Why is model theory needed?

[olcott]

Well-formed formulas have meaning only when an interpretation is given
for the symbols. Mendelson

No one seems to know why model theory is needed.
A ∧ B → A is known to be true on the basis of the meaning of the
symbols, no model theory needed.

--
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]

Well, there's "the applied", and, also "the pure".

[olcott]

Applied what and pure what and how do they make any difference?

[Ross Finlayson]

Also there's a great complement to induction, analytical deduction.

Also there's not confusing "follow instructions" and "make judgments".

Mapping domain objects to formalisms at all, it's modeled in model theory.

It's "the working meta-theory" for many who don't have "a working pure theory",
what is "their working theory" and "their working logic".

Psychology doesn't work on everybody, but everybody has one.

Model theory is to theory as theory is to theory.

Abstraction as its fundamental operation, then there's arithmetization and
geometrization as very strong algebraizations, besides of course that there
are models of indeterminism besides these strong models of determinism.
Besides determinism, there's also "the closed", with regards to a usual theory
that's "an open system", like models of physics, vis-a-vis conservation and continuity.

The "pure" and the "applied" theory, is for the logical and the elements of an axiomatic
domain the non-logical, not to confuse flow machines with objective cognition.

The "working theory" is both the pure and the applied.

A real pure theory of everything in the West is often seen invoked a
"Leibniz' monadology" and a "Plato's ideals" and a "monism", which is
a holism that is a wholism, about the holonomic, in the monodromic.

Then, also "pure theory" is several things, about the realm of deduction,
where, deduction is altogether grander than induction, about whether
"this pure theory is the domain of all the following theory" like set theory
is for descriptive set theory the today's formalisms called foundations,
vis-a-vis the necessities and desiderata of a formalism called foundations,
a foundation. There's a pure theory that's a model of everything thus
a universal theory, then also "pure theories" that are only any models,
abstractly not applied, that "the applied" is the application of what was
abstraction to what results objectively so, for various working theories
that do or don't include philosophical foundations like rigor logically and
mathematically, critical reasoning vis-a-vis un-advised induction or
rule hysteresis, and these kinds of things in cognitive and numerical and
logical resources, the resources of which are the inner and outer or
esoteric and exoteric machinations of theories in work, which result
belief or knowledge or phenomena for the noumena, these types things.


These types things, ....

[André G. Isaak]

On 2023-04-08 13:16, olcott wrote:
> Well-formed formulas have meaning only when an interpretation is given
> for the symbols. Mendelson
>
> No one seems to know why model theory is needed.
> A ∧ B → A is known to be true on the basis of the meaning of the
> symbols, no model theory needed.
>

((A ∧ B) → A) is a statement which is true in *all* models, so of course
you don't need to specify the model. And this is properly true on the
basis of the meaning of the *connectives*, not the meaning of the
symbols themselves.

((A ∧ B) → A) can be evaluated without reference to the symbols A or B.

The same cannot be said for (A ∧ B). How would you evaluate (A ∧ B) as
true or false without some model?

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

[olcott]

On 4/8/2023 3:27 PM, André G. Isaak wrote:
> On 2023-04-08 13:16, olcott wrote:
>> Well-formed formulas have meaning only when an interpretation is given
>> for the symbols. Mendelson
>>
>> No one seems to know why model theory is needed.
>> A ∧ B → A is known to be true on the basis of the meaning of the
>> symbols, no model theory needed.
>>
>
> ((A ∧ B) → A) is a statement which is true in *all* models, so of course
> you don't need to specify the model. And this is properly true on the
> basis of the meaning of the *connectives*, not the meaning of the
> symbols themselves.
>
> ((A ∧ B) → A) can be evaluated without reference to the symbols A or B.
>
> The same cannot be said for (A ∧ B). How would you evaluate (A ∧ B) as
> true or false without some model?
>
> André
>

The same way that this is done in the syllogism.

[olcott]

On 4/8/2023 3:27 PM, André G. Isaak wrote:

> On 2023-04-08 13:16, olcott wrote:
>> Well-formed formulas have meaning only when an interpretation is given
>> for the symbols. Mendelson
>>
>> No one seems to know why model theory is needed.
>> A ∧ B → A is known to be true on the basis of the meaning of the
>> symbols, no model theory needed.
>>
>
> ((A ∧ B) → A) is a statement which is true in *all* models, so of course
> you don't need to specify the model. And this is properly true on the
> basis of the meaning of the *connectives*, not the meaning of the
> symbols themselves.
>
> ((A ∧ B) → A) can be evaluated without reference to the symbols A or B.
>
> The same cannot be said for (A ∧ B). How would you evaluate (A ∧ B) as
> true or false without some model?
>
> André
>

*That was a very superb answer*
My reply is to handle (A ∧ B) exactly the way that the syllogism
handles them thousands of years before anyone thought of model theory.

Since we didn't need model theory to do this thousands of years ago why
do we need it now?

[André G. Isaak]

On 2023-04-08 15:04, olcott wrote:
> On 4/8/2023 3:27 PM, André G. Isaak wrote:
>> On 2023-04-08 13:16, olcott wrote:
>>> Well-formed formulas have meaning only when an interpretation is given
>>> for the symbols. Mendelson
>>>
>>> No one seems to know why model theory is needed.
>>> A ∧ B → A is known to be true on the basis of the meaning of the
>>> symbols, no model theory needed.
>>>
>>
>> ((A ∧ B) → A) is a statement which is true in *all* models, so of
>> course you don't need to specify the model. And this is properly true
>> on the basis of the meaning of the *connectives*, not the meaning of
>> the symbols themselves.
>>
>> ((A ∧ B) → A) can be evaluated without reference to the symbols A or B.
>>
>> The same cannot be said for (A ∧ B). How would you evaluate (A ∧ B) as
>> true or false without some model?
>>
>> André
>>
>
> *That was a very superb answer*
> My reply is to handle (A ∧ B) exactly the way that the syllogism
> handles them thousands of years before anyone thought of model theory.
>
> Since we didn't need model theory to do this thousands of years ago why
> do we need it now?

And what do syllogisms have to do with this? How exactly do you evaluate
(A ∧ B) in absence of a model using syllogisms?

[olcott]

Syllogisms never allowed semantics to be removed from logic thus there
is no need to put semantics back in.

With the syllogism referring to A and B without having already defined
them is simply flatly wrong.

[André G. Isaak]

Apparently you have a very idiosyncratic definition of 'syllogism'.

> With the syllogism referring to A and B without having already defined
> them is simply flatly wrong.

Aristotelian logic (from which we get the term 'syllogism') explicitly
deals with symbolic terms independent of their semantics. He defines
which forms constitute valid syllogisms *without* reference to the
meanings of the terms involved. That's the entire foundation of logic --
to focus on the *form* an argument takes rather than the meanings of the
terms involved.

You should probably stop using the term 'syllogism' until you've
actually *read* the works of Aristotle and his successors.

[Julio Di Egidio]

On Saturday, 8 April 2023 at 23:34:41 UTC+2, André G. Isaak wrote:
> On 2023-04-08 15:23, olcott wrote:
<snip>

> > With the syllogism referring to A and B without having already defined
> > them is simply flatly wrong.
>
> Aristotelian logic (from which we get the term 'syllogism') explicitly
> deals with symbolic terms independent of their semantics. He defines
> which forms constitute valid syllogisms *without* reference to the
> meanings of the terms involved. That's the entire foundation of logic --
> to focus on the *form* an argument takes rather than the meanings of the
> terms involved.

Except that to recognize the logical form of a statement you must
be able to understand its meaning to begin with.

> You should probably stop using the term 'syllogism' until you've
> actually *read* the works of Aristotle and his successors.

Try and also learn the limitations of it.

Julio

[olcott]

Try and provide an example of a correct syllogism that lacks sufficient
semantics to evaluate whether it is true or false.

Basic structure
A categorical syllogism consists of three parts:

Major premise
Minor premise
Conclusion

Each part is a categorical proposition, and each categorical proposition
contains two categorical terms.[13] In Aristotle, each of the premises
is in the form "All A are B," "Some A are B", "No A are B" or "Some A
are not B", where "A" is one term and "B" is another:

https://en.wikipedia.org/wiki/Syllogism#Basic_structure

[André G. Isaak]

All As are Bs.
X is an A.
Therefore X is a B.

That's a perfectly valid syllogism despite the fact that I have provided
no semantics for either A, B, or X.

> Basic structure
> A categorical syllogism consists of three parts:
>
> Major premise
> Minor premise
> Conclusion
>
> Each part is a categorical proposition, and each categorical proposition
> contains two categorical terms.[13] In Aristotle, each of the premises
> is in the form "All A are B," "Some A are B", "No A are B" or "Some A
> are not B", where "A" is one term and "B" is another:

And you'll note that neither 'A' nor 'B' are defined in the above,
because they don't have to be as we are focusing on the form of
arguments rather than their contents.

[olcott]

You failed to meet the requirements:

Try and provide an example of a correct syllogism that

*lacks sufficient semantics to evaluate whether it is true or false*
*lacks sufficient semantics to evaluate whether it is true or false*
*lacks sufficient semantics to evaluate whether it is true or false*

>> Basic structure
>> A categorical syllogism consists of three parts:
>>
>> Major premise
>> Minor premise
>> Conclusion
>>
>> Each part is a categorical proposition, and each categorical
>> proposition contains two categorical terms.[13] In Aristotle, each of
>> the premises is in the form "All A are B," "Some A are B", "No A are
>> B" or "Some A are not B", where "A" is one term and "B" is another:
>
> And you'll note that neither 'A' nor 'B' are defined in the above,
> because they don't have to be as we are focusing on the form of
> arguments rather than their contents.
>
>

--

[Julio Di Egidio]

On Sunday, 9 April 2023 at 01:05:21 UTC+2, André G. Isaak wrote:

> On 2023-04-08 16:51, olcott wrote:

> > Try and provide an example of a correct syllogism that lacks sufficient
> > semantics to evaluate whether it is true or false.

Wrong question.

> All As are Bs.
> X is an A.
> Therefore X is a B.
>
> That's a perfectly valid syllogism despite the fact that I have provided
> no semantics for either A, B, or X.

How do you know that it is "perfectly valid" if you do not at least try
it in a couple of instances? Blind faith? Indeed, on the contrary,
how/when to apply it is what there is to learn!

> And you'll note that neither 'A' nor 'B' are defined in the above,
> because they don't have to be as we are focusing on the form of
> arguments rather than their contents.

Except that *to recognize the logical form of a statement you must
be able to understand its meaning to begin with*. Then, and only
then, you have a "form" to possibly play with.

Consider this:
- Balls are spheres.
- This gala is a ball.
- Hence, this gala is a sphere.

Julio

[Julio Di Egidio]

Corollary: You don't even know your Aristotle!

Julio

[André G. Isaak]

Syllogisms are neither true nor false. Propositions are true or false.
Syllogisms are valid or invalid.

