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Substitutional quantification is an alternative to the objectual or referential interpretation of the quantifiers $\forall$ and $\exists$. The truth-conditions for objectual quantifiers are given in terms of Tarskian satisfaction and the variables bound by the quantifiers are understood as ranging over objects. In contrast, the substitutional approach turns on the truth of sentences formed by replacing the variables by certain terms of the language.

The earliest reference I can find to substitutional quantification is Quine's 1969 Ontological Relativity and Other Essays. It mentions nothing about earlier origins of the approach. I am wondering if Quine invented substitutional quantification (or was the first to call it such) and if not, am looking for references to earlier accounts.

UPDATE:

Quine allegedly took the notion from Leśniewski (https://plato.stanford.edu/entries/lesniewski/#Qua) but there is some controversy about whether that is the correct interpretation of the latter's work. In 1962, Ruth Barcan Marcus, referring to the substitutional approach, writes: "Recently, one of the fruitful interpretations of quantification seems to have been abandoned or at least submerged" (https://doi.org/10.1080/00201746208601353), suggesting a much early origin of the notion.

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  • $\begingroup$ Do you have a link to that paper? $\endgroup$ – evaristegd Jul 24 at 5:44
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    $\begingroup$ In modern language, is the objectual interpretation more or less Tarski-style semantics and the substitutional quantification more or less the usual rules of inference for quantifiers? $\endgroup$ – Andrej Bauer Jul 24 at 7:28
  • $\begingroup$ @AndrejBauer Yes, I think that is accurate. $\endgroup$ – Mallik Jul 24 at 12:57
  • $\begingroup$ In that case, it doesn't seem accurate to describe them as "alternatives" -- they are complementary. One is syntax, the other is semantics. $\endgroup$ – Mike Shulman Jul 24 at 15:35
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If I remember correctly, Shoenfield's 1967 text "Mathematical Logic" defined Tarskian truth in a substitutional way: Add to the vocabulary new constants to name all the elements of a structure, and then define a sentence $\exists x\,\phi(x)$ to be true (in that structure) if at least one substitution instance $\phi(c)$ is true. So truth is defined only for sentences, not for formulas with free variables, and there is no mention of assignments of values to variables. Yet the result is equivalent to Tarski's.

(My impression is that a substitutional view of quantification is much older, even older than Tarski's definition, but that it was probably only implicitly used, not explicitly defined.)

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I think this goes all the way back to Frege. Frege's logic allowed quantification over concepts. His "concepts" are supposed to be abstract versions of linguistic predicates in the same way that "objects" are abstract versions of noun phrases. You can combine a noun phrase and a predicate to make a sentence, and you can combine an object and a concept to make a proposition. (Russell's original version of his paradox, in his letter to Frege, involved "a predicate which cannot be predicated of itself".)

"The problem is that there is no explanation of the meaning of $\forall X$ and $\exists X$ intelligible to one who speaks only English (or German), and not Begriffsschrift $\ldots$ that is compatible with regarding $X$ as standing strictly for a verb phrase or predicate and not a noun phrase $\ldots$ To me, at least, this simple grammatical problem seems to make Fregean second-order logic as much nonsense as the non-word `everyso'." (Burgess, Fixing Frege, p. 212)

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