Hi Brent On Wed, Jun 3, 2015 at 1:17 PM, meekerdb <[email protected]> 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
