Going back to your first post:

In S4 or S5, or a derivative therefrom, can an individual exist in every possible world without contradiction? — Banno

I don't know what you mean by that. There is mention of systems and contradiction, which is syntactic, and mention of worlds, which is semantic. It's not clear to me what relationship between them you are asking about. Also, 'possible world' is relative to a given world and the accessibility relation, so what do yo mean by 'in every possible world' without reference a single world as the "base" from which other worlds are accessible to it? Or maybe you just mean just "in all worlds", so that we could delete the word 'possible'?

But more fundamentally, S4 and S5 are not predicate modal logics. So there is no semantics about individuals.

I'm interested in your question, but I wonder whether you might reformulate it.

take LNC as a given in order to derive RAA — Banno

Right, that was my Task (2). But I mentioned considerations about in my last paragraph.

The natural way to do it is first to derive the deduction theorem:

Where G is a set of formulas and p and q are formulas, then:

If G u {p} |- q, G |- p -> q

Then make that a derived rule. Then deriving RAA as a rule is only a few steps away.

It also occurred to me that someone who thinks LNC is always primary could argue:

Even given the explanations about deducibility, LNC entails (entailment is semantic) a subset of the instances of Axiom A4 and Axiom A4 entails LNC, so they are logically equivalent. That's true, but it doesn't advance any argument, because trivially all the validities are logically equivalent anyway.

I realized that a natural deduction proof of LNC is also trivial. This one has different notation, but it is essentially the same as in Kalish, Montague, and Mar and other common systems:

1. P & ~P assumption {1}
2. P simplification {1}
3. ~P simplification {1}
4. ~(P & ~P) RAA [1, 2, 3]

The ease of getting LNC from RAA illustrates the closeness between RAA and LNC. So we are tempted to think LNC and RAA are conceptually "the same". But from LNC to RAA is only one direction. The other direction is from LNC to RAA. So we ask how easy is to get to RAA from LNC. That depends on the particular Hilbert style system we use. A Hilbert style system is a combination of axioms and rules, and RAA in natural deduction systems, unlike LNC, is a rule, not a theorem nor an axiom. So we would be deriving a rule from a combination of axioms and rules.

So there are two tasks: (1) Derive LNC from a Hilbert style system that does not have LNC as an axiom, and (2) Devise a system, without RAA, in which LNC is an axiom, and derive RAA.

Task (1):

A common Hilbert style system (call it 'H1') is given by:

Capital letters are meta-variables ranging over formulas.

Axioms:

A1. P -> (Q -> P)
A2. (P -> (Q -> R)) -> ((P -> Q) -> (P -> R))
A3. (~P -> ~Q) -> (P -> Q)

Rules:

R1. Any axiom can be put on a new line.

R2. If P is on a line and also (P -> Q) is on a line, then Q may be put on a new line.

Definition:

D1. P & Q stands for ~(P -> ~Q).

[end description of H1]

From D1, LNC is:

~(P -> ~P)

And that is what we need to derive. To reduce tedium, the natural suggestion is to first derive RAA as a rule, then apply RAA as previously in this post. Deriving RAA as a rule is a lot of steps, but it is fairly straightforward to do and I think it's fairly common in textbooks. So, I'm not going to type it here.

But since LNC is couched with & and ~, an even clearer approach would be to take & and ~ as the primitives. Then put the axioms in those terms for system H2:

D2. P -> Q stands for ~(P & ~Q)

So A1 - A3 become:

A4. ~(P & ~~(Q & ~P))

A5. ~(P & ~~(Q & ~R)) -> ~(~(P & ~Q) & ~~(P & ~R))

A6. ~(~(~P & ~~Q) & ~~(P & ~Q))

Take an instance of A4:

~(P & ~~(P & ~P))

Eliminate double negation:

~(P & (P & ~P))

Apply associativity:

~((P & P) & ~P)

Apply idempotency:

~(P & ~P)

So, lo and behold, there's LNC.

So we might be tempted to say that H2 itself is a Hilbert style system with LNC as a "subschema" of one of the axiom schemas; that LNC was there all the time, hidden but implicit. Ah, but not so fast there, pardner. First we have to derive rules for double negation, associativity, and idempotency.

A more elegant argument is simply to point to the completeness theorem for H2:

H2 is entailment complete.
LNC is a validity.
Therefore, LNC is a theorem of H2.

But to be convinced of the conclusion, we need to witness the proof that H2 is entailment complete and witness the proof that LNC is a validity.

Task (2)

Offhand, I don't know of a Hilbert style system that has LNC as an axiom. We could add LNC to the axioms of H2, but that would result in system with a non-independent axiom set. That's logically permissible, but it is inelegant and it reduces the challenge, which might not want to do. So the interesting question is to find an independent and entailment complete set of axioms that includes LNC.

it's the one you are refering to here: — Amalac

Yes, it's the proof of a theorem about propositional logic. And we prove not just that LNC is true in a particular model but moreover that it is true in all models (i.e. that it is a validity).

Could you please show me a proof of the Law of Contradiction that didn't have it as an axiom — Amalac

That might be tedious for me to type out, and if you are not familiar with proof calculi for propositional logic, then it wouldn't be of much use to you.

I recommend 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar. Within about a chapter you could assign yourself the easy exercise of deriving LNC in the natural deduction system there.

forgot to add that other negation — Amalac

Yes, I mentioned it as a possibility. It was ungenerous of me to wonder whether you were trolling when Occam's razor would better suggest that you merely made a typo.

