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Definition of mathematics again.
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Zuhair
Aug 6, 2012, 7:12:08 AM8/6/12
to
Mathematics is all of what can be "faithfully" interpreted in a
consistent formal system extending Logic.
Emphasis is put on "faithfully" which is meant to copy the properties
of interpreted concept, for example to say that for example "All
apples are Juicy" and not say anything else about what constitute an
apple and what Juicy means, is actually not different from saying "All
bananas are sweet" without saying anything else about bananas and what
sweet means, both are just statements composed of objects fulfilling a
predicate, so both alone are not faithful to the intended
interpretation of those statements. So although they are formalized
(put in a formal language) yet the formalization is not faithful. We
need not only put matters in formal symbols, we need enough
formalization necessary to ensure that the intended interpretation is
captured by the formal system and that the later is not speaking of a
different concept.
I actually think only mathematics can be interpreted faithfully in a
consistent formal system extending logic, and it doesn't matter if
that part of mathematics is proved to exist in reality and thus be a
part of physics also (i.e. an overlap of mathematics and physics) or
whether it doesn't, in either case it is mathematics!
This is clearly a logicist definition of mathematics which I think it
to be the nearest one to the truth of what mathematics is.
I think "ordinary mathematics" is all of what can be interpreted in a
consistent, categorical and effectively generated formal system
extending logic.
Zuhair
consistent formal system extending Logic.
Emphasis is put on "faithfully" which is meant to copy the properties
of interpreted concept, for example to say that for example "All
apples are Juicy" and not say anything else about what constitute an
apple and what Juicy means, is actually not different from saying "All
bananas are sweet" without saying anything else about bananas and what
sweet means, both are just statements composed of objects fulfilling a
predicate, so both alone are not faithful to the intended
interpretation of those statements. So although they are formalized
(put in a formal language) yet the formalization is not faithful. We
need not only put matters in formal symbols, we need enough
formalization necessary to ensure that the intended interpretation is
captured by the formal system and that the later is not speaking of a
different concept.
I actually think only mathematics can be interpreted faithfully in a
consistent formal system extending logic, and it doesn't matter if
that part of mathematics is proved to exist in reality and thus be a
part of physics also (i.e. an overlap of mathematics and physics) or
whether it doesn't, in either case it is mathematics!
This is clearly a logicist definition of mathematics which I think it
to be the nearest one to the truth of what mathematics is.
I think "ordinary mathematics" is all of what can be interpreted in a
consistent, categorical and effectively generated formal system
extending logic.
Zuhair
Nam Nguyen
Aug 6, 2012, 10:05:19 AM8/6/12
to
On 06/08/2012 5:12 AM, Zuhair wrote:
> Mathematics is all of what can be "faithfully" interpreted in a
> consistent formal system extending Logic.
>
> Emphasis is put on "faithfully" which is meant to copy the properties
> of interpreted concept
> Mathematics is all of what can be "faithfully" interpreted in a
> consistent formal system extending Logic.
>
> Emphasis is put on "faithfully" which is meant to copy the properties
> of interpreted concept
> So although they are formalized
> (put in a formal language) yet the formalization is not faithful. We
> need not only put matters in formal symbols, we need enough
> formalization necessary to ensure that the intended interpretation is
> captured by the formal system and that the later is not speaking of a
> different concept.
FOL axiomatic formalization already does that: for example, PA is
> (put in a formal language) yet the formalization is not faithful. We
> need not only put matters in formal symbols, we need enough
> formalization necessary to ensure that the intended interpretation is
> captured by the formal system and that the later is not speaking of a
> different concept.
supposed to more "faithfully" capture the concept of "the natural
numbers" than Q, where PA is extended from.
>
> I actually think only mathematics can be interpreted faithfully in a
> consistent formal system extending logic
FOL already provisions that: consistent formal systems are supposed to
be about "mathematics".
>
> I think "ordinary mathematics" is all of what can be interpreted in a
> consistent, categorical and effectively generated formal system
> extending logic.
(a) Model theoretically, there are mathematical statements that
_it's impossible_ to assert them as true or false; and the
impossibility is non-transient, foundational, (permanently)
impossible in principle - _hence_ also in practicality.
(b) FOL Provability-wise, by definition of syntactical consistency,
_it's impossible_ to assert the consistency of a formal system,
if it's genuinely so. ("Genuine" as opposed to "assumed").
So "ordinary mathematics" is subjective and is an untenable notion.
--
----------------------------------------------------
There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI
----------------------------------------------------
Zuhair
Aug 6, 2012, 4:14:22 PM8/6/12
to
On Aug 6, 5:05 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 06/08/2012 5:12 AM, Zuhair wrote:
>
> > Mathematics is all of what can be "faithfully" interpreted in a
> > consistent formal system extending Logic.
>
> > Emphasis is put on "faithfully" which is meant to copy the properties
> > of interpreted concept
> > So although they are formalized
> > (put in a formal language) yet the formalization is not faithful. We
> > need not only put matters in formal symbols, we need enough
> > formalization necessary to ensure that the intended interpretation is
> > captured by the formal system and that the later is not speaking of a
> > different concept.
>
> FOL axiomatic formalization already does that: for example, PA is
> supposed to more "faithfully" capture the concept of "the natural
> numbers" than Q, where PA is extended from.
Yes PA is a consistent formal system extending FOL. For the example
> On 06/08/2012 5:12 AM, Zuhair wrote:
>
> > Mathematics is all of what can be "faithfully" interpreted in a
> > consistent formal system extending Logic.
>
> > Emphasis is put on "faithfully" which is meant to copy the properties
> > of interpreted concept
> > So although they are formalized
> > (put in a formal language) yet the formalization is not faithful. We
> > need not only put matters in formal symbols, we need enough
> > formalization necessary to ensure that the intended interpretation is
> > captured by the formal system and that the later is not speaking of a
> > different concept.
>
> FOL axiomatic formalization already does that: for example, PA is
> supposed to more "faithfully" capture the concept of "the natural
> numbers" than Q, where PA is extended from.
that you gave FOL will represent what I call logic. However generally
speaking logic is not necessarily FOL, it can be any system dealing
with tautologies that is consistent and complete and most importantly
admits provability for examaple it can be infinitary FOL like L(w1,w),
or on the other hand it can be finitary logic.
>
>
>
> > I actually think only mathematics can be interpreted faithfully in a
> > consistent formal system extending logic
>
> Setting aside what you really meant by "formal system extending logic",
> FOL already provisions that: consistent formal systems are supposed to
> be about "mathematics".
>
Not necessarily there are lots of formal systems in FOL that even if
>
>
> > I actually think only mathematics can be interpreted faithfully in a
> > consistent formal system extending logic
>
> Setting aside what you really meant by "formal system extending logic",
> FOL already provisions that: consistent formal systems are supposed to
> be about "mathematics".
>
assumed consistent yet they are not winning the notion of being
mathematical by many mathematicians, they might be reasoned about as
part of "formal" knowledge but not of mathematics, some actually
regard a theory like ZF itself as not being part of mathematics. So
this definition of mine is not something that is agreeable by
mathematicians, but it would be agreeable my mathematician who agree
to logicism. What I'm saying here is that I think that logicism is the
correct school of philosophy of mathematics.
Zuhair
Nam Nguyen
Aug 6, 2012, 4:50:52 PM8/6/12
to
On 06/08/2012 2:14 PM, Zuhair wrote:
> On Aug 6, 5:05 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> On 06/08/2012 5:12 AM, Zuhair wrote:
>>
>>
>> Setting aside what you really meant by "formal system extending logic",
>> FOL already provisions that: consistent formal systems are supposed to
>> be about "mathematics".
>>
> Not necessarily there are lots of formal systems in FOL that even if
> assumed consistent yet they are not winning the notion of being
> mathematical by many mathematicians, they might be reasoned about as
> part of "formal" knowledge but not of mathematics, some actually
> regard a theory like ZF itself as not being part of mathematics. So
> this definition of mine is not something that is agreeable by
> mathematicians, but it would be agreeable my mathematician who agree
> to logicism. What I'm saying here is that I think that logicism is the
> correct school of philosophy of mathematics.
Ok, then I really don't have much of an opinion to present, mathematics
> On Aug 6, 5:05 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> On 06/08/2012 5:12 AM, Zuhair wrote:
>>
>>
>> Setting aside what you really meant by "formal system extending logic",
>> FOL already provisions that: consistent formal systems are supposed to
>> be about "mathematics".
>>
> Not necessarily there are lots of formal systems in FOL that even if
> assumed consistent yet they are not winning the notion of being
> mathematical by many mathematicians, they might be reasoned about as
> part of "formal" knowledge but not of mathematics, some actually
> regard a theory like ZF itself as not being part of mathematics. So
> this definition of mine is not something that is agreeable by
> mathematicians, but it would be agreeable my mathematician who agree
> to logicism. What I'm saying here is that I think that logicism is the
> correct school of philosophy of mathematics.
being relative to each human being, and "correct school of philosophy of
mathematics" is really anyone's subjective notion.
At least there's always a good chance that we talk about the same FOL
framework, where we'd know or could verify what is or isn't a
_logically valid_ statement.
MoeBlee
Aug 6, 2012, 7:04:54 PM8/6/12
to
On Aug 6, 3:14 pm, Zuhair <zaljo...@gmail.com> wrote:
> logic is not necessarily FOL, it can be any system dealing
> with tautologies that is consistent and complete
Second order logic is not complete. You don't consider it logic?
> logic is not necessarily FOL, it can be any system dealing
> with tautologies that is consistent and complete
Also, what does "DEALING with tautologies" mean exactly?
> logicism is the
> correct school of philosophy of mathematics.
mathematics from a system of system with no non-logical rules or
axioms.
I don't see how it could be done.
Bertram Russell, the most famous logicist, himself admitted that his
attempt used axioms that are non-logical.
MoeBlee
MoeBlee
Aug 6, 2012, 7:06:21 PM8/6/12
to
On Aug 6, 6:04 pm, MoeBlee <modem...@gmail.com> wrote:
> Bertram Russell
I meant 'Bertrand'.
MoeBlee
> Bertram Russell
I meant 'Bertrand'.
MoeBlee
Aatu Koskensilta
Aug 6, 2012, 8:47:41 PM8/6/12
to
Nam Nguyen <namduc...@shaw.ca> writes:
> FOL axiomatic formalization already does that: for example, PA is
> supposed to more "faithfully" capture the concept of "the natural
> numbers" than Q, where PA is extended from.
People do sometimes say things like this. There's nothing to these
> FOL axiomatic formalization already does that: for example, PA is
> supposed to more "faithfully" capture the concept of "the natural
> numbers" than Q, where PA is extended from.
reflections beyond the trivial observation that PA proves everything Q
does and then some more. PA proves, to mention one example, that
addition is associative, while Q does not.
> Setting aside what you really meant by "formal system extending logic",
> FOL already provisions that: consistent formal systems are supposed to
> be about "mathematics".
consistent or not, shouldn't be about apples or beer or the olympic
games.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Nam Nguyen
Aug 6, 2012, 8:58:47 PM8/6/12
to
On 06/08/2012 6:47 PM, Aatu Koskensilta wrote:
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> FOL axiomatic formalization already does that: for example, PA is
>> supposed to more "faithfully" capture the concept of "the natural
>> numbers" than Q, where PA is extended from.
>
> People do sometimes say things like this. There's nothing to these
> reflections beyond the trivial observation that PA proves everything Q
> does and then some more. PA proves, to mention one example, that
> addition is associative, while Q does not.
I'm not sure if we have to argue something here. (After all it's just
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> FOL axiomatic formalization already does that: for example, PA is
>> supposed to more "faithfully" capture the concept of "the natural
>> numbers" than Q, where PA is extended from.
>
> People do sometimes say things like this. There's nothing to these
> reflections beyond the trivial observation that PA proves everything Q
> does and then some more. PA proves, to mention one example, that
> addition is associative, while Q does not.
a comment about something Zuhair tried to do, which in the end I had
no further comment).
>
>> Setting aside what you really meant by "formal system extending logic",
>> FOL already provisions that: consistent formal systems are supposed to
>> be about "mathematics".
>
> No they're not. There's nothing in "FOL" to say formal systems,
> consistent or not, shouldn't be about apples or beer or the olympic
> games.
can't be formalized as _mathematical_ formal systems?
I mean, years ago "complex numbers" might have been an odd or
much-hated concept, but that can be formalized as a _mathematical_
formal system these days. Right?
Zuhair
Aug 7, 2012, 4:11:41 AM8/7/12
to
On Aug 7, 2:04 am, MoeBlee <modem...@gmail.com> wrote:
> On Aug 6, 3:14 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> > logic is not necessarily FOL, it can be any system dealing
> > with tautologies that is consistent and complete
>
> Second order logic is not complete. You don't consider it logic?
>
> Also, what does "DEALING with tautologies" mean exactly?
>
> > logicism is the
> > correct school of philosophy of mathematics.
>
> My understanding of the notion of logicism is that of deriving
> mathematics from a system of system with no non-logical rules or
> axioms.
>
> I don't see how it could be done.
>
No logicism is about reducing mathematics to a formal system
> On Aug 6, 3:14 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> > logic is not necessarily FOL, it can be any system dealing
> > with tautologies that is consistent and complete
>
> Second order logic is not complete. You don't consider it logic?
>
> Also, what does "DEALING with tautologies" mean exactly?
>
> > logicism is the
> > correct school of philosophy of mathematics.
>
> My understanding of the notion of logicism is that of deriving
> mathematics from a system of system with no non-logical rules or
> axioms.
>
> I don't see how it could be done.
>
"extending" logic, definitely this formal system can use extra-logical
symbols and rules.
> Bertram Russell, the most famous logicist, himself admitted that his
> attempt used axioms that are non-logical.
>
> MoeBlee
Zuhair
Aug 7, 2012, 6:27:40 AM8/7/12
to
On Aug 7, 2:04 am, MoeBlee <modem...@gmail.com> wrote:
> On Aug 6, 3:14 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> > logic is not necessarily FOL, it can be any system dealing
> > with tautologies that is consistent and complete
>
> Second order logic is not complete. You don't consider it logic?
Not in the sense of the definition here, I mean not in the sense that
>
> > logic is not necessarily FOL, it can be any system dealing
> > with tautologies that is consistent and complete
>
> Second order logic is not complete. You don't consider it logic?
mathematics can be seen as an extension of.
Anyhow generally speaking second order logic belong to the study of
logic, i.e. investigating logical structures, whether it is part of
logic or not? that's something that needs to be answered? I'm
personally not sure of it. Of course I'm speaking about
SOL with semantics that do not support a notion of provability,
otherwise SOL with semantics that support provability are
sane and complete so they qualify as "logic" the definition refers
to..
Zuhair
Aug 7, 2012, 6:34:44 AM8/7/12
to
On Aug 7, 3:47 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
"faithful" manner in FOL or other logical systems especially if we
require the formalization to be effectively generated, just a guess of
course, but say that they can be formalized as such then I'd be
inclined to name the resulting formalization as mathematics of apples,
beer, etc...
