A poster was quoted, "Possibility is defined as "not necessary", and something is necessary if it is true in all possible worlds."

It should not be overlooked that he reformulated that to this (I'm using 'P' for 'possibly' and 'N' for necessarily'):

""possible" is defined as "not necessarily not": Pq <-> ~N~q". Therefore if something is true then it is possibly true: q -> Pq."

The definition there is correct. And "q -> Pq" is correct, but not merely from the definition but from axioms.

/

Another poster claimed that defining "possibly P" as "not necessarily not P" is circular because "necessarily" is defined in terms of possible worlds. No, "necessarily" is not defined at all; it is primitive. Moreover, "possible worlds" is semantic and is not involved in syntactical definitions. Moreover, while words such as "possible worlds" suggest intuitive motivations, mention of "possible worlds" is not needed for the semantics, as the semantics can be given in full formality without nicknames such as "possible worlds".

/

I don't know any person who would say this:

"Possibly the book is in the room. So the book is not in the room."

If someone told me that, then I would consider them incapable of coherent conversation and incapable of shedding any light on where the book is or might be.

/

Semantically (a simplified chart):

q is true or not true (but not both) in any given model ("world")

q is necessary iff q is true in every model

q is possible iff q is true in at least one model

q is contingent iff (q is true in at least one model and false in at least one model)

q is actually true iff (there is a certain model designated as the "actual model" and q is true in that model)

("the actual model" may refer to the world of observable facts or whatever explanation one would like to give for a notion of the "real world", "actual world", etc. When context is clear, we just say 'q is true'.)

I should have said, "we move to modal logic of some other appropriate system more involved than mere first order logic". Of course there is no limitation on systems and semantics that may be devised.

For treatment of the existence predicate in modal logic, see the common textbook, Hughes & Cresswell.

"Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze

That makes it appear that I said that we can't expect that posts have good outcomes, etc.

But what I posted:

Posters don't ordinarily think "Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze

That is to say that I don't expect that ordinarily posters ask those questions before posting.

In acting virtuously, virtuous action becomes habit — Kuro

I understand that view.

'utterance' means speaking out loud. Or do you have a different sense in mind?

Yes, "G |= F" means G semantically entails F; and "G |- F" means G proves F.

But, due to completeness and soundness, G |= F iff G |-F.

So you don't advance any point by switching from one to the other mid-proof.

For that matter, due to the deduction theorem, you only need the implication sign, not any turnstile.

/

Depending on the context, 'proposition' stands for something different from 'sentence'. But you use 'p' for a sentence (you negate it, so it's a sentence). I don't see how one would figure out anything about platonism or anti-realism from your argument.

T(q) ≔ q is a true proposition
P(q) ≔ q is a proposition — Michael

You didn't use them in the proof.

The semantic turnstile as opposed to the proof turnstile is not important in this context. You don't even need any turnstile.

I did say that "maybe it's simpler to just understand T(q) as 'q is a true proposition'." — Michael

Yes, after I audited both your original and revised arguments. Of course, I have no problem with emending your argument again now.

Since "proposition" and "true proposition" are not in your argument itself, this would work:

1. Tq <-> p ... premise
2. Ax(Tx -> Px) ... premise
3. Tq -> ExTx
4. ExTx -> ExPx
4. p -> ExPx
5. ~ExPx -> ~p

That's all fine, but the more general point I mentioned is that we need to move to modal logic to have existence as a predicate.

That some x is true semantically entails that some x is a proposition, given that truth is predicated of (and only of) propositions. — Michael

Truth is semantic. My point is that you are missing the premise:

Ax(Tx -> Px)

Without claiming that you do or don't commit to the example as being of a substantive theory (though literally you did say that, it's reasonable to take you now as clarifying that you didn't mean it literally), I asked, "Is that a proposed formulation somewhere?" and went on to say why it doesn't work.

My mistake: The logic is not correct. Line 3 (whether original or reviesd) is a non sequitur.