But again it boils down to:

When I resolve the double negative, we get "It is the case that the law is provable by the rule of putting an axiom on a line", which is what I said in the first place. — TonesInDeepFreeze

So I wonder why you still haven't recognize it, as I had already mentioned it twice.

what's this purely syntactical definition of “axiom” you speak of? — Amalac

A theory is a set of sentences closed under deduction.

(Some authors define a theory to be any set of sentences. And Enderton says a theory is a set of sentences closed under entailment, which is not syntactical. But they are equivalent with the completeness theorem, so, in that context, no harm done by saying 'closed under deduction' rather than 'closed under entailment'.)

A set of formulas S is an axiomatization of a theory T if and only if all members of T are provable from S. For a given axiomatization S of T, an axiom P is a member of S.

But they must still accept it implicitly without proof (that's what I mean by “accept it as an axiom”). — Amalac

That's a quite broad notion of axiom. It's not what 'axiom' usually means in a context such as formal logic, or as far as I know, even very much, if at all, in an informal discussion about reasoning.

It's not the case that sets of axioms are always independent, but independence of axioms is something we ordinarily desire.

As Aristotle said: It is utterly impossible to prove everything. — Amalac

Of course, I understand that in the everyday sense. But there is also a technical (though granted, pedantic) sense in which even axioms are provable. Also, for decidable theories, it is a decidable axiomatization where the set of axioms and the set of theorems are the same set.

Someone who doubted that the Law of Contradiction was true, would not accept any proof that assumed, without proof, that the Law of Contradiction is true — Amalac

That is incorrect. To accept a proof does not require accepting the truth of the premises. To accept a proof may be merely to audit that each step adheres to the rules of the proof system (the logic system). Thus, if the proof system is sound, then the assumption of the premises (no matter whether they are true of false) entails the conclusion.

it is trivially true that if you have the Law as an axiom, then you can prove that the Law is true. — Amalac

We can prove LNC is valid (even stronger than true). So we have the consequent of your hypothetical, so we don't need the antecedent. And we prove LNC is valid by proving that it is true in every model. That does not require proof in the logic system itself. However, its validity is entailed by proof in any logic system that that proves only validities (the soundness theorem).

if I have “God exists” as an axiom, I can prove that God exists. — Amalac

First, sentences of the form 'x exists' are not clear. One way to make them clear is with an existence predicate, but that is usually an advanced topic in predicate modal logic.

So I'll use this instead: G = "there exists a being that is omnipresent, omniscient, omnipotent, and omnibenevolent".

Then, yes, taking G as an axiom proves G as a theorem, but it doesn't prove that G is true. It only proves G is true if all the axioms used in the proof are true. So, since G is an axiom used in the proof, if it is false, then, though we have proved G, we have not proved that G is true.

It is fine to have it as a logical axiom, since it is logically true. Sceptics should learn that it is logically true.
— TonesInDeepFreeze

That's just an assertion, the LNC sceptic will demand a proof for it. They want to learn why it's true. — Amalac

No, it's not just an assertion. It's a theorem about propositional logic. And it is reducible, in a sense, to a theorem about Boolean algebra. And its proof is reducible to finitary operations, which are reducible to auditing the execution of an algorithm. So (heuristically speaking) we may say that at the root of the question is ability to audit the execution of an algorithm. Of course, it's hard to imagine such an ability in a person who was so delusional that they claimed to witness '0' and '1' written in the same space when only one of them was written in that space. But that is not the same as going all the way back up the chain of reductions I just described to say that LNC must be an axiom.

Logic is about the rules of language. Language is about how the world is. — Banno

Logic is about entailment and inference. Logic concerns both syntax of language and meaning with language. Meaning includes denotation, which concerns individuals in the world. And meaning includes evaluation of truth of sentences, which concerns states-of-affairs in the world. ('the world' may be taken in such senses as the real world, possible worlds, fictional worlds, or mathematical worlds.) Generally speaking, logic doesn't say what is the case in worlds, but logic is not just rules of language unless included in those rules are the means by which we relate language to objects and what is the case in the world. Maybe said this way: Logic does not concern what is the case in the world, but rather logic does include concern of HOW language relates to what is the case in the world.

How do you know that they also don't prove it as a theorem? — Amalac

That's a very silly question. I don't know they don't prove it as a theorem, since it is not the case that they don't prove it as a theorem, and I can't know that which is not the case. Are you trolling me?

Or maybe you meant to type: How do you know that they also don't prove it is not a theorem?

And I answered that in a previous post. When we remove the double negative we get: How do you know that they prove is a theorem?

By accepting the Law of Contradiction as an axiom? — Amalac

We may agree with LNC and use LNC without LNC being an axiom.

By definition, axioms are accepted as true without proof. — Amalac

That's one notion. But another definition of 'axiom' is purely syntactical.

If your point is that LNC is endemic in reasoning, then I agree with you on that point. But that doesn't entail that LNC must be an axiom. (And I'm putting aside the matter of paraconsistent logic.)

It cannot be proven — Amalac

Now you're just reasserting a claim that I refuted.

Why should a LNC sceptic accept that axiom in the first place? — Amalac

It is fine to have it as a logical axiom, since it is logically true. Sceptics should learn that it is logically true.

My point is that your proofs only work if someone accepts the Law as an axiom — Amalac

That's false. The proofs can be mechanically audited whether the auditer knows of LNC as an axiom or not. Indeed, even for everyday reasoning, probably most people haven't even heard of LNC, especially the notion of it is an axiom. And that does not contradict that good reasoning (other than dialethistic) conforms to LNC and sometimes uses it - either as an explicit or implicit principle.

your proofs only work if someone accepts the Law as an axiom, if they don't, then your “proofs” are circular. — Amalac

You are reasserting a claim that I refuted.

