That's a different question. A possible world comes about as the result of a "what if..."; then we can see if that "what if..." leads to a consistent story or not. If it is inconsistent, then there can be no such possible world.

That is, logic gives us a grammar with which to judge our statements. — Banno

Ok, so we more or less agree then.

But my point was that a statement such as “It is not the case that the sun both was and was not a star, at the same time and in the same sense ” doesn't seem to be merely a truth about “grammar”, but also about the world, and it would seem like that proposition was true before any human being verbally constructed it or thought of it.

I am aware this leads to other difficult questions, related to philosophies like psychologism and the disputes between the advocates of that philosophy and people like Frege and Husserl, so I won't talk about it anymore here to avoid derailing the thread.

I won't talk about it anymore here to avoid derailing the thread. — Amalac

Oh, I think it pertinent.

At issue in this thread is whether one can deduce the existence of a being from logic alone. I think that it should be in principle not possible. I think this because logic is about what we can say, and not about the way things are.

It is not the case that the sun both was and was not a star, at the same time and in the same sense ” doesn't seem to be merely a truth about “grammar”, but also about the world, — Amalac

If we found a situation in which there was an apparent contradiction, what we would do is to re-think how we set out contradictions. Consider "It is both a wave and a particle"; the prima facie contradiction dissolves in the mathematics. Is Pluto both a planet and not a planet?

Ok then, do you accept that the Law of Contradiction is necessarily true? For there sake of this discussion, I'll maintain that it is.

If so, you must admit that the proposition:

"It is both a wave and a particle"; — Banno

... can't be true if you can replace “not a wave” with “a particle” without changing the meaning of the sentence.

If that proposition is true, then particle cannot mean “not a wave”, where both of those words “wave” in that sentence mean exactly the same thing. If they don't, then that proposition does not assert that A and ¬A, since the word wave (in the sentence resulting after you replace the meaning of “a particle”) would not mean “something that is not a particle”. It would have changed meaning in that case, and thus no longer have the same sense.

That's why the Law asserts that you can't assert A and ¬A, unless the second A has a different meaning/sense.

W1={egg, bacon} — fdrake

A world = a sub-domain?

A world = a sub-domain? — bongo fury

Of the set of possible worlds under consideration, I think so.

Within worlds, you still have the base logic and whatever domain of discourse is in that world.

I visualise possible worlds and accessibility relations as a graph whose nodes are sets (worlds) and whose links are the accessibility relation.

Possibility of X in a world is having a neighbour world (which can include itself) in which X evaluates as true.
Necessity of X in a world means having all neighbour worlds of that world evaluate X as true.

Possibility of X is "can I transition in one step to a world where this is true?"
Necessity of X is "if I step one world away, is X true no matter where I step?"

That means you can flesh out statements about possibility and necessity in terms of graph connections.

So a claim like "X is possible at world W" means "W has at least one neighbour where X evaluates as true". If W is a neighbour of a world where X is true, then X is possible at W. If all of W's neighbours have X as true, then X is necessary at W.

For @Banno, what S5 does is make the graph a partition/equivalence relation. It means that for all worlds:

(1) Every world is its own neighbour - that means if X is true in all worlds connected to W (X is necessary), then since W is connected to W (reflexivity), X is true in W. Necessarily A implies A.

(2) If W1 has another world W2 as its neighbour, then W2 has W1 as its neighbour. That means if X is true in W1, X must be possible in W2, so X is necessarily possible in W1 - every time X is true at W1, it will force all of W1's neighbours to have X as a possibility, which means possibly X is true in all of W1's neighbours, which means X => necessarily possibly X.

(3) If W1 is a neighbour of W2, and W2 is a neighbour of W3, then W1 is a neighbour of W3 (transitivity). That means that if X is true at W1, and W3 is a neighbour of W2, and W2 is a neighbour of W1, then X is true at W3 through the chain of links . This is a "walking" condition, it makes being a neighbour the same idea as being connected. In terms of modality, what this means is that if X is necessary at W1, and W1 is a neighbour of W2, then X is necessary at W2. Why? Well in order to be a neighbour of W2, it would also have to be a neighbour of W1, and we know all of W1's neighbours have X evaluate to true since X is necessary.

