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."
>