1

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

A) take an assignment function $s$ that maps the set $Var = \{ v_1, v_2, ... \}$ of free variables into the domain $D$ of the interpretation and consider the resulting truth-value of the sentence $A(v_1)[s]$

B) take a name $\overline{c}$ for each object $c \in D$, where $D$ is the domain of the interpretation and consider the resulting truth-value of the sentence $A(v_1/ \overline{c})$.

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

Question 2) are the following the "correct" reading?

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 (the substitutional one) needs to be "corrected" [according to BBJ, Computability & Logic (5th ed - 2007), pag.116], in order to take care of uncountable domains. What happens with the approach A), where we have countable many variables, so the assignment function $s$ can "take care of" only countable many elements of the domain ?

Community
  • 1
Mauro ALLEGRANZA
  • 99,247
  • 1
    A) and B) do not necessarily express the objectual and substitutional approaches to quantification. For B), note that model-theoretically we don't need to take constants to be linguistic objects in the sense of a sign one could in principle write down. For example Hodges in his A Shorter Model Theory takes the objects themselves as constants which denote themselves. The distinction expressed in your first quote is discussed more thoroughly in this question: http://math.stackexchange.com/q/481577/7235 – Benedict Eastaugh Jan 09 '14 at 15:26
  • Thanks a lot : i will read it. But my concern is not with the mathematica correctness of both definitions. It is more a subtlety about exposition. Signs of the language (variables or names) stay for objects of the world: so that the operation of assignement is a "bridge" between the two environments, that give meaning to an expression. If I take a formula (an expression of the language) and I substitute something in place of a variable, I'm making a operation into the language: this is why I'm thinking in term of substitution of a name for the object and not handling the object itself. – Mauro ALLEGRANZA Jan 09 '14 at 15:34
  • I have found this notes from John MacFalane (http://johnmacfarlane.net/142/substitutional-quantifiers.pdf) but I'm not able to identify the difference you are pointing at ("A and B do not necessarily express the objectual and substitutional approaches to quantification"). – Mauro ALLEGRANZA Jan 09 '14 at 16:05
  • @Benedict Eastaugh - 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 $\mathbb{R}$) we can "run out" of names (if we stay with "usual" countable languages). – Mauro ALLEGRANZA Jan 13 '14 at 10:19
  • That's what I was alluding to, yes. There's a large literature on substitutional quantification in philosophy of language and logic (with applications to philosophy of mathematics), since nominalistically-inclined philosophers are keen to eliminate the ontological commitment that follows from objectual quantification. – Benedict Eastaugh Jan 13 '14 at 10:45

0 Answers0