it's logical

[Rich D]

I) "This sentence contains seven words."

False.
Therefore its negation must be true:

II) "This sentence does not contain seven words."


--
Rich

[Dan Christensen]

It seems you need a function that maps strings of characters to N. Something like:

Words("This sentence contains seven words." ) = 5

Words("This sentence does not contain seven words." ) = 7

It's MORE logical!

Dan

[Mitchell Smith]

One reason logicians had to develop formal methods is because negations do not have a standard position in natural language. Outer negation would be more like,

"It is not the case that this sentence contains seven words."

Indeed, the most stirling example of illogic I can imagine is logic.

mitch

[Jeff Barnett]

John McCarthy in the late 1960s ran around expressing suspicion about
doing logic with anything that contained quote marks. One of his
favorite examples was the sentence: "The queen asked whether `Was Sir
Walter Scott the author of Waverley?'" One needed to avoid the deduction
that this sentence was equivalent to "The queen asked whether true?"
Neither you or your unworthy rival, Peter, are far from this sort of
nonsense. Don't argue just yet. Rather, sketch an algorithm that you
think might deal with examples of the above sort while avoiding mistakes
of a six year old. These supposed conundrums written in natural language
are complete nonsense unless you 1) formalize the supposed logic rules
being used or 2) have knowledge approaching that of Smullyan to
anticipate and avoid deep mistakes.
--
Jeff Barnett

[Ross Finlayson]

In usual predicates negation just sort of multiplies out the simpler match terms,
in terms of AND and OR or about conjunction and disjunction. Then, the
placement of NOT in usual natural languages where ordering is what indicates
for stems of terms in dependency grammars, may be unambiguous.

It's similar with ranges or BETWEEN, about how there's read out a bunch of
match terms then that a Sure/No/Yes algorithms works out both positive and
negative terms, where negative terms have a linear constant, thus making for
constant, and range terms exponentiate and thus work down under the factorial,
that a Sure/No/Yes algorithm makes things like a collection of search term
predicates both linear in size, constant-time in composition or modification,
and resulting a linear-time predicate in the number of predicates, in match terms.

I see this as a fundamental computing unit right above the usual AND and OR trees,
what makes for a data structure that accommodates especially the composition
of collections of filters that result, then that Sure/No/Yes a.k.a. Yes/No/Maybe results
Boolean predicates over collections of match terms in parameters, and their composition
thusly.

[Mitchell Smith]

Are you trying to make me laugh?

You will find non-propositional inference rules on pages 12 and 13 of the document at the link

https://drive.google.com/file/d/15AZO0UZpR_i9wDS9nZyL6kYn8EJnr6FC/view?usp=drivesdk

Those same inference rules can be found on pages 5 and 6 of the document at the link,

https://drive.google.com/file/d/15IdSKGmmHgUcC3-TopCkE4khF4XHZwm7/view?usp=drivesdk

The first document addresses an aspect of Skolem's paper on arithmetic that has gone ignored because of the influence of the Hilbert school and Goedel.

Speaking to Russell's analysis of descriptions, Skolem writes:

"... This conception [Russell's] does not seem to me to be beyond doubt; but even if it were to be correct for the descriptive functions of ordinary language, we would not need to adopt it for the descriptive functions of arithmetic."

The only reason that the Hilbert school took interest in the paper is because it had been "quantifier-free" and, the received view of the period had been that quantifiers were likely sources for the introduction of paradox. No one had even considered a different syntactic formulation for descriptions from that of Russell. Moreover, it is unlikely that they would have had success. The mathematical understanding of "logic" had not progressed sufficiently to develop alternatives.

Formal derivations can be found from page 15 to page 45.

First-order logic is a logic of existential import. Consequently, the syntax of a definite description conveys the existence of the object described. The use of descriptions in ordinary mathematics does not work this way. Definitions are given. Then, they are substantiated as non-vacuous (usually with an example). If what is defined is meant to be an individual, then a uniqueness proof is also provided.

