Hi Brent On Wed, Jun 3, 2015 at 1:17 PM, meekerdb <meeke...@verizon.net> wrote:
> On 6/3/2015 7:16 AM, Brian Tenneson wrote: > > > > On Wednesday, June 3, 2015 at 2:16:31 AM UTC-7, Bruno Marchal wrote: >> >> >> On 02 Jun 2015, at 20:10, Brian Tenneson wrote: >> >> Grammatical systems just might be the type of thing Tegmark is looking >> for that is a framework for all mathematical structures... or at least a >> large class of them. >> >> I am still exploring the idea of grammatical system induction. >> >> >> I am not sure what you mean by grammatical induction. Is that not >> equivalent with the omega-induction principles, like with PA? >> > > It is described in this document: > > https://docs.google.com/document/d/1amDb4Yti4egpKfcO2oLcnGAH8UpC8_tKb7ivuH3AT7A/edit?usp=sharing > > > A couple of points I don't understand. First, G is a set of sentences. > I'm not sure what "any" means. > It means that G is a subset of the set of utterances. > Does it mean G is all grammatical sentences? > G is the set of grammatically-correct utterances for the formal system (A,G,X,I). Yes, all of them. That does not mean that G needs to be the entire set of utterances though. > Is G assumed finite, or countable? > There are no assumptions on G other than it is a subset of the set of all utterances using symbols in the alphabet A. > Second, why is H defined as an element of G^n (Cartesian product of sets) > instead of just a subset of G? > H is a *subset* of a Cartesian power of G, not an *element* of a Cartesian power of G. It is possible that a rule of inference is not defined for all of G^n, so H is the domain of the rule of inference in question. Modus ponens, for instance, in the FOL (first order logic) formal system is only defined so that it is this function: Modus ponens = {( (p, p-->q) , q ) : p is in element of G, p-->q is an element of G, and q is an element of G}. Modus ponens is not defined for all of G^2. For instance, (p,q-->p) is not in the domain of Modus Ponens. In first order logic, n is usually 1 or 2. > Third, if [H->G] is a function doesn't that implies that T(H) ends with a > unique G, which is not generally true of inferences. > > Well, I checked this list of some rules of inference http://en.wikipedia.org/wiki/List_of_rules_of_inference ALL of them have a single conclusion, which is an element of G. Which inference rules have multiple conclusions? We can make the minor adjustment that T is in [H-->G^m] instead of T is in [H-->G]. But I don't see why we have to. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
