Relevance logic and the relevance condition

[Ross Finlayson]

This sort of "relevant logic" seems much more how I see things.

https://en.wikipedia.org/wiki/Alan_Ross_Anderson#Relevance_logic

I'm a big fan of Dana Scott, and from what I've just read of Alan Anderson,
a big fan of Alan Anderson.


" A entails B only if A and B share at least one non-logical constant.

As simple as this idea appears, implementing it in a formal system requires
a radical departure from the semantics of classical logic. Anderson and Belnap
(with contributions from J. Michael Dunn, Kit Fine, Alasdair Urquhart, Robert K. Meyer,
Anil Gupta, and others) explored the formal consequences of the relevance condition
in great detail in their influential Entailment books (see references below), which are
the most frequently cited works in the field of relevance logic.

Anderson and Belnap were quick to observe that the concept of relevance
had been central to logic since Aristotle, but had been unduly neglected
since Gottlob Frege and George Boole laid the foundations for what would
come to be known, somewhat ironically, as "classical" logic. (For an example
of classical logic's failure to satisfy the relevance condition, see the article
on the principle of explosion.) "


So, I guess Dan will have to be retroactively revised to say "most mathematicians,
except those who believe in relevance conditions".

"Anderson was known for being a Platonist (or realist, or monist) about logic;
he believed in "The One True Logic," and he believed that it was a relevance logic. "

[Dan Christensen]

On Saturday, August 19, 2023 at 9:32:06 PM UTC-4, Ross Finlayson wrote:
> This sort of "relevant logic" seems much more how I see things.
>
> https://en.wikipedia.org/wiki/Alan_Ross_Anderson#Relevance_logic
>
> I'm a big fan of Dana Scott, and from what I've just read of Alan Anderson,
> a big fan of Alan Anderson.
>
>
> " A entails B only if A and B share at least one non-logical constant.
>
> As simple as this idea appears, implementing it in a formal system requires
> a radical departure from the semantics of classical logic. Anderson and Belnap
> (with contributions from J. Michael Dunn, Kit Fine, Alasdair Urquhart, Robert K. Meyer,
> Anil Gupta, and others) explored the formal consequences of the relevance condition
> in great detail in their influential Entailment books (see references below), which are
> the most frequently cited works in the field of relevance logic.
>
> Anderson and Belnap were quick to observe that the concept of relevance
> had been central to logic since Aristotle, but had been unduly neglected
> since Gottlob Frege and George Boole laid the foundations for what would
> come to be known, somewhat ironically, as "classical" logic. (For an example
> of classical logic's failure to satisfy the relevance condition, see the article
> on the principle of explosion.) "
>
>

What problems that now eludes classical logic do you hope solve with this new and "relevant" system? I'm guessing none. IIUC you will actually have FEWER methods at your disposal, and MORE restrictions on those remaining -- just a "working" subset of the methods of classical logic IIUC.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[Ross Finlayson]

I think it might actually be simpler, there's a bit more usual book-keeping
and pro-forma work involved to check the corners and work out the edges,
but, at least a naive theorem-prover doesn't automatically hit infinite loops.

Also it's a lot more satisfying when absurd non-conditions don't imply anything.

[Ross Finlayson]

Also it's sort of part of a program for axiomless natural deduction and some "true
logical principles", various goals or first-order-formalizability and such. For example,
if David Hilbert had a programme to try to formalize mathematics, one sort of looks
at John Corcoran and axiomless natural deduction for (all of) logic.

Less traps, less tricks, less nonsense, ... a more "classical" "classical" logic, ...,
anyways it's not losing much saying "you can't introduce contradictions and
say they prove something unrelated/ir-relevant" and "nobody much will miss
the principle of absurd explosion given they'll cover cases with the principle of
combinatorial explosion".

[Ross Finlayson]

It's really a matter of contingencies, not stipulations, the propositions, the logic,
each standing up for themselves and where they disagree a model of contention.

The "material implication" really is a poor choice of words for "it's said so".

This is also called "science", it's to be learned, that science works on logic just great,
and of course real logic and mathematics is ubiquitously successful in science.
(... Not necessarily that mathematical models are correct for the physical interpretation,
but, that's all there are for it.)