André

[Ross Finlayson]

Multi-valent logics include inner and outer products where
tertium-non-datur or law-of-excluded middle do not satisfy.

It's a dialectic what results resolution of logical paradox
under necessary distinction of terms.

If you have a logical paradox in your theory,
it's not done.

[Julio Di Egidio]

> Multi-valent logics include inner and outer products where
> tertium-non-datur or law-of-excluded middle do not satisfy.

Bullshit. If you actually study Aristotle, i.e. the theory of
syllogisms, you learn that the validity is *a consequence of*
(and, post facto, is equivalent to) a specific relationship of
meaning between the terms. Whence a proper example is
*not* "A->B" but rather "Socrates is Mortal", i.e. with what that
entails in terms of what we (need to) know about the world.

Julio

[Ross Finlayson]

Sure, that's included.

It's, ..., weak, and a fragment, and simple enough for usual digestion,
it's included.

This way though "paradox" is not included, and, is excluded.
This is where, how that's so, is it's discovered, to not exist.

"Socrates is Mortal", is plainly proper. "Socrates is Mortal", though
in this use/mention distinction that is a use/mention distinction,
is plainly logical.

"When is model theory not needed?" ... When you forgot why not.

This is for the important facility of "corrective memory",
that what you forget just isn't that important, because you
know that "Socrates is Mortal" is both, plainly, proper and logical.

[Julio Di Egidio]

On Sunday, 9 April 2023 at 02:12:23 UTC+2, Ross Finlayson wrote:

> This way though "paradox" is not included, and, is excluded.
> This is where, how that's so, is it's discovered, to not exist.

Bullshit ad nauseam...

> because you
> know that "Socrates is Mortal" is both, plainly, proper and logical.

I don't *know* any such thing...

EOD.

Julio

[olcott]

Major premise: All humans are mortal.
Minor premise: All Greeks are humans.
Conclusion: All Greeks are mortal.

The typical syllogism anchors its semantics in defined sets, thus its
conclusion can be evaluated as true or false.

https://en.wikipedia.org/wiki/Syllogism#Basic_structure

[olcott]

Great reply, I agree. You said it better than I could.

[André G. Isaak]

And the definitions of those sets are what constitutes a model.

The above syllogism is *valid*, but whether it is sound (thereby
allowing you to conclude that the conclusion is true) is outside the
scope of formal logic.

There is nothing in logic which tells you whether the statement 'all
humans are mortal' is true or not -- observation tells us the all humans
who are already *dead* are mortal, but assumption that currently living
humans are mortal is not something that we know with any certainty (the
first immortal human may already be among us). We can, however, posit
models in which all humans are indeed mortal just as we can posit models
in which this claim is not actually true.

Similarly, your minor premise is only true in some models. In a model
which includes greek centaurs, greek dogs, greek pegasi, etc. it is false.

Formal logic studies those aspects of arguments which logic *can* tell
us about, namely whether a given conclusion follows from a given set of
premises. That is, whether the form of the argument is valid. Evaluating
the truth or falsity of individual statements must, except in the case
of tautologies and contradictions, rely on extra-logical information. It
is often convenient to make assumptions about that extra-logical
information, and we refer to that set of assumptions as a model.

[Julio Di Egidio]

On Sunday, 9 April 2023 at 02:46:55 UTC+2, André G. Isaak wrote:

> Formal logic studies those aspects of arguments which logic *can* tell
> us about, namely whether a given conclusion follows from a given set of
> premises. That is, whether the form of the argument is valid. Evaluating
> the truth or falsity of individual statements must, except in the case
> of tautologies and contradictions, rely on extra-logical information.

I disagree. You are still missing the point that a syllogism
is *invalid* if any of its constituents is false.

Julio

[André G. Isaak]

Apparently you do not understand the meaning of 'valid'. The following
is a perfectly valid syllogism:

All men are geese.
All chairs are men.
Therefore all chairs are geese.

It isn't *sound* as both its premises and the conclusion are false, but
it is valid.

[Julio Di Egidio]

It is NOT, that is called lying with logic!
You don't know what a syllogism is...

Julio

[olcott]

So what is there to model theory besides the definitions of sets being
applied to terms in a formal expression?

> The above syllogism is *valid*, but whether it is sound (thereby
> allowing you to conclude that the conclusion is true) is outside the
> scope of formal logic.
>

Not when you examine Tarski's formal logic.

> There is nothing in logic which tells you whether the statement 'all
> humans are mortal' is true or not

It is an axiom.

> -- observation tells us the all humans
> who are already *dead* are mortal, but assumption that currently living
> humans are mortal is not something that we know with any certainty (the
> first immortal human may already be among us). We can, however, posit
> models in which all humans are indeed mortal just as we can posit models
> in which this claim is not actually true.
>
> Similarly, your minor premise is only true in some models. In a model
> which includes greek centaurs, greek dogs, greek pegasi, etc. it is false.
>
> Formal logic studies those aspects of arguments which logic *can* tell
> us about, namely whether a given conclusion follows from a given set of
> premises. That is, whether the form of the argument is valid. Evaluating
> the truth or falsity of individual statements must, except in the case
> of tautologies and contradictions, rely on extra-logical information. It
> is often convenient to make assumptions about that extra-logical
> information, and we refer to that set of assumptions as a model.
>
> André
>
>

--

[Julio Di Egidio]

Or, logic as opposed to mathematics.

Compare these two arguments:

- All men are geese,
- All chairs are men,
- Therefore all chairs are geese.

- If all men are geese,
- If all chairs are men,
- Then all chairs are geese.

The latter is "valid", but it is not a *syllogism*.

Indeed, from the former we would conclude that
"all chairs are geese" (if it where *valid*, in the sense
of the premises being reciprocally true, see my initial
conter-example, not just the form being correct):
IOW, when and only when *validly in that form*, it is
an *inference rule*; while the second conclusion
remains conditional and just "vacuously valid".

Julio

[olcott]

Even this has the connected semantics of shared terms.
The principle of explosion is not possible with syllogisms.

[Ross Finlayson]

I enjoyed "Shaffer's Laser" for "ex falso nihilum".

http://www.jonathanschaffer.org/laser.pdf

[Julio Di Egidio]

On Sunday, 9 April 2023 at 04:12:08 UTC+2, Ross Finlayson wrote:

> I enjoyed "Shaffer's Laser" for "ex falso nihilum".

Already debunked, you.
You piece of shit are a fucking calamity,
systematically shitting on everything that is not braindead.
Eat your own shit and die, you piece of shit!

*Plonk*

Julio

[Mostowski Collapse]

While the syntactic consequence is handy to show that
something is "generally valid", the semantic consequence
is handy to show that something is "not generally valid".

It only requires soundness of the logical calculus:

/* Soundness */
|-A => |= A
https://en.wikipedia.org/wiki/Soundness

The approach to show that something is "not generally valid"
is then to exhibit a counter model M', such that M'[A]=0.
We then have |/= A, since it is not anymore M[A]=1 for

all M. And then by contraposition we have |/- A. What has
puzzled logicians, such as Gödel etc, as well, whether the
converse direction |=A => |-A also holds.

olcott schrieb am Samstag, 8. April 2023 um 21:17:04 UTC+2:
> Well-formed formulas have meaning only when an interpretation is given
> for the symbols. Mendelson
>
> No one seems to know why model theory is needed.
> A ∧ B → A is known to be true on the basis of the meaning of the
> symbols, no model theory needed.
>

[Mitchell Smith]

On Saturday, April 8, 2023 at 2:17:04 PM UTC-5, olcott wrote:
> Well-formed formulas have meaning only when an interpretation is given
> for the symbols. Mendelson
>
> No one seems to know why model theory is needed.
> A ∧ B → A is known to be true on the basis of the meaning of the
> symbols, no model theory needed.
>
> --
> Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
> hits a target no one else can see." Arthur Schopenhauer

Peter,

A similar question had been asked on MSE two years ago. I have no reason to join the auspicious minds of MSE (who practice censorship), but, I have occasionally answered as a guest.

I gave a two-part answer to the question at the link,

https://math.stackexchange.com/questions/3970711/what-is-the-point-of-model-theory

under 'mls'.

You may find it informative. The original poster did accept my answer.

[Mostowski Collapse]

Quizz: And when did Model Theory start? In its modern form?
I mean mentioning Kant and Newton is Ok, but does it help to
give a picture of **modern** Model Theory. When did it start?

[olcott]

I looked at this. What I really need to start with is the gist of the
purpose of model theory. It looks like it simply defines sets that the
variables range over, just like the syllogism.

[Mostowski Collapse]

Carnap should not be left out, he formed much of modern "semantics",
and had even some purposeful thoughts about his venture:

Elimination of Metaphysics through Logical Analysis of Language
(Überwindung der Metaphysik durch Logische Analyse der Sprache)
https://philarchive.org/archive/TEOv1

LoL

[olcott]

On 4/9/2023 10:15 AM, Mostowski Collapse wrote:
> Carnap should not be left out, he formed much of modern "semantics",
> and had even some purposeful thoughts about his venture:
>

Carnap Meaning Postulates form the basis of Montague Semantics.
https://liarparadox.org/Meaning_Postulates_Rudolf_Carnap_1952.pdf

[Mitchell Smith]

On Sunday, April 9, 2023 at 9:56:50 AM UTC-5, Mostowski Collapse wrote:

The best answer, Jan, is probably with de Morgan and Peacock. I have not read Peacock, but, I can provide quotes from de Morgan's work on double algebra. Among other things, he explicitly discusses how an untrained intellect discovering a formal algebra is likely to apply an interpretation of the symbols independent from the intended interpretations of the original authors.

One can probably relate this to the inscrutability of reference argument attributed to Quine.

De Morgan's problem had been that signs for operations appeared to have the same functions across different types of numbers. Prior to complex numbers and quaternions, no one had actually noticed the problem. Thus, the meaning of operation symbols became relegated to the specification of a domain of discourse.

The algebraic picture is re-introduced into "foundations" when Skolem points out that Zermelo's set theory cannot be categorical. Skolem also recognizes that the same is true of arighmetic and is credited with the first recognitikn of non-standard models.

Gotta go.... can say more later.

mitch

[Mostowski Collapse]

But when did modern model theory start? What one would
nowadays hear in an university course about model theory?
It surely not Skolem paradox and non-categoricity of set theory.

In first-order logic, only theories with a finite model can
be categorical. So what is usually meant by categoricity in
FOL is something weaker, and it can be defined without

model theory, namely. A theory T is complete:

For every formula A either T |- A or T |- ~A.

And incomplete theory is automatically non-categorical.
How would you show that an incomplete theory is
non-categorical, with what peopel would learn today

in an university course about model theory?

[Mitchell Smith]

Peter,

Ask yourself, "What is the role of truth in a proof?"

Truth governs the transition between statements. This is why one speaks of a rule of detachment. A true conclusion is obtained from true premises.

