Interpretation(s) of AxPx

[apoorv]

AxPx is a symbol in FOL for a sentence. Now in sentential
logic,sentence symbols can be interpreted to mean any sentence or
proposition whatsoever.Is it that AxPx should always have the
interpretation 'For All things x in the Domain' or other
interpretations are posible? If so how far is it justified to take
AxPx-->Pc for every constant c in the language as a logical axiom?
-apoorv

[William Elliot]

On Sun, 22 Apr 2012, apoorv wrote:

> AxPx is a symbol in FOL for a sentence.

AxPx is not a symbol, it's a formula or sentence.
A, P and x are symbols.

> Now in sentential logic,sentence symbols can be interpreted to mean any
> sentence or proposition whatsoever.Is it that AxPx should always have
> the interpretation 'For All things x in the Domain' or other

> interpretations are possible? If so how far is it justified to take

[Frederick Williams]

apoorv wrote:
>
> [...] Is it that AxPx should always have the

> interpretation 'For All things x in the Domain' or other
> interpretations are posible?

There is also the substitutional interpretation. Look up 'truth value
semantics'. There's a conference proceedings edited by Hugues Leblanc
titled... um... something.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[apoorv]

On Monday, April 23, 2012 7:12:50 PM UTC+5:30, Frederick Williams wrote:
> apoorv wrote:
> >
> > [...] Is it that AxPx should always have the
> > interpretation 'For All things x in the Domain' or other
> > interpretations are posible?
>
> There is also the substitutional interpretation. Look up 'truth value
> semantics'. There's a conference proceedings edited by Hugues Leblanc
> titled... um... something.

Thanks for that useful tip.
In the tautology A&B->A of sentential logic,
A could have any possible interpretation;
A could be 'Ron is intelligent',
'Ron is not intelligent'
'Pis true for all x'
'P is true for no x'
'Pis true for some x'
'P is true foe all x some of the time'
and so on.
What I am trying to understand is why in the logical axiom
AxPx->Pc
AxPx have the sole interpretation 'P is true for all x'
and not any of the following:
'P is not true of any x'
'P is true of some x'
'P is true of all x but c'
'P is true of only one x'
'P is true of all x some of the time'
etc etc.
-apoorv

[Frederick Williams]

apoorv wrote:

> What I am trying to understand is why in the logical axiom
> AxPx->Pc
> AxPx have the sole interpretation 'P is true for all x'
> and not any of the following:
> 'P is not true of any x'
> 'P is true of some x'
> 'P is true of all x but c'
> 'P is true of only one x'
> 'P is true of all x some of the time'
> etc etc.

In those cases AxPx->Pc wouldn't be true.

[apoorv]

On Tuesday, April 24, 2012 3:54:42 AM UTC+5:30, Frederick Williams wrote:
> apoorv wrote:
>
> > What I am trying to understand is why in the logical axiom
> > AxPx->Pc
> > AxPx have the sole interpretation 'P is true for all x'
> > and not any of the following:
> > 'P is not true of any x'
> > 'P is true of some x'
> > 'P is true of all x but c'
> > 'P is true of only one x'
> > 'P is true of all x some of the time'
> > etc etc.
>
> In those cases AxPx->Pc wouldn't be true.

It would be true in each case if Pc were true OR each of the interpretation of AxPx were false.
-apoorv

[Frederick Williams]

Yes, sorry. While lying abed this morning, I wondered if you meant, why
is P variable in its meaning, while A is fixed? I.e., why is A a
logical constant? What a logical constant _is_ (supposing that the
logical constants aren't just chosen arbitrarily), is a question I have
asked myself here at least once.