Without assuming the Law as true without proof, for all I know you may have both asserted and denied everything you've said this far. — Amalac

It is possible for one to assert and deny a proposition. And it is even ubiquitous that people assert propositions that are inconsistent with other propositions. So probably what you mean is that it is not possible to be correct while both asserting and denying a proposition. Then your question seems to be how do we know that contradictions are not the case. But the question of how we know things is different from the question of what axioms we choose. We may know that a statement is true by reasoning from different axioms sets that each yield the statement as a theorem. It is not required that LNC be one of the axioms. If, as we ordinarily do, we require a system that is complete in the sense of proving all validities then it is only required that LNC at least be a theorem even if not an axiom.

they must assume that it is certain implicitly — Amalac

They prove it as a theorem. Of course, our motivation for the system would include proving it as a theorem.

otherwise it is not even possible to talk meaningfully — Amalac

"Talk meaningfully" is a large and undefined rubric.

Only if you assume that it is not also the case that the Law is also not provable by the rule of putting an axiom on a line — Amalac

When I resolve the double negative, we get "It is the case that the law is provable by the rule of putting an axiom on a line", which is what I said in the first place.

the “proof” circular — Amalac

It is petitio principii. But ordinary logic allows it, otherwise you could never put a premise onto a line.

And the same applies for the rest of the “proofs”. — Amalac

First, I don't know why you put 'proofs' in what I surmise to be scare quotes.

Second, (1) is not petitio principii, unless you hold that proof from logical axioms is petitio principii. (2) Is basically an example of the principle of explosion itself closely related to LNC.

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).

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" ).

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.

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

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'.

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.

I reread the thread in question and I agree with myself. I stand by every word I wrote. — fishfry

You continued to claim that I didn't write 'onto itself'. You even quoted the post where I wrote 'onto itself' and said you do not see it, and yet it was there right in the quote. And you went on. Then another poster referred to one of your posts about it and quoted me yet again showing that indeed I did write 'onto itself'. Then nothing from you.

And you say you stand by your part in the discussion! But, even more crazy, now you are talking about my having written 'onto itself'! You are contradicting yourself right here!

your use of onto was regarding a "claim," and not a mapping. — fishfry

No, it was about a formula.

How would anyone take from that, that you are referring to an onto mapping rather than meaning a "claim unto itself" — fishfry

I wasn't referring to a mapping. The statement I made about the consistency of the formula had nothing to do with a mapping.

The FORMULA onto itself is not inconsistent. I meant that the formula taken alone is not inconsistent. And I even followed up to you to explain exactly that the formula alone is not inconsistent but is inconsistent with one of the axioms. And you even argued about THAT with me.

Note: Reflecting on this, I think it is possible that 'onto itself' in the way I used it is not standard English, and if that is the case, then my use could be unclear. But, there are two prongs here (1) That I did write it and you continued to say that I didn't write it, even though you quoted it yourself, yet now you do recognize that I wrote it, which makes it bizarre to say you still stand by what you wrote in the thread. (2) Even if my use is not standard, I still did illustrate my point in the thread: The formula is not inconsistent but rather it is inconsistent with the other axiom. Moreover even if I had not qualified with 'onto itself' or that 'onto itself' is not clear or standard, then my original statement is still correct. The formula is not inconsistent, though, of course, it is inconsistent with the other axiom.

/

Meanwhile, still you can't even recognize that you asked me about 'particle' and I immediately replied that it is primitive, though you falsely claim that you had to ask me twice.

/

I'm wondering whether what is going on with you is that you are so determined to think that you are right that you are willing to make the most preposterously false statements to do it.

You are not a crank in the sense of cranks who argue ignorantly and illogically about Cantor, Godel, et. al, but you share some traits: (1) terrible confusions,(2) blatant illogic, (3) skipping rebuttals to you, (4) persistently misconstruing posts and then posting as if something was claimed that was not claimed - so effectively ongoing strawman. Of all the people I've met on on the Internet, you're among the worst.

I recall the discussion I mentioned. It wasn't about the supposed absence of 'not' (I think perhaps that was another incident) but the bizarrely incorrect claimed absence of 'onto itself' and other baffling reading errors by fishfry.

I mention it because fishfry was so bizarre; it's an object lesson.

Starting here, looking at the my posts (I'm GrandMinnow) and fishfry's posts::



and ending with the third party here:

And, let's take this one point again:
— TonesInDeepFreeze

Like the girls in junior high school used to say: Let's not and say we did. — fishfry

Of course not with you. You don't wish to respond to the point that you made a false claim about me: So clearly false that all one has to do is look at the two posts.

When someone makes a false claim about someone, then waves off even responding to being shown that it is false, then that is included in the rubric of 'lying'.

You started out by agreeing that your exposition was unclear — fishfry

I didn't mean in this particular argument. Granted, I could have been made it explicit that I recognize that I can be clearer sometimes but that I am not saying that I was unclear in this argument.

and that I was asking clarifying questions. — fishfry

I did not say that. You are making that up out of thin air. Again, bizarre.

And, let's take this one point again:

You said that you had to ask me twice. But you asked me and then I IMMEDIATLY answered.

Sure, one can make a mistake like that innocently, but with you it's a dominant pattern and you won't recognize such instances when they are pointed out to you.

Then you go back to the personal insults. — fishfry

When your posting is so bizarrely off-base and patently illogical, through so many different conversations, then it's reasonable to point that out and to wonder aloud what is the source of your problem.