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon man nicht sprechen kann, darüber muss man schweigen"
> Nam Nguyen <namducngu...@shaw.ca> writes:
> > FOL axiomatic formalization already does that: for example, PA is
> > supposed to more "faithfully" capture the concept of "the natural
> > numbers" than Q, where PA is extended from.
>
> People do sometimes say things like this. There's nothing to these
> reflections beyond the trivial observation that PA proves everything Q
> does and then some more. PA proves, to mention one example, that
> addition is associative, while Q does not.
>
> > Setting aside what you really meant by "formal system extending logic",
> > FOL already provisions that: consistent formal systems are supposed to
> > be about "mathematics".
>
> No they're not. There's nothing in "FOL" to say formal systems,
> consistent or not, shouldn't be about apples or beer or the olympic
> games.
Agreed, but I don't really think that those can be formalized in a
> > FOL axiomatic formalization already does that: for example, PA is
> > supposed to more "faithfully" capture the concept of "the natural
> > numbers" than Q, where PA is extended from.
>
> People do sometimes say things like this. There's nothing to these
> reflections beyond the trivial observation that PA proves everything Q
> does and then some more. PA proves, to mention one example, that
> addition is associative, while Q does not.
>
> > Setting aside what you really meant by "formal system extending logic",
> > FOL already provisions that: consistent formal systems are supposed to
> > be about "mathematics".
>
> No they're not. There's nothing in "FOL" to say formal systems,
> consistent or not, shouldn't be about apples or beer or the olympic
> games.
"faithful" manner in FOL or other logical systems especially if we
require the formalization to be effectively generated, just a guess of
course, but say that they can be formalized as such then I'd be
inclined to name the resulting formalization as mathematics of apples,
beer, etc...
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon man nicht sprechen kann, darüber muss man schweigen"
Zuhair
Aug 7, 2012, 6:39:03 AM8/7/12
to
Mathematics is all of what can be "faithfully" interpreted in a
Ordinary Mathematics: is all of what can be "faithfully" interpreted
in a consistent effectively generated categorical formal system
extending logic.
Zuhair
LudovicoVan
Aug 7, 2012, 6:48:22 AM8/7/12
to
"Zuhair" <zalj...@gmail.com> wrote in message
news:34e516d5-b0ad-4d89...@h5g2000vbl.googlegroups.com...
> , just a guess of
> course, but say that they can be formalized as such then I'd be
> inclined to name the resulting formalization as mathematics of apples,
> beer, etc...
-LV
news:34e516d5-b0ad-4d89...@h5g2000vbl.googlegroups.com...
> On Aug 7, 3:47 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>> Nam Nguyen <namducngu...@shaw.ca> writes:
>> > FOL axiomatic formalization already does that: for example, PA is
>> > supposed to more "faithfully" capture the concept of "the natural
>> > numbers" than Q, where PA is extended from.
>>
>> People do sometimes say things like this. There's nothing to these
>> reflections beyond the trivial observation that PA proves everything Q
>> does and then some more. PA proves, to mention one example, that
>> addition is associative, while Q does not.
>>
>> > Setting aside what you really meant by "formal system extending logic",
>> > FOL already provisions that: consistent formal systems are supposed to
>> > be about "mathematics".
>>
>> No they're not. There's nothing in "FOL" to say formal systems,
>> consistent or not, shouldn't be about apples or beer or the olympic
>> games.
>
> Agreed, but I don't really think that those can be formalized in a
> "faithful" manner in FOL or other logical systems especially if we
> require the formalization to be effectively generated
Modal logic extends classical logic and can capture those notions.
>> Nam Nguyen <namducngu...@shaw.ca> writes:
>> > FOL axiomatic formalization already does that: for example, PA is
>> > supposed to more "faithfully" capture the concept of "the natural
>> > numbers" than Q, where PA is extended from.
>>
>> People do sometimes say things like this. There's nothing to these
>> reflections beyond the trivial observation that PA proves everything Q
>> does and then some more. PA proves, to mention one example, that
>> addition is associative, while Q does not.
>>
>> > Setting aside what you really meant by "formal system extending logic",
>> > FOL already provisions that: consistent formal systems are supposed to
>> > be about "mathematics".
>>
>> No they're not. There's nothing in "FOL" to say formal systems,
>> consistent or not, shouldn't be about apples or beer or the olympic
>> games.
>
> Agreed, but I don't really think that those can be formalized in a
> "faithful" manner in FOL or other logical systems especially if we
> require the formalization to be effectively generated
> , just a guess of
> course, but say that they can be formalized as such then I'd be
> inclined to name the resulting formalization as mathematics of apples,
> beer, etc...
MoeBlee
Aug 7, 2012, 10:21:13 AM8/7/12
to
On Aug 7, 3:11 am, Zuhair <zaljo...@gmail.com> wrote:
> On Aug 7, 2:04 am, MoeBlee <modem...@gmail.com> wrote:
> > My understanding of the notion of logicism is that of deriving
> > mathematics from a system [...] with no non-logical rules or
> On Aug 7, 2:04 am, MoeBlee <modem...@gmail.com> wrote:
> > My understanding of the notion of logicism is that of deriving
> > axioms.
> No logicism is about reducing mathematics to a formal system
> "extending" logic, definitely this formal system can use extra-logical
> symbols and rules.
own definition, but what you just said is not how the word is
ordinarily used in the philosophy of mathematics.
> > Bertra[nd] Russell, the most famous logicist, himself admitted that his
> > attempt used axioms that are non-logical.
>
> You mean the axiom of reducibility.
Aside from any opinion of Russell, also the axioms of infinity and
>
> You mean the axiom of reducibility.
choice are non-logical.
> Still this doesn't abort logicism.
doesn't rule out that there might be some other system to uphold the
claim. But what system would that be? Again, logicism is NOT the mere
claim (or hope) to develop mathematics in a formal system extending
logic, but rather the claim (or hope) to develop mathematics without
recourse to non-logical axioms or rules.
MoeBlee
Michael Stemper
Aug 7, 2012, 1:25:21 PM8/7/12
to
In article <f24861c3-1762-4735...@c25g2000yqa.googlegroups.com>, MoeBlee <mode...@gmail.com> writes:
>On Aug 7, 3:11=A0am, Zuhair <zaljo...@gmail.com> wrote:
--
Michael F. Stemper
#include <Standard_Disclaimer>
Life's too important to take seriously.
>On Aug 7, 3:11=A0am, Zuhair <zaljo...@gmail.com> wrote:
>> On Aug 7, 2:04=A0am, MoeBlee <modem...@gmail.com> wrote:
>> > My understanding of the notion of logicism is that of deriving
>> > mathematics from a system [...] with no non-logical rules or
>> > axioms.
>
>> No logicism is about reducing mathematics to a formal system
>> "extending" logic, definitely this formal system can use extra-logical
>> symbols and rules.
>
>No, that is not what is meant by 'logicism'. You may stipulate your
>own definition, but what you just said is not how the word is
>ordinarily used in the philosophy of mathematics.
>
>> > Bertra[nd] Russell, the most famous logicist, himself admitted that his
>> > attempt used axioms that are non-logical.
>>
>> You mean the axiom of reducibility.
>
>Aside from any opinion of Russell, also the axioms of infinity and
>choice are non-logical.
What criteria determine whether an axiom is "logical" or "non-logical"?
>> > My understanding of the notion of logicism is that of deriving
>> > mathematics from a system [...] with no non-logical rules or
>> > axioms.
>
>> No logicism is about reducing mathematics to a formal system
>> "extending" logic, definitely this formal system can use extra-logical
>> symbols and rules.
>
>No, that is not what is meant by 'logicism'. You may stipulate your
>own definition, but what you just said is not how the word is
>ordinarily used in the philosophy of mathematics.
>
>> > Bertra[nd] Russell, the most famous logicist, himself admitted that his
>> > attempt used axioms that are non-logical.
>>
>> You mean the axiom of reducibility.
>
>Aside from any opinion of Russell, also the axioms of infinity and
>choice are non-logical.
--
Michael F. Stemper
#include <Standard_Disclaimer>
Life's too important to take seriously.
MoeBlee
Aug 7, 2012, 2:42:00 PM8/7/12
to
On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
wrote:
> What criteria determine whether an axiom is "logical" or "non-logical"?
Semantically, it is logical iff it is true in all structures for the
language.
MoeBlee
wrote:
> What criteria determine whether an axiom is "logical" or "non-logical"?
language.
MoeBlee
Frederick Williams
Aug 7, 2012, 3:09:16 PM8/7/12
to
language one needs to know what a logical constant is. It is quite
arbitrary as to whether element-of is a logical constant or not. So the
answer to Michael Stemper (excellent) question is: you decide, it is
quite arbitrary.
--
The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.
Frederick Williams
Aug 7, 2012, 3:17:16 PM8/7/12
to
Michael Stemper wrote:
>
> What criteria determine whether an axiom is "logical" or "non-logical"?
Oh it's quite simple. Not.
>
> What criteria determine whether an axiom is "logical" or "non-logical"?
This: P v ~P is a (candidate to be a) logical axiom because it is true
in all interpretations. I.e. whether P is true or false, P v ~P comes
out true. What you may not do is fiddle with the interpretation of v
and read it as "and"? Why not? Because v is a logical constant. So to
decide if a formula is a logical axiom you need to know what a logical
constant is. And that is purely a matter of whim.
MoeBlee
Aug 7, 2012, 3:28:47 PM8/7/12
to
On Aug 7, 2:09 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
given the standard semantics for equality. (Of course, the
interpretation of the quantifiers, connectives, and the role of
variables are also "fixed" too.)
> It is quite
> arbitrary as to whether element-of is a logical constant or not.
'e' does not have a fixed standard semantics in the way that '=' does,
so 'e' is not (ordinarily) considered a logical constant, and
pesonally I've never seen a treatment of languages or logic in which
'e' is specified as a logical constant.
> So the
> answer to Michael Stemper (excellent) question is: you decide, it is
> quite arbitrary.
The notion of a logical axiom is well enough understood and is not
arbitrary. One might present certain objections with the notion, but
it is not a merely arbitrary classification.
MoeBlee
wrote:
> MoeBlee wrote:
> > Semantically, it is logical iff it is true in all structures for the
> > language.
>
> To determine whether a formula is true in all structures for the
> language one needs to know what a logical constant is.
It is usually allowed that the only logical constant is '=' when it is
> > Semantically, it is logical iff it is true in all structures for the
> > language.
>
> To determine whether a formula is true in all structures for the
> language one needs to know what a logical constant is.
given the standard semantics for equality. (Of course, the
interpretation of the quantifiers, connectives, and the role of
variables are also "fixed" too.)
> It is quite
> arbitrary as to whether element-of is a logical constant or not.
so 'e' is not (ordinarily) considered a logical constant, and
pesonally I've never seen a treatment of languages or logic in which
'e' is specified as a logical constant.
> So the
> answer to Michael Stemper (excellent) question is: you decide, it is
> quite arbitrary.
arbitrary. One might present certain objections with the notion, but
it is not a merely arbitrary classification.
MoeBlee
MoeBlee
Aug 7, 2012, 3:34:12 PM8/7/12
to
On Aug 7, 2:17 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
> What you may not do is fiddle with the interpretation of v
> and read it as "and"? Why not?
Yes, obviously, we speak in a context in which the interpretation of
the connectives is fixed.
We don't dispute that it is arbitary that we take the typographic
shape 'v' to have the fixed interpretation of disjunction.
But that is not what is at stake in the matter. Of course, we speak in
the context of certain typographic conventions. We could, with more
bother, couch our classifications by saying such as: "IF 'v' is agreed
upon as standing for disjunction, THEN such and such ..." But there's
no need for that since a reasonable person may well enough understand
that we've already made clear that the context is one in which 'v' is
being used for disjunction.
MoeBlee
wrote:
> What you may not do is fiddle with the interpretation of v
> and read it as "and"? Why not?
the connectives is fixed.
We don't dispute that it is arbitary that we take the typographic
shape 'v' to have the fixed interpretation of disjunction.
But that is not what is at stake in the matter. Of course, we speak in
the context of certain typographic conventions. We could, with more
bother, couch our classifications by saying such as: "IF 'v' is agreed
upon as standing for disjunction, THEN such and such ..." But there's
no need for that since a reasonable person may well enough understand
that we've already made clear that the context is one in which 'v' is
being used for disjunction.
MoeBlee
Aatu Koskensilta
Aug 7, 2012, 6:44:26 PM8/7/12
to
mind.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
Aatu Koskensilta
Aug 7, 2012, 6:45:46 PM8/7/12
to
MoeBlee <mode...@gmail.com> writes:
> The notion of a logical axiom is well enough understood and is not
> arbitrary. One might present certain objections with the notion, but
> it is not a merely arbitrary classification.
What counts as logical and what not, in the sence of logicism, is not
> The notion of a logical axiom is well enough understood and is not
> arbitrary. One might present certain objections with the notion, but
> it is not a merely arbitrary classification.
at all well understood.
MoeBlee
Aug 7, 2012, 7:30:11 PM8/7/12
to
On Aug 7, 5:45 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
In, for example, classical first order logic, it's well enough
understood what a logical axiom is. I don't opine as to vagaries,
problems, or disagreements as to the notion in all systems of logic
whatsoever.
MoeBlee
> MoeBlee <modem...@gmail.com> writes:
> > The notion of a logical axiom is well enough understood and is not
> > arbitrary. One might present certain objections with the notion, but
> > it is not a merely arbitrary classification.
>
> What counts as logical and what not, in the sence of logicism, is not
> at all well understoo.
> > The notion of a logical axiom is well enough understood and is not
> > arbitrary. One might present certain objections with the notion, but
> > it is not a merely arbitrary classification.
>
> What counts as logical and what not, in the sence of logicism, is not
In, for example, classical first order logic, it's well enough
understood what a logical axiom is. I don't opine as to vagaries,
problems, or disagreements as to the notion in all systems of logic
whatsoever.
MoeBlee
MoeBlee
Aug 7, 2012, 7:38:04 PM8/7/12
to
On Aug 7, 5:44 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
the modern context that we have subsequent to Frege and Russell. Of
course, granted, when Frege and Russell were on the case, the notion
of a structure for a language, such as in the way of Tarski, had not
yet been developed. However, the notion of "logical" was not merely,
as proposed by zuhair, that of "formalized", and I think the notion of
"true in all structures" does reflect in, more modern terms, the
general sense of "logically true" as opposed to "contingently true".
MoeBlee
> MoeBlee <modem...@gmail.com> writes:
> > On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
> > wrote:
>
> >> What criteria determine whether an axiom is "logical" or "non-logical"?
>
> > Semantically, it is logical iff it is true in all structures for the
> > language.
>
> This is not the conception of logic that Russell and Frege had in
> mind.
It's the best I can summarize the conception as couched technically in
> > On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
> > wrote:
>
> >> What criteria determine whether an axiom is "logical" or "non-logical"?
>
> > Semantically, it is logical iff it is true in all structures for the
> > language.