[apoorv]

On Tuesday, April 24, 2012 7:12:53 PM UTC+5:30, Frederick Williams wrote:
> apoorv wrote:
> >
> > On Tuesday, April 24, 2012 3:54:42 AM UTC+5:30, Frederick Williams wrote:
> > > apoorv wrote:
> > >
> > > > What I am trying to understand is why in the logical axiom
> > > > AxPx->Pc
> > > > AxPx have the sole interpretation 'P is true for all x'
> > > > and not any of the following:
> > > > 'P is not true of any x'
> > > > 'P is true of some x'
> > > > 'P is true of all x but c'
> > > > 'P is true of only one x'
> > > > 'P is true of all x some of the time'
> > > > etc etc.
> > >
> > > In those cases AxPx->Pc wouldn't be true.
> > It would be true in each case if Pc were true OR each of the interpretation of AxPx were false.
>
> Yes, sorry. While lying abed this morning, I wondered if you meant, why
> is P variable in its meaning, while A is fixed? I.e., why is A a
> logical constant? What a logical constant _is_ (supposing that the
> logical constants aren't just chosen arbitrarily), is a question I have
> asked myself here at least once.

I have been trying to understand the relationship between sentential logic and FOPL.Consider the sentences Pa, Pb,Pc and so on in FOPL where a, b, c etc are constants. These sentences can have different interpretations depending on the domain for the chosen structure. For example, Pa could be interpreted as P is true of John (a maps onto john) or as P is true of Ron (a maps onto Ron).
Now 'P is true of John' or 'P is true of Ron'are surely sentences of sentential logic. Pa itself, then, is a parameter of the language of sentential logic (sentence symbol) which can be interpreted to mean different sentences (although in a limted manner).Now we augment our set of parameters by introducing S such that
S->Pa , S->Pb ,S->Pc and so on are claimed to be tautologies. It can be seen that such an S will be identical to the logical constant 'F'.
What puzzles me is that for the sentence AxPx of FOPL ( which can be considered as a parameter or sentence symbol in sentential logic), we have
AxPx->Pa etc
-apporv

[Frederick Williams]

apoorv wrote:

> What puzzles me is that for the sentence AxPx of FOPL ( which can be considered as a parameter or sentence symbol in sentential logic), we have
> AxPx->Pa etc

Because it means 'if every individual has property P, then individual a
has property P.' But that's obvious, so your question must go deeper,
and I'm sorry I've failed to grasp it. Meanwhile the pros--who maybe
could help you--are busy arguing with Peter Olcott.

From the point of view of sentential logic, AxPx -> Pa is not valid.
It's validity depends on (among other things) the quantifier.

[apoorv]

On Monday, April 30, 2012 7:42:19 PM UTC+5:30, Frederick Williams wrote:
> apoorv wrote:
>
> > What puzzles me is that for the sentence AxPx of FOPL ( which can be considered as a parameter or sentence symbol in sentential logic), we have
> > AxPx->Pa etc
>
> Because it means 'if every individual has property P, then individual a
> has property P.' But that's obvious, so your question must go deeper,
> and I'm sorry I've failed to grasp it.

> From the point of view of sentential logic, AxPx -> Pa is not valid.
> It's validity depends on (among other things) the quantifier.

I am looking at AxPx purely syntactically as one sees sentence symbols of sentential logic.
The wffs of FOPL can be mapped into sentence symbols (parameters) of sentential logic;
For example Pa could map into A, Pb could map into B and so on.
Suppose we were looking for a wff of sentential logic R such that
R-> A and R->B; then R would be A&B&S, where S is some arbitrary wff of sentential logic.
But if we want R->A,R->B,R->C and so on for an infinite number of sentence symbols, then
R can only be the logical constant F. In which case the wff AxPx of FOPL would map into F.That's what puzzling me.

>Meanwhile the pros--who maybe
> could help you--are busy arguing with Peter Olcott.

Maybe there is some justifiable exasperation at some of my earlier posts. On the other hand,there are some issues to which I never received a satisfactory response.
1)How can a finite string of symbols such as AxPx ever convey an infinite amount of information as it is presumed to do at least in some cases?