Is it possible that you're not always as clear in your meaning as you think you are? — fishfry

Not only do I think it is possible, but I bet it's true. I am keenly aware that (1) It is difficult to write about these topics on-the-fly and in the confines of posts, and therefore, no matter how hard I try, I'm bound to sometimes falter, and (2) Looking back at some of my posts, I see that what I wrote could have been clearer or needs certain corrections.

But I don't think that is the main problem with you. With you, even things that really are quite clear get misconstrued. There was even an incident in which you continued to insist that I made a certain claim, but I actually wrote the explicit negation of that claim. You had overlooked the word 'not' in my post. Sure, such a mistake can happen to anyone, but it was remarkable that you persisted even after it had been pointed out to you and even claimed the word 'not' did not appear! And then, after another poster also alerted you that the word appeared, as I recall, you still did not post that you recognized it finally.

And, time after time, even when I state clear and simple things, chronically, you read into them things not there nor implied.

And there have been even a couple of bizarre incidents (even lately) where you conflated what you said with what I said. Again, it can happen to anyone - but you persist in such incidents even after your error has been pointed out to you.

Meanwhile, I respond on point to you, and make every reasonable effort not to misconstrue you or mischaracterize your remarks, and I'm happy to correct myself if I did.
— TonesInDeepFreeze

Can you see that it's possible that this is not my perception? — fishfry

Indeed I do. But I don't recall an instance in which I unintentionally misconstrued you, then refused to recognize it when pointed out, let alone went on and on doing it over the same point, as you do with me. And, though I can't recall the specific incidents, I think there were one or perhaps two times when I misunderstood you and posted that I recognized that when it was pointed out to me.

You said "The number of particles is finite." Now you proposed that as an axiom to be added to the standard axioms of set theory. — fishfry

Just to be clear, I wrote it as a report of what an author wrote. I made clear that I don't personally propose any particular axioms for physics.

Being familiar with the latter, and not knowing what a "particle" is, I assumed you meant mathematical sets, or mathematical points. In which case your formulation would indeed be in contradiction with the axiom of infinity. — fishfry

No! You're doing it again! You're ignoring what I posted. I wrote explicitly about the alternatives (1) particles are sets or (2) particles are not sets but instead are urelements.

https://thephilosophyforum.com/discussion/comment/563405

[Note: I just now marked some edits to that post. But even with the pre-edited version, it was explicit that that there are two approaches, and in one approach particles are urelements not sets.]

And even if particles were sets, having the set of them being finite still would not violate the axiom of infinity. You're repeating the mistake on the point you conceded just a few posts ago!

Even if particles were sets (which, as I mentioned, we can avoid anyway) it is not inconsistent with the axiom of infinity to have the set of particles as a finite set.

So you blew right past my earlier post. I can see at least four possibilities (1) You skip reading a pretty good amount of the posts or (2) You read them but have a comprehension or retention problem or (3) You are nuts or (4) You willfully mangle the conversation for some kind of trolling effect. My guess is that it's a combination of (1) and (2).

whatever grave offense I may have caused — fishfry

I duly note what I take to be your sarcasm. But there's no grave offense. Hardly even offense. More like a feeling that it's too bad that someone who is not altogether uninformed sometimes reads so poorly, reasons so abysmally, and is so characteristically recalcitrant about it.

And you THEN -- after I challenged you on this point -- admitted that "particle" is a primitive, — fishfry

To ZFC, add a 1-place predicate symbol — TonesInDeepFreeze

When we add a symbol without definition, then it is clear that we are adding a primitive.

And I said over and over in various posts, that we add primitives and axioms for physics.

Suppes combines infinitistic set theory with added physics primitives [emphasis added] and a definition of a system of particle mechanics such that the set of particles is finite. — TonesInDeepFreeze

And even if 'particle' weren't meant as a primitive, but were defined, it would not detract from my point of giving you examples of axioms.

A reasonable conversation would be:

F: Is 'particle' primitive or defined?

T: Primitive.

F: Okay, now it's clear.

or

F: Is 'particle' primitive or defined:

T. Defined.

F. Then what is its definition?

T. There is a chain of definitions leading up to it. I can't practically type it all out here. But my point is not to convince you of the Suppes's cogency, but rather just to give an example of an axiom.

Even if I hadn't earlier mentioned 'particle' as added, that would not have been misleading you, since any lack of details can still be supplied on request. There is nothing I posted or didn''t post that was my fault of you misconstruing me.

Alice: C is the case.

Betty: That's wrong [or, That's problematic, or whatever criticism].

Alice: No, it's right, because [fill in S, which is support for C, here].

Betty: You withheld the information S.

Alice: I didn't include S when I said C, but that doesn't justify claiming that C is wrong, especially as It is not the case that C is wrong. Instead, you could have said, "You claimed C, but have not supported it", to which I could respond (1) "Even if I don't personally support it, it is still the case that P" or (2) "Here is the support S".

What you are arguing essentially is that there is something wrong with my arriving at (2). That's illogical.

And anyway, I did mention previously that 'particle' is primitive.

Without that information, your statement that there are only finitely many particles makes no sense. — fishfry

Wrong. You make no sense. Even if I didn't give any information about the set of particles other than it is finite, that doesn't entail that saying that it is finite makes no sense. You are again abysmally illogical.

After you said that, it was clear to me that there could indeed be only finitely many of them without creating a contradiction. — fishfry

Again, the consistency doesn't depend on the predicate being a new primitive. Your notion is ridiculous. Even if the set where made without a new primitive, it still is not inconsistent to have a finite set while other sets are infinite.

Your illogic is stunning.