1. Tq <-> p ... premise
2. Tq -> ExTx ... EG
3. ExTx -> ExPx ... non sequitur (prob you have an unstated premise in mind)
4. p -> ExPx ... sentential logic, but relies on non sequitur in step 3
5. ~ExPx -> ~p ... sentential logic, but relies on non sequitur in step 3

'7+5=12' is true iff '7+5' is a theorem

is the case because both sides of the biconditional are true.

But that is not an instance of a substantial theory of truth.

There is one premise there:

Tq <-> p

Following that, I don't see a problem with the logic. But you use vacuous quantification with

ExTq

and

ExPq

So, though there is no mistake in the logic, I don't see any point in it.

Of course it is.

But in any "adequate" system, there are statements such that neither the statement nor its negation is derivable. So derivability doesn't work for defining 'is true'.

more substantial theories of truth [...] “7 + 5 = 12” is true iff “7 + 5 = 12” follows from the axioms of maths — Michael

Is that a proposed formulation somewhere?

It doesn't work in ordinary mathematics. A sentence is either true or false but not both. And a sentence is true if and only if its negation is false. But with our ordinary mathematical axiomatizations, there are sentences such that neither the sentence nor its negation are derivable.

I added that modal logic is the main arena for this.

Yes, of course, we can do that.

[You know all this; I'm writing it for benefit of those who don't:]

You can have the predicate 'is John', which is something different from just the name 'John'.

Simplest example from mathematics:

'0' is an operation symbol, defined

0 = x <-> Ay ~yex.

But we don't write:

Ex 0

And we don't write:

Ex x

They are not well formed. A quantifier is concatenated with a formula, and a mere term is not a formula.

Meanwhile, for any term T whatsoever, it's a logical theorem:

Ex x = T

which includes:

Ex x = x

To put it another way, there is not "existence predicate" in predicate logic. We go to modal logic for that.

But we could define a predicate symbol:

Mx <-> x = 0

And we can say:

ExMx

Classical logic is 2.5k years old — Agent Smith

Classical logic is about 140 years old.

Yet again you shoot your mouth off not knowing what you're talking about.

what logic am I using when I say that if John is bald then John exists? — Michael

"John exists" is not expressed in mere predicate logic. You need modal logic for it.

Yes, since Tarski's context is formal, he is not opining on the intensional senses.

Back to the point:

"snow" denotes snow because "snow" denotes snow — RussellA

That unedifying tautology has nothing to do with understanding Tarski.

One poster coemmented that P is true IFF P is a definition — Agent Smith

No, I did not.

I said:

'P' is true iff P

is a definition of 'is true'.

Leaving off the quote marks is ruinous.

As far as I can tell, you are unfamiliar with use-mention.

this is a chicken-and-egg situation — Agent Smith

Saying "''P' is true iff P" does not preclude that one cannot investigate P without first establishing that 'P' is true: (1) Establish P (such as looking at snow), then (2) infer 'P' is true. That is not circular.

If you actually READ Tarski's papers, you would see how he goes on to explicate that.

I'm not inferring that you are claiming that Tarski made that comment. I'm pointing out that the comment is irrelevant to understanding Tarski.

Meanwhile, it would be helpful if you'd recognize the 15 instances of 'truth' or 'true'. And take note that 'is' is not a standalone denoter in this context.

I asked previously, what textbook in mathematical logic is your main reference for the subject of mathematical logic. The context of Tarski here is overwhelmingly mathematical logic.

"is" names is — RussellA

That's not how it works in the context of Tarski. Rather, 'is white' is an undivided unit.

"snow" denotes snow because "snow" denotes snow. — RussellA

Tarski mentioned no such pointless tautology.

The statement P is true IFF P. — Agent Smith

Where did Tarski write that?

The quote marks are crucial.

'P' is true iff P.

Doesn't this lead to a chicken-and-egg situation? — Agent Smith

It's a biconditional. Formal definitions are biconditionals. He's not saying how we know that 'snow is white' is true. He's only giving a definition of 'is true'.

Moreover, if you want to know that 'snow is white' is true, then you can look at snow to find out.

The mere fact that a formulation is a biconditional doesn't entail that we can't move from right to left (or from left to right) to make our inferences.

You really really need to read a book on the subject of formal logic. Otherwise, you will continue to incessantly spin out in your own confusions.