>
> This is not the conception of logic that Russell and Frege had in
> mind.
the modern context that we have subsequent to Frege and Russell. Of
course, granted, when Frege and Russell were on the case, the notion
of a structure for a language, such as in the way of Tarski, had not
yet been developed. However, the notion of "logical" was not merely,
as proposed by zuhair, that of "formalized", and I think the notion of
"true in all structures" does reflect in, more modern terms, the
general sense of "logically true" as opposed to "contingently true".
MoeBlee
Aatu Koskensilta
Aug 7, 2012, 8:23:30 PM8/7/12
to
MoeBlee <mode...@gmail.com> writes:
> However, the notion of "logical" was not merely, as proposed by
> zuhair, that of "formalized", and I think the notion of "true in all
> structures" does reflect in, more modern terms, the general sense of
> "logically true" as opposed to "contingently true".
Sure. But Frege and Russell didn't think of logical truths in terms of
> However, the notion of "logical" was not merely, as proposed by
> zuhair, that of "formalized", and I think the notion of "true in all
> structures" does reflect in, more modern terms, the general sense of
> "logically true" as opposed to "contingently true".
structures or interpretations at all. A different sort of explanation is
needed, if we are to understand logicism (as it once was).
MoeBlee
Aug 7, 2012, 7:47:10 PM8/7/12
to
On Aug 7, 6:38 pm, MoeBlee <modem...@gmail.com> wrote:
> It's the best I can summarize the conception as couched technically in
> the modern context that we have subsequent to Frege and Russell.
But I can't say that this should withstand criticism for potentially
> It's the best I can summarize the conception as couched technically in
> the modern context that we have subsequent to Frege and Russell.
being a misleading oversimplification.
MoeBlee
Nam Nguyen
Aug 7, 2012, 9:17:39 PM8/7/12
to
Axioms are purely syntactical hence the criteria of determining what
is a logical or non-logical axiom is also syntactical.
"A logical axiom is a formula which is a propositional axiom,
a substitution axiom, an identity axiom, or an equality axiom"
(Shoenfield, Mathematical Logic, pg. 21).
LudovicoVan
Aug 7, 2012, 10:57:33 PM8/7/12
to
"Michael Stemper" <mste...@walkabout.empros.com> wrote in message
news:jvrj20$rvp$1...@dont-email.me...
> What criteria determine whether an axiom is "logical" or "non-logical"?
Logic deals with validity, i.e.*logical*
necessity/possibility/self-contradiction, as resulting from the form itself
of statements, regardless of factual matters. Mathematics can then pick the
valid (non self-contradictory) statements and is interested in matters of
*factual* truth, i.e. in theories about mathematical objects. Logic is the
language of mathematics, mathematics is the content of mathematics.
-LV
news:jvrj20$rvp$1...@dont-email.me...
> What criteria determine whether an axiom is "logical" or "non-logical"?
necessity/possibility/self-contradiction, as resulting from the form itself
of statements, regardless of factual matters. Mathematics can then pick the
valid (non self-contradictory) statements and is interested in matters of
*factual* truth, i.e. in theories about mathematical objects. Logic is the
language of mathematics, mathematics is the content of mathematics.
-LV
Bill Taylor
Aug 8, 2012, 2:43:16 AM8/8/12
to
On Aug 8, 12:23 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> But Frege and Russell didn't think of logical truths in terms of
> structures or interpretations at all. A different sort of explanation
> is needed, if we are to understand logicism (as it once was).
The original thread here seems to invite this last proviso:
> But Frege and Russell didn't think of logical truths in terms of
> structures or interpretations at all. A different sort of explanation
> is needed, if we are to understand logicism (as it once was).
"as it once was". And indeed, wherever I have read anything
much about Logicism, it seems to come with an at least implicit
such proviso, as if it were largely a thing of the past.
Is this so - can such a view be reasonably maintained?
Is logicism "a thing of the past"? Or has it, perhaps in
some modified or diluted sense, "taken over" the field
of math foundations, at least to the extent that it has
forced it to become very formal.
I know there are said to be people who still call themselves
"logicists" today, though I forget who they are. Would they
have been regarded as the genuine article by Frege & Russell?
-- Wondering Willy
** "Let us not pretend to doubt in philosophy
** what we do not doubt in our hearts." - Pierce
William Elliot
Aug 8, 2012, 3:32:18 AM8/8/12
to
The logic of a mathematical definition of mathematics is ill.
Aatu Koskensilta
Aug 8, 2012, 3:37:50 AM8/8/12
to
William Elliot <ma...@panix.com> writes:
> The logic of a mathematical definition of mathematics is ill.
Your mother is ill.
> The logic of a mathematical definition of mathematics is ill.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
Zuhair
Aug 8, 2012, 3:59:00 AM8/8/12
to
On Aug 8, 1:45 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
analytic", and that it can be derived from the basic three rules of
thought (Identity, non contradiction, excluded middle) and they tried
to reduce all of mathematics to those, I think as far as this view
"extending" logic with some of what we call know as "non logical"
symbols and rules that are "consistent" with a logical language that
is sane, and complete and what is even more important "implicationally
complete" this would actually render those true "extensions" of logic.
Anyhow it is somehow arbitrary to call e or p (the mereological part-
hood primitive) as non logical.
Zuhair
> MoeBlee <modem...@gmail.com> writes:
> > The notion of a logical axiom is well enough understood and is not
> > arbitrary. One might present certain objections with the notion, but
> > it is not a merely arbitrary classification.
>
> What counts as logical and what not, in the sence of logicism, is not
> at all well understood.
>
> --
I think what was occupying their minds is that mathematics is "priori-
> > The notion of a logical axiom is well enough understood and is not
> > arbitrary. One might present certain objections with the notion, but
> > it is not a merely arbitrary classification.
>
> What counts as logical and what not, in the sence of logicism, is not
> at all well understood.
>
> --
analytic", and that it can be derived from the basic three rules of
thought (Identity, non contradiction, excluded middle) and they tried
to reduce all of mathematics to those, I think as far as this view
"extending" logic with some of what we call know as "non logical"
symbols and rules that are "consistent" with a logical language that
is sane, and complete and what is even more important "implicationally
complete" this would actually render those true "extensions" of logic.
Anyhow it is somehow arbitrary to call e or p (the mereological part-
hood primitive) as non logical.
Zuhair
Zuhair
Aug 8, 2012, 3:48:24 AM8/8/12
to
On Aug 7, 10:28 pm, MoeBlee <modem...@gmail.com> wrote:
> On Aug 7, 2:09 pm, Frederick Williams <freddywilli...@btinternet.com>
> wrote:
>
> > MoeBlee wrote:
> > > Semantically, it is logical iff it is true in all structures for the
> > > language.
>
> > To determine whether a formula is true in all structures for the
> > language one needs to know what a logical constant is.
>
> It is usually allowed that the only logical constant is '=' when it is
> given the standard semantics for equality. (Of course, the
> interpretation of the quantifiers, connectives, and the role of
> variables are also "fixed" too.)
>
> > It is quite
> > arbitrary as to whether element-of is a logical constant or not.
>
> 'e' does not have a fixed standard semantics in the way that '=' does,
> so 'e' is not (ordinarily) considered a logical constant, and
> pesonally I've never seen a treatment of languages or logic in which
> 'e' is specified as a logical constant.
Tarski once proposed considering e as a constant in order to resolve
> On Aug 7, 2:09 pm, Frederick Williams <freddywilli...@btinternet.com>
> wrote:
>
> > MoeBlee wrote:
> > > Semantically, it is logical iff it is true in all structures for the
> > > language.
>
> > To determine whether a formula is true in all structures for the
> > language one needs to know what a logical constant is.
>
> It is usually allowed that the only logical constant is '=' when it is
> given the standard semantics for equality. (Of course, the
> interpretation of the quantifiers, connectives, and the role of
> variables are also "fixed" too.)
>
> > It is quite
> > arbitrary as to whether element-of is a logical constant or not.
>
> 'e' does not have a fixed standard semantics in the way that '=' does,
> so 'e' is not (ordinarily) considered a logical constant, and
> pesonally I've never seen a treatment of languages or logic in which
> 'e' is specified as a logical constant.
set theoretic paradoxes, I have vague memory of that, I don't know if
he suggested it as a logical constant?
Aatu Koskensilta
Aug 8, 2012, 5:38:14 AM8/8/12
to
Bill Taylor <wfc.t...@gmail.com> writes:
> The original thread here seems to invite this last proviso:
> "as it once was". And indeed, wherever I have read anything
> much about Logicism, it seems to come with an at least implicit
> such proviso, as if it were largely a thing of the past.
Logicism (of the Frege-Russell sort) is entirely a thing of the
> The original thread here seems to invite this last proviso:
> "as it once was". And indeed, wherever I have read anything
> much about Logicism, it seems to come with an at least implicit
> such proviso, as if it were largely a thing of the past.
past. The principles and notions of Frege's original system were
perfectly logical -- provided we accept concepts (or classes) and their
basic properties part of logic -- save for the unfortunate Axiom V. And
alas, without V it's not possible to derive Hume's law. (Hume's Law
alone suffices for everything else in Frege's development.)
> Is logicism "a thing of the past"? Or has it, perhaps in some
> modified or diluted sense, "taken over" the field of math foundations,
> at least to the extent that it has forced it to become very formal.
days, to be sure, in the sense that all manner of formal theories are
involved. The situation probably would not have pleased Frege, since
these formalities are mere objects of mathematical study, and not
vehicles for mathematical and logical reasoning.
There's also neo-logicism, of course, a purely academic exercise. It
started off with great stuff, e.g. the discovery that everything goes
just swimmingly if we replace Axiom V with Hume's Law in
_Grundlagen_. But as that's pretty much everything sensible there's to
say about it, and since naturally no one is actually in any need of
logicist reassurance about arithmetic, the rest is predictably nothing
but your customary philosophical waffle, bringing in its wake as usual
insightful, technically sophisticated, and detailed detached analysis of
every possible way one could conceivably waffle about and around all
this.
> I know there are said to be people who still call themselves
> "logicists" today, though I forget who they are. Would they
> have been regarded as the genuine article by Frege & Russell?
Aatu Koskensilta
Aug 8, 2012, 5:51:41 AM8/8/12
to
Zuhair <zalj...@gmail.com> writes:
> I think what was occupying their minds is that mathematics is "priori-
> analytic", and that it can be derived from the basic three rules of
> thought (Identity, non contradiction, excluded middle) and they tried
> to reduce all of mathematics to those
Not all of it, not Frege anyway. Frege was with Kant on geometry,
> I think what was occupying their minds is that mathematics is "priori-
> analytic", and that it can be derived from the basic three rules of
> thought (Identity, non contradiction, excluded middle) and they tried
> to reduce all of mathematics to those
declaring it synthetic, in no need and indeed incapable of reduction to
logic. Unlike Kant he thought arithmetic (and analysis etc.) was
analytic. That is, Frege believed the relevant basic notions could be
defined in purely logical terms, and the relevant basic principles (when
formulated using these definitions) established by purely logical proof,
as he understood logic.
> I think as far as this view "extending" logic with some of what we
> call know as "non logical" symbols and rules that are "consistent"
> with a logical language that is sane, and complete and what is even
> more important "implicationally complete" this would actually render
> those true "extensions" of logic.
Zuhair
Aug 8, 2012, 6:26:48 AM8/8/12
to
On Aug 8, 12:51 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
complete". Now take the class L* of all systems extending FOL, those
of course would contain non logical symbols and axioms but are written
according to the rules of FOL and are consistent. Now L* reflect the
"Capacity" of FOL; now to me L* is actually logical, it is priori and
analytic, it is the class of all "stories" written in FOL, so to me I
see it as a part of logic in the "larger" sense, and since I think
that mathematics can be interpreted in some of the elements of L*,
then mathematics is nothing but an extension of logic really and is to
be regarded as part of Logic in the Larger sense. So for example if
one is trying to say interpret all mathematics in ZF and suppose that
was successful and ZF was proved consistent, then to me this suits
logicism, perhaps not in Russell/Frege sense, but it does go with the
spirit of logicism in the larger sense. But anyhow I'm myself growing
suspicious of my own definition since it might be the case that other
fields can also be interpreted faithfully in FOL for example, not only
that there seems to be some mathematics in the base logic itself, even
FOL, and propositional logic do use the natural numbers apparatus for
dealing with complex situations within them, on the other hand I think
perhaps mathematics is much smaller than what I think, possibly fairly
simple systems extending FOL are enough to interpret faithfully all of
math. but "simple" needs to be characterized. Anyhow.
Zuhair
> Zuhair <zaljo...@gmail.com> writes:
> > I think what was occupying their minds is that mathematics is "priori-
> > analytic", and that it can be derived from the basic three rules of
> > thought (Identity, non contradiction, excluded middle) and they tried
> > to reduce all of mathematics to those
>
> Not all of it, not Frege anyway. Frege was with Kant on geometry,
> declaring it synthetic, in no need and indeed incapable of reduction to
> logic. Unlike Kant he thought arithmetic (and analysis etc.) was
> analytic. That is, Frege believed the relevant basic notions could be
> defined in purely logical terms, and the relevant basic principles (when
> formulated using these definitions) established by purely logical proof,
> as he understood logic.
>
> > I think as far as this view "extending" logic with some of what we
> > call know as "non logical" symbols and rules that are "consistent"
> > with a logical language that is sane, and complete and what is even
> > more important "implicationally complete" this would actually render
> > those true "extensions" of logic.
>
> I'm afraid you've lost me here.
>
Take for example FOL which is sane, complete and "implicationally
> > I think what was occupying their minds is that mathematics is "priori-
> > analytic", and that it can be derived from the basic three rules of
> > thought (Identity, non contradiction, excluded middle) and they tried
> > to reduce all of mathematics to those
>
> Not all of it, not Frege anyway. Frege was with Kant on geometry,
> declaring it synthetic, in no need and indeed incapable of reduction to
> logic. Unlike Kant he thought arithmetic (and analysis etc.) was
> analytic. That is, Frege believed the relevant basic notions could be
> defined in purely logical terms, and the relevant basic principles (when
> formulated using these definitions) established by purely logical proof,
> as he understood logic.
>
> > I think as far as this view "extending" logic with some of what we
> > call know as "non logical" symbols and rules that are "consistent"
> > with a logical language that is sane, and complete and what is even
> > more important "implicationally complete" this would actually render
> > those true "extensions" of logic.
>
> I'm afraid you've lost me here.