You can see that this is a fertile ground for ontological arguments - if god is possible and god is possibly necessary, then god is necessary, then god exists...

So, W1 = some world (among others) whose domain is {egg, bacon}?

So, W1 = some world (among others) whose domain is {egg, bacon}? — bongo fury

Yes.

Set of all possible worlds there is:

{W1, W2}

W1 is {egg, bacon}
W2 is {egg}

. I think this because logic is about what we can say, and not about the way things are — Banno

Not always. There are things that Logic is undeniable cause it comes simply from truth. When logic is based in pure truth facts it is indeed the way things are!

Ah, so not some one among several with the same domain.

So worlds are not in general to be identified by their domain?

That is just a nice thing about your example?

So worlds are not in general to be identified by their domain? — bongo fury

I don't see how I suggested that? Explain it to me please.

Well... equals?

I imagine that equality works on worlds too. I'd say that two worlds are equal when they consist of (all and only) the same elements / when they evaluate the same for all stuff/statements within 'em.

If you have a clock, the set of seconds in a minute form a set of worlds, and the ticking of one second to another forms an accessibility relation - t -> t+1 for all seconds t.

In that situation, two worlds would be equal if they were the same time instant.

Would you please tell me in what book or article I can read the stipulation of semantics for quantified modal logic you use?

I don't think you even need S5 for it? Given that you can choose world elements.

W1={egg, bacon}

W2={egg}

The statement E: "At least one entity in this world is an egg" — fdrake

What are W1 and W2? I would guess they are domains for two different worlds, since you refer to "world elements".

A world is not ordinarily just a domain, but rather (intuitively, as the full definition is more technical) a domain for the world and a set of relations on the domain for the world. Or, if the semantics stipulates just one overall domain, then a world is just a set of relations on the domain.

So does 'egg' stand for an individual? Or does it stand for the 1-place relation 'is an egg'?

I haven't yet read the rest of the posts, so maybe I'll find out more. I have more questions, but this is a start.

Would you please tell me in what book or article I can read the stipulation of semantics for quantified modal logic you use? — TonesInDeepFreeze

I can't because I'm mostly making it up from SEP and university memories. It is quite possible that what I said was entirely wrong!

I was envisioning the set of worlds:

{W1, W2}

With W1={egg}, and W2={bacon, egg}. The accessibility relation was just R = { (W1,W2), (W2,W1) }. I suppose more formally each of these has a hierarchy of statements regarding eggs, bacon, and the whole underlying logic thrown into them.

So, W1 = some world (among others) whose domain is {egg, bacon}?
— bongo fury

Yes. — fdrake

Set of all possible worlds there is:

{W1, W2}

W1 is {egg, bacon}
W2 is {egg} — fdrake

So {e b} and {e} are domains. So W1 and W2 are domains. But you say that W1 and W2 are worlds. As far as I can tell, that is conflating 'world' with 'domain for a world'.

So {e b} and {e} are domains. So W1 and W2 are domains. But you say that W1 and W2 are worlds. As far as I can tell, that is conflating 'world' with 'domain for a world'. — TonesInDeepFreeze

Could you explain the difference so it is very clear to me please?

Quantified modal logic is pretty technical. And I am rusty in my brief study of it long ago. So I might not state some things correctly, but I'll do my best. Also, the subject is complicated by the fact that there are various equivalent and pretty technically complicated ways of describing the semantics, while also there are alternatives to choose from. One of the choices is whether there is just one domain for the model or whether there are different domains for different worlds.

I'll say this in an intuitive way (it can be made more formally rigorous):

In predicate logic, a model has a domain, which is a set of individuals, and a set of relations on the domain. So a model is a "state of affairs". In predicate logic, models don't have worlds. Rather, the model is the world. A model (a world, or a state of affairs) is not just a set of individuals, but rather it is a set of individuals and facts about those individuals. The facts are captured as relations (predicates).