Can you see that I had to ask you twice in order to dig out your hidden assumption that "particle" is a brand new primitive in set/physics theory? — fishfry

Nothing was "hidden". (1) Even if I didn't mention it at a particular juncture, that doesn't entail that I am "hiding" it. (2) I did mention in previous posts that we add primitives and axioms. (3) In an earlier post, I did mention that we add 'particle' as a predicate, and without giving it a definition; so of course it would be primitive. (4) I did mention that Suppes himself adds primitives and axioms (except with an inessential technical qualification that I explained). So one may allow that it is primitive in his axiom, or if one doesn't want to take that as granted, then one can ask. But there was no misleading you about it.

And you failed to count to to even the number one. When you FIRST asked me about primitives, answered you IMMEDIATLY in the next post:

What's a particle? What's mass?
— fishfry

They are primitives.* — TonesInDeepFreeze

And you make the false claim that you had to ask me twice, not as a causal matter of fact, but rather while claiming that I was "hiding" and you had to "smoke it out". Yeah, you had to "smoke it out" by asking and then receiving my immediate reply. And even IF you had to ask twice that's not so bad really. Meanwhile, so many questions and points I've made to you that you have ignored; you continually make arguments that I rebut and then you ignore the rebuttal but still go on repeating your more basic misconstruing and strawmen. You are bizarre.

If you want to add new primitives to the theory, and you don't bother to tell me that, then it's perfectly understandable that I would have confusion about your meaning. — fishfry

It is perfectly reasonable to say I didn't mention it or to ask about it. It is very unreasonable to claim that I was "hiding" or that you had to "smoke it out" (especially as I answered you about it immediately). And I did mention 'particle' as an added to the language in an earlier post, and I did mention at least a few times that we add primitives and axioms, and one might take from context, in Suppes's formulation that it is primitive or, if not taking it from context, ask before falsely claiming that without the information you are justified to claim there is a contradiction. And, even more basically, even if it were not primitive, then it is still ludicrously illogical to claim there is a contradiction between stating a given set is finite and having the axiom of infinity.

So if you think you have an idea, or if you even claim it's logically possible, the burden is on you to be crystal clear in your thoughts; because nobody in 120 years has axiomatized physics, let alone unified it with set theory, which seems logically contradictory on its face (to me at least). — fishfry

I have been clear. (I noticed today that I botched a formulation in an earlier post, but it is not material to the particular argument we are now having.) Also, what I claimed to be consistent is merely the initial setup of adding a 1-place predicate, and adding that there is a unique set all and only those objects that have the predicate and adding that that set is finite. And it is consistent with ZFC. It's trivial that is consistent. Anyone can trivially see it for themself.

Meanwhile, the first claim on this subject was YOUR claim that the axiom of infinity contradicts physics. You have not supported that claim. And, again, here's what I said about that:

If there is not some formalization of physics in mind, then it is not clear what it means for a formal axiom to be in contradiction with a set of unformalized statements. It might mean that any conceptualization of the meaning of the axiom of infinity is incompatible with the concepts of physics, or something along those lines. I don't make a claim pro or con about such informal senses of 'contradiction', but I am interested in the question whether any possible reasonably sufficient formal axiomatization of physics would entail the negation of the axiom of infinity. If such a theory can't formulate the axiom of infinity in the language of the theory, then, a fortiori, there is not a contradiction with the axiom of infinity, so that would settle the question. On the other hand, if the theory includes set theory or any variant of set theory (such as set theory with urelements) that includes the theorem "there exists an infinite set", then the question is whether it is possible to have a consistent system that combines set theory or such variant with a reasonably sufficient set of physics axioms. (I'll leave tacit henceforth that we might need a variant such as set theory with urelements. But moving to a variant would not vitiate my point, since the variant would still include the "there exists an infinite set", which is the supposed source of inconsistency. Also, I'll take as tacit "reasonably sufficient".)

There are two questions: (1) Can there be a consistent set of axioms for physics? I don't opine, especially since I have no expertise in physics. (2) If there can be an axiomatization for physics that combines with set theory, then would any such axiomatization be inconsistent? My point is that I have not seen an argument that it is not at least plausible that there might be a consistent axiomatization that combines physics with set theory, and that I do think it might be plausible. — TonesInDeepFreeze

I'll concede that point. — fishfry

I am truly curious why you even disputed it to begin with, and then persisted in yet another post. Especially as this is typical with you. You weren't reading correctly? Your weren't reading correctly because you mostly only skim? A mental lapse? A mental lapse because you have a continual preconception that when I disagree with you or question whether your claim is supported that I am bound to be wrong about it?

We tend to talk past each other and I'm content to leave it at that. — fishfry

No, you regularly ignore and misconstrue, sometimes even to the point of posting as if I said the bald negation of what I actually said. Meanwhile, I respond on point to you, and make every reasonable effort not to misconstrue you or mischaracterize your remarks, and I'm happy to correct myself if I did.

That is not an equivalence.

But if you adopt as an axiom claims that are subject to experiment and investigation, your science won't get you very far. — fishfry

Meanwhile, your original point that the axiom of infinity combined with physics is inconsistent has not been sustained. I don't know whether you recognize that now.

Next, as to your new point, perhaps I don't understand what you're saying. Axioms can be interpreted and then the interpretations subjected to experiment, so that either the experiments support the axiom as interpreted or refute the axiom as interpreted, in which case the theory would need to be reformulated, if possible, with different axioms. I don't see a problem with that.

In any case, again (since so much effort was spent to get to this juncture): It has not been shown here that the axiom of infinity is inconsistent with possible axiomatizations of physics.

The claim that "the set of particles is finite" contradicts the axiom of infinity is shockingly wrong.