2) If the Godel sentence for PA is Phi, then
Phi <->There is no proof of Phi<->There is no proof of ‘There is no proof of Phi’
Suppose we want to check if a given sentence Theta is the Godel sentence.
We try to see if there is a proof of Theta. If we find one, Theta is not what we are looking for.
However, if Theta were indeed the Godel sentence, we would never prove that it has no proof And hence is the Godel sentence.
That is, given a sentence Theta, we can prove that it is not the Godel sentence, but never that it is indeed the Godel sentence.
So, how can one ever express the Godel sentence explicitly as a wff of the language of PA?
3) The diagonal in Cantor’s argument is defined as For all i, ~Di=Xii. But we know that there are i for which there is no proof that Xii=1 and no proof that ~Xii=1. In which case we would never know what Di is .So, does ~Di=Xii constitute a complete definition of the anti- diagonal? Can the anti- diagonal be finitely defined at all?
-apoorv

[Graham Cooper]

ALL(x) RETURNFALSE(x) -> RETURNFALSE(c)

This is not consistently written regarding ALL()

A(x) A(c) RF(x) -> RF(c)

This is a TAUTOLOGY for any Predicate
whether RF(x) returns true or false for any value.

Herc
--
1 X ^ NOT(X)
2 G = NOT(PRV(G))
3 S > INF
4 R = {X | NOT(X e X)}
5 IF HALT() GOTO 5
6 ALL(F) MAX(F)
=
THE 6 DEAD ENDS IN MATHEMATICS
but only 4 are recognised contradictions

[Graham Cooper]

On Apr 24, 3:48 am, apoorv <sudhir...@hotmail.com> wrote:
> ALL(x) P(x)
> 'P is true for some x'

In this case the formula ALL(x) P(x) would not hold, though P could
still be well defined like ISEVEN(x)

ALL(n):N ISEVEN(n)

is a FALSE formula

Herc

[George Greene]

On Apr 22, 1:51 pm, apoorv <sudhir...@hotmail.com> wrote:
> Is it that AxPx should always have the
> interpretation 'For All things x in the Domain'

That is the standard/classical/default semantics, for first-order
logic.

> or other interpretations are posible?

Of course other interpretations are possible, but they are non-
standard.

> If so how far is it justified to take AxPx-->Pc for every constant c in the language as a logical axiom?

This is not a good use of the word "far". Taking x to be able to be
instantiated to any constant c in the language is taking things LESS
far than to everything in the domain, the point being that, usually,
standardly, EVERY CONSTANT c IN THE LANGUAGE, ALSO NEEDS to be
interpreted as SOMETHING IN THE DOMAIN.
It is possible for the domain to have "anonymous" elements, i.e.,
elements that are NOT the thing-to-which-some-term-in-the-language-has-
been-interpreted.
The converse, however -- letting a first-order language have a
constant that is NOT interpretable as some element of the domain -- is
FAR WORSE in terms of deviation or substandardness, compared to what
is usual. THAT basically CAN'T happen. So, in other words, the class
of terms in the language is USUALLY SMALLER than the domain (unless
you for some odd reason needed 5 terms for the same object -- which,
while you often get it in math, you get mainly in the context of
domains that are infinite to begin with, so having 5 more copies of
everything does NOT make the collection "bigger"), so instantiating to-
and-only-to the terms would be going LESS far, not more.

Rather than talking of going "farther", you arguably are talking
about SHRINKING the domain to contain only the terms of the language.
The question then arises, what exactly do constant-letters-in-a-
language LOOK like: in particular, are they finely-articulated enough
-- are they squiggly enough -- for there to exist MORE-than-countably-
many of them? If this were the case then one could indeed talk about
going "farther" since first-order languages are standardly/normally
countable.

If you choose to identify the domain specifically with the terms in
the language (exactly), all&only the terms in the language, then you
get "substitutional quantification" or "truth-value semantics" for the
quantifiers (please wiki or google either term).

