
Re: Interdeducibility of properties of an abstract consequence relation.
Posted:
Mar 6, 2013 6:57 PM


On Mar 7, 9:21 am, Frederick Williams <freddywilli...@btinternet.com> wrote: > Please don't change the groups posted to. It's clearly a sci.logic > question. > > > > > > > > > > Graham Cooper wrote: > > > On Mar 7, 6:56 am, Frederick Williams <freddywilli...@btinternet.com> > > wrote: > > > Frederick Williams wrote: > > > > [blah] > > > > This is slightly improved (oh really?): > > > > Let there be a countable stock of predicates each of some arity n > > > (n = 0, 1, 2, ...), and an infinite but countable stock of variables. > > > ~, &, U, ( and ) are used autonomously. > > > > An atomic formula is a predicate followed by a number of variables equal > > > to the predicate's arity. Just what predicates and variables are > > > doesn't matter much, but one can tell that a string of them is or is not > > > an atomic formula, and if it is an atomic formula, just which atomic > > > formula it is. So, for example, if P is a predicate of arity two, the > > > variable _won't_ include b and bb because then Pbbb would be ambiguous. > > > This kind of fussy detail won't be mentioned further. > > > it's pretty standard to use a delimiter, > > Indeed so, commas and brackets are often used but they are avoidable. > For an explanation of "my" choice of language conventions, see the last > paragraph of this reply. > > > > > > > > > > > or data structure for > > arguments > > > > A formula is any one of the following: > > > > * alpha, where alpha is an atomic formula, > > > * ~alpha, where alpha is a formula, > > > * (alpha & beta), where alpha and beta are formulae, > > > * Ux alpha, where x is a variable and alpha is a formula. > > > > If x is a variable, Ux is called a quantifier. (alpha > beta) > > > abbreviates ~(alpha & ~beta) > > > > Below alpha, beta, ... (maybe decorated with primes) are formulae; and > > > x, y, ... are variables. > > > > 'x occurs free in alpha' is defined as follows: > > > * each occurrence of a variable in an atomic formula is a free > > > occurrence, > > > * x occurs free in ~alpha iff x occurs free in alpha, > > > * x occurs free in (alpha & beta) iff x occurs free in alpha or in beta, > > > * x occurs free in Uy alpha iff x occurs free in alpha and x is not y. > > > > An occurrence of a variable that is not free is said to be bound. > > > Clearly a variable that is bound is bound because of some quantifier. > > > Such a variable is said to be in the scope of that quantifier. > > > Axioms refer to formula's as variables. > > Nothing in my post has got anything to do with axioms. It's about > something else. See the word "abstract" in the subject header, see the > "Note,..." sandwiched between ***** and ***** below. > > > > > > > > > > > or(X , Y) < X > > or(X, Y) < Y > > > You Still can't formulate an AXIOM that refers to arguments, unless > > your syntax is more convoluted again! > > > ALL(a, b, c....) > > P(x,y, a, b, c, ...) > > A(x) .... P.... > > > where a, b, c might me sub arguments of sub arguments of sub arguments > > of P. > > > > Let M be a set of formulae. 'x occurs free in M' means x occurs free in > > > at least one member of M. Below M, N, ... are sets of formulae (not > > > excluding the empty set). > > > > Informally, Sub(alpha,x,y,beta) means 'beta is derived from alpha by > > > substituting y for x in all places where x is not in the scope of Ux'. > > > 'in the scope of' was defined only so that that informal explanation of > > > Sub(alpha,x,y,beta) could be given. I think the phrase has no further > > > use so far as this post is concerned. Precisely: > > > > * if alpha is atomic, Sub(alpha,x,y,beta) iff beta is obtained from > > > alpha by replacing all occurrences of x with y, > > > * Sub(~alpha,x,y,beta) iff there is a gamma such that > > > Sub(alpha,x,y,gamma) and beta is ~gamma, > > > * Sub((alpha & beta),x,y,gamma) iff there are alpha' and beta' such > > > that Sub(alpha,x,y,alpha'), Sub(beta,x,y,beta') and gamma is > > > (alpha' & beta'), > > > * Sub(Uz alpha,x,y,beta) iff one of: > > > ** x is not free in Uz alpha and beta is Uz alpha, > > > ** x is free in Uz alpha, y is not z, and there is a gamma for which > > > Sub(alpha,x,y,gamma) and beta is Uz gamma. > > > this is very fine grain low level > > > there are simple variable renaming schemes that do away with > > substitutions. > > > INFERENCES = 0 > > > For Each New Formula, increment and append INFERENCES to every > > variable name. > > > e.g. > > > Predicate > > vert( line ( pnt( X , Y ) , pnt( X , Z ))) > > > Goal? > > vert( line ( pnt( 1 , 1 ) , pnt( 1, 5 ))) ? > > > Match > > vert( line ( pnt( X1 , Y1 ) , pnt( X1 , Z1 ))) > > > X1 = 1 > > Y1 = 1 > > X1 = 1 > > Z1 = 5 > > > SUCCESS! > > > > Sub(M,x,y,N) means that for every alpha in M there is a beta in N such > > > that Sub(alpha,x,y,beta) and every formula in N can be obtain thus from > > > one in M. > > > Set at a time variable renaming? > > > Again I don't see what the utility is. > > It enables me to state P9 below. > > > > > > > > > > > > > Molecular Parallel > > Computers? > > > Variables have arbitrary names just as each formula may be arbitrary > > itself. > > What is the point of changing all X,Y,Zs in a dozen formula to P,Q,Rs? > > > When you use a Set Specification Axiom then you can then reference > > Sets without worrying about which parameter to index Qx p(...x...) > > > e( A, nats ) <> isNat( A ) > > > Axioms deal with the SET nats > > which has an implicit parameter. > > > > Now consider a relation = between sets of formulae and formulae having > > > the following properties. > > > > P1 If alpha in M then M = alpha. > > > P2 If M = alpha then M union N = alpha. > > > P3 If {alpha} = beta and {beta} = gamma, then {alpha} = gamma. > > > P4 If M = alpha and N = beta, them M union N = (alpha & beta). > > > P5 If M = (alpha > beta) and N = alpha, then M union N = beta > > > P6 M = (alpha > beta) iff M union {alpha} = beta. > > > P7 Ux alpha = alpha. > > > P8 If x is not free in M union {beta} and if M union {alpha} = beta, > > > then M union {Ux alpha} = beta. > > > P9 If Sub(M,x,y,N), Sub(alpha,x,y,beta) and M = alpha, then N = beta. > > > > Now to my question: can any of P1 through to P9 be deduced from the > > > others? > > ***** > > > Note, I make no reference to axioms, interpretations or truth, = is > > > just what this posts says it is. (Except for typos. :) > ***** > > > Use (alpha > beta) abbreviates ~(alpha & ~beta) > > > P5 If M = (alpha > beta) and N = alpha, then M union N = beta > > > might get P4. > > Might it? I wouldn't mind seeing the details. > > > Most of them follow easily by letting > > Follow from what? The question is to do with some (maybe) following > from the others. 'Follow' except in the context of 'follow from ...' I > don't understand. > > > M = p1 & p2 & p3 ... & pn > > M need not be finite. > > > and using M > > > rather than M = > > This is the last paragraph I referred to above. As I remarked in my OP, > P1P9 (and thus the preceding gubbins) comes from a text. I am > currently reading that text. Were I reading something else I might have > written a different post, as it is my post reflects (to some degree) the > contents of that text. So your suggestion that I might do something > else altogether gets me nowhere. Nevertheless, thank you for your > interest. > >  > When a true genius appears in the world, you may know him by > this sign, that the dunces are all in confederacy against him. > Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
dopey idiot! Not a clue what he is reading about (dysfunctional syntax) and quoting verbatim.
Herc  www.BLoCKHEADS.com