>
complete". Now take the class L* of all systems extending FOL, those
of course would contain non logical symbols and axioms but are written
according to the rules of FOL and are consistent. Now L* reflect the
"Capacity" of FOL; now to me L* is actually logical, it is priori and
analytic, it is the class of all "stories" written in FOL, so to me I
see it as a part of logic in the "larger" sense, and since I think
that mathematics can be interpreted in some of the elements of L*,
then mathematics is nothing but an extension of logic really and is to
be regarded as part of Logic in the Larger sense. So for example if
one is trying to say interpret all mathematics in ZF and suppose that
was successful and ZF was proved consistent, then to me this suits
logicism, perhaps not in Russell/Frege sense, but it does go with the
spirit of logicism in the larger sense. But anyhow I'm myself growing
suspicious of my own definition since it might be the case that other
fields can also be interpreted faithfully in FOL for example, not only
that there seems to be some mathematics in the base logic itself, even
FOL, and propositional logic do use the natural numbers apparatus for
dealing with complex situations within them, on the other hand I think
perhaps mathematics is much smaller than what I think, possibly fairly
simple systems extending FOL are enough to interpret faithfully all of
math. but "simple" needs to be characterized. Anyhow.
Zuhair
Frederick Williams
Aug 8, 2012, 7:07:49 AM8/8/12
to
MoeBlee wrote:
>
> On Aug 7, 5:45 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> > MoeBlee <modem...@gmail.com> writes:
> > > The notion of a logical axiom is well enough understood and is not
> > > arbitrary. One might present certain objections with the notion, but
> > > it is not a merely arbitrary classification.
> >
> > What counts as logical and what not, in the sence of logicism, is not
> > at all well understoo.
>
> In, for example, classical first order logic, it's well enough
> understood what a logical axiom is.
Indeed it is, but there is nothing necessary about it, it is just a
>
> On Aug 7, 5:45 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> > MoeBlee <modem...@gmail.com> writes:
> > > The notion of a logical axiom is well enough understood and is not
> > > arbitrary. One might present certain objections with the notion, but
> > > it is not a merely arbitrary classification.
> >
> > What counts as logical and what not, in the sence of logicism, is not
> > at all well understoo.
>
> In, for example, classical first order logic, it's well enough
> understood what a logical axiom is.
convention that says v is a constant and P a variable.
Frederick Williams
Aug 8, 2012, 7:16:18 AM8/8/12
to
Bill Taylor wrote:
>
> I know there are said to be people who still call themselves
> "logicists" today, though I forget who they are. Would they
> have been regarded as the genuine article by Frege & Russell?
Quine might have been the last of them, or Rosser.
>
> I know there are said to be people who still call themselves
> "logicists" today, though I forget who they are. Would they
> have been regarded as the genuine article by Frege & Russell?
Frederick Williams
Aug 8, 2012, 7:23:36 AM8/8/12
to
Nam Nguyen wrote:
>
> On 07/08/2012 12:42 PM, MoeBlee wrote:
> > On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
> > wrote:
> >
> >> What criteria determine whether an axiom is "logical" or "non-logical"?
> >
> > Semantically, it is logical iff it is true in all structures for the
> > language.
>
> Incorrect.
>
> Axioms are purely syntactical hence the criteria of determining what
> is a logical or non-logical axiom is also syntactical.
>
> "A logical axiom is a formula which is a propositional axiom,
> a substitution axiom, an identity axiom, or an equality axiom"
>
> (Shoenfield, Mathematical Logic, pg. 21).
That's a daft response to Moe Blee's point. According to that, things
>
> On 07/08/2012 12:42 PM, MoeBlee wrote:
> > On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
> > wrote:
> >
> >> What criteria determine whether an axiom is "logical" or "non-logical"?
> >
> > Semantically, it is logical iff it is true in all structures for the
> > language.
>
> Incorrect.
>
> Axioms are purely syntactical hence the criteria of determining what
> is a logical or non-logical axiom is also syntactical.
>
> "A logical axiom is a formula which is a propositional axiom,
> a substitution axiom, an identity axiom, or an equality axiom"
>
> (Shoenfield, Mathematical Logic, pg. 21).
called 'axioms' in books other than Shoenfield's--if they differ--will
not be axioms, even though the authors of those books will call them
such. The question is this: by what criteria do Shoenfield and other
authors choose their axioms? Among the criteria will be that the axioms
are true: true in all structures for the logical axioms and true in all
intended structures for the non-logical ones.
Frederick Williams
Aug 8, 2012, 7:25:20 AM8/8/12
to
LudovicoVan wrote:
>
> "Michael Stemper" <mste...@walkabout.empros.com> wrote in message
> news:jvrj20$rvp$1...@dont-email.me...
>
> > What criteria determine whether an axiom is "logical" or "non-logical"?
>
> Logic deals with validity, i.e.*logical*
> necessity/possibility/self-contradiction, as resulting from the form itself
> of statements, regardless of factual matters.
Indeed so, but deciding what "form" is is problematic.
>
> "Michael Stemper" <mste...@walkabout.empros.com> wrote in message
> news:jvrj20$rvp$1...@dont-email.me...
>
> > What criteria determine whether an axiom is "logical" or "non-logical"?
>
> Logic deals with validity, i.e.*logical*
> necessity/possibility/self-contradiction, as resulting from the form itself
> of statements, regardless of factual matters.
Alan Smaill
Aug 8, 2012, 7:37:56 AM8/8/12
to
Bill Taylor <wfc.t...@gmail.com> writes:
> On Aug 8, 12:23 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
>> But Frege and Russell didn't think of logical truths in terms of
>> structures or interpretations at all. A different sort of explanation
>> is needed, if we are to understand logicism (as it once was).
>
> The original thread here seems to invite this last proviso:
> "as it once was". And indeed, wherever I have read anything
> much about Logicism, it seems to come with an at least implicit
> such proviso, as if it were largely a thing of the past.
>
> Is this so - can such a view be reasonably maintained?
> Is logicism "a thing of the past"? Or has it, perhaps in
> some modified or diluted sense, "taken over" the field
> of math foundations, at least to the extent that it has
> forced it to become very formal.
>
> I know there are said to be people who still call themselves
> "logicists" today, though I forget who they are. Would they
> have been regarded as the genuine article by Frege & Russell?
There are subscribers to "neologicism":
> On Aug 8, 12:23 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
>> But Frege and Russell didn't think of logical truths in terms of
>> structures or interpretations at all. A different sort of explanation
>> is needed, if we are to understand logicism (as it once was).
>
> The original thread here seems to invite this last proviso:
> "as it once was". And indeed, wherever I have read anything
> much about Logicism, it seems to come with an at least implicit
> such proviso, as if it were largely a thing of the past.
>
> Is this so - can such a view be reasonably maintained?
> Is logicism "a thing of the past"? Or has it, perhaps in
> some modified or diluted sense, "taken over" the field
> of math foundations, at least to the extent that it has
> forced it to become very formal.
>
> I know there are said to be people who still call themselves
> "logicists" today, though I forget who they are. Would they
> have been regarded as the genuine article by Frege & Russell?
http://mally.stanford.edu/Papers/neologicism2.pdf
>
> -- Wondering Willy
>
> ** "Let us not pretend to doubt in philosophy
> ** what we do not doubt in our hearts." - Pierce
Alan Smaill
Aatu Koskensilta
Aug 8, 2012, 8:38:43 AM8/8/12
to
Frederick Williams <freddyw...@btinternet.com> writes:
> Bill Taylor wrote:
>
>> I know there are said to be people who still call themselves
>> "logicists" today, though I forget who they are. Would they have
>> been regarded as the genuine article by Frege & Russell?
>
> Quine might have been the last of them, or Rosser.
Might have, but wasn't. Recall his radical and thoroughgoing holism,
> Bill Taylor wrote:
>
>> I know there are said to be people who still call themselves
>> "logicists" today, though I forget who they are. Would they have
>> been regarded as the genuine article by Frege & Russell?
>
> Quine might have been the last of them, or Rosser.
extending all the way to the very laws of logic, which we might one day
decide to ditch should better ones come along. Randall Holmes, I think,
came out years ago as a logicist (or not, depending on points of
semantics) on FOM.
Nam Nguyen
Aug 8, 2012, 9:34:15 AM8/8/12
to
On 08/08/2012 5:23 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 07/08/2012 12:42 PM, MoeBlee wrote:
>>> On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
>>> wrote:
>>>
>>>> What criteria determine whether an axiom is "logical" or "non-logical"?
>>>
>>> Semantically, it is logical iff it is true in all structures for the
>>> language.
>>
>> Incorrect.
>>
>> Axioms are purely syntactical hence the criteria of determining what
>> is a logical or non-logical axiom is also syntactical.
>>
>> "A logical axiom is a formula which is a propositional axiom,
>> a substitution axiom, an identity axiom, or an equality axiom"
>>
>> (Shoenfield, Mathematical Logic, pg. 21).
>
> That's a daft response to Moe Blee's point. According to that, things
> called 'axioms' in books other than Shoenfield's--if they differ--will
> not be axioms, even though the authors of those books will call them
> such.
Shoenfield didn't invent axioms I'm almost certain; and if there are
> Nam Nguyen wrote:
>>
>> On 07/08/2012 12:42 PM, MoeBlee wrote:
>>> On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
>>> wrote:
>>>
>>>> What criteria determine whether an axiom is "logical" or "non-logical"?
>>>
>>> Semantically, it is logical iff it is true in all structures for the
>>> language.
>>
>> Incorrect.
>>
>> Axioms are purely syntactical hence the criteria of determining what
>> is a logical or non-logical axiom is also syntactical.
>>
>> "A logical axiom is a formula which is a propositional axiom,
>> a substitution axiom, an identity axiom, or an equality axiom"
>>
>> (Shoenfield, Mathematical Logic, pg. 21).
>
> That's a daft response to Moe Blee's point. According to that, things
> called 'axioms' in books other than Shoenfield's--if they differ--will
> not be axioms, even though the authors of those books will call them
> such.
books that are wrong or "glossing over" on some technical issues I
wouldn't be too surprised.
> The question is this: by what criteria do Shoenfield and other
> authors choose their axioms?
or _inconsistent_ .
> Among the criteria will be that the axioms
> are true: true in all structures for the logical axioms and true in all
> intended structures for the non-logical ones.
_a logical axiom_ in PA + {~x=x} for example, the system having no
structure at all.
I'm not blind to the "criteria" as opposed to the definition.
But one shouldn't explain criteria _at the expense of definition_ .
As alluded to above, the criteria is _logical axioms_ are those
that _can be used for inference in ALL formal systems: consistent
or otherwise. Iow, logical axioms reflect _common patterns_ of
inferences.
Frederick Williams
Aug 8, 2012, 10:50:34 AM8/8/12
to
Nam Nguyen wrote:
>
> On 08/08/2012 5:23 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >> On 07/08/2012 12:42 PM, MoeBlee wrote:
> >>> On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
> >>> wrote:
> >>>
> >>>> What criteria determine whether an axiom is "logical" or "non-logical"?
> >>>
> >>> Semantically, it is logical iff it is true in all structures for the
> >>> language.
> >>
> >> Incorrect.
> >>
> >> Axioms are purely syntactical hence the criteria of determining what
> >> is a logical or non-logical axiom is also syntactical.
> >>
> >> "A logical axiom is a formula which is a propositional axiom,
> >> a substitution axiom, an identity axiom, or an equality axiom"
> >>
> >> (Shoenfield, Mathematical Logic, pg. 21).
> >
> > That's a daft response to Moe Blee's point. According to that, things
> > called 'axioms' in books other than Shoenfield's--if they differ--will
> > not be axioms, even though the authors of those books will call them
> > such.
>
> Shoenfield didn't invent axioms I'm almost certain;
Indeed. I think his are Sch\"utte's.
>
> On 08/08/2012 5:23 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >> On 07/08/2012 12:42 PM, MoeBlee wrote:
> >>> On Aug 7, 12:25 pm, mstem...@walkabout.empros.com (Michael Stemper)
> >>> wrote:
> >>>
> >>>> What criteria determine whether an axiom is "logical" or "non-logical"?
> >>>
> >>> Semantically, it is logical iff it is true in all structures for the
> >>> language.
> >>
> >> Incorrect.
> >>
> >> Axioms are purely syntactical hence the criteria of determining what
> >> is a logical or non-logical axiom is also syntactical.
> >>
> >> "A logical axiom is a formula which is a propositional axiom,
> >> a substitution axiom, an identity axiom, or an equality axiom"
> >>
> >> (Shoenfield, Mathematical Logic, pg. 21).
> >
> > That's a daft response to Moe Blee's point. According to that, things
> > called 'axioms' in books other than Shoenfield's--if they differ--will
> > not be axioms, even though the authors of those books will call them
> > such.
>
> Shoenfield didn't invent axioms I'm almost certain;
> and if there are
> books that are wrong or "glossing over" on some technical issues I
> wouldn't be too surprised.
>
> > The question is this: by what criteria do Shoenfield and other
> > authors choose their axioms?
>
> By inference-ability from them: in _all_ formal systems, consistent
> or _inconsistent_ .
choose inconsistent axioms and easily infer all one wishes. Axioms and
rules should be sound, i.e. axioms true of the structures in question,
and rules transmitting truth.
Marshall
Aug 8, 2012, 11:04:56 AM8/8/12
to
On Wednesday, August 8, 2012 12:37:50 AM UTC-7, Aatu Koskensilta wrote:
> William Elliot <ma...@panix.com> writes:
>
> > The logic of a mathematical definition of mathematics is ill.
>
> Your mother is ill.
+1
> William Elliot <ma...@panix.com> writes:
>
> > The logic of a mathematical definition of mathematics is ill.
>
> Your mother is ill.
Your mama is ill.
Yo mama is ill.
Yo mama be illin'.
Marshall
MoeBlee
Aug 8, 2012, 12:38:46 PM8/8/12
to
On Aug 7, 9:57 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> valid (non self-contradictory) statements
The word 'valid' gets different meanings (note, in particular,
Shoenfield uses 'valid' in a way different from other authors). But
usually, and in this context, 'valid' does not mean merely "non-self-
contradictory" but rather, it means "true in every structure for the
language".
Sentences are classified as either
valid (logically true) iff true in every structure
contingent iff true in some structures while false in others
logically false iff false in every structure.
In that terminology, an invalid sentence is one that is either
contingent or logcially false.
MoeBlee
> valid (non self-contradictory) statements
The word 'valid' gets different meanings (note, in particular,
Shoenfield uses 'valid' in a way different from other authors). But
usually, and in this context, 'valid' does not mean merely "non-self-
contradictory" but rather, it means "true in every structure for the
language".
Sentences are classified as either
valid (logically true) iff true in every structure
contingent iff true in some structures while false in others
logically false iff false in every structure.
In that terminology, an invalid sentence is one that is either
contingent or logcially false.
MoeBlee
Zuhair
Aug 8, 2012, 12:46:00 PM8/8/12
to
On Aug 8, 3:38 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
he doesn't have a particular philosophical stance on this issue, but
he is more inclined to Realism, and said if he is to take sides then
he'd prefer to be a realist or if not then possibly a logicist but
never a formalist. He generally opposes my definitions of mathematics
(though not vehemently) and say that the difference between math and
other fields doesn't lie in capability of being formalized (as I say)
but the difference is in the CONTENT and not the form. he thinks that
in principle many other fields of knowledge can be formalized and yet
that does't make them mathematical. When I asked him about what may be
his definition of mathematics? he gave the following answer:
"Mathematics is the study of necessary truths about formal
structures".