[apoorv]

On Sunday, May 6, 2012 12:34:12 AM UTC+5:30, apoorv wrote:
> On Monday, April 30, 2012 7:42:19 PM UTC+5:30, Frederick Williams wrote:
> > apoorv wrote:

> 1)How can a finite string of symbols such as AxPx ever convey an infinite amount of information as it is presumed to do at least in some cases?

The postulates of Physics are drawn from observation of the real world. Not so for maths where they could be any construct of the mind.Yet,it somehow seems to me that the information argument provides some connect between the logical axioms and the real world and allows us to test whether are axioms are robust or not.The following is a vey unusual way of looking at logic.

Sentences A and B are independent of each other if given A, a priori B could be either True or B could be False. As an example, if A is ‘The sun rises in the east’ and B is ‘John is intelligent’
And if indeed (we know that ) John is intelligent, then A->B is true, but surely we cannot infer B from A. A priori, there is some probability that A could be true or A could be false and so also for B. Given some a priori probabilities for A, B etc. we could find the probability for any wff of sentential logic.
Now consider an infinite number of sentences Pa, Pb, Pc, etc. of FOPL. To be more specific we can take them to be ‘a is intelligent’, ‘b is intelligent’, ‘c is intelligent’ and so on. If all of these are independent of each other and if a priori, none is either true in all interpretations or false in all interpretations (that is there is a finite a priori probability of them being true) then the
Probability (AxPx is true) is zero i.e. AxPx is definitely False (or equivalently the assertion ‘AxPx is True’ has an infinite information content).

As an example, consider the sentences Ey y=S0 and then Ey y=SS0 and so on in the language of set theory. Without assuming the axiom AxEy y=Sx None of them is necessarily true and nothing prevents us from choosing ~Ey y=SS0 as an axiom in place of Ey y=SS0. In other words, each of the assertions Ey y=S0, Ey y=SS0 is an independent assertion which has a finite probability of being true. Then the assertion
AxEy y=Sx describes the infinite number of choices we made, conveys an infinite amount of information, has a zero probability of being true; it is false.

[apoorv]

On Sunday, May 6, 2012 11:14:58 AM UTC+5:30, George Greene wrote:
> On Apr 22, 1:51 pm, apoorv <sudhir...@hotmail.com> wrote:
> > Is it that AxPx should always have the
> > interpretation 'For All things x in the Domain'
>
> That is the standard/classical/default semantics, for first-order
> logic.
>
> > or other interpretations are posible?
>
> Of course other interpretations are possible, but they are non-
> standard.

Ok. So the axiom AxPx->Pc is a codification of the standard semantics. What bothers me is that one could map this axiom into sentential logic to get a sentence that implies an infinite number of independent sentences.And that is not possible in sentential logic.

[Frederick Williams]

So just conclude that the mapping is not X preserving. I don't know
what X is.

[Frederick Williams]

Some work has been done on this. Popper claimed that the probability of
the universal statement is always zero, Bar-Hillel (I think it was he)
claimed otherwise: that from Pa, Pb, Pc, ... having non-zero probability
one could, granted some further conditions, conclude that (Ax)Px had
probability non-zero.

[alan.den...@gmail.com]

The FOPL sentence AxPx represents a class of sentences with a particular
kind of structure.
The symbols of sentential logic, however, represent ANY sentence whatsoever.

Another way of saying that is: the symbols of sentential logic are totally
generic, whereas the sentences of FOPL are not.

Your map is bothering you because you are, essentially, mapping structured
symbols onto unstructured ones and then reasoning about the unstructured
symbols. By forgetting the original structure you allow yourself conclusions
that would not otherwise be allowed.

I hope that helped.

[George Greene]

On May 7, 3:33 pm, apoorv <sudhir...@hotmail.com> wrote:

> Ok. So the axiom AxPx->Pc is a codification of the standard semantics.