Then in foundations I really care about this "The One True Logic" idea, though when
it comes to "platonism, realism, monism", there are various objective notions of
the realism, including where and how the platonist monism is more real than us.

Entailment as coincidence is not entailment as causation. (Or correlation isn't causation.)

Don't get me wrong, there's nothing that "classical logic with ir-relevant contradictions"
can _prove_ that relevance logic can't: in terms of factual things that are real, about logic.
Agreeably there are some algorithms that are succinct which show some of the same things,
but just like your "Lawmaker in a world of Peons", or "Drunk in a Bar plying others with Drugs"
or what it is, those are non-sensical aberrations that thusly "relevance logic" doesn't arrive at.

Though, of course it's simple to arrive at how such non-senses are _not_ so.

"If you can't go making sense of it: don't go making non-sense of it."

Don't get me wrong, I don't just level such criticism at you, but fairly to the system.

[Dan Christensen]

On Saturday, August 19, 2023 at 11:40:01 PM UTC-4, Ross Finlayson wrote:
> On Saturday, August 19, 2023 at 8:25:54 PM UTC-7, Dan Christensen wrote:
> > On Saturday, August 19, 2023 at 9:32:06 PM UTC-4, Ross Finlayson wrote:
> > > This sort of "relevant logic" seems much more how I see things.
> > >
> > > https://en.wikipedia.org/wiki/Alan_Ross_Anderson#Relevance_logic
> > >
> > > I'm a big fan of Dana Scott, and from what I've just read of Alan Anderson,
> > > a big fan of Alan Anderson.
> > >
> > >
> > > " A entails B only if A and B share at least one non-logical constant.
> > >
> > > As simple as this idea appears, implementing it in a formal system requires
> > > a radical departure from the semantics of classical logic. Anderson and Belnap
> > > (with contributions from J. Michael Dunn, Kit Fine, Alasdair Urquhart, Robert K. Meyer,
> > > Anil Gupta, and others) explored the formal consequences of the relevance condition
> > > in great detail in their influential Entailment books (see references below), which are
> > > the most frequently cited works in the field of relevance logic.
> > >
> > > Anderson and Belnap were quick to observe that the concept of relevance
> > > had been central to logic since Aristotle, but had been unduly neglected
> > > since Gottlob Frege and George Boole laid the foundations for what would
> > > come to be known, somewhat ironically, as "classical" logic. (For an example
> > > of classical logic's failure to satisfy the relevance condition, see the article
> > > on the principle of explosion.) "
> > >
> > >

> > What problems that now eludes classical logic do you hope will be solved with this new and "relevant" system? I'm guessing none. IIUC you will actually have FEWER methods at your disposal, and MORE restrictions on those remaining -- just a "working" subset of the methods of classical logic.

> >

> I think it might actually be simpler, there's a bit more usual book-keeping
> and pro-forma work involved to check the corners and work out the edges,

You will have FEWER methods at your disposal, and MORE restrictions on those remaining, and you think it MIGHT ACTUALLY be simpler to use??? You really must get some practice actually applying logic, e.g. in mathematical proofs.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com <------- This may help you understand

[Dan Christensen]

On Sunday, August 20, 2023 at 12:10:47 AM UTC-4, Ross Finlayson wrote:
[snip]

>
> Entailment as coincidence is not entailment as causation. (Or correlation isn't causation.)
>

Aha, there's the problem!!!

"If it is raining (R), then it is cloudy (C)" does NOT mean that rain CAUSES cloudiness. It means only that, at the moment, it is not both raining and not cloudy.

R C R=>C
T T T
T F F
F T T
F F T

There is no causality in mathematics. There are no simple rules for determining causality. See how difficult it was for scientists to determine that smoking causes cancer. Or that human activity is causing a global warming crisis.