One problem with addressing this situation is that syntax must fit together like a crossword puzzle. If you have to modify inference rules, a great deal of syntax must also be modified. So, implementing WHAT SKOLEM ACTUALLY WROTE does not admit of a minor adjustment to Goedel's blatant and unsubstantiable platonism.

Simply recreating this sequent form, however, is not enough. Authors like Wittgenstein offered legitimate concerns over the idea of a "defined existent."

The aspect of a "defined existent" must be resolved by differentiating definite descriptions into attributive, referential, and demonstrative uses. The descriptions in the first document are a "proposal" for what constitutes the syntactic structure for a demonstrative description. The definiens is characterized by an existential form with a nested universal asserting syntax needed for a unuqueness proof.

The way to understand these classifications is to recognize Frege as having used descriptions referentially. Because this impacted classical bivalence, Russell took exception. His analysis of descriptions became known as the attributive use. Strawson (like Skolem) had not been convinced and reasserted a referential use for descriptions. A large portion of the literature on descriptions will include the debate between referential and attributive uses.

Tangentially, Russell had objected to Strawson by claiming that Strawson had ignored Russell's work on demonstratives. In more modern treatments, Kaplan's logic of demonstratives takes center stage. Like Frege's semantics (and the theory of partial functions) there is a "null object" intended as the denotation for meaningless expressions.

Demonstrative descriptions come up in a separate context, though. The debate over the nature of descriptions impacts "fact theory." The problem lies with "slingshot arguments":

https://en.m.wikipedia.org/wiki/Slingshot_argument

https://plato.stanford.edu/entries/facts/slingshot-argument.html

These arguments are understood with respect to the distinction between attributive and referential uses. What I had noticed in Neale's discussion of facts and the slingshot argument is that a third notion --- a demonstrative description --- was being presented as unproblematic.

The problem then became one of sorting out syntax that could fit together properly.

There is also a problem involving individuation.

What is presupposed when mindlessly asserting that the sign of equality means "identity" is that "oneness" is implicit. This is "true in a paradigm," perhaps.

I had to learn about Skolem's work the hard way. Claiming that "set theory is the foundation of mathematics" is a logicist claim. Skolem's analysis of the fact that Zermelo's set theory could not be made categorical constitutes a shift to universal algebra. What goes unnoticed, however, is that this is better traced back to Augustus de Morgan. According to de Morgan, the sign of equality can never be made "formal" because substitutivity in calculi requires justification.

That means, in turn, that the metaphysical (logicism) and ontological (first-order inference) bases for the necessary truth of reflexive equality statements are paradigmatic.

At best, "identity" is a warrant for self-substitutions in the calculi associated with paradigms.

If you are going to claim that I am making "mistakes," you had better take me to your "burning bush."

The second document above also has derivations. These involve the pair of relations I refer to as distinctness and discernibility.

Denied distinctness conveys existential import because its assertion involves nested universal quantifiers.

Denied discernibility is without existential import, as is discernibility itself. It is characterized without any nested quantifiers.

The second document analyzes the situation in relation to metric axioms. These relate directly to geometric intuitions associated with analysis from ordinary mathematics. Most students of mathematics who are not indoctrinated with the philosophy of formal systems will interpret the metric axioms as an application of the principle of the identity of indiscernibles. Analytic philosophers and first-order logicians have done a great deal of work excluding this principle as a "logical principle."

Although it may seem unrelated, the interpretation of Goedel's incompleteness theorems as distinguishing between truth and provability is related to this interpretation of metrics. Homotopy type theory is "verificationist" foundation. One reason for this is that "contractibility" to a point undermines the distinction betwen open and closed sets found in point-set topology. My original idea had been based upon Cantor's nested set theorem for closed sets of vanishing diameter. Contractibility is a property of *continuous* maps. It assumes that there is always a "point" where distances become infinitesimal. That confuses continuity with completeness.

And, in general, category theory is assuming the consistency of mathematics.

As for comparing me with Mr. Olcott, I have never claimed that established results like the incompleteness theorems or the halting problem are in any way invalid.

