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." >
