# Necessarily, the number of planets is nine. ?!

kennethamy

Thu 28 Jan, 2010 04:55 pm
@Angra Mainyu,
Angra Mainyu;123286 wrote:

Now, that "could have been false", and "could have not existed" may mean "if the past had been different", or "even with the same past"; but I'm not sure what else that might mean; so, I'm trying to figure that out.

P could have been false= The negation of P does not imply a contradiction.

X could not have existed= The negation of the proposition that X exists does not imply a contradiction.

Of course, as both Hume and Kant pointed out, every proposition of the form that X exists is contingent, since existence is not a predicate.

Emil

Thu 28 Jan, 2010 07:02 pm
@kennethamy,
kennethamy;123343 wrote:
P could have been false= The negation of P does not imply a contradiction.

X could not have existed= The negation of the proposition that X exists does not imply a contradiction.

Of course, as both Hume and Kant pointed out, every proposition of the form that X exists is contingent, since existence is not a predicate.

I agree with the conclusion but what does existence being/not being a predicate have to do with it?

Angra Mainyu

Fri 29 Jan, 2010 12:14 am
@Emil,
kennethamy wrote:
Angra Mainyu wrote:

Now, that "could have been false", and "could have not existed" may mean "if the past had been different", or "even with the same past"; but I'm not sure what else that might mean; so, I'm trying to figure that out.

P could have been false= The negation of P does not imply a contradiction.

X could not have existed= The negation of the proposition that X exists does not imply a contradiction.

I understand that that is the definition you provided.

What I'm suggesting is that that definition does not seem to match the usage of "could have been false", or "could have not existed" when those expressions are used to characterize contingency.

The counter-examples that come to mind are P1: "Water is H2O", or P2: "The atomic number of gold is 79". The negation of P1, or of P2, does not imply a contradiction. Yet, it seems to be a more or less common view (even among professional philosophers) that said propositions are not contingent.

kennethamy wrote:

Of course, as both Hume and Kant pointed out, every proposition of the form that X exists is contingent, since existence is not a predicate.

What do you think of statements like "An infinite set exists", or "The number 3 exists"? Are they contingent, or necessary?

---------- Post added 01-29-2010 at 03:16 AM ----------

Emil wrote:

It is impossible to keep giving non-circular definitions. This is called the dictionary problem. I'm fine with PWS being a framework to understand logic but you should read Swartz and Bradley (1979) for an introduction to logic using PWS.

It's not the logic what I find obscure. It's the framework itself. I don't think the dictionary problem is really an issue, since I'm not asking to keep giving definition ad infinitum, but just give an explanation of a framework that seems at least obscure.
Emil wrote:

No? I think it matches exactly what the common meaning is. Again see the work referred to before and this site. All of them agree in essence with the two definitions that I mentioned above.

While it seems that that site provides definitions matching yours, I don't think it matches a more common philosophical usage; the examples of water and gold were meant to illustrate that usage, but see (for instance) this page, or this article on rigid designators.

For instance, someone could claim "The atomic number of gold is not 79", or "Water is not H2O". The could say that maybe physicists made a mistake. Or that an evil superpowerful entity deceived them, altering the results of their experiments, or whatever.

While those claims would be unreasonable, they would not be contradictory. In other words, it's not contradictory to deny "Water is H2O" (or "The molecular composition of water is H2O", if you like), or "The atomic number of gold is 79".

So, by the definitions that you and kennethamy provide, "Water is H2O", and "The atomic number of gold is 79" would be contingent. However, many philosophers maintain that they're necessary, even though they're aware of the fact that their negations are not contradictory; in fact, they often use them to illustrate the distinction between logical possibility and metaphysical possibility.

That shows that they're not using "contingent" and "necessary" the way you and kennethamy are using them.

Given that this other usage seems to be the most common by far (at least in my experience) on discussions on the internet, I would say it's likely that Owen used the words in that sense when he said "Necessarily".

Emil wrote:

That's not what they commonly mean. Where'd you get this whole past idea? Though the past might be different and if it was, then certain contingent things would not exist and others would, etc.