I would say logical rather than mathematical or scientific. — RussellA

First, my point stands that you said the word 'true' is not in the paragraph you quoted, but you intentionally ignored that the words 'truth' or 'true' are mentioned 15 times in the surrounding paragraphs. My point stands: He was talking about 'truth' having different meanings to different people, not about 'denotation' having different meanings. One might think that, for purpose of clear communication, you would admit this now. Instead, where I gave you the actual quoted context, with 15 mentions of 'truth' or 'true', you just go on to post as if it doesn't exist.

About formal languages, mathematics, and the sciences, read the papers! And read the IEP and SEP articles about them. It is a plain fact that Tarski's overriding concern is with languages for mathematics and the sciences. It's throughout the papers.

it is clear from his article that his definition of "true" is more relevant to the formal language of linguistics than either mathematics or science. — RussellA

No, it is overwhelmingly clear that his primary concern is formal languages for mathematics and the sciences. Tarski takes pains to point out that we can't expect to nail the definition of 'true' for informal contexts and that his proposal is directed at mathematics and the sciences. Moreover, a great amount of the paper goes on to actual mathematical formulations of the definition of 'true'. The papers are steeped in it. The primary area of concern is model theory, which is a subject of mathematical logic. Man, you need to read the papers, and for even more much needed background, get a textbook in mathematical logic so that you can appreciate the fruit of Tarski's work.

The word "denote" may be used in different ways, but as there is no substantial difference in meaning between the "ordinary" sense of the word "denote" and a formal sense of the word "denote", he cannot have moved from an "ordinary" sense to a formal sense. — RussellA

But he did! He showed the method of models, mathematically. Read the papers!

My points stand: (1) Tarski's overriding concern is with formal languages for mathematics and the sciences. (2) In the passage you quoted, Tarski is not about the fact that 'denotation' has different senses but about the fact that 'truth' ('true' also) has different senses. Either merely blithely or dishonestly, you misrepresented Tarski with your original 1-4 decoction, and then again by choosing just one paragraph from the middle of the other paragraphs. (3) Whether or sentence is analytic or not is not part of Tarski's concern in the particular regard of his schema.

But you will keep replying with your confused, uninformed and dogmatic misinterpretations. I'm running out of time to keep correcting you.

The complete paragraph containing item 4) is:
It seems to me obvious that the only rational approach to such problems would be the
following: We should reconcile ourselves with the fact that we are confronted, not with one
concept, but with several different concepts which are denoted by one word; we should try to make these concepts as clear as possible (by means of definition, or of an axiomatic procedure, or in some other way); to avoid further confusions, we should agree to use different terms for different concepts; and then we may proceed to a quiet and systematic study of all concepts involved, which will exhibit their main properties and mutual relations.

How is this paragraph about the meaning of true. The word "true" isn't mentioned. ? — RussellA

It is remarkable that you say that. It is quite an example of willfully ignoring a mass of context. 'truth' or 'true' are mentioned 15 times in the section in which that paragraph is in the middle.

[begin quote; bold and italics added]

14. IS THE SEMANTIC CONCEPTION OF TRUTH THE "RIGHT" ONE?

***The subject of this section is whether the semantic conception of truth is the only right one.

I should like to begin the polemical part of the paper with some general remarks.

I hope nothing which is said here will be interpreted as a claim that the semantic conception of truth is the "right" or indeed the "only possible" one. I do not have the slightest intention to contribute in any way to those endless, often violent discussions on the subject: "What is the right conception of truth?" I must confess I do not understand what is at stake in such disputes; for the problem itself is so vague that no definite solution is possible. In fact, it seems to me that the sense in which the phrase "the right conception" is used has never been made clear. In most cases one gets the impression that the phrase is used in an almost mystical sense based upon the belief that every word has only one "real" meaning (a kind of Platonic or Aristotelian idea), and that all the competing conceptions really attempt to catch hold of this one meaning; since, however, they contradict each other, only one attempt can be successful, and hence only one conception is the "right" one.

*** We're not getting in arguments premised on the view that there is only one right concept of truth.

Disputes of this type are by no means restricted to the notion of truth. They occur in all domains where - instead of an exact, scientific terminology - common language with its vagueness and ambiguity is used; and they are always meaningless, and therefore in vain.