[olcott]

On 8/19/2023 8:32 PM, Ross Finlayson wrote:
>
>
> This sort of "relevant logic" seems much more how I see things.
>
> https://en.wikipedia.org/wiki/Alan_Ross_Anderson#Relevance_logic
>
> I'm a big fan of Dana Scott, and from what I've just read of Alan Anderson,
> a big fan of Alan Anderson.
>
>
> " A entails B only if A and B share at least one non-logical constant.
>
> As simple as this idea appears, implementing it in a formal system requires
> a radical departure from the semantics of classical logic. Anderson and Belnap
> (with contributions from J. Michael Dunn, Kit Fine, Alasdair Urquhart, Robert K. Meyer,
> Anil Gupta, and others) explored the formal consequences of the relevance condition
> in great detail in their influential Entailment books (see references below), which are
> the most frequently cited works in the field of relevance logic.
>
> Anderson and Belnap were quick to observe that the concept of relevance
> had been central to logic since Aristotle, but had been unduly neglected
> since Gottlob Frege and George Boole laid the foundations for what would
> come to be known, somewhat ironically, as "classical" logic. (For an example
> of classical logic's failure to satisfy the relevance condition, see the article
> on the principle of explosion.) "
>

[Establishing a better foundation for logic]
My new post redefines the foundation of HOL such that the
relevance logic of the syllogism is re-established in HOL.


--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

[Ross Finlayson]

That's just "data", and all "correlation".

Statistics is a very useful methodology in science, establishing that according
to the design of statistical experiment, and various laws of numbers, that a
null hypothesis can be invalidated, given that there's a causal logic to that.

There's only causality in mathematics.

Here though is for that there's a greater surrounds of studies in logic, that
"classical" logic was always the "connexives" before Boole's "classical 'truth-functional'"
(or, dysfunctional), that this "relevance logic" or "relevant logic" is exactly about
eliminating the "paradoxes" of the false antecedent or false consequent.

Dana Scott and these relevance logic adherents and on to Graham Priest
and so on are pretty great.

"Finally, to complete his system Boole added the logical concepts of "everything" (the "universe")
and "nothing": 1 and 0 respectively. The notation 1-x thus provided a symbolic representation
of all things not in the category described by x."

"Through basic mathematics, Boole had arrived at a key principle that Aristotle took great pains
to show in Book 4 of his Metaphyiscs: "'It is impossible for the same attribute at once to belong
and not to belong to the same thing and in the same relation' ... This is the most certain of all
principles ... [and] is by nature the source of all the other axioms." For Boole, however, the
principle of contradiction was not the foundation of other logical rules but merely a byproduct
of x^2 = x". -- Cohen, "Equations from G-d, Pure Mathematics and Victorian Faith"

So, there are some attributes that are compound or reflect for the modal logic and the
monotonicity of logic and the temporality of logic that it's modal and monotic and temporal,
that help reflection that not all binary propositions are either true or false, by themselves,
but rather as in the sense of the relevant, where they are or aren't.

That is, the "same relation" involves all the possible stipulations, that in a universe of mathematical
objects, for example, and logical objects, things are either causal or they aren't, and if they aren't
they don't exist, or they're non-logical stipulations.

Anyways lots of logicians look up to Dana Scott and Alan Anderson and these types who quite
definitely neither need nor want "material implication", which is an oxymoron, and who have
a different definition of entailment, and I'm glad to hear there's quite a school of this relevance
logic as as it reflects some of what I say about it, then here that mathematical proofs are as
of the direct implication and the causal.

[olcott]

On 8/19/2023 8:32 PM, Ross Finlayson wrote:
>
>

> This sort of "relevant logic" seems much more how I see things.
>
> https://en.wikipedia.org/wiki/Alan_Ross_Anderson#Relevance_logic
>
> I'm a big fan of Dana Scott, and from what I've just read of Alan Anderson,
> a big fan of Alan Anderson.
>
>
> " A entails B only if A and B share at least one non-logical constant.
>
> As simple as this idea appears, implementing it in a formal system requires
> a radical departure from the semantics of classical logic. Anderson and Belnap
> (with contributions from J. Michael Dunn, Kit Fine, Alasdair Urquhart, Robert K. Meyer,
> Anil Gupta, and others) explored the formal consequences of the relevance condition
> in great detail in their influential Entailment books (see references below), which are
> the most frequently cited works in the field of relevance logic.
>
> Anderson and Belnap were quick to observe that the concept of relevance
> had been central to logic since Aristotle, but had been unduly neglected
> since Gottlob Frege and George Boole laid the foundations for what would
> come to be known, somewhat ironically, as "classical" logic. (For an example
> of classical logic's failure to satisfy the relevance condition, see the article
> on the principle of explosion.) "

*Exactly my same view* See my post on

[Establishing a better foundation for logic]

to see how I propose to correct his.

