Whose website is that?

Suspicious page alert Banno.

He doesn't care, and he should be respected because of that.

The Marcus Family site. It's been useful for years, I'm not too concerned, but do as you see fit.

Well, whoever the site owner is, he has an impressive Curriculum Vitae. A bit too focused on one side of the Continental-Analytic debate, in humble opinion, but whatever.

Sorry to interrupt your Thread, Banno. I honestly don't know what to do to get you to treat me as one of your colleagues, but no worries. I'm accustomed to such type of intellectual scorn.

Carry on.

Alternat link: Internet Archive.

This is the whole text of the book, 10.3MB

Then give people some options...?

https://dn790009.ca.archive.org/0/items/FromALogicalPointOfView/

EDIT: For example, the .djvu file is 3.3 MB.

Lots to ponder in this essay. Just as a place to start:

Quine contrasts two statements (pp. 147-8):

(1) (∃x)(x is necessarily greater than 7)

and

(2) Necessarily (∃x)(x is greater than 7)

(1) is an existential generalization of a modal statement, and is either incoherent or false. (2) is unproblematic. To explain the difference, Quine makes the analogy with a game that must have a winner and a loser: “It is necessary that some one of the players will win, but there is no one player of whom it may be said to be necessary that he win.” Likewise, there must certainly be a number greater than 7, but we cannot say that any given number is necessarily that number. As Quine puts it,

Necessary greaterness than 7 makes no sense as applied to a number x; necessity attaches only to the connection between ‛x > 7’ and the particular method . . . of specifying x. — Quine, 148

Let’s rephrase (2) in ordinary English: “It has to be the case that some number is greater than 7” or perhaps “It has to be the case that 7 is not the highest number”. Why is it true? In what does the necessity lie? As Quine points out, the necessity here does not concern any attribute of the number 7. It is nothing like “A bachelor is an unmarried male”, where synonymy is supposed to result in analyticity or tautology. Synonymy is not the issue here. But we want to say that (2) is analytically true – that is, true by virtue of its logical form – since it doesn’t matter what number we plug in; it has to come out true. If Quine allows (2) to be an example of a logical principle, then he would agree.

But have we really freed ourselves from any (questionable) definitional analyticity? Don’t we need the concept/definition of “number” in order to know that there is no greatest number?

I believe we ought to say that Quine’s (2) is really shorthand for:

(3) Necessarily (∃x)(x is a number) & (x is greater than 7)

Or does this commit us to the existence of some number? We could rephrase it, then (and I’ll use English for simplicity):

(4) Necessarily, if some number exists, there’s another one that is greater.

This preserves the analyticity we desire: Granted a number – and what “number” means – we know it can’t be the greatest number.

But coming back now to tautology, if I write “x is a number” and then write “There is a number greater than x”, have I written a tautology? Are the logical constants alone what make the statements tautologous, regardless of what we plug in? I don’t think so. We need “number” to have attributes, one of which is “always exceeded” or “cannot be highest” or some such. The logical form alone can’t give us this. So “greaterness” is not about “7”, as Quine says, but it is about “number”. You can’t understand “number” without knowing what to do with “greaterness”. Is this analytic/definitional necessity?

Notice that this is not at all the same thing as saying, "You can't understand 'water' without knowing that water is composed of H20". Necessity, as Kripke shows us, may be a feature of either analytic or synthetic statements. So what gives "number" its peculiar type of analyticity? If statements like (3) are not true by tautology, but nor is math empirical . . . what's the best account? Would we be better off, for instance, with an argument that shows that any number x can't be the greatest number because there is no such thing?

This is a complex issue. I'm still working through it. First some comments on syntax, then on semantics.

Necessity and possibility quantify over complete propositions. Folk will be familiar with propositional logic, the p's and q's of p⊃q and so on. "Normal" modal logic (the system K) allows us to write ☐p and ◇p, so that the whole of the proposition is inside the scope of the modal operator.