As for your demand for some sort of algorithm, I have little interest in the pseudoproblems arising within the philosophy of formal systems. The altered inference rules are best understood with respect to the "givenness" of Markov's constructive mathematics. Skolem had advocated for a "recursive mode of thought" and Markov's system is Turing complete. I am not the one who is befuddled by Goedel's platonism.

Quoting conventions arise, in part, to avoid semantic paeadoxes. Great. So, you use a metalanguage to talk about truth in an object language.

Are you really talking about truth?

You need a metametalanguage to talk about truth in the metalanguage.

Are you really talking about truth?

Shall I go on?

Sadly, scientists and philosophers engaged with religious dogma at the end of the 19th century with the expectation that "logic, mathematics, and evidence" would "prove" their beliefs to be superior to those they chose to ridicule.

Whatever they got from these debates, what they did not get is any conception of truth that can be differentiated from their personal beliefs (unless they have the infinite wisdom of an unending hierarchy of metalanguages).

And, I will be happy to talk about elements of Smullyan's book on the continuum hypothesis.

Yes. I did the work you have demanded.

I have every expectation that you will not look at it.

[Ross Finlayson]

That's a pretty great post there, Mitch, that's pretty great, I enjoyed that,
and, to some extent, appreciate it.

There's much or maybe not-so-much to be said about fundamentally a
difference "numbering" and "counting".

There's though that I read Cohen's stipulation that CH is independent ZF that everybody
accepts a long time ago.

I don't know anything due Smullyan, given that for example I reject "material" implication as unsound.

There's a big difference between "domains" and "universe". "Universe" is singular. A "domain" may
be a closed "world", and a "little private universe", but it is not "the" "universe".

It's agreeable that "meta" is "relative" while "absolute" includes some, "absolute relativity",
or "objective subjectivity" as it were. This is where also the "higher-order" is essentially contrived
in terms of for example "higher order types" making for expansion and collapse or extension
and collapse, like when you mention Skolem, for Levy and Skolem, for "first-order formalization",
and where "there's a pure logic and it's first-order, no meta", or even "zeroeth-order: ordering in theory".


About continuity, there's at least three definitions of continuity:
line-reals, ran(EF), sweep, after the spiral space-filling curve, Aristotle's, half Peano's,
field-reals, Eudoxus, Archimedean, Cauchy/Weierstrass, Dedekind, algebra + LUB + measure 1.0,
signal-reals, Nyquist and Shannon, signal analysis' discrete analog, smooth grain, huge rationals',
where these are book-ended by the long-line reals
long-line-reals,
..., Du Bois-Reymonds' infinitarcalcul's, all real functions', ...
long-line-reals,
that that's THREE definitions of continuity not just ONE that implies that for NUMBERING
and CLOCK ARITHMETIC and SIGNAL RECONSTRUCTION there are THREE definitions of continuity,
at least, and not just ONE, the usual "a conservative extension of ZF's, plus LUB and measure being 1.0
being axiomatized, where category theory and HoTT's univalency or the illative is an unsatisfying, unrigorous
tacking on to the strength of ZF instead of showing why numbering means there are some functions
that aren't Cartesian in terms of their domains, when for example the world is a spiral space-filling curve".


So, you can find a camp of Skolemites here, also.


I recorded this monologue on "descriptive differential dynamics" since reading your post,
for "symmetries, submersions, sheaves: descriptive differential dynamics".

https://www.youtube.com/watch?v=RtdXHM6k07Y




I identify with being a logical positivist but have to reject what calls itself "logical positivism"
kind of like rejecting "humanism" which is really "animalism" or "platonism" which is really "obscurantism".
So, I'm unwilling to abandon "strong mathematical platonism: a conscientious logical positivism"
to "obscurantist if not oxy-moronic conservative but inconsistent extensions of the concluded".

It's logical.

[Rich D]

On April 2, Jeff Barnett wrote

>> I) "This sentence contains seven words."
> False.
>> Therefore its negation must be true:
>> II) "This sentence does not contain seven words."
>
> John McCarthy in the late 1960s ran around expressing suspicion about
> doing logic with anything that contained quote marks.