But, rules of detachment can be declared to be merely syntactic. At issue, then, is the distinction between a semantic conception of truth and an analytical conception of truth. Many of your statements are compatible with the historical development of analytical truth. But, this is relegated to intensional logic. The traditional interpretation of Goedel's incompleteness theorem involves distinguishing between truth and provability, and, in turn, this distinction depends upon a semantic conception of truth interpreted as a correspondence theory. Meaning postulates do not --- and cannot --- be used to assert that a term denotes an object.

I agree that all one can eventually conclude from "the received view" is that words are being used to explain words. So, there is something not quite right about all of this "truth talk." The word describing this problem is "deflation," and, it goes all of the way back to Aristotle.

The link,

https://drive.google.com/file/d/1EUTImanTxstmDEBBNr_sJGq5m-Rh9JkG/view?usp=drivesdk

contains a translation from Aristotle's book Categories which I have augmented with my own interpretations relative to my personal knowledge of modern topics.

At the top of page 3, Aristotle speaks of "existence" and "the truth of statements" being reciprocal. Then, he observes that the truth of a statement does not imply existence. So, the reciprocity corresponds with the deflationary nature of truth while the residual asymmetry justifies studying truth as a correspondence principle.

It is difficult to see how these can be studied "univocally" within the same paradigm. And, deflationism, itself, has numerous philosophical aspects,

https://plato.stanford.edu/entries/truth-deflationary/

So, while I can find a quote from Aristotle interpretable with respect to these difficulties, I certainly cannot find a formalization for every nuance that philosophers can imagine.

The first-order paradigm is "a way of studying" mathematics which presupposes a correspondence theory of truth relative to fixed domains of discourse. If you are going to claim that a different paradigm accounts for mathematics, you must accommodate the fact that first-order logic is a part of mathematics.

This cannot be done by any method which equates truth with provability. This very problem is seen with homotopy type theory and its promoters seem only to concern themselves with winning an argument.

And, for what this might be worth, whereas mathematicians communicate with proofs, those who apply mathematics are not under a similar constraint. So, how truth is understood relative to proofs is an aspect of mathematics which introduces these problems involving semantics. I may choose to study mathematics differently from a first-order logician, but, I have an understanding of why I must respect how first-order logicians apply their paradigm. I only take issue with "foundational" claims. It took many years until I could defend the first-order paradigm in spite of not being convinced of its wider claims.

[Mitchell Smith]

On Sunday, April 9, 2023 at 12:15:07 PM UTC-5, Mostowski Collapse wrote:

Jan,

If I open Chang and Keisler, there is much more than first-order logic. In fact, they describe model theory as universal algebra combined with logic. One cannot only look at the logic component. Skolem's contribution pushes foundational studies back to the algebraic perspective.

I would have to do some serious digging to try to answer your specific question.

A modern model theory class would probably speak of Tarski. It might mention something like "East coast" and "West coast" approaches to the subject. If I recall correctly, the East coast approach is associated with Abraham Robinson (Tarski taught on the West coast of the United States).

But, whether spoken of in a class on model theory or not, first-order model theory rests on portraying a dichotomy between syntax and semantics.

Ewald's sourcebook attributes the first notions of a different approach to algebra to Peacock and Gregory. Moreover, this is a development within British mathematics. To Peacock, he attributes the idea of a purely symbolic algebra. To Gregory, he attributes the idea that algebraic symbols could equally well represent operations in addition to numbers. Ewald then says of de Morgan:

"He was the first mathematician to appreciate the importance of the new algebra for the analysis of logic, and the first to provide a reasonably complete and explicit description of a formal calculus."

In his paper, "On the foundations of algebra," de Morgan writes:

"Algebra now consists of two parts, the technical and the logical. Technical algebra is the art of using symbols under regulations which, when this part of the subject is considered independently of the other, are prescribed as the definition of the symbols. Logical algebra is the science which investigates the method of giving meaning to the primary symbols, and of interpreting all subsequent symbolic results."

Our modern language uses different words, but what de Morgan is describing is our distinction between syntax and semantics.

This distinction is more dramatically emphasized in his paper, "Trigonometry and double algebra" where he describes the scenario:

"The proficient in a symbolic calculus would naturally demand a supply of meaning. Suppose him left without the power of obtaining it from without: his teacher is dead, and he must invent meanings for himself. His problem is: Given symbols and laws of combination, required meanings for the symbols of which the right to make those combinations shall be a logical consequence. He tries, and succeeds;he invents a set of meanings which satisfy the conditions. Has he then supplied what his teacher would have given, if he had lived? In one particular, certainly: he has turned his symbolic calculus into a significant one. But, it does not follow that he has done it in a way which his teacher would have taught him had he lived."

So, imagine that Zermelo says, "This means that" while Skolem says "Not so fast." Augustus de Morgan had already identified the situation.

That this is associated with British mathematics is important because Russell had distinguished between material implication and formal implication in "Principles of Mathematics." Tarski had famously abandoned Lesniewski because he saw Russell's work as more fruitful. If I recall correctly, Lesniewski is attributed with identifying the importance of syntactic categories. So, Tarski's work on the semantic conception of truth combines the many facets needed to use the dichotomy between syntax and semantics for the notion of "truth in a model" and the sense of "logical consequence" as a model-theoretic construct.

There is another aspect to de Morgan's scenario given above. Meaningfulness conveyed through instruction may be distinguished from abstract interpretation. So, what is apparently involved is a trichotomy between syntax, semantics, and pragmatics. This trichotomy had been introduced/popularized by Carnap, where pragmatics is intended to refer to meaningfulness as it relates to a language user. This is why I now choose to understand semantics only with respect to the use of truth in derivations. One does not really find this aspect mentioned in typical courses on "mathematical logic." And, analytic philosophers now tend to portray pragmatics so that it no longer is actually bound to human agency. But, at some point an agent must interpret, eh?

There is something else from the second paper mentioned above:

"It is most important that the student should bear in mind that, with one exception, no word nor sign of arithmetic has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct symbolic algebras. [...]

"The one exception above noted, which has some share of meaning, is the sign = placed between two symbols, as in A=B. It indicates that the two symbols have the same resulting meaning, by whatever different steps attained."

Relative to Tarski's semantic conception of truth, the arguments to a sign of equality *must* denote. The necessary truth of reflexive equality is not "given." Both logicism and the first-order paradigm introduce this on the basis of principles different from the claimed semantic theory. Under Tarski's semantic theory of truth, it is possible for reflexive equality statements to be false. That is something else one will not find in a modern class on model theory. And, it is why "foundational" claims reducing mathematics to algebra fail to encompass analysis.

I know this is not the answer for which you had hoped. But, unless you think God has told your teachers a secret, what we have to work from is the documented literature.

mitch

[Mostowski Collapse]

Strictly speaking Algebraic Logic is not the same as Model Theory.
Thats quite a fallacy to identify Model Theory with Algebraic Logic.
I hope you don't commit this fallacy?

Some times this fallacy is based on the confusion of the word
"algebra". The word "algebra" in "Universal Algebra" has not the same
meaning as the word "Algebraic" in "Algebraic Logic".

You can think about it in terms of "truth makers":
- Algebraic Logic: Algebra of truth values of the truth makers
- Model Theory: Algebra of situations the truth makers refer to

But you can combine the two, there is no doubt. And you can
obtain currious results, like in Set Theory and the Continuum Problem,
by Smullyan and Fitting.

[Mostowski Collapse]

Currently it seems Model Theory and Algebraic Logic gets
replaced by Truth Maker Theory? Not sure. There is a little
surge in papers right now, isn't it?

"Truthmaker semantics is a novel formal semantic framework
which has been recently developed in a series of publications
((Fine, forthcoming), (Fine, 2017), (Fine, 2016)) by Kit Fine
starting from the work done by Van Fraassen’s in (Van Fraassen, 1969)

Or its just a combination of Model Theory and Algebraic Logic?
In this paper see Theorem 1, the link is made. If you watch
it carefully, you can see both:

A Truthmaker Semantics Approach to Modal Logic
Giuliano Rosella - 28. April 2021
https://eprints.illc.uva.nl/id/eprint/1792/1/MoL-2019-28.text.pdf

[Mostowski Collapse]

Typically in modern model theory, universal algebra is only
in as far touched, as to get rid of it. LoL If you look at term
models, and Hebrand models, you get rid of the functions

and equality, and keep the relations. So that you have
only the algebra of the relational model, and are not
bothered by functions and equality.

Although Chang & Keisler might claim:

model theory = universal algebra + logic

But its rather the following:

model theory = semantics - universal algebra

There are some books that do it better than Chang & Keisler.
But then there is a certain drive to get Algebraic Logic
on board, which resulted in Continuous Model Theory:

Introduction to Continuous Model Theory
https://faculty.math.illinois.edu/~henson/ASLBoiseMar2017slides.pdf

And people start talking about logic topology.

LoL

[Mitchell Smith]

Absolutely correct. This had been another aspect which I have had to sort out through the years. And, it is a very important one.

For example, in "Sets for Mathematics," Lawvere and Rosebrugh mildly advocate for Tarski's cylindric algebra. This is, of course, an algebraization of first-order logic. But, this algebraization has consequences for inference rules. To my knowledge this is not made specific in any sense comparable with the symbolic methods of first-order logic. Yet, the consequences --- or, more accurately, the diffetences --- are outlined in Appendix C of "Algebraizable Logics" by Blok and Pigozzi.

I would portray my non-propositional inference rules as the comparable symbolic realization assiciated with algebraization. They are motivated by the transitivity axiom of cylindric algebra.

But, to make sense of it in relation to first-order logic, one must recognize the relationship of order to truth persistence in first-order model theory. Existentials are preserved for model extensions (upward). Universals are preserved for submodels (downward).

When I found the Pavicic and Megill paper proving non-categoricity of propositional logic, I knew that it had been an algebraic result. Nevertheless, I had to learn more about orthomodular logic and ortholattices. In 2006, Eric Schechter published a syntactic form of this "hexagon interpretation." More precisely, it is a non-distributive interpretation.

I certainly cannot reproduce the entire analysis from his book, but the link,

https://drive.google.com/file/d/1IkOcACAMIir01yTTNILgsJTxfpluupV9/view?usp=drivesdk

contains examples of his tables. They are 6x6 Cayley tables. The usual truth table relations are to be found in subtables formed from the four corner cells. Schechter uses the symbol for "empty set" as falsity and a capitalized Greek Omega for truth.

The tables are derived from a partial power set on 3 symbols. The file also contains my deliberations on how this partiality may be explained relative to pedagogy from physics.

My personal work also contains a characterization of truth table forms relative to Cayley tables. My Cayley tables are 16x16. My Cayley tables arise from treating truth-functional compositionality independently from Boolean polynomials.

An interesting difference occurs when this abstract compositionality is applied to Schechter's tables. As you are well aware, the join of Boolean atoms will yield one of the 6 central elements of the Boolean lattice. One can define exclusive disjunction and the biconditional well enough with Schechter's methods. But, when you start compsing values according to my methods, Schechter's tables become representatives of classes of tables which share valuations in the corner cells. I would need to write code in order to exhaustively generate the possibilities.

