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Topic: Interdeducibility of properties of an abstract consequence relation.
Replies: 11   Last Post: Mar 8, 2013 8:40 PM

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Graham Cooper

Posts: 4,495
Registered: 5/20/10
Re: Interdeducibility of properties of an abstract consequence relation.
Posted: Mar 8, 2013 7:02 PM
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On Mar 9, 5:46 am, Frederick Williams <>
> Graham Cooper wrote:

> > > > On Mar 7, 9:21 am, Frederick Williams <>
> > > > wrote:

> > > > > P9 If Sub(M,x,y,N), Sub(alpha,x,y,beta) and M |= alpha, then N |= beta.
> > Again I don't see what P9 is for.
> > formula1 [X/Y] =  formula2
> > is standard substitution syntax.
> My (or rather, Hermes's) notation Sub(alpha,x,y,beta) signifies that
> alpha, x, y and beta stand in a certain relation to one another.
> Furthermore that relation is carefully defined.  (If it isn't carefully
> defined _in my post_, then the fault is mine and I shall rectify it.)
> What does the "standard" definition mean?  Standard may be good,
> understandable is better.

You've never seen

f(X+1) [Y/X] = f(Y+1) ?

Making SUBS a predicate has no purpose and neither does formalising
variable renaming in Quantified logic.

As opposed to Lambda Calculus where substituting [Y/X] is used in the


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