Zuhair
> Frederick Williams <freddywilli...@btinternet.com> writes:
> > Bill Taylor wrote:
>
> >> I know there are said to be people who still call themselves
> >> "logicists" today, though I forget who they are. Would they have
> >> been regarded as the genuine article by Frege & Russell?
>
> > Quine might have been the last of them, or Rosser.
>
> Might have, but wasn't. Recall his radical and thoroughgoing holism,
> extending all the way to the very laws of logic, which we might one day
> decide to ditch should better ones come along. Randall Holmes, I think,
> came out years ago as a logicist (or not, depending on points of
> semantics) on FOM.
>
> --
No Holmes is not, I personally asked him about this. He replied that
> > Bill Taylor wrote:
>
> >> I know there are said to be people who still call themselves
> >> "logicists" today, though I forget who they are. Would they have
> >> been regarded as the genuine article by Frege & Russell?
>
> > Quine might have been the last of them, or Rosser.
>
> Might have, but wasn't. Recall his radical and thoroughgoing holism,
> extending all the way to the very laws of logic, which we might one day
> decide to ditch should better ones come along. Randall Holmes, I think,
> came out years ago as a logicist (or not, depending on points of
> semantics) on FOM.
>
> --
he doesn't have a particular philosophical stance on this issue, but
he is more inclined to Realism, and said if he is to take sides then
he'd prefer to be a realist or if not then possibly a logicist but
never a formalist. He generally opposes my definitions of mathematics
(though not vehemently) and say that the difference between math and
other fields doesn't lie in capability of being formalized (as I say)
but the difference is in the CONTENT and not the form. he thinks that
in principle many other fields of knowledge can be formalized and yet
that does't make them mathematical. When I asked him about what may be
his definition of mathematics? he gave the following answer:
"Mathematics is the study of necessary truths about formal
structures".
Zuhair
MoeBlee
Aug 8, 2012, 12:49:01 PM8/8/12
to
On Aug 8, 5:26 am, Zuhair <zaljo...@gmail.com> wrote:
> since I think
> that mathematics can be interpreted in some of the elements of L*,
> then mathematics is nothing but an extension of logic really and is to
> be regarded as part of Logic in the Larger sense.
Whatever the merits of that view, it is not logicism.
> since I think
> that mathematics can be interpreted in some of the elements of L*,
> then mathematics is nothing but an extension of logic really and is to
> be regarded as part of Logic in the Larger sense.
> So for example if
> one is trying to say interpret all mathematics in ZF and suppose that
> was successful and ZF was proved consistent, then to me this suits
> logicism,
definition. Merly representing mathematics in a proven consistent ZF
definitely does not fulfill the logicist goal.
> perhaps not in Russell/Frege sense, but it does go with the
> spirit of logicism in the larger sense.
out the heart of the meaning of 'logicism'.
You seem to want to jam the notion of "logicism" where it doesn't fit.
You'd do better to find a different word for your philosophy.
MoeBlee
MoeBlee
Aug 8, 2012, 12:49:53 PM8/8/12
to
On Aug 8, 6:07 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:
MoeBlee
wrote:
> is nothing necessary about it, it is just a
> convention that says v is a constant and P a variable.
In previous posts I already addressed that point of yours.
> convention that says v is a constant and P a variable.
MoeBlee
MoeBlee
Aug 8, 2012, 12:43:25 PM8/8/12
to
On Aug 8, 2:59 am, Zuhair <zaljo...@gmail.com> wrote:
> I think what was occupying their minds is that mathematics is "priori-
> analytic", and that it can be derived from the basic three rules of
> thought (Identity, non contradiction, excluded middle) and they tried
> to reduce all of mathematics to those, I think as far as this view
> "extending" logic with some of what we call know as "non logical"
> symbols and rules that are "consistent" with a logical language that
> is sane, and complete and what is even more important "implicationally
> complete" this would actually render those true "extensions" of logic.
Logicism is definitely not the philosophy merely that mathematics can
> I think what was occupying their minds is that mathematics is "priori-
> analytic", and that it can be derived from the basic three rules of
> thought (Identity, non contradiction, excluded middle) and they tried
> to reduce all of mathematics to those, I think as far as this view
> "extending" logic with some of what we call know as "non logical"
> symbols and rules that are "consistent" with a logical language that
> is sane, and complete and what is even more important "implicationally
> complete" this would actually render those true "extensions" of logic.
be formalized as you described.
> Anyhow it is somehow arbitrary to call e or p (the mereological part-
> hood primitive) as non logical.
not fixed, as opposed to the context in which the semantics of '=' is
fixed. (Of course, one may prefer to take '=' without fixed semantics.
But in the context in which the semantics of '=' is fixed, it is taken
as a logical symbol).
MoeBlee
> Zuhair
MoeBlee
Aug 8, 2012, 12:33:43 PM8/8/12
to
On Aug 7, 8:17 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> Axioms are purely syntactical hence the criteria of determining what
> is a logical or non-logical axiom is also syntactical.
Of course, axioms, being formulas, are syntactical objects. However we
> Axioms are purely syntactical hence the criteria of determining what
> is a logical or non-logical axiom is also syntactical.
can still speak of semantical aspects. One of these aspects is whether
sentence (including axioms) is logically true (i.e. true in every
structure for the language).
MoeBlee
Michael Stemper
Aug 8, 2012, 1:15:46 PM8/8/12
to
In article <502169BC...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
The flood of posts that mine appears to have kicked off supports you there.
>This: P v ~P is a (candidate to be a) logical axiom because it is true
>in all interpretations. I.e. whether P is true or false, P v ~P comes
>out true.
Okay. If there's a difference between this and "tautology", I'm too
dim to pick up on it.
But, in a related vein, why would somebody create an *axiom* that
asserts something that's directly provable from definitions? Especially
since it has no meaning without those axioms?
(N.B. If the answer gets into model theory or something equally abstruse,
feel free to pat me on the head and suggest that I check back after I'm
in grad school.)
--
Michael F. Stemper
#include <Standard_Disclaimer>
Life's too important to take seriously.
>Michael Stemper wrote:
>> What criteria determine whether an axiom is "logical" or "non-logical"?
>
>Oh it's quite simple. Not.
>> What criteria determine whether an axiom is "logical" or "non-logical"?
>
The flood of posts that mine appears to have kicked off supports you there.
>This: P v ~P is a (candidate to be a) logical axiom because it is true
>in all interpretations. I.e. whether P is true or false, P v ~P comes
>out true.
Okay. If there's a difference between this and "tautology", I'm too
dim to pick up on it.
But, in a related vein, why would somebody create an *axiom* that
asserts something that's directly provable from definitions? Especially
since it has no meaning without those axioms?
(N.B. If the answer gets into model theory or something equally abstruse,
feel free to pat me on the head and suggest that I check back after I'm
in grad school.)
--
Michael F. Stemper
#include <Standard_Disclaimer>
Life's too important to take seriously.
Michael Stemper
Aug 8, 2012, 1:33:46 PM8/8/12
to
In article <jvu6s2$cke$1...@dont-email.me>, mste...@walkabout.empros.com (Michael Stemper) writes:
>In article <502169BC...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
>>Michael Stemper wrote:
>>> What criteria determine whether an axiom is "logical" or "non-logical"?
>In article <502169BC...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
>>Michael Stemper wrote:
>>> What criteria determine whether an axiom is "logical" or "non-logical"?
>>This: P v ~P is a (candidate to be a) logical axiom because it is true
>>in all interpretations. I.e. whether P is true or false, P v ~P comes
>>out true.
>
>Okay. If there's a difference between this and "tautology", I'm too
>dim to pick up on it.
>
>But, in a related vein, why would somebody create an *axiom* that
>asserts something that's directly provable from definitions? Especially
>since it has no meaning without those axioms?
Sorry, the last bit should have said "without those definitions".
>>in all interpretations. I.e. whether P is true or false, P v ~P comes
>>out true.
>
>Okay. If there's a difference between this and "tautology", I'm too
>dim to pick up on it.
>
>But, in a related vein, why would somebody create an *axiom* that
>asserts something that's directly provable from definitions? Especially
>since it has no meaning without those axioms?
MoeBlee
Aug 8, 2012, 1:34:23 PM8/8/12
to
On Aug 8, 6:23 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:
> by what criteria do Shoenfield and other
> authors choose their axioms?
The question of why particular human beings choose certain axioms is
one matter, but meanwhile, every formal sentence is an axiom in
infinitely many systems.
> Among the criteria will be that the axioms
> are true: true in all structures for the logical axioms and true in all
> intended structures for the non-logical ones.
If one wishes a (say, first order) system for proving things regarding
certain structures of interest, then:
(1) One chooses (or it's already agreed upon) axioms that are true in
every structure for the language (and complete in the sense that every
logically true sentence is provable by those axioms with also the
truth preserving rules). These axioms and rules provide what we call
the "logic" itself of the system.
(2) One chooses axioms that are not true in every structure for the
language but that are true in every one of the certain structures of
interest. These are what we call the "non-logical" (or sometimes, the
"mathematical") axioms.
MoeBlee
wrote:
> by what criteria do Shoenfield and other
> authors choose their axioms?
one matter, but meanwhile, every formal sentence is an axiom in
infinitely many systems.
> Among the criteria will be that the axioms
> are true: true in all structures for the logical axioms and true in all
> intended structures for the non-logical ones.
certain structures of interest, then:
(1) One chooses (or it's already agreed upon) axioms that are true in
every structure for the language (and complete in the sense that every
logically true sentence is provable by those axioms with also the
truth preserving rules). These axioms and rules provide what we call
the "logic" itself of the system.
(2) One chooses axioms that are not true in every structure for the
language but that are true in every one of the certain structures of
interest. These are what we call the "non-logical" (or sometimes, the
"mathematical") axioms.
MoeBlee
MoeBlee
Aug 8, 2012, 1:37:53 PM8/8/12
to
On Aug 8, 6:25 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:
undecidable.
MoeBlee
wrote:
> LudovicoVan wrote:
> > Logic deals with validity, i.e.*logical*
> > necessity/possibility/self-contradiction, as resulting from the form itself
> > of statements, regardless of factual matters.
>
> Indeed so, but deciding what "form" is is problematic.
By Church's theorem we know that the set of valid formulas is
> > Logic deals with validity, i.e.*logical*
> > necessity/possibility/self-contradiction, as resulting from the form itself
> > of statements, regardless of factual matters.
>
> Indeed so, but deciding what "form" is is problematic.
undecidable.
MoeBlee
MoeBlee
Aug 8, 2012, 1:46:19 PM8/8/12
to
On Aug 8, 8:34 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> > Among the criteria will be that the axioms
> > are true: true in all structures for the logical axioms and true in all
> > intended structures for the non-logical ones.
>
> That's why I said it is incorrect: it ignores that x=x _is still_
> _a logical axiom_ in PA + {~x=x} for example, the system having no
> structure at all.
PA is usually understood to include identity theory, which usually is
> > Among the criteria will be that the axioms
> > are true: true in all structures for the logical axioms and true in all
> > intended structures for the non-logical ones.
>
> That's why I said it is incorrect: it ignores that x=x _is still_
> _a logical axiom_ in PA + {~x=x} for example, the system having no
> structure at all.
axiomatized so that Ax x=x is an axiom. The fact that PA+"~x-x" is
inconsistent so it has no model, does not detract from the fact that
Ax x=x is a logical axiom for PA.
Again, a sentence S is a logical sentence iff S is true in every
structure for the language. Given that we're using the fixed semantics
for '=', Ax x=x is true in every structure for the language of PA
(which follows a fortiori since Ax s=x is true in any structure for
any language). So Ax x=x is a logical axiom.
MoeBlee
Zuhair
Aug 8, 2012, 1:52:19 PM8/8/12
to
I agree with all of what you wrote here actually, you gave correct
information.
According to Holmes if those axioms are describing "necessary truths"
about a "formal structure" then they are mathematical, but is it the
case that any necessary truth about a formal structure must be
mathematical, what if that formal structure was not about mathematics?
or is it the case that "every" formal structure is the subject of
mathematics?
Zuhair
Zuhair
Aug 8, 2012, 2:13:54 PM8/8/12
to
structure of the language).
That's why I said that logic deals with tautologies. Anyhow logic can
also be understood
in a wider sense (My version, I know it doesn't possible conform with
what is known)
When I said "logic" in my definition I meant what you are saying but
of course I add to it other attributes like being sane, complete and
implicationaly complete (at least for theories with countably many
axioms) in order for it to found mathematics. But "logicism" as I
perceive it (call it Z-Logicism) is about being "compatible with logic
(in the first sense)".
I think (I'm not sure) that Russell and Frege where not using "logic"
in this technical sense as you are referring to (i.e. validity or
being true in every structure of the language), I think they where
willing to accept some extra-logical symbols like membership among the
logical "in their sense" symbols. Anyhow yet I think they demanded
very simple axiomatization about it that they couldn't arrive at.
Anyhow I agree that I was wrong about my claim that my definition fits
"logicism" in the usual sense, you are right it doesn't.
I'm trying to figure out Holmes's definition of mathematics, I'm
coming to think that my definition is too wide as to make a lot of
what is not of mathematical interest be called mathematical, which is
not correct.
Zuhair
MoeBlee
Aug 8, 2012, 2:57:33 PM8/8/12
to
On Aug 8, 1:13 pm, Zuhair <zaljo...@gmail.com> wrote:
> S is a logical sentence iff S is a tautology (always true in every
> structure of the language).
Where I'm using "logical" in the sense of "logically true". Probably,
> S is a logical sentence iff S is a tautology (always true in every
> structure of the language).
to avoid any chance of misunderstanding I should say "logically
true".
And where 'tautology' is in the sense you just mentioned. Some authors
use 'tautology' in the sense of a 'valdity' or 'logically true', while
some authors reserve 'tautology' for 'logically true by evaluation of
the sentence connectives' (i.e. logically true by "truth table
evaluation").
> But "logicism" as I
> perceive it (call it Z-Logicism) is about being "compatible with logic
> (in the first sense)".
> I think (I'm not sure) that Russell and Frege where not using "logic"
> in this technical sense as you are referring to (i.e. validity or
> being true in every structure of the language),
far from being a kind of modern day counterpart to the older 19th and
early 20th century notions.
> I think they where
> willing to accept some extra-logical symbols like membership among the
> logical "in their sense" symbols.
to (or is related closely with) assertion as to a property. So where
we would say "Fx" to mean "x has property F" (which is logical (though
not necessarily logically true) in the sense that it involves only
predication) they would say x e {x | x has property F} = {x | Fx}, or
x is in (or is a member of) the "extension" of the property. So a
class is the extension of a property. [But I'm giving this in rough
outline; maybe this needs to revision to be more precisely correct.]
> Anyhow yet I think they demanded
> very simple axiomatization about it that they couldn't arrive at.
axioms that, at least to me, do not seem logical (though I'm less sure
of the technicalities of Reducibility): Infinity, Choice,
Reducibility.
