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MoeBlee · Guest

On Jul 11, 9:29 am, translogi <wilem...@googlemail.com> wrote:
[quote]On Jul 9, 6:59 pm, MoeBlee <jazzm...@hotmail.com> wrote:

"T is negation complete" does not say that for every
formula (even if it has free variables) A in the language of T, we
have T |- A or T |- ~A.

Rather, "T is negation complete" says that for every SENTENCE A in the
language of T, we have T |- A or T |- ~A.

But then what is then a sentence?
Not every wff (well formed formula )
[/quote]
A well formed formula with no free variables.

[quote]Or more precisely
what makes

1) P -> P a sentence
and
2) (P-> P) -> P not a sentence?
[/quote]
If 'P' is a sentence, then they>re both sentences (of some suitable
language). Who said one is a sentence but not the other?

[quote]or should i read it as

1a) for every P [P->P}
but then the second one should be read as
2a) for every P [(P->P)->P]
and so it is a sentence (although a false one)
[/quote]
I don>t know what difficulty you find in the simple rule:

If P is a sentence and Q is a sentence, then (P -> Q) is a sentence,
which entails (with informal convention of dropping outer parentheses)
that if P is a sentence then P -> P is a sentence and (P -> P) -> P is
a sentence.

[quote]Or maybe if a sentence defind as a well formed formula that is allways
true or always false in every model.
[/quote]
No, simply a sentence is a well formed formula with no free variables.

[quote](Given that we are talking about completeness we can talk about true
and false setences i guess, it isn>t obviously circular, but i would
prefer a definition of what a sentence is without referencing to Truth
or falsehood)
[/quote]
Such a definition is in just about any textbook in mathematical logic.
A sentence is a well formed formula with no free varialbes.

[quote]Maybe it is just painfully obvious for all of you
But I am just having a bit of philosophical struggle with this at the
moment.
[/quote]
It>s hardly philosophical. Just look at the definitions of the syntax.

MoeBlee

translogi · Guest

On Jul 14, 7:36 pm, MoeBlee <jazzm...@hotmail.com> wrote:
[quote]On Jul 11, 9:29 am, translogi <wilem...@googlemail.com> wrote:

On Jul 9, 6:59 pm, MoeBlee <jazzm...@hotmail.com> wrote:
"T is negation complete" does not say that for every
formula (even if it has free variables) A in the language of T, we
have T |- A or T |- ~A.

Rather, "T is negation complete" says that for every SENTENCE A in the
language of T, we have T |- A or T |- ~A.
But then what is then a sentence?
Not every wff (well formed formula )

A well formed formula with no free variables.

Or more precisely
what makes

1) P -> P a sentence
and
2) (P-> P) -> P not a sentence?

If 'P' is a sentence, then they>re both sentences (of some suitable
language). Who said one is a sentence but not the other?

or should i read it as

1a) for every P [P->P}
but then the second one should be read as
2a) for every P [(P->P)->P]
and so it is a sentence (although a false one)

I don>t know what difficulty you find in the simple rule:

If P is a sentence and Q is a sentence, then (P -> Q) is a sentence,
which entails (with informal convention of dropping outer parentheses)
that if P is a sentence then P -> P is a sentence and (P -> P) -> P is
a sentence.

Or maybe if a sentence defind as a well formed formula that is allways
true or always false in every model.

No, simply a sentence is a well formed formula with no free variables.

(Given that we are talking about completeness we can talk about true
and false setences i guess, it isn>t obviously circular, but i would
prefer a definition of what a sentence is without referencing to Truth
or falsehood)

Such a definition is in just about any textbook in mathematical logic.
A sentence is a well formed formula with no free varialbes.

Maybe it is just painfully obvious for all of you
But I am just having a bit of philosophical struggle with this at the
moment.

It>s hardly philosophical. Just look at the definitions of the syntax.

MoeBlee
[/quote]
The problems start when it is not given that P is an sentence.

Would it then mean that
P-> P isn>t a sentence either?

