🚀 go-pugleaf

-*----
In article <1992Nov16.091653.1@woods.ulowell.edu> cotera@woods.ulowell.edu writes:
> Can anyone explain why we use logic? Unfortunately, any logical 
> arguments to support logic, are unfortunately, invalid.

Unfortunately, I have no notion of what Ray Cote means by
"invalid."  Perhaps he could explain this.  (Those who notice
that such explanation is meta-logical discussion get an A.)

Russell

In article <1992Nov16.091653.1@woods.ulowell.edu> cotera@woods.ulowell.edu writes:
>Can anyone explain why we use logic? Unfortunately, any logical arguments to
>support logic, are unfortunately, invalid.

For the same reason that courts accept only two pleas: guilty or not
guilty.  It is quite hard enough to get a jury to agree on that one bit
without asking them also to agree on details of the witnesses'
stories.

Once the high order bit is settled, the lower order bits (severity of
sentence and such) can be decided more casually according to the
position of that bit and with the aid of information left over from the
determinations of the higher order bits.

Voting tends to be organized along similar lines, with questions
preferably framed to have yes/no answers in order to simplify
discussion and to maximize the representativeness of the winning
outcome.

Pretrial hearings and primaries serve to sharpen the question, not to
refine the answer.

This is the principle of the most significant bit, MSB.  Over the
millennia we have acquired an intuitive appreciation of the MSB
principle, not just in the courtroom and the legislature but
everywhere, and we have evolved and refined tools for dealing with that
bit.  These tools constitute logic.

Our modern view of these tools recognizes a library of predicates, each
producing a bit that for certain applications is the appropriate high
order bit.

In addition there is the collection of all n-ary operations on a
2-valued set, better known as the Boolean operations, giving a complete
library of operations for manipulating bits.

But this modern view in terms of predicates and Boolean connectives is
as much a stereotype of natural logic as the Pentagon is pentagonal,
and any simple modern account can bear only a superficial resemblance
to the full gamut of tools of natural logic.  In particular there has
been no convincing demonstration that the weighty matters contemplated
in courts and the legislature are judged exclusively, or even
primarily, according to the rules of the first order predicate
calculus.  The logic of the court would appear to center more on
catching smugglers of argumentum ad hominem and other logical
contraband, while that of Congress is the same with the additional
requirement of alienating the fewest number weighted by alliance.  In
such settings the MSB principle plays a critical role in conferring the
benefit of logical rigor on a situation whose natural state is logical
chaos.

Formal logic presumably did not appear until well after natural logic
had evolved and been adopted for everyday use.  The first two millennia
of formal logic to make it into today's historical record focused on
the correct application of an abstract view of the extant tools,
without raising questions of their soundness or completeness.  Peirce
in the 1870's was the first to explicitly incorporate the two-valued
nature of truth into the methodology of formal logic, though Boole had
earlier come very close when he tied truth to sets.

Today formal logic continues to focus on correctness of argument, with
two-valued truth playing a central though by no means solo role as the
importance of the first bit comes to be seen, if only subconsciously,
in its proper light: a very good but by no means essential heuristic.

As nonlogical principles are assimilated along with logical rules of
inference into single decision methods and conditional rules of
calculation, the boundary of logic and nonlogic blurs, leaving us with
formal reasoning systems containing a blend of rules whose origins in
their respective logical and nonlogical calculi grow less discernible
with every passing decade.
-- 
Vaughan Pratt	  		A fallacy is worth a thousand steps.

Can anyone give me a counterexample of the following:

Let T a consistent first order theory with equality which
decidable set of quatifier-free logical consequences:

  QF(T) = { B  |   t |= B, B quantifier-free }

Let A be a T-consistent quantifier-free formula.
Then (T,A) is also a consistent first order logic with
equality and QF(T,A) is ALSO DECIDABLE.

M. Ayala
Universitaet Kaiserslautern
FRG
ayala@informatik.uni-kl.de

In article <1992Nov16.091653.1@woods.ulowell.edu> cotera@woods.ulowell.edu writes:

   Can anyone explain why we use logic? Unfortunately, any logical arguments to
   support logic, are unfortunately, invalid.

Because if you don't use logic we'll kill you.