You haven't convinced me of your point in the least. — fishfry

What possibly could refute that it is consistent that some sets are finite and other sets are infinite?

discussion with you is hopeless.
— TonesInDeepFreeze

Perhaps we can mutually agree on at least this. — fishfry

What I wrote is:

If you can't see that [it is not inconsistent that some sets are finitie and some sets are infinite], then discussion with you is hopeless. — TonesInDeepFreeze

You continually ignore and terribly misconstrue what I write, and now you can't even see that the finititue of a particular set does not contradict the axiom of infinity. So if you persist that way, then discussion with you is pointless.

The set of particles is finite.
— TonesInDeepFreeze

Contradicts axiom of infinity. — fishfry

No it does not. If you can't see that, then discussion with you is hopeless.

That some sets are finite does not contradict that some sets are infinite.

What's a particle? What's mass? — fishfry

They are primitives.*

Everything's defined in terms of a single primitive, ∈ — fishfry

No, we add primitives for physics. I covered that very clearly in previous posts. You are just skipping the explanations given you.

I did not say that physics can be formulated using only the axioms of set theory. I explicitly said that we take the axioms of set theory and add primitives

* Technical note: Strictly speaking, Suppes doesn't add primitives, but instead he defines a 'system for particle mechanics' as a tuple with certain properties. The tuple is a structure or sometimes called a 'system', in the same way as an algebraic structure or other structures in mathematics and science. Anyway, this is not an essential difference from adding primitives. His definition of a certain kind of structure can be easily transformed into adding primitives. Moreover, defining a certain a kind of tuple adds even less to set theory. Moreover, the physics axioms also could be conveyed instead as properties of the structures.

You have ignored and outrageously misconstrued what I wrote, yet again. I didn't want to comment on the discussion itself again, but your reading confusions, as seen in this thread and other threads, are quite remarkable.

I'm going to address your question. But I have some other remarks first, and I'll quote from some earlier posts that are important still in the context of this post.

My main interest is this claim:

The axiom of infinity is in contradiction with known physics — fishfry

If there is not some formalization of physics in mind, then it is not clear what it means for a formal axiom to be in contradiction with a set of unformalized statements. It might mean that any conceptualization of the meaning of the axiom of infinity is incompatible with the concepts of physics, or something along those lines. I don't make a claim pro or con about such informal senses of 'contradiction', but I am interested in the question whether any possible reasonably sufficient formal axiomatization of physics would entail the negation of the axiom of infinity. If such a theory can't formulate the axiom of infinity in the language of the theory, then, a fortiori, there is not a contradiction with the axiom of infinity, so that would settle the question. On the other hand, if the theory includes set theory or any variant of set theory (such as set theory with urelements) that includes the theorem "there exists an infinite set", then the question is whether it is possible to have a consistent system that combines set theory or such variant with a reasonably sufficient set of physics axioms. (I'll leave tacit henceforth that we might need a variant such as set theory with urelements. But moving to a variant would not vitiate my point, since the variant would still include the "there exists an infinite set", which is the supposed source of inconsistency. Also, I'll take as tacit "reasonably sufficient".)

There are two questions: (1) Can there be a consistent set of axioms for physics? I don't opine, especially since I have no expertise in physics. (2) If there can be an axiomatization for physics that combines with set theory, then would any such axiomatization be inconsistent? My point is that I have not seen an argument that it is not at least plausible that there might be a consistent axiomatization that combines physics with set theory, and that I do think it might be plausible.

Some earlier points bear on the question and need to be kept in mind:

The axiom of infinity is inconsistent with known physics since there is no principle of modern physics that stipulates the existence of any infinite set,
— fishfry

That doesn't entail inconsistency. Just because a theory doesn't have a certain principle doesn't entail that adding that principle causes inconsistency. But if physics had a principle that it is not the case that there exists an infinite set, then yes, there would be inconsistency. But even if physics had a principle that there are not infinitely many particles, that is not itself inconsistent with the existence of infinite sets, such as infinite sets of numbers if numbers are not axiomatized to be particles. — TonesInDeepFreeze

contemporary physics can not accept the axiom of infinity as a physical principle.
— fishfry

I never said that it would be a physical principle. It would be a mathematical theorem to which are added primitives and axioms for theorems of physics. — TonesInDeepFreeze

Physics has not been axiomatized at all. — fishfry

I don't know that there are not axiomatizations of any part of physics or even of a large part of it. At the very least, Suppes provides an axiomatization of particle mechanics. Granted, that's not an axiomatization of modern physics. But at least the question of infinity is addressed, as Suppes combines infinitistic set theory with added physics primitives and a definition of a system of particle mechanics such that the set of particles is finite. I was along those lines when I said that the existence of infinite sets is not inconsistent with having a finite set of particles. Also, not claiming an axiomatization of physics, but arguing for the plausibility that certain questions in physics might be affected by set theory: http://logic.harvard.edu/EFI_Magidor.pdf at page 10.

/

What kind of axiom would we add to set theory that would be an axiom for physics? — fishfry

I will not suggest any particular axioms, as I am not expert in physics. It is better anyway that anyone may nominate, at their own will, any postulates of physics (or any formulas of physics deemed fundamental and productive enough) for axioms. These can be, for example, the postulates of special relativity. (Nota bene: Again, I am not claiming that this project would be successful. I am saying only that it might be plausible.) Examples from Suppes for particle mechanics:

The set of particles is finite.

The mass of a particle is a positive real number.

For particles p and q, and elapsed time t, the force of (p q t) = - the force of (q p t).

I won't defend any particular formulations from possible criticism. The particulars are not my point, on an assumption that any details that raise objection could be adjusted to suit whatever formulation the physicist more prefers.