I got it by observing non-philosophical usage; I'm trying to approximate philosophical usage of "could have been false", "could have not existed", "could have been different", "could have happened", and all those "could have", but since it seems unclear, I'm trying to approximate folk usage, and propose those alternatives to test them against philosophical usage.

But that might not work (the usages might be different), in which case, some other alternative would be required. In any case, elucidating philosophical usage (at least, the most common one) is what I'm trying to do and if that's successful, we can apply that to the planets if you like.

Owen phil

Fri 29 Jan, 2010 04:12 am
@ACB,
ACB;123214 wrote:

Originally Posted by Owen (the number of planets) = 9, is a contingent truth.
(the number of planets) []= 9, is a contingent truth.
(the number of planets) > 7, is a contingent truth.
(the number of planets) []> 7, is a contingent truth.

Can you please explain why you think the second and fourth statements above are true?

Ex, means, there is some x such that.
Ax, means, for all x such that.

Definition 1. []p =df It is necessary that p is true.
Definition 2. <>p =df It is possible that p is true.
Definition 3. (x []>y) =df [](x > y).
Definition 4. G(the x:Fx) =df Ey(Ax(x=y <-> Fx) & Gy).

(the number of planets is 9) <-> Ey(Ax(x=y <-> x numbers the planets) & y=9). By Definition 4.

Ax(x=9 <-> x numbers the planets) <-> Ey(Ax(x=y <-> x numbers the planets) & y=9). By, Fa <-> Ey(Fy & y=a).

(the number of planets is necessarily greater than seven) <->
Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)). By: Definition 4, Def 3.

(the number of planets is 9) -> (the number of planets is necessarily greater than seven).

Proof:

1. Ax(x=9 <-> x numbers the planets) -> (Ax(x=9 <-> x numbers the planets) & [](9 > 7)).
Because [](9 > 7) is tautologous.

2. (Ax(x=9 <-> x numbers the planets) & [](9 > 7)) -> Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)).
By, Fa -> EyFy.

therefore,

3. Ax(x=9 <-> x numbers the planets) -> Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)).
By, 1, 2, MP.

QED.

That is, it cannot be the case that (the number of planets is nine) is true and (the number of planets is necessarily greater than seven) is false.

(the number of planets is nine) is a contingent truth, because it is possibly true and possibly false.
(the number of planets is necessarily greater than seven) is a contingent truth, because it is possibly true and possibly false also.

If, the number of planets is six, were true then the number of planets is necessarily greater than seven is false, because [](6 >7) is contradictory.

That the number of planets is eight, does not change the argument.
ie. The number of planets is eight, implies the number of planets is necessarily greater than seven, is still true.

The second statement is proven in the same way.

kennethamy

Fri 29 Jan, 2010 06:39 am
@Emil,
Emil;123375 wrote:
I agree with the conclusion but what does existence being/not being a predicate have to do with it?

It follows from the premise that existence is not a predicate, that there are no necessarily existential truths. For to say there are, would be to say that existence was a property of the object that (supposedly) necessarily exists. For instance, the proposition that God necessarily exists (The Ontological Argument) implies that existence is a property of God. That is why the refutation of the Ontological Argument is that existence is not a predicate.

---------- Post added 01-29-2010 at 07:43 AM ----------

Angra Mainyu;123402 wrote:

What I'm suggesting is that that definition does not seem to match the usage of "could have been false", or "could have not existed" when those expressions are used to characterize contingency.

The counter-examples that come to mind are P1: "Water is H2O", or P2: "The atomic number of gold is 79". The negation of P1, or of P2, does not imply a contradiction. Yet, it seems to be a more or less common view (even among professional philosophers) that said propositions are not contingent.

.

Some professional philosophers. (Kripke and Putnam). And, they have offered it as an intuition rather than an argument, so far as I can see. It is not my intuition. Nor, was it Hume's.

Angra Mainyu

Fri 29 Jan, 2010 07:16 am
@kennethamy,
kennethamy wrote:

It follows from the premise that existence is not a predicate, that there are no necessarily existential truths.