[Dan Christensen]

On Sunday, August 20, 2023 at 1:01:04 PM UTC-4, Ross Finlayson wrote:
> On Saturday, August 19, 2023 at 10:00:41 PM UTC-7, Dan Christensen wrote:
> > On Sunday, August 20, 2023 at 12:10:47 AM UTC-4, Ross Finlayson wrote:
> > [snip]
> > >
> > > Entailment as coincidence is not entailment as causation. (Or correlation isn't causation.)
> > >
> > Aha, there's the problem!!!
> >
> > "If it is raining (R), then it is cloudy (C)" does NOT mean that rain CAUSES cloudiness. It means only that, at the moment, it is not both raining and not cloudy.
> >
> > R C R=>C
> > T T T
> > T F F
> > F T T
> > F F T
> >
> > There is no causality in mathematics. There are no simple rules for determining causality. See how difficult it was for scientists to determine that smoking causes cancer. Or that human activity is causing a global warming crisis.

> That's just "data", and all "correlation".
>

There is more to causality than data and correlation. In the case of smoking causing cancer, each step from exposure to smoke to the growth of tumors had to be verified at the molecular and cellular level. There is no simple truth table or formula for causality if that is what you are hoping for -- your elusive Holy Grail???

> Statistics is a very useful methodology in science, establishing that according
> to the design of statistical experiment, and various laws of numbers, that a
> null hypothesis can be invalidated, given that there's a causal logic to that.
>
> There's only causality in mathematics.
>

Wrong again. Again, there is no causality in mathematics.

> Here though is for that there's a greater surrounds of studies in logic, that
> "classical" logic was always the "connexives" before Boole's "classical 'truth-functional'"
> (or, dysfunctional), that this "relevance logic" or "relevant logic" is exactly about
> eliminating the "paradoxes" of the false antecedent or false consequent.
>

[snip]

There are no such "paradoxes." If you want to avoid using vacuous truth in proofs, there is nothing requiring you to do so, but others will continue to use it in proofs to no ill effects. Must be frustrating as hell for you!

[Ross Finlayson]

P.F. Strawson also has quite a bit going on in trying to rehabilitate logic
to excise the paradoxes.

Heh, "Dan: there's no causality in mathematics". Say goodbye, sound inference.

So, Scott and Strawson and Anderson and the relevance logic set, et alia,
at best look askance at your false/false antecedents/consequents,
and note for example in his "classical truth-functional Boole's algebraization"
that there's a world, where anytime any scratches his own nose,
he slaps his own face.

It's like they say, ..., "quit hitting yourself, and don't play that cruel trick
of making someone slap themself and saying 'quit hitting yourself'".

One reason they're called "paradxoes" is because there are alternate inferences
in the same logic, that contradict the outputs of "the Drinker", contradicting it,
that it's either ignorant, or wrong.

Again, by applying such as a relevant condition up-front, such "results" vanish.

Also, don't confuse "vacuity" with "assumity".

[Dan Christensen]

On Sunday, August 20, 2023 at 4:20:02 PM UTC-4, Ross Finlayson wrote:
> On Sunday, August 20, 2023 at 11:13:41 AM UTC-7, Dan Christensen wrote:
[snip]

> > There are no such "paradoxes." If you want to avoid using vacuous truth in proofs, there is nothing requiring you to do so, but others will continue to use it in proofs to no ill effects. Must be frustrating as hell for you!

> P.F. Strawson also has quite a bit going on in trying to rehabilitate logic
> to excise the paradoxes.
>

And if they can't find any, just make some up?

> Heh, "Dan: there's no causality in mathematics". Say goodbye, sound inference.
>

You seem to be conflating "implies" and "causes."

> So, Scott and Strawson and Anderson and the relevance logic set, et alia,
> at best look askance at your false/false antecedents/consequents,

Maybe they didn't really want to do math proofs anyway.

[Ross Finlayson]

I'm of the opinion they'd reject that. (Strawson, Scott, Anderson, et alia, Aristotle.)

If you can prove two contrary things, then, something's wrong with it.

For example, if you can't prove that there are individual qualities, attributes,
uniqueness, then besides that you have a bar where everybody's name is Dan,
either that's a singleton or it's not, and if it's not, then it contradicts that you
don't have an evil twin.