In predicate modal logic, models are more complicated. A model for predicate modal logic (I'm leaving out some other stuff here) is a set of worlds and an accessibility relation on the set of worlds. Again, a world is a domain, which is a set of individuals, and a set of relations on the domain. We can choose two different stipulations:

(1) There is only one domain for the model. It is the domain for all the worlds.

(2) Each world has its own domain and that domain may be different from domains for other worlds.

in discussions about existence in worlds, I think there could be a lot riding on which of those two contexts we are in, so we should be clear as to which of the two we mean.

/

By the way, through my conversation with Snakes Alive and looking more closely at some textbooks, I am starting to get a better idea of how an existence predicate works and also how evaluation of truth can work when some of the constants don't map to any member of certain of the domains in (2) above.

/

The textbooks I'm selectively reading now are:

A New Introduction To Modal Logic - Hughes & Cresswell

Logic, Language, And Meaning Volume 2 - Gamuit

You know, like in the movie trailer when the voiceover guy says, "In a world [he says the word 'world' in that overly dramatic way] where salamanders are smarter than humans ...", the world is not just the humans and salamanders and all the other objects, but also the facts about them.

You know, like in the movie trailer when the voiceover guy says, "In a world [he says the word 'world' in that overly dramatic way] where salamanders are smarter than humans ...", the world is not just the humans and salamanders and all the other objects, but also the facts about them. — TonesInDeepFreeze

:up:

in discussions about existence in worlds, I think there could be a lot riding on which of those two contexts we are in, so we should be clear as to which of the two we mean. — TonesInDeepFreeze

Thank you, I see. Let me see if I can rephrase the issue.

Imagine a world at which God exists, this world has a domain, and one of the entities in that domain is God. The phrase "God exists" is true in this world, but what set is that existential quantifier quantifying over?

"There exists at least one God" could be quantifying over the domain of the world in particular, in which case "There exists" is a perhaps a merely possible proposition - in a world where there wasn't a God, it would be false.

But it could be that "There exists" is quantifying over the union of the domain of all worlds, in which case if something exists in one world, it exists in all of them - since "exists" is only looking at the shared domain of entities which are distributed over all the worlds. That would make existence necessary existence for everything (not just God).

I think that's a separate issue from the issue with my equivocating between worlds and world domains? The appropriate scope of the existential quantifier "within world" so to speak is distinct from what a world is "made of" - my construal of it as a domain of objects missed out that it's actually a formal language structure within the world as well as there being a domain in the world which the formal language structure takes its (at least actual) truth-value cues from.

Yes - if it looks like a contradiction, first response is that you have said it wrong.

Looks good. Thanks.

But doesn't that mean that the Law of Contradiction reflects some kind of a priori, given structure of the world, such that the world must always follow that Law, and doesn't merely stablish a rule of grammar or a statement about what we must believe?

Do you disagree with any of this?:

When we have seen that a tree is a beech, we do not need to look again in order to ascertain whether it is also not a beech; thought alone makes us know that this is impossible. But the conclusion that the law of contradiction is a law of thought is nevertheless erroneous. What we believe, when we believe the law of contradiction, is not that the mind is so made that it must believe the law of contradiction. This belief is a subsequent result of psychological reflection, which presupposes the belief in the law of contradiction. The belief in the law of contradiction is a belief about things, not only about thoughts. It is not, e.g., the belief that if we think a certain tree is a beech, we cannot at the same time think that it is not a beech; it is the belief that if the tree is a beech, it cannot at the same time be not a beech. Thus the law of contradiction is about things, and not merely about thoughts; and although belief in the law of contradiction is a thought, the law of contradiction itself is not a thought, but a fact concerning the things in the world. If this, which we believe when we believe the law of contradiction, were not true of the things in the world, the fact that we were compelled to think it true would not save the law of contradiction from being false; and this shows that the law is not a law of thought. — Russell

Imagine a world at which God exists, this world has a domain, and one of the entities in that domain is God. The phrase "God exists" is true in this world, but what set is that existential quantifier quantifying over? — fdrake

I wish we had a specific formal semantics that together we reference. Otherwise, we risk getting lost in the twists and turns of an analysis bereft of a road map.