This does not happen with my tables because of an order-isomorphism between the 16-element Boolean lattice and the Cartesian product of 4-element de Morgan lattices. Each such de Morgan lattice has two fixed points in the corresponding de Morgan involution. Consequently, the Cartesian product has four fixed points. These four fixed points coincide with the Boolean generators and their negations.

Whatever I am doing, Jan, I am doing my best to respect extant results in mathematical logic. Part of that involves understanding that algebraic logic and symbolic logic are different.

[Mitchell Smith]

I understand that there are different opinions on model theory. Relative to more modern developments (stability theory, I guess) Hodges summarizes model theory as logic combined with algebraic geometry.

That is somewhat humorous, however. Traditionally, geometry had been eschewed because of "the crisis in geometry" arising from the development of non-Euclidean geometries. And, this is the basic problem: everyone wants to run around being "innovative" without tying their work to previous work (beyond the circle of experts of which they are a member).

Markov's constructive mathematics explicitly recovers classical reasoning by introducing "givenness" relative to what he calls strengthened implication. Russell explicitly discusses "givenness" among alternatives to logicism in Chapter 19 of "Principles of Mathematics." Homotopy type theorists are introducing distinct notions of identity for defined symbols and asserted substitutions. All of this hearkens back to Fregean identity puzzles and Russell's response.

The only thing professionals appear to be doing is to generate publications in the furtherance of personal careers. No students are served by people talking in circles, and, the general public (or at least those wealthy enough to pay taxes) are paying for this nonsense. Our universities are full of people who simply do not know how to cooperate with one another.

But, there is progress. I will not deny that.

By the way, I actually took a course with Henson. My academic career effectively ended at a college level because of illness. I did, however, attempt graduate school at the University of Illinois. It had been reasonably close to my home, and, I wanted to be close in case the medical problem returned. It did. I did not even finish the first semester.

Henson has another paper you would find humorous. Who knew that logic and complex analysis might be related to one another?

https://www.ams.org/journals/tran/1984-282-01/S0002-9947-1984-0728700-X/S0002-9947-1984-0728700-X.pdf

When you speak of Herbrand logic, remember that it is a constructive logic that is not the same as first-order logic in spite of similar syntactic form. Mr. Greene would also mention Herbrand logic at times. Along similar lines, Skolemization in logic programming is different from the mention of Skolem functions in a first-order model. In the latter, one speaks of whether "enough Skolem functions" are available in the same one one speaks of "enough regularity" for certain functions to be applicable.

The idea that mathematics is reducible to logic programming is "another way of studying mathematics." Its appeal, of course, lies with the fact that it is effective.

mitch

[Mostowski Collapse]

The exiting thing right now seems to be truth makers.
So you have to fast forward 100 years, from 1922 Skolem,
to what happens right now in the present, namely 2023.

Some people cauche truth makers in dependent type theory.
Like they use record types as little situation capsules. Leaving
competely behind the Kripke idea of possible worlds.

In as far they might also encounter Howards 1969 weak and
strong existential quantifier again. I find:

Weak sums (existential quantifiers)
In higher-order logic (or simple type theory) as used
in Montague’s semantics, where there is an impredicative
type t of all formulas, it can be either directly introduced
or defined by means of the universal quantifier as in (2).
(2) ∃x.P(x) = ∀X:t. (∀x.(P(x) ⇒ X)) ⇒ X

Strong sums (Σ-types)
Besides being useful mechanisms to organise structures in
various applications, Σ-types may also play other roles. For
example, in Martin-Lof’s type theory, Σ also plays the role of
existential quantifier in its logic.
https://aclanthology.org/2021.cstfrs-1.5.pdf

Again this nothing for my philosophy teacher, who already
missed the lambda expression bandwagon, and is longging
for his retirement. But it could nevertheless fit people, even if

they look like grand grand parents, if they stay young in the mind?

Have Fun!

[Mostowski Collapse]

Is this difficult to do in DC Proof?

/* donkey anaphora */
Every farmer who owns a donkey beats it.

Lets say we want to model such a meaning
postulate? And then derive that the farmer "donald"
who has a donkey "sugar", beats it? Does he?

[olcott]

On 4/10/2023 11:11 AM, Mostowski Collapse wrote:
> Is this difficult to do in DC Proof?
>
> /* donkey anaphora */
> Every farmer who owns a donkey beats it.
>

∀x ∈ Farmer (owns_donkey(x) → beats_donkey(x))

> Lets say we want to model such a meaning
> postulate? And then derive that the farmer "donald"
> who has a donkey "sugar", beats it? Does he?
>

"donald" ∈ Farmer (owns_donkey("donald") → beats_donkey("donald"))
"sugar" ∈ donkey

As long as the meaning postulate's assumptions are true then "donald"
beats "sugar" is true.

[olcott]

Since my goal is to correctly mathematically formalize the notion of
analytic truth I always go the opposite direction:

What is the role of formal proof in the notion of analytic truth?

> Truth governs the transition between statements. This is why one speaks of a rule of detachment. A true conclusion is obtained from true premises.
>

Since {true} is a semantic concept the correct evaluation of the truth
of expressions of language cannot correctly be made apart form
semantics.

> But, rules of detachment can be declared to be merely syntactic.

Semantic can be specified syntactically as Rudolf Carnap meaning
postulates have proved.

> At issue, then, is the distinction between a semantic conception of truth and an analytical conception of truth.

They are the same unless terms of the art overrides the conventional
meaning of these terms.

That cats are animals is analyzed on the basis of the semantic meaning
of these terms.

> Many of your statements are compatible with the historical development of analytical truth. But, this is relegated to intensional logic. The traditional interpretation of Goedel's incompleteness theorem involves distinguishing between truth and provability, and, in turn, this distinction depends upon a semantic conception of truth interpreted as a correspondence theory. Meaning postulates do not --- and cannot --- be used to assert that a term denotes an object.
>
> I agree that all one can eventually conclude from "the received view" is that words are being used to explain words. So, there is something not quite right about all of this "truth talk."

The entire body of the analytic side of the analytic syntactic
distinction is entirely specified as relations between otherwise
meaningless terms.

ALL of the meaning of every natural language or formal language term is
entirely encapsulated in its relation to other terms.

> The word describing this problem is "deflation," and, it goes all of the way back to Aristotle.
>
> The link,
>
> https://drive.google.com/file/d/1EUTImanTxstmDEBBNr_sJGq5m-Rh9JkG/view?usp=drivesdk
>
> contains a translation from Aristotle's book Categories which I have augmented with my own interpretations relative to my personal knowledge of modern topics.
>
> At the top of page 3, Aristotle speaks of "existence" and "the truth of statements" being reciprocal. Then, he observes that the truth of a statement does not imply existence. So, the reciprocity corresponds with the deflationary nature of truth while the residual asymmetry justifies studying truth as a correspondence principle.
>

That sound incoherent to me.

> It is difficult to see how these can be studied "univocally" within the same paradigm. And, deflationism, itself, has numerous philosophical aspects,
>
> https://plato.stanford.edu/entries/truth-deflationary/
>
> So, while I can find a quote from Aristotle interpretable with respect to these difficulties, I certainly cannot find a formalization for every nuance that philosophers can imagine.
>
> The first-order paradigm is "a way of studying" mathematics which presupposes a correspondence theory of truth relative to fixed domains of discourse.

The correspondence theory of truth only applies to empirical truth.
A set of physical sensations corresponds to "there is a TV in my living
room right now"

Analytic truth is purely axiomatic.

> If you are going to claim that a different paradigm accounts for mathematics, you must accommodate the fact that first-order logic is a part of mathematics.
>

*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

> This cannot be done by any method which equates truth with provability.

I just showed the actual inherently correct precise relation between
truth and provability.

The sequence of truth preserving inference steps from (a) to X <is> the
relation between truth and provability.

> This very problem is seen with homotopy type theory and its promoters seem only to concern themselves with winning an argument.
>
> And, for what this might be worth, whereas mathematicians communicate with proofs, those who apply mathematics are not under a similar constraint. So, how truth is understood relative to proofs is an aspect of mathematics which introduces these problems involving semantics. I may choose to study mathematics differently from a first-order logician, but, I have an understanding of why I must respect how first-order logicians apply their paradigm. I only take issue with "foundational" claims. It took many years until I could defend the first-order paradigm in spite of not being convinced of its wider claims.

[Mostowski Collapse]

Where do you derive beats "sugar"? And where
do you even state that "donald" owns "sugar"?

Maybe DC Poop can step in? Oh no, its busy with
finding waldo, ehm, I mean finding falsum:

https://blog.recrutainment.de/wp-content/uploads/2019/03/Waldo_Strand.jpg

[olcott]

On 4/11/2023 2:20 AM, Mostowski Collapse wrote:
> Where do you derive beats "sugar"? And where
> do you even state that "donald" owns "sugar"?
>

Maybe if you didn't top post you would see this.

[olcott]

On 4/8/2023 3:27 PM, André G. Isaak wrote:

> On 2023-04-08 13:16, olcott wrote:
>> Well-formed formulas have meaning only when an interpretation is given
>> for the symbols. Mendelson
>>
>> No one seems to know why model theory is needed.
>> A ∧ B → A is known to be true on the basis of the meaning of the
>> symbols, no model theory needed.
>>
>

> ((A ∧ B) → A) is a statement which is true in *all* models, so of course
> you don't need to specify the model. And this is properly true on the
> basis of the meaning of the *connectives*, not the meaning of the
> symbols themselves.
>
> ((A ∧ B) → A) can be evaluated without reference to the symbols A or B.
>
> The same cannot be said for (A ∧ B). How would you evaluate (A ∧ B) as
> true or false without some model?
>
> André
>

How do we evaluate (A ∧ B) with a model?

[Mostowski Collapse]

Am pretty sure in english grammar "own" and "beat"
are verbs that take a subject and an object.

See also:

"In linguistics, transitivity is a property of verbs that
relates to whether a verb can take objects and how many
such objects a verb can take. It is closely related to valency,
which considers other verb arguments in addition to direct objects."
https://en.wikipedia.org/wiki/Transitivity_(grammar)

But you only show unary predicates owns_donkey/1
and beats_donkey/1. There is surely something wrong!

Also donkey refers to a common noun, and not
to the singular "sugar", i.e. the donkey name "sugar".

Another error.

[olcott]

On 4/11/2023 12:21 PM, Mostowski Collapse wrote:
> Am pretty sure in english grammar "own" and "beat"
> are verbs that take a subject and an object.
>
> See also:
>
> "In linguistics, transitivity is a property of verbs that
> relates to whether a verb can take objects and how many
> such objects a verb can take. It is closely related to valency,
> which considers other verb arguments in addition to direct objects."
> https://en.wikipedia.org/wiki/Transitivity_(grammar)
>
> But you only show unary predicates owns_donkey/1
> and beats_donkey/1. There is surely something wrong!
>
> Also donkey refers to a common noun, and not
> to the singular "sugar", i.e. the donkey name "sugar".
>
> Another error.
>

Owns_Donkey(x) has Donkey as a constant, thus not needed as an argument.