MoeBlee · Guest

On Jul 15, 10:25 am, translogi <wilem...@googlemail.com> wrote:

[quote]The problems start when it is not given that P is an sentence.
[/quote]
I don>t know what context in which it is not given that something is
or is not a sentence that you see a problem.

[quote]Would it then mean that
P-> P isn>t a sentence either?
[/quote]
If P is not a sentence, the P->P is not a sentence.

That is ensured by the recursive definition of 'occurs free in'.

I don>t see whatever difficulty you think there is.

MoeBlee

translogi · Guest

On Jul 15, 7:31 pm, MoeBlee <jazzm...@hotmail.com> wrote:
[quote]On Jul 15, 10:25 am, translogi <wilem...@googlemail.com> wrote:

The problems start when it is not given that P is an sentence.

I don>t know what context in which it is not given that something is
or is not a sentence that you see a problem.

Would it then mean that
P-> P isn>t a sentence either?

If P is not a sentence, the P->P is not a sentence.

That is ensured by the recursive definition of 'occurs free in'.

I don>t see whatever difficulty you think there is.

MoeBlee
[/quote]
But then negation completeness means nothing (much) at all.

Because it only talks about sentences.
and sentences are only formulas without (free) variables.

Then there is not a lot to talk about

In propositional logic you are onlu talking about formula>s with
constants.
(truth and falsehood)

Most (or even all) axioms are not sentences in this view.

MoeBlee · Guest

On Jul 15, 11:44 am, translogi <wilem...@googlemail.com> wrote:

[quote]But then negation completeness means nothing (much) at all.

Because it only talks about sentences.
and sentences are only formulas without (free) variables.
[/quote]
No, sentences are a lot to talk about. A theory is a set of sentences
closed under entailment. In logic we deal with both open formulas and
sentences, but in certain respects, it>s really sentences that we>re
most concerned about.

[quote]Then there is not a lot to talk about

In propositional logic you are onlu talking about formula>s with
constants.
(truth and falsehood)

Most (or even all) axioms are not sentences in this view.
[/quote]
No, in many (most?) treatments, the axioms are sentences. When they>re
stated as open formulas we are to take it as tacit that the actual
axioms are closures of the open formulas. And in pure propositional
logic, there are no open formulas, since there are no free variables
anywhere, since there are no variables at all.

MoeBlee

MoeBlee · Guest

On Jul 15, 12:11 pm, Balthasar <nomail@invalid> wrote:

[quote]Moreover there are treatments of FOPL where no open formulas appear,
only sentences. (Actually, motivated by a certain philosophical view,
I>m preferring this approach.)
[/quote]
What philosophical view do you mean?

MoeBlee

Balthasar · Guest

On Tue, 15 Jul 2008 11:44:27 -0700 (PDT), translogi
<wilemien@googlemail.com> wrote:

[quote]
Most (or even all) axioms are not sentences in this view.

Huh? Could you elaborate on that?[/quote]

G.

Balthasar · Guest

On Tue, 15 Jul 2008 12:02:05 -0700 (PDT), MoeBlee
<jazzmobe@hotmail.com> wrote:

[quote]
Because it only talks about sentences.
and sentences are only formulas without (free) variables.

No, sentences are a lot to talk about. A theory is a set of sentences
closed under entailment. In logic we deal with both open formulas and
sentences, but in certain respects, it>s really sentences that we>re
most concerned about.

Moreover there are treatments of FOPL where no open formulas appear,[/quote]
only sentences. (Actually, motivated by a certain philosophical view,
I>m preferring this approach.)

B.

Jan Burse · Guest

translogi schrieb:
[quote]In propositional logic you are onlu talking about formula>s with
constants.
(truth and falsehood)

Most (or even all) axioms are not sentences in this view.
[/quote]
Yes, propositional logic consists only of closed
formulas. All formulas are closed in SAT.

