Prev: [ An AD(n) = (L(n,n) + 1) mod 9 -> An AD(n) =/= L(n,n) ] -> Higher Infinities
Next: -> Higher Infinities
From: Frederick Williams on 20 Jun 2010 14:40
Frederick Williams wrote:
>
> "David C. Ullrich" wrote:
>
> > No, it is not true that FOL includes inference rules.
>
> Because you are taking a "logic" to be a set of formulae (or perhaps
> sentences)?

What I have in mind of course is a set of formulae which contains some
"initial" formulae (axioms) and is closed under some rules of
inference. But will that do? One definition of formula may allow
predicate symbols, function symbols and constants; another may allow
predicate symbols but no function symbols or constants. So there are at
least two FOLs and hence it cannot be the case that FOL is *a* set of
formulae. I don't know.

--
I can't go on, I'll go on.
From: Christopher Menzel on 20 Jun 2010 18:01
On 2010-06-19, George Greene <greeneg(a)email.unc.edu> wrote:
> On Jun 19, 11:07 am, Frederick Williams
><frederick.willia...(a)tesco.net> wrote:
>> "David C. Ullrich" wrote:
>> > No, it is not true that FOL includes inference rules.
>>
>> Because you are taking a "logic" to be a set of formulae (or perhaps
>> sentences)?
>
> No, because DCU is JUST LYING.

Don't be silly. At most he's just saying something false. He is not
trying to deceive the readers of sci.logic. And, unqualified at least,
the quote above does appear to be false. Certainly one way (of two
standard ways, it seems to me) to define a logic is to specify a formal
language and a deductive apparatus for it, and a deductive apparatus
requires inference rules.

> One way to define ANY logic is by its inference rules.
> Even if these are not the totality of the definition, they are ALWAYS
> a CRUCIAL part of it.

Now that's not true either, although I will not accuse you of lying
about it. ;-) The other standard way of specifying a logic is model
theoretic. That is, one specifies a language and a notion of an
interpretation for the language in terms of which one defines a semantic
notion of logical consequence. A corresponding deductive apparatus for
such a logic may or may not be defined. In general, of course, it is
impossible to provide a complete deductive apparatus for a given model
theoretic logic -- consider standard second-order logic, for example.
And, depending on one's purposes, there may not be any point in doing
so.

There is a well known collection of essays entitled _Model Theoretic
Logics_, edited by Barwise and Feferman, that explores this notion of a
logic in great technical detail.

From: George Greene on 20 Jun 2010 20:38
On Jun 20, 6:01 pm, Christopher Menzel <cmen...(a)philebus.tamu.edu>
wrote:
> On 2010-06-19, George Greene <gree...(a)email.unc.edu> wrote:
> > One way to define ANY logic is by its inference rules.
> > Even if these are not the totality of the definition, they are ALWAYS
> > a CRUCIAL part of it.  
>
> Now that's not true either, although I will not accuse you of lying
> about it. ;-)  The other standard way of specifying a logic is model
> theoretic.

The way you are about to describe IS NOT "other".

>  That is, one specifies a language and a notion of an
> interpretation for the language in terms of which one defines a semantic
> notion of logical consequence.

ANY notion of logical consequence IS a rule of inference.
To say that any wff is a logical consequence of some others IS TO SAY
that it can BE INFERRED from them. If you have rules for determining
what
is a logical consequence OF what, then you have rules of inference.
The fact that they are defined semantically rather than syntactically
(and are therefore what(GUFFAW), rules "of CONSEQUENCE" ?)
DOES NOT STOP them from being rules of inference.

>  A corresponding deductive apparatus for
> such a logic may or may not be defined.  In general, of course, it is
> impossible to provide a complete deductive apparatus for a given model
> theoretic logic -- consider standard second-order logic, for example.

To the extent that the syntactic rules cannot be codified, the model-
theoretic ones HAVE NOT BEEN specified. It is hardly relevant to say
that X WOULD be a semantic consequence of Y when YOU CAN'T TELL
whether it is, or not.

The semantic definition IS NOT an alternative to the syntactic one.

> And, depending on one's purposes, there may not be any point in doing so.  

If THOSE are one's purposes then there MOST CERTAINLY is "not any
point"
in EVEN *CALLING* the result "a logic"!
CLUE:
http://www.gwu.edu/~philosop/news_events/documents/Pedeferri1.pdf

> There is a well known collection of essays entitled _Model Theoretic
> Logics_, edited by Barwise and Feferman, that explores this notion of a
> logic in great technical detail.

I'm sure, but this does not change the fact that if you cannot tell at
least APPROXIMATELY
what is a consequence of what, then you have completely defeated the
purpose. Standard classical FOL is
standard BECAUSE (among many other reasons) THERE IS a complete
syntactic characterization
of its semantic consequence relation. But to the extent that ANY
semantic consequence relations
ARE UNDERSTOOD, AT ALL,
that understanding is CODIFIED *VIA*RULES*, EVEN when the rules are
not explicitly
syntactic or complete.


From: Bill Taylor on 21 Jun 2010 01:15
> I've often wondered if, when you're hammering out those capitals, you
> also foam at the mouth.

NICE ONE! Joke of the week!!

_ _____ _ _
| | | ___ | | | | |
_\|/_ | | | | | | | | |_| _\|/_
/|\ | |_ | |__| | | |_ _ /|\
|___| |_____| |___| |_|

_ _____ _ _
| | | _ | | | | |
_\|/_ | | | | | | | | |_| _\|/_
/|\ | |_ | |_| | | |_ _ /|\
|___| |_____| |___| |_|
From: Frederick Williams on 21 Jun 2010 06:57
Christopher Menzel wrote:
>
> On 2010-06-19, George Greene <greeneg(a)email.unc.edu> wrote:
> > On Jun 19, 11:07� am, Frederick Williams
> ><frederick.willia...(a)tesco.net> wrote:
> >> "David C. Ullrich" wrote:
> >> > No, it is not true that FOL includes inference rules.
> >>
> >> Because you are taking a "logic" to be a set of formulae (or perhaps
> >> sentences)?
> >
> > No, because DCU is JUST LYING.
>
> Don't be silly. At most he's just saying something false. He is not
> trying to deceive the readers of sci.logic. And, unqualified at least,
> the quote above does appear to be false. Certainly one way (of two
> standard ways, it seems to me) to define a logic is to specify a formal
> language and a deductive apparatus for it, and a deductive apparatus
> requires inference rules.

It may be that what David Ullrich has in mind is that a logic is a set
of formulae, expressed in a certain language, closed under certain rules
of inference. It would, I suppose, be odd just to be "given" the set
without knowing (or at least pretending not to know) what the rules for
forming formulae were, and what the rules of inference were. It has,
perhaps, this point: if one considers that the only important things
about FOL are the answers to such questions as:

Is phi a theorem?

and

Does phi follow from psi?

Then the responses are

Is phi in the set?

and

Is psi -> phi in the set?

[In the later case it may not be 'psi -> phi', of course, it depends on
the language.]
--
I can't go on, I'll go on.
First  |  Prev  |  Next  |  Last
Pages: 1 2 3 4
Prev: [ An AD(n) = (L(n,n) + 1) mod 9 -> An AD(n) =/= L(n,n) ] -> Higher Infinities
Next: -> Higher Infinities