Date: Feb 22, 2013 8:32 AM Author: fom Subject: Re: Matheology § 222 Back to the root<br> s On 2/21/2013 7:08 PM, William Hughes wrote:

>

> So

>

>

> A) For every natural number n, P(n) is true.

>

> implies

>

> B) For any n: There does not exist a natural number

> between 1 and n such that P(n) is false

>

> However, we cannot conclude

>

> B') There does not exist a natural number

> m such that P(m) is false

>

This may be a repost. The first one did not

seem to arrive on the server.

> Rather than "implies" you should use the

> term "interpreted".

>

> Since classical partial logics would not

> lead to this situation, it appears to be

> a free logic.

>

> In free logic, the usual principle of

> specification, namely,

>

> AxP(x) -> P(t)

>

> is replaced by

>

> Ay(AxP(x) -> P(y))

>

> The semantics for such a logic is given

> by an ordered triple

>

> <D_I,D_O,f>

>

> where D_I and D_O are disjoint sets

> whose union is non-empty and f is an

> interpretation function.

>

> Interpretation is given by

>

> 1) f(t) is a member of U{D_I,D_O}, where t is a name

>

> 2) f(P) where P is an n-adic predicate is a set of

> n-tuples of members from U{D_I,D_O}

>

> 3) every member of U{D_I,D_O} has a name assignment.

>

> Truth is classical expect for the provision,

>

> Ax(P(x)) is true in a model just in case P(t)

> is true for all names t such that f(t) is a

> member of D_I.

>

> ============

>

>

> It is easier to see what is going on when one

> speaks of existential import.

>

> The existential quantifier in free logic has

> two forms. There is the usual existential

> quantifier that applies to transformation

> rules in relation to 'Ax' and then there

> is the definition

>

> E!t <->df Ex(x=a)

>

> Which permits one to show the restricted

> universal specification as

>

> AxA -> (E!t -> A(t/x))

>

>

> ============

>

> Unfortunately WM neither knows these matters nor

> understands his obligation to debate these

> matters in terms of historical context.

>

> Instead he chooses to be insulting to everyone

> who does not believe what he believes and

> accuses them of error.

>

> Classically, this does go back to the history

> of description theory. Frege introduced the

> problem of definiteness. The formula

>

> x+3=5

>

> has no truth value. The formula

>

> 2+3=5

>

> does. Names are important.

>

> However, for reasons not involving mathematical

> statements, Russell rejected Frege's description

> theory and introduced a theory which has resulted

> in a questionable model theory for mathematics.

>

> The Fregean issue is revisited by Abraham Robinson

> in discussing his own objections to Russell's

> description theory. Once again, the role of names

> (in the sense of symbols corresponding to descriptions)

> in establishing the interpretation of the

> sign of equality becomes central.

>

> Using fairly standard ideas from model

> theory, he writes:

>

>

> "Now let M be a first-order structure

> and suppose we are given a (many-one)

> map C from a set V of individual

> constants and of relation and function

> symbols in L onto individuals and

> functions and relations with the

> corresponding number of places in

> M."

>

>

>

> He goes on to observe,

>

>

> "Thus, the (meta-) relation |= depends

> not only on M and on X [X is a sentence, fom]

> but also on C although this is not apparent

> in our notation. In particular, C induces

> a correspondence or map also from the

> set of terms which occur in S_0(V) [S_0(V)

> is the set of sentences, fom], T_0(V) say,

> onto the set of individuals of M; in other

> words, it defines a *denotation* in M

> for any t in T_0(V)."

>

>

>

> The fundamental issue in this construction

> of course, has to do with identity.

> Robinson offers his opinion on this

> as well,

>

>

> "We still have to clarify the role of

> identity. One correct definition of

> the identity from the point of view

> of first-order model theory is undoubtedly

> to conceive of it as the set of diagonal

> elements of MxM, i.e., as the set of

> ordered pairs from M whose first and

> second pairs coincide. The symbol "="

> then denotes this relation and it is

> correct that (M |= a=b) if "a" and "b"

> are constants which denote the same

> individual in M, or, more generally,

> that (M |= s=t) if "s" and "t" are terms

> which denote the same individual in

> M. But, the identity may also be

> *introduced* by this condition so that

> (M |= s=t), *by definition* if "s"

> and "t" denote the same individual

> under the correspondence C, which is

> again assumed implicitly, and this

> seems more apposite in connection

> with the discussion of sentences which

> involve both descriptions and

> identity."

>