> Anyhow I agree that I was wrong about my claim that my definition fits
> "logicism" in the usual sense
originality about various subjects you discuss, and I'm especially
encouraged when also you're willing to retract a mistake.
MoeBlee
Frederick Williams
Aug 8, 2012, 3:11:58 PM8/8/12
to
Michael Stemper wrote:
>
> In article <502169BC...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
> >Michael Stemper wrote:
>
> >> What criteria determine whether an axiom is "logical" or "non-logical"?
> >
> >Oh it's quite simple. Not.
>
> The flood of posts that mine appears to have kicked off supports you there.
>
> >This: P v ~P is a (candidate to be a) logical axiom because it is true
> >in all interpretations. I.e. whether P is true or false, P v ~P comes
> >out true.
>
> Okay. If there's a difference between this and "tautology", I'm too
> dim to pick up on it.
Yes, P v ~P is a tautology, but there are logical axioms which aren't.
>
> In article <502169BC...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
> >Michael Stemper wrote:
>
> >> What criteria determine whether an axiom is "logical" or "non-logical"?
> >
> >Oh it's quite simple. Not.
>
> The flood of posts that mine appears to have kicked off supports you there.
>
> >This: P v ~P is a (candidate to be a) logical axiom because it is true
> >in all interpretations. I.e. whether P is true or false, P v ~P comes
> >out true.
>
> Okay. If there's a difference between this and "tautology", I'm too
> dim to pick up on it.
For example, axioms for the quantifiers. (Though some people use the
word 'tautology' in a broader sense that includes all logical truths.)
> But, in a related vein, why would somebody create an *axiom* that
> asserts something that's directly provable from definitions? Especially
> since it has no meaning without those axioms?
then each row of the truth tables is an interpretation.
Frederick Williams
Aug 8, 2012, 3:13:26 PM8/8/12
to
not clear.
LudovicoVan
Aug 8, 2012, 3:23:19 PM8/8/12
to
"MoeBlee" <mode...@gmail.com> wrote in message
news:f9af9f97-d86c-49aa...@b10g2000vbj.googlegroups.com...
That is totally beside the point. Logic is *precisely* the study of the
*form* of statements (and we are in the *informal* here, not yet in the
formal) and their "logical validity", and its starting point is the notion
of *self-contradiction*, itself defined on the underlying proto-notion of
mutual incompatibility of predicates (this can be depicted with Ven-like
diagrams). -- Again, all this is informal, it *defines* logic, and is
preliminary to any formalization. Of course, there is no possible *sense*,
of logic or of anything else, to those who deny that there must be an
informal to found any formal.
P. F. Strawson talks about this in his great Introduction to Logical
heory. -- Is it only by chance that he belongs to the analytic school?
-LV
news:f9af9f97-d86c-49aa...@b10g2000vbj.googlegroups.com...
*form* of statements (and we are in the *informal* here, not yet in the
formal) and their "logical validity", and its starting point is the notion
of *self-contradiction*, itself defined on the underlying proto-notion of
mutual incompatibility of predicates (this can be depicted with Ven-like
diagrams). -- Again, all this is informal, it *defines* logic, and is
preliminary to any formalization. Of course, there is no possible *sense*,
of logic or of anything else, to those who deny that there must be an
informal to found any formal.
P. F. Strawson talks about this in his great Introduction to Logical
heory. -- Is it only by chance that he belongs to the analytic school?
-LV
MoeBlee
Aug 8, 2012, 4:02:17 PM8/8/12
to
On Aug 8, 12:52 pm, Zuhair <zaljo...@gmail.com> wrote:
> the problem is what makes the "non-logical" axioms mathematical?
My own personal feeling is not to worry about such a question, but
> the problem is what makes the "non-logical" axioms mathematical?
rather to accept that mathematics includes (though is not necessarily
confined to) any formal axiomatization.
If we're doing the semantics formally in set theory, then we never
talk about things like elephants and apples anyway.
Meanwhile, at least in first order, the distinction between "logical
axioms" and "non-logical" (or "mathematical") axioms is cleary defined
even though, by Church's theorem, it's not decidable.
> According to Holmes if those axioms are describing "necessary truths"
> about a "formal structure" then they are mathematical,
formal structure" but I don't doubt it is interesting. Meanwhile, I do
understand that some sentences are true in all structures (these are
the logically true sentences) and some sentences are true only in some
but not all structures (these are contingent sentences).
> what if that formal structure was not about mathematics?
there's nothing like elephants and apples mentioned.
Meanwhile, you might want to consider the philsopophy of If-Then-ism
(I think also called 'consequentialism'?) I think it is related to
some of the ideas you have.
MoeBlee
Frederick Williams
Aug 8, 2012, 4:16:10 PM8/8/12
to
MoeBlee wrote:
>
> Meanwhile, at least in first order, the distinction between "logical
> axioms" and "non-logical" (or "mathematical") axioms is cleary defined
> even though, by Church's theorem, it's not decidable.
Do you include, say, x=y -> y=x among the logical axioms?
>
> Meanwhile, at least in first order, the distinction between "logical
> axioms" and "non-logical" (or "mathematical") axioms is cleary defined
> even though, by Church's theorem, it's not decidable.
Frederick Williams
Aug 8, 2012, 4:26:04 PM8/8/12
to
MoeBlee wrote:
>
> Meanwhile, at least in first order, the distinction between "logical
> axioms" and "non-logical" (or "mathematical") axioms is cleary defined
Does that mean anything other than 'each author has clearly defined
>
> Meanwhile, at least in first order, the distinction between "logical
> axioms" and "non-logical" (or "mathematical") axioms is cleary defined
it'? Note that for many authors = is a logical symbol and thus enters
into the axioms, but for some--Abraham Robins comes to mind--it is an
extra-logical symbol.
> even though, by Church's theorem, it's not decidable.
Zuhair
Aug 8, 2012, 4:38:48 PM8/8/12
to
Thanks Moe for all of your helpful information here.
Zuhair
MoeBlee
Aug 8, 2012, 5:13:54 PM8/8/12
to
On Aug 8, 2:13 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
> All I meant was, the meaning of the word "form" as in "logical form" is
> not clear.
In a certain sense, we could say "syntax" instead.
MoeBlee
wrote:
> All I meant was, the meaning of the word "form" as in "logical form" is
> not clear.
MoeBlee
MoeBlee
Aug 8, 2012, 5:15:11 PM8/8/12
to
On Aug 8, 2:23 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> >> Indeed so, but deciding what "form" is is problematic.
>
> > By Church's theorem we know that the set of valid formulas is
> > undecidable.
>
> That is totally beside the point.
It was beside the point, since I misread what Mr. Williams had
> >> Indeed so, but deciding what "form" is is problematic.
>
> > By Church's theorem we know that the set of valid formulas is
> > undecidable.
>
> That is totally beside the point.
written.
MoeBlee
MoeBlee
LudovicoVan
Aug 8, 2012, 5:22:08 PM8/8/12
to
"MoeBlee" <mode...@gmail.com> wrote in message
news:cde5d11c-e252-4bd5...@i11g2000yqf.googlegroups.com...
Frederick was replying to me, the whole thing snipped by you.
Unnecessarily,
-LV
LudovicoVan
Aug 8, 2012, 5:22:31 PM8/8/12
to
"MoeBlee" <mode...@gmail.com> wrote in message
news:e81e7019-fdb5-4cad...@z11g2000yqa.googlegroups.com...
-LV
MoeBlee
Aug 8, 2012, 5:54:21 PM8/8/12
to
On Aug 8, 3:26 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
by (or equivalent with) a number of authors.
> Note that for many authors = is a logical symbol and thus enters
> into the axioms, but for some--Abraham Robins comes to mind--it is an
> extra-logical symbol.
I've already addressed this in previous posts.
The definition is:
x is a logically true sentence <-> x is true in every structure
Now, if '=' is given the fixed standard semantics for '=', then "true
in the structure" is evaluated per that semantics, and such sentences
as Ax x=x are true in every structure.
But, if '=' is not given the fixed standard semantics for '=' but
instead is treated just like any 2-place predicate symbol so that for
a given structure, '=' might or might not stand for the equality
relation on the structure, then it Ax x=x is NOT true in every
structure.
But that doesn't change that the definition:
x is a logical sentence <-> x is true in every structure
is clear and determinate, as long as we agree on whether '=' is or is
not to be given the fixed standard semantics.
It's just that "true in the structure" itself has a different meaning
depending on whether '=' is used or is not used with the fixed
standard semantics.
> > even though, by Church's theorem, it's not decidable.
>
> Could you expand on that please?
Church's theorem (not to be confused with Church's thesis) is (stating
it as precisely as I can think in the moment) that the set of Godel
numbers of the valid (loigically true) formulas of a language is not
recursive. That is, the characteristic function for the set of Godel
numbers of valid formulas is not a recursive function.
And Church's theorem is a fairly easy corollary of the incompleteness
theorem. (What I don't understand, and perhaps Aatu or someone can
help, is why Church's theorem deserves even to be credited or named
for Church and is dated at 1936, since the proof is so obvious and
could be seen easily by anyone who understood the incompleteness
theorem as early as 1931.)
MoeBlee
wrote:
> MoeBlee wrote:
>
> > Meanwhile, at least in first order, the distinction between "logical
> > axioms" and "non-logical" (or "mathematical") axioms is cleary defined
>
> Does that mean anything other than 'each author has clearly defined
> it'?
I mean that the definition I gave is clear and it's a definition used
>
> > Meanwhile, at least in first order, the distinction between "logical
> > axioms" and "non-logical" (or "mathematical") axioms is cleary defined
>
> Does that mean anything other than 'each author has clearly defined
> it'?
by (or equivalent with) a number of authors.
> Note that for many authors = is a logical symbol and thus enters
> into the axioms, but for some--Abraham Robins comes to mind--it is an
> extra-logical symbol.
The definition is:
x is a logically true sentence <-> x is true in every structure
Now, if '=' is given the fixed standard semantics for '=', then "true
in the structure" is evaluated per that semantics, and such sentences
as Ax x=x are true in every structure.
But, if '=' is not given the fixed standard semantics for '=' but
instead is treated just like any 2-place predicate symbol so that for
a given structure, '=' might or might not stand for the equality
relation on the structure, then it Ax x=x is NOT true in every
structure.
But that doesn't change that the definition:
x is a logical sentence <-> x is true in every structure
is clear and determinate, as long as we agree on whether '=' is or is
not to be given the fixed standard semantics.
It's just that "true in the structure" itself has a different meaning
depending on whether '=' is used or is not used with the fixed
standard semantics.
> > even though, by Church's theorem, it's not decidable.
>
> Could you expand on that please?
it as precisely as I can think in the moment) that the set of Godel
numbers of the valid (loigically true) formulas of a language is not
recursive. That is, the characteristic function for the set of Godel
numbers of valid formulas is not a recursive function.
And Church's theorem is a fairly easy corollary of the incompleteness
theorem. (What I don't understand, and perhaps Aatu or someone can
help, is why Church's theorem deserves even to be credited or named
for Church and is dated at 1936, since the proof is so obvious and
could be seen easily by anyone who understood the incompleteness
theorem as early as 1931.)
MoeBlee
MoeBlee
Aug 8, 2012, 6:01:47 PM8/8/12
to
On Aug 8, 4:22 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "MoeBlee" <modem...@gmail.com> wrote in message
I don't believe so.
MoeBlee
> "MoeBlee" <modem...@gmail.com> wrote in message
MoeBlee
MoeBlee
Aug 8, 2012, 6:01:01 PM8/8/12
to
On Aug 8, 4:22 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "MoeBlee" <modem...@gmail.com> wrote in message
you. However, in the usual sense in which people speak of form as in
"the form of statements", especially with regard to formal languages,
we may refer to syntax.
MoeBlee
LudovicoVan
Aug 8, 2012, 6:11:14 PM8/8/12
to
"MoeBlee" <mode...@gmail.com> wrote in message
news:d4dc29ae-f6b9-4aef...@h11g2000yqk.googlegroups.com...
> On Aug 8, 4:22 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "MoeBlee" <modem...@gmail.com> wrote in message
>> news:cde5d11c-e252-4bd5...@i11g2000yqf.googlegroups.com...
>> > On Aug 8, 2:13 pm, Frederick Williams <freddywilli...@btinternet.com>
>> > wrote:
>>
>> >> All I meant was, the meaning of the word "form" as in "logical form"
>> >> is
>> >> not clear.
>>
>> > In a certain sense, we could say "syntax" instead.
>>
>> Indeed, in the opposite sense in which I meant it.
>>
>> Frederick was replying to me, the whole thing snipped by you.
>
> I surely do not mean to speak for you nor to steal the spotlight from
> you.
Suck my socks. You have just missed the point and that's all.
>> "MoeBlee" <modem...@gmail.com> wrote in message
>> news:cde5d11c-e252-4bd5...@i11g2000yqf.googlegroups.com...
>> > On Aug 8, 2:13 pm, Frederick Williams <freddywilli...@btinternet.com>
>> > wrote:
>>
>> >> All I meant was, the meaning of the word "form" as in "logical form"
>> >> is
>> >> not clear.
>>
>> > In a certain sense, we could say "syntax" instead.
>>
>> Indeed, in the opposite sense in which I meant it.
>>
>> Frederick was replying to me, the whole thing snipped by you.
>
> I surely do not mean to speak for you nor to steal the spotlight from
> you.
> However, in the usual sense in which people speak of form as in
> "the form of statements", especially with regard to formal languages,
> we may refer to syntax.
very limited and in may ways the less interesting part of logic and, more
generally, of systematics.
-LV
Nam Nguyen
Aug 8, 2012, 10:26:08 PM8/8/12
to
On 08/08/2012 10:33 AM, MoeBlee wrote:
> On Aug 7, 8:17 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
>> Axioms are purely syntactical hence the criteria of determining what
>> is a logical or non-logical axiom is also syntactical.
>
> Of course, axioms, being formulas, are syntactical objects. However we
> can still speak of semantical aspects.
Sure. I've never said otherwise: we can speak of a lot aspects of
> On Aug 7, 8:17 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
>> Axioms are purely syntactical hence the criteria of determining what
>> is a logical or non-logical axiom is also syntactical.
>
> Of course, axioms, being formulas, are syntactical objects. However we
> can still speak of semantical aspects.
formulas, semantic aspect, finiteness aspect, model-theoretical
truth aspect, ...
> One of these aspects is whether
> sentence (including axioms) is logically true (i.e. true in every
> structure for the language).
op's request is about logical, non-logical, _axioms_ .
Iow, there's a good reason why can claim an axiom is a formula
but _not_ the other way around: on meta level, the semantic of
axiom can't be separated from those of FOL formal system,
and provability, where inconsistency is abound.
--
----------------------------------------------------
There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI
----------------------------------------------------
Nam Nguyen
Aug 8, 2012, 11:10:47 PM8/8/12
to
On 08/08/2012 8:26 PM, Nam Nguyen wrote:
> On 08/08/2012 10:33 AM, MoeBlee wrote:
>> On Aug 7, 8:17 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>
>>> Axioms are purely syntactical hence the criteria of determining what
>>> is a logical or non-logical axiom is also syntactical.