*** The concept of truth is not the only subject over which there are such arguments. When the subject matter is not exact enough, there are such arguments.

It seems to me obvious that the only rational approach to such problems would be the following: We should reconcile ourselves with the fact that we are confronted, not with one concept, but with several different concepts which are denoted by one word; we should try to make these concepts as clear as possible (by means of definition, or of an axiomatic procedure, or in some other way); to avoid further confusions, we should agree to use different terms for different concepts; and then we may proceed to a quiet and systematic study of all concepts involved, which will exhibit their main properties and mutual relations.

*** In situations where a word is used to denote more than one concept, we should agree to use different words.

Referring specifically to the notion of truth, it is undoubtedly the case that in philosophical discussions - and perhaps also in everyday usage - some incipient conceptions of this notion can be found that differ essentially from the classical one (of which the semantic conception is but a modernized form). In fact, various conceptions of this sort have been discussed in the literature, for instance, the pragmatic conception, the coherence theory, etc.

*** There have been a lot of different concepts of truth.

It seems to me that none of these conceptions have been put so far in an intelligible and unequivocal form. This may change, however; a time may come when we find ourselves confronted with several incompatible, but equally clear and precise, conceptions of truth. It will then become necessary to abandon the ambiguous usage of the word "true," and to introduce several terms instead, each to denote a different notion. Personally, I should not feel hurt if a future world congress of the "theoreticians of truth" should decide - by a majority of votes - to reserve the word "true" for one of the non-classical conceptions, and should suggest another word, say, "frue," for the conception considered here. But I cannot imagine that anybody could present cogent arguments to the effect that the semantic conception is "wrong" and should be entirely abandoned.

*** There could be competing concepts of truth that are all just as rigorous. In that case, we would oblige by using a word other than 'truth'.

[end quote]

Yes, the concept of denote itself is one of those that has differing views. But that's not Tarski's point. Rather his point is that, among concepts having differing views, the concept of truth is in particular one of them. The section is not about what we mean by 'denote' but about the concept of truth and the word 'truth' and the fact that the word 'truth' denotes different things for different people. Especially, Tarski is not at all saying he uses the word 'denote' in different ways.

/

I have never said that Tarski was concerned with literary criticism. — RussellA

Yes, and I didn't say that you did. It's my point that he wasn't discussing literary criticism; and dragging Umberto Eco into this is quite aside understanding Tarski.

Within the article he wrote:
Semantics is a discipline which, speaking loosely, deals with certain relations between
expressions of a language and the objects (or "states of affairs") "referred to" by those expressions. As typical examples of semantic concepts we may mention the concepts of
designation, satisfaction, and definition as these occur in the following examples:
the expression "the father of his country" designates (denotes) George Washington; snow satisfies the sentential function (the condition) "2 is white"; the equation "2 . x = 1" defines (uniquely determines) the number 1/2. — RussellA

Yes, and 'denotes' there is in the sense you've been told about in this thread.

I haven't said that Tarski was not concerned with mathematical logic. I pointed out that Tarski had a concern with the semantic conception of truth, and the semantic conception of truth is not the same as the mathematical conception of truth. — RussellA

Wow. You miss the very central point of his articles. Tarski is concerned with providing a rigorous mathematical formulation of the adjective 'is true'. Read the articles.

Are you saying that the ordinary sense of the word "denote" is the mathematical sense of the word "denote" ? — RussellA

When Tarski is using everyday examples, he uses the everyday sense of 'denote'. Then he goes on to make it even more rigorous mathematically.

It is a simple idea until one considers how "a unicorn" maps to a unicorn, or "beauty" maps to beauty. — RussellA

Tarski talks about the fact that in everyday situations we don't have precision. He says that he does not claim to provide an explication of the concept of truth that can withstand all the vagaries of natural language. Nor does he claim to explicate the notion of 'denote' that can withstand whatever disagreements there may be among different settings.

I am pointing out, as Umberto Eco pointed out, that the meaning of "denote" is far more complex than as used in the ordinary sense of "a cat" denotes a cat. — RussellA

Of course it can be as complex as one wants it to be. But Tarski starts with an ordinary intuitive sense and then goes on to pin it down for more rigorous contexts. Read his articles.