[olcott]

On 4/11/2023 12:21 PM, Mostowski Collapse wrote:

> Am pretty sure in english grammar "own" and "beat"
> are verbs that take a subject and an object.
>
> See also:
>
> "In linguistics, transitivity is a property of verbs that
> relates to whether a verb can take objects and how many
> such objects a verb can take. It is closely related to valency,
> which considers other verb arguments in addition to direct objects."
> https://en.wikipedia.org/wiki/Transitivity_(grammar)
>
> But you only show unary predicates owns_donkey/1
> and beats_donkey/1. There is surely something wrong!
>

Owns_Donkey(x) has Owns and Donkey as constants, thus not needed as
arguments.

> Also donkey refers to a common noun, and not
> to the singular "sugar", i.e. the donkey name "sugar".
>

"sugar" ∈ donkeys // as I already said

[Mostowski Collapse]

Yes thats an error here:

> Owns_Donkey(x) has Owns and Donkey as constants

Well since when is Donkey constant in the problem.
"donald" might own "sugar", but he might
not own "atlass". You can also read the paper:

To see how binary predicates are used.
https://aclanthology.org/2021.cstfrs-1.5.pdf

Although there is one solution that is much
much simpler than the solutions in the paper.
But also this much much simpler solutions

works with predications that are binary. The
straight forward solutions, putting aside being
a farmer or being a donkey, which is irrelevant

if anway owns ⊆ famers x donkeys and
beat ⊆ formers x donkeys is as simple as follows:

∀x∀y(owns(x,y) => beats(x,y))
owns(donald,sugar)
==============================
beats(donald,sugar)

Should be provable in DC Poop.

[olcott]

On 4/11/2023 1:26 PM, Mostowski Collapse wrote:
> Yes thats an error here:
>> Owns_Donkey(x) has Owns and Donkey as constants
>
> Well since when is Donkey constant in the problem.
> "donald" might own "sugar", but he might
> not own "atlass". You can also read the paper:
>
> To see how binary predicates are used.
> https://aclanthology.org/2021.cstfrs-1.5.pdf
>
> Although there is one solution that is much
> much simpler than the solutions in the paper.
> But also this much much simpler solutions
>
> works with predications that are binary. The
> straight forward solutions, putting aside being
> a farmer or being a donkey, which is irrelevant
>
> if anway owns ⊆ famers x donkeys and
> beat ⊆ formers x donkeys is as simple as follows:
>
> ∀x∀y(owns(x,y) => beats(x,y))
> owns(donald,sugar)
> ==============================
> beats(donald,sugar)
>

Yet that includes things that are excluded.
∀x∀y((owns(x,y) ∧ Farmer(x) ∧ Donkey(y)) => beats(x,y))

[Mostowski Collapse]

This works also:

∀x∀y(owns(x,y) ∧ farmer(x) ∧ donkey(y) => beats(x,y))
owns(donald,sugar)
farmer(donald)
donkey(sugar)
==============================
beats(donald,sugar)

[Julio Di Egidio]

On Tuesday, 11 April 2023 at 20:37:35 UTC+2, Mostowski Collapse wrote:
> This works also:
>
> ∀x∀y(owns(x,y) ∧ farmer(x) ∧ donkey(y) => beats(x,y))
> owns(donald,sugar)
> farmer(donald)
> donkey(sugar)
> ==============================
> beats(donald,sugar)

I don't understand what is interesting about that
(even after trying to read about it), apart maybe
from the fact that I cannot find a propositional way
to say it, hence at least 1st order: but especially
since it is a contingent statement, so what...?

Julio

[Ross Finlayson]

Write it there without "material implication" or else,
you can re-write that with the ass before the cart as it were.

[Julio Di Egidio]

Besides, I'd write:
∀x∀y (farmer(x) ∧ donkey(y) -> (owns(x,y) -> beats(x,y)).

Julio

[Julio Di Egidio]

Then compare these two (and there I have stopped):

```coq

(* Donkey anaphora:
"Every farmer who owns a donkey beats it." *)

Module FarmerAndDonkey_typed.

(* If "so it is". *)
Parameter Farmer : Type.
Parameter Donkey : Type.

(* For two binary predicates on the above. *)
Parameter owns : Farmer -> Donkey -> Prop.
Parameter beats : Farmer -> Donkey -> Prop.

Axiom farmer_and_donkey :
forall (x : Farmer),
forall (y : Donkey),
owns x y -> beats x y.

End FarmerAndDonkey_typed.

Module FarmerAndDonkey_untyped.

(* If "he says so". These can also be
defined/axiomatized as identically True. *)
Parameter farmer : Type -> Prop.
Parameter donkey : Type -> Prop.

(* For any two binary predicates. *)
Parameter owns : Type -> Type -> Prop.
Parameter beats : Type -> Type -> Prop.

Axiom farmer_and_donkey :
forall (x : Type),
forall (y : Type),
farmer x /\ donkey y ->
owns x y -> beats x y.

End FarmerAndDonkey_untyped.

```

Julio

[Julio Di Egidio]

On Wednesday, 12 April 2023 at 08:36:31 UTC+2, Julio Di Egidio wrote:

> Module FarmerAndDonkey_untyped.
>
> (* If "he says so". These can also be
> defined/axiomatized as identically True. *)

Sorry, scratch the comment, just <<If "he says so".>>
Axiomatizing those would be emulating the typed way
in the untyped way, i.e. intermediate between the two
solutions I have shown.

Julio

[Julio Di Egidio]

Eh, intermediate and wrong in itself: the typed way
has farmer/donkey as parameters, while axiomatically
locking the semantics of farmer/donkey would actually
not be the same statement. So, just the two ways I have
shown, that's what I (basically) get.

Julio

[Julio Di Egidio]

Here is a propositional version, maybe:

```Coq

(* Donkey anaphora:
"Every farmer who owns a donkey beats it." *)

Module FarmerAndDonkey_prop.

(* If "s/he is so". *)
Parameter ToBeAFarmer : Prop.
Parameter ToOwnADonkey : Prop.
Parameter ToBeatTheDonkey : Prop.

Axiom farmer_and_donkey :
ToBeAFarmer ->
ToOwnADonkey -> ToBeatTheDonkey.

End FarmerAndDonkey_prop.

```

The "problem" with it is that the anaphoric
function, i.e. applying it correctly, is all
in the *naming* and remains up to the user
(notice that "ToBeatADonkey" would NOT be
a correct formalization, though indeed not
in a syntactic sense).

Julio

[Mostowski Collapse]

What could be interesting, is running it through a Montague
parser. Montague grammars aspire to be compositional.
So for example if you say:

every human is mortal

A Montague grammar might give an intensional logic meaning
A to "every human" and a intensional logic meaning B to "mortal",
and then there is even an intensional logic combinator "is",

A = every human
B = mortal
A is B

so that the sentence gets its meaning by the composition
of A and B by the combinator is. Who can intensional logic do that?
I speculate that the Donkey anaphora poses a similar problem.

See also:

3.2 Compositionality
For Montague the principle of compositionality did not seem
to be a subject of deliberation or discussion, but the only way
to proceed. In effect he made compositionality the core part of his
‘theory of meaning’ (Montague 1970c, 378), which was later
summed up in the slogan: ‘Syntax is an algebra, semantics is
an algebra, and meaning is a homomorphism between
them’ (Janssen 1983, 25).

I guess for Coq it should be that difficult to do some
of the intensional logic from Montague, since it offers
higher order logic as well? Can we give the Donkey anaphora

a compositional reading?

[Mostowski Collapse]

Sorry link missing:

3.2 Compositionality
https://plato.stanford.edu/entries/montague-semantics/#Com

[olcott]

On 4/12/2023 9:49 AM, Mostowski Collapse wrote:
> What could be interesting, is running it through a Montague
> parser. Montague grammars aspire to be compositional.
> So for example if you say:
>
> every human is mortal
>

▷ UML inheritance symbol is used to mean <is a type of>
∀x ∈ human (x ▷ mortal)

> A Montague grammar might give an intensional logic meaning
> A to "every human" and a intensional logic meaning B to "mortal",
> and then there is even an intensional logic combinator "is",
>
> A = every human
> B = mortal
> A is B
>
> so that the sentence gets its meaning by the composition
> of A and B by the combinator is. Who can intensional logic do that?
> I speculate that the Donkey anaphora poses a similar problem.
>
> See also:
>
> 3.2 Compositionality
> For Montague the principle of compositionality did not seem
> to be a subject of deliberation or discussion, but the only way
> to proceed. In effect he made compositionality the core part of his
> ‘theory of meaning’ (Montague 1970c, 378), which was later
> summed up in the slogan: ‘Syntax is an algebra, semantics is
> an algebra, and meaning is a homomorphism between
> them’ (Janssen 1983, 25).
>

Natural language is formalized using Rudolf Carnap (1952) Meaning
Postulates. I agree with everything that you said so far.

There is in my opinion no important theoretical difference between
natural languages and the artificial languages of logicians; indeed I
consider it possible to comprehend the syntax and semantics of both
kinds of languages with a single natural and mathematically precise
theory. (Montague 1970c, 373)
https://plato.stanford.edu/entries/montague-semantics/

The sum total of all human knowledge that can be expressed using
language can be encoded in a single formal system
https://en.wikipedia.org/wiki/Ontology_(computer_science)
This is essentially a form of type theory.

> I guess for Coq it should be that difficult to do some
> of the intensional logic from Montague, since it offers
> higher order logic as well? Can we give the Donkey anaphora
>
> a compositional reading?
>
> Julio Di Egidio schrieb am Mittwoch, 12. April 2023 um 06:19:44 UTC+2:
>> On Tuesday, 11 April 2023 at 20:37:35 UTC+2, Mostowski Collapse wrote:
>>> This works also:
>>>
>>> ∀x∀y(owns(x,y) ∧ farmer(x) ∧ donkey(y) => beats(x,y))
>>> owns(donald,sugar)
>>> farmer(donald)
>>> donkey(sugar)
>>> ==============================
>>> beats(donald,sugar)
>> I don't understand what is interesting about that
>> (even after trying to read about it), apart maybe
>> from the fact that I cannot find a propositional way
>> to say it, hence at least 1st order: but especially
>> since it is a contingent statement, so what...?
>>
>> Julio

[Mostowski Collapse]

For UML, this would be enough., your ∀x ∈ rather
introduces a category error. Its enough to say:

human ▷ mortal

Since is_a is between classes, and not between an instance and a class.
See also, inheritance belongs to Class-level relationships

Class-level relationships
Generalization/Inheritance
(subclass) _______▻ (superclass)
https://en.wikipedia.org/wiki/Class_diagram#Class-level_relationships

[Mostowski Collapse]

But with UML, you cannot so much make a distinction between:

every human is mortal
some human is mortal
https://en.wikipedia.org/wiki/Term_logic

The ▷ might pass for "every human is mortal", in that we say
"human ▷ mortal". But what about "some human is mortal"?
In a Montague parser, the "is" would combine both forms

and give a correct result. How can this be done?