(Lets not consider QSAT, where one can quantify over
propositional variables, i.e. exists p.p, and one
might have prepositional constants and variables.
Example for a constant being f (bottom, _|_))

But there are formuals and schemas, even for propositional
logic.

A formula (small cap):

p -> p

A schema (upper cap):

P -> P

Stands for the following formulas (small cap):

p -> p
q -> q
(p -> q) -> (p -> q)
etc..

Bye

Jan Burse · Guest

Jan Burse schrieb:
[quote]translogi schrieb:
In propositional logic you are onlu talking about formula>s with
constants.
(truth and falsehood)

Most (or even all) axioms are not sentences in this view.

Yes, propositional logic consists only of closed
formulas. All formulas are closed in SAT.

(Lets not consider QSAT, where one can quantify over
propositional variables, i.e. exists p.p, and one
might have prepositional constants and variables.
Example for a constant being f (bottom, _|_))

But there are formuals and schemas, even for propositional
logic.

A formula (small cap):

p -> p

A schema (upper cap):

P -> P

Stands for the following formulas (small cap):

p -> p
q -> q
(p -> q) -> (p -> q)
etc..

Bye
[/quote]
Considering first order predicate logic,
unfortunately schemas are not the same as
quantified formulas. Take for example the
following schema:

A -> forall x A

Where A is a schema variable. I can
"instantiate" A with the formula p(x),
resulting in:

p(x) -> forall x p(x).

Although there is a variable clash! On
the other hand quantified, higher order,
we have:

forall q(q -> forall x.q)

When we want to instantiate the above formula,
we are forced to do a alfa conversion,
i.e. replace x by a new variable y:

p(x) -> forall y p(x)

Bye

herbzet · Guest

Jan Burse wrote:
[quote]herbzet schrieb:
[quoting translogi]:

1) P -> P a sentence
and
2) (P-> P) -> P not a sentence?
or should i read it as

1a) for every P [P->P}
but then the second one should be read as
2a) for every P [(P->P)->P]
and so it is a sentence (although a false one)

When you write:

P -> P

and allow P to stand for any propositional
formula, then this is a schema, and not
a formula.
[/quote]
[etc.]

Just wanted to say, nice post.

--
hz

herbzet · Guest

MoeBlee wrote:
[quote]translogi wrote:
[/quote]
[...]

[quote]In propositional logic you are only talking about formula>s with
constants.
(truth and falsehood)
[/quote]
Eh?

[...]

[quote]And in pure propositional
logic, there are no open formulas, since there are no free variables
anywhere, since there are no variables at all.
[/quote]
What??? What do you call p, q, r, etc.?

--
hz (killing two birds with one stone.)

herbzet · Guest

Jan Burse wrote:
[quote]translogi schrieb:

In propositional logic you are onlu talking about formula>s with
constants.
(truth and falsehood)

Most (or even all) axioms are not sentences in this view.

Yes, propositional logic consists only of closed
formulas. All formulas are closed in SAT.
[/quote]
Would someone please explain to me why propositional logic
does not have variables? Or does not have free variables?
Or whatever? I must have missed the memo.

Your propositional formulae at the bottom of this post
sure look to me like they contain free variables.

[quote](Lets not consider QSAT, where one can quantify over
propositional variables, i.e. exists p.p, and one
might have prepositional constants and variables.
Example for a constant being f (bottom, _|_))

But there are formuals and schemas, even for propositional
logic.

A formula (small cap):

p -> p

A schema (upper cap):

P -> P

Stands for the following formulas (small cap):

p -> p
q -> q
(p -> q) -> (p -> q)
etc..
[/quote]
--
hz

MoeBlee · Guest

On Jul 15, 10:31 pm, herbzet <herb...@gmail.com> wrote:
[quote]MoeBlee wrote:

And in pure propositional
logic, there are no open formulas, since there are no free variables
anywhere, since there are no variables at all.

What??? What do you call p, q, r, etc.?
[/quote]
I call them 'sentence letters'.

