by J. L. S.
SO, I PROPOSE they are identical. We should find out if Carnap had another name than Rudolf, since I have my System GHP to read, a 'hopefully plausible' or 'highly powerful' version of Myro's System G (based on Grice's System Q, for Quine).
So I propose they are identical.
Once Virgil and Dante speak the same lingo, they can raise to the City!
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More on this later. My point is really a hint to think of a specific quote by Carnap -- from his book he wrote to 'replace' Principia Mathematica (seeing how inflation was killing Germany) --. Basically System C, like System G, then would contain:
vocabulary
syntax -- inference rules
introduction and elimination of operators. I have of late concentrated on "~", but the points can be made more generally about logical operators in general, notably conectives.
semantics
pragmatics
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The sketch for Grice's "System Q" in his "Vacuous Names" I have expanded at the Grice Club. I should provide specific quotes from that essay by Grice, I suppose, and I may!
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By Roger Bishop Jones, for The City of Eternal Truth
ReplyDeleteIf I could actually get sight of G-HP then I could probably do something with it.
If you want C-R to be the same then the thing to do would be to subject G-HP to formalisation, i.e. do some formal metatheory, and maybe an embedding into HOL so that you could use ProofPower as a proof tool for G-HP.
I do have a system myself, but its probably too far removed from either Grice or Carnap to have a place in the conversation. It is interesting, so simple I can explain it in a few words.
It is a first order set theory. simplified down to the barest essentials, except that it is infinitary.
There are no constants, only variables.
Any well-founded set can count as a variable (this is an abstract syntax, so we don't worry about whether the things we use are very syntactic). So we have available a huge number of variables (an inaccessible cardinality of variables even). And you can have as many as you like in a single formula.
There is only one primitive relation, which is membership (no equality). There is only one way of forming a compound sentence. This is a generalised infinitary scheffer stroke.
It works like this. You apply this sentence constructor (say PI) to a set of variables and a set of formulae (any cardinality) and the result is true iff at least one of the formulae is false for some value of the variables. (this is the generalised NAND, its possible I used NOR)
Logically, this is like Witters in the Tractatus, he has only the one connective, but he writes about it so obscurely its not clear (to me, I'm sure some scholar can tell us) whether Witters intended it to be infinitary or finitary. Probably finitary, so mine's more powerful.
This is defined formally in one of my "PDFs" and there are reams of formal proofs about it, because I was trying to investigate non-well-founded sets with it (one of Forster's predelictions).
If you like gobbledygook its here:
http://www.rbjones.com/rbjpub/pp/doc/t027.pdf
(and other versions in other documents).
But putting that aside, I need to see G-HP, and I have yet to find it anywhere. How about scan or a photocopy?
Russell wrote out 35 pages of definitions and theorems for Carnap when Carnap solicited his help in obtaining a cut price copy of Principia!
G-HP is surely shorter than that, and the photocopier has been invented.
RBJ
Sure. I love your generosity in displaying your system J-rb, ie. system Jones sub roger bishop. Surely I should have my System Sjl. But life has found me as an exegeta (as I think they say in Latin) for Grice! So I conceived this idea by reading Grice. His system Q (for Quine) he developed for the Quine festschrift (in Davidson/Hintikka). Of course he could have called it "System My-Lovely". The name is UNimportant. It is a few pages so I will try and make it available to you from that book.
ReplyDeleteCan you believe that Ostertag reprinted PART of this "Vacuous Names" (for that is the title of Grice's article, or contribution to the Davidson/Hintikka book, "Words and objections", Reidel, 1969) in his "Definite descriptions" volume -- with only, as I say, the SECOND part of "Vacuous Names", i.e. WITHOUT "System G". Otherwise, people would be knowing MORE about it. We do know that, you can quote him on that, that Quine found the system "forbiddingly complex" yet on the whole he was "all for it" ("Reply" (to Grice)) in Davidson/Hintikka, one page long).
I was mentioning the System G-hp to Jones because in some publications of mine (one, principally, not to count my PhD dissertation), I do make some 'revisited' use of it. And since I am still elaborating on these matters, I thought of letting Jones know so that whenever I drop things about "System G", or "System G-hp" we know what we are talking about! (Actually I love to REFER to System G-hp rather than use it!).
I'm fascinated that Russell sent so many copies to Carnap!
One point of contact for the System G-hp and the System C-r where 'r' should stand for 'revisited', since, say, it's Jones -- or JonesSperanza -- revisiting Carnap (or CarnapGrice) -- may be the topic of 'analytic'. Since Grice does not REALLY consider the notion when he does System Q (being what everybody was expecting he would tackle, he having co-written the "Defense of a dogma" to which Quine had replied in "Word and object" and this was "Words and objections". Instead, he just plays with "System Q" vis a vis scope-indications, which is totally secondary for the _essence_ of System Q itself -- and which was what Quine found 'forbiddingly complex' but on the whole 'all for it' (Quine had other scope notations in use at was slightly unimpressed -- Grice's notational device in his System Q is subscript -- for something like the arrival of a symbol in a formula.