NO, IT ISN'T. As I said at the end of the reply, THAT axiom is a NON-
standard semantics, "truth-value" semantics or substitutional
quantification.
If that axiom is all there is.
Under the standard semantics, what you just posted IS AN INFERENCE
RULE called "Universal instantiation", and it does NOT apply JUST to
"P" ( a unary predicate ) but TO ANY unary formula-schema, and what is
on the right side IS NOT "c" (some specific constant) but rather ANY
term in the language.
Moreover, the actual semantics implies that AxPx ->P(ANYTHING
WHATSOEVER IN THE *DOMAIN*), NOT just P(the few things THAT HAPPEN TO
HAVE NAMES THAT ARE TERMS). You almost CAN'T "codify" the standard
semantics WITHIN the language itself -- the semantics IS "META" -- is
BEYOND -- the language itself.

> What bothers me is that one could map this axiom into sentential logic
> to get a sentence that implies an infinite number of independent sentences.

Well, OF COURSE it implies an infinite number of independent
sentences. If you say that something is true of all x and the domain
of x's happens to include all the natural numbers, then you have just
implied infinitely many independent statements about the natural
numbers. The classic example is where P means "Provable".
You assert the consistency of the system by alleging Ax[~P(x,"0=1")].

> And that is not possible in sentential logic.

Of COURSE it isn't! Sentential logic IS *0th*-order logic! Ax[Px] is
from *1st*-order logic!
This is not something that *bothers* anybody -- the systems are just
DIFFERENT! One is higher than the other!

[George Greene]

On May 8, 6:04 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:

> > Ok. So the axiom AxPx->Pc is a codification of the standard semantics.
> What bothers me is that one could map this axiom into sentential logic
> to get a sentence that implies an infinite number of independent sentences.

Well, really, NO you couldn't. This sentence IS NOT IN 0th-order
logic; it's in 1st-order logic.
Under the substitutional/non-standard/truth-value semantics, this
would map into AN INFINITARY conjunction
with one conjunct for each term in the language.

> And that is not possible in sentential logic.

OF COURSE it's not possible; infinitely long sentences are not
standardly part of either kind of language.
But the point is, SOMETIMES YOU *MEAN* the infinitely long
conjunction; sometimes you really are trying to talk about all members
of an infinite class and say that all of them have some property.
That's what 1st-order logic IS FOR (talking about infinite domains).
If your domain is finite then you don't need quantifiers AT ALL.

[Graham Cooper]

What do you mean AxPx->Pc

A(x)A(c) P(x)->P(c) is a TAUTOLOGY

It's just X->X
A(x)P(x) -> A(c)P(c)

A(x)E(c) P(x)->P(c) does not always hold.

P() could be: LIVED_PAST_1000()

ALL person, LIVED_PAST_1000(person) -> SOMEBODY(a_person)
LIVED_PAST_1000(a_person)
is FALSE, but

ALL(x) THE_UNIVERSE_IS_BIG(x) -> EXIST(c) THE_UNIVERSE_IS_BIG(c)
is TRUE

it depends on what P is.

---------------------------

You can GENERALISE given a WITNESS to the opposite occurring.

EXIST(ROCKET) LAUNCH_FAIL(ROCKET)

NOT ALL ROCKETS LAUNCH == a true statement about all rockets


Herc

[Frederick Williams]

George Greene wrote:
>
> On May 8, 6:04 am, Frederick Williams <freddywilli...@btinternet.com>
> wrote:
>
> > > Ok. So the axiom AxPx->Pc is a codification of the standard semantics.
> > What bothers me is that one could map this axiom into sentential logic
> > to get a sentence that implies an infinite number of independent sentences.

No, I didn't.