>>
>> Of course, axioms, being formulas, are syntactical objects. However we
>> can still speak of semantical aspects.
>
> Sure. I've never said otherwise: we can speak of a lot aspects of
> formulas, semantic aspect, finiteness aspect, model-theoretical
> truth aspect, ...
>
>> One of these aspects is whether
>> sentence (including axioms) is logically true (i.e. true in every
>> structure for the language).
>
> Then you're talking _only_ about formulas, _not_ axiom. And the
> op's request is about logical, non-logical, _axioms_ .
And here, in your model-theoretical criteria, you can't specifically
> On 08/08/2012 10:33 AM, MoeBlee wrote:
>> On Aug 7, 8:17 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>
>>> Axioms are purely syntactical hence the criteria of determining what
>>> is a logical or non-logical axiom is also syntactical.
>>
>> Of course, axioms, being formulas, are syntactical objects. However we
>> can still speak of semantical aspects.
>
> Sure. I've never said otherwise: we can speak of a lot aspects of
> formulas, semantic aspect, finiteness aspect, model-theoretical
> truth aspect, ...
>
>> One of these aspects is whether
>> sentence (including axioms) is logically true (i.e. true in every
>> structure for the language).
>
> Then you're talking _only_ about formulas, _not_ axiom. And the
> op's request is about logical, non-logical, _axioms_ .
stipulate which formulas, or how many of them, are "logical", since
nobody can fathom what "every" "structure for the language" might
mean: since we can not even know exactly what the purportedly most
well-known structure, i.e. the natural numbers, be.
Btw, the word "logic/logical" and the model-theoretically-centric
"interpretation" (as in "truth interpretation") don't really rhyme.
Which is why language model and formal system aren't of the same
definition. In fact one isn't even a derivative of the other.
Virgil
Aug 9, 2012, 12:42:10 AM8/9/12
to
In article <jvuecl$1fs$1...@dont-email.me>,
>
--
"LudovicoVan" <ju...@diegidio.name> wrote:
> "MoeBlee" <mode...@gmail.com> wrote in message
> news:f9af9f97-d86c-49aa...@b10g2000vbj.googlegroups.com...
> > On Aug 8, 6:25 am, Frederick Williams <freddywilli...@btinternet.com>
> > wrote:
> >> LudovicoVan wrote:
> >
> >> > Logic deals with validity, i.e.*logical*
> >> > necessity/possibility/self-contradiction, as resulting from the form
> >> > itself
> >> > of statements, regardless of factual matters.
> >>
> >> Indeed so, but deciding what "form" is is problematic.
> >
> > By Church's theorem we know that the set of valid formulas is
> > undecidable.
>
> That is totally beside the point.
Everything is "beside" whatever point LV claims to be making.
> "MoeBlee" <mode...@gmail.com> wrote in message
> news:f9af9f97-d86c-49aa...@b10g2000vbj.googlegroups.com...
> > On Aug 8, 6:25 am, Frederick Williams <freddywilli...@btinternet.com>
> > wrote:
> >> LudovicoVan wrote:
> >
> >> > Logic deals with validity, i.e.*logical*
> >> > necessity/possibility/self-contradiction, as resulting from the form
> >> > itself
> >> > of statements, regardless of factual matters.
> >>
> >> Indeed so, but deciding what "form" is is problematic.
> >
> > By Church's theorem we know that the set of valid formulas is
> > undecidable.
>
> That is totally beside the point.
>
--
Zuhair
Aug 9, 2012, 4:32:27 AM8/9/12
to
On Aug 8, 11:02 pm, MoeBlee <modem...@gmail.com> wrote:
> On Aug 8, 12:52 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> > the problem is what makes the "non-logical" axioms mathematical?
>
> My own personal feeling is not to worry about such a question, but
> rather to accept that mathematics includes (though is not necessarily
> confined to) any formal axiomatization.
>
> If we're doing the semantics formally in set theory, then we never
> talk about things like elephants and apples anyway.
No those terms can seep through as "defined" terms, ZF for example
>
> > the problem is what makes the "non-logical" axioms mathematical?
>
> My own personal feeling is not to worry about such a question, but
> rather to accept that mathematics includes (though is not necessarily
> confined to) any formal axiomatization.
>
> If we're doing the semantics formally in set theory, then we never
> talk about things like elephants and apples anyway.
which is generally regarded as a formal system in which mathematics is
faithfully interpreted (founded) is too strong that it can interpret
any possible physical theory that can be faithfully interpreted in a
logical system (if such should exist), you see physics is of the
strength of 2nd or at most 3rd order arithmetic (Holmes) so if you
imagine something like TOE (theory of everything) that is formal then
this would be interpretable in ZF and you can bring all the physical
terminology herein. So restriction of primitives in the basic language
doesn't make us avoid apples, elephants, etc...
Frederick Williams
Aug 9, 2012, 8:07:56 AM8/9/12
to
distinction between "logical axioms" and "non-logical" (or
"mathematical") axioms is clearly defined'?
> And Church's theorem is a fairly easy corollary of the incompleteness
> theorem. (What I don't understand, and perhaps Aatu or someone can
> help, is why Church's theorem deserves even to be credited or named
> for Church and is dated at 1936, since the proof is so obvious and
> could be seen easily by anyone who understood the incompleteness
> theorem as early as 1931.)
theorem applies to second order logic, or first order Peano arithmetic
(and similar).
Frederick Williams
Aug 9, 2012, 8:10:06 AM8/9/12
to
Zuhair wrote:
> No those terms can seep through as "defined" terms, ZF for example
> which is generally regarded as a formal system in which mathematics is
> faithfully interpreted (founded)
Really? Despite Skolem's "paradox"?
> No those terms can seep through as "defined" terms, ZF for example
> which is generally regarded as a formal system in which mathematics is
> faithfully interpreted (founded)
Zuhair
Aug 9, 2012, 8:25:22 AM8/9/12
to
On Aug 9, 3:10 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
Skolem's paradox is a shortcoming of first order logic, it pops up
with un-intended models of ZF, then intended model of ZF doesn't
suffer from this paradox, and there is not clear argument against its
existence. Anyhow Skolem's paradoxical state of affairs is not a
genuine paradox like Russell's and the like. Skolem's theorems renders
any first order logic (finitary) with infinite model as non
Categorical, something that infinitary first order logic L(w1,w)
remedies.
Zuhair
wrote:
with un-intended models of ZF, then intended model of ZF doesn't
suffer from this paradox, and there is not clear argument against its
existence. Anyhow Skolem's paradoxical state of affairs is not a
genuine paradox like Russell's and the like. Skolem's theorems renders
any first order logic (finitary) with infinite model as non
Categorical, something that infinitary first order logic L(w1,w)
remedies.
Zuhair
MoeBlee
Aug 9, 2012, 11:22:13 AM8/9/12
to
On Aug 8, 3:16 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
> Do you include, say, x=y -> y=x among the logical axioms?
Given the standard semantics for '=', we have that Axy(x=y -> y=x) is
logically true. So, if it is included in the axioms, then it is a
logical axioms.
MoeBlee
wrote:
> Do you include, say, x=y -> y=x among the logical axioms?
logically true. So, if it is included in the axioms, then it is a
logical axioms.
MoeBlee
MoeBlee
Aug 9, 2012, 11:23:46 AM8/9/12
to
On Aug 8, 5:11 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> Suck my socks.
Unless they're made out of butterscotch parfait, it's not going to
happen.
MoeBlee
> Suck my socks.
Unless they're made out of butterscotch parfait, it's not going to
happen.
MoeBlee
MoeBlee
Aug 9, 2012, 11:28:27 AM8/9/12
to
On Aug 8, 9:26 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 08/08/2012 10:33 AM, MoeBlee wrote:
> > One of these aspects is whether
> > sentence (including axioms) is logically true (i.e. true in every
> > structure for the language).
>
> Then you're talking _only_ about formulas, _not_ axiom. And the
> op's request is about logical, non-logical, _axioms_ .
A logically true sentence that is an axiom of system S is a logical
> On 08/08/2012 10:33 AM, MoeBlee wrote:
> > One of these aspects is whether
> > sentence (including axioms) is logically true (i.e. true in every
> > structure for the language).
>
> Then you're talking _only_ about formulas, _not_ axiom. And the
> op's request is about logical, non-logical, _axioms_ .
axiom of system S.
> Iow, there's a good reason why can claim an axiom is a formula
> but _not_ the other way around:
an axiom for infinitely many systems.
If a formula P is valid (satisfied by all assignments for the
variables in all interpretations) and P is an axiom of system S, then
P is a logical axiom of system S.
MoeBlee
MoeBlee
Aug 9, 2012, 11:47:30 AM8/9/12
to
On Aug 9, 3:32 am, Zuhair <zaljo...@gmail.com> wrote:
> On Aug 8, 11:02 pm, MoeBlee <modem...@gmail.com> wrote:
> > If we're doing the semantics formally in set theory, then we never
> > talk about things like elephants and apples anyway.
>
> No those terms can seep through as "defined" terms, ZF for example [...]
> On Aug 8, 11:02 pm, MoeBlee <modem...@gmail.com> wrote:
> > If we're doing the semantics formally in set theory, then we never
> > talk about things like elephants and apples anyway.
>
Fine, then they're defined entirely from the two abstract notions of
equality and membership. So talking about elephants will reduce to
some statement purely about equality and membership. That doesn't seem
really like talking about elephants to me ....
MoeBlee
MoeBlee
Aug 9, 2012, 11:43:39 AM8/9/12
to
On Aug 8, 10:10 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 08/08/2012 8:26 PM, Nam Nguyen wrote:
> > On 08/08/2012 10:33 AM, MoeBlee wrote:
> >> On Aug 7, 8:17 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
> >>> Axioms are purely syntactical hence the criteria of determining what
> >>> is a logical or non-logical axiom is also syntactical.
>
> >> Of course, axioms, being formulas, are syntactical objects. However we
> >> can still speak of semantical aspects.
>
> > Sure. I've never said otherwise: we can speak of a lot aspects of
> > formulas, semantic aspect, finiteness aspect, model-theoretical
> > truth aspect, ...
>
> >> One of these aspects is whether
> >> sentence (including axioms) is logically true (i.e. true in every
> >> structure for the language).
>
> > Then you're talking _only_ about formulas, _not_ axiom. And the
> > op's request is about logical, non-logical, _axioms_ .
>
> And here, in your model-theoretical criteria, you can't specifically
> stipulate which formulas, or how many of them, are "logical",
As I said, the property of being logically true is determinate but not
> On 08/08/2012 8:26 PM, Nam Nguyen wrote:
> > On 08/08/2012 10:33 AM, MoeBlee wrote:
> >> On Aug 7, 8:17 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
> >>> Axioms are purely syntactical hence the criteria of determining what
> >>> is a logical or non-logical axiom is also syntactical.
>
> >> Of course, axioms, being formulas, are syntactical objects. However we
> >> can still speak of semantical aspects.
>
> > Sure. I've never said otherwise: we can speak of a lot aspects of
> > formulas, semantic aspect, finiteness aspect, model-theoretical
> > truth aspect, ...
>
> >> One of these aspects is whether
> >> sentence (including axioms) is logically true (i.e. true in every
> >> structure for the language).
>
> > Then you're talking _only_ about formulas, _not_ axiom. And the
> > op's request is about logical, non-logical, _axioms_ .
>
> And here, in your model-theoretical criteria, you can't specifically
> stipulate which formulas, or how many of them, are "logical",
decidable. Usually, though, an author will specify the axioms of
system S and specify that certain of them are the logical axioms, then
prove (in the course of proving the soundness theorem) that each of
those specified as a logical axiom actually is a logical truth.
> Btw, the word "logic/logical" and the model-theoretically-centric
> "interpretation" (as in "truth interpretation") don't really rhyme.
the remark below?:
> Which is why language model and formal system aren't of the same
> definition. In fact one isn't even a derivative of the other.
word?
In any case, nothing I've said contradicts that the formal system and
models for the language of the system are not the same thing.
MoeBlee
MoeBlee
Aug 9, 2012, 11:54:25 AM8/9/12
to
On Aug 9, 7:07 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:
> What I meant was, what is the relevance of Church's theorem to your 'the
> distinction between "logical axioms" and "non-logical" (or
> "mathematical") axioms is clearly defined'?
Church's theorem does not impinge on the fact that the definition of
the set of logically true sentences is clear. I was merely qualifying
that even though the definition is clear, it does not provide for a
DECIDABLE set.
> > And Church's theorem is a fairly easy corollary of the incompleteness
> > theorem. (What I don't understand, and perhaps Aatu or someone can
> > help, is why Church's theorem deserves even to be credited or named
> > for Church and is dated at 1936, since the proof is so obvious and
> > could be seen easily by anyone who understood the incompleteness
> > theorem as early as 1931.)
>
> Church's theorem applies to first order logic. The incompleteness
> theorem applies to second order logic, or first order Peano arithmetic
> (and similar).
The incompleteness theorem applies to any system that is consistent,
recursively axiomatized, and provides for the certain amount of
arithmetic.
But Church's theorem (that validity in pure first order logic is not
decidable) can be obtained as corollary of the incompleteness theorem.
MoeBlee
wrote:
> What I meant was, what is the relevance of Church's theorem to your 'the
> distinction between "logical axioms" and "non-logical" (or
> "mathematical") axioms is clearly defined'?
the set of logically true sentences is clear. I was merely qualifying
that even though the definition is clear, it does not provide for a
DECIDABLE set.
> > And Church's theorem is a fairly easy corollary of the incompleteness
> > theorem. (What I don't understand, and perhaps Aatu or someone can
> > help, is why Church's theorem deserves even to be credited or named
> > for Church and is dated at 1936, since the proof is so obvious and
> > could be seen easily by anyone who understood the incompleteness
> > theorem as early as 1931.)
>
> Church's theorem applies to first order logic. The incompleteness
> theorem applies to second order logic, or first order Peano arithmetic
> (and similar).
recursively axiomatized, and provides for the certain amount of
arithmetic.
But Church's theorem (that validity in pure first order logic is not
decidable) can be obtained as corollary of the incompleteness theorem.
MoeBlee
Zuhair
Aug 9, 2012, 12:44:43 PM8/9/12
to
I do think (possibly I'd be wrong) that matters like biology,
sociology, history, arts , etc.. cannot be formalized "faithfully" by
a system extending logic. But on the other hand one can say in
principle that some disciplines other than mathematics like "physics",
"linguistics", "ethics" , "music", etc... may be formalized
"faithfully" in systems extending logic, I'm not saying they are, I'm
only saying they "might" be, and according to Holmes if a theory in
physics of everything "TOE" should be there, then TOE would be
interpretable in ZF, since TOE would not be stronger than 3rd order
arithmetic and so all kinds of non mathematical semantics like atom,
energy, electrons, etc.... would be used as "defined" entirely from
membership and identity, and here is the problem, we didn't really
eliminate those from our vocabulary. So even if we only use membership
and identity still the non logical axioms may not be solely
mathematical since stuff other than mathematics might be faithfully
interpreted in them, so I don't know if it is relevant to call them
"mathematical" axioms. I actually had this intent that of those non
logical axioms being mathematical even if other stuff can be
interpreted in them and I used to view such matters
as overlaps between mathematics and those fields, and by then calling
them "mathematical" axioms would be justified, some kind of view that
I'm a little bit doubting nowadays.