Am I right in concluding that your view is that mathematical existential statements (like "There exists an infinite set", or "There exist infinite even numbers", or "There exists an empty set", etc.), are contingent?

kennethamy wrote:
Some professional philosophers. (Kripke and Putnam). And, they have offered it as an intuition rather than an argument, so far as I can see. It is not my intuition. Nor, was it Hume's.

Some philosophers, yes. But since they offer those cases as cases of necessary statements, it seems that they reject the definitions that you and Emil propose. And this seems to be a fairly common view, not just that of Kripke and Putnam.

In particular, it seems to be the most common view on internet discussions, so I think that Owen probably meant to make that distinction, rather than the one that matches your definition.

That said, what do you think about those examples?

Do you think "Water is H2O" and "The atomic number of gold is 79" are contingent? (your definition seems to imply so, but I just want to be sure that that's your position).

Owen phil

Fri 29 Jan, 2010 07:17 am
@kennethamy,
kennethamy;123422 wrote:
It follows from the premise that existence is not a predicate, that there are no necessarily existential truths. For to say there are, would be to say that existence was a property of the object that (supposedly) necessarily exists. For instance, the proposition that God necessarily exists (The Ontological Argument) implies that existence is a property of God. That is why the refutation of the Ontological Argument is that existence is not a predicate.

What premise implies that existence is not a predicate?

God exists, has God as its subject and exists as its predicate.

Exists is not a primary predicate like 'x is red', but it is clearly a predicate.
Rather, it is the logical sum of primary predicates.

x has the property G, implies there is some property that x has.
Gx -> EF(Fx), ie., Gx -> x exists.

Necessarily existent truths...

It is necessary that x=x is true for all x's.

x=x <-> AF(Fx <-> Fx), x=x <-> Ey(x=y),
ie. x exists is necessarily true for all x.

Angra Mainyu

Fri 29 Jan, 2010 07:55 am
@Owen phil,
Owen;123413 wrote:
Ex, means, there is some x such that.
Ax, means, for all x such that.

Definition 1. []p =df It is necessary that p is true.
Definition 2. <>p =df It is possible that p is true.
Definition 3. (x []>y) =df [](x > y).
Definition 4. G(the x:Fx) =df Ey(Ax(x=y <-> Fx) & Gy).

(the number of planets is 9) <-> Ey(Ax(x=y <-> x numbers the planets) & y=9). By Definition 4.

Ax(x=9 <-> x numbers the planets) <-> Ey(Ax(x=y <-> x numbers the planets) & y=9). By, Fa <-> Ey(Fy & y=a).

(the number of planets is necessarily greater than seven) <->
Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)). By: Definition 4, Def 3.

(the number of planets is 9) -> (the number of planets is necessarily greater than seven).

Proof:

1. Ax(x=9 <-> x numbers the planets) -> (Ax(x=9 <-> x numbers the planets) & [](9 > 7)).
Because [](9 > 7) is tautologous.

2. (Ax(x=9 <-> x numbers the planets) & [](9 > 7)) -> Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)).
By, Fa -> EyFy.

therefore,

3. Ax(x=9 <-> x numbers the planets) -> Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)).
By, 1, 2, MP.

QED.

That is, it cannot be the case that (the number of planets is nine) is true and (the number of planets is necessarily greater than seven) is false.

(the number of planets is nine) is a contingent truth, because it is possibly true and possibly false.
(the number of planets is necessarily greater than seven) is a contingent truth, because it is possibly true and possibly false also.

If, the number of planets is six, were true then the number of planets is necessarily greater than seven is false, because [](6 >7) is contradictory.

That the number of planets is eight, does not change the argument.
ie. The number of planets is eight, implies the number of planets is necessarily greater than seven, is still true.

The second statement is proven in the same way.

While I acknowledge that I find the concepts of contingency and necessity problematic, using them for the sake of the argument, I would say that the error is in "That is, it cannot be the case that (the number of planets is nine) is true and (the number of planets is necessarily greater than seven) is false."; that does not seem to be what the conclusion means.

The conclusion is:

3. Ax(x=9 <-> x numbers the planets) -> Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)).

Correcting for the actual number of planets, we get:

3'. Ax(x=8 <-> x numbers the planets) -> Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)).