Yes, I deleted my reply because immediately after posting it, I read about the Euler and Riemann sums.

It does NOT follow that physics uses or is formalized by infinite sets. — fishfry

I didn't say it does. And I am not saying that it would be consistent if it did. I am saying that I haven't seen an argument that it would be inconsistent if we added to set theory, primitives and axioms for physics, and that it seems plausible that we might be able to do so.

Did you think I argued that physics uses or is formalized by infinite sets? If you did, then that would be yet another example, from different threads, in which you read into what I posted claims that I did not make in those posts.

there's no ontological commitment in physics to infinite sets. — fishfry

And using set theory doesn't entail that physics would take on an ontological commitment. Again, I am asking about a possible axiomatization. That is syntactical. Consistency is syntactical.

I think I might have had an incorrect premise that modal semantics evaluates truth in these stages: first per world and then per model. Rather, perhaps I can answer my own questions if I dispense with that premise and view semantics as done "top-down" for the model overall. That premise disallowed me from better understanding your posts. I'm going go back to studies I had forgotten a long time ago to see whether I can correct myself now. (If you have a textbook to recommend, then it would be appreciated.)

I see now that I face an obstacle in talking about constant symbols and terms with you. That obstacle is that I am using Hughes & Cresswell, but their quantified modal system has no constants, no terms other than the variables themselves, and '=' is introduced only in a later chapter. Their language has only: universal quantifier (with existential quantifier defined), individual variables, sentential connectives, relation symbols, and the necessity operator.

Of course, that is enough, since in theories, constants and operation symbols can be defined from relation symbols.

But it makes it difficult for me to attend to the details vis-a-vis your remarks, because I lack a reference for how terms are dealt with semantically in a definitive textbook.

So, which textbook guides you the most in this subject? I can see whether I can get it cheap enough.

if a = b in w, then in all w', a = b. — Snakes Alive

I take that mean that the letters 'a' and 'b' are variables in the meta-language and not constant symbols in the object language. And with that, the above sentence seems to be the way we should understand the semantics.

∃x[x = a], to mean 'a exists,' — Snakes Alive

But there 'a' is used as constant symbol in the object language.

For me, 'Ex x=a' is just a trivial theorem of identity theory. For any term t, we have the theorem:

Ex x=t

Then if you have '∃' quantifying over the domain of individuals, independent of the domain of worlds, then it will have the same value at any world – either it will be necessarily true, or necessarily false. — Snakes Alive

As a theorem of identity theory, isn't it true in all models for a language for quantified+identity modal logic?

'∃x[x = a]' is true in w iff there is an individual x in the sub-domain associated with w that is identical to a. — Snakes Alive

I don't see how you can use 'a' as a symbol in the object language when you write 'Ex x=a', but also as a symbol in the meta-language when you talk about 'a being in a domain'. Also with 'x', but there I can reword in my own mind to make it work, while, in this particular situation, I can't do that with the way 'a' is being used, and as I don't think this is merely pedantic but rather it confuses me as to what really is being said.

[EDIT: Disregard this post, except the last quote and my response to it. I think I had a misconception in this matter.]

Across worlds, not models. A model has a set of worlds, in its frame. — Snakes Alive

Right, my mistake, we're talking about modal logic. But make the correct substitution, and my remarks still pertain both to domains for models with predicate logic and to the domains for the worlds in models for modal logic.

What do you mean, 'ordinary?' — Snakes Alive

The method of semantics for first order languages as described in any textbook in mathematical logic (or as they describe methods with inessential differences). And am I wrong that also the most basic methods for modal logic in textbooks follow suit? Doesn't basic modal logic stipulate worlds recursively, based first on the ordinary predicate logic clauses and adding the clause for the modal operator?

Obviously in a strict sense you cannot reconcile them, since ordinary predicate logic has no notion of a constant that doesn't refer to anything. There's just a domain, and then the interpretation function maps each constant to a member of that domain. — Snakes Alive

Exactly. But isn't that method also used in modal predicate logic too for worlds?

So if 'b' refers to Bob, then it might refer to Bob in all worlds where Bob exists, but to * in all worlds in which he doesn't exist. — Snakes Alive

Then there has to be an object that is in the intersection of the domains of all worlds. That seems to be a big requirement. Also, there are domains that might have nothing to do with the intuitive intent.

{0} and {1}. Two domains and empty intersection between them. And a constant symbol 'c' with no intuitive referent. I don't see how what you describe is supposed to work.

[EDIT: The following is okay but perhaps no longer relevant anyway.]

No, it just requires some distinguished object in the domain, like say *. — Snakes Alive

I was referring to the syntactical side of things for definite descriptions, not to your semantical method.

I can specify world in which Donovan doesn't exist — Banno

I'm not addressing philosophy of language or more advanced modal logic, but in ordinary predicate logic, 'Donovan' is a plain name (essentially, it's a constant symbol). An interpretation of the language assigns a member of the universe to the name.

That is different from a definite description. In some theories there might or not be a theorem "There exists a unique individual that wrote the song "Mellow Yellow"". If we don't have that theorem, then see my post above as to what we can do about the definite description "The unique individual who wrote "Mellow Yellow".

(I'm assuming you mean the Scottish singer and not the main character in the movie 'Donovan's Reef' nor the main character in the TV show 'Ray Donovan'.)

I seem to be again on a different page from you.

he point of a non-logical constant is that its value is invariant across worlds. — Snakes Alive

I'm referring to constant symbols. It is not the case that the point is to have the value for a constant symbol to be invariant across models. A model assigns to each constant some member of the universe of the model. There is no requirement that all models agree on what they assign to the constant symbol. Not all models have the same universe, so it's not even possible that they all agree on what they assign to a constant symbol.