(2) Necessarily (∃x)(x is greater than 7) — J

This can be parsed as ☐∃(x)(fx) were "f" is "greater than seven". This is well-formed, since ∃(x)(fx) is complete.

(1) (∃x)(x is necessarily greater than 7) — J

The apparent parsing here is ∃(x)☐(fx). But "fx" is incomplete. The "x" is a variable, not an individaul constant. It's not that "x" could stand for anything - that'd be U(x)(fx). It's that we just do not know what x might be. It does not say that something is f, nor that nothing is f. That is, it is not a whole proposition. Hence it cannot be replaced by the p's and q's of propositional calculus, and cannot take a modal operator in normal modal logic. But the situation is more complex than that.

To a large extent this is a modern version of the de re/de dicto distinction. but we can be much clearer here using modal first order language than was possible in medieval times.

Despite these syntactic misgivings there may well be interpretations in possible world semantics in which ∃(x)☐(fx) can be understood. "There is something that in every possible world is f". The question then becomes how this is to be understood, and if it can be made consistent. From what I have been able to glean, if we step from K to S5 and permit ∃(x)☐(fx), then modal collapse follows.

Part one of the article is a study of some now fairly typical examples of opaque contexts. That is, contexts in which one cannot swap names around without losing the truth of what one is saying.

The examples include quotation, what we might now call propositional attitude, and modality. It's worth noting that these are three distinct issues, and they might (do....) need to be addressed in different ways.

The thesis is summed up in the last sentence:

What is important is to appreciate that the contexts ‘Necessarily . . .’ and ‘Possibly . . .’ are, like quotation and ‘is unaware that . . .’ and ‘believes that . . . referentially opaque.

The thesis is summed up in the last sentence:
What is important is to appreciate that the contexts ‘Necessarily . . .’ and ‘Possibly . . .’ are, like quotation and ‘is unaware that . . .’ and ‘believes that . . . referentially opaque. — Banno

Yes. And this is amplified as follows:

If to [any] referentially opaque context of a variable we apply a quantifier, with the intention that it govern that variable from outside the referentially opaque context, then what we commonly end up with is unintended sense or nonsense . . . — Quine, 148

I added "any" to Quine's statement because we can now appreciate that referential opacity characterizes (at least) three different situations: quotation, "belief"-type statements, and modality; as you say, they are three distinct issues.

Staying with modality for the moment, I'm curious how we should handle this idea from Kripke concerning what he calls "strongly rigid designators":

A rigid designator of a necessary existent can be called strongly rigid. — Naming & Necessity, 48

We know that a rigid designator has to designate the same object in every possible world. Thus, no one other than Nixon can be "Nixon". But Kripke is clear that Nixon, as such, did not have to exist. "Nixon" is not a strongly rigid designator.

Is a number strongly rigid? Kripke uses Quine's example from "Reference and Modality":

What's the difference between asking whether it's necessary that 9 is greater than 7 or whether it's necessary that the number of planets is greater than 7? — N&N, 48

Kripke suggests that the answer, "intuitively," would be:

Well, look, the number of planets might have been different from what it in fact is. It doesn't make any sense, though, to say that nine might have been different from what it in fact is. — N&N, 48

So, does "9" rigidly and strongly designate nine? That is, is there something about the number nine which makes us want to say that it must necessarily exist? We could do without Nixon, but not nine. This type of necessity seems problematic. Can it ever be anything other than stipulative? If there are insights from modal logic that would help here, please share.

To a large extent this is a modern version of the de re/de dicto distinction — Banno

Can you say more about this?

but we can be much clearer here using modal first order language than was possible in medieval times. — Banno

Isn't that what Quine doubts?

Is he wrong? How?

How does possible world semantics restore coherence in the face of referential opacity?

Asking for a friend.

Isn't that what Quine doubts? — bongo fury

Yep. He was pushing back against formal modal logic.

I'm not sure of the time line here. According to the forward, there were substantial revisions to the present article in around 1960-61. Kripke's A Completeness Theorem for Modal Logic was published in 1959. The Princeton Lectures, which became Naming and Necessity, were in 1970, well after Quine's paper.

