fatalism.
Suppose it’s true that excluded middle (EM) holds—viz., that any proposition p, it’s necessary that p or not-p. Symbolically:
(EM) (p ∨ ~p)
Now suppose that bivalence holds—viz., that for any proposition p, it’s necessary that either p is true or p is false. Symbolically:
(B) (Tp ∨ Fp)
Let’s also suppose EM and B are necessarily equivalent. Symbolically:
`
(EM/B) [(p ∨~p) ^ (Tp ∨ Fp) ^ (p ↔Tp) ^ (~p ↔Fp)]
Okay now for fatalism. Fatalism (as I understand it) is the thesis that the future is fixed—viz., that anything that might happen in the future, what will happen must happen such that there is only one possible future.
Some have argued that from EM/B fatalism follows. Here’s the argument:
Consider the proposition
(1) There will be sea battle tomorrow.
By EM/B it follows that (1) is either true or false right now. Furthermore, if (1) is true right now, then no one can prevent tomorrow’s sea battle, for it impossible for anyone to make a true time-indexed proposition false. And vice-versa. If (1) is false right now, then no one can prevent the non-occurrence of tomorrow’s sea battle, for it is impossible for anyone to make a false time-indexed proposition true. Since (1) was arbitrarily chosen, this argument can be generalized for any future-tensed proposition. That is, let pf pick out any future-tensed proposition. It follows from EM/B that pf is either true or false right now. Furthermore, if pf is true right now, then no one can prevent it, for it is impossible for anyone to make a true time-indexed proposition false. And the same goes if pf is false. If pf is false right now, then no one can prevent it, for it is impossible for anyone to make a false time-indexed proposition true. And this is just the thesis of fatalism, which said that the future is fixed in such a way that there is only one possible way the future can be.
(LE) (Tp ⊃ Tp) ^ (Fp ⊃ Fp)
(LE) is equivalent to
(LE’) (Tp ⊃ ~◊Fp) ^ (Ftp ⊃ ~◊Tp)
Here’s a way of symbolizing the argument for fatalism from EM/B and LE:
(EM/B) [(p ∨~p) ^ (Tp ∨ Fp) ^ (p ↔Tp) ^ (~p ↔Fp)]
(2) Any future-tensed proposition pf is a proposition.
(3) ∴ [(pf∨~pf) ^ (Tpf∨ Fpf) ^ (Tpf ↔ pf) ^ (Fpf ↔ ~pf)]
(LE’) (Tp ⊃ ~◊Fp) ^ (Ftp ⊃ ~◊Tp)
(4) (Tpf ⊃ ~◊Fpf) ∨ (Fpf ⊃ ~◊Tpf)
(5) ∴ ~◊Fpf ∨ ~◊Tpf
(EM) (p ∨ ~p)
Now suppose that bivalence holds—viz., that for any proposition p, it’s necessary that either p is true or p is false. Symbolically:
(B) (Tp ∨ Fp)
Let’s also suppose EM and B are necessarily equivalent. Symbolically:
`
(EM/B) [(p ∨~p) ^ (Tp ∨ Fp) ^ (p ↔Tp) ^ (~p ↔Fp)]
Okay now for fatalism. Fatalism (as I understand it) is the thesis that the future is fixed—viz., that anything that might happen in the future, what will happen must happen such that there is only one possible future.
Some have argued that from EM/B fatalism follows. Here’s the argument:
Consider the proposition
(1) There will be sea battle tomorrow.
By EM/B it follows that (1) is either true or false right now. Furthermore, if (1) is true right now, then no one can prevent tomorrow’s sea battle, for it impossible for anyone to make a true time-indexed proposition false. And vice-versa. If (1) is false right now, then no one can prevent the non-occurrence of tomorrow’s sea battle, for it is impossible for anyone to make a false time-indexed proposition true. Since (1) was arbitrarily chosen, this argument can be generalized for any future-tensed proposition. That is, let pf pick out any future-tensed proposition. It follows from EM/B that pf is either true or false right now. Furthermore, if pf is true right now, then no one can prevent it, for it is impossible for anyone to make a true time-indexed proposition false. And the same goes if pf is false. If pf is false right now, then no one can prevent it, for it is impossible for anyone to make a false time-indexed proposition true. And this is just the thesis of fatalism, which said that the future is fixed in such a way that there is only one possible way the future can be.
(LE) (Tp ⊃ Tp) ^ (Fp ⊃ Fp)
(LE) is equivalent to
(LE’) (Tp ⊃ ~◊Fp) ^ (Ftp ⊃ ~◊Tp)
Here’s a way of symbolizing the argument for fatalism from EM/B and LE:
(EM/B) [(p ∨~p) ^ (Tp ∨ Fp) ^ (p ↔Tp) ^ (~p ↔Fp)]
(2) Any future-tensed proposition pf is a proposition.
(3) ∴ [(pf∨~pf) ^ (Tpf∨ Fpf) ^ (Tpf ↔ pf) ^ (Fpf ↔ ~pf)]
(LE’) (Tp ⊃ ~◊Fp) ^ (Ftp ⊃ ~◊Tp)
(4) (Tpf ⊃ ~◊Fpf) ∨ (Fpf ⊃ ~◊Tpf)
(5) ∴ ~◊Fpf ∨ ~◊Tpf
Labels: it's written in the stars.
posted by Derek at 4:06 PM
1 Comments:
I'm guessing you, like me, would deny the conditional "if is true right now, then no one can prevent tomorrow’s sea battle" for lack of respecting modes of "can"?
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