Then, we can add true premise:

4. Ax(x=8 <-> x numbers the planets)

And from 3' and 4, by MP, we get:

5. Ey(Ax(x=y <-> x numbers the planets) & [](y > 7))

However, that does not imply that necessarily, the number of planets is greater than seven. 5. means that there exists some Y such that Y is the number of planets (but 5. does not state that such Y necessarily is the number of planets) and that Y is necessarily greater than 7.

To make it simpler without the notation.

The number of coins in my pocket is 2. Necessarily, 2 is greater than 1. So, there exists some number Z such that Z is (but not necessarily is) the number of coins in my pocket, and Z is necessarily greater than 1.

From "The number 2 is the number of coins in my pocket, and 2 is necessarily greater than 1", it does not follow "The number of coins in my pocket is necessarily greater than 1".

Emil

Fri 29 Jan, 2010 07:59 am
@Angra Mainyu,
Angra Mainyu;123402 wrote:
I understand that that is the definition you provided.

What I'm suggesting is that that definition does not seem to match the usage of "could have been false", or "could have not existed" when those expressions are used to characterize contingency.

I disagree.

Quote:
The counter-examples that come to mind are P1: "Water is H2O", or P2: "The atomic number of gold is 79". The negation of P1, or of P2, does not imply a contradiction. Yet, it seems to be a more or less common view (even among professional philosophers) that said propositions are not contingent.
But these are used with metaphysical impossibility/possibility, not logical impossibility/possibility. I defined contingency/necessity in terms of logical possibility. It is possible to define an analogous term in metaphysical possibility (whatever that means, if anything). Let's call them logical contingency and metaphysical contingency respectively. Which of them are you having problems with? I have no problems with the first and I think the latter is obscure.

Quote:
What do you think of statements like "An infinite set exists", or "The number 3 exists"? Are they contingent, or necessary?
I haven't thought about it much, but I'll go with logical contingent because I don't see any contradiction.

Quote:
It's not the logic what I find obscure. It's the framework itself. I don't think the dictionary problem is really an issue, since I'm not asking to keep giving definition ad infinitum, but just give an explanation of a framework that seems at least obscure.
Which framework; PWS or metaphysical contingency/necessity?

I think the dictionary problem is relevant. You keep asking for clarification for concepts. I cannot keep giving it to you. Besides the amount of vagueness that you can tolerate is extremely low. Most of metaphysics is obscure to you, isn't that right?

Quote:
While it seems that that site provides definitions matching yours, I don't think it matches a more common philosophical usage; the examples of water and gold were meant to illustrate that usage, but see (for instance) this page, or this article on rigid designators.
I will have to read up on those. No comments on them so far.

Quote:
For instance, someone could claim "The atomic number of gold is not 79", or "Water is not H2O". The could say that maybe physicists made a mistake. Or that an evil superpowerful entity deceived them, altering the results of their experiments, or whatever.

While those claims would be unreasonable, they would not be contradictory. In other words, it's not contradictory to deny "Water is H2O" (or "The molecular composition of water is H2O", if you like), or "The atomic number of gold is 79".

So, by the definitions that you and kennethamy provide, "Water is H2O", and "The atomic number of gold is 79" would be contingent. However, many philosophers maintain that they're necessary, even though they're aware of the fact that their negations are not contradictory; in fact, they often use them to illustrate the distinction between logical possibility and metaphysical possibility.
Ok. But I don't know what "metaphysical possibility means". I agree so far that they are contingent.

Quote:
That shows that they're not using "contingent" and "necessary" the way you and kennethamy are using them.
I think that shows that there are at least two different concepts of contingent and necessary. Not surprising. I could mention a few more analogous terms if you like. (Think of ones defined in terms of other possibilities: physical, practical, willful, epistemic...)

Quote:
Given that this other usage seems to be the most common by far (at least in my experience) on discussions on the internet, I would say it's likely that Owen used the words in that sense when he said "Necessarily".
The most common? I nearly only see it with you.