It appears that the modal logic that Quine was addressing was mostly that prior to what we might be using now. And much, much clearer than Medieval modal logic.

So a new issue is how earlier problems might be parsed in formal modal terms, what those problems then look like and if there are ensuing issues.

I'm not too up on the de dicto/de re distinction, but it should be one of those that is amenable to formal description. Maybe @Count Timothy von Icarus will weigh in. For my part I understand that de dicto modalities have the whole proposition within the scope of the modal operator, as in K, applying across all possible worlds, while de re modalities apply to the properties of some individual - and here the terminology becomes ambiguous - and not always in every possible world. So in terms of syntax, de dicto is most similar to ☐∃(x)f(x) and de re, to ∃(x)☐(fx), while in terms of semantics de dicto understands necessity as "true in every possible world" while de re might understand necessity as "true in this (or some) world", a cumbersome notion incompatible with S5.

Others understand this stuff in more detail than I.

I'm not too up on the de dicto/de re distinction, ↪J but it should be one of those that is amenable to formal description. — Banno

OK. I quickly read through the SEP article and remembered why I'd never completely understood it in the first place. :smile: Glad to have some help from @Count Timothy von Icarus or anyone else.

Literally, de dicto is "of the said/expressed" and de re "of the matter/thing." For example, suppose some little girl says: "when I grow up, I want to marry the richest man in the world!"

De dicto, we could interpret this as the girl having perhaps a bit of an avaricious streak. She wants to marry the richest man in the world, whoever this happens to be. De re, this would be equivalent to "when I grow up, I want to marry Elon Musk!" bespeaking a fondness for our glorious DOGE master.

Re modality, clearly Elon Musk is not "the richest man in the world" by necessity. This is subject to change. The predication of "richest man in the world" of Musk is per accidens as opposed to per se.

You could also consider: "The number of states in the USA is necessarily evenly divisible by five."

De re, the sentence is equivalent with:"Fifty is necessarily evenly divisible by five," which I think most people would allow is true. De dicto this is clearly false. In the past, the US often had a number of states that was not evenly divisible by five, and once our glorious orange Augustus annexes Canada and Greenland it seems we will also be left with a number of states that is not divisible by five.

I am most familiar with how this ties into modality in terms of how it relates to the apparent conflict between divine foreknowledge and free will (or contingency as a whole). A key discussion here is St. Thomas's Summa Theologiae, Q.14 A. 13, .

If something is a fact, then to report that it is the case is to report that it is necessarily true. If Socrates is sitting, "Socrates is currently sitting" is true by necessity, but this is necessitas per accidens. By contrast, "man is an animal" is necessitas per se, de re (assuming for the sake of the example that all men are necessarily animals.)

That said, I don't know helpful this will be since modality is here considered in terms of necessary and contingent being/beings and causes (obviously a much broader notion of cause/principle than mechanism or constant conjunction alone), as opposed to possible worlds. I imagine the distinction would have different implications under different assumptions. I seem to vaguely recall Quine eventually rejecting the distinction on the grounds that it would imply essentialism.

But, if you're interested in the context:

In evidence of this, we must consider that a contingent thing can be considered in two ways; first, in itself, in so far as it is now in act: and in this sense it is not considered as future, but as present; neither is it considered as contingent (as having reference) to one of two terms, but as determined to one; and on account of this it can be infallibly the object of certain knowledge, for instance to the sense of sight, as when I see that Socrates is sitting down.In another way a contingent thing can be considered as it is in its cause; and in this way it is considered as future, and as a contingent thing not yet determined to one; forasmuch as a contingent cause has relation to opposite things: and in this sense a contingent thing is not subject to any certain knowledge. Hence, whoever knows a contingent effect in its cause only, has merely a conjectural knowledge of it.