You could, of course, have a modal logic where individual terms like constants have different denotations relative to different worlds. — Snakes Alive

Yes, that is what I imagine is the default.

And you could then allow that they refer to 'nothing,' say, at worlds where the relevant individual doesn't exist. — Snakes Alive

That I don't understand. As I explained, if we fail to assign a member of the universe to a constant symbol, then the methods of evaluation for satisfaction and truth for formulas and sentences per a model falls apart.

And I don't know what 'the relevant individual' refers to in your remark. For domains with cardinality greater than 1, there is no particular individual that must be mapped to from a constant symbol. But every constant symbol must map to some member of the domain.

How you want to represent this formally is up to you – one old formal trick is to use a dummy object, say *, to which the value of all terms that have nothing satisfying them at the world map to. — Snakes Alive

That seems to me to be about a separate question.

That seems to be a semantical version of the syntactical Fregean method of "the scapegoat" for failed definite descriptions. I understand that method for handing conditional definitions of constant symbols (and can be expanded to operation symbols). This handles those definite descriptions that fail because either the existence or uniqueness condition fails. The method requires a theory in which at least one constant symbol 's' is either primitive or already defined:

If we want a definition

c = the_unique x P

but have not derived the theorem E!xp

then we revise to

(If E!xP -> c = the_unique x P) & (~E!xP -> c = s).

You would then need to make a semantics that deals with the dummy object — Snakes Alive

But the problem of failed definite descriptions needs to be dealt with syntactically first (as I did above) or we cannot ensure that definitions uphold the criteria of eliminability.

Anyway, the question I have is not what happens when a definite description fails, but rather, how do we reconcile ordinary semantics for either predicate logic or modal predicate logic with dangling non-denotating constants?

As I understand, you suggest perhaps marking any formula with such a constant as "not satisfied" or with a third truth value, or some other accommodation we would stipulate. But that would mess up the whole context of addressing the first post in this thread and its corollary questions.

Not to say you couldn't construct such a notion, of unique existence — Snakes Alive

Sure, but I wonder what it would be.

Do you have any thoughts on my question: How can we have a method of models in which, for certain models, there are constant symbols such that the model does not assign a member of the domain of the model? It throws off the way we evaluate satisfaction and truth in models.

Quantifiers are predicates of formulae. — Snakes Alive

That's not a usage I have happened to have seen. But that doesn't in itself disqualify it.

But at least now it is clear that 'E' as a predicate is a very different kind of animal from 'E!' as a quantifier, which was not clear, except in stages, in your previous explanations. Anyway, I do appreciate your explanations in general; they have helped. Thank you.

E!v[p] is true at w iff there is exactly one individual x in the domain such that p is true at w on any assignment that maps v to x — Snakes Alive

That is the ordinary uniqueness quantifier, not a predicate.

So, thus far, we have:

'E' as a predicate symbol before a term: where 't' is a term, Et.

'E' as the existential quantifier: where 'x' is a variable and P is a formula, ExP.

'E!' as a uniqueness quantifier, where 'x' is a variable and P is a formula, E!xP.

But not yet another thing that is a uniqueness predicate (whatever that would mean).

The unique existence predicate is second-order — Snakes Alive

Where can I read more about a uniqueness predicate in a second order modal logic?

it has a different syntax, and occurs alongside a variable and a formula — Snakes Alive

I understand a uniqueness quantification symbol that is followed by a variable and then a formula: E!xP.

But I am not familiar with a syntax, even in second order logic, that has predicate symbol preceding a variable then a formula. Where can I read about that?

It's like an existential quantifier, except only one individual in the domain is allowed to satisfy the formula. — Snakes Alive

The uniqueness quantifier works that way. But where I can see a specification of the syntax and semantics of a second order modal logic that has a uniqueness predicate?

Sometimes, the exclamation point signals unique existence — Snakes Alive

Then that opens yet another can of worms. I understand what it means for there to be a unique individual having a certain property: E!xP. And in this thread I'm starting to understand an existence predicate: Ea. And I could understand writing 'E!a' to mean the same as 'Ea'. But I don't know what it would mean to say "individual a has unique existence and not merely existence". What would be the semantics for that? (And then I'd have to see what modifications for the logic would be required.)

Hughes & Cresswell uses just 'E', not 'E!'.

And the semantics for 'E' are fixed (in the manner described by Snakes Alive). (I would say that the semantics for 'E' is fixed in the same sense of 'fixed' when we say the semantics for '=' is fixed.)

And 'E' is not just a predicate symbol with no supporting axioms. Rather Hughes & Cresswell describes a logic system that modifies first order logic and also has axioms mentioning the special primitive predicate symbol 'E'.

Also, Hughes & Cresswell addresses the question I asked about how a model can excuse a constant from having a referent in the domain of the model. I'll understand better as I study more carefully.

Then I understand it this way: It's a primitive symbol, but there are no special logical axioms for it, and it can be interpreted differently in different models. But we are particularly interested in those interpretations in which it is interpreted as you described.

But that raises the question: What would be axioms that would entail that any model of the theory evaluates E!a as true only in the way you described?

But, even more basically, in ordinary predicate logic, for any given model for the language, the referent of a variable or constant exists in the domain for that model. That follows from the definition of 'model for the language'. So I don't understand how that can be different with interpretations of a modal language.

PS. I do find discussion about an existence predicate in Hughes & Cresswell. If explanation is too complicated for the confines of posts, I'll try to figure it out from that textbook.