Quote:
I got it by observing non-philosophical usage; I'm trying to approximate philosophical usage of "could have been false", "could have not existed", "could have been different", "could have happened", and all those "could have", but since it seems unclear, I'm trying to approximate folk usage, and propose those alternatives to test them against philosophical usage.
Folk usage is vague. It is not possible to extract a non-vague concept from folk usage. That is the way with general language.

Angra Mainyu

Fri 29 Jan, 2010 08:34 am
@Owen phil,
Emil wrote:
But these are used with metaphysical impossibility/possibility, not logical impossibility/possibility.

True.

Emil wrote:

I defined contingency/necessity in terms of logical possibility.

Then, my point is that contingency/necessity are at least more often (if not nearly always) used in terms of metaphysical impossibility/possibility, and that that was more likely to be Owen's usage as well.

Emil wrote:

It is possible to define an analogous term in metaphysical possibility (whatever that means, if anything). Let's call them logical contingency and metaphysical contingency respectively. Which of them are you having problems with? I have no problems with the first and I think the latter is obscure.

Similarly, I have no problems with the first and I think the second is at best obscure. However, I think the most common usage by far (at least, on the internet, and probably among today's philosophers) is the second.

Emil wrote:

Which framework; PWS or metaphysical contingency/necessity?

The metaphysical contingency/necessity (which I think is the usual in philosophy), and rather than possible world semantics, the concept of possible world itself (i.e., we can make a formal system that has no contradictions at all, yet it's useless).

Emil wrote:

I think the dictionary problem is relevant. You keep asking for clarification for concepts. I cannot keep giving it to you.

I think that's exaggerated. I keep asking for definitions of the same concepts: contingent, necessary, and, if you want to use PWS, possible world; of course, I want definitions that match usage among philosophers.

Emil wrote:

Besides the amount of vagueness that you can tolerate is extremely low.

Not really. Vagueness is all over the place. Terms like tall, short, fat, big, heap, etc., are vague. But even terms like "dog", or "pig" are to some extent (e.g., start at some point about 1 billion years ago, move forward one minute at a time and try to find the first minute where dogs exist).

Emil wrote:

Most of metaphysics is obscure to you, isn't that right?

Yes, but that is more problematic than vagueness in daily concepts, because the vagueness has a zillion times be used to equivocate and reach conclusions like "God exists", etc. (and others not related to theism, but also unsupported).

Moreover, given that the terms are not folk terms, definitions seem to be the way to introduce them.

Emil wrote:

Ok. But I don't know what "metaphysical possibility means".

Neither do I, but I think this is what Owen means, and what most people who debate here mean by "possible", "necessary", "contingent", etc. (i.e., metaphysical, not logical). The same for many if not most professional philosophers who use the terms "necessary", "contingent", etc., these days.

Emil wrote:

I agree so far that they are contingent.

Emil wrote:

I think that shows that there are at least two different concepts of contingent and necessary.

Perhaps, but even so, my point was that the people in question were not using the one you're using.

But if that is so, then don't you think we should clarify which one we're using, before discussing whether something is contingent or necessary?

Otherwise, we risk talking past each other.

Emil wrote:

The most common? I nearly only see it with you.

I disagree. It's also the one used by Putnam and Kripke, and the one you see in all arguments for the existence of God that use the necessary/contingent distinction.

In addition, it's the distinction I get the impression people use when debating on-line, except for you, kennethamy and maybe others.

Emil wrote:

Folk usage is vague. It is not possible to extract a non-vague concept from folk usage. That is the way with general language.

It depends on the concept, but often, this is true (not always because when technical concepts are defined, they have to be defined in terms of already known concepts, some of which have to be folk concepts).

However, if the philosophical concept is not defined, then it's difficult to see where else to look for them.

Owen phil

Fri 29 Jan, 2010 08:39 am
@Angra Mainyu,
Angra Mainyu;123432 wrote:
While I acknowledge that I find the concepts of contingency and necessity problematic, using them for the sake of the argument, I would say that the error is in "That is, it cannot be the case that (the number of planets is nine) is true and (the number of planets is necessarily greater than seven) is false."; that does not seem to be what the conclusion means.