So, the definition of "entailment" you use has others that are more apropos in their
contexts, which are logical, your "implies" is a narrow view of correlation, _after the
fact of having taken a census of all the correlated entries_, your "proofs" are little more
than detailing which of those you pick, and "causality" is generally deemed inviolable
in systems of cause-and-effect, like after axiomatics, for example, ALL the derivations
what follow, for example the one where you're the King Drunk and the one where you're not.

So, you don't have models of first-class causality, you don't have models of completion
of alternatives, you don't have models of unambiguous uniqueness, but, what-you-say
is what-you-say.

I'm reading this Otto Frisch, "What little I remember".

"My father had a strong feeling for right and wrong. ... I remember one occasion when
he startled me by telling a visitor that I knew Pali [the language of the Buddha], which
of course I didn't. He then opened a copy of the speeches of the Buddha in the original
language, which the visitor had brought along, and showed me the first three words
of a chapter, asking me what they meant. He had judged me rightly. I had previously
glanced into a German translation of Buddha's speeches and knew that every chapter
began with the same words; so I was confident in saying 'Evam me sutam', meant
'This I have heard!'. Fortunately the visitor did not probe my knowledge of Pali
any further."

[Dan Christensen]

On Sunday, August 20, 2023 at 5:31:37 PM UTC-4, Ross Finlayson wrote:
> On Sunday, August 20, 2023 at 2:05:15 PM UTC-7, Dan Christensen wrote:

[snip]

>
> If you can prove two contrary things, then, something's wrong with it.
>

There is nothing wrong obtaining a contradiction as an intermediate result in a proof. It just means that you have previously introduced a premise/assumption that turns out to be false. It's called proof by contradiction.

> For example, if you can't prove that there are individual qualities, attributes,
> uniqueness, then besides that you have a bar where everybody's name is Dan,
> either that's a singleton or it's not, and if it's not, then it contradicts that you
> don't have an evil twin.
>
> So, the definition of "entailment" you use has others that are more apropos in their
> contexts, which are logical, your "implies" is a narrow view of correlation

[snip]

It has nothing to do with correlation (or causality). P implies Q means only that, at the moment, it is not that case that P is true and Q is false.

[Ross Finlayson]

Ring around the rosie /
pocket full of posies /
ashes, ashes /
all fall down.

That's all that correlation _is_.

Then, again, where individuals have at all unique attributes,
that there is King Drunk but he has no sway,
is a contradiction, that any can arrive at, against anything you "prove",
a counter, "proving" the entire thing self-contradictory.

I.e., anything you write, any can follow with a King Drunk "proof",
and make both false. Also there's a quick case for induction that
there are infinitely many ways to add false/false antecedents/consequents
to your system.

I know, it doesn't interest you to follow out all conclusions,
and you just pick and choose what you want to make consequent,
or rather false/false antecedent/consequent.

I suppose that's what I've said about it for quite a while.

Long live the King: his own little world.

Finding a contradiction doesn't establish a negation,
so much as indicating any intermediary derivation is negated,
or ambiguous, or poorly defined, or subject quantifier ambiguity,
establishing that the proposition is "false", not that it concludes "negation".

So anyways, "the relevance condition" is a very useful filter against
the improper use of such methods as mutual contradiction. It's pretty
much what I said "no contradiction indicates anything about otherwise
free variables".

Just so it's clear, "entailment" and "implies" have different definitions,
than "Boole's algebraized neo-classical truth-dysfunctional".

[Dan Christensen]

On Sunday, August 20, 2023 at 10:42:44 PM UTC-4, Ross Finlayson wrote:
> On Sunday, August 20, 2023 at 7:02:18 PM UTC-7, Dan Christensen wrote:

[snip]

>
> Finding a contradiction doesn't establish a negation...

[snip]

OMG, Ross here seems never to have heard of proof by contradiction! I guess we shouldn't be surprised, but REALLY!!!!

See: https://en.wikipedia.org/wiki/Proof_by_contradiction

[Julio Di Egidio]

On Monday, 21 August 2023 at 05:02:03 UTC+2, Dan Christensen wrote:
> On Sunday, August 20, 2023 at 10:42:44 PM UTC-4, Ross Finlayson wrote:
>
> > Finding a contradiction doesn't establish a negation...
>

> OMG, Ross here seems never to have heard of proof by contradiction!

