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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: 4,273
Registered: 5/20/10
Re: Number of variables needed to axiomatize first order predicate calculus?
Posted: Mar 15, 2013 4:46 AM
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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





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