Summa Theologiae, Q.14 A. 13

Perhaps also helpful:

Necessity, in a general way, denotes a strict connection between different beings, or the different elements of a being, or between a being and its existence. It is therefore a primary and fundamental notion, and it is important to determine its various meanings and applications in philosophy and theology.

In Logic, the Schoolmen, studying the mutual relations of concepts which form the matter of our judgments, divided the judgments or propositions into judgments in necessary matter (in materia necessaria), and judgments in contingent matter (in materia contingenti). (Cf. S. Thom., I Perihermen, lect. xiii.) The judgments in necessary matter were known as propositiones per se; they are called by modern philosophers "analytic", "rational", "pure", or "a priori" judgments. The propositio per se is defined by the Schoolmen as one the predicate of which is either a constitutive element or a natural property of the subject...

When we consider the divers beings, not from the point of view of existence, but in relation to their constitution and activity, necessity may be classified as metaphysical, physical, and moral.

- Metaphysical necessity implies that a thing is what it is, viz., it has the elements essential to its specific nature. It is a metaphysical necessity for God to be infinite, man rational, an animal a living being. Metaphysical necessity is absolute.

- Physical necessity exists in connection with the activity of the material beings which constitute the universe. While they are contingent as to their existence, contingent also as to their actual relations (for God could have created another order than the present one), they are, however, necessarily determined in their activity, both as to its exercises and its specific character. But this determination is dependent upon certain conditions, the presence of which is required, the absence of one or the other of them preventing altogether the exercise or normal exercise of this activity. The laws of nature should always be understood with that limitation: all conditions being realized. The laws of nature, therefore, being subject to physical necessity are neither absolutely necessary, as materialistic Mechanism asserts, nor merely contingent, as the partisans of the philosophy of contingency declare; but they are conditionally or hypothetically necessary. This hypothetical necessity is also called by some consequent necessity.

MERCIER, Ontologie (Louvain, 1902), ii, 3; RICKABY, First Principles of Knowledge (London, 1902), I, v; IDEM, General Metaphysics (London, 1901), I, iv.

Thank you, this is very clear. I think @Banno is right that the issue being raised in the Quine essay strongly resembles de re / de dicto. Referential opacity is the connecting link.

If something is a fact, then to report that it is the case is to report that it is necessarily true. If Socrates is sitting, "Socrates is currently sitting" is true by necessity, but this is necessitas per accidens. By contrast, "man is an animal" is necessitas per se, de re (assuming for the sake of the example that all men are necessarily animals.) — Count Timothy von Icarus

A question about this, though. "Necessity by accident" has an odd ring. Is the idea that, if "Socrates is currently sitting" is true, then as long as it remains true, it is necessarily the case that Socrates is sitting? The necessity would arise from the fact that there is only one way (allegedly) for a statement to be true, and that is by its stating something that is the case? I'm struggling to phrase the necessity in some understandable way -- maybe you can help.

"Man is an animal," in contrast, would be a good example of a Kripkean synthetic necessity. There is nothing analytic about the notion; it so happens, though, that we have discovered it to be true. And Kripke would go on to point out that we don't need this necessary truth in order to designate "man" -- we were able to do this quite well before we knew any science. Had it turned out that humans were not in fact animals, we would not have said, "Oh, we we were wrong in our identification of what a human is. We'd better call them doomans instead" Rather, we would have said, "We thought humans might be angelic or unique, but that is not so. They're still humans, just different from what we thought." (This is Kripke's "gold" example in a slightly different wording.)

So in terms of syntax, de dicto is most similar to ☐∃(x)f(x) and de re, to ∃(x)☐(fx), [...] — Banno

Agreed.

[...] while in terms of semantics de dicto understands necessity as "true in every possible world"... — Banno

Agreed, e.g.

Note that [problematic statements] (30) and (31) are not to be confused with:

Necessarily (∃x) (x > 7),

Necessarily (∃x) (if there is life on the Evening Star then there is life on x),

which present no problem of interpretation comparable to that presented by (30) and (31). The difference may be accentuated by a change of example: in a game of a type admitting of no tie it is necessary that some one of the players will win, ... — Quine p.147

(There is a winner in each play of the game, there is a richest man in each world, there is always a number greater than 7, or etc.)