A1 = every human
A2 = some human
B = human
A1 is B
A2 is B

[Julio Di Egidio]

On 12/04/2023 16:49, Mostowski Collapse wrote:
> Julio Di Egidio schrieb am Mittwoch, 12. April 2023 um 06:19:44 UTC+2:
>> On Tuesday, 11 April 2023 at 20:37:35 UTC+2, Mostowski Collapse wrote:
>>> This works also:
>>>
>>> ∀x∀y(owns(x,y) ∧ farmer(x) ∧ donkey(y) => beats(x,y))
>>> owns(donald,sugar)
>>> farmer(donald)
>>> donkey(sugar)
>>> ==============================
>>> beats(donald,sugar)
>>
>> I don't understand what is interesting about that
>> (even after trying to read about it), apart maybe
>> from the fact that I cannot find a propositional way
>> to say it, hence at least 1st order: but especially
>> since it is a contingent statement, so what...?
>

> What could be interesting, is running it through a Montague
> parser. Montague grammars aspire to be compositional.
> So for example if you say:
>
> every human is mortal
>
> A Montague grammar might give an intensional logic meaning
> A to "every human" and a intensional logic meaning B to "mortal",
> and then there is even an intensional logic combinator "is",
>
> A = every human
> B = mortal
> A is B
>
> so that the sentence gets its meaning by the composition
> of A and B by the combinator is. Who can intensional logic do that?
> I speculate that the Donkey anaphora poses a similar problem.

<snip>

No no, that syllogism is perfectly fine in propositional
logic, indeed formally it's just an instance of the
transitivity (indeed, composability) of application:

ToBeAMan -> ToBeMortal, and
ToBeSocrates -> ToBeAMan, therefore
ToBeSocrates -> ToBeMortal.

And then maybe you can also see how Wittgenstein was right
all along, that the very notion of _atomic proposition_
was messed up to begin with...

Julio

[Mostowski Collapse]

Troll alert! LoL

[olcott]

On 4/12/2023 12:10 PM, Mostowski Collapse wrote:
> But with UML, you cannot so much make a distinction between:
>
> every human is mortal
> some human is mortal
> https://en.wikipedia.org/wiki/Term_logic
>

human ▷ mortal
∴ ∀x ∈ human (x ∈ mortal)
∴ ∃x ∈ human (x ∈ mortal)

> The ▷ might pass for "every human is mortal", in that we say
> "human ▷ mortal". But what about "some human is mortal"?
> In a Montague parser, the "is" would combine both forms
>
> and give a correct result. How can this be done?
>
> A1 = every human
> A2 = some human
> B = human
> A1 is B
> A2 is B
>
> between for example term logic Mostowski Collapse schrieb am Mittwoch, 12. April 2023 um 19:02:27 UTC+2:
>> For UML, this would be enough., your ∀x ∈ rather
>> introduces a category error. Its enough to say:
>>
>> human ▷ mortal
>>
>> Since is_a is between classes, and not between an instance and a class.
>> See also, inheritance belongs to Class-level relationships
>>
>> Class-level relationships
>> Generalization/Inheritance
>> (subclass) _______▻ (superclass)
>> https://en.wikipedia.org/wiki/Class_diagram#Class-level_relationships
>> olcott schrieb am Mittwoch, 12. April 2023 um 17:22:30 UTC+2:
>>> ▷ UML inheritance symbol is used to mean <is a type of>
>>> ∀x ∈ human (x ▷ mortal)

[Julio Di Egidio]

> Troll alert! LoL

Too much even for you? That's a good sign...

Julio

[Ross Finlayson]

Socrates' wisdom outlived him, and is less mortal than prototypical man.

Socrates is a school, of a man.


Relations are reflexive, all of them together, even when
inverses fall into the generalized and not closures.

Closures are tractable, to be sure, but the "sensible",
"fungible", and "tractable" are three different things,
to arrive at the "tractable".

[Julio Di Egidio]

And the spammers systematically diluting it all...

ESAD, really.

Julio

[olcott]

ToBeAnEasterEgg -> ToBeAMan
is also implied when the consequent is true and the antecedent is false.

Semantic Necessity operator: ⊨□
ToBeAnEasterEgg ⊨□ ToBeAMan is false because there is no semantic
connection between the antecedent and the consequent

[Ross Finlayson]

Socrates lives, in the hearts, of men, or man. Man is immortal, while a man yet lives.

This is where natural language excels, and there are classical concepts.

[Julio Di Egidio]

On Wednesday, 12 April 2023 at 21:12:57 UTC+2, olcott wrote:
> On 4/12/2023 12:48 PM, Julio Di Egidio wrote:

<snip>

> > ToBeSocrates -> ToBeMortal.
> >
> > And then maybe you can also see how Wittgenstein was right
> > all along, that the very notion of _atomic proposition_
> > was messed up to begin with...
>

> ToBeAnEasterEgg -> ToBeAMan
> is also implied when the consequent is true and the antecedent is false.

You piece of demented ubiquitous spamming shit, I have explained
just yesterday the difference between a conditional and a syllogism.

You pieces of nazi-retaded spamming shit!

Sure, keep going...

Julio

[Ross Finlayson]

If you think "material" implication is a "valid classical inference rule",
there's another one where it's _not_, a valid classical inference rule.
(Ex nihilum nihilum, vis-a-vis, ex nihilum binihilum, scusi.)

There, when you put, "ass(y)", do you mean the ass that gets beat?

Just wondering that otherwise the drover gets beat, and, the old,
..., "the ass|u|me made an ass of u and me", if you don't know that one.

It's a most usual warning note on the logic bottle label why there are derivations
after definitions, explaining mistakes, "when you _assumed_, you made
an _ass_, of _u_ (you) and _me_, both".

It's a very common expression in English for example about material implication
and the risks of forgetfulness or non-faithful derivations of terms, that,
when you thus _assume_, the warning goes, you made an _ass_ of _u_ and _me_, both.

It's one of the most common expressions in logical warning about entailment and
the risks and dangers of material implication and ex falso quodlibet,
you're an ass, both.

Material implication and ex falso quodlibet, is a sodomite's.

The ass is the humble donkey and the drover's responsibility,
whereas, butt-stuff is unhygienic, and inherently unsafe, you filthy animals.

So, "material" implication is like dropping the soap.

"There were no anti-cruelty laws."

Once upon a time, this guy got so pissed, he found an old beat-to-death
donkey, took its jawbone, and slew a bunch of heathen infidels. It slayed.



Rules, stipulations, are exceptions, to none.

Life, just isn't fair. Though, if you try, you just might find, you get what you need.



Happy Trails, ....

[Julio Di Egidio]

On Wednesday, 12 April 2023 at 21:28:36 UTC+2, Ross Finlayson wrote:

> Life, just isn't fair. Though, if you try, you just might find, you get what you need.
>
> Happy Trails, ....

LOL, the last resort of a vile insane retarded cunt.

Nice villa you have there, by the way, you piece of blood-sucking polluting shit.

ESAD, you and your shameless insanity.

*Plonk*

Julio

[Ross Finlayson]

Hm, it's like "I've never been there, they tell me it's nice".

It's like "The Cave", you can check out, but, no-one will know you've gone except by your absence.

So, "classical concepts" are these things we arrive at _above_, classical logic,
from the, mutual conception, of concepts, that in the West is monist and reductive
or productive, and in the East is nihilist and productive or reductive.

So, "The Cave", or for example, "The Chinese Room", are about the same.

There are requirements.

It's not easy being made, ....

Anyways classical logic is strong not its links, but its weave.

[Mostowski Collapse]

Ask Rossy Boy about Tarksi, Herbrand and RDF. He is
totally clueless. Like spongebob from bikini bottom.

LoL

[Mostowski Collapse]

Rossy Boy is like the little frog in the pond, the frog
which got overrun by car, but somehow it can still croak,
although his brain got splattered over the asphalt.

[Ross Finlayson]

How is "Living Dead Girl" like "Bell Bottom Blues"?



We mustn't forget the circles of competitive advantage, and the facts of dissimulation,
about why a logic dedicated to lies and maintaining their belief, is a calculus of lies,
their costs and risks, their benefits and rewards, that though it's only a game of
"what's the truth" and "where's the truth". In a lyrical example like Depeche Mode's
"Policy of Truth", which is a good album for macking, it's warned that there are
those who will or won't collude, in what and where is the truth, that their whole-sum
game, is exacting it.

Some do, ....

For something like "Nine Inch Nails" "Happiness in Slavery" or "Everything in this
world is blue", is that most people do _not_ have the propensity to spend enough
useful time getting to foundations, because, Earthly needs and limits, disallow it.

Paints a pretty picture, ....

Then, there are secrets, according to the Akashic Record there are none, or there's
the "Three books: a book in a book in a book", for Trinitarians and Unitarians, it's
rather like U2's "a secret: is something, you only tell, one other person",
public secrets and private secrets.

Paint's a pretty picture, ....

Of course, there is the opaque.

Truth and lies, secrets and not-secrets.

[Ross Finlayson]

On Wednesday, April 12, 2023 at 2:06:31 PM UTC-7, Mostowski Collapse wrote:

What's red and green and goes 100 miles an hour?

How did the punk cross the road?
He safety-pinned a chicken to his face.

[Mitchell Smith]

On Monday, April 10, 2023 at 1:12:45 PM UTC-5, olcott wrote:
> On 4/9/2023 7:51 PM, Mitchell Smith wrote:
> > On Sunday, April 9, 2023 at 10:01:47 AM UTC-5, olcott wrote:
> >> On 4/9/2023 9:01 AM, Mitchell Smith wrote:
> >>> On Saturday, April 8, 2023 at 2:17:04 PM UTC-5, olcott wrote:
> >>>> Well-formed formulas have meaning only when an interpretation is given
> >>>> for the symbols. Mendelson
> >>>>
> >>>> No one seems to know why model theory is needed.
> >>>> A ∧ B → A is known to be true on the basis of the meaning of the
> >>>> symbols, no model theory needed.

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

> >>> Peter,
> >>>
> >>> A similar question had been asked on MSE two years ago. I have no reason to join the auspicious minds of MSE (who practice censorship), but, I have occasionally answered as a guest.
> >>>
> >>> I gave a two-part answer to the question at the link,
> >>>
> >>> https://math.stackexchange.com/questions/3970711/what-is-the-point-of-model-theory
> >>>
> >>> under 'mls'.
> >>>
> >>> You may find it informative. The original poster did accept my answer.
> >> I looked at this. What I really need to start with is the gist of the
> >> purpose of model theory. It looks like it simply defines sets that the
> >> variables range over, just like the syllogism.