(Can we start a group for bozos who want to propose dusty old paradoxes
as if they were somehow something we hadn't all seen a thousand times
before?  Something like alt.logic.stupid-tricks?)
--
		Charles R. Martin/(Charlie)/martinc@cs.unc.edu
     Dept. of Computer Science/CB #3175 UNC-CH/Chapel Hill, NC 27599-3175
	    3611 University Dr #13M/Durham, NC 27707/(919) 419 1754
     "Oh God, please help me be civil in tongue, pure in thought, and able
      to resist the temptation to laugh uncontrollably.  Amen." -- Rob T

In article <1992Nov17.164246.18220@uklirb.informatik.uni-kl.de> ayala@uklirb.informatik.uni-kl.de (Mauricio Ayala R.) writes:

>Can anyone give me a counterexample of the following:
>
>Let T a consistent first order theory with equality which
>decidable set of quatifier-free logical consequences:
>
>  QF(T) = { B  |   t |= B, B quantifier-free }
>
>Let A be a T-consistent quantifier-free formula.
>Then (T,A) is also a consistent first order logic with
>equality and QF(T,A) is ALSO DECIDABLE.
>
>M. Ayala
>Universitaet Kaiserslautern
>FRG
>ayala@informatik.uni-kl.de
>

Solution follows.










Take the theory over the language with one constant  0 , one function  f(x) ,
one unary predicate  P(x), and a binary relation  R(x,y), as follows:

For all  x  R(0,f(x))
R(0,0)
and for each  n  in a non-recursive r.e. set, the axioms:

[ There exists an  x  such that  not R(0,x) ]  -->  [ P(n) ]

(where  n  is the abbreviation of  f(x)  applied  n  times to  0 )

Then one can check that the set of quantifier free sentences that are logical
consequences of this axioms is recursive.

However if one expands the language with one more constant  c , and add the
sentence   not R(0,c), then that set becomes non-recursive.  [if you dont allow
an expansion of the language, then it can be shown that there is no counter-
example].

|>    Can anyone explain why we use logic? Unfortunately, any logical arguments to
|>    support logic, are unfortunately, invalid.
|> 
|> Because if you don't use logic we'll kill you.

This may actually have happened once !!!!!


From that grand old corny western, "Gunfight at O.K. Coral".....

Doc:   Where's your logic, Wyatt ?
Wyatt: To hell with logic; that's my brother lying there !

In article <1992Nov16.091653.1@woods.ulowell.edu>, cotera@woods.ulowell.edu writes:
|> Can anyone explain why we use logic? Unfortunately, any logical arguments to
|> support logic, are unfortunately, invalid.
|> --Ray Cote

Simple: to get from 'true' statements about some universe of discourse to other
true statements of the same universe. Why this has turned out to be the preferred
way of doing science should be obvious.

Klaus
-- 
    __o   
  _`\<,_  
 (*)/ (*) Claudio Caputti
+++++++++++++++++++++++++++++++++++++++^++++++++++++++++++++++++++++++++++++++
Free will - the result of chaotic amplification of quantum events in the brain.
------------------------------------------------------------------------------

In article <MARTINC.92Nov17151735@hatteras.cs.unc.edu>, martinc@hatteras.cs.unc.edu (Charles R. Martin) writes:
|> In article <1992Nov16.091653.1@woods.ulowell.edu> cotera@woods.ulowell.edu writes:
|> 
|>    Can anyone explain why we use logic? Unfortunately, any logical arguments to
|>    support logic, are unfortunately, invalid.
|> 
|> Because if you don't use logic we'll kill you.
|> 
|> (Can we start a group for bozos who want to propose dusty old paradoxes
|> as if they were somehow something we hadn't all seen a thousand times
|> before?  Something like alt.logic.stupid-tricks?)
|> --
|> 		Charles R. Martin/(Charlie)/martinc@cs.unc.edu
|>      Dept. of Computer Science/CB #3175 UNC-CH/Chapel Hill, NC 27599-3175
|> 	    3611 University Dr #13M/Durham, NC 27707/(919) 419 1754
|>      "Oh God, please help me be civil in tongue, pure in thought, and able
|>       to resist the temptation to laugh uncontrollably.  Amen." -- Rob T

This kind of response is what you expect from the dogmatic only, and being
dogmatic is a simple reason NOT to use logic. Also your response suggest you
don't like to think too hard about what you're doing,  and why you're doing it.
It must be the power of commercials: "Just Do It!"