The conclusion is:
3. Ax(x=9 <-> x numbers the planets) -> Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)).
Correcting for the actual number of planets, we get:
3'. Ax(x=8 <-> x numbers the planets) -> Ey(Ax(x=y <-> x numbers the planets) & [](y > 7)).

Then, we can add true premise:
4. Ax(x=8 <-> x numbers the planets)
And from 3' and 4, by MP, we get:
5. Ey(Ax(x=y <-> x numbers the planets) & [](y > 7))

However, that does not imply that necessarily, the number of planets is greater than seven. 5. means that there exists some Y such that Y is the number of planets (but 5. does not state that such Y necessarily is the number of planets) and that Y is necessarily greater than 7.

To make it simpler without the notation.
The number of coins in my pocket is 2. Necessarily, 2 is greater than 1. So, there exists some number Z such that Z is (but not necessarily is) the number of coins in my pocket, and Z is necessarily greater than 1.

From "The number 2 is the number of coins in my pocket, and 2 is necessarily greater than 1", it does not follow "The number of coins in my pocket is necessarily greater than 1".

5. Ey(Ax(x=y <-> x numbers the planets) & [](y > 7))

5. means, the number of planets is necessarily greater than 7, is true.
It does not mean, Necessarily the number of planets is greater than 7.

The number of planets is necessarily greater than 7, means, Ey(Ax(x=y <-> x numbers the planets) & (8 []> 7)), by definition.

Necessarily, the number of planets is greater than 7... is false.

Angra,
The number of coins in my pocket is 2. Necessarily, 2 is greater than 1. So, there exists some number Z such that Z is (but not necessarily is) the number of coins in my pocket, and Z is necessarily greater than 1.
From "The number 2 is the number of coins in my pocket, and 2 is necessarily greater than 1", it does not follow "The number of coins in my pocket is necessarily greater than 1".

Wrong, It does not follow that, (Necessarily, the number of coins in my pocket is greater than 1) but it does follow that, (The number of coins in my pocket is necessarily greater than 1).

Angra Mainyu

Fri 29 Jan, 2010 08:51 am
@Owen phil,
Owen wrote:

5. Ey(Ax(x=y <-> x numbers the planets) & [](y > 7))

5. means, the number of planets is necessarily greater than 7, is true.
It does not mean, Necessarily the number of planets is greater than 7.

5. means "There exists some Y, such that Y is the number of planets (as a matter of fact, not of necessity), and Y is necessarily greater than 7".

Owen wrote:

Wrong, It does not follow that, (Necessarily, the number of coins in my pocket is greater than 1) but it does follow that, (The number of coins in my pocket is necessarily greater than 1).

Only if by "The number of coins in my pocket" you mean "the actual number at this moment".

However, that expression is ambiguous, and I was using it in the sense you seemed to be using a similar one about planets, since your conclusion was "Necessarily, the number of planets is 9" (that's the thread's title), so I thought you meant the same by "The number of planets is necessarily 9" (or rather 8).

If you let the actual, present-day number of planets fix the referent of "The number of planets", then "The number of planets is necessarily 8" is true, but if you mean "Necessarily, the number of planets is 8", that is usually accepted as false (though we'd have to discuss what necessity and contingency mean before we can assess that better, but let's say it's false).

Still, if your only conclusion is "The number of planets is necessarily 8", and you let the present-day, actual number of planets fix the referent of "The number of planets", then I'm not sure why this is problematic. The conclusion "Necessarily, the number of planets is 9" (or 8) can't be derived in that manner.

Owen phil

Fri 29 Jan, 2010 10:03 am
@Angra Mainyu,
Angra Mainyu;123452 wrote:
5. means "There exists some Y, such that Y is the number of planets (as a matter of fact, not of necessity), and Y is necessarily greater than 7".

Only if by "The number of coins in my pocket" you mean "the actual number at this moment".

However, that expression is ambiguous, and I was using it in the sense you seemed to be using a similar one about planets, since your conclusion was "Necessarily, the number of planets is 9" (that's the thread's title), so I thought you meant the same by "The number of planets is necessarily 9" (or rather 8).

