> 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", 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.
> 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.
> >> > 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.
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
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...
> 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,
> I'm not familar with his definition of "necessary truth regarding a
> 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?
> If you do all your semantics (structures, et. al) in set theory then
> 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.
> On Aug 8, 3:26 pm, Frederick Williams <freddywilli...@btinternet.com>
> 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
> 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.
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'?
> 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 animated figures stand Adorning every public street And seem to breathe in stone, or Move their marble feet.
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"?
-- The animated figures stand Adorning every public street And seem to breathe in stone, or Move their marble feet.
On Aug 9, 3:10 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
> 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"?
> --
> The animated figures stand
> Adorning every public street
> And seem to breathe in stone, or
> Move their marble feet.
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.
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.
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
axiom of system S.
> Iow, there's a good reason why can claim an axiom is a formula
> but _not_ the other way around:
Every formula is an axiom for some systems. Indeed, every formula is
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.
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 [...]
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 ....
> 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
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.
I don't know the signficance you have in mind with that remark. Is it
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.
Maybe you mean something by "rhyme" other than the usual sense of the
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.
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.
On Aug 9, 6:47 pm, MoeBlee <modem...@gmail.com> wrote:
> 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 [...]
> 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
Well possibly not elephants, since elephants are biological beings and
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.
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
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.
> 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.
>> 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.
> > 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
No of course there is no need for any additional axioms.
> 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 <zaljo...@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.
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.
-- "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
> 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.
> > > > 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!
Jesse just explained to you that sometimes the word 'tautology' is
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.
> 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:
> 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!
> Jesse just explained to you that sometimes the word 'tautology' is
> 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
Yes but not always used like that, some authors use the words "valid"
and "tautology" in synonymous manner and I personally agree to that.
Zuhair <zaljo...@gmail.com> writes:
> On Aug 17, 5:29 pm, MoeBlee <modem...@gmail.com> wrote:
>> 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:
>> 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!
>> Jesse just explained to you that sometimes the word 'tautology' is
>> 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
> Yes but not always used like that, some authors use the words "valid"
> and "tautology" in synonymous manner and I personally agree to that.
As long as it is clear how you intend to use the term, there should be
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