Zuhair
MoeBlee
Aug 9, 2012, 3:53:34 PM8/9/12
to
On Aug 9, 11:44 am, Zuhair <zaljo...@gmail.com> wrote:
> all kinds of non mathematical semantics like atom,
> energy, electrons, etc.... would be used as "defined" entirely from
> membership and identity,
I don't know how you'd do that, though I'm not informed about this
> all kinds of non mathematical semantics like atom,
> energy, electrons, etc.... would be used as "defined" entirely from
> membership and identity,
kind of thing. It is an interesting subject that I wish I knew more
about.
I do understand that using additional primitive predicates (such as
"electron" or whatever) we could provide first order axiomatizations
for various subject matter. But I didn't know it was proposed to do
this purely set theoretically, with no other primitives.
MoeBlee
LudovicoVan
Aug 9, 2012, 4:33:39 PM8/9/12
to
"Zuhair" <zalj...@gmail.com> wrote in message
news:a79dc6e3-4dd4-479c...@w14g2000vbx.googlegroups.com...
> according to Holmes if a theory in
> physics of everything "TOE" should be there, then TOE would be
> interpretable in ZF, since TOE would not be stronger than 3rd order
> arithmetic and so all kinds of non mathematical semantics like atom,
> energy, electrons, etc.... would be used as "defined" entirely from
> membership and identity, and here is the problem, we didn't really
> eliminate those from our vocabulary.
But, even if they were defined from membership and identity, they'd need
their own additional axioms, wouldn't they? A physical formal theory is
just not a mathematical formal theory, rather they both use (extend, in your
terms) logic.
-LV
news:a79dc6e3-4dd4-479c...@w14g2000vbx.googlegroups.com...
> according to Holmes if a theory in
> physics of everything "TOE" should be there, then TOE would be
> interpretable in ZF, since TOE would not be stronger than 3rd order
> arithmetic and so all kinds of non mathematical semantics like atom,
> energy, electrons, etc.... would be used as "defined" entirely from
> membership and identity, and here is the problem, we didn't really
> eliminate those from our vocabulary.
their own additional axioms, wouldn't they? A physical formal theory is
just not a mathematical formal theory, rather they both use (extend, in your
terms) logic.
-LV
LudovicoVan
Aug 9, 2012, 4:35:53 PM8/9/12
to
"LudovicoVan" <ju...@diegidio.name> wrote in message
news:k016rf$goo$1...@dont-email.me...
Should read: formal logic.
-LV
news:k016rf$goo$1...@dont-email.me...
-LV
Zuhair
Aug 9, 2012, 4:52:10 PM8/9/12
to
On Aug 9, 11:33 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message
No of course there is no need for any additional axioms.
Zuhair
> "Zuhair" <zaljo...@gmail.com> wrote in message
Zuhair
Pubkeybreaker
Aug 9, 2012, 6:42:07 PM8/9/12
to
On Aug 6, 7:12 am, Zuhair <zaljo...@gmail.com> wrote:
> Mathematics is all of what can be "faithfully" interpreted in a
> consistent formal system extending Logic.
>
> Emphasis is put on "faithfully" which is meant to copy the properties
> of interpreted concept, for example to say that for example "All
> apples are Juicy" and not say anything else about what constitute an
> apple and what Juicy means, is actually not different from saying "All
> bananas are sweet" without saying anything else about bananas and what
> sweet means, both are just statements composed of objects fulfilling a
> predicate, so both alone are not faithful to the intended
> interpretation of those statements. So although they are formalized
> (put in a formal language) yet the formalization is not faithful. We
> need not only put matters in formal symbols, we need enough
> formalization necessary to ensure that the intended interpretation is
> captured by the formal system and that the later is not speaking of a
> different concept.
>
> I actually think only mathematics can be interpreted faithfully in a
> consistent formal system extending logic, and it doesn't matter if
> that part of mathematics is proved to exist in reality and thus be a
> part of physics also (i.e. an overlap of mathematics and physics) or
> whether it doesn't, in either case it is mathematics!
>
> This is clearly a logicist definition of mathematics which I think it
> to be the nearest one to the truth of what mathematics is.
>
> I think "ordinary mathematics" is all of what can be interpreted in a
> consistent, categorical and effectively generated formal system
> extending logic.
>
> Zuhair
A constructive suggestion:
Join the discussion groups at www.mersenneforum.org
You may find a more favorable reception there.
> Mathematics is all of what can be "faithfully" interpreted in a
> consistent formal system extending Logic.
>
> Emphasis is put on "faithfully" which is meant to copy the properties
> of interpreted concept, for example to say that for example "All
> apples are Juicy" and not say anything else about what constitute an
> apple and what Juicy means, is actually not different from saying "All
> bananas are sweet" without saying anything else about bananas and what
> sweet means, both are just statements composed of objects fulfilling a
> predicate, so both alone are not faithful to the intended
> interpretation of those statements. So although they are formalized
> (put in a formal language) yet the formalization is not faithful. We
> need not only put matters in formal symbols, we need enough
> formalization necessary to ensure that the intended interpretation is
> captured by the formal system and that the later is not speaking of a
> different concept.
>
> I actually think only mathematics can be interpreted faithfully in a
> consistent formal system extending logic, and it doesn't matter if
> that part of mathematics is proved to exist in reality and thus be a
> part of physics also (i.e. an overlap of mathematics and physics) or
> whether it doesn't, in either case it is mathematics!
>
> This is clearly a logicist definition of mathematics which I think it
> to be the nearest one to the truth of what mathematics is.
>
> I think "ordinary mathematics" is all of what can be interpreted in a
> consistent, categorical and effectively generated formal system
> extending logic.
>
> Zuhair
A constructive suggestion:
Join the discussion groups at www.mersenneforum.org
You may find a more favorable reception there.
Aatu Koskensilta
Aug 15, 2012, 3:59:35 AM8/15/12
to
Zuhair <zalj...@gmail.com> writes:
> No Holmes is not, I personally asked him about this. He replied that
> he doesn't have a particular philosophical stance on this issue, but
> he is more inclined to Realism, and said if he is to take sides then
> he'd prefer to be a realist or if not then possibly a logicist but
> never a formalist.
Well, it was years ago:
http://www.cs.nyu.edu/pipermail/fom/1998-October/002248.html
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
> No Holmes is not, I personally asked him about this. He replied that
> he doesn't have a particular philosophical stance on this issue, but
> he is more inclined to Realism, and said if he is to take sides then
> he'd prefer to be a realist or if not then possibly a logicist but
> never a formalist.
Well, it was years ago:
http://www.cs.nyu.edu/pipermail/fom/1998-October/002248.html
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Jesse F. Hughes
Aug 17, 2012, 8:35:24 AM8/17/12
to
mste...@walkabout.empros.com (Michael Stemper) writes:
> In article <502169BC...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
>>Michael Stemper wrote:
>
>>> What criteria determine whether an axiom is "logical" or "non-logical"?
>>
>>Oh it's quite simple. Not.
>
> The flood of posts that mine appears to have kicked off supports you there.
>
>>This: P v ~P is a (candidate to be a) logical axiom because it is true
>>in all interpretations. I.e. whether P is true or false, P v ~P comes
>>out true.
>
> Okay. If there's a difference between this and "tautology", I'm too
> dim to pick up on it.
In many contexts, "tautology" refers to those formulas which are
universally true due to propositional structure. That is, if we take a
propositional formula, P v ~P, say, and substitute first order formulas
for the propositional variables, the result is a tautology. Thus,
(Ax)Px v ~(Ax)Px
is a tautology. But,
(Ax)Px v (Ex)~Px
is not a tautology, even though it is true under every interpretation.
It is not the result of a substitution of some propositional tautology.
--
"It seems to me that in wartime Americans shouldn't be attacking each
other in this way on a *worldwide* forum. Then again, I know I'm an
American, but I have no way of knowing that you are, which would
explain a lot." --James Harris, on why Yanks should accept his proof
> In article <502169BC...@btinternet.com>, Frederick Williams <freddyw...@btinternet.com> writes:
>>Michael Stemper wrote:
>
>>> What criteria determine whether an axiom is "logical" or "non-logical"?
>>
>>Oh it's quite simple. Not.
>
> The flood of posts that mine appears to have kicked off supports you there.
>
>>This: P v ~P is a (candidate to be a) logical axiom because it is true
>>in all interpretations. I.e. whether P is true or false, P v ~P comes
>>out true.
>
> Okay. If there's a difference between this and "tautology", I'm too
> dim to pick up on it.
In many contexts, "tautology" refers to those formulas which are
universally true due to propositional structure. That is, if we take a
propositional formula, P v ~P, say, and substitute first order formulas
for the propositional variables, the result is a tautology. Thus,
(Ax)Px v ~(Ax)Px
is a tautology. But,
(Ax)Px v (Ex)~Px
is not a tautology, even though it is true under every interpretation.
It is not the result of a substitution of some propositional tautology.
--
"It seems to me that in wartime Americans shouldn't be attacking each
other in this way on a *worldwide* forum. Then again, I know I'm an
American, but I have no way of knowing that you are, which would
explain a lot." --James Harris, on why Yanks should accept his proof
Zuhair
Aug 17, 2012, 9:17:47 AM8/17/12
to
On Aug 17, 3:35 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> mstem...@walkabout.empros.com (Michael Stemper) writes:
Zuhair
> mstem...@walkabout.empros.com (Michael Stemper) writes:
> > In article <502169BC.9BC50...@btinternet.com>, Frederick Williams <freddywilli...@btinternet.com> writes:
> >>Michael Stemper wrote:
>
> >>> What criteria determine whether an axiom is "logical" or "non-logical"?
>
> >>Oh it's quite simple. Not.
>
> > The flood of posts that mine appears to have kicked off supports you there.
>
> >>This: P v ~P is a (candidate to be a) logical axiom because it is true
> >>in all interpretations. I.e. whether P is true or false, P v ~P comes
> >>out true.
>
> > Okay. If there's a difference between this and "tautology", I'm too
> > dim to pick up on it.
>
> In many contexts, "tautology" refers to those formulas which are
> universally true due to propositional structure. That is, if we take a
> propositional formula, P v ~P, say, and substitute first order formulas
> for the propositional variables, the result is a tautology. Thus,
>
> (Ax)Px v ~(Ax)Px
>
> is a tautology. But,
>
> (Ax)Px v (Ex)~Px
>
> is not a tautology, even though it is true under every interpretation.
> It is not the result of a substitution of some propositional tautology.
>
> --
This *IS* a tautology!
> >>Michael Stemper wrote:
>
> >>> What criteria determine whether an axiom is "logical" or "non-logical"?
>
> >>Oh it's quite simple. Not.
>
> > The flood of posts that mine appears to have kicked off supports you there.
>
> >>This: P v ~P is a (candidate to be a) logical axiom because it is true
> >>in all interpretations. I.e. whether P is true or false, P v ~P comes
> >>out true.
>
> > Okay. If there's a difference between this and "tautology", I'm too
> > dim to pick up on it.
>
> In many contexts, "tautology" refers to those formulas which are
> universally true due to propositional structure. That is, if we take a
> propositional formula, P v ~P, say, and substitute first order formulas
> for the propositional variables, the result is a tautology. Thus,
>
> (Ax)Px v ~(Ax)Px
>
> is a tautology. But,
>
> (Ax)Px v (Ex)~Px
>
> is not a tautology, even though it is true under every interpretation.
> It is not the result of a substitution of some propositional tautology.
>
> --
Zuhair
MoeBlee
Aug 17, 2012, 10:29:06 AM8/17/12
to
On Aug 17, 8:17 am, Zuhair <zaljo...@gmail.com> wrote:
> On Aug 17, 3:35 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> > mstem...@walkabout.empros.com (Michael Stemper) writes:
> > > In article <502169BC.9BC50...@btinternet.com>, Frederick Williams <freddywilli...@btinternet.com> writes:
> > >>Michael Stemper wrote:
>
> > >>> What criteria determine whether an axiom is "logical" or "non-logical"?
>
> > >>Oh it's quite simple. Not.
Yes, it is simple, at least with these definitions:
> On Aug 17, 3:35 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> > mstem...@walkabout.empros.com (Michael Stemper) writes:
> > > In article <502169BC.9BC50...@btinternet.com>, Frederick Williams <freddywilli...@btinternet.com> writes:
> > >>Michael Stemper wrote:
>
> > >>> What criteria determine whether an axiom is "logical" or "non-logical"?
>
> > >>Oh it's quite simple. Not.
A sentence S is logically valid iff S is true in every interpretation
(for the language).
A sentence S is a logical axiom of system Y iff (S is an axiom of
system Y & S is a logically true).
> > > > The flood of posts that mine appears to have kicked off supports you there.
>
> > >>This: P v ~P is a (candidate to be a) logical axiom because it is true
> > >>in all interpretations. I.e. whether P is true or false, P v ~P comes
> > >>out true.
>
> > > Okay. If there's a difference between this and "tautology", I'm too
> > > dim to pick up on it.
>
> > In many contexts, "tautology" refers to those formulas which are
> > universally true due to propositional structure. That is, if we take a
> > propositional formula, P v ~P, say, and substitute first order formulas
> > for the propositional variables, the result is a tautology. Thus,
>
> > (Ax)Px v ~(Ax)Px
>
> > is a tautology. But,
>
> > (Ax)Px v (Ex)~Px
>
> > is not a tautology, even though it is true under every interpretation.
> > It is not the result of a substitution of some propositional tautology.
>
> > --
>
> This *IS* a tautology!
reserved for a certain kind of validity. In that sense the AxPx v
Ex~Px is not a tautology. It is a validity, but it is not a tautology.
MoeBlee
Zuhair
Aug 17, 2012, 11:54:02 AM8/17/12
to
and "tautology" in synonymous manner and I personally agree to that.
Zuhair
Jesse F. Hughes
Aug 17, 2012, 12:16:54 PM8/17/12
to
little confusion. But there would be less confusion if you would use
the term "validity" instead, since sometimes people jump into the middle
of a thread and they may not understand which meaning of "tautology" you
have in mind.
--
Jesse F. Hughes
"It is a clear sign that something is very, very, very wrong, as human
beings are, well human. Maybe some people think that mathematicians
are not, but I disagree. They are human beings." -- James S. Harris
MoeBlee
Aug 17, 2012, 5:51:04 PM8/17/12
to
That's why I used the word "sometimes" in my post.
> > some authors use the words "valid"
> > and "tautology" in synonymous manner
Of course.
MoeBlee
> > some authors use the words "valid"
> > and "tautology" in synonymous manner
MoeBlee
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