Your kind of flame is simply pathetic.

KGa
-- 
    __o   
  _`\<,_  
 (*)/ (*) Claudio Caputti
+++++++++++++++++++++++++++++++++++++++^++++++++++++++++++++++++++++++++++++++
Free will - the result of chaotic amplification of quantum events in the brain.
------------------------------------------------------------------------------

In article <1992Nov18.144719.24323@nntp.nta.no> klaus@hal.nta.no (Klaus Gaarder FNI) writes:

   In article <MARTINC.92Nov17151735@hatteras.cs.unc.edu>, martinc@hatteras.cs.unc.edu (Charles R. Martin) writes:
   |> In article <1992Nov16.091653.1@woods.ulowell.edu> cotera@woods.ulowell.edu writes:
   |> 
   |>    Can anyone explain why we use logic? Unfortunately, any logical arguments to
   |>    support logic, are unfortunately, invalid.
   |> 
   |> Because if you don't use logic we'll kill you.
   |> 
   |> (Can we start a group for bozos who want to propose dusty old paradoxes
   |> as if they were somehow something we hadn't all seen a thousand times
   |> before?  Something like alt.logic.stupid-tricks?)

   This kind of response is what you expect from the dogmatic only, and
   being dogmatic is a simple reason NOT to use logic. 

There's a difference between being dogmatic and being deep.  Assume for
purposes of argument only that I was being deep: what did I mean then?

   Also your response suggest you don't like to think too hard about
   what you're doing, and why you're doing it.  It must be the power of
   commercials: "Just Do It!"

Wrong: just tired of re-seeing silly paradoxes that were old hat when
Jesus was a teenybopper.

   Your kind of flame is simply pathetic.

Nope, you just didn't understand it.

--
		Charles R. Martin/(Charlie)/martinc@cs.unc.edu
     Dept. of Computer Science/CB #3175 UNC-CH/Chapel Hill, NC 27599-3175
	    3611 University Dr #13M/Durham, NC 27707/(919) 419 1754
     "Oh God, please help me be civil in tongue, pure in thought, and able
      to resist the temptation to laugh uncontrollably.  Amen." -- Rob T

In article <1992Nov16.091653.1@woods.ulowell.edu> cotera@woods.ulowell.edu writes:

   Can anyone explain why we use logic? Unfortunately, any logical arguments to
   support logic, are unfortunately, invalid.
   --Ray Cote

On the contrary, logical arguments may be not only valid, but also informative.

For instance, to explain why we use classical first order logic (for many
important purposes), logical theorems such as soundness and completeness are
very relevant.  So are fancier theorems such as the non-enumerability of
consequence in 2nd order logic, or Lindstrom's theorem.  

Of course it depends what the argument is intended to accomplish.  A truly
illogical person (perhaps a child) must be trained, not convinced.  A logical
person who wants to gain insight into what he's doing and why can learn a lot
from mathematical logic.  

I believe that this point is originally due to Michael Dummett, "Justification
of Deduction", in _Truth_and_Other_Enigmas_.

	Josh

|> >Can anyone give me a counterexample of the following:
|> >
|> >Let T a consistent first order theory with equality which
|> >decidable set of quatifier-free logical consequences:
|> >
|> >  QF(T) = { B  |   t |= B, B quantifier-free }
|> >
|> >Let A be a T-consistent quantifier-free formula.
|> >Then (T,A) is also a consistent first order logic with
|> >equality and QF(T,A) is ALSO DECIDABLE.

|> Solution follows.
|> 
|> Take the theory over the language with one constant  0 , one function  f(x) ,
|> one unary predicate  P(x), and a binary relation  R(x,y), as follows:
|> 
|> For all  x  R(0,f(x))
|> R(0,0)
|> and for each  n  in a non-recursive r.e. set, the axioms:
|> 
|> [ There exists an  x  such that  not R(0,x) ]  -->  [ P(n) ]
|> 
|> (where  n  is the abbreviation of  f(x)  applied  n  times to  0 )
|> 
|> Then one can check that the set of quantifier free sentences that are logical
|> consequences of this axioms is recursive.
|> 
|> However if one expands the language with one more constant  c , and add the
|> sentence   not R(0,c), then that set becomes non-recursive.  [if you dont allow
|> an expansion of the language, then it can be shown that there is no counter-
|> example].