Evidently Quine is ok with the kind of reading you (and Wiki) are calling de dicto.

However, not so sure about:

[...] while de re might understand necessity as "true in this (or some) world", a cumbersome notion incompatible with S5. — Banno

Whereas (I think) Quine's objection is to a typical de re reading, that there should be

... one player of whom it may be said to be necessary that he win. — Quine p.147

Not because such a reading (there existing a winner of all possible plays of the game or a richest in all worlds or a greater than 7 in all worlds) is self-evidently non-sensical but because it has arisen through referential opacity, and hence behaves incoherently. E.g.

What is this number which, according to ["(∃x)(x is necessarily is greater than 7"], is necessarily greater than 7? According to ["9 is necessarily greater than 7"], from which ["(∃x)(x is necessarily is greater than 7"] was inferred, it was 9, that is, the number of planets; but to suppose this would conflict with the fact that ["the number of planets is necessarily greater than 7"] is false. — Quine p.148

Does this objection hold up? If not why not?

... Hmm, chapter 6 of this book is called "Quine on de re and de dicto modality". :nerd:

Cheers.

It is perhaps becoming clear how two somewhat different uses of "necessity" are at work here. One has necessity as opposed to analyticity, the other has necessity as opposed to possibility. Early philosophy did not make this distinction, leading to difficulty. Aristotelian essentialism apparently does not differentiate analyticity from possibility.

'If Socrates is sitting, "Socrates is currently sitting" is true by necessity' would now be understood in terms of accessibility. There are possible worlds in which Socrates is not sitting. And in every possible world in which Socrates is sitting, Socrates is sitting. Putting this another way, if we consider only those possible worlds in which Socrates is sitting, then in every one of those worlds, Socrates is indeed sitting. So in a way of speaking, in those words in which Socrates is sitting, "necessarily" Socrates is sitting.

This would not be valid in S5, since every world is accessible. Indeed, "Socrates is sitting, therefore necessarily socrates is sitting" is invalid in K and S4.

It is perhaps becoming clear how two somewhat different uses of "necessity" are at work here. One has necessity as opposed to analyticity, the other has necessity as opposed to possibility. — Banno

To which we can add a third wrinkle, as I referred to earlier: necessity as opposed to tautology. '9 is greater than 7' is presumably analytic and likely necessary, but is it a tautology?

Banno pretends to talk about things he does not understand, such as the de re/de dicto distinction:

To a large extent this is a modern version of the de re/de dicto distinction. — Banno

But:

I'm not too up on the de dicto/de re distinction — Banno

Or Aristotelian logic:

Aristotelian essentialism apparently does not differentiate analyticity from possibility. — Banno

Or Medieval philosophy:

but we can be much clearer here using modal first order language than was possible in medieval times. — Banno

---

How does possible world semantics restore coherence in the face of referential opacity? — bongo fury

A good question.

---

It appears that the modal logic that Quine was addressing was mostly that prior to what we might be using now. And much, much clearer than Medieval modal logic. — Banno

Quine was as ignorant of Medieval logic as you are. He is not responding to it.

Aristotelian essentialism apparently does not differentiate analyticity from possibility. — Banno

Good resources showing that Aristotelian essentialism is more robust than anything the moderns have stumbled upon are as follows:

If that is right, you may be interested in Gyula Klima's "Contemporary 'Essentialism' vs. Aristotelian Essentialism," where he compares a Kripkean formulation of essentialism to an Aristotelian formulation of essentialism, and includes formal semantics for signification and supposition, which involves the notion of inherence. Paul Vincent Spade also has an informal piece digging into the metaphysical differences between the two conceptions, "The Warp and Woof of Metaphysics: How to Get Started on Some Big Themes."