/

In sum:

(1) Tarski's overriding concern is with defining 'is true' in context of formal languages for mathematics and the sciences.

(2) He uses an ordinary sense of 'denote' (or cognates of 'denote), but then moves on to instead specify the method of formal modals, where 'denote' is subsumed by certain kinds of functions from linguistic objects to model theoretic objects. This is the movement from informal semantics to formal semantics that Tarski provides.

(3) Whether 'snow is white' is analytic or not is not part of Tarski's concern in the two articles. Moreover, as pointed out: Whether 'snow is white' is analytic depends on which definition of 'snow' we're looking at. There are common enough definitions in with 'white' is not in the differentia.

I can't help but think that the current "rigorous systemization of mathematics for the science that does not abide by thesis C" is inconsistent. I cannot prove that formally, but I can discuss the infinity paradoxes. — keystone

Here is my careful philosophical response: Bull.

All you did there is argument by mere repeated assertion, as you ignored the rebuttals I've already given.

What in the world? Your comment about Peano systems is ludicrously ignorant. So I corrected you. There's no "looping back" by me.
— TonesInDeepFreeze

I'm not in a position to argue that Peano systems are inconsistent so I'd like to set this aside for now.
— keystone — keystone

Yet you went on to ignorantly argue about it!:

I can accept it as an algorithm for generating the set, but not as a completed set....but we've been here before.... — keystone

You don't get say that you don't want to argue about it, then argue about it anyway, then blame your interlocutor from replying. Not on this planet at least.

Hey, I'm all for you dropping whatever you want to drop. Saves me from cleaning up your messes. But "setting aside" doesn't Orwellian mean "I continue to remark on it, then say I want to set it aside, then if you reply you're "looping" back"."

I also don't think you want to discuss the intuition further since it's not formal — keystone

Wrong and contradicts what I've said about that, and contradicts the fact that I (unlike you) DO read in the philosophy of mathematics about different intuitions, including the range of finitism. And I( (unlike you) DO read about formalizations of those different intuitions.

I have nothing against expressing intuitions. But you do more than that. You compare your intuitions with set theory, including criticism of set theory, but you don't know anything about set theory so your criticisms are woefully ill-premised. That's where I come in and say, "Whoa whoa whoa there, pardner, it's fine to have criticisms of set theory, but you better know what you're talking about."

Look at the history, I've tried to end the discussion on this point but you keep looping it back it! — keystone

What in the world? Your comment about Peano systems was ludicrously ignorant and so wrong it's not even coherent. So I corrected you. There's no "looping back" by me.

I'm going to let you have the last word on this point for now since I'm trying to keep the discussion focused on Thomson's lamp and continuous objects vs. points. — keystone

Well, you asked me.

And, very likely you'll have the last word anyway, since I really am out of time for a while.

So do you agree that the = in an infinite summation means something different than the = in a finite summation? — keystone

Absolutely I do not agree.

'=' stands for equality. Period.

So, now I have to give you a free lesson from the first chapter of Calculus 1.

The infinite sum here is:

Let f(0) = 1. Let f(k+1) = f(k)/2

df. SUM[k e N] f(k) = the lim of f(k) [k e N]

thm. the lim of f(k) [k e N] = 2

/

Did you not take Calculus 1 when you were in school?

I can accept it as an algorithm for generating the set, but not as a completed set — keystone

It's not an algorithm. It's not "generating" a set. It is a certain tuple.

It's amazing to me that cranks are FULL of criticisms to mathematics but they know nothing about it!

It seems that this fundamental particle of set theory needs to be defined then. — keystone

From the axioms, we prove that there is a unique x such that x has no members:

E!xAy ~y e x

Then we define :

0 = x <-> Ay ~ y e x.

And, informally, we nickname that "the empty set".

Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).
— TonesInDeepFreeze

Ok, I get what you're saying now. — keystone

Mark that as one of the very rare times a light goes on in a crank's mind. Alas, though, even when it happens, the crank will later double back to commit the error again.

also cannot say that 1+1/2+1/4+1/8+... = 2 — keystone

WRONG. Yours is the typical claim of someone who knows not even the first week of Calculus 1.