[apoorv]

well, there is just one wff of sentential logic, namely a
contradiction that
does imply an infinite number of independent sentences. If we are
given
an unknown wff X such that X->A,X->B,X->C and so on are tautologically
true, then purely in the framewok of sentential logic, the inference
is that
X is a contradiction.So how do we get over this in FOPL which kind of
subsumes sentential logic or is an extension of sentential logic?
-apoorv

[apoorv]

On May 13, 2:45 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On May 8, 5:33 am, apoorv <sudhir...@hotmail.com> wrote:
>
> > On Sunday, May 6, 2012 11:14:58 AM UTC+5:30, George Greene wrote:
> > > On Apr 22, 1:51 pm, apoorv <sudhir...@hotmail.com> wrote:
> > > > Is it that AxPx should always have the
> > > > interpretation 'For All things x in the Domain'
>
> > > That is the standard/classical/default semantics, for first-order
> > > logic.
>
> > > > or other  interpretations are posible?
>
> > > Of course other interpretations are possible, but they are non-
> > > standard.
>
> > Ok. So the axiom AxPx->Pc is a codification of the standard semantics. What bothers me is that one could map this axiom into sentential logic to get a sentence that implies an infinite number of independent sentences.And that is not possible in sentential logic.
>
> What do you mean AxPx->Pc
>
> A(x)A(c) P(x)->P(c) is a TAUTOLOGY
>
> It's just  X->X
> A(x)P(x) -> A(c)P(c)
>
> A(x)E(c) P(x)->P(c) does not always hold.

I suppose you mean AxPx->EyPy. That would hold for all non empty
domains.

[apoorv]

Thanks. Agreed that the wff Pa of FOPL could not be translated into
any
sentence of sentential logic, but it still represents a class of and
not a definite
sentence. Pa could be 'John is intelligent' or 'Tom is intelligent'.
So it could be true
in one interpretation and false in another.
-apoorv

[Graham Cooper]

Right, then it only holds when x is free, not as a general subformula.



Herc

[George Greene]

On May 20, 1:17 pm, apoorv <sudhir...@hotmail.com> wrote:
> > Another way of saying that is: the symbols of sentential logic are totally
> > generic, whereas the sentences of FOPL are not.

That doesn't really have to be the case. They CAN be thought of as
totally generic -- they CAN be generalized -- but it's not absolutely
mandatory in all contexts.
In any particular context of their actual USE, some of them are going
to be true and some of them are going to be false, and some things are
going to follow and other things are not.

>
> > Your map is bothering you because you are, essentially, mapping structured
> > symbols onto unstructured ones and then reasoning about the unstructured
> > symbols. By forgetting the original structure you allow yourself conclusions
> > that would not otherwise be allowed.
>
> > I hope that helped.

Not really.

> Thanks. Agreed that  the wff  Pa of FOPL could not be translated into any
> sentence of sentential logic,

This IS NOT true! It translates JUST FINE into ANY ATOMIC sentence of
0th-order logic!
It just simply has a truth-value! And it is not conjoint or disjoint
or denied or anything.
IT IS ATOMIC. In 0th-order logic, that also makes it "generic", but
"generic" IS OPTIONAL, NOT mandatory.

> but it still represents a class of and not a definite sentence.

THAT IS WRONG. Pa *IS* a definite sentence, if a is a constant.

> Pa could be 'John is intelligent' or 'Tom is intelligent'.
> So it could be true
> in one interpretation and false in another.

ALL sentences except tautologies and contradictions can be true in one
interpretation and false in another! THAT MEANS *NOTHING*!!
Validities and contradictions ARE THE EXCEPTION, NOT the rule! They
are RARE!!

[George Greene]

On May 20, 1:05 pm, apoorv <sudhir...@hotmail.com> wrote:
> well, there is just one wff of sentential logic, namely a
> contradiction that
> does imply an infinite number of independent sentences.

Well, YOU were the one who originally said there were none!
Don't correct me for agreeing with you!

The contradiction is a degenerate case anyhow. We are talking about
consistent formal theories.

[apoorv]

Do we know for sure that the theories we are talking about are consistent?
If in sentential logic only a contadiction implies an infinite number of independent sentences, how do we assume as an axiom in FOPL the existence
of such a sentence?That is what puzzles me.
-apoorv