Note that Banno's whole logical horizon is bound up with the bare particulars of predicate logic, so I'm not sure it is possible to easily convey an alternative semantics to someone who who has never been exposed to an alternative paradigm. — Leontiskos

Klima spends more time with Kripke and Spade more time with Quine. It's no coincidence that those who do not know the past do not progress beyond it.

I really don't see why you need to make this yet another thread about me.

Can we please have mod attention to this persistent failure on Leon's part to address the topic at hand, and to indulge in personal insults directed at me?

Meh. I'll take this to PM in an attempt to keep the thread on topic.

- Yes, "derailment for thee but not for me." But it's good to know that no one takes your historical claims seriously. Try reading a book written before the 20th century. It will help you understand your very limited and naive perspective.

Can we please have mod attention to this persistent failure on Leon's part to address the topic at hand, and to indulge in personal insults directed at me? — Banno

They should also go back and address all the crap that Banno littered my thread with.

I feel this has, perhaps something to do with "Ordinals"... I've been listening to Quine's Pursuit of Truth, which brings up Modalities and I believe also Substitutivity, I will transcribe what he says in each section and add to them here word for word.

But in my paradoxes and Infinities course, I'm currently going over the different modalities of Ordinals which order Infinities in certain ways... and further, there is an intersection here with linguistics. The powerset of words is greater than the set of words because there are more sets of words (sentences) than there are individual words.

Modalities and referencing in math as it is with language... make a biforking chart for example to row 3... what happens down line 0 0 0 is linear and the modality is linear there is a transitive property that a is proceeded by b and b by c and the a is proceeded by c ...logically... but when we try to reference point 000 and point 110 or even 111 as if they were the same as point 000 simply because they're on the same row doesn't mean it will create a bijection from points 000 to 111 or 110 if we're declaring a linear modality while referencing things outside of the modality.

- This thread is obviously an extension of your anti-Medieval tack from my thread on St. Anselm's Proof. An anti-Medieval tack is not problematic in itself, but it needs to be more than posturing. Here is what you said in your penultimate post from that thread:

Again, possible world semantics shows us were we have been led astray. — Banno

If you think that possible world semantics solves some problem that the Medievals could not solve without it, then you have to set out the problem, their solution, and the alternative possible worlds solution. Or as Bongo said in response:

but we can be much clearer here using modal first order language than was possible in medieval times. — Banno

How does possible world semantics restore coherence in the face of referential opacity?

Asking for a friend. — bongo fury

Now it is fairly well established that Quine's understanding of Aristotelian essentialism was highly superficial. But you are welcome to try to do the work that Quine failed to do: set out a robust and well-referenced account of the solution that you find lackluster, and then show how your own possible worlds solution is supposed to be an improvement. As Bongo alludes to, I don't recall Quine seeing possible worlds semantics as being especially promising or advancing. Ironically, much of the recent neo-Aristotelianism flows from a growing dissatisfaction with the artificiality of possible worlds semantics. We are slowly correcting modern errors, first with Kripke's modal form of essentialism, and then moving with Fine and Klima towards more traditional and robust forms of essentialism, that do not rely on the overrated device of possible worlds. You seem to be stuck in positivistic decades that have been largely superseded by a hearkening back to richer philosophical traditions.

-

Quine occupies a curious position in the history of philosophy, with antecedents in Pragmatism but with sympathies very similar to the linguistic philosophy of Russell and Wittgenstein, and an attitude not so far from that of the Vienna Circle. — Banno

No, not the history of philosophy, but rather the history of 20th century Anglo philosophy. Again, one must read books written before 1900 if they are to make claims about the history of philosophy, and this very recent tradition you are immersed in is virtually unknown outside of the English-speaking world. If you have no knowledge of Medieval philosophy it's hard to understand why you make so many claims about it. It's a bit like the cat-lover who has never seen a dog and yet goes around telling everyone how much bigger and better cats are than dogs.