You are so full of shit, indeed a perfect representative of the generalized
fraud and lobotomy.

*Nazi-Spammer Alert*

Julio

[Python]

Julio, Dan is not the sharpest knife in the drawer, neither are you.

But he is *not* a "nazi", why could you think he is?

You basically call "nazi" everyone who disagree with you, while you
are posting a HUGE mass of blunders.

How come, Julio? Could you get some help locally?

[Ross Finlayson]

Contradiction establishes a broken link, or, a non-binary condition,
not necessarily negation at the end, where you'd have it.

Don't you recall just the few months ago when I introduced constructivism
and explained how they don't put much stock in proofs by contradiction?

Don't you think other people have their own recall of same?

Otherwise don't you think that others reading this would feel that
you have insulted, not my, but their intelligence, and that the insult
leads to contempt?

Now, we'll all heard of relevance logic, and constructivism, and,
know that some adherents of relevance logic, are even strong
monist platonists, and, know that some constructivists reject
proof by contradiction and demand constructive inference.

[Dan Christensen]

On Sunday, August 20, 2023 at 11:19:58 PM UTC-4, Ross Finlayson wrote:
> On Sunday, August 20, 2023 at 8:02:03 PM UTC-7, Dan Christensen wrote:
> > On Sunday, August 20, 2023 at 10:42:44 PM UTC-4, Ross Finlayson wrote:
> > > On Sunday, August 20, 2023 at 7:02:18 PM UTC-7, Dan Christensen wrote:
> > [snip]
> >
> > >
> > > Finding a contradiction doesn't establish a negation...
> >
> > [snip]
> >
> > OMG, Ross here seems never to have heard of proof by contradiction! I guess we shouldn't be surprised, but REALLY!!!!
> >
> > See: https://en.wikipedia.org/wiki/Proof_by_contradiction

> Contradiction establishes a broken link, or, a non-binary condition,

Wrong in both cases.

> not necessarily negation at the end, where you'd have it.
>

Proof by contradiction can be used several times in mid proof. See, for example, lines 28 and 35 in https://dcproof.com/LiarParadox2.htm (44 lines).

[Ross Finlayson]

One way to look at the Liar paradox is that in a language where only
truisms are well-formed formulas, and rather affirmatory, that there's
only a sort of prototype of fallacy which the Liar fits, and, furthermore
that after it's, "false", that it doesn't even exist.

[Ross Finlayson]

Says "snip".

[olcott]

On 8/19/2023 8:32 PM, Ross Finlayson wrote:
>
>

> This sort of "relevant logic" seems much more how I see things.
>
> https://en.wikipedia.org/wiki/Alan_Ross_Anderson#Relevance_logic
>
> I'm a big fan of Dana Scott, and from what I've just read of Alan Anderson,
> a big fan of Alan Anderson.
>
>
> " A entails B only if A and B share at least one non-logical constant.
>
> As simple as this idea appears, implementing it in a formal system requires
> a radical departure from the semantics of classical logic. Anderson and Belnap
> (with contributions from J. Michael Dunn, Kit Fine, Alasdair Urquhart, Robert K. Meyer,
> Anil Gupta, and others) explored the formal consequences of the relevance condition
> in great detail in their influential Entailment books (see references below), which are
> the most frequently cited works in the field of relevance logic.
>

*THIS IS MY OWN KEY POSITION*
DIVERGENCE FROM THE RELEVANCE LOGIC OF THE SYLLOGISM WAS AN ERROR

> Anderson and Belnap were quick to observe that the concept of relevance
> had been central to logic since Aristotle, but had been unduly neglected
> since Gottlob Frege and George Boole laid the foundations for what would
> come to be known, somewhat ironically, as "classical" logic. (For an example
> of classical logic's failure to satisfy the relevance condition, see the article
> on the principle of explosion.) "
>

With (possibly an extended notion of) relevance logic Gödel
incompleteness is impossible.

It is stated this way:
*unless a conclusion is a semantically necessary*
*consequence of all of its premises the argument is deemed invalid*

Unprovable expressions are then rejected as invalid thus are no longer
available cannot be used to show incompleteness.

[Ross Finlayson]

This thread on relevance logic is largely a rejection of quasi-modal
logic. Yet, it is so by pointing out how relevant relevance logic, is.