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

> > Peter,
> >
> > Ask yourself, "What is the role of truth in a proof?"
> >
> Since my goal is to correctly mathematically formalize the notion of
> analytic truth I always go the opposite direction:
>
> What is the role of formal proof in the notion of analytic truth?
> > Truth governs the transition between statements. This is why one speaks of a rule of detachment. A true conclusion is obtained from true premises.
> >
> Since {true} is a semantic concept the correct evaluation of the truth
> of expressions of language cannot correctly be made apart form
> semantics.
> > But, rules of detachment can be declared to be merely syntactic.
> Semantic can be specified syntactically as Rudolf Carnap meaning
> postulates have proved.
> > At issue, then, is the distinction between a semantic conception of truth and an analytical conception of truth.
> They are the same unless terms of the art overrides the conventional
> meaning of these terms.
>
> That cats are animals is analyzed on the basis of the semantic meaning
> of these terms.
> > Many of your statements are compatible with the historical development of analytical truth. But, this is relegated to intensional logic. The traditional interpretation of Goedel's incompleteness theorem involves distinguishing between truth and provability, and, in turn, this distinction depends upon a semantic conception of truth interpreted as a correspondence theory. Meaning postulates do not --- and cannot --- be used to assert that a term denotes an object.
> >
> > I agree that all one can eventually conclude from "the received view" is that words are being used to explain words. So, there is something not quite right about all of this "truth talk."
> The entire body of the analytic side of the analytic syntactic
> distinction is entirely specified as relations between otherwise
> meaningless terms.
>
> ALL of the meaning of every natural language or formal language term is
> entirely encapsulated in its relation to other terms.
> > The word describing this problem is "deflation," and, it goes all of the way back to Aristotle.
> >
> > The link,
> >
> > https://drive.google.com/file/d/1EUTImanTxstmDEBBNr_sJGq5m-Rh9JkG/view?usp=drivesdk
> >
> > contains a translation from Aristotle's book Categories which I have augmented with my own interpretations relative to my personal knowledge of modern topics.
> >
> > At the top of page 3, Aristotle speaks of "existence" and "the truth of statements" being reciprocal. Then, he observes that the truth of a statement does not imply existence. So, the reciprocity corresponds with the deflationary nature of truth while the residual asymmetry justifies studying truth as a correspondence principle.
> >
> That sound incoherent to me.
> > It is difficult to see how these can be studied "univocally" within the same paradigm. And, deflationism, itself, has numerous philosophical aspects,
> >
> > https://plato.stanford.edu/entries/truth-deflationary/
> >
> > So, while I can find a quote from Aristotle interpretable with respect to these difficulties, I certainly cannot find a formalization for every nuance that philosophers can imagine.
> >
> > The first-order paradigm is "a way of studying" mathematics which presupposes a correspondence theory of truth relative to fixed domains of discourse.
> The correspondence theory of truth only applies to empirical truth.
> A set of physical sensations corresponds to "there is a TV in my living
> room right now"
>
> Analytic truth is purely axiomatic.
> > If you are going to claim that a different paradigm accounts for mathematics, you must accommodate the fact that first-order logic is a part of mathematics.
> >
> *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
> > This cannot be done by any method which equates truth with provability.
> I just showed the actual inherently correct precise relation between
> truth and provability.
>
> The sequence of truth preserving inference steps from (a) to X <is> the
> relation between truth and provability.
> > This very problem is seen with homotopy type theory and its promoters seem only to concern themselves with winning an argument.
> >
> > And, for what this might be worth, whereas mathematicians communicate with proofs, those who apply mathematics are not under a similar constraint. So, how truth is understood relative to proofs is an aspect of mathematics which introduces these problems involving semantics. I may choose to study mathematics differently from a first-order logician, but, I have an understanding of why I must respect how first-order logicians apply their paradigm. I only take issue with "foundational" claims. It took many years until I could defend the first-order paradigm in spite of not being convinced of its wider claims.

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

Peter,

First, let me compliment you on your grasp of an analytic conception of truth. I accept many of your responses.

I reject the idea that you are unable to comprehend a correspondence theory of truth. So, avoiding an admission of different paradigms with the claim that you are in possession of knowledge of "correct reasoning" appears to me as outright dishonesty.

As for what you described as incoherent, I will gladly take some blame. I would not be the first peron to express a difficult statement with intractable syntax. But, it makes me think of three statements:

1) I presume that you are using an electronic device capable of computation to read my words.

2) The phrase, "the electronic device you are using to read my words," is, therefore, a meaningful phrase.

3) The actual, material object allowing you to read my words is not the phrase, "the electronic device you are using to read my words.

With regard to a different issue, the classical division between syntax and semantics is often described as avoiding circularities which must exist in dictionaries providing meanings.

It would seem, then, that your system must be grounded on circular syntactic forms in some way.

I have never seen any such syntax in any of your posts.

Can you demonstrate such a "starting point" or explain how your system avoids circular syntax?

mitch

[Julio Di Egidio]

On Thursday, 13 April 2023 at 04:15:29 UTC+2, Mitchell Smith wrote:
> On Monday, April 10, 2023 at 1:12:45 PM UTC-5, olcott wrote:
>
> > Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
> > hits a target no one else can see." Arthur Schopenhauer
>
> Peter,
> First, let me compliment you on your grasp of an
> analytic conception of truth. I accept many of your responses.

You too keep confirming you are *completely full of shit*.

Congratulations, way to go...

Julio

[olcott]

If we are a brain-in-a-bottle then all empirical "truth" is false.
If the universe began five minutes ago then all memory is false.
None-the-less 2 + 3 = 5 remains infallibly true.

> As for what you described as incoherent, I will gladly take some blame. I would not be the first peron to express a difficult statement with intractable syntax. But, it makes me think of three statements:
>
> 1) I presume that you are using an electronic device capable of computation to read my words.
>
> 2) The phrase, "the electronic device you are using to read my words," is, therefore, a meaningful phrase.
>
> 3) The actual, material object allowing you to read my words is not the phrase, "the electronic device you are using to read my words.
>
> With regard to a different issue, the classical division between syntax and semantics is often described as avoiding circularities which must exist in dictionaries providing meanings.
>
> It would seem, then, that your system must be grounded on circular syntactic forms in some way.
>

Not at all. I realized that the set of all analytical knowledge is
inherently tree structured for forty years.

One poster understands Montague Grammar (of natural language semantics).

I am aiming at specifying the architecture of the formal system of all
analytic knowledge where analytic means anything that can be expressed
using language that can be verified as completely true without any sense
data from the sense organs.

This may or may not address Quine's objections. I really don't care to
carefully analyze the mistakes of others. Instead I start from scratch
and reverse engineer the requirements of the correct system.
https://en.wikipedia.org/wiki/Ontology_(computer_science)

The system that I am referring to would have an enormous number of
natural langugae axioms so that it cold analyze things in the world
through language.

For example it could compute all of the ways to prove that the election
fraud claims are lies. It could take on each individual on social media
simultaneously and make their lies look ridiculous even to themselves as
well as everyone else.

To begin to do this we must refute the Tarski Undefinability theorem and
establish a the analytical foundation of truth.

> I have never seen any such syntax in any of your posts.
>
> Can you demonstrate such a "starting point" or explain how your system avoids circular syntax?
>
> mitch
>
>

[Mitchell Smith]

Your statement that "2 + 3 = 5" sounds blatantly platonistic, leading to the paradigms you wish to avoid.

So, what makes it infallibly true?

But, let me move on.

When you refer to "one poster understands Montague grammar," I assume that you are referring to Mr. Burse' quoted passage from the SEP link. The previous passage lies within the larger quote,

"For Montague the principle of compositionality did not seem to be a subject of deliberation or discussion, but the only way to proceed. In effect he made compositionality the core part of his ‘theory of meaning’ (Montague 1970c, 378), which was later summed up in the slogan: ‘Syntax is an algebra, semantics is an algebra, and meaning is a homomorphism between them’ (Janssen 1983, 25). Yet although Montague used the term ‘Frege’s functionality principle’ for the way in which extension and intension are compositionally intertwined, he did not have a special term for compositionality in general. Later authors, who identified the Principle of Compositionality as a cornerstone of Montague’s work, also used the term ‘Frege’s Principle’ (originating with Cresswell 1973, 75); Thomason 1980 is an early source for the term ‘compositional’."

Frege described his "discovery" of compositional form at one point if his paper, "On Sinn and Bedeutung." Beaney prefers not to translate the paper title. I believe it is usually translated as "On sense and reference." From reading it, I am inclined toward "On sense and referents."

Frege writes:

"One might also say that judgements are distinctions of parts within truth values.Such distinction occurs by return to the thought. To every sense attaching to a truth value would correspond its own manner of analysis. However, I have here used the word 'part' in a special sense. I have in fact transferred the relation between the parts and whole of the sentence to its referent, by calling the referent of a word part of the referent of a sentence, if the word itself is part of the sentence.This way of speaking can certainly be attacked because the whole referent and one part of it do not suffice to determine the remainder, and, because the word 'part' is already used of bodies in another sense. A special term would need to be invented."

Elsewhere, as in "Function and Concept," Frege is clear about taking truth values as objects. I would like to say that one must understand an object being a part of itself in view of the previous quote. I expect you would have a problem with that. What Frege actually says involves syntax with no variables or uninterpreted parameters:

"Here I can only say briefly: an object is anything that is not a function, so an impression for it does not contain any empty place.

"A statement contains no empty place, and therefore we must take its referent as an object."

So, something like,

TRU(x,y)

would be a function because of the indeterminate arguments. But,

TRU(NTRU,NTRU)

would be an object because the argument places are filled by constants.

Take a moment to scroll through the appendix at page 96 of the file,

https://drive.google.com/file/d/15087p1Io6K8xOZO2NUFzv_m7UAks1rTq/view?usp=drivesdk

One could arrange the constants as an algebra signature, thereby getting Montague's "semantics is an algebra."

Shramko and Wansing are authors who take "truth objects" seriously. They authored the SEP link on truth values,

https://plato.stanford.edu/entries/truth-values/

There own work involves 16 truth values decorating projections of tesseracts,

https://plato.stanford.edu/entries/truth-values/generalized-truth-values.html

The tables I constructed are motivated by several sources, including Frege's writings. You might find them informative with respect to "analytically stipulating" truth objects that can be parts of truth objects. This is compositionality without Boolean polynomials That is, parts of referents derived from parts of statements.

Maybe,... maybe not.

mitch

[olcott]

I like to examine these things at the level of their foundational
architecture and proceed inward from there.

*I will start with the pinnacle level*
The entire body of analytical truth is simply an interdependent semantic
tautology.

*Here is the next level*
(a) Some expressions of language are stipulated to have the semantic
property of Boolean true.

(b) Some expressions of language are a semantically necessary
consequence of others.

[Ross Finlayson]

Hey, thanks again Mitch, though, about Platonism, there are different shades of Platonism,
that "the number 5 the sum of 2 and 3 exists", isn't a "distinct" thing so much as a "coherent" thing.