Ironically, much of the recent neo-Aristotelianism flows from a growing dissatisfaction with the artificiality of possible worlds semantics. We are slowly correcting modern errors, first with Kripke's modal form of essentialism, and then moving with Fine and Klima towards more traditional and robust forms of essentialism, that do not rely on the overrated device of possible worlds. — Leontiskos

Any good references?

- I gave two in the post above with this quote. Click to the thread and links will be included:

If that is right, you may be interested in Gyula Klima's "Contemporary 'Essentialism' vs. Aristotelian Essentialism," where he compares a Kripkean formulation of essentialism to an Aristotelian formulation of essentialism, and includes formal semantics for signification and supposition, which involves the notion of inherence. Paul Vincent Spade also has an informal piece digging into the metaphysical differences between the two conceptions, "The Warp and Woof of Metaphysics: How to Get Started on Some Big Themes." — Leontiskos

David Oderberg also writes a fair bit on this topic, e.g. "How to Win Essence Back from Essentialists." Banno also has an old thread on a paper of Kit Fine's, which I believe to be too conservative.

Edit: Banno's claim that the Medievals lacked a "modal first order language" betrays a very curious form of ignorance. At bottom is the fact that Medievals were explicitly uncomfortable separating natural language from logic in the way that someone like Frege, Russell, or Quine was wont to do, and therefore they did not arrive at artificial constructs like possible worlds semantics. Such artificial constructs (and their weaknesses) flow from the idea that logic and natural language can be separated. To take one example, modal logic was highly developed by the late Medieval period, but it was not reified into rigidly formal constructs. Cf. "Natural Logic, Medieval Logic and Formal Semantics."

I'm afraid this is over my head, but I appreciate your response.

mispost

probably not, most of its probably your head there from other sources or even intuition. Just, I'm not shining a light well enough to connect the dots on how they align.

In math when we say 1 is less than 2, and 2 is less than 3 and then say 1 is less than 3, we're showing a transitive property in logic... according to a linear modality of referencing the points 1 2 and 3. We can say 1 is lesser in relation to 3...

The biforking model top of the model is a fork... ^ each left branch from a fork is 0 and each right fork is a 1

So row 1 would be 0 on the left side of for, 1 on the right row 2 would have a fork coming from side 0 and a fork coming from side 1 both are labeled the same 0 to 1 left to right...

You end up with a pyramid of forks... fork 0 0 0 would follow all the left forks, and all spots on that path are linked by a line upon the forking branch from the tip of the pyramid to far left extreme of the pyramid base, if you took the path 1 1 1 youd take the right forking path all the way to the right extreme of the pyramid base... traveling down the points 1 1 1 in row 1 then 2 then 3 all follow a path on a line and all reference each other with transitivity between row 1 and row 3... such that whats in row 2 proceeded row 1 and what's in row 3 proceeded row 2 ... but if you go down the far left, even though you're using linear modality the third spot on 0 0 0 the far left base of the pyramid, doesn't mean we can cross reference between spot 3 on path 000 and spot 3 on path 111 at the far right base... because there's no transitivity betwen the spot at 111 with the spot at 000. I can make a picture if needed, would probably make it way easier to understand hat I'm saying.

What follows when we cross reference say a word, using the wrong modality might be like a categorical error or fallacy of equivocation...if say row 3 ended up as 3 different definitions of the same word ...

Which is to say... say you used X logic to get to a definition of a word... a word that had 8 ways to be used across the different parts of speach it could cover...

All 8 definitions would rest in row 3 of this pyramid we just constructed...

That doesn't mean each definition can be used as a reference for the word in the sentence.

...necessity as opposed to tautology — J

Yeah, it's a good point. I'm not sure where to go with that, so will give it some more thought.

I've been listening to Quine's Pursuit of Truth — DifferentiatingEgg

Have you a link?

The powerset of words is greater than the set of words because there are more sets of words — DifferentiatingEgg

There are arguments that the number of sentences in a natural language can be indenumerable. There was a thread on that a few years back. I'll see if I can locate it. It might be of interest to your course.

Yes, how tautology fits is part of the subtext.