[FOM] Axioms of reducibility and infinity
kremer at uchicago.edu
kremer at uchicago.edu
Mon Aug 8 21:25:34 EDT 2011
This is somewhat confused. "Order" in the context of ramified type theory doesn't mean what you think it means.
In ramified type theory, propositional functions are assigned a type and an order. The type is determined recursively by the type of arguments to the function (so individuals are of lowest type, functions of individuals are of higher type etc). But order is determined by the items quantified over in the function (in the function's expression? -- this is a matter of some unclarity).
This means we can have two functions of the same type that are materially equivalent (true of the same things) but are of different orders. So for example suppose that we have three people in a room, Joe, Bob and Bill, and Joe is 6 feet tall while Bob and Bill are 5 feet tall. Suppose also that Joe is a Republican and Bob and Bill are Democrats.
Now consider the following two functions:
(a) x is a Democrat in the room
(b) x is in the room and is shorter than someone in the room
These two functions are materially equivalent (they both are true of Bob and Bill)
But the second function quantifies over individuals whereas the first does not, so the second function is of higher order than the first.
Reducibility then says that for every function of whatever order, there is an equivalent function of the same type (taking the same type of arguments) of lowest order compatible with that type (so here for (a) there is (b) -- and we are guaranteed some such (b) even if we don't know a property like "Democrat" shared by only Bob and Bill but not Joe).
The whole logic remains "higher-order" in the sense that you have in mind, but the effect of reducibility is to claim that as far as extensions of functions is concerned, the ramification introduced by "orders" in the sense just explained makes no difference (the motivation for introducing this ramification is tied to solving paradoxes like the liar which were held by Russell to involve an illegitimate form of quantification).
On Wittgenstein: one thing he objected to in the Axiom of Reducibility was its seeming quantification across all types. This was supposed to be avoided by the idea of "typical ambiguity" which Wittgenstein saw as a dodge. (This criticism is made by him in a letter to Russell in something like 1914, as I recall.) In the Tractatus Wittgenstein says that even if Reducibility were true this would only be a kind of accident -- meaning that there is no logical guarantee that we can always find a function like (b) to correspond to any higher-order function like (a).
Hope this helps.
---- Original message ----
>Date: Mon, 8 Aug 2011 14:06:32 -0700 (PDT)
>From: fom-bounces at cs.nyu.edu (on behalf of steve newberry <stevnewb at att.net>)
>Subject: Re: [FOM] Axioms of reducibility and infinity
>To: Foundations of Mathematics <fom at cs.nyu.edu>
>My understanding of the Axiom of Reducibility is that it was
>intended to state that:
>To every proposition of higher-order, there is an equivalent
>proposition of First-order,
>or more precisely, to every entity definable in Higher-order
>logic there is an equivalent
>such entity definable in First-0rder logic.
>If AXIOMATICALLY true, then there is no ontological
>difference between First- and
>Higher- order logic, which is now well known to be untrue,
>and Wittgenstein may
>well have intuited that fact.
>--- On Sun, 8/7/11, Alasdair Urquhart
><urquhart at cs.toronto.edu> wrote:
> From: Alasdair Urquhart <urquhart at cs.toronto.edu>
> Subject: Re: [FOM] Axioms of reducibility and infinity
> To: "Foundations of Mathematics" <fom at cs.nyu.edu>
> Date: Sunday, August 7, 2011, 10:59 AM
> Wittgenstein's reasons for rejecting the axiom of infinity
> are quite clear. As stated by Whitehead and Russell
> in Principia Mathematica, it says that there are infinitely
> many individuals (i.e. objects of the lowest type).
> Clearly there is no reason to think this is true a priori
> of the world (the Tractatus is an attempt to describe
> the a priori logical structure of the world).
> In other words, in the construal of Whitehead, Russell
> and the early Wittgenstein, the Axiom of Infinity
> is an empirical postulate -- there is no reason to think
> it is a logical truth.
> I have never understood Wittgenstein's reasons for
> rejecting the Axiom of Reducibility, and always found
> his discussions of it quite obscure.
> On Sun, 7 Aug 2011, Francisco Gomes Martins wrote:
> > I´m working on Tractatus; Wittgenstein rejects the axiom
> of reducibility (see
> the axiom of infinity (5.535) and even the even the set
> theory (6.031). First, I ´d like to know more about those
> axioms. Second, I´d like to know why/how does Wittgenstein
> reject all of them?
> > Francisco
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