Archive for October, 2010

“[speaking of going back in time] I’d particularly like to go back to my own childhood, see if things look as I recall them.”

Ira blinked. “Run a chance of running into yourself?”

“Why not?”

“Well . . there are paradoxes, are there not?”

“How? If I’m going to, then I did. That old cliche about shooting your grandfather before he sires your father, then going fuff! like a soap bubble—and all descendants, too, meaning both of you among others—is nonsense. The fact that I’m here and you’re here means that I didn’t do it—or won’t do it; the tenses of grammar aren’t built for time travel—but it does not mean that I never went back and poked around. I haven’t any yen to look at myself when I was a snot-nose; it’s the era that interests me. If I ran across myself as a young kid, he—I—wouldn’t recognize me; I would be a stranger to that brat. He wouldn’t give me a passing glance; I know, I was he.”" -Heinlein, Time Enough for Love, p. 358.

“If I must wait a thousand years to understand that word,” Hamadryad replied, “then I probably never will. Minerva says that it cannot be defined in Galacta and even when I speak Classic English, I find that I think in Galacta, which means that I do not really grasp English. Since the word love’ occurs so frequently in ancient English literature, I thought my failure to understand that word might be the block that keeps me from thinking in English.”

“Well, let’s shift to Galacta and take a swing at it. In the first place, very little thinking was ever done in English; it is not a language suited to logical thought. Instead, it’s an emotive lingo beautifully adapted to concealing fallacies. A rationalizing language, not a rational one. But most people who spoke English had no more idea of the meaning of the word ‘love’ than you have, even though they used it all the time.”

Heinlein, Time Enough for Love, p. 120

Some of you may have noticed that I have removed the links from the blogs and moved them to the front page. I am in the middle of major overhaul of the website including creating separate non-Wordpress sub-sites for my different projects (listed on the front page). Expect further changes in the future.

Classical predicate logic and empty domains

Introduction

In this essay I answer the question in the title with a very confident “Yes”, I give multiple arguments for this answer.

For the purposes of this essay a pluralistic proposition theory of truth carriers with sentences as secondary truth carriers is assumed.

The background and the reason for writing this essay is that many people deny that they understand sentences that are contradictory. They deny to understand sentences such as:

“Emil is sitting in his favorite char and Emil is not sitting in his favorite chair.”

Or the contracted version:

“Emil both is and is not sitting in his favorite chair.”

They claim that:

P1. For any sentence, if that sentence is contradictory, then it is meaningless.

But still the very same people believe that such sentences are false. This is inconsistent.

My claim is that P1 is false and that P2 and P3 are true:

P2. For any sentence, if that sentence is contradictory, then it is meaningful.

P3. There is a sentence such that it is contradictory and meaningful.

For each argument used, there is a proof of that argument’s validity in the appendix.

Are contradictions meaningful

schwitzsplinters.blogspot.com/2010/03/kant-on-killing-bastards-on.html

Have a read yourself. Thanks to Ben (blog∧¬blog) for the link.

“There are multiple reasons that people give. One is that some people do not think that a world filled only with cattle and pig farms and other such animals, but no other kinds, is a good place to live. This may be thought of as viewing the world as art, rather than as a factory. What do you want to see out of your window, and what do you want to see when you go on vacation? We could, if we wanted, divert the Colorado river and use the Grand Canyon as a giant landfill for our garbage. Is that a good idea?

Another thought is that we sometimes develop medicines and other useful things from different species, and without those animals, those new developments will not take place. And since it is impossible to know whether or not some species will prove useful in the future, preserving them all, insofar as one can do that, is the best way to have the most options in the future.

There is also the argument that is easily seen from thinking about an analogy. In the past, miners used to take birds down with them into the mines, and if the bird died (which was more sensitive to contaminants in the air than people), the miners would leave the mine. Without the birds, they did not know there was a problem until a person dropped off, and then it would likely be too late for them. Thus, the birds were an early warning system for danger in the mine. Likewise, animals may be viewed as an early warning system for life on planet earth; as we pollute the planet, the more sensitive animals die off first. And, of course, there are health consequences before one dies, so there is reason to keep the planet cleaner than just clean enough for humans to live. If we preserve the planet in a condition that saves the more sensitive animals, it will be safer for us.

