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Topic: Number of variables needed to axiomatize first order predicate calculus?
Replies: 6   Last Post: Mar 15, 2013 11:25 PM

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Graham Cooper

Posts: 4,237
Registered: 5/20/10
Re: Number of variables needed to axiomatize first order predicate calculus?
Posted: Mar 15, 2013 11:25 PM
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On Mar 16, 10:55 am, me <me154...@gmail.com> wrote:
> On Mar 15, 1:46 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
>
>

> > On Mar 13, 5:54 pm, George Greene <gree...@email.unc.edu> wrote:
>
> > > On Mar 8, 6:02 pm, Frederick Williams <freddywilli...@btinternet.com>
> > > wrote:

>
> > > > It is well known that at least three variables are needed to axiomatize
> > > > sentential calculus, by which I mean that at least three letters are
> > > > needed to write down the axioms and rules.

>
> > > Come ON!  You can't take your own question seriously until
> > > and UNLESS YOU FIRST *specify* a *specification*LANGUAGE*!!

>
> > > The specification-language you used to specify the axioms of
> > > propositional calculus
> > > is one that NO ONE HAS *EVER* seen used IN ANY other context!
> > > It basically just copies the grammar of the underlying language AS
> > > ASSUMED
> > > and then INTRODUCES A CONCEPT from a COMPLETELY HIGHER ORDER,
> > > namely first-order, without ANY PRETEXT of explication!  It
> > > universally quantifies
> > > over the things that it is trying to "specify", even though universal
> > > quantification
> > > is EXPLICITLY PROHIBITED in the object-language and even though this
> > > specification
> > > language itself does NOT explicitly contain any quantifier-symbols!
> > > It is EASY to
> > > "get small" if you just LEAVE OUT WHOLE CHUNKS of explication of what
> > > you
> > > are doing!

>
> > > The use of ","(comma) and "/"(slash) for clarifying that from some
> > > finite collection
> > > of input sentences, you can infer some other finite collection of
> > > output sentences,
> > > IS (thank god) used elsewhere and I suppose that could be tolerated as
> > > being a
> > > paradigm-wide linguisitic convention for the specification of logics-
> > > as-collections-
> > > of-rules-in-general ANYway.  You could certainly begin this quest by
> > > saying that you
> > > would like to state all the inference rules of first-order logic using
> > > this comma/slash
> > > framework.

>
> > > But in general it is just going to be MUCH HARDER because arguably THE
> > > MOST
> > > important inference rule in FOL is NOT an inference rule simpliciter,
> > > but RATHER
> > > a META-inference rule, or, literally, a SECOND-ORDER*FIRST-ORDER*-
> > > inference-rule,
> > > namely, UNIVERSAL GENERALIZATION, in which you infer a sentence in FOL
> > > (a universally
> > > quantified sentence) NOT from some prior collection of sentences but
> > > rather FROM A *PROOF*!
> > > Basically, from [ B |- phi(x) ], where B is a finite set of sentences
> > > with x not occurring in them,
> > > you infer Ax[phi(x)].  The mere DARING of this NESTING of "|-" -- not
> > > to mention, HOW MANY
> > > VARIABLES does it take TO EVEN STATE the non-occurrence clause?? --
> > > the rule could
> > > basically be written [ B |- phi(x) ] |- Ax[phi(x)]  -- oh, and HOW
> > > MANY VARIABLES does it
> > > take to explain what x *as*parameter*of*phi(.)* means, as OPPOSED to
> > > what variables
> > > mean in other contexts -- remember, in your propositional/0th-order
> > > treatment, variables
> > > ONLY meant PROPOSITIONS -- is certainly BEYOND the pale of ANY
> > > specification language that you might have been trying to use FOR ANY
> > > OTHER purpose!
> > > If the language in which you are doing this is not CONSTANT (or at
> > > least constant in
> > > paradigm or parametric profile) across the various languages&levels-of-
> > > logics that
> > > you are investigating, then "the number of variables it takes" is just
> > > not going to be
> > > meaningful as a metric.  Oh, yeah, and then you also have to ask how
> > > many variables
> > > it takes to explain what kind of thing "phi(.)" is.  Phi(.) is after
> > > all ITSELF a universally
> > > quantified VARIABLE.

