This was bothering me, but I think I found the logical flaw:
Quote:
NOT (route FAST is taken)
And then from this it deduced
IF route FAST is taken THEN I will arrive at 3pm
This, I’m afraid dear reader, is also permitted by the laws of logic.
The statements “not P” and “not P OR Q” might have the same truth value, but they are not logically equivalent. At this point in the proof, saying “THEN I will arrive at 3pm” is arbitrary, and could have been “THEN pigs will fly.” I think that’s what’s known as the principle of explosion, but I could be wrong about that term.
The statements “not P” and “not P OR Q” might have the same truth value, but they are not logically equivalent. At this point in the proof, saying “THEN I will arrive at 3pm” is arbitrary, and could have been “THEN pigs will fly.” I think that’s what’s known as the principle of explosion, but I could be wrong about that term.
OK I agree this is very much the crux of the issue. However:
First, yes, I agree, it could have been anything, including “THEN pigs will fly”.
Second, the way that “IF .. THEN” is defined in propositional or first order seems not to capture quite what we mean by those words in ordinary language. I think this is part of what you are pointing out.
But taking the rules of first order logic as a given, it really is valid to derive from “P” to “P or Q”, because in all the worlds where “P” is true, “P or Q” is also true, which is what it means for it to be valid to derive from one thing to another. And for the same reason it really is valid to derive from “not P” to “if P then Q”, for an arbitrary Q.
In some previous discussions of the 5-and-10 problem, some people have concluded that what needs to be done is to rework the definition of “IF … THEN” in first order logic. I think it is quite illuminating to attempt something like this, but I don’t think this ultimately resolves the issue.
I haven’t yet seen any completely satisfying resolution of this problem, but I do think that it’s a productive problem to work at, and also I find it fun, so thank you for doing it with me :)
Second, the way that “IF .. THEN” is defined in propositional or first order seems not to capture quite what we mean by those words in ordinary language. I think this is part of what you are pointing out.
I feel like the confusion between propositional logic and ordinary language is the only reason Lob’s theorem is even being discussed in the first place. The car’s programmers used IF X THEN Y to represent the statement “If X, then Y happens”, which means something quite different. Other than the incidental similarity of these statements in the English language, why is this more relevant than any other programming error?
3. Make a list of all possible logical sentences of the form IF route 1 is taken THEN I will arrive at such-and-such a time AND IF route 2 is taken THEN I will arrive at such-and-such a time AND ...
Because the algorithm was created without including the additional assumption (used later on in the “proof”) that if route 1 route is taken, then route 2 would NOT be taken (and vice versa). If you include only that additional piece of information, then the statements generated in step 3 are “logically” equivalent to:
“IF route 1 is not taken AND IF route 2 is taken THEN I will arrive at Time” (or “IF route 1 is taken I will arrive at Time AND route 2 is not taken”).
And that (again from our route 1 XOR route 2 assumption) is equivalent to a list of :
IF route # is taken, THEN I will arrive at time
for all possible combinations of route and time, with no conjunctions at all.
This was bothering me, but I think I found the logical flaw: Quote: NOT (route FAST is taken)
And then from this it deduced
IF route FAST is taken THEN I will arrive at 3pm This, I’m afraid dear reader, is also permitted by the laws of logic.
The statements “not P” and “not P OR Q” might have the same truth value, but they are not logically equivalent. At this point in the proof, saying “THEN I will arrive at 3pm” is arbitrary, and could have been “THEN pigs will fly.” I think that’s what’s known as the principle of explosion, but I could be wrong about that term.
OK I agree this is very much the crux of the issue. However:
First, yes, I agree, it could have been anything, including “THEN pigs will fly”.
Second, the way that “IF .. THEN” is defined in propositional or first order seems not to capture quite what we mean by those words in ordinary language. I think this is part of what you are pointing out.
But taking the rules of first order logic as a given, it really is valid to derive from “P” to “P or Q”, because in all the worlds where “P” is true, “P or Q” is also true, which is what it means for it to be valid to derive from one thing to another. And for the same reason it really is valid to derive from “not P” to “if P then Q”, for an arbitrary Q.
In some previous discussions of the 5-and-10 problem, some people have concluded that what needs to be done is to rework the definition of “IF … THEN” in first order logic. I think it is quite illuminating to attempt something like this, but I don’t think this ultimately resolves the issue.
I haven’t yet seen any completely satisfying resolution of this problem, but I do think that it’s a productive problem to work at, and also I find it fun, so thank you for doing it with me :)
I feel like the confusion between propositional logic and ordinary language is the only reason Lob’s theorem is even being discussed in the first place. The car’s programmers used IF X THEN Y to represent the statement “If X, then Y happens”, which means something quite different. Other than the incidental similarity of these statements in the English language, why is this more relevant than any other programming error?
Ah, so the error is back here:
3. Make a list of all possible logical sentences of the form
IF route 1 is taken THEN I will arrive at such-and-such a time AND IF route 2 is taken THEN I will arrive at such-and-such a time AND ...
Because the algorithm was created without including the additional assumption (used later on in the “proof”) that if route 1 route is taken, then route 2 would NOT be taken (and vice versa). If you include only that additional piece of information, then the statements generated in step 3 are “logically” equivalent to:
“IF route 1 is not taken AND IF route 2 is taken THEN I will arrive at Time” (or “IF route 1 is taken I will arrive at Time AND route 2 is not taken”).
And that (again from our route 1 XOR route 2 assumption) is equivalent to a list of :
IF route # is taken, THEN I will arrive at time
for all possible combinations of route and time, with no conjunctions at all.