Obviously, some people find some arguments more compelling than others.” (source)

This is a set of fairly good reasons. I haven’t come across good reasons for it before. Mostly what one gets is a bunch of appeals to nature (fallacy).

I found the original paper on the paradox. Enjoy.

en.wikipedia.org/wiki/Preface_paradox

David Makinson, The paradox of the preface

“Suppose that there is a trolley that is headed toward five people tied to the tracks ahead. There is no switch, but there is a bridge with you and someone else on it. But in this case, this is no wimpy trolley; a mere man in its way will not significantly affect its movement; it will easily crush flesh and bone. However, in this case, both you and the other person are in cars on the bridge. You know that a car is enough to stop the trolley, and you would like to drive yours off the bridge and stop the trolley, but the other car is in the way of you doing that, so you cannot simply drive off the bridge. You can, however, use your car to ram the other car, which will knock it off the bridge and onto the tracks, and thus stop the trolley from running over the five people. For stopping the trolley, it makes no difference whether someone is in the car or not; it is the car, and the car by itself, that will stop the trolley. But the other person is in the other car and will die if you push his car off the bridge, but you must act immediately, or the trolley will be past the bridge, so you do not have time to talk with the other driver or pull him out of his car; you either ram his car immediately, knocking it in the way of the trolley, or the trolley goes by and kills the five people. What should you do?

Just in case these details matter to you, I will now provide three variants:

1. If the car is knocked off the bridge, the car lands on the tracks, and when the trolley hits the car, the man in it dies.

2. If the car is knocked off the bridge, the man dies on impact with the tracks, so he is dead before the trolley hits the car.

3. The car is a convertible, and if the car is knocked off the bridge, the man will fall out of the car to the side of the tracks, but onto some rocks, which will kill him, but not immediately, so that he will still be alive at the time that the trolley comes to a complete stop from hitting his car, and then he dies within a minute, without being able to speak or do anything.”

The idea of these examples is to deal with people that think that the practical imperative shows why we think that which we think (or that which most people’s intuitions are). Clearly, if that was the case, then it would be permissible to push the car off the bridge because one does not use the man as a means to end end, he just happens to be in the car that we use. But still, most people’s intuitions disagree with this. Thus, the practical imperative does not explain (or is the cause of) our intuitions.

1. There is a thing, x, such that it is a contingent proposition.

∃xCx

2. For any thing, x, if x is a contingent proposition then there is a possible world, w, where x is true, and there exists a possible world, w’, where x is false.

∃xCx→∃x∃w∃w’Pxaw∧Pxbw’

3. Thus, there is a thing, x, and there is a possible world, w, and there is a possible world, w’, such that x has the property true in world w and x has the property false in w’.

⊢ ∃x∃w∃w’Pxaw∧Pxbw’ [1, 2, MP]

4. For any thing, x, and for any possible world, w, if x has the property true or has the property false in w, then x exists in w.

∃x∃wPxaw→∃x∃wExw

5. Thus, there is a thing, x, and there is a possible world, w, and there is a possible world, w’, such that x exists in w and x exists in w’ and

⊢∃x∃w∃w’Exw∧Exw’∧Pxaw∧Pxbw’ [3, 4, MP]

6. For any thing, x, for any possible world, w, x has the property true iff x does not hasve the property false.

∀x∀w(Pxaw↔¬Pxbw)

7. For any thing, x, and for any thing, y, for any possible world, w, and for any possible world, w’, if x exists in w and y exists in w’, then (x and y are identical iff for any property, z, if x has it in w, then y has it in w’).

∀x∀y∀w∀w’(Exw∧Eyw’)→(x=y↔(∀zPxzw↔Pyzw’))

This set is inconsistent (5∧6→¬7)1. But it seems clear to me that we ought to give up 7.

Notes

1I thought about giving the proof tree but it would be awfully long. Until I have an automatic way of doing this, I won’t prove such complex argument beyond formalizing them.