>
> > Eventually you should arrive at a n-OL logic with syntax complex
> > enough to encapsulate n-OL.

>
> > But 2OL needn't be MORE complex SYNTAX, just fewer restrictions.
>
> > e.g.  lowercase = terms, uppercase = variables
>
> > 1OL
> > add( M, 0, M )
> > add( M, s(N), (S) ) <-  add( M, N, S ).

>
> > 2OL
> > P(s(0)) <- ( P(X) -> P(s(X) ) & P(0) )

>
> > this is HIGH ORDER PROLOG where predicates may be VARIABLES and match
> > to generic prolog clauses.  e.g. P could match with notsmallest(X),
> > where notsmallest(X)->notsmallest(X+1)

>
> > -------------------------------
>
> > Now all the AXIOMS of the theory may be written in a SUBSET of the
> > full 1OL.

>
> > e.g.  HORN CLAUSES using only
>
> > p1(...) & p2(...) & p3(...)  ->   pa(...)
>
> > -----------------------------
>
> > What you want to look for is a SUBSET of the (1OL) language that can
> > perform some computational feat - e.g. be Turing Machine Equivalent.

>
> > Horn Clauses have this property since
>
> > UNIFY(  f(a,b,c...)  , f2(z,y,x,...) )
>
> > which finds a COMMON predicate between 2
> > is Turing Equivalent.

>
> > Then this computation method must be utilised to EXPAND on the limited
> > 2OL syntax.

>
> > -------------------------
>
> > The way I do this is using Naive Set Definition to specify which
> > variable in each predicate a SET is referencing.

>
> > e( A, evens ) :-  even(A).
>
> > *this example is simple, even() only has one argument.
>
> > Then I can process ALL() quantifiers via Set References, rather than
> > explicitly stating which term to Quantify over in Quantified Logic
> > style..  A(x), p(...x...) >=...
> > which is rather messy with partial formula insides hanging out like
> > that!.

>
> > ---------------------------
>
> > So yes you can INCREASE the complexity of 2OL syntax, but it's going
> > to get convoluted with string processing and looking inside predicate
> > types, or you can use a subset of 1OL syntax to write the axioms which
> > work on a simpler processing system which can still do the computing
> > work necessary.

>
> > e.g.   Not All Numbers Are Evens
>
> > ~A(X) e( X, evens )
>
> > I bypass referencing internals by using Set Referencing
>
> > !(ss( nats, evens ))
>
> > Not all members of nats are members of evens.
>
> > -------------------------
>
> > which is in Horn Clause format same as 2OL axioms and computable!
>
> > !(ss( S1, S2 )  <-   intersect( S1, !(S2)).
>
> > A 2OL Axiom.
>
> > (which will return the 1st ODD when invoked)
>
> > Herc
>
> > --www.BLoCKPROLOG.com
>
> Oix
>


IOW, its quite possible to embed a more complicated grammar
in a simpler grammar!

Then you can use a computer to evaluate the language..


LOGIC SYSTEM (wff - 10L)
--------------------------------------
HORN CLAUSE (reverse inference rules - 2OL)
======================
3GL INTERPRETER (+ Database)
________________________
COMPUTER


Then you can merely TRANSLATE the wff
to any Logic Language you like.

INPUT> ALL(N):EVENS N e NATS ?
V
<PRE-PARSER>
V
PROLOG> ss( EVENS, NATS)
PROLOG> e(@A, EVENS) , e(A, NATS)
PROLOG> YES
V
<POST_PARSER>
V
RESULT> ALL(N):EVENS N e NATS

Parsing into QUANTIFIER LINGO is not required for problem solving
though.

Herc
--
www.BLoCKPROLOG.com



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