McCarthy's exercise regimen, and paranoia, have no bearing
on the paradox above.

> One of his favorite examples was the sentence:
> "The queen asked whether `Was Sir Walter Scott the author of Waverley?'"
> One needed to avoid the deduction that this sentence was equivalent
> to "The queen asked whether true?"

um yeah, let's avoid that deduction, because the sentence doesn't
parse. Trivially.

A valid, meaningful, sentence would be:
"The queen asked whether Sir Walter Scott was the author of Waverley."
or: "The queen asked: `Was Sir Walter Scott the author of Waverley?'"

But McCarthy's "favorite example" is scatterbrain, as is your
memo, and doesn't at all address the paradox above.

> Rather, sketch an algorithm that you think might deal with
> examples of the above sort while avoiding mistakes
> of a six year old.

OK, here's the algorithm:

P --> ~(~P)

This algorithm/tautology fails in the example given.
Pop quiz, dude: is first order logic refuted?

--
Rich

[Jeff Barnett]

It seems you have lost complete track of what logic is supposed to do
(be useful for). I presumed you know who John was and you are the very
first one I've ever heard call him paranoid. I've heard him called many
other things - some true, some not - but paranoid isn't one of them. I
suppose you don't know that at the time he was saying the above that
many computational linguist and logicians (looking to improve
linguistics) made the mistake indicated by the above in sketches of
proposed solutions to various problems.

The people he was talking to were much like PO and you and as PO has
told you, many many times, pay attention!
--
Jeff Barnett

[Rich D]

On April 3, Jeff Barnett wrote:
>>>> I) "This sentence contains seven words."
>>> False.
>>>> Therefore its negation must be true:
>>>> II) "This sentence does not contain seven words."
>
>>> John McCarthy in the late 1960s ran around expressing suspicion about
>>> doing logic with anything that contained quote marks.
>
>> McCarthy's exercise regimen, and paranoia, have no bearing
>> on the paradox above.
>
>>> One of his favorite examples was the sentence:
>>> "The queen asked whether `Was Sir Walter Scott the author of Waverley?'"
>>> One needed to avoid the deduction that this sentence was equivalent
>>> to "The queen asked whether true?"
>
>> um yeah, let's avoid that deduction, because the sentence doesn't parse.

>> A valid, meaningful, sentence would be:
>> "The queen asked whether Sir Walter Scott was the author of Waverley."
>> or: "The queen asked: `Was Sir Walter Scott the author of Waverley?'"
>> But McCarthy's "favorite example" is scatterbrain, as is your
>> memo, and doesn't at all address the paradox above.
>
>>> Rather, sketch an algorithm that you think might deal with
>>> examples of the above sort
>

>> OK, here's the algorithm:
>> P --> ~(~P)
>> This algorithm/tautology fails in the example given.
>> Pop quiz, dude: is first order logic refuted?
>

> It seems you have lost complete track of what logic is supposed to do.

uh, a logic problem, which contains an apparent contradiction
(a/k/a paradox), seems to you a loss of track of the purpose of logic?

> I presumed you know who John was and you are the very
> first one I've ever heard call him paranoid.

I responded to YOUR remark on his behavior and irrational
fear; which is paranoia, by definition.

> I suppose you don't know that at the time he was saying the above that

> many computational linguist and logicians made the mistake indicated by

> the above in sketches of proposed solutions to various problems.

um, scatterbrain errors from 50 years ago have no bearing on
the paradox presented above.

Try again.  Do try to focus your powerful brain on the problem at hand:

I) "This sentence contains seven words."
False.

Negate (I), thereby creating a true statement:

II) "This sentence does not contain seven words."

Either answer the challenge, or admit that you can't.  Or continue
your scatterbrain tap dancing, which offers some amusement value -

> The people he was talking to were much like PO and you and as PO has
> told you, many many times, pay attention!

PO?

--
Rich

[olcott]

Hence a good reason for Tarski's metalanguage.
Wordcount("This sentence contains seven words") == 7 is false
Wordcount("This sentence does not contain seven words") != 7 is false.