For the record: Myro. Myro does use "System G", I think by name, in his contribution to the Grandy/Warner festschrift for Grice. Anyway, more on this later, I hope!
Russell didn't do 35 copies, he did one 35 page extract of the definitions and proofs in volume 1 of principia. (Principia v1 with proofs and minor lemmas omitted).
ReplyDeleteI have other systems, but I am semantically oriented, so I don't worry about concrete syntax, and usually play with the semantics or the ontology. Thus I have ontologies which would yield logical systems similar in strength to set theory but in which the things which exist are not sets. e.g. of functions, or even of categories.
Words and Objections I am never likely to acquire, because I have "Words and Objects" on my shelf and it is one of those books which I am unable to read.
I get compulsions to throw it in the bin about half way through chapter 1 which I can only resist by putting it back on the shelf.
If the main feature of G-HP is a novel syntax to indicate scope of variables, then it would probably be hard to treat formally. I usually make things as abstract as possible and avoid concrete syntax because the concrete syntax is complex to reason about, and there isn't a lot of point, concrete syntax belongs to pragmatics (in the ordinary sense of the term). My inclination would be to look at the stuff underneath the syntax and see whether there is anything of interest there.
RBJ
"Russell "did one 35-page extract of the definitions and proofs in volume 1 of principia. (Principia v1..."
ReplyDelete--- Thanks. Sorry for my lose wording. That was so kind of Russell. It all transpired in the Carnap's abridgement I would gather. I suppose the proofs and lemmas, as you say, you CAN omit, which leaves us with the definitions. So one may think of System PM as THE system -- as the archetypical system here. Will find out, I hope, to see if "System" is used for this and whether there are minor variations of System PM that received similar names -- or not. I suppose we can speak of System CW of Frege ("Conceptual Writing") as yet another System. I wonder if Hilbert worked with a particular system which he called some way or other. Etc. I suppose the idea of naming systems like that started with Kripke and his "System S"?
Discarded the proofs and the minor lemmas, leaving the definitions and the important theorems.
ReplyDeletePrincipia consists of a logical system (which is essentially that described by Russell in his 1908 paper) and a body of logical and mathematical knowledge (theorems) together with formal proofs for them in that system.
Russell's extract probably didn't include the system at all, since Carnap already knew about that (Principia was in the library at Jena).
So it consisted in a concise account of what he did with that system in volume 1. (there is a published paperback "Principia to *56" which probably covers similar ground, or not quite as much, but does include all the detail).
In terms of variations, the first thing people wanted to do was drop the ramifications and the axiom of reducibility. Ramsey was one of these, but he didn't work out the details. Another, according to Reck, was Leon Chwistek. But Reck seems to count Carnap's "Abriss der Logik" as the first publication in which the simple theory of types is fully presented and used.
The next development is Church's in which he produces a simple theory of types based on the simply typed lambda calculus (published 1940), and this is then augmented into a polymorphic type theory by the Computer Scientist Mike Gordon (1985) for use in the HOL theorem prover of which ProofPower is a variant.
There are very many other type-theories, some much more complex, which have been studied and used, notably introducing dependent types and the "propositions as types" paradigm, but it is moot whether these should be regarded as descendents of Russell's type theory.
RBJ
Thanks. Interesting you should mention the 1908 paper since I often wondered how much of PM (by Whitehead and Russell, with Whitehead as first author, even) came from Whitehead! I suppose for Grice, Russell's theory of Types is the epitome of something VERY complex. How else would you explain that out of ALL possible illustrations of some 'synthetic' proposition, he comes up, in his joint essay with Strawson (now in WoW:204):
ReplyDelete"We might take as our example[...] the natural impossibility of a child of three's understanding Russell's Theory of Types." -- They add that 'natural impossiblity' be read as to include 'causal' impossibility. So we hear,
A: My neihgbour's three-year-old undestands Russell's Theory of Types.
B: You mean the child is a particularly bright lad.
A: No. I mean that I say. He really does understand it.
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"One might be inclined to reply, "I don't believe you. The thing's impossible"".
"But if the child were then produced, and did (as one knows he would not) expound the theory correctly, answer questions on it, criticise it, and so on, one would in the end be forced to acknowledge that he claim was literally true and that the child was a prodigy."
I'm liberal enough to use 'prodigy' to apply to Carnap who understood the thing so clearly! (at whatever age!).