Re: Semantics of First-Order Languages

On 26 mar, 15:33, malc...@xxxxxxxxx wrote:
On 14 mar, 16:56, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:





On Mar 14, 2:45 am, malc...@xxxxxxxxx wrote:

On 13 mar, 16:59, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
With an interpretation, we then apply a recursive function that takes
sentences as input and, for each sentence, gives an output of true or
an output of false.
I do not agree. An interpretation provides us with a condition to test
the truth or falsity of a given sentence under that interpretation.

And what determines the truth or falsity is the recursive function
that is based on the interpretation.

But this condition is helpless unless our model (the structure) allows
us to effectively test it,

Do you mean 'effective' in the technical sense of 'computable'?

 > i.e. it could happen that the structure

doesn't determine the truth or falsity of the given condition (and I
think this should not be regarded as a matter of calculation, but as a
problem posited by the nature of the structure itself.

We admit that it is not always computable whether a given sentence is
true per a given interpretation. But the truth or falsity is
nevertheless determined. The recursively defined function is total on
the set of sentences of the language. Given any sentence, it has a
value as an argument for the recursively defined function.

Of course, this is classical, non-constructive mathematics. If you
don't accept non-constructive reasoning, or require that every
function is computable, then of course, you may object. But that is a
much deeper objection. If you're going to study mathematical logic in
a classical reasoning context, then it's not going to help you to get
hung up, in a stop-start way, on such objections that can rather be
noted all-in-one from the git-go.

MoeBlee
The recursively defined function is total on
the set of sentences of the language. Given any sentence, it has a
value as an argument for the recursively defined function.

I admit that the recursively defined function, given any sentence,
would have a value if every condition used in the definition could be
evaluated (not necessarily effectively), but this evaluation is
carried by means of sets (and functions which are also sets). Imagine
we choose our universe to contain some consecutive alephs and that our
language is enough expressive to state at least one of the identities
of the GCH. How could that identity be, even in principle, evaluated?

I am sure I must be terribly misunderstanding something, but I cannot
guess what it is...- Ocultar texto de la cita -

- Mostrar texto de la cita -

Let our (non-logical) language L consist of only two constants a and
b.
'a=b' is a formula of our language.

Let X be the L-structure (M,aX,bX). Where M is a set with the
following elements: the least cardinal greater than aleph0 and the
power set of the set of natural numbers. Let aX be the least cardinal
greater than aleph0 and bX the power set of the set of natural numbers
to b.

Now let S be any interpretation (i. e. any mapping from the set of
variables into M).
Is 'a=b' true under S?

.

Relevant Pages

  • Re: Semantics of First-Order Languages
    ... the truth or falsity of a given sentence under that interpretation. ... the set of sentences of the language. ... evaluated, but this evaluation is ...
    (sci.logic)
  • Re: Why an inconsistent ZF may be desirable, and should be welcome.
    ... is true under an interpretation M of L if, and only if, the interpreted ... individual, within a symbolic language (reasonably, these would include ... if we define a mathematical object and a set constructively as ... and without any loss of generality - I consider mathematics ...
    (sci.logic)
  • Re: Basic programming Question?
    ... Indeed I am not aware of any programming language that would ... The unnecessary evaluation in all these cases derives, I think, from ... the structure of a simple stack machine, such as is used to interpret ... Henceforth, then, the Party directs that all computers shall ...
    (comp.programming)
  • Thai Instructor $31K to $45 K, Must be US Citizen; job in Monterey, Califonia
    ... Defense Language Institute Foreign Language Center, Monterey, ... an evaluation of degreeobtained outside the U.S. must be mailed to ... Applicants whose self-nominations are received by the cut-off date will ...
    (soc.culture.thai)
  • Re: Enderton problem
    ... So A appears to have a descending chain. ... It's also not elementarily equivalent to N, ... have language with usual truths about N expressible in this ... It depends on the interpretation of "<". ...
    (sci.logic)