If you let the actual, present-day number of planets fix the referent of "The number of planets", then "The number of planets is necessarily 8" is true, but if you mean "Necessarily, the number of planets is 8", that is usually accepted as false (though we'd have to discuss what necessity and contingency mean before we can assess that better, but let's say it's false).

Still, if your only conclusion is "The number of planets is necessarily 8", and you let the present-day, actual number of planets fix the referent of "The number of planets", then I'm not sure why this is problematic. The conclusion "Necessarily, the number of planets is 9" (or 8) can't be derived in that manner.

1. (the number of planets)=8.
2. Necessarily, 8 > 7.
therefore,
3. Necessarily, (the number of planets) > 7.

This argument has true premises and a false conclusion, in spite of Leibniz's Law.

Leibniz and Russell claim that: (the number of planets) = 8 -> (F(the number of planets) <-> F(8)), for all F.

The problem is to show where it is faulty.

(the number of planets) > 7, is contingent because it is both possibly true and possibly false. p is analytic if it is not contingent.

Necessary(p) is true iff p is a logical truth.
If p is contingent then Necessary(p) is contradictory.

Necessary((the number of planets) > 7), is a contradiction.
Necessary((the number of planets) = 8), is a contradiction.

Quine, who introduced this problem in 1962, claims that referential opacity is to blame. That is, he does not accept modal predications as values of predicate variables.

I have shown that we can accept modal contexts where descriptions are present, if we make a distinction between Necessarily((the number of planets) is greater than 7) and ((the number of planets) is necessarily greater than 7).

Angra Mainyu

Fri 29 Jan, 2010 10:40 am
@Owen phil,
Owen wrote:

1. (the number of planets)=8.
2. Necessarily, 8 > 7.
therefore,
3. Necessarily, (the number of planets) > 7.

Okay, so the argument in question does reach the conclusion that necessarily, the number of planets > 7.

Let's consider what the premises mean:

Premise 1 (the number of planets)=8 means "There exist exactly 8 planets in our planetary system".

Premise 2 seems to mean "The number 8 is greater than the number 7, and the number 8 couldn't have been not greater than the number 7" (that "couldn't have been" is unclear in my view, as is the contingent/necessary distinction, but we don't need to settle that here).

The conclusion seems to mean "Therefore, there exists more than 7 planets in our planetary system, and it couldn't have been the case that no more than 7 planets existed in our planetary system".

So, when we clarify the meaning of the statements, what we get is the following:

[INDENT]
1': There exist exactly 8 planets in our planetary system.

2': The number 8 is greater than the number 7, and the number 8 couldn't have been not greater than the number 7.

3': Therefore, there exists more than 7 planets in our planetary system, and it couldn't have been the case that no more than 7 planets existed in our planetary system. [/INDENT] But the conclusion does not follow from the premises.

Owen wrote:

I have shown that we can accept modal contexts where descriptions are present, if we make a distinction between Necessarily((the number of planets) is greater than 7) and ((the number of planets) is necessarily greater than 7).

Right, but ((the number of planets) is necessarily greater than 7), making that distinction, is like saying that the number of planets is 8, and 8 is necessarily greater than 7.

That's okay, but others might interpret it differently, risking miscommunication.

kennethamy

Fri 29 Jan, 2010 11:35 am
@Angra Mainyu,
Angra Mainyu;123427 wrote:
Am I right in concluding that your view is that mathematical existential statements (like "There exists an infinite set", or "There exist infinite even numbers", or "There exists an empty set", etc.), are contingent?

Some philosophers, yes. But since they offer those cases as cases of necessary statements, it seems that they reject the definitions that you and Emil propose. And this seems to be a fairly common view, not just that of Kripke and Putnam.

In particular, it seems to be the most common view on internet discussions, so I think that Owen probably meant to make that distinction, rather than the one that matches your definition.

That said, what do you think about those examples?

Do you think "Water is H2O" and "The atomic number of gold is 79" are contingent? (your definition seems to imply so, but I just want to be sure that that's your position).

I think that mathematics is an entirely different matter. And has to be treated differently. Outside of mathematics all existential propositions are contingent truths. As Hume said, what can be thought of as existing can be thought of as not existing.

