On 3/22/2013 9:09 PM, Virgil wrote: > In article <FImdnfw3_ePXa9HMnZ2dnUVZ_q-dnZ2d@giganews.com>, > fom <fomJUNK@nyms.net> wrote: > >> On 3/22/2013 6:40 PM, Virgil wrote: >>> In article <1-KdnWUae4shRdHMnZ2dnUVZ_s-dnZ2d@giganews.com>, >>> fom <fomJUNK@nyms.net> wrote: >>> >>>> On 3/22/2013 4:49 PM, WM wrote: >>>>> >>>>> Fools stay together. >>>>> >>>> >>>> As observed before: >>>> >>>> Ex(phi(x)) -> Ax(phi(x)) >>>> >>>> is true in Wolkenmuekenheim >>> >>> And, far too often, so is >>> >>> Ax(phi(x)) -> Ex(phi(x)). >>> >> >> >> Is that one not always true? >> >> AxP(x) -> P(t) > > Couldn't Ax(phi(x)) be true when Ex(phi(x)) isn't? > E.G., > All four sided triangles have less than three sides. > and > There is a four sided triangle having less than three sides. >
This is a historical problem. It seems to begin with a medieval translation of logic that creates an issue involving reference to non-existent objects.
You can find a much better account than I can give here:
What you want to focus on is the existential import associated with the 'O' form in the square of opposition.
The modern criticism becomes essential to the realist philosophies of the late nineteenth century. Just as Aristotle conflated "substance" with the "essence" distinguishing definitions from distinguishing properties, Frege insisted that "truth" is the business of logic. Because of his realist positions, reference to non-existent objects such as "round squares" or "four-sided triangles" had to be inadmissible. Thus "truth" and "existence" are inseparably bound in the presuppositions of what is taught as "mathematical logic".
That is the general origin of the sense of those particular axioms.
From here, the story takes one immediate turn and one later turn.
The immediate turn is Bertrand Russell. For Frege, it had been sufficient to map all nonsensical sentences to falsity and direct all non-existent reference to the empty class as the "extension of concepts" having self-contradictory definitions (or, self-ambiguating ontology [not(x=x)]). Russell objected to both of these strategies. This led to Russellian description theory.
It is a very successful theory. If the topic of a sentence does not have an existing reference, the sentence still has a classical truth value (T or F).
Because of Russell's paradox, the first edition of Principia Mathematica uses a "no-classes" principle. But this ultimately required the infamous axiom of reducibility.
When he explains the axiom of reducibility, he observes that assuming tangible set existence makes the axiom of reducibility unnecessary for set theory. With the assumption of set existence, the membership relation is a "predicative relation" and unproblematic.
Because of his description theory, he developed the entire theory of types without concern for tangible linguistic reference implied by the word "denotation". This is something to think about when using words like "constant" and "parameter".
The later turn of events comes with free logic. In the link above, you will see the phrase "existential import". Free logic has motivation from several places, but what is fundamental is that it does not assume that all singular terms denote. It thus distinguishes between those terms with "existential import" and those terms without.