In this context, by 'variable' we (I) mean 'individual
variable' (i.e., the ones used in predicate logic that are, e.g., free
or bound). Now, some authors also call sentence letters 'variables'.
But those are not the KIND of variables that are at stake when we
refer to 'free and bound'. Ordinarily (there may be exceptions), an
author takes a sentence letter standing alone, or any formula built up
from just sentence letters and connectives to be a SENTENCE since it
does not have free variables (i.e., INDIVIDUAL variables that are not
bound). Moreover, as to what sentence letters are if they are not
considered variables of any kind, it is common to consider them to be
0-place predicate symbols.

/

For example, the symbols of a propositional system may be categorized
as:

sentence letters: p q r, etc.
connectives: ~ -> & v, etc.
parentheses (not needed if we use Polish notation)

and we may regard 't' and 'f' (for 'truth' and 'falsehood') as 0-ary
connectives.

/

Symbols for a first order predicate system may be catergorized as:

n-place predicate symbols (n a natural number, and when n=0, we have a
sentence letter)

n-place function symbols (n a natural number, and when n=0, we have an
individual constant)

n-ary connectives: ~ -> & v etc. (and we may have 't' and 'f' as 0-ary
connectives)

quantifiers: A E

parentheses (not needed if we use Polish notation)

/

My own prefernce for first order language is:

n-place predicate symbols

n-place function symbols

connectives - ~ -> (with & v <-> | 'dagger' t f defined)

quantifier - A (with E defined)

no parentheses, as officially Polish notation, but in practice use
parentheses for infix notation as informal

For identity theory:

The 2-place predicate symbol '=' (with the fixed semantics).

Define E!

Use the iota definite description operator infomally but in a clear
manner (the most obvious completely rigorous definition would require
double induction to define the sets of terms and formulas, then
simultaneous recursion would be required to define certain of the
syntactical operations such as, if I recall correctly, 'term
substituted for variable').

For set theory:

Extension of identity theory.

Add the 2-place predicate symbol 'e'.

Define set abstraction notation from the iota operator.

MoeBlee

translogi · Guest

On Jul 16, 8:13 pm, Jan Burse <janbu...@fastmail.fm> wrote:
[quote]herbzet schrieb:

Would someone please explain to me why propositional logic
does not have variables? Or does not have free variables?
Or whatever? I must have missed the memo.

Your propositional formulae at the bottom of this post
sure look to me like they contain free variables.

No, they are zero arity predicate constants.

Bye
Do I now still understand it[/quote]
(Or am i just getting totally confused)

p-> p

p is a sentence letter
(or equivalent an 0 -place predicate symbol)
(or a zero arity predicate constants
(or a propositional variable)
[ any other name for this concept ;-) ]

Is therefore p->p a sentence?

But if p->p is a sentence then also (p->p)->p is a sentence.

(if what a sentence is is only defined by syntax)

Back to Completeness.

negation completeness says

for every sentence A

|- A or |- ~A

But still neither |- (p->p)->p nor |- ~((p->p)->p)

therefore:
- (p->p) ->p is not a sentence
- propositional logic is not negation complete
or
something else all together????

replacing sentence letter p with Q

where Q stands for any propositional
formula, (then this is a schema, and not
a formula.)

Q -> Q
and
(Q->Q) ->Q

Makes it all just more complicated.
are they both sentences?
only the first one?
neither of them?
(that only the second is a sentence nobody will have that opinion)

Answer to Balthazar and Herbzet
If
Axiom A has sentence letters,

and
Sentences are formula>s without sentence letters (Or propositional
variables)

Then
Axiom A is not a sentence.

But it was to test if the definition
Sentences are formula>s without sentence letters cut any wood.
(Still think it doesn>t)

Defining axioms as sentences feels a bit the wrong way round.
(Completeness means that if formula A is true then formula A is
provable)
so starting with defining axioms as sentences looks at least weird.

So it all goes back to the question

What is a Sentence?
(Other than what is a well formed formula)

still confused


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