Water is H20 is not an existential statement. It does not even imply that water exists.

Angra Mainyu

Fri 29 Jan, 2010 12:25 pm
@kennethamy,
kennethamy wrote:

I think that mathematics is an entirely different matter. And has to be treated differently. Outside of mathematics all existential propositions are contingent truths. As Hume said, what can be thought of as existing can be thought of as not existing.

I agree, but the fact is that in metaphysics, they don't seem to be treated differently - talk of contingency and necessity, for instance, applies to mathematical statements as well.

Outside of mathematics, some people make existential claims like "the rules of logic exist", etc.

kennethamy wrote:
Water is H20 is not an existential statement. It does not even imply that water exists.

That's true, but unrelated to my question about them.

My question is: Do you think "Water is H2O" and "The atomic number of gold is 79" are contingent? (your definition of "contingent" seems to imply so, but I just want to be sure that that's your position).

kennethamy

Fri 29 Jan, 2010 01:04 pm
@Angra Mainyu,
Angra Mainyu;123493 wrote:
I agree, but the fact is that in metaphysics, they don't seem to be treated differently - talk of contingency and necessity, for instance, applies to mathematical statements as well.

Outside of mathematics, some people make existential claims like "the rules of logic exist", etc.

That's true, but unrelated to my question about them.

My question is: Do you think "Water is H2O" and "The atomic number of gold is 79" are contingent? (your definition of "contingent" seems to imply so, but I just want to be sure that that's your position).

I would suppose they are, but I am not sure. Kripke thinks they are necessary a posteriori truths, and I can see an argument for that. But, again, all existential statements are contingent.

Angra Mainyu

Fri 29 Jan, 2010 01:13 pm
@kennethamy,
kennethamy;123497 wrote:
I would suppose they are, but I am not sure. Kripke thinks they are necessary a posteriori truths, and I can see an argument for that.

However, according to your definition of contingency, they are contingent, since denying them does not entail a contradiction (you said earlier "But to say that a statement is contingent, is to say that it is logically possible for it to be false. It's negation does not imply a contradiction.").

For instance, someone can claim that physicists got it wrong, or that some evil demon deceived them, and that water is not H2O, and the atomic number of gold is not 79.

Those claims would be silly, but they're not contradictory.

kennethamy

Fri 29 Jan, 2010 01:45 pm
@Angra Mainyu,
Angra Mainyu;123500 wrote:
However, according to your definition of contingency, they are contingent, since denying them does not entail a contradiction.

For instance, someone can claim that physicists got it wrong, or that some evil genius deceived them, and that water is not H2O, and the atomic number of gold is not 79.

Those claims would be silly, but they're not contradictory.

Yes. I agree. Here, I am nearly betwixt-and-between. But my inclination is to say they are contingent truths.

ACB

Fri 29 Jan, 2010 02:26 pm
@Angra Mainyu,
Angra Mainyu;123500 wrote:
However, according to your definition of contingency, they are contingent, since denying them does not entail a contradiction (you said earlier "But to say that a statement is contingent, is to say that it is logically possible for it to be false. It's negation does not imply a contradiction.").

For instance, someone can claim that physicists got it wrong, or that some evil demon deceived them, and that water is not H2O, and the atomic number of gold is not 79.

Those claims would be silly, but they're not contradictory.

As I see it, a statement is contingent if it is actually true but could have been false (i.e. it is logically possible for it to have been false). Thus "the capital of France is Paris" is true, but could have been false; the capital could have been (say) Lyon. That is to say, there is one (actual) world in which the capital is Paris, and simultaneously another (possible) world in which it is not. Hence "the capital of France is Paris" is contingently true. Note that this has nothing to do with error.

Now, "water is H2O" and "the atomic number of gold is 79" could logically be false in the actual world (if the physicists are wrong). But that is not the point. The point is: could they be true in the actual world while at the same time being false in a possible world? That, I think, is the test for contingency.

The question to ask is not "Could it logically be false that water is H2O", but "If it is true that water is H2O, could it logically have been false?" And that remains an open question.

Kennethamy - I wrote the above before I saw your post #59.