In predicate logic, "what does the quantifier range over?" has a simple answer. But in predicate modal logic, it's not immediately clear to me what "the quantifier ranges over" means. The more pertinent question instead might be, "how is the truth value of a quantified statement evaluated?" At least I can say that I don't find mentions of "the quanitifer ranges over" in stipulating the semantics. Instead:

(1) There are two methods: (a) just one domain per model or (b) domains for each world.

(2) Evaluation of truth (per a model) of a sentences with quantifiers.

"There exists at least one God" — fdrake

In the context you set up, I don't like that sentence. First you used 'God' as a name. But here you use 'God' like a predicate ("There exists at least one [g]od" I would take to mean "There exists an x such that x has the property of being a god" ).

I think there are real problems with what Russell says here. He was working with a faulty model of language as a seperate domain to the world - hence the theory of descriptions.

But perhaps the best way to proceed would be for you to set out exactly what you think Russell's argument is in the piece you quote; because I don't see an argument there.

But perhaps the best way to proceed would be for you to set out exactly what you think Russell's argument is in the piece you quote; because I don't see an argument there — Banno

There is an argument, it's just brief:

1. Either the Law of Contradiction states merely what we must believe, and what we can't believe, or it also asserts how the world necessarily is and must continue to be like.

2. It is impossible for the Law of Contradiction to
ever be false (interpreted as an assertion about the world, and not merely about our thoughts).

3. If the Law of Contradiction merely stated that we can't help believing that a thing cannot have a property X and a property ¬X at the same time and in the same sense, then the Law could be false in spite of the fact that we can't help believing in it, thus it is possible that the Law of Contradiction (in the sense in which it is applicable to the world) is false.

4. 3 contradicts 2.

5. Therefore, the Law of Contradiction must be a fact about the world. ( From 1, 2, 3 and 4)

Seems to me that (2) assumes your conclusion.

Well yes, since all proofs assume the Law, it itself cannot be proven, as Aristotle pointed out.

But it's just blindingly obvious, is it not? I mean, if we can't be certain about that, we can't be certain about anything.

But it's just blindingly obvious, is it not? — Amalac

Well...

That's the point in question.

Something philosophically quite odd happens when someone uses a name. There's a way in which the use ceases to be words and becomes the thing. If I say here that I am replying to Amalac, it doesn't mean that I am replying to a word, but to a person.

So there is a way of using words that stops being about language and becomes the world.

Perhaps it would help to think of it like this: Russell supposes that either the law of noncontradiction is a fact int he world or a figment of language. But actually, it is both. Even as Amalac is both a word and you.

all proofs assume the Law — Amalac

Not all proofs use the law. Indeed, the law is not even usually one of the logical axioms.

it itself cannot be proven — Amalac

Yes it can. (Here I'm taking the law in the sense of a single instance. For a schema we could adjust):

(1) Trivial. If the law is an axiom then it also provable by the rule of putting an axiom on a line.

(2) If it is not an axiom, then it is still provable from any set of logical axioms for a system that is complete in the sense of proving all validities.

(3) Trivial. It is provable from any inconsistent axiomatization.

Examples for all three cases include all the most common Hilbert style and natural deduction style systems (excepting those that stipulate that all tautologies are axioms, such as Enderton).

Perhaps it would help to think of it like this: Russell supposes that either the law of noncontradiction is a fact int he world or a figment of language. — Banno

I don't think so, here he says:

The belief in the law of contradiction is a belief about things, not only about thoughts. — Russell

His “only” implies that he holds that the belief in the Law of Contradiction is both about thoughts and about things.

Even as Amalac is both a word and you. — Banno

A word is not the same thing as that to which the word refers.

So if Amalac, in the following sentence, means “the word Amalac” then I am not Amalac.

If it means “the person writing this right now”, then I am Amalac.

The only way you can assert that “Amalac” is both a person and a word is through either denying the Law of Contradiction, or through the fallacy of equivocation.