I.e., a "strong mathematical platonism" doesn't necessarily ontologically commit to "2 + 3 = 5, in
a vacuum", instead that "all the objects of mathematics exist" then about why for example
"it must be only all the 'true' mathematics".

I've been thinking about referents and I'd like to relate your "geometrical metamathematics"
thread and my "Question Words, and what's an answer" thread. You mention "Referent" and
it's a pretty key thing where there is "template" and there is "instance", about that essentially
"quantification is a first-class act and a higher-order act to result a lower-order act", these,
"outer products and their inner products these co-products".

Then, the key here for "context" is that: "in the universe, the object: is everything that it is not".
This is the sort "default reflection, absolute and total", which helps advise the relative and global,
in terms of the point and local, and hierarchically and orderedly while dispersively and free.


Separability and composability or "compositionality and deconstructionality" are usual proper
features of any object its attributes its elements. It's part of "default reflections".

Then, I look at logics as "these are digital models of flow that implement flow machines, the
deterministic sort that implement these logical computing atoms, which for example model how
other usual flow machines work even those nondeterministic ones". Where it's as well the
other way, "there is no determinism", it still makes the same system for "axiomatics".

So, there are different shades of Platonism and since antiquity and there are different grasps
(or, lack thereof...) of platonism that "strong mathematical platonism" can be the same thing
as "a theory, a logic, a mathematics, a science / according to chance, a physics".

[olcott]

I am familiar with many of the details that you discuss above most
importantly the Frege's principle of compositionality. I have discussed
these things for many years on sci.lang. The Richard Montague grammar of
natural language semantics would have to be exhaustively elaborated
before my architecture of the foundation of correct reasoning could be
fully implemented.

None-the-less it might be best to fully elaborate the essence of the
notion of truth itself before we can possibly fully understand what it
means for an expression of the formal language of a formal system to
be true.

All mathematical truth is analytical truth, thus there is no need to
look at the synthetic side of the analytic / synthetic distinction.
This also seems to require no reference to the correspondence theory of
truth. Tarski refers to "snow is white" is true because {snow is white}.

The entire body of analytic truth is really nothing more than a set of
interdependent semantic tautologies: Expressions of language that are a
semantically necessary consequence of others expressions of language
that are stipulated to be true.

How do we know that a {cat} <is an> {animal} and not a type of {ten
story office building} ? The semantic properties that have been assigned
to these two categories are mutually exclusive. There are no cats with
elevators and no ten story office buildings that cough up hairballs.

[olcott]

The correspondence theory of truth as it applies to the synthetic
(empirical) side of the analytic synthetic distinction is simply a
unidirectional mapping from sensory stimulation to abstract models of
the world.

> So, avoiding an admission of different paradigms with the claim that you are in possession of knowledge of "correct reasoning" appears to me as outright dishonesty.
>

Try and find any counter example of analytic truth that is not a
necessary consequence of expressions of language that have been
stipulated to be true.

> As for what you described as incoherent, I will gladly take some blame. I would not be the first peron to express a difficult statement with intractable syntax. But, it makes me think of three statements:
>
> 1) I presume that you are using an electronic device capable of computation to read my words.
>
> 2) The phrase, "the electronic device you are using to read my words," is, therefore, a meaningful phrase.
>
> 3) The actual, material object allowing you to read my words is not the phrase, "the electronic device you are using to read my words.
>
> With regard to a different issue, the classical division between syntax and semantics is often described as avoiding circularities which must exist in dictionaries providing meanings.
>
> It would seem, then, that your system must be grounded on circular syntactic forms in some way.
>

Analytic truth (except for the unknown) is a tree of knowledge.

> I have never seen any such syntax in any of your posts.
>
> Can you demonstrate such a "starting point" or explain how your system avoids circular syntax?
>
> mitch
>
>

[Mostowski Collapse]

So after discussing for a few days, in another thread,
Culio seems to be the same moron like back then.
Not the fainthest clue what the paper is about?

Julio Di Egidio schrieb am Mittwoch, 12. April 2023 um 06:19:44 UTC+2:

> On Tuesday, 11 April 2023 at 20:37:35 UTC+2, Mostowski Collapse wrote:
> > This works also:
> >
> > ∀x∀y(owns(x,y) ∧ farmer(x) ∧ donkey(y) => beats(x,y))
> > owns(donald,sugar)
> > farmer(donald)
> > donkey(sugar)
> > ==============================
> > beats(donald,sugar)

> I don't understand what is interesting about that
> (even after trying to read about it), apart maybe
> from the fact that I cannot find a propositional way
> to say it, hence at least 1st order: but especially
> since it is a contingent statement, so what...?
>

> Julio

[olcott]

On 4/11/2023 11:19 PM, Julio Di Egidio wrote:
> On Tuesday, 11 April 2023 at 20:37:35 UTC+2, Mostowski Collapse wrote:
>> This works also:
>>
>> ∀x∀y(owns(x,y) ∧ farmer(x) ∧ donkey(y) => beats(x,y))
>> owns(donald,sugar)
>> farmer(donald)
>> donkey(sugar)
>> ==============================
>> beats(donald,sugar)
>
> I don't understand what is interesting about that
> (even after trying to read about it), apart maybe
> from the fact that I cannot find a propositional way
> to say it, hence at least 1st order: but especially
> since it is a contingent statement, so what...?
>
> Julio

The question is: Why is model theory needed?
By assigning values to x and y we can see that
beats(donald,sugar) is true.

[Mitchell Smith]

On Saturday, April 8, 2023 at 4:34:41 PM UTC-5, André G. Isaak wrote:
> On 2023-04-08 15:23, olcott wrote:
> > On 4/8/2023 4:15 PM, André G. Isaak wrote:
> >> On 2023-04-08 15:04, olcott wrote:
> >>> On 4/8/2023 3:27 PM, André G. Isaak wrote:

> >>>> On 2023-04-08 13:16, olcott wrote:
> >>>>> Well-formed formulas have meaning only when an interpretation is given
> >>>>> for the symbols. Mendelson
> >>>>>
> >>>>> No one seems to know why model theory is needed.
> >>>>> A ∧ B → A is known to be true on the basis of the meaning of the
> >>>>> symbols, no model theory needed.
> >>>>>
> >>>>

> >>>> ((A ∧ B) → A) is a statement which is true in *all* models, so of
> >>>> course you don't need to specify the model. And this is properly
> >>>> true on the basis of the meaning of the *connectives*, not the
> >>>> meaning of the symbols themselves.
> >>>>
> >>>> ((A ∧ B) → A) can be evaluated without reference to the symbols A or B.
> >>>>
> >>>> The same cannot be said for (A ∧ B). How would you evaluate (A ∧ B)
> >>>> as true or false without some model?
> >>>>
> >>>> André
> >>>>
> >>>
> >>> *That was a very superb answer*
> >>> My reply is to handle (A ∧ B) exactly the way that the syllogism
> >>> handles them thousands of years before anyone thought of model theory.
> >>>
> >>> Since we didn't need model theory to do this thousands of years ago why
> >>> do we need it now?
> >>
> >> And what do syllogisms have to do with this? How exactly do you
> >> evaluate (A ∧ B) in absence of a model using syllogisms?
> >>
> >> André
> >>
> >
> > Syllogisms never allowed semantics to be removed from logic thus there
> > is no need to put semantics back in.
> Apparently you have a very idiosyncratic definition of 'syllogism'.
> > With the syllogism referring to A and B without having already defined
> > them is simply flatly wrong.
> Aristotelian logic (from which we get the term 'syllogism') explicitly
> deals with symbolic terms independent of their semantics. He defines
> which forms constitute valid syllogisms *without* reference to the
> meanings of the terms involved. That's the entire foundation of logic --
> to focus on the *form* an argument takes rather than the meanings of the
> terms involved.
>
> You should probably stop using the term 'syllogism' until you've
> actually *read* the works of Aristotle and his successors.
> André
>
> --
> To email remove 'invalid' & replace 'gm' with well known Google mail
> service.

You have made a false statement.

Aristotelian logic is described over several books. The book, "Prior Analytic," does, indeed focus entirely on the syntactic forms he believes to exemplify "argumentation."

Like a number of philosophical analyses (Kant and Machiavelli come to mind) he then employs a "divide an conquer" strategy. General argumentation is divided into "argument from principles" (demonstrative argument) and "argument from common belief" (dialectical argument). The notion of "first principles" and that knowledge may be organized into an order lies with demonstrative argument. Philosophers have focused upon epistemology because of the relationship to knowledge, but one might also subsume pedagogy under Aristotelian demonstration. Comparison, rhetoric, and forensics (legal uses) all fall under dialectical argumentation.

Aristotle describes demonstrative argumentation in "Posterior Analytic." Epistemology arises from this book because Aristotle had attempted to relate this book to his metaphysics through the notion of essence. Kant's notion of analyticity had been an attempt to identify "essence" as a "first prjnciple" construct on the basis of the law of contradiction. Modern philosophy has largely retained the relationship to the law of contradiction for the word 'analytic.' The analytic conception of truth is not spoken about very often, but, Mr. Olcott correctly locates it with Carnap's meaning postulates.

He describes dialectical argumentation in "Topics." Subdivisions of dialectical argumentation different from its use for comparison are given further elaboration in other books.

This top-level division reflects modern debate over (object language) definitions because definitions can be challenged or undermined by counterexample. Aristotle locates this activity within dialectal argumentation. More generally, he describes dialectical reasoning in terms of "both sides of a contradiction." By contrast, his demonstrative argumentation is described as reasoning from one side of a contradiction or the other. As ordered knowledge, definitions are important to demonstrative reasoning. Also important is the distinction between a definition and a principle asserting an existent. I forget the terms used in my translations. One of the two is referred to as a "thesis"; I believe that it is the existential assertion. He (or my translator) reserves "axiom" for universally applied principles like excluded middle.

There are other books --- "De Interpretatione," "Categories," and possibly one more which are also significant to his explanation of logic. It is in "Categories" the notion of order is directly associated with demonstrative argumentation and compared with other notions of order such as time, counting, exposition, and syntactic structure.

Aristotle surely does what you claim. Under no circumstances, other than personal belief, however, can one accept that your foundational claim is supported from reading Aristotle. Indeed, his discussion of parts of speech cannot be found in "Prior Analytics," and, so the very notion of subjects and predicates required for syllogistic analyses is not addressed under your claim.

Form must be discerned and differentiated before it can be categorized. It must be categorized before it is explained.

Modern analytic philosophy speaks in terms of "stands for."

Aristotle speaks in terms of differences "making a stand."

Methinks you speak with a forked tongue.

mitch

[Julio Di Egidio]

On Tuesday, 18 April 2023 at 13:57:48 UTC+2, Mitchell Smith wrote:

> Modern analytic philosophy speaks in terms of "stands for."
> Aristotle speaks in terms of differences "making a stand."
> Methinks you speak with a forked tongue.

Methinks you work for the nazi enemy, systematically
spouting upside down bestialities.

People like you should be shot in the face.

-LV