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The basic clause in the semantic definition of satisfaction for quantifiers in f-o logic can be stated in two alternative forms: (for simplicity I assume a formula A(x))

A) take an assignment function s that maps the set Var = { v1, v2, ... } of free variables into the domain D of the interpretation and consider the resulting truth-value of the sentence A(v1)[s]

B) take a name for each object c ∈ D, where D is the domain of the interpretation and consider the resulting truth-value of the sentence A(v1 / c̅).

Question 1) are there respectively : A) the objectual and B) the substitutional approach to quantification ?

Question 2) are the following the respective "correct" reading of them ?

for 1) Through s we assign a denotation (an object) to the term x (a variable) so that the formula becomes a sentence with a fixed meaning (i.e.it becomes meaningful);

for 2) I perform a substitution of a term (a linguistic entity) into the formula so that the formula becomes a sentence with a fixed meaning.

Question 3) The B) approach (substitutional) needs to be "corrected" (ref.BBJ, Computability & Logic (5th ed - 2007), pag.116), in order to take care of uncountable domains ?

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(B) is not quite what is usually referred to as substitutional quantification. In the substitutional approach, we say that ∀xφ(x) is true if for every name c in the language, φ(c) is true. This is a problem for uncountable domains since we usually think of languages as being countable.

But, in (B) as it is here, you are extending the language, and there is nothing saying that you can't add uncountably many names -- indeed, if the domain is uncountable, then you will have to add uncountably many names. Extending a language in such a way is standard in the mathematical field of model theory, and the two definitions are in fact equivalent -- the interpretations of your new names do essentially the same thing as the variable assignment function.

  • You are right, thanks ! - I've found in Benson Mates, Elementary Logic (1972) [pag.62-63], the distinction and in W.V.Quine, Philosophy of Logic (1970), a comment about it : the above case B) must be corrected using only the names already existing in the language. That is the reason why, with uncountable domains (like R, the real line) we can "run out" of names (if we stay with "usual" countable languages). – Mauro ALLEGRANZA Jan 13 '14 at 10:21
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I'm fairly sure you've got all three questions right. IIRC, two problems with the substitutional approach have to do with uncountable domains, as you mention, and then again in the semantics of quantified modal logics.

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