necessary because modal operators appear to make extensional language intensional
Each system makes different statements about which sorts of modal inferences are legitimate • The existence of non-equivalent systems is problematic • Our intuitions don’t help us determine which one is correct • But Kripke does
Truth Function and Validity in M M takes as its axioms all the truth-functional tautologies and the following two modal formulas: 1. □ (A ⊃ B) ⊃ (□ A ⊃ □ B) 2. □ A ⊃ A
Semantics for M The M-model structure is G, K, R • K is the set of all possible worlds • G is the actual world • R is relative possibility/accessibility
Accessibility/Relative Possibility • Informally -- A world is accessible to ours in some way if at least some of the things that can or must happen here can or must happen there • Formally, a world W is possible relative to/accessible to a world W’ just in case every situation that IN FACT obtains in W is POSSIBLE in W’
Truth Functions in M • R is defined over K from the outset; we must suppose that it is antecedently fixed just which worlds are possible relative to which. • To get an M-model structure, the relation of accessibility has to be reflexive. That is, it has to be the case that every world is accessible/possible relative to itself.
To determine truth value in M we start with the truth of atomic sentences at specified worlds We build more complex sentences out of the atomic sentences
Each of the following must be T/F 1. ~A is true in W if and only if A is false in W. 2. (AvB) is true in W if and only if either A is true in W or B is true in W. 3. ◊A is true in W if and only if there is at least one possible world, W’, such that W’ is accessible to W and A is true in W’. 4. □ A is true in W if and only if for every world, W’, such that W’ is accessible to W, A is true in W’
Remember this example If Obama got more electoral college votes than McCain, then Obama is proclaimed to be the 44th President of the United States. Possibly Obama got more electoral college votes than McCainObama is proclaimed to be the 44th President of the United States
A new example • Our World, W • W1 is accessible to our world nomologically but not historically – in W1 the British won the American Revolutionary War, but all the physical laws of the universe are the same – • W2 that is historically accessible but not nomologically accessible to our world W.
1. If ~A is read as “no America” then this atomic sentence of M is false in our world W, true in W1, and false in W2.
2. If (AvB) is read as “America won the revolution or Britain won the revolution” then it is true in all three worlds W, W1, and W2.
3. If ◊A is read as “possibly America could have lost the Revolutionary War” then we can say this is true in W, our world, because we’ve specified a nomologically accessible world, W1, in which Britain won the Revolutionary War, and this jibes with our intuitions about what “possibly” means and about what is indeed possible.
we can postulate some world Wn that is nomologically accessible to W2 but historically accessible to W1 such that in Wn the America that’s made out of N-Rays lost the Revolutionary war, and this allows us to say that “possibly America lost the Revolutionary War” is true in W2 as well.
4. If □ A is read as “necessarily America won the Revolutionary war” then this is false according the examples given of W, W1, W2, and Wn above.
Truth Value in Longer M-Statements The statement “Possibly Obama got more electoral college votes than McCain” is true in a world W1 that is nomologically accessible to our world W but where McCain received more votes. But is it true in our world?
◊A does not imply A (that’s the whole problem!). • We might be tempted to say it’s true in our world W because ~A is false here in W, and because ◊A ≡ ◊~A. • But this doesn’t solve the problem. Surely it’s not the case that A ≡~◊~A, because that would mean everything that is the case is necessarily the case
Validity in M • reflexivity is going to be the essential condition for judging a formula/sentence to be M-valid
4. □ A is true in W if and only if for every world, W’, such that W’ is accessible to W, A is true in W’ So apparently □ A can be true in W even if A isn’t …but, but, but, : □ A ⊃ A is an axiom of M, so…
So □ A ⊃ A only comes out false in one case: where □A is true in W, but A itself is false in W • this can only happen if W is not possible relative to itself.
□A is M-valid if it is true not only in all worlds accesible to an arbitrary world W, but also in W itself. • To ensure validity for □ (A ⊃ B) ⊃ (□ A ⊃ □ B) and for all formulas derivable from it, we need merely make the relation of accessibility reflexive.
