In a message dated 3/13/2002 1:17:30 PM Central Standard Time, jjllambias@hotmail.com writes:What do you mean by "actual quantification"? Quantity and quality (universal-particular, affirmative-negative) as well as import. <Right, except I'm not sure what you mean with those {..}. You cannot insert another term in there for the equality to hold.> In case broda has arguments attached. <Which one is the traditional system?> All + with the assumption that all classes mentioned as subject are non-null (and maybe a few less certain things as well). Using {ro broda} and {ro da poi} as the usual quantifiers of course already guarantees the assumption -- in true sentences anyhow. <If you mean (A-,E-,I+,O+),> I never would, since it is unlojbanic in its first two members and lacks subalternation. < >If you check other pages, you will find that Helman is not dealing with >restricted quantification as such but using the notation as a stage in the >process of translating English into symbols in ordinatry first-order logic. On the contrary, he specifically defines restricted universal quantification and unrestricted universal quantification, and then gives the following "principle of equivalence": (Ax: Sx) Px =||= Ax (Sx -> Px). (where should be inverted). What's more, in the chapter about existential quantification he also has the equivalence between the restricted and unrestricted forms: (Ex: Sx) Px =||=Ex (Sx & Px) and also the principle of obversion: ¬ (Ax: Sx) Px =||= (Ex: Sx) ~Px which works only if the restricted universal is A-. >Once the block attached to the quantifier is correctly filled in, the whole >can then be correctly moved into the formula in the usual way. But the >"restricted quantifier" (as the regular use of "thing" suggests) is just a >passing phase of translation, not a part of the logic.> Well, I didn't read the whole book, just a few sections that talked about restricted quantification. I never saw any evidence that it was developed as a separate system. In fact all the cases I saw were parts of translation exercises, like the one you sent me to originally: "(AxFx)Gx" asa more or less Englishy sentence that could then be converted into "Ax(Fx => Gx)", but not used in proofs or derivations. Obversion is just a device for making "not every" a bit more readable, as I read him. But I will look at some more (and of course it works for an all positive set as well -- under the standard condition -- no empty subjects). Of course, the restricted quantifier is -, since it just is the ultimate form in a minorly gussied up way. Part of the gussying is, alas, to hide the real subject of the of the final quantifier, namely the universal class. <Saying "you can work it out" is unconvincing. Maybe if you actually did work it out it would be more convincing, but since in the end it is just a matter of definitions...> Every universal quantifier (in a non-empty universe) entails every instanceof its matrix, every matrix with a free term entails its particular closure on that term: AxFx therefore Fa therefore ExFx. That is about as thorough a workingout as I can think of. |