--
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]

You mean "trust but verify"?

Is there, "does this sentence satisfy the maxims of Grice"?

What kind of Quine atom is that?

Do many naive axiomatizations of ZF's well-foundedness admit
any set containing the empty set?

What is the operation of the group of all groups?
Are there more than one?

I wonder that if you replace "uh" with "a..." an indefinite article,
and "um", with "om", say, things read easily.

What's regularity in Aczel's non-well-founded?

What if there are no logical paradoxes?

[Ross Finlayson]

"What's the Lie of all Lies?" "There's a Truth."
"What's the Truth of all Truths?" "There's a Truth."

Truth or Lies, take your pick.

[olcott]

On 4/5/2023 12:28 AM, Ross Finlayson wrote:
> On Tuesday, April 4, 2023 at 4:09:04 PM UTC-7, olcott wrote:
>> On 3/31/2023 1:05 PM, Rich D wrote:
>>> I) "This sentence contains seven words."
>>>
>>> False.
>>> Therefore its negation must be true:
>>>
>>> II) "This sentence does not contain seven words."
>>>
>>>
>>> --
>>> Rich
>> Hence a good reason for Tarski's metalanguage.
>> Wordcount("This sentence contains seven words") == 7 is false
>> Wordcount("This sentence does not contain seven words") != 7 is false.
>>
>>
>>
>> --
>> Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
>> hits a target no one else can see." Arthur Schopenhauer
>
> You mean "trust but verify"?
>

I don't mean anything like that. I mean that every English sentence is
fully translated into some form of higher order logic.


*Introducing the foundation of correct reasoning*
(the actual way that analytical truth really works)

Just like with syllogisms conclusions are a semantically necessary
consequence of their premises

Semantic Necessity operator: ⊨□

(a) Some expressions of language L are stipulated to have the semantic
property of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
P is a subset of expressions of language L
T is a subset of (a)

Provable(P,X) means P ⊨□ X
True(X) means X ∈ (a) or T ⊨□ X
False(X) means T ⊨□ ~X

> Is there, "does this sentence satisfy the maxims of Grice"?
>
> What kind of Quine atom is that?
>
> Do many naive axiomatizations of ZF's well-foundedness admit
> any set containing the empty set?
>
> What is the operation of the group of all groups?
> Are there more than one?
>
> I wonder that if you replace "uh" with "a..." an indefinite article,
> and "um", with "om", say, things read easily.
>
> What's regularity in Aczel's non-well-founded?
>
> What if there are no logical paradoxes?
>

[Ross Finlayson]

How about what is the ordinal of all ordinals?
How about when there were only finite ordinals,
how is that any different than Russell's extra-ordinary ordinal?

(The ordinal of all ordinals is named Ord,
the group of all groups is named Grp.)

What we can "intuit" and what we can "infer" mostly arises
from what we can "deduce".

I'm quite a believer that natural language admits a logical
form, and in the thread about "Question Words" ("interrogatives")
there's a reference to where "natural language is translated
to first-order logic". Then the idea is that there are duals
as about statements and questions posed, about connectives,
which make for flexible formal knowledge representations,
then that there are some linear statements that don't require
recursion to validate, and others that do.

This changes the closed Chomsky hierarchy, into a higher-order
hierarchy, of those that do and those that don't, require recursion
(to validate).

Then natural languages are better expressed in dependency grammars,
then for as along the lines of the "Question Words" thread.

[Mitchell Smith]

Well, Ross, while I am glad you liked it somewhat, it is actually a bad post.

I am still getting used to the interface I am using and mistook a post by Mr. Barnett as being a reply to me. I probably owe him some kind of apology.

Still, his posting had been of variety with which I am familiar.

It contained the demand that any (perceived) critic of received views formulate a logic of "equal rigor," if such an expression can carry any significance.

mitch

[Ross Finlayson]

At least we might agree that "Hilbert's Infinite Living Museum" never empties.

It does get full, though.

I enjoyed your post, there, I'm sort of a fanatic about logicism and
also have learned some things from you about lattices.