An infinite summation is a LIMIT, not a final term in the sequence, as the sequence has no final term.

This is at the very heart of the paradox puzzles. The final state of the lamp is NOT a limit. There is no convergence between alternating "On" and "Off". So not only is there not a final state, but there's not even a limit of the sequence. The mathematics can't help, because the "realm" is impossible even on its own.

On the other hand, with Zeno's puzzle, the mathematics can offer that there is a limit to the sequence, thus infinite summation IS defiined.

Why don't you learn at least the first chapter in a Calculus 1 book?

I'm not in a position to argue that Peano systems are inconsistent — keystone

In that context, I don't mean 'system' in the sense of axioms and a theory. I mean it in the sense of a tuple with a carrier set with a distinguished object and an operation, like an algebra. In that sense, 'consistency' or 'inconsistency' do not even apply.

if ZFC is inconsistent then you can prove that infinite sets are empty and you can prove that infinite sets are infinite? — keystone

Of course.

nested sets of sets containing no objects — keystone

There is only one set that has no members. That it is called a 'set' is extraneous to the formal theory. The formal theory doesn't even need to mention the word 'set'. We could just as well say "the object that has no members". And we don't even have to say "object". We could just say "There is unique x such that x has no members".

If I stop using the word continuum and instead say that a ruler is a continuous object does that sit better with you? — keystone

'continuous' is defined in mathematics. I don't know what you mean by it.

Does that mean that the table (and your set theoretic description) only describes the state of the lamp as time approaches 2? — keystone

That's close enough to what I already said. But, again, keep in mind, the mathematics does not describe the imaginary scenario in every respect - indeed, because the imaginary scenario is not coherent while the mathematics is consistent.

I know it's not physically possible due to the physical limitations related to flicking a switch but I think we can set that detail aside. — keystone

That's not even the issue. The problem is as I mentioned in my remarks.

Set Theory fails to provide mathematics for the 'sciences' of this fictitious realm. — keystone

There is only an incoherent description of something that can't even be a fictitious or abstract model of anything, because it can't be the case that there is a final state that is a successor state where, for each state, there is a successor state.

Especially a finitist would see that immediately. For a finitist there is no such realm, and for an infinitist too.

I'm out of time, probably for a while.

But one more tidbit.

Some remarks here made my wonder how (using only first order PA (and theorems I already know proven in first order PA)) to prove:

There are no natural numbers m and n such that m < n < m+1.

I got stuck, so I looked it up. It's a bit trickier than one might think.

Toward a contradiction, suppose m < n < m+1.

Since ~ n < 0, let n = m+p, for some p, with ~ p = 0

Since n < m+1, let m+1 = n+q for some q, with ~ q = 0

So n+q+p = m+1+p = m+p+1 = n+1

So p+q = 1

Since ~ p = 0, for some j, let p = j+1

Since ~ q = 0, for some k, let q = k+1

So j+k+2 = 1

So j+k+1 = 0, which is impossible

I just can't envision any computer holding even just the natural numbers without exploding. — keystone

Of course, one may adopt a thesis that mathematics should only mention what can happen with a computer (call it 'thesis C'). Then, go ahead and tell us your preferred rigrorous systemization for mathematics for the sciences that still abides by thesis C.

And one can reject thesis C. And there is a rigorous systemization of mathematics for the science that does not abide by thesis C.

I got on an airplane that flied well, getting me from proverbial point A to point B. Show me your better airplane.

And one more detail:

And often we say that the naturals are defined as nested sets of sets. — keystone

The von Neumann characterization, which has been standard for a long time now is:

The set of natural numbers is the least successor inductive set.

df. Sn = n u {n}

thm. n is a natural number <-> (n = 0 or Ek(k is a natural number & n = Sk))

But now matter how we define the set of natural numbers, starting element, the successor operation and the starting element, as long as it is a Peano system*, then we get distinct natural numbers.

Distinctness does not depend on a particular characterization of the natural numbers (such as von Neumann's).

* And recall that all Peano systems are isomorphic with one another.