I don't quite get this comment. If the constant c is in the original language,
but with no extra axioms, isn't the original theory still decidable for
quantifier-free formulas ? And thus isn't the example now a counterexample
of the type originally required ?

Help sort me out please, somebody.
------------------------------------------------------------------------------
              Bill Taylor              wft@math.canterbury.ac.nz
------------------------------------------------------------------------------
                       Kleeneness is next to Godelness.
------------------------------------------------------------------------------
 

In article <MARTINC.92Nov18221201@hatteras.cs.unc.edu>
Date: 19 Nov 92 03:12:01 GMT
In article <1992Nov18.144719.24323@nntp.nta.no> klaus@hal.nta.no (Klaus Gaarder FNI) writes:

   In article <MARTINC.92Nov17151735@hatteras.cs.unc.edu>, martinc@hatteras.cs.unc.edu (Charles R. Martin) writes:
   |> In article <1992Nov16.091653.1@woods.ulowell.edu> cotera@woods.ulowell.edu writes:
   |> 
   |>    Can anyone explain why we use logic? Unfortunately, any logical arguments to
   |>    support logic, are unfortunately, invalid.
   |> 
   |> Because if you don't use logic we'll kill you.
   |> 
   |> (Can we start a group for bozos who want to propose dusty old paradoxes
   |> as if they were somehow something we hadn't all seen a thousand times
   |> before?  Something like alt.logic.stupid-tricks?) 


   This kind of response is what you expect from the dogmatic only, and
   being dogmatic is a simple reason NOT to use logic. 

|>There's a difference between being dogmatic and being deep.  Assume for
|>purposes of argument only that I was being deep: what did I mean then?

|>Wrong: just tired of re-seeing silly paradoxes that were old hat when
|>Jesus was a teenybopper.

   Your kind of flame is simply pathetic.

|>Nope, you just didn't understand it.

Sorry, but I completely fail to see what's so deep about your comment, I also fail
to see what it has to with " re-seeing silly paradoxes".

I thus consider my response to your flame as valid as ever before.

Klaus@NTR
-- 
    __o   
  _`\<,_  
 (*)/ (*) Claudio Caputti
+++++++++++++++++++++++++++++++++++++++^++++++++++++++++++++++++++++++++++++++
Free will - the result of chaotic amplification of quantum events in the brain.
------------------------------------------------------------------------------

In article <1992Nov20.101850.7674@nntp.nta.no> klaus@hal.nta.no (Klaus Gaarder FNI) writes:

    Sorry, but I completely fail to see what's so deep about your
   comment.


I don't doubt it.
--
		Charles R. Martin/(Charlie)/martinc@cs.unc.edu
     Dept. of Computer Science/CB #3175 UNC-CH/Chapel Hill, NC 27599-3175
	    3611 University Dr #13M/Durham, NC 27707/(919) 419 1754
     "Oh God, please help me be civil in tongue, pure in thought, and able
      to resist the temptation to laugh uncontrollably.  Amen." -- Rob T

In article <GUTTMAN.92Nov19120734@circe.mitre.org> guttman@circe.mitre.org (Joshua D. Guttman) writes:
>In article <1992Nov16.091653.1@woods.ulowell.edu> cotera@woods.ulowell.edu writes:
>
>   Can anyone explain why we use logic? Unfortunately, any logical arguments to
>   support logic, are unfortunately, invalid.
>   --Ray Cote
>
>On the contrary, logical arguments may be not only valid, but also informative.
>
>For instance, to explain why we use classical first order logic (for many
>important purposes), logical theorems such as soundness and completeness are
>very relevant.  So are fancier theorems such as the non-enumerability of
>consequence in 2nd order logic, or Lindstrom's theorem.  
>
>Of course it depends what the argument is intended to accomplish.  A truly
>illogical person (perhaps a child) must be trained, not convinced.  A logical
>person who wants to gain insight into what he's doing and why can learn a lot
>from mathematical logic.  

This seems to fall far short of addressing the basic question of why we
use logic.  Why not do all our thinking with just the equations of
arithmetic?  Or the equations of binary relations under composition and
converse as suggested by Tarski and Givant in "A Formalization of Set
Theory without Variables."

To me the chief characteristic of logic is its emphasis on sorting out
situations into *two* cases, whether always successfully (Boolean
logic) or sometimes not (intuitionistic logic).  See my message of Nov.
16 arguing this viewpoint in considerable detail.
-- 
Vaughan Pratt	  		A fallacy is worth a thousand steps.

Hmmm... rather to my astonishment, it appears that the word "logic" here
is becoming polymorphous.  On at least one hand, we have a very general
idea of logic as "the study of the principles of reasoning" or "valid
reasoning as opposed to invalid reasoning".  On another hand, we have
the much more constrained definition of formal or mathematical logic.
(An appropriate Peano axiomatization leading to the generation of
infinite hands is left as an exercise.)

I believe that the original question was really in the first domain, so
it isn't clear that the mathematical arguments -- while interesting in
themselves -- are very informative.  Considering the question from the
more general position, it appears that the original question

   Can anyone explain why we use logic? Unfortunately, any logical
   arguments to support logic, are unfortunately, invalid.  --Ray Cote

is clearly an attempt to attack the use of "logic" by paradox, by
claiming that there is no "logical" reason to use logic.  However, as
someone else pointed out, the question presumes an appeal to logic on
its face, because the distinction between a "valid" and "invalid"
argument is itself a logical one.  One might argue therefore that the
best argument for using logic is the fact that the attempt to challenge
the use of logic must itself appeal to "logic", that is, to valid
reasoning.  This can lead to the idea of a Platonic Form for "valid
reasoning" that pre-exists our attempts to use that valid reasoning,
making "logic" into a sort of natural-science search for a better and
better description of this Ideal of Valid Reasoning.

More pragmatically, there appear to be only two forms of reaching
agreement available: either we appeal to an idea of "valid argument"
under which, by a dialectic or discursive process, we arrive at a
conclusion acceptable to all parties; or we deny that such a process can
exist, in which case we are reduced to the more direct method of hitting
one another over the head with large rocks until one side or the other
gives in.

--
		Charles R. Martin/(Charlie)/martinc@cs.unc.edu
     Dept. of Computer Science/CB #3175 UNC-CH/Chapel Hill, NC 27599-3175
	    3611 University Dr #13M/Durham, NC 27707/(919) 419 1754
     "Oh God, please help me be civil in tongue, pure in thought, and able
      to resist the temptation to laugh uncontrollably.  Amen." -- Rob T

pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
   guttman@circe.mitre.org (Joshua D. Guttman) writes:
   >cotera@woods.ulowell.edu writes:
   >
   >   Can anyone explain why we use logic? Unfortunately, any logical arguments to
   >   support logic, are unfortunately, invalid.
   >   --Ray Cote
   >
   >On the contrary, logical arguments may be not only valid, but also informative.
   >(etc.)

   This seems to fall far short of addressing the basic question of why we
   use logic.  Why not do all our thinking with just the equations of
   arithmetic?  Or the equations of binary relations under composition and
   converse as suggested by Tarski and Givant in "A Formalization of Set
   Theory without Variables."

Are you asking why we use a syntax of variables and predicates (or rather a
natural language grammar in which these can be discerned)?  Or are you asking
why a particular logic, quantified predicate logic, is more beneficial than
another, say purely equational logic?  

As for why we do not do all our thinking with equations of arithmetic, I'd
answer, because numbers are not the only things we want to think about, and
quantifier-free formulas are not the only ones we want to assert.  I have not
looked at Tarski/Givant.

As to why we use any sort of deductive reasoning, I'd answer, because we have
beliefs and we examine assumptions and we want to know what their consequences
are.  Indeed, it would hard to imagine that we could attribute beliefs to a
creature